Sonochemistry

Abstract

Sonochemical splitting of thermodynamically very stable water molecule provides the evidence for drastic conditions inside the cavitation bubble. Kinetics of OH• radicals or H2O2 molecules formation during sonolysis of water can be used for quantification of acoustic power delivered to the system. This chapter focuses on the influence of several fundamental parameters, such as ultrasonic frequency, saturating gas, and some soluble nitrogen compounds on chemical reactivity of multibubble cavitation in homogeneous aqueous media in connection with the recent data on multibubble sonoluminescence.

Full text

SPRINGER BRIEFS IN MOLECULAR SCIENCE
ULTRASOUND AND SONOCHEMISTRY
Rachel Pflieger
Sergey Nikitenko
Carlos Cairós
Robert Mettin
Characterization
of Cavitation
Bubbles and
Sonoluminescence

SpringerBriefs in Molecular Science
Ultrasound and Sonochemistry
Series editors
Bruno G. Pollet, Faculty of Engineering, Norwegian University of Science
and Technology, Trondheim, Norway
Muthupandian Ashokkumar, School of Chemistry, University of Melbourne,
Melbourne, VIC, Australia
SpringerBriefs in Molecular Science: Ultrasound and Sonochemistry is a series of
concise briefs that present those interested in this broad and multidisciplinary field
with the most recent advances in a broad array of topics. Each volume compiles
information that has thus far been scattered in many different sources into a single,
concise title, making each edition a useful reference for industry professionals,
researchers, and graduate students, especially those starting in a new topic of
research.
More information about this series at http://www.springer.com/series/15634

About the Series Editors
Bruno G. Pollet is a full Professor of Renewable Energy at the Norwegian
University of Science and Technology (NTNU) in Trondheim. He is a
Fellow of the Royal Society of Chemistry (RSC), an Executive Editor of
Ultrasonics Soncohemistry and a Board of Directors’member of the
International Association of Hydrogen Energy (IAHE). He held Visiting
Professorships at the University of Ulster, Professor Molkov’s HySAFER
(UK) and at the University of Yamanashi, Professor Watanabe’s labs
(Japan). His research covers a wide range of areas in Electrochemical
Engineering, Electrochemical Energy Conversion and Sono-electr
ochemistry (Power Ultrasound in Electrochemistry) from the develop-
ment of novel materials, hydrogen and fuel cell to water treatment/
disinfection demonstrators & prototypes. He was a full Professor of
Energy Materials and Systems at the University of the Western Cape
(South Africa) and R&D Director of the National Hydrogen South Africa
(HySA) Systems Competence Centre. He was also a Research Fellow and
Lecturer in Chemical Engineering at The University of Birmingham
(UK) as well as a co-founder and an Associate Director of The University of Birmingham Centre for Hydrogen
and Fuel Cell Research. He has worked for Johnson Matthey Fuel Cells Ltd (UK) and other various industries
worldwide as Technical Account Manager, Project Manager, Research Manager, R&D Director, Head of R&D
and Chief Technology Officer. He was awarded a Diploma in Chemistry and Material Sciences from the
UniversitéJoseph Fourier (Grenoble, France), a B.Sc. (Hons) in Applied Chemistry from Coventry University
(UK) and an M.Sc. in Analytical Chemistry from The University of Aberdeen (UK). He also gained his Ph.D. in
Physical Chemistry in the field of Electrochemistry and Sonochemistry under the supervision of Profs. J. Phil
Lorimer & Tim J. Mason at the Sonochemistry Centre of Excellence, Coventry University (UK). He undertook
his PostDoc in Electrocatalysis at the Liverpool University Electrochemistry group led by Prof.
David J. Schiffrin. Bruno has published many scientific publications, articles, book chapters and books in
the field of Sonoelectrochemistry, Fuel Cells, Electrocatalysis and Electrochemical Engineering. Bruno is
member of editorial board journals (International Journal of Hydrogen Energy/Electrocatalysis/Ultrasonics
Sonochemistry/Renewables-Wind, Water and Solar/Electrochem). He is also fluent in English, French and
Spanish. Current Editorships: Hydrogen Energy and Fuel Cells Primers Series (AP, Elsevier) and Ultrasound
and Sonochemistry (Springer).
Prof. Muthupa ndian Ashokkumar (Ashok) is a Physical Chemist who
specializes in Sonochemistry, teaches undergraduate and postgraduate
Chemistry and is a senior academic staff member of the School of
Chemistry,University of Melbourne. Ashok is a renowned sonochemist, with
more than 20 years of experience in this field, and has developed a number of
novel techniques to characterize acoustic cavitation bubbles and has made
major contributions of applied sonochemistry to the Materials, Food and
Dairy industry. His research team has developed a novel ultrasonic processing
technology for improving the functional properties of dairy ingredients.
Recent research also involves the ultrasonic synthesis of functionalnano- and
biomaterials that can be used in energy production, environmental remedia-
tion and diagnostic and therapeutic medicine. He is the Deputy Director of an
Australian Research Council Funded Industry Transformation Research Hub
(ITRH; http://foodvaluechain.unimelb.edu.au/#research;IndustryPartner:
Mondelez International) and leading the Encapsulation project (http://
foodvaluechain.unimelb.edu.au/research/ultrasonic-encapsulation). He has received about $ 15 million research
grants to support his research work that includes several industry projects. He is the Editor-in-Chief of Ultrasonics
Sonochemistry, an international journal devoted to sonochemistry research with a Journal Impact Factor of 4.3). He
has edited/co-edited several books and special issues for journals; published *360 refereed papers (H-Index: 49) in
high impact international journals and books; and delivered over 150 invited/keynote/plenary lectures at interna-
tional conferences and academic institutions. Ashok has successfully organised 10 national/international scientific
conferences/workshops and managed a number of national and international competitive research grants. He has
served on a number of University of Melbourne management committees and scientific advisory boards of external
scientific organizations. Ashok is the recipient of several prizes, awards and fellowships, including the Grimwade
Prize in Industrial Chemistry. He is a Fellow of the RACI since 2007.

Rachel Pflieger •Sergey I. Nikitenko •
Carlos Cairós•Robert Mettin
Characterization
of Cavitation Bubbles
and Sonoluminescence
123

Rachel Pflieger
Marcoule Institute for Separation Chemistry
ICSM UMR5257, CEA, CNRS
University of Montpellier, ENSCM
Bagnols-sur-Cèze Cedex, France
Sergey I. Nikitenko
Marcoule Institute for Separation Chemistry
ICSM UMR5257, CEA, CNRS
University of Montpellier, ENSCM
Bagnols-sur-Cèze Cedex, France
Carlos Cairós
Department of Analytical Chemistry
University of La Laguna
La Laguna, Tenerife, Spain
Robert Mettin
Third Institute of Physics
Georg-August-University Göttingen
Göttingen, Germany
ISSN 2191-5407 ISSN 2191-5415 (electronic)
SpringerBriefs in Molecular Science
ISSN 2511-123X ISSN 2511-1248 (electronic)
Ultrasound and Sonochemistry
ISBN 978-3-030-11716-0 ISBN 978-3-030-11717-7 (eBook)
https://doi.org/10.1007/978-3-030-11717-7
Library of Congress Control Number: 2018967437
©The Author(s), under exclusive licence to Springer Nature Switzerland AG 2019
This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part
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Preface
Acoustic cavitation is a peculiar phenomenon that occurs in liquids irradiated with
strong acoustic fields: Bubbles appear that are subsequently driven by the field to
undergo strong volume pulsations. Since the conditions in and near the bubbles can
become really extreme during their implosion, a variety of surprising and intriguing
phenomena can be observed in cavitating liquids. And right after discovery, people
tried to use such effects and employ cavitation as a tool—usually with ultrasound.
Possibly, the most prevalent and best-known application is ultrasonic cleaning, but
another branch has gained significant attention and importance over the last dec-
ades: cavitation chemistry, better known as sonochemistry. Although acceptance
and transfer of sonochemical methods to industrial processing might have started
more reserved than it was the case in ultrasonic cleaning, nowadays, cavitation
chemistry is a strong and recognized chemistry branch, and its exotic flavor has
ceased. Nevertheless, cavitation, bubble dynamics, and the processes in collapsing
bubbles are complex. And in particular, the interdisciplinary character of these
subjects requires an open mind to learn and to train—even for the specialists in one
of the connected fields, as there are chemistry, chemical engineering, material
science, acoustics, hydrodynamics, and more. There are in the meantime excellent
monographs, article collections, and review papers available for a thorough study of
virtually every relevant topic. However, possibly the chance for a rapid overview
with a final up-to-date picture of sonochemistry and the related topics is missing.
Therefore, the aim of this little book is not to supply a complete reference nor to
replace established textbooks, but to give the reader a concise and modern intro-
duction at hand for a qualified overview. The book is part of the series Springer
Briefs in Molecular Science: Ultrasound and Sonochemistry, and the other titles
can perfectly complement our text.
The book focuses on the characterization of cavitation bubbles with respect to
multibubble systems, either through direct observations of the bubbles or through
measurements of sonoluminescence spectra and of sonochemical activity. Chapter 1
gives a short background on nucleation and dynamics of individual bubbles in a
sound field, proceeds with bubble instabilities and interactions, and finally examines
bubble ensembles. Chapter 2introduces to sonoluminescence, its interpretation, and
v

spectral analysis of the hot plasma core of collapsed bubbles. Methods for derivation
of pressures and plasma analysis are presented, and pathways for the chemistry
behind the emissions are discussed, along with bubble sizing methods based on
sonoluminescence. Chapter 3discusses sonochemistry in terms of dosimetry,
chemical reactions, and dependences on parameters like ultrasonic frequency and
important chemical additives.
We hope the reader finds the text useful and profits from the topics selected. The
field is huge, and we are aware that many important aspects remain only touched or
even untold. Thus, we encourage the reader to follow the literature hints and
references of his or her interest to deepen and complete the knowledge on ultra-
sound, cavitation, sonoluminescence, and sonochemistry.
The authors would like to thank Bruno Pollet and Muthupandian Ashokkumar
for encouraging this contribution to the Springer Briefs series.
R. P. and S. N. thank Tony Chave, Matthieu Virot and all actual and former
students and postdocs of the sonochemistry group at ICSM for their valuable
contributions in the studies performed at ICSM. Obviously, many thanks go to
people from whom we learned a lot during enriching collaborations: Muthupandian
Ashokkumar (University of Melbourne), Robert Mettin, Carlos Cairós, and Thomas
Kurz (Georg-August-University Göttingen), Micheline Draye (Savoie Mont Blanc
University), Thierry Belmonte (Lorraine University) and their teams.
R. M. and C. C. thank all actual and former members of the cavitation and
bubble dynamics group at Drittes Physikalisches Institut, Georg-August-University
Göttingen. Without their contributions over decades, the map of acoustic cavitation
would still contain many more white spots, and it is a pleasure to report on results
from the group. Particular thanks go to Werner Lauterborn, Thomas Kurz, Andrea
Thiemann, Till Nowak, Fabian Reuter, Philipp Frommhold, Reinhard Geisler, Max
Koch, and Christiane Lechner, to name just a few of the many important people.
Especially, we like to thank the Christian Doppler Forschungsgemeinschaft and
Lam Research AG Villach (Austria) for continuous and generous support in the
framework of the Christian Doppler Laboratory for Cavitation and Micro-Erosion
and particularly Frank Holsteyns, Alexander Lippert, and Harald Okorn-Schmidt.
Finally, the authors would like to thank all staff at Springer for their qualified
support and patience.
Bagnols-sur-Cèze, France Rachel Pflieger
Bagnols-sur-Cèze, France Sergey I. Nikitenko
La Laguna, Spain Carlos Cairós
Göttingen, Germany Robert Mettin
vi Preface

Contents
1 Bubble Dynamics ........................................ 1
1.1 Introduction ........................................ 1
1.2 The Bubble Collapse ................................. 2
1.3 Bubble Size ........................................ 2
1.4 Nuclei and Nucleation ................................ 3
1.5 Bubble Oscillations, Nonlinearity, and the Rayleigh Collapse .... 5
1.6 The Rayleigh–Plesset Equation .......................... 7
1.7 Acoustic Cavitation: Bubble Types ....................... 9
1.8 The Blake Threshold ................................. 10
1.9 Bubble Populations and Response Curves .................. 11
1.10 Toward Realistic Bubble Systems ........................ 12
1.11 Spherical Stability ................................... 14
1.12 Gas Diffusion ....................................... 16
1.13 Bjerknes Forces and Bubble Translation ................... 18
1.14 Viscous Drag Force .................................. 21
1.15 Phase Diagrams ..................................... 22
1.16 Structure Formation .................................. 23
1.17 Bubble Traps ....................................... 24
1.18 Non-spherical Bubble Dynamics and Jetting Collapse .......... 26
1.19 Jetting at Solid Objects ................................ 27
1.20 Translation-Induced Jetting ............................. 28
1.21 Activity Mapping .................................... 30
1.22 Beyond Adiabatic Compression ......................... 32
References .............................................. 33
2 Sonoluminescence ........................................ 39
2.1 Introduction ........................................ 39
2.2 Experimental Evidence of the Formation of a Plasma in
Cavitation Bubbles ................................... 40
2.3 Plasma Diagnostics to Derive Information on the Plasma
from SL Spectra ..................................... 42
vii

2.3.1 Temperature Determination ....................... 42
2.3.2 Current Limitations of the Fitting of SL Emissions ...... 47
2.3.3 Pressure/Density Determination .................... 48
2.3.4 Electron Densit y and Electron Temperature ............ 49
2.4 Emission of Non-volatile Solutes in SL .................... 51
2.4.1 Alkali Metals ................................. 52
2.4.2 Lanthanides and Uranyl .......................... 54
2.5 Bubble Size Estimation by SL Intensity Measurement Under
Pulsed Ultrasound ................................... 55
References .............................................. 58
3 Sonochemistry .......................................... 61
3.1 Introduction ........................................ 61
3.2 Sonochemical Dosimetry .............................. 62
3.3 Effect of Ultrasonic Frequency .......................... 63
3.4 Effect of Oxygen .................................... 64
3.5 Effect of Nitrogen and Ammonia ........................ 65
3.6 Effect of CO and CO
2
................................ 67
References .............................................. 69
Conclusion ................................................... 73
viii Contents

Chapter 1
Bubble Dynamics
1.1 Introduction
Bubble dynamics and cavitation have been recognized as a relevant topic of physics
and engineering for more than 100 years. Starting with erosion problems at ship
propellers end of the nineteenth century [1,2], experimental and theoretical
research went on to intense ultrasound fields in liquids after World War I [3].
However, the phenomena are intrinsically difficult to investigate since the involved
spatial scales span many orders of magnitude, the timescales are partly extremely
fast, and the behavior of bubbles includes important nonlinearities. Thus, it is not
surprising that the development of the subject was fostered significantly by
advances in technical equipment, namely high-speed imaging devices and modern
computers. In the meantime, there exist several excellent review articles and
monographs on cavitation. Starting with the earlier work by Flynn [4] and
Rozenberg [5] from the 1960s and 1970s, they proceed within the 1980s with
Neppiras [6] and Young [7], to be followed by Leighton [8] and Brennen [9] in the
1990s. Till today, an amazing progression of research activity in the whole field can
be found, boosted by the strong interest in sonoluminescence (see the overview by
Young [10]) and the perspective of non-classical chemistry (see for instance
Mason’s and Lorimer’s books on sonochemistry [11,12]). Due to the growth and
spreading of the topic, many review articles and specialized overviews have
appeared meanwhile, and an adequate and complete reference is difficult, if not
impossible. From the perspective of the quite active group at Göttingen University,
several recent reviews and collections of previous work should be mentioned; see
[13,14–18]. The following chapter does not intend to replace a deeper study of the
indicated references on cavitation, but wants to give a rapid and useful overview of
characteristics of acoustic cavitation bubbles. This serves as a certain basis for the
subsequent chapters on sonoluminescence and sonochemistry in multibubble sys-
tems, but also as a general briefing, facilitating further studies.
©The Author(s), under exclusive licence to Springer Nature Switzerland AG 2019
R. Pflieger et al., Characterization of Cavitation Bubbles and Sonoluminescence,
Ultrasound and Sonochemistry, https://doi.org/10.1007/978-3-030-11717-7_1
1

1.2 The Bubble Collapse
The key phenomenon that leads to the most spectacular effects of cavitation is the
rapid and strong compression of gas phase in a bubble, also termed a bubble
collapse. This leads to dramatic pressure and temperature peaks that in turn cause
acoustic shock waves and erosion of hardest materials outside the bubble and
chemical reactions and light emission in the interior. Rapid compression and
heating of gas or plasma is known in various contexts and on quite different scales,
e.g., supernova explosions and interstellar shock waves, atmospheric re-entrance
and supersonic flight, combustion engines or inertial confinement fusion. In cavi-
tation, the peculiar agent of compression is rather simple and usually small: a
bubble, i.e., a “void”in a liquid.
1.3 Bubble Size
Bubbles are not really empty, but usually contain vapor of the host liquid and
non-condensable gas. While the ambient liquid is usually to a good approximation
incompressible, the gaseous content of the bubble can be well expanded or
squeezed, which leads to the potential of substantial changes in bubble volume. It is
convenient to start considerations of dynamics with an idealized, spherical bubble
embedded in a three-dimensional infinite domain of liquid. The bubble radius is
denoted by RtðÞand time by t. The liquid pressure far away from the bubble is
named p1tðÞ. It contains the hydrostatic pressure p0and any acoustic pressure
caused by an additional sound field. The external pressure variations govern the
oscillating bubble behavior, but let us first have a look onto the static case. For
constant p1tðÞ¼p0, there exists an equilibrium; i.e., the bubble is at rest at its
equilibrium radius R0, also called rest radius. Then, the outside liquid pressure is
compensated by the interior gas pressure pb, composed of the pressure of
non-condensable gas pg(e.g., air) plus the vapor pressure pvthat also counteracts
from the bubble inside. To consider the surface tension rbetween liquid and gas
phase, an additional outside pressure psRðÞ¼2r=R, the Laplace pressure, comes
into play [9]. The equilibrium radius including these effects results from the pres-
sure balance of inside and outside pressures: pgR0
ðÞþpv¼psR0
ðÞþp0. We denote
pgR0
ðÞ¼pg;0and assume that the interior bubble pressure is solely determined by
the actual bubble radius.
1
This follows, for instance, for a given amount of
homogeneous non-condensable ideal gas with the equation of state pV ¼NRgT,
with the gas constant Rg, the temperature T, and the bubble volume V¼4pR3=3.
At the equilibrium radius and an ambient temperature T0, one obtains
1
More realistic descriptions of bubbles might consider non-equilibrium conditions like heat con-
duction, inhomogeneous bubble interior, or dynamics of evaporation/condensation of liquid/vapor
at the bubble wall.
2 1 Bubble Dynamics

pg;04pR3
0=3¼NRgT0. Then, the number of moles Nof the gas is related to the
equilibrium radius via the equation N¼4pR3
0p0pv
ðÞþ2rR2
0

=3RgT0

. Note
here that due to the surface tension, the inside bubble pressure is higher than the
ambient pressure p0, which can cause diffusion of gas out of the bubble.
Since in general cases the bubble can undergo large variations of its volume in
time, it is important to specify what exactly is meant whenever one is talking about
the bubble “size”. The equilibrium radius R0from the static case is usually (and in
the following) employed to measure the “equilibrium size”of a bubble (equiva-
lently one could, for instance, refer to the static equilibrium volume or to the
amount of non-condensable gas molecules in the bubble). Other important bubble
size measures are the maximum radius Rmax and the minimum bubble radius Rmin
during a volume oscillation. These can be quite distinct from the equilibrium value,
and determine the expanded and compressed state.
1.4 Nuclei and Nucleation
Where do the bubbles come from? The formation of a bubble can occur under
various conditions. Physically speaking, distinct cases are boiling, i.e., a local
energy deposition and evaporation of a certain liquid volume under ambient
pressure conditions, and cavitation, where the liquid evaporates due to a tension
(negative pressure) in the liquid under ambient temperature. If the tensile stress is
caused by bulk liquid flows, one speaks of hydrodynamic cavitation, and if it is
generated by a sound wave, we study acoustic cavitation. The fundamental phe-
nomena of the two types of cavitation are mostly the same, but it is the latter
mechanism that we will focus on.
It is important to note that as long as one is dealing with so-called real liquids,in
particular water, there typically exist stabilized (sub)micron-sized entities of
non-condensable gas (e.g. air), even under silent conditions [19–21]. Such small
pre-existing bubbles are termed nuclei, and their abundance and size distribution
depend on parameters like dissolved gas content or temperature, but also on the
history of the liquid sample.
2
As an example, Fig. 1.1a shows measured nuclei
statistics of untreated, of degassed, and of filtered tap water.
The first occurrence of cavitation bubbles in a liquid under tensile excitation is
called nucleation, and any parameter threshold that is passed to cause nucleation is
termed cavitation threshold. However, in all cases where no extremely clean and
degassed liquid is used and no special pretreatment has been undertaken, cavitation
bubbles occur in a process that might rather be termed “activation”than “nucle-
ation”: Some of the previously passive nuclei, stabilized in the bulk liquid, at
2
Free submicron bubbles should dissolve quite rapidly because of surface tension, as suggested
above. However, bubbles might be stabilized in crevices of solid particles [8] or be stabilized
statically or dynamically when covered partly with hydrophobic material; see [8,22].
1.3 Bubble Size 3

contaminants or container walls, are turned into expanding (and subsequently
collapsing) cavitation bubbles by the tensile stress (caused by the acoustic wave in
acoustic cavitation). In this sense, we deal usually with heterogeneous nucleation.
This explains the experience that cavitation thresholds in terms of tensile strength or
acoustic pressure amplitudes are typically much lower than their values expected
for a tensile rupture of the pure liquids (which would be “real”cavitation in the
sense of homogeneous nucleation; compare [9]). The reason is that for expansion of
a pre-existing void of radius R, a tension of the order of the Laplace pressure 2r=R
is needed, and “pre-existing voids”in a pure liquid, caused by random fluctuations,
would have radii Rnot much larger than typical intermolecular distances—leading
to extreme tension values. The presence of nuclei in non-treated real liquids (like
tap or sea water, Fig. 1.1a) reduces the needed tension to the order of 2r=ð1lmÞ,
which in the case of water yields about 150 kPa since r0:075 N/m. In effect,
cavitation thresholds and also the numbers of nucleated cavitation bubbles in real
liquids will depend on liquid impurities, dissolved gas, or even the vessel walls.
Such factors, which are not always easy to control, give rise to some degree of
sensitivity and variability of cavitating systems, experienced as a certain amount of
randomness. In addition, a cavitating system will change the nuclei distribution by
itself due to gas diffusion out of or into the oscillating bubbles and merging or
splitting bubble events. Assessment of equilibrium bubble radii under sonication
conditions is, however, not an easy task, since the visible bubble sizes are per-
manently changing. Results from a recent approach based on the statistics of
Fig. 1.1 a Histograms of nuclei populations in treated and untreated tap water (reproduced from
[23] and [9], with kind permission). Toward smaller sizes, the number density of nuclei typically
increases. However, also, the measurements get more difficult, and thus, the statistics shown
are cut off and do not include nuclei radii below about 1 µm. Such smaller nuclei should as well
exist abundantly. bEquilibrium bubble radius distribution obtained from a cavitating system under
25 kHz sonication in tap water Req ¼R0

. Letters (colors online) indicate different cavitation
bubble structures. Reproduced from [24], ©Elsevier 2018
4 1 Bubble Dynamics

momentary bubble sizes and a single-bubble model [24] are shown in Fig. 1.1b.
Comparison of these distributions of cavitation bubble equilibrium radii with the
distributions of still nuclei reveals that the decaying shape and the order of absolute
magnitude (several µm) essentially remain the same. Differences occur in the
details; e.g., less large bubbles are found in the strong sound field since they split
and break up.
1.5 Bubble Oscillations, Nonlinearity, and the Rayleigh
Collapse
If we imagine a gentle variation of the bubble radius from its equilibrium size, it
gets clear that bubbles can oscillate around R0: Expansion leads to a decrease of
interior pressure and thus to the tendency to restore the original bubble size. Vice
versa, a compression will increase the gas pressure and lead to a rebound. The
bubble volume finally forms an oscillatory system, and its own resonance frequency
can be derived as
fres ¼1
2pR
3cp0
q

1=2
;Rres ¼1
2pf
3cp0
q

1=2
ð1:1Þ
This frequency fres is the linear resonance of a bubble, also called the Minnaert
frequency [9,25]. The value depends on the ambient pressure p0, the liquid density
q, and the polytropic exponent cof the ideal gas. On the right side, we have
inverted the formula to arrive at an expression for a linear resonant radius Rres. This
is of importance if a fixed driving frequency acts on a population of bubbles, a case
usually met in technical applications of ultrasound and in particular in
sonochemistry.
Resonant bubble oscillations are, for instance, responsible for audible sound of
running or splashing water [25]. It turns out, however, that this bubble oscillator is
nonlinear: While one could imagine an arbitrary large expansion of a bubble, its
compression below zero radius is unphysical. This is one reason for the restoring
force being asymmetric with respect to expansion and compression, which finally
leads to anharmonic oscillations [26]—including the implosive “collapse”type of
motion. In fact, the most extreme case of a bubble collapse would happen if the
spherical cavity was empty; i.e., no gas (or vapor) pressure would ever counteract
the inflowing liquid (and thus no rebound would occur). Historically, Rayleigh [2]
was the first to investigate the spherical implosion into empty space, which is why it
is called Rayleigh collapse. The speed of the spherically inrushing liquid front as
well as the pressure in the liquid (assumed to be incompressible) can be calculated
analytically for this case. It is found that an infinite speed and infinite pressure in the
liquid at the bubble would be approached at the cavity closing moment [2,7].
This result serves to illustrate the enormous energy focusing of a bubble collapse by
1.4 Nuclei and Nucleation 5

the three-dimensional influx and in particular the erosive potential by the pressure
peak. However, for a description of the internal conditions of realistic bubbles
during their compression phase, consideration of the gas is additionally required
(and Rayleigh did so as well in his paper [2]). Then, the collapse is cushioned to
finite—but potentially still quite high—wall velocities, and finally, the inward
liquid motion is stopped and reversed. The gas is rapidly heated up during the
implosion, and high pressure and temperature peaks occur at the minimum bubble
size, which facilitates chemical reactions and light emission from the gas—sono-
chemistry and sonoluminescence. In Fig. 1.2a and b, a Rayleigh collapse of an
empty bubble is illustrated, showing the bubble radius and the diverging bubble
wall speed. If the same bubble contains a noble gas following an adiabatic ideal gas
equation of state, the results for radius–time and bubble wall velocity versus time
look like Fig. 1.2c, d if the amount of gas corresponds to an equilibrium radius of
R0¼5lm. Additionally, gas temperature and gas pressure are plotted versus time
in Fig. 1.2e, f. From the adiabatic law and neglecting vapor and surface tension, one
obtains T¼T0R0=RðÞ
3j1ðÞ
and p¼p0R0=RðÞ
3jwith the adiabatic exponent j(see
below) and the reference values T0¼293 K and p0¼105Pa. The peaks at
Fig. 1.2 Bubble dynamics in water. The radius–time curve of a Rayleigh collapse of an empty
bubble is shown in (a). Initially, the radius is 50 µm with a resting bubble wall; the ambient
pressure is 100 kPa. Plot b shows the modulus of the bubble wall velocity on a logarithmic scale
versus time for the empty bubble. The same collapse conditions for an argon-filled bubble with
adiabatic heating are shown in (c). In the collapse, the gas is compressed and heated, and then, it
expands again. Several decaying rebound oscillations occur with less violent collapses. The bubble
wall velocity (modulus) is shown in (d); note the inversion of direction at the bubble radius
minima and maxima, visible as vertical lines in the logarithmic scale. The temperature and the
pressure in the bubble, according to ideal adiabatic compression, are shown in plots (e) and (f)
6 1 Bubble Dynamics

15,000 K and almost 20 kbar are rather high, but such extreme values occur only
during a short moment of time—here several nanoseconds. More realistic views of
the bubble would consider in particular heat conduction of the hot gas toward the
liquid, and thus, the given values are upper bounds.
1.6 The Rayleigh–Plesset Equation
A quite good model of the dynamics of a spherical bubble is obtained by the
Rayleigh–Plesset (RP) equation. This ordinary differential equation of second
order describes the evolution of the bubble radius R(t) over time when descrip-
tions of the internal bubble pressure pbRðÞand the pressure far from the bubble
p1tðÞare given [8]:
R€
Rþ3
2
_
R2þ4l
q
_
R
Rþ2r
qR¼1
qpbRðÞp1tðÞðÞð1:2Þ
Effects of surface tension rand dynamic viscosity lare included, qis the
density of the liquid, and the dots denote differentiation with respect to time as
usual. Following our choice above of an ideal gas in the bubble, we derive the
internal gas pressure as
pb¼p0þ2r=R0pv
ðÞR0=RðÞ
3cþpvð1:3Þ
with vapor pressure pvand polytropic exponent c. A value c¼1 represents
isothermal compression, while c¼j¼cp=cvcorresponds to adiabatic heating of
the gas with the adiabatic exponent j, derived from the ratio of the specific heats at
constant pressure and constant volume, cpand cv, respectively. For an ideal gas, this
ratio can be expressed by the number of available degrees of freedom fgof the gas
molecules: j¼cp=cv¼fgþ2

=fg. For noble gases, one obtains j¼5=31:67
which results in most effective heating, and this has been used in Fig. 1.2. For air,
containing mainly two-atomic gases, the value is close to j¼7=5¼1:40. In
simple models that take heat conduction into account, ccan be approximately tuned
to an effective value lying between 1 and j, depending on the momentary bubble
wall speed [27]. The term p1tðÞcomprises the static pressure p0and the acoustic
pressure pac tðÞ that is usually assumed to be harmonic with frequency fand
amplitude pa:p1tðÞ¼p0þpac tðÞ¼p0þpasin 2pftðÞ.
From the RP equation, one obtains analytically via linearization the Minnaert
frequency fres from (1.1) and the small amplitude bubble response in form of linear
sinusoidal oscillations around R0in dependence on paand f[7,8]. However, when
bubble dynamics gets more involved than small harmonic oscillations or the
Rayleigh collapse, analytic results are difficult to obtain and become involved [28],
and investigations strongly rely on numerical solutions of the equations [26].
1.5 Bubble Oscillations, Nonlinearity, and the Rayleigh Collapse 7

Figure 1.3a exemplifies how the sinusoidal bubble wall motion develops into the
collapse type for increased driving amplitude. The accompanying part in Fig. 1.3b
depicts the bubble displacement if the incident sound field is a traveling wave,
which is discussed in detail below in Sect. 1.20.
To reach a more accurate description of strong collapses and the bubble interior,
extensions of the standard RP model exist (such models are often termed
“Rayleigh–Plesset-like”). For instance, liquid compressibility is taken into account
by employing terms with M¼_
R=c,Mbeing the Mach number of the bubble wall
with respect to the sound velocity in the liquid c[29]. Furthermore, the simple laws
of the gas in the bubble (e.g., adiabatic or isothermal ideal gas) can be replaced by
more sophisticated choices with excluded volume (e.g., van der Waals gas) and heat
Fig. 1.3 a Radius versus time for a bubble of 5 µm rest radius and sinusoidally driven at 25 kHz.
The radius history is shown for increasing driving pressure amplitudes. Up to 90 kPa, the
oscillations remain nearly sinusoidally, as in a linear approximation. At 110 kPa, first collapse
oscillations occur with characteristic peak form around the minima of the radius. At 130 kPa, after
an extended expansion phase to about tenfold the rest radius, a main collapse happens at 22.5 µs.
Extreme temperature and pressure conditions occur in this driven collapse, similar to those in the
free collapse shown in Fig. 1.2c–f. The sharp qualitative transition in the collapse behavior for
only small change in pressure amplitude is characteristic for the nonlinearity of oscillating bubbles
and connected with the so-called Blake threshold (see Sect. 1.8). bSpatial translation of the bubble
from awhen driven by a plane traveling wave in positive x-direction. At lower excitation (90 and
110 kPa), the bubble is moving forth and back, and only a small mean displacement in direction of
wave propagation occurs. At 130 kPa driving, the bubble undergoes a significant and fast jump in
forward direction during the main collapse around 22.5 µs, and the net displacement is of the order
of the maximum bubble radius before collapse. This is further discusses in Sect. 1.20
8 1 Bubble Dynamics

conduction models [27]. To include liquid evaporation and condensation as well as
chemical reactions, additional equations can be coupled to the radial dynamics [30,
31]. The equations used for Figs. 1.2 and 1.3 and also for the following results
follow actually the Keller–Miksis model, an RP variant including liquid com-
pressibility [16,26,32]:
R€
R1MðÞþ
3
2
_
R21M
3

¼1þM
qplþR
qc
dpl
dtð1:4Þ
pl¼p0þ2r
R0

R0
R

3c
2r
R4l_
R
Rp0pac tðÞ ð1:5Þ
Results presented are based on the numerical solution of this model for water
under standard conditions (air pressure and room temperature), either adiabatic
c¼jor isothermal c¼1.
1.7 Acoustic Cavitation: Bubble Types
Let us assume the above introduced pre-existing “nuclei”bubbles as spherical
non-condensable gas bubbles of very small radii Req (although the geometry might
be more complicated if they are attached to a microscopic crevice of a dust particle
or a wall), and let them be exposed to an external pressure variation. To become
“cavitation bubbles”that show relevant effects, some nuclei should expand
significantly, and indeed, the characterization of cavitation bubbles by their col-
lapse dynamics is common and convenient (although not necessarily unique in the
literature). For instance, the expansion ratio a¼Rmax =R0or the compression ratio
b¼R0=Rmin are reasonable dynamical measures for bubble activity in the sense of
cavitation effects. From a physical point of view, the bubble wall acceleration is
dominated by the inertia of the liquid for a>2…3[6], which leads to a pro-
nounced collapse. Therefore, the term “inertial cavitation”is used frequently to
characterize such bubbles with stronger implosion. In contrast, the weaker oscil-
lating bubbles are dominated by the internal gas pressure and are sometimes termed
“stable cavitation”(a<2…3).
3
With respect to the occurrence of specific cavi-
tation effects, the threshold value of amight have to be somehow larger than 3,
depending on the effect under question. For instance, in the context of single-bubble
sonochemistry, a reaction threshold value of about 4 has been observed [33], and
for sonoluminescence in xenon sparged sulfuric acid, avalues beyond 6 have been
3
The term “stable”for gas dominated bubble dynamics is somehow unfortunate since less strong
collapsing bubbles can nevertheless exhibit instabilities (e.g., develop non-spherical shapes and
splitting), while inertial cavitation bubbles can well oscillate in stable regimes. The older notion of
“transient”cavitation for inertial cavitation is misleading in the same sense.
1.6 The Rayleigh–Plesset Equation 9

found to be consistent with observations of light emission [34]. However, since the
dynamical quantities like aor bdo not fully determine temperature and pressure
peaks during realistic bubble collapses, they should be seen rather as rough indi-
cators than as exact threshold measures.
1.8 The Blake Threshold
The response of a bubble to the external pressure signal depends crucially on its
equilibrium radius R0. To turn a nucleus into a transient cavitation bubble, its
surface tension pressure has to be counteracted by the tensile part of the acoustic
pressure pac that together with the static pressure p0forms the pressure p1far from
the bubble. The application of a quasi-static tension pac ¼p\0 will lead to a
shift of the bubble size, and the condition for the new equilibrium bubble radius R
0
reads pbR
0

þpv¼psR
0

þp0p. Closer analysis shows that from a critical
value of tension p¼p
cr on, a bubble of given equilibrium radius R0cannot
find a new stable equilibrium R
0anymore and will expand without bound!
4
This
phenomenon was described by Blake [35] and Neppiras and Noltingk [36] and is
termed the Blake threshold. Neglecting vapor pressure, the relation of critical
tensile pressure pBlake and bubble rest radius R0reads [28,37]
pBlake ¼p
cr ¼p01þ4
27
a3
s
1þaS
ðÞ

1=2
"#
;aS¼2r
p0R0
ð1:6Þ
Stating this result slightly differently: For a given quasi-static tension p, there
exists a critical nucleus or bubble equilibrium size R0;cr beyond which an “infinite”
expansion occurs. This radius is found by solving (1.6) for R0and placing pfor
p
cr. The Blake threshold phenomenon is encountered as well for the time-varying
pressure fields pac tðÞin acoustic cavitation, e.g., the sinusoidal pressure of a sound
wave pasin xtðÞintroduced above, here with the angular frequency x¼2pf. Then,
the critical tension p
cr will occur in the negative-going phase if the driving
pressure amplitude surpasses the critical value pa;cr ¼p
cr. Although the quasi-static
consideration is not fully justified anymore, the values of critical acoustic pressures
or critical bubble sizes remain essentially valid for the sinusoidally driven bubble (if
not too high driving frequencies are considered). However, different to the static
case, any applied tension pac \0 will now change to overpressure þpac [0 after
half an acoustic cycle, and the otherwise, “infinite”expansion will be reverted
toward a collapse.
4
The unlimited expansion occurs theoretically in an unbounded liquid volume. In a real situation,
the nucleus expansion will be stopped by boundary conditions, but it can reach a “macroscopic”
bubble size.
10 1 Bubble Dynamics

1.9 Bubble Populations and Response Curves
The Blake threshold is crucial in acoustic cavitation since for higher driving
pressures, it separates small “passive”bubbles (that oscillate only weakly) from the
larger “active”ones (that undergo strong collapse). Indeed, the highest bubble
expansion and compression ratios can occur directly beyond that threshold, as can
be seen from bubble response diagrams like those given in Fig. 1.4. The top graphs
show the quantity Rmax R0
ðÞ=R0¼a1 for varying bubble sizes under constant
driving frequency and three different fixed pressure amplitudes in water. The “low”
ultrasonic frequency of 20 kHz on the left is contrasted to the “high”-frequency
ultrasound of 500 kHz on the right. The form of response curves is very similar for
both frequencies, and in particular the jump in relative expansion at the small radius
side is apparent—the Blake threshold. The transition from small to rather large
expansion ratios occurs very sharply for increasing R0, and the bubbles of strongest
relative expansion are only slightly larger than the largest passive ones. It is
remarkable that still larger bubbles decrease again in their relative expansion
(although considerable avalues persist). In particular, the bubbles of resonant size,
i.e., those with fres matching the driving frequency f, do not peak significantly
between the other bubble sizes for higher driving amplitudes. This fact is contra-
dictory to former common notions that mainly resonant bubbles would be activated
by ultrasound; one finds rather that usually much smaller bubbles, just beyond the
Blake threshold, are main actors in acoustic cavitation at higher acoustic pressures.
Fig. 1.4 Response curves of spherical bubbles driven at 20 kHz (left) and at 500 kHz (right). In
each plot, the normalized maximum bubble expansion during one driving period, i.e.
Rmax R0
ðÞ=R0¼a1, is shown versus R0. Three curves with indicated driving pressures are
given for both frequencies. If higher periodic or aperiodic (chaotic) oscillations occur, the distinct
maxima are plotted, which results in multiple points related to R0and partly in a gray-shaded
broadening of the curves. Hysteresis (i.e., more than one stable solution) is included in the same
way. Resonances are partly numbered: 1/1 refers to the linear resonance where R0Rres , 10/1 and
20/1 indicate the nonlinear resonances at R0Rres =10 and R0Rres=20 (Keller–Miksis model,
water at ambient conditions). Reproduced from [38], ©Kluwer Academic 1999
1.9 Bubble Populations and Response Curves 11

A closer comparison of both diagrams reveals that the absolute values are shifted to
smaller radii and higher pressure amplitudes for the higher frequency, which is a
consequence of the faster driving oscillation with shorter tensile phase. A synopsis of
spherical bubble response in a larger plane of driving frequency fand equilibrium
radius R0is given in the graphs of Fig. 1.5, depicting the calculated and grayscale
coded values of a1. It can be seen that at the moderate driving pressure of 100 kPa
(Fig. 1.5a), the strongest response for a given driving frequency occurs still near the
respective linear resonance radius Rres which is connected with the frequency via
Rres fres 3 (m/s) (a rule of thumb for water at ambient conditions). For the loga-
rithmic axes, this relation shows up as a bright diagon al line running from small radii at
high frequencies to larger radii at low frequencies. At the lower frequencies, however,
already nonlinear resonances appear at smaller bubble sizes, manifesting itself in
brighter stripes parallel to and to the left of the main diagonal. The higher amplitude of
200 kPa (Fig. 1.5b) reveals clearly the importance of the Blake threshold (arrows),
and the linear resonance ceases to be a prominent feature.
1.10 Toward Realistic Bubble Systems
Solutions of spherical bubble models can give primary information on the behavior
of driven bubbles in an ultrasonic field. Refinements of such models with respect to
the interior gas and vapor, heat conduction, and chemical reactions can improve the
validity of results obtained; see [27,30,31]. Now, we want to draw the attention
onto the fact that in realistic active acoustic cavitation environments, like that
shown in Fig. 1.6, an individual spherical bubble driven in a homogeneous sound
Fig. 1.5 Response diagrams of spherical bubbles driven at pa= 100 kPa (a) and at 200 kPa (b).
The normalized expansion Rmax R0
ðÞ=R0¼a1 is shown grayscale coded versus rest radius
R0and driving frequency f. In the right plot, a1 was cut above 9 for better visibility of the
structures. The arrows indicate the jump line connected with the Blake threshold of surface tension
that occurs at pa= 200 kPa above R00.6 µm
12 1 Bubble Dynamics

Fig. 1.6 Example of typical acoustic cavitation in a resonator device, driven at 90 kHz.
aSonoluminescence image from an aqueous sodium chloride solution sparged with argon,
recorded by a long-term (30 s) exposure digital color photograph (color online) [46]. Frame width
is 5 cm. Due to the standing wave distribution of the sound field, bright stripes are visible.
Furthermore, parts of the emissions show orange color, while other parts appear blue. This
indicates differences in the bubble dynamics, possibly liquid injection into collapsing bubbles in
the orange regions (see Sect. 1.20). bTypical short-term exposure (1 µs) of a cavitating region
where sonoluminescence occurs (part of a high-speed series, frame width 2 mm). On this scale,
many individual bubbles of various smaller sizes are visible and a few larger bubbles that attract
and collect the smaller bubbles. cDetail of the high-speed series (turned 90°, size of each frame ca.
0.3 mm 1 mm, time between frames 143 µs, exposure time 1 µs). Typical bubble velocities are
in the range of 10–20 cm/s. Bubbles appear small (near collapsed state) on the first frames and
then larger on the later frames (near maximum expansion state). This is result of a beat between
recording rate (7000 frames/s) and ultrasonic driving frequency (90,000 Hz). See also the
examples shown in [18]. Figures aand breproduced from [47], ©VCH Weinheim 2018
1.10 Toward Realistic Bubble Systems 13

field and far from disturbances like other bubbles or boundaries is an exception.
Most important deviations from this idealization are inherent instabilities of the
spherical bubble shape, gradients in the acoustic driving field, interactions with
neighboring bubbles and objects, and potentially a bulk liquid flow. Such factors
can lead to bubble deformation and bubble movement in the liquid. Furthermore,
bubble collisions and splitting are quite common in multibubble systems. This is
illustrated in Fig. 1.6c where a sequence of a high-speed recording is reproduced
frame by frame. Bubble motion and coalescence can be traced this way, and the
very dynamic bubble life might be perceived, taking into account the fast timescales
(the sequence in Fig. 1.6c covers 2.85 ms). Nevertheless, between collisions, the
bubbles are more or less well separated, and many appear at least roughly as
spherical entities. Thus, we discuss in the following several important aspects for
multibubble systems again on basis of a single bubble. This has to be seen as a
starting point, since a full description of a larger ensemble of oscillating and
moving, merging, and splitting bubbles like in Fig. 1.6, under relevant inclusion of
the modified sound propagation and induced turbulence, remains a very demanding
and delicate task. Thus, the “bottom-up”approach [15] is seen as an applicable way
to learn more about cavitating systems piece by piece, while other aspects might
stay aside in the discussion.
5
1.11 Spherical Stability
The spherical shape that is typically assumed by individual bubbles minimizes
surface energy; i.e., it is caused by surface tension. This spherical shape might
become unstable under volume oscillations and/or systematically deformed if the
symmetry is externally perturbed, for instance, due to a relative motion of the
bubble in the liquid or neighbored objects. In case of extreme deformations, the gas
domain finally splits and the bubble disintegrates. Further down the systematic
shape deformation of bubbles due to asymmetric flow fields and jetting will be
considered. Now, the instability of spherical bubble shape is discussed. Shape
instabilities can occur in principal without external trigger via strong amplification
of minute disturbances. The instability of a plane boundary between heavier and
lighter fluid when accelerated toward the heavier fluid is known as Rayleigh–
Taylor instability [48]. In the case of a spherical bubble, it turns out that essentially
a rapid collapse phase can be Rayleigh-Taylor unstable, while the expansion is
stabilizing [8,49–53]. Furthermore, a spherical (“breathing”) mode of bubble
oscillation can be accompanied by surface oscillation modes, as illustrated in
Fig. 1.7 [8,53–56]. The modes are ordered with an integer nthat essentially
5
In a way as a contrast, “top-down”descriptions of cavitation start from multiphase flow of liquid
and vapor (for hydrodynamic cavitation, see [9,39]) or from sound propagation in bubbly media
(see [40–45]).
14 1 Bubble Dynamics

indicates the symmetry. Such modes lead to less focused collapse and potentially to
ejected satellite bubbles. As well, droplets of liquid phase may be ejected into the
bubble as a consequence of modal oscillations.
When unstable, surface modes can grow strongly in amplitude and finally disin-
tegrate the bubble. Excitation of surface modes is a param etric resonance phenomenon
and appears prominently for the smaller nand if the modal eigenfrequency f
n
falls
close to an integer multiple of half of the driving frequency f:
fn¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n21ðÞnþ2ðÞr=qR3
ðÞ
p=2pmf=2;m¼1;2;.... This estimate is valid
only for small oscillations when RR0. In case of stronger volume oscillations of the
bubble, deviations from this rule occur, as numerical calculations and experiments
show [56]. Since the viscous damping is stronger for higher modal order, the modes of
smaller nare generally the most unstable ones. For illustration, Fig. 1.8 shows cal-
culated stability regions of the lowest three modes for driven bubbles in the parameter
plane of driving amplitude and bubble equilibrium size. Note that the modal number
nstarts with n= 2 since n= 0 is the spherical (volume) mode, and n= 1 corresponds
to bubble translation. The gross picture shows that both for smaller bubble sizes and
for lower driving amplitudes, the surface modes become stable, while an increase of
both quantities tends to destabilize the spherical shape. Closer inspection shows
unstable peaks and stripes that reflect the resonances where the modal and the half
driving frequency match as discussed above. Additionally, the expansion ratio is
indicated by the lines of a¼2 and a¼3. Bubbles of high expansion ratios that
remain stable occur only at the small radii and higher driving within a stripe close to
the Blake threshold. Furthermore, it can be seen that larger bubbles become para-
metrically unstable already before a large volume expansion can be reached.
An important consequence of shape instabilities is the limitation of active bubble
sizes in ultrasonically driven cavitation systems and thus of the reachable collapse
Fig. 1.7 Surface modes of a bubble moderately driven at 18 kHz in partly degassed water.
Frames from left to right and top row to bottom row, frame height is about 500 µm, and the mean
bubble radius is about 110 µm. Interframe time is 6.7 µs, and roughly, the eight frames of a row
correspond to one acoustic driving period. Clearly, modal oscillations of n= 4 are visible with
some addition of n= 3, and the bubble size indeed falls close to the resonance condition for the
fourth mode stated in the text (image C. Cairós and A. Troia)
1.11 Spherical Stability 15

conditions. After nucleation, the bubbles may grow by gas diffusion and by
merging events, but the final size will be determined by the instabilities. Only in
regions of lower acoustic driving, e.g., near pressure nodes, the larger bubbles can
stably exist. Since the instability threshold is crossed both for increasing bubble
radius and for increasing pressure amplitude, in real systems, the energy concen-
tration by the bubble collapse, reflected roughly by the expansion ratio a, cannot be
amplified arbitrarily by an increase in driving. From Fig. 1.8, one reads that an
“escape”from spherical shape instability appears most possible for very small
bubbles. However, there the Blake threshold of surface tension can block bubble
expansion (the lines a= 2 and 3 are indicated). Furthermore, additional limitations
in form of gas diffusion and acoustic forces occur. Both aspects are briefly outlined
in the following to complete the discussion of system parameters where single
spherical bubbles can stably exist (sometimes such a parameter region is termed
“bubble habitat”[16]).
1.12 Gas Diffusion
For a complete discussion of the effects acting on the cavitation bubble population,
gas diffusion has to be considered. The amount and type of dissolved gas in the
liquid are relevant not only for the content of the bubbles, but also for their
nucleation, growth, or dissolution. With respect to nucleation, typically governed
Fig. 1.8 Regions of parametric surface mode instabilities of a driven spherical bubble in water at
20 kHz (a) and 500 kHz (b), calculated with the equations given in [15]. The parameter plane of
bubble equilibrium size R0and driving pressure amplitude pais colored according to instability of
modes n=2,n= 3, and n= 4: white: stable; light gray: one unstable mode; middle tone: two
unstable modes; dark gray: three unstable modes. The lines indicate boundaries where the
expansion ratio a¼Rmax=R0reaches 2 (dashed line) and 3 (solid line). Above the lines, alpha gets
larger. Zones of stable, strongly collapsing bubbles appear to the upper left
16 1 Bubble Dynamics

by the nuclei present, it is expected that the dissolved gas concentration has an
effect on them: It has been observed that cavitation inception is easier at higher
concentrations of dissolved air and that degassing and pre-pressurizing lowers the
cavitation threshold in terms of the tension needed for cavitation bubbles to occur
[8]. A free small bubble of non-condensable gas should dissolve due to the Laplace
overpressure psRðÞ¼2r=Rinside, if the liquid is not supersaturated with the gas.
This pressure will push out the gas into solution, and even more so for the smaller
getting bubble, until the bubble vanishes. This is somehow not happening for the
nuclei that are potentially fully or partly covered by hydrophobic substances, and
several explanations are employed for this observation (static and dynamic ones;
see, for instance, the discussion by Yasui et al. [22] and the related literature).
Many of the explanations, however, include the background concentration of gas in
the liquid. Thus, it is concluded that the amount of dissolved gas has—at least—an
indirect influence on the nucleation of cavitation bubbles, namely via the stabi-
lization of nuclei. In essence, the more gas dissolved, the more stable and abundant
are the nuclei, and cavitation is facilitated.
Interestingly, the tendency of small bubbles to dissolve can be counteracted by
volume oscillations. This phenomenon is known as rectified diffusion [7,8,57].
Since the gas diffusion through the bubble interface is proportional to the surface
area, an expanded bubble has higher inflow per time than outflow in collapsed state.
Furthermore, the gas concentration gradient across the bubble wall turns out to be
higher in the expanded phase than in the contracted phase, which acts in the same
direction. Additionally, nonlinear oscillations can result in longer times of large
than of small radii—again supporting inflow against outflow of gas. These effects
can outbalance the high Laplace pressure, and an acoustically driven bubble can
finally accumulate gas from the liquid and grow in size if it is oscillating strongly
enough. The boundary beyond which the bubbles grow is termed rectified diffusion
threshold, and it can be crossed by increasing the driving pressure, the bubble size,
or the liquid saturation level (for a more detailed quantitative discussion, see for
instance [8,58]). In conclusion, the rectified gas diffusion in driven bubble systems
leads to a parameter regime where single bubbles grow in size, as opposed to the
complementary parameter region where they dissolve. The rectified diffusion
threshold thus constitutes a parameter boundary for active cavitation bubbles: Only
beyond it, one expects relevant bubble populations. The growing bubbles, of
course, will finally encounter shape instabilities and therefore be limited in size, as
discussed above. Let us further note that for stronger driving pressures and small
bubbles, the rectified diffusion boundary approaches the Blake threshold, as shown
by Louisnard and Gomez [37]. Then, only active bubbles grow by diffusion.
Another important mechanism of bubble growth is by collision with other bubbles.
This is consequence of acoustic forces and discussed in the following.
1.12 Gas Diffusion 17

1.13 Bjerknes Forces and Bubble Translation
Acoustic cavitation bubbles are typically moving. The reason behind this lies in
acoustic forces, i.e., forces resulting from the interaction of bubbles and sound field.
Such forces have been derived first by Bjerknes [59] and are nowadays called
Bjerknes forces. An acoustic wave propagates variations of pressure pand density qin
space via the oscillatory motion of the medium and thus is described by spatially and
temporally varying quantities ~
pac x;tðÞ,~
qx;tðÞ, and ~
ux;tðÞ. The particle velocity is
indicated by u, and the tilde indicates that the variations are around an equilibrium
value p0,q0, and u0(the latter equals zero in the absence of a bulk flow). While the
local pressure p xb;tðÞat the position xbof the bubble drives its volume pulsation, it is
the pressure gradient rpxb;tðÞthat is responsible for an acceleration of the bubble
and therefore its spatial motion. Summing up the pressure forces over the bubble wall
leads to the simple expression FtðÞ¼VtðÞrpxb;tðÞas the resulting momentary
net force on the bubble of volume VtðÞif the pressure variation on the scale of the
bubble size can be assumed linear (long acoustic wavelength approximation:
Rk¼c=f)[7,8]. A simple application is the buoyancy force acting in gravity.
Then, rp¼qgis the (constant) hydrostatic pressure gradient in a liquid of density q,
gbeing the gravitational acceleration acting downwards. The resulting buoyancy
force is directed upwards and results as Fbuoy ¼Vqg. In acoustic fields, both
pressure and pressure gradient are oscillating, and also the bubble volume varies with
time.
6
Thus, the time average of FtðÞis considered which is termed primary Bjerknes
force:FB1 ¼VtðÞrpac xb;tðÞ
hi
T. The temporal average is denoted by the brackets

hi
Tand is taken over the driving period T¼1=fsince a periodic variation with this
period is assumed (otherwise, longer time averages have to be taken). It is important to
note that this average and thus the Bjerknes force does typically not vanish, even if the
gradient (and possibly also the bubble volume variation) is a sinusoidal function of
time with zero time average. The reason is that bubble volume and gradient do not
oscillate independently, but rather with a fixed phase relation. The textbook case
considers a linearized bubble oscillation RtðÞ¼R01þsin xtþ/0
ðÞ½driven by
small amplitude harmonic plane waves, leading to analytical expressions (see for
instance [7,8]). The result is that for a plane standing acoustic wave field pac x;tðÞ¼
pacos kxðÞcos xtðÞwith the wave number k¼x=c, the bubbles larger than the linear
resonance size Rres are forced toward the pressure nodes, while bubbles smaller than
Rres are pushed toward the pressure antinodes. In plane traveling waves of the form
pac x;tðÞ¼pacos xtkxðÞ, all bubbles are pushed forwards (in þx-direction), and
those close to Rres feel the largest force. Important modifications of this picture appear
for the case of higher pressure amplitudes that lead to larger, non-sinusoidal bubble
volume excursions and stronger collapse oscillations. Figure 1.9 shows an illustration
6
Pressure gradients of the sound field are typically much larger than the hydrostatic pressure
gradient, and therefore buoyancy can often be neglected in the discussion of acoustic cavitation
bubbles. Only for larger bubbles and weak driving, buoyancy might supersede acoustic forces
which leads to a rise of the bubble to the surface.
18 1 Bubble Dynamics

of primary Bjerknes force direction including higher pain a standing wave field.
Notably, also bubbles smaller than resonance size experience a repulsive force away
from the antinode at sufficiently high pressure amplitudes (“Bjerknes force reversal”
[60,61]). This is an additional phenomenon that limits access to very strongly driven
bubble collapse: Exceedingly expanded bubbles tend to leave the high pressure zones
in standing waves.
The secondary Bjerknes force FB2 is similar to the primary one, but results from
the scattered sound field psc from a neighbored bubble instead of the incident sound
field pac:FB2 ¼VtðÞrpsc xb;tðÞ
hi
T. To a certain approximation, the force of
bubble 2 at position xb2 exerted on bubble 1 at location xb1 can be expressed as
FB2 ¼4pq _
V1tðÞ_
V2tðÞ

Txb1 xb2
ðÞ=d3which means that its strength decays
with the reciprocal squared bubble–bubble distance d¼jxb1 xb2 j[62]. If sizes
and oscillations of the neighbored bubbles are similar, the force leads typically to
mutual attraction. Only for bubbles oscillating in antiphase (like one larger and one
smaller than the resonance size), the force is repulsive [8,63]. Again, nonlinear
bubble oscillations in the stronger driving regime as well as coupling between the
bubbles generate more complicated and often larger secondary Bjerknes forces
[62]. In the most cases, however, the secondary Bjerknes forces lead to strong
attraction of adjacent bubbles and finally to coalescence. Figure 1.10 is illustrating
such a case for two bubbles that oscillate such strongly that sonoluminescence
flashes occur in their collapse. The blurry darker silhouettes track the bubble
maximum size, while the flashes mark the points of bubble collapse. Both outline
the bubble paths, and the bending of the trajectories toward each other as well as the
strong bubble acceleration right before coalescence can be perceived.
Figure 1.11 illustrates the partly complicated behavior of the secondary Bjerknes
force at higher driving pressures. At low driving amplitudes, the parameter plane of
Fig. 1.9 Primary Bjerknes forces in a standing acoustic wave of 20 kHz (left) and 1 MHz (right).
The colors show the direction of the force for variation of sound amplitude paand bubble
equilibrium radius R0: White indicates a force toward the high pressure regions, dark areas a force
toward the pressure nodes. The regions are separated on the R0-axis at the resonance radii
(Rres 140 lm for 20 kHz and Rres 3:2lm for 1 MHz; water at normal conditions, isothermal
calculation with j¼1). Reproduced from [15], ©Universitätsverlag Göttingen 2007
1.13 Bjerknes Forces and Bubble Translation 19

Fig. 1.10 Collision of two sonoluminescing bubbles under the action of the secondary Bjerknes
force. After merging, the flashes continue and get brighter (xenon bubble in phosphoric acid driven
at 36.5 kHz, recording at 5000 frames/s). Reproduced Fig. 2a from [64], ©American Physical
Society
Fig. 1.11 Sign distribution of the secondary Bjerknes force between two bubbles of equilibrium
radius R01 and R02 for fixed pressure amplitude pa. White areas indicate an attractive force between
the bubbles, and dark areas indicate mutual repulsion. Top row: f= 20 kHz, pa= 10, 50, and
200 kPa; bottom row: f= 1 MHz, pa= 10, 100, and 200 kPa (Rres as in Fig. 1.9; water under
normal conditions; j¼1). Reproduced from [15], ©Universitätsverlag Göttingen 2007
20 1 Bubble Dynamics

both bubble sizes is well divided into attraction and repulsion rectangles, as
described by the textbook case of linear bubble oscillations [8]. For higher driving,
the nonlinear resonances of the bubble oscillation lead to parallel (self-similar)
repulsive stripes, since rather different oscillation phases are found as well around
these resonances. At even higher excitation, additionally, the delay in collapse time
at higher nonlinear expansion comes into play: The pattern returns to roughly the
initial structure, but now with the Blake threshold radius playing the role of the
linear resonance radius.
1.14 Viscous Drag Force
The moving bubbles experience a counteracting force due to viscosity of the liquid,
the drag force FD. For this discussion, one usually employs the Reynolds number
Re ¼Rjubjq=l. From measurements at cavitating systems like that shown in
Fig. 1.6, one obtains typical bubble speeds of cm/s up to several m/s and typical
(average or maximum) bubble radii in the range of a few up to about 100 µm. For
water l¼103Pa sðÞ, this means that Reynolds numbers up to a few hundred have
to be considered.
Supposing first a non-oscillating bubble with a bubble centroid velocity ub
relative to the fluid, one finds for small Reynolds numbers Re the drag force
FD¼4plRub[9]. Here, it is assumed that the bubble surface is “free”in the
sense that the fluid molecules do not encounter shear stress. If the bubble is covered
with a hydrophobic substance, a no-slip boundary condition would be more
appropriate at the bubble wall, and the drag force at small Re increases to
FD¼6plRub. This is the classical result of Stokes’drag of a moving sphere in
fluids. In many cases, bubbles in real liquids (in particular water) are better
described by this no-slip formula [9,65]. This fact is apparently due to frequently
given hydrophobic contaminations. The drag on stationary bubbles at higher Re
becomes more involved, and corrections to the given expressions occur. For
example, Brennen [9] cites Klyachko [66] for a formula that fits data well up to
Re 1000: FD¼6plRub1þRe2=3=6

. If we now consider moving and oscil-
lating bubbles, the analysis becomes even more complicated, and one partly has to
rely on numerical calculations. An extended discussion has been given by
Magnaudet and Legendre [67] with a derivation of low and high Re limits, the latter
being FDtðÞ¼12plRtðÞub. It is noted by the authors that this expression also
applies for lower Re if the bubble wall velocity is sufficiently high, since the viscous
dissipation is then dominated by the oscillation (vs. the translation). Thus, the
formula is a good starting point to model the drag on strongly oscillating and
faster-moving cavitation bubbles, and it has been shown to work well for experi-
mental data [68].
1.13 Bjerknes Forces and Bubble Translation 21

1.15 Phase Diagrams
Going back to the parameter regions where active bubbles are to be expected in
ultrasonically driven cavitating systems, we put together the boundaries of shape
stability (SI), rectified diffusion (RD), and Bjerknes force reversal (BJ). For a plane
standing wave, one obtains a synopsis like those presented in Fig. 1.12, shown
there for 20 kHz and 1 MHz driving frequency (compare also the diagrams given
by Apfel [69] or Church [70]). In the plane of bubble radius R0and driving pressure
amplitude pa, the corresponding lines are depicted. The SI boundary is shown only
for n¼2, and the RD threshold is given for two different levels of relative gas
saturation (1.0 corresponds to full saturation at ambient pressure of 1 atm, 0.1 to
only 10% saturation). Circles indicate points of accumulation for the non-degassed
water, i.e., values of stable bubble sizes and driving pressures that are reached under
diffusional growth and the action of the primary Bjerknes force. Of course, not all
bubbles will finally have the indicated radius and be located at a position of the
indicated driving pressure amplitude, since the real cavitating liquid constitutes a
dynamic multibubble system of translating, interacting, merging, and splitting
bubbles. However, the values are a reasonable estimation of an “average”bubble in
the system. For instance, the 20 kHz case suggests equilibrium radii in the range of
5µm, and the measured distributions in Fig. 1.1b (there at 25 kHz) essentially
spread around this value. Note that this result is not trivial, although the nuclei
distribution shown in Fig. 1.1a for filtered tap water gives a similar size range. The
nuclei statistics is obtained under silent and static conditions, while the cavitation
bubble equilibrium sizes are result of complicated bubble and sound field inter-
action, as described above. Thus, the bubble population of a cavitating system is
based on a dynamical process. Often, it is justified to assume a stationary situation
of this dynamical system which allows a quasi-static analysis, as proposed in the
phase diagrams of Fig. 1.12. Note further that non-active bubbles of larger size,
Fig. 1.12 Phase diagrams for bubbles in a standing wave, indicating the lines of surface
instability (SI) and Bjerknes force inversion (BJ). Also, the thresholds of rectified diffusion are
given for water at gas saturation (RD 1.0) and degassed to 10% of saturation (RD 0.1). Left:
f= 20 kHz, right: f= 1 MHz (water at normal conditions, j¼1). Points of accumulation are
marked by a circle. Reproduced from [15], ©Universitätsverlag Göttingen 2007
22 1 Bubble Dynamics

complying R0[Rres, can be trapped in low driving pressure zones, e.g., near
standing wave antinodes. Their parameter region in the phase diagram stretches
along the pa¼0 axis from the linear resonance radius on to the right: beyond
R0140 lm for 20 kHz (not visible in Fig. 1.11) and beyond R03:2lm for
1 MHz. Since rectified diffusion into the bubble works only for larger volume
pulsations, these weakly oscillating bubbles tend to dissolve due to surface tension.
However, they can collect gas through merging processes with other bubbles that
are driven to the antinodes as well and grow nevertheless. Then, these passive
bubbles can rise due to buoyancy once large enough to overcome the trapping
primary Bjerknes forces.
1.16 Structure Formation
From the discussion of secondary Bjerknes forces above, it becomes clear that a
spatially homogeneous distribution of acoustic cavitation bubbles should be
unstable. Since similar neighboring bubbles attract each other, they will develop
clusters, somehow like stars in the universe form galaxies.
7
Furthermore, the spatial
distribution depends on the characteristics of the driving ultrasonic field. In par-
ticular, spatial modulations of pressure amplitude, as found in standing or decaying
wave fields, will cause bubble motion on scales of the acoustic wavelength via
primary Bjerknes forces and potentially separation of larger and smaller bubble
sizes. Indeed, acoustically cavitating systems usually develop bubble structures, and
specific patterns of spatial bubble locations are found. A variety of them has been
described in [71], and here, we show just a few typical ones in Fig. 1.13. Within a
structure, the bubbles undergo frequent collisions, merging, and splitting processes,
but overall the multibubble system stays in a stationary state that is recognizable as
a certain structure. In clustering patterns, strongly oscillating bubbles travel
inwards, attracted first by primary and later by secondary Bjerknes forces toward
other, already accumulated bubbles. After coalescence and growth, they become
shape unstable and split off tiny bubbles that remain passive due to their surface
tension pressure. These small, very weakly oscillating bubbles feel low Bjerknes
forces and are transported outside the structure again by liquid convection. The
liquid flow, which is considered as turbulent in and near the bubble agglomerates,
thus controls the motion and redistribution of these microbubbles. They are occa-
sionally perceived as a misty cloud around the bubble structure. Later, they dissolve
or serve as nuclei again.
7
While secondary Bjerknes forces indeed decay with the squared distance like gravitational forces,
there are differences in that stars move without friction and do typically not collide. Furthermore,
the secondary Bjerknes force changes for very close or far distances, and the “mass”of a bubble
depends on the driving pressure at its position. Nevertheless, partly interesting similarities exist
visually between bubble structures and galactic structures.
1.15 Phase Diagrams 23

1.17 Bubble Traps
While the bubble structures as a whole are quite stationary, the individual bubbles
within the structure are not. They are normally transient in space and time, and thus,
observations of details of their dynamics over a longer time interval are difficult. This
means that many theoretical considerations and models are hard to verify on a
single-bubble level, in particular the large radial oscillations and the rapid hard
collapse. However, it is possible to isolate strongly driven bubbles and drive them
under stationary conditions in so-called bubble traps. Such devices use a
non-cavitating acoustic standing wave field in a resonator of one or few acoustic
wavelengths size. Then, a single bubble is seeded that runs toward the pressure
antinode, where it is caught. If the bubble is small enough, primary Bjerknes forces
will ensure a capture against buoyancy. The peculiar point is here that the bubble
permanently performs extreme volume pulsations with strong collapses and can
show light emission—sonoluminescence—and chemistry—sonochemistry.
8
An
example for a rectangular resonator cell and the observed bubble dynamics of the
trapped, light-emitting bubble are shown in Fig. 1.14. Other geometries like cylin-
ders or spheres work as well. Seeding can be accomplished by electrolysis, by a
focused laser pulse, or by air entrainment of a droplet falling onto the liquid surface.
The liquid is usually degassed to avoid bulk cavitation, but also to provide for a
diffusionally stable trapped bubble that otherwise would grow by rectified diffusion
Fig. 1.13 Some typical cavitation bubble structures generated by strong ultrasonic fields in the
frequency range of 20–40 kHz. aDouble layers around nodal planes of a horizontal standing wave
field. bFilament in a traveling wave field in front of a transducer. cSnapshot detail of a filament.
dSmall bubble cluster at different phases of bubble oscillation: nearly collapsed at 8 ms, expanded
at 9 ms, intermediate bubble sizes at 10 ms. eFlare structure near a submerged transducer
(placed to the left) (Images cand dfrom [15], ©Universitätsverlag Göttingen 2007)
8
Inactive larger bubbles can be trapped at pressure nodes of a standing acoustic wave.
24 1 Bubble Dynamics

and rise. It turns out that under variation of the main external parameters, namely
acoustic pressure amplitude, dissolved gas type and content, and ambient pressure,
several stable regimes of bubble dynamics can be reached and observed (the driving
frequency is usually fixed by the trap’s geometry). Many important findings with
respect to sonoluminescence (SL) have been accomplished in bubble traps, the
phenomenon then termed single-bubble sonoluminescence (SBSL). See, for
instance, the seminal article by Gaitan et al. [72], experiments and discussions by
Putterman and coworkers [73,74] and the rich literature in Young’s book [10] and
Crum’s resource paper [75]. Results comprise the determination of sonolumines-
cence flash duration [76], spectral time traces of the flash [77], and many substance
and parameter studies on brightness and emission spectra [78,79], including evi-
dence of a plasma in the collapsing bubble [80,81]. Further, important experiments
with “levitated”(trapped) bubbles consider the observation of chemistry from a
single bubble [33,82–85] and bubble dynamics and stability studies [55]. Also, the
Bjerknes force reversal has been shown experimentally in a bubble trap [86,87].
Bubble traps constitute a unique environment to prepare stable bubble regimes
with extreme collapse phenomena, and they are ideally suited for experimental
observations. Several results discussed later in Chap. 2rely on trapped bubbles. On
the other hand, the traps are usually designed for individual bubbles, and
Fig. 1.14 Left: Rectangular bubble trap with a piezoelectric transducer glued to the bottom (cell
width 5 cm). The bluish-white spot in the middle of the upper part of the water-filled cuvette is the
light emitted by a stably oscillating bubble, visible on this photograph of 20 min exposure. Right:
Series of photographic short-term exposures of a similar trapped sonoluminescing bubble (line by
line, time between frames 500 ns, frame size 160 lm160 lm). The bubble appears dark in
front of the bright background. Radius–time dynamics correspond roughly to the curve of 130 kPa
in Fig. 1.3a, and the according large volume variations can be perceived. Driving frequency is
21.4 kHz, and thus, the 100 frames cover roughly one full acoustic cycle (both images courtesy of
R. Geisler, see also [16])
1.17 Bubble Traps 25

phenomena like translation, coalescence, or splitting are intentionally not captured.
Thus, it remains to be explored how far the captured bubbles represent members of
the multibubble environments. For instance, it is well known that optical emission
spectra can differ between multibubble sonoluminescence systems (MBSL) and
single SBSL bubbles. In particular, spectral features like prominent emission lines
are partly absent under SBSL conditions [88], most probably linked with the
absence of disturbances and instabilities that are present in MBSL environments.
Some aspects thereof are presented in the following. There is still a bridge to close
between the isolated active bubbles and a large bubble ensemble like in the
structures illustrated above. Experiments with few-bubble systems or individual
unstable bubbles might offer advances here [33], and this question remains a field of
active research.
1.18 Non-spherical Bubble Dynamics and Jetting Collapse
Above, we have treated instabilities of the spherical shape of an oscillating bubble
in form of unstable surface modes. These are triggered by incidental events from
pervasive fluctuations, and the exact bubble shape dynamics has a random or
non-reproducible component. A different source of non-spherical bubble dynamics
is given by an asymmetric environment of the bubble: Any deviation from isotropic
geometry can cause a systematic and reproducible bubble deformation by imposing
a gradient in pressure. Relevant cases are adjacent boundaries from walls, particles,
other bubbles, or a free surface. Furthermore, the time-varying pressure gradient of
the sound field and a hydrostatic pressure gradient due to gravity (buoyancy) will
disturb the symmetry. Last but not least, the pure motion of the bubble relative to
the liquid represents an asymmetry. However, an answer to the question to what
extent an initially spherical bubble shape will be significantly affected under the
asymmetries is not straightforward. It will depend on the bubble size, the relative
strength of disturbance, on the oscillatory bubble motion and on parameters like
surface tension and viscosity. For example, a non-oscillating bubble rising under
buoyancy deforms into an oblate ellipsoid at small velocities and into a spherical
cap at higher speeds [9]. An expanding and collapsing bubble under translation can
develop a liquid jet in direction of the relative motion [89]. This is illustrated in
Fig. 1.15 and described in more detail below. The jet is a peculiar phenomenon of
collapsing cavities that significantly affects the bubble and its vicinity: By piercing
the bubble and hitting the opposite bubble wall, the liquid jet changes the bubble
topology from a sphere to a torus. The subsequent bubble collapse therefore pro-
ceeds toward a ring, not toward a point. The interior gas phase of a jetting bubble
becomes less compressed than in a spherical collapse, and peak temperatures are
lower to some extent. One reason is that part of the collapse energy goes to liquid
kinetic energy of the jet flow and not to gas compression [90]. If one assumes an
inner structure of the gas like ingoing focusing waves or shocks [91,92], such
temperature enhancing mechanisms would also work less in a non-spherical
26 1 Bubble Dynamics

collapse. Additionally, the jetting process can lead to liquid phase being injected
into the heated gas which would have a cooling effect, but as well has impact on
potential chemical reactions [34,93,94].
9
Furthermore, circulation is introduced in
the liquid [89,96], which manifests itself in vortex flows around the bubble [97]
and small-scale turbulence. This is important for mixing and transport processes in
the liquid and can also increase shear forces exerted by the bubble oscillation [98].
1.19 Jetting at Solid Objects
Similar jets occur for bubbles collapsing close to a solid boundary, for example,
near objects [16,99,100]; see Fig. 1.16. Then, the jet develops toward the
boundary, and the bubble also approaches the object. Jet impact and flow, bubble
gas phase, and the induced vortex flows can directly interact with the boundary,
which is particularly relevant for erosion of material by cavitation [101] and for
ultrasonic cleaning applications [5,102–104]. In both cases, bubbles collapsing in
direct vicinity of the surfaces are responsible for the observed effects of damage or
dirt removal [101,105]. The detailed mechanisms comprise jet impact and shock
waves for material damage, and additionally shear forces and sweeping of the
Fig. 1.15 Schematic sketch of a jetting bubble collapse induced by relative motion of bubble and
liquid (not to scale; variants of shapes and timings are possible). The lines indicate the bubble wall
in an axisymmetric cross section; the arrows represent liquid flow connected with the jet. For
simplicity, the inward flow during collapse (1–5) and outward flow during rebound (6) are not
shown. The jet hits the opposite bubble wall at instant (4), and the afterward torus-like bubble
collapses further (5). In the re-expansion (6), the jet flow persists and entrained gas forms a
characteristic bulge or peak (“nose”)
9
Details of liquid injection are still subject of investigation. At least, three scenarios could take
place: (I) During re-expansion of the bubble, the spherical shape is roughly restored, and remnants
of the jet might disintegrate into droplets, remaining in the gas phase until the next collapse
happens. (II) The jet impact onto the opposite bubble wall can cause nanosplashes that disintegrate
into droplets [95]. (III) The rear side of the bubble might become unstable and split off droplets. In
this context, note the non-smooth bubble backside in Fig. 1.17.
1.18 Non-spherical Bubble Dynamics and Jetting Collapse 27

contact line (gas/liquid/solid phase boundary) for cleaning. Jets occurring at solid
interfaces are very well investigated due to their technical relevance, and a large
literature exists on experiments and modeling (see, for instance, [14,101,106,107]
as a starting point). The main parameter for the dynamics is the normalized standoff
distance D¼D=Rmax where Dis the distance between the center of the expanded
bubble and the solid wall, and Rmax is the maximum bubble radius before collapse.
Although the bubble behavior scales rather well with Din a large range of
parameters and irrespective of the real bubble size, the phenomena are still very
complex, since the flows depend partly sensitively on the initial conditions [108].
Therefore, many details of bubble collapses at walls are still under investigation,
and interesting findings could be reported recently with improving experimental
and numerical tools. For instance, the vortex flow induced by the jet is inverted if
the bubble collapses relatively close to the wall, i.e., Dbeing smaller than about
1.35 [97]. Other work is concerned with details of jet impact and thereby produced
secondary “nanojets”that can cause liquid transport into the gas phase [95]. Further
actual research in this field deals with scaling approaches [109], more complicated
surface geometries [110], or jets in microchannels [111] where viscosity and sur-
face tension have to be taken into account.
1.20 Translation-Induced Jetting
The jet developed in the collapse of a moving bubble can be understood as a
consequence of conservation of momentum [89]. If we consider a one-dimensional
translation of a spherical bubble, the bubble position xtðÞ can be approximately
described by the following equation that is solved simultaneously with the RP or
Keller–Miksis equation (1.4), (1.5) given above:
Fig. 1.16 Jetting bubble near a solid boundary for D1:4. The boundary is at the bottom,
visible via the reflections. The black silhouette is from an experimental recording where a laser
pulse-seeded bubble collapses. The lighter gray (online: blue) part corresponds to the central cross
section of an axisymmetric numerical calculation with the finite volume/volume of fluids method
(see Koch et al. [112]). Due to the cut through the bubble, its interior toroidal structure is visible in
the simulated data. The dark feature appearing atop of the experimental bubble after collapse
consists of secondary cavitation bubbles. They are induced by the toroidal collapse shock wave
(“counter jet”[113]) and is not captured by the simulations
28 1 Bubble Dynamics

mb€
xþ2p
3qd
dtR3_
x

¼4p
3R3d
dxp1x;tðÞþFdR;ub
ðÞ:
Here, mbis the gas mass in the bubble and Fdis a viscous drag force as described
before with the relative bubble velocity ub¼_
xuxðÞ. The term dp1x;tðÞ=dxis the
pressure gradient that invokes the translational motion. Its value is taken at the
bubble position as if the bubble would be absent, as is the value of the liquid velocity
uxðÞ. This means that the acoustic wavelength should be much larger than the bubble
size, similar to the Bjerknes force formulas given above. Figure 1.3b in Sect. 1.6
exemplifies bubble motion for increasing the driving pressure amplitude (which as
well increases the pressure gradient amplitude) in a plane traveling wave running in
positive x-direction. It is seen that weakly oscillating bubbles just gently move forth
and back, while a stronger collapse leads to a significant acceleration and a forward
jump of the bubble. This can be understood by momentary conservation of the
Kelvin impulse. The Kelvin impulse (or quasi-momentum) is a quantity related (but
not identical) to the momentum of the liquid displaced by the moving bubble, and it
is relevant for any motion of bodies through a fluid [114,115]. In our case, we can
assume that the product of relative bubble center velocity uband virtual mass mvof
the bubble is approximately conserved. The virtual mass (also added mass) repre-
sents inertia of the flowing liquid and adds to the gas mass. For a spherical body like
the expanded bubble, the virtual mass corresponds to half of the displaced liquid
mass, mv¼2pqR3=3, and thus is proportional to the bubble volume [9]. Since the
liquid density is much higher than the gas density (apart from the ultimate collapse
peak when the gas is extremely compressed), the virtual mass is the dominant part
responsible for the moving bubble’s inertia. During implosion, the bubble volume
and therefore the virtual mass shrink, and the bubble velocity has to speed up
accordingly (roughly by R3). This acceleration becomes prominent for a strong
collapse and is clearly visible in Fig. 1.3b as the jump of the strongest driven bubble
at the main collapse (further leaps appear at the subsequent after bounce collapses).
For very strong volume shrinkage and fast acceleration, the Kelvin impulse cannot
be sustained by the accelerated virtual mass of the spherical bubble alone, and the
forward liquid jet flow through the bubble occurs [89]. This jet is faster than the
bubble center velocity and carries significant momentum, thus overtaking (and
slowing down) the collapsed bubble [115,116]. Kelvin impulse is transferred into a
circulation (vortex flow) after the jet impacts the other bubble side, and thus, it can be
conserved by the topological change to a torus bubble [89]. Of course, spherical
bubble models cease to be valid at this moment, but the essential bubble motion until
the jetting is captured quite well by RP-like models, as exemplified in Fig. 1.3b.
Analysis of the detailed flow dynamics around jetting bubbles, as well as the interior
gas heating and potential liquid injection, requires much more sophisticated models,
and such questions are topic of actual research.
Some experimental images of jetting bubbles in ultrasonic fields are presented in
Fig. 1.17. These bubbles, far from boundaries, are moving relatively fast due to
primary Bjerknes forces, and the jet is induced by the described mechanism. It is
1.20 Translation-Induced Jetting 29

important to note that acoustic cavitation bubbles are virtually always moving and
thus are prone to undergo jetting collapses. However, the parameter thresholds for
the transition from a still spherical collapse (in the sense of simply connected gas
volume) to a torus collapse are not easily found, and one relies on numerical work.
Resonant bubble sizes and/or increased driving amplitude will lead to larger
expansion ratios and higher bubble velocities, finally causing a jet. For a traveling
wave, Calvisi et al. show some results based on the boundary integral method, and
it can be seen that indeed at sufficiently high driving, a jetting collapse will always
develop [90]. In the context of sonoluminescence of different colors at different
regions of the cavitation cloud (see [93,118] and compare Fig. 1.6a), there are
strong indications that bubble populations with and without jetting collapse are
causing the difference [34]: Jetting can inject liquid that contains non-volatile
components that are in turn responsible for the change of color.
1.21 Activity Mapping
We have seen that an acoustically cavitating system is composed of a distribution of
bubbles, and not all are active in a specific, desired sense. For instance, sonolu-
minescence measurements from a bubble cloud will always monitor emission
averaged over all emitters, and it remains unclear if spectral features are shared by
the full population or just come from relatively few but relatively strong sources.
Fig. 1.17 High-speed recordings of moving and jetting acoustic cavitation bubbles. Series aand
bshow air bubbles in water below an ultrasonic horn, driven at 20 kHz (from [117], recordings at
250,000 frames/s). Collapse and jet take place between frames 1 and 3 (from left). The
re-expansion of the bubbles is rather unstable, and shape deformations and bubble split-off are
triggered by the jet. Series cshows selected frames of a high-speed recording of a xenon bubble in
sulfuric acid (at 100,000 frames/s). The rebound is much more stable here, probably due to the
high viscosity of the acid. On the other hand, the rear side of the bubble ejects microbubbles on its
path [34], a feature not occurring in water here. Gas ejections on the backside of moving bubbles
have, however, been observed for bubbles in argon sparged aqueous NaCl solution; see [64].
Figures aand breproduced from [117], ©American Physical Society 2014. Figure creproduced
from [34], ©Elsevier 2017
30 1 Bubble Dynamics

Fig. 1.18 Collapse events and light emissions of an accelerating, relatively large bubble of xenon
in phosphoric acid. The bubble starts to develop stronger and stronger jetting during collapse, and
the light emission ceases (driving at 23 kHz, 150,000 frames/s, timing given in ls, frame width
470 lm). Reproduced from [64], ©American Physical Society 2017
Fig. 1.19 Sample images from high-speed recordings at different frame rates and magnifications
(xenon in phosphoric acid; top rows: 36.5 kHz; bottom rows: 23 kHz). Scales and recording
speeds indicated. It can be seen that some moving bubbles emit light in every collapse and also
that sonoluminescence can occur together with pronounced jetting. Reproduced from [64], ©
American Physical Society 2017
1.21 Activity Mapping 31

The same holds in principle for chemical reaction yields. Ideally, one would like to
resolve emissions or reactions in space and time and relate activity to specific
bubbles and their dynamics. This is yet not really possible in multibubble systems
since the light (or yield) is usually much too weak to be detected in the timespan of
a bubble oscillation or even bubble lifetime. Only in the case of the extremely
bright SL emission from xenon bubbles in sulfuric or phosphoric acid, a first step in
this direction has been demonstrated. By simultaneous high-speed monitoring of
bubble dynamics and light flashes, various aspects of light-emitting bubbles in
MBSL environment could be explored [64], for instance, the collision of lumi-
nescing bubbles, as shown in Fig. 1.10, or the dependence of flash brightness on
jetting strength, given in Fig. 1.18. A collection of different scenarios, observed at a
variety of exposure times, is presented in Fig. 1.19. As a perspective, a spatial
mapping of bubbles and their light emissions within a larger cloud or structure
might be provided, even spectrally resolved, if sensitivity and speed of technical
devices advance further in the future—which is not an unrealistic idea.
1.22 Beyond Adiabatic Compression
The most interesting part of bubble dynamics—the collapse—is the most difficult to
observe in direct imaging. Since scales are extremely fast and close to optical
resolution limits, further advancing steps in imaging techniques and in technology
have to be taken. Apart from an improvement of speed, sensitivity, and resolution
of standard optical imaging sensors, potentially as well novel approaches with
X-ray and electron microscope imaging are promising candidates. However, in any
case, a lot of information about the conditions in the collapsing bubbles can be
obtained from other measures: light emission spectra and chemical reactions. These
topics are treated in the following chapters, and anticipating the results there, we
will learn that the phenomena during the hot collapse phase can be even more
complex and intriguing than the rest of the bubble life. In our discussion of bubble
dynamics up to now, we did not go beyond a rather simple interior bubble model,
namely an adiabatically and homogeneously heated ideal gas. For many aspects like
shape instabilities or acoustic forces, this does not mean a severe shortcoming, since
the energy exchanged via light emission or chemical reactions is rather low and
often has no dramatic influence on the gross dynamics. However, if a realistic
picture of the few extreme nanoseconds is required, one has to go beyond adiabatic
heating. We have mentioned above models that include heat conduction, evapo-
ration and condensation, and chemical reactions. More information on such models
that employ ordinary differential equations is contained, for instance, in Yasui’s
booklet within this series [119]. Another method that by construction captures all
these additional issues is a molecular dynamics simulation. An example from [92]is
32 1 Bubble Dynamics

shown in Fig. 1.20. In the future, more information from such quite expensive
calculations and other advanced models is to be expected, giving insight into the
otherwise unresolved hot spot of collapsing bubbles.
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38 1 Bubble Dynamics

Chapter 2
Sonoluminescence
2.1 Introduction
The previous chapter presented physical characterizations of cavitation bubbles on
the microscopic scale, looking, e.g., on the bubble shape, on its stability and
evolution, and on the way bubble dynamics can explain energy focusing that leads
to sonochemistry and sonoluminescence. These latter two phenomena are macro-
scopic manifestations of acoustic cavitation and can also serve to characterize
bubbles and their activity. Micro- and macroscopic measurements are fully com-
plementary in the study of cavitation: While the former will bring information on
each type of bubble separately, in a time-resolved manner, the latter will deliver an
overall spatially averaged picture of the effects of the whole bubble population.
The present chapter focuses on the light emitted by cavitation bubbles at col-
lapse, the so-called sonoluminescence (SL), and in particular on its measurements
and on the information that can be derived from them. The various theories
developed to explain SL emission will not be detailed here. The reader is referred to
the book Sonoluminescence [1] by Young which also provides an excellent review
of the knowledge on SL up to 2005.
Afirst part presents experimental evidence that allowed to affirm that a plasma
forms in cavitation bubbles at collapse. The various information (temperatures,
density, and electron density) that can be derived from SL spectra using tools from
plasma spectroscopy are then described. A third part deals with the way mea-
surements of SL intensity under pulsed ultrasound can be used to estimate some
bubble size distribution.
Focus will be put on aqueous solutions and multibubble sonoluminescence since
these systems are the most relevant for sonochemical applications.
©The Author(s), under exclusive licence to Springer Nature Switzerland AG 2019
R. Pflieger et al., Characterization of Cavitation Bubbles and Sonoluminescence,
Ultrasound and Sonochemistry, https://doi.org/10.1007/978-3-030-11717-7_2
39

2.2 Experimental Evidence of the Formation of a Plasma
in Cavitation Bubbles
It is now widely recognized, both by sonochemists and by the plasma community
[2], that a plasma forms in cavitation bubbles at collapse. First irrefutable experi-
mental evidence of it was the observation in single-bubble sonoluminescence
(SBSL) spectra of sulfuric acid in the presence of 67 mbar of a rare gas of the
emission from electronically excited ions (Ar
+
,Kr
+
,Xe
+
, and O
2
+
)[3,4] and from
high-energy excited rare gas atoms (Fig. 2.1). The energy of the emitting excited
states is too high for their population to have a thermal origin. For instance, an
energy of >13 eV (translating into temperatures >150,000 K) is necessary to
populate the observed emitting states of Ar atoms. These emissions can only be
explained by excitation via high-energy particle (e.g., electrons) collisions origi-
nating from a hot plasma core [3]. Further evidence of the plasma formation was
given by O
2
+
emission. Indeed, the energy required to dissociate O
2
molecule
(5.1 eV) is lower than its ionization energy (12.1 eV), which means that in a
thermal process, O
2
molecules would dissociate, and O
2
+
would not be formed.
Though often considered, the plasma origin of the MBSL emission from
aqueous solutions was not that straightforward since no emission from excited ions
that would be formed in the plasma was observed. It is to be noted that this absence
does not mean that, e.g., Ar
+*
or H
2
O
+*
ions do not form but more probably that
they are non-radiatively de-excited (“quenched”) by collisions with water molecules
that are abundant in cavitation bubbles due to the high vapor pressure of water.
It was recently that definite evidence of a plasma formation during multibubble
cavitation in water saturated with a rare gas was provided by the observation of
emission from highly excited OH (C
2
R
+
) radicals (Fig. 2.2)[5]. The emission of
OH (C
2
R
+
–A
2
R
+
) cannot be accounted for in a thermal model, since it would
Fig. 2.1 SBSL spectrum
from 85% H
2
SO
4
with
67 mbar and an acoustic
pressure of 1.7 bar.
Reprinted from [4] with
permission; ©American
Physical Society
40 2 Sonoluminescence

require 16.1 eV to excite water molecules at the right level. On the contrary, this
emission has been reported in spectra of electrical discharges [6,7] through water
vapor and OH (C
2
R
+
) formation can be readily explained by electron impacts. The
emission of OH (C
2
R
+
–A
2
R
+
) was reported [5] to be very strong at high US
frequency, and to increase when the gas was changed from Ar to Kr to Xe, cor-
relating with the decrease in the ionization potential of the gas.
(a)
(b)
Fig. 2.2 a SL spectra of water under different rare gases at 20 kHz (adapted from [5]; ©Wiley);
bsimplified energy diagram of OH radical and wavelength range of its emissions
2.2 Experimental Evidence of the Formation of a Plasma …41

2.3 Plasma Diagnostics to Derive Information
on the Plasma from SL Spectra
Usual plasma diagnostics tools allow to estimate several plasma parameters, such as
temperatures, density, and electron density. Some of these tools have been used
coupled with sonoluminescence spectroscopy, though mostly in some particular
cases due to the peculiarities of the “sonochemical plasma”that will be detailed in
the following.
2.3.1 Temperature Determination
Determining the maximum temperature reached at collapse has been a long-time
quest. Since sonoluminescence emission occurs at the last stage of collapse, its
spectrum should reflect the most extreme conditions reached. The first adopted
approach (particularly used in SBSL measurements) was to derive a temperature
from the SL continuum shape, assuming that the origin of SL would be, e.g.,
blackbody or Bremsstrahlung emission. The main drawbacks of this approach are
that the calculated temperature is highly model-dependent and that one sole con-
tribution is considered to shape the SL spectrum.
On the contrary, molecular emissions provide a more direct approach since their
shape reflects the relative populations of excited levels. The following paragraph
does not intend to be a course on spectroscopy but to recall the most useful facts to
interpret spectra. In the wavelength range measured in SL (UV to near-IR), the
observed emissions correspond to electronic transitions e0!e00
ðÞ(see Fig. 2.2b).
Each electronic level comprises several vibrational levels v0
ðÞand each of the latter
several rotational J0
ðÞlevels (Fig. 2.3). The transition e0
;v0
;J0!e00
;v00
;J00
ðÞpro-
duces a spectral line of intensity I(k) given by:
Fig. 2.3 Scheme of the electronic, vibrational, and rotational levels
42 2 Sonoluminescence

IðkÞ¼ne0v0J0Ae0v0J0e00v00 J00 hm0Ukk0
ðÞ ð2:1Þ
where ne0v0J0is the population of the emitting rovibronic level, Ae0v0J0e00v00 J00 is the
transition probability between the two levels (tabulated), hm
0
is the energy differ-
ence between the two levels and U(k−k
0
) the line broadening function. It is
generally assumed that the electronic, vibrational, and rotational levels all follow a
Boltzmann law, each at the corresponding temperature (electronic temperature T
e
,
vibrational T
v
, and rotational T
r
). The population of the emitting rovibronic level is
then given by:
ne0v0J0¼ge0eEe0
kTe
Qe
gv0eEv0
kTv
Qv
gJ0eEJ0
kTr
Qr
ntot ð2:2Þ
where ntot is the total population of the considered species, kis Boltzmann constant,
ge0;gv0and gJ0are the (tabulated) degeneracies of the states, Qeis the electronic
partition function defined as (vibrational, Qv, and rotational, Qr;partition functions
are defined similarly):
Qe¼X
e
geeEe
kTeð2:3Þ
Thus, the population of each emitting level and, consequently, the intensity of
each spectral line and the global shape of the transition are determined by the three
temperatures T
e
,T
v
, and T
r
. The general case in a non-equilibrium plasma is
T
e
>T
v
>T
r
T
gas
where T
gas
is the gas temperature, i.e., the translation temper-
ature. If time scales are long enough, thermal equilibrium can be reached and all
temperatures have the same value.
Spectroscopic codes exist that calculate the emission shape for a given species
and given temperatures, like for instance Lifbase [8] or Specair [9]. They also take
into account broadening of the emissions by several parameters including the
experimental broadening and the pressure broadening.
Such an approach has been used on C
2
Swan (d
3
P
g
−a
3
P
u
) bands in the
pioneering works of Flint and Suslick [10] in silicone oil and of Didenko et al. [11]
in an aqueous 10
−3
M benzene solution, both sonicated at 20 kHz under Ar. Both
assumed unicity of the temperature, but considering the way it was determined (it
was chosen to tally with the relative intensities of C
2
bands) it corresponds to a
vibrational temperature. Fitting of the spectra resulted in the temperatures
5075 ±156 K for silicone oil and 4300 ±200 K for the aqueous benzene solu-
tion. The same approach was applied some years later on the emissions of Fe, Cr,
and Mo in the MBSL 20 kHz spectra of metal carbonyls in silicone oil [12] under
Ar, leading to similar temperatures: 4700–5100 ±400 K.
Concentrated acids were also studied because of their very bright sonolumi-
nescence that allows high-resolution spectroscopy. MBSL spectra of concentrated
phosphoric acid sonicated at 20 kHz under He [13] show the molecular emissions
2.3 Plasma Diagnostics to Derive Information on the Plasma …43

of OH (A
2
R
+
–X
2
P) and PO (B
2
R
+
–X
2
P), the fitting of which gave temperatures of
9500 and 4000 K, respectively. This difference in temperature was explained by the
emission arising from symmetrically collapsing bubbles for OH and by fast
moving bubbles for PO. In concentrated sulfuric acid sonicated at 20 kHz under Ar,
the emission of excited Ar atoms (4p-4 s manifold) indicates a temperature of
8000 K [14].
Interestingly, OH (A
2
R
+
–X
2
P) emission spectrum in MBSL [13] in concen-
trated phosphoric acid is quite similar to its emission in SBSL spectra of a rapidly
moving single bubble [15] in 65% H
3
PO
4
regassed with 67 mbar Ar (see Fig. 2.4)
and temperatures derived from simulations are very close. The moving SBSL study
allows to investigate the effect of the acoustic pressure: As expected, its increase
leads to a net increase in OH temperature. As for a change in the rare gas nature,
going over the series from He to Xe leads to an increase in temperature from
6000 to >10,000 K. Xu and Suslick [15] attributed this effect to the strong decrease
in thermal conductivity of the gas limiting the heat losses during bubble collapse.
Another explanation would be the strong decrease in the gas ionization potential
that makes plasma formation easier and increases its electron energy.
In all these studies, the unicity of temperature was assumed in the fitting pro-
cedure, and in most cases, this approach lead to consistent results. More recently,
however, a study [16] of the influence of the ultrasonic frequency on SL spectra of
aqueous tert-butanol solutions under Ar showed that fitting with a single temper-
ature was not possible in this case and that T
v
>T
r
. For instance, temperatures of
T
v
= 6300 ±1000 K and T
r
= 4800 ±1000 K were obtained at 20 kHz under Ar
in the concentration range 0.05–0.4 M, where C
2
Swan bands were highest. At high
frequency (204, 362, and 613 kHz), higher vibrational temperatures were obtained
[16], indicating that the plasma had a higher electron temperature. For example, the
simulation of Swan bands lead to T
v
= 8000 ±1000 K and T
r
= 4000 ±1000 K
Fig. 2.4 a Emission of OH (A
2
R
+
–X
2
P) in SBSL spectra of a rapidly moving bubble driven at
different acoustic pressures in 65% H
3
PO
4
regassed with 67 mbar Ar (reproduced with permission
from [15]; ©American Physical Society); bemission of OH (A
2
R
+
–X
2
P) in SBSL spectra of a
rapidly moving bubble driven at 2.4 bar in 65% H
3
PO
4
regassed with 67 mbar of a rare gas
(reproduced with permission from [15]; ©American Physical Society)
44 2 Sonoluminescence

at 362 kHz under Ar in the tert-butanol concentration range 1.10
−3
–5.10
−3
M.
Besides, replacing Ar by Xe at 20 kHz leads to a very high T
v
(14,000 K). This
effect was explained by the relatively low ionization potential of Xe providing a
higher electron temperature in the non-equilibrium plasma generated at bubble
collapse. Another interesting finding was that an increase in tert-butanol concen-
tration induced a decrease in T
v
and in the intensity of the SL continuum, up to a
certain concentration range where it then stayed constant (and where C
2
emission
has its maximum intensity). Obviously, an increase in concentration of this volatile
solute leads to an increase in the number of tert-butanol molecules inside cavitation
bubbles, molecules that cushion the bubble collapse, leading to a lower energy
concentration, and consume this energy. Hence, the lower SL intensity and T
v
.
Much higher concentrations of tert-butanol were needed at low US frequency to
quench SL, which was attributed to the smaller bubble size (i.e., to the higher
bubble surface/volume ratio) at high ultrasonic frequency (see Figs. 1.3 and 1.11),
favorable to vaporization of volatile molecules into bubbles, and to the fact that
high-frequency bubbles remain active for many more cycles than 20 kHz ones, thus
accumulating more hydrocarbon decomposition products.
Fig. 2.5 Molecular emission of OH (A
2
R
+
–X
2
P) and NH (A
3
P–X
3
R
−
) in SL spectra of 0.1 M
ammonia solutions sonicated under Ar at 20 and 359 kHz and their simulations with Specair [9].
Reproduced with permission from [17]; ©Royal Society of Chemistry
2.3 Plasma Diagnostics to Derive Information on the Plasma …45

Similar temperatures were estimated by fitting of OH (A
2
R
+
–X
2
P)andNH
(A
3
P–X
3
R
−
) emissions inSL spectra of 0.1 M ammonia solutions under Ar (Fig. 2.5).
The non-equilibrium state of theplasma was confirmed (T
v
>T
r
), as well as the reaching
of higher vibrational temperatures at high US frequency.
Table 2.1 compares the temperatures estimated from molecular emissions during
the sonolysis of aqueous solutions. Obtained temperatures are in general in a rela-
tively good agreement. From Table 2.1, it can be seen that higher vibrational tem-
peratures (more extreme conditions) are reached in the presence of Xe and at high US
frequency. The temperatures determined from Swan band emissions in aqueous
benzene and tert-butanol solutions at 20 kHz under Ar show some difference, indi-
cating that the nature of the solute impacts the conditions reached at collapse but also
the chemical reactions that lead to the formation of the emitting species and conse-
quently the obtained excited state [16]. Interestingly, all these temperatures measured
for aqueous solutions are not very far from the temperatures derived from OH (A
2
R
+
–
X
2
P) emission in MBSL spectra of concentrated phosphoric acid sonicated at
20 kHz under He (9500 K) [13], or from the emission of excited Ar atoms (4p-4s
manifold) in sulfuric acid sonicated at 20 kHz under Ar (8000 K) [14]. This may
appear surprising since such media are known to offer very bright SL due to their low
volatility and the high solubility of sonolysis products that make bubble collapse
particularly efficient [18]. This apparent inconsistency may be due to the formation of
Table 2.1 Rovibronic temperatures estimated from the simulations of molecular emissions in SL
spectra from aqueous solutions
Solution Gas US
frequency,
kHz
Transition T
v
,K T
r
, K References
Benzene
10
−3
M
Ar 20 C
2
Swan
(d
3
P
g
–a
3
P
u
)
4300 ±200 Didenko
et al. [11]
tert-
butanol
0.05–
0.4 M
Ar 20 C
2
Swan
(d
3
P
g
–a
3
P
u
)
6300 ±1000 4800 ±1000 Pflieger
et al. [16]
tert-
butanol
1.10
−3
–
5.10
−3
M
Ar 204/362/
613
C
2
Swan
(d
3
P
g
–a
3
P
u
)
5800/8000/
5000 (±1000)
5800/4000/
4000
(±1000)
Pflieger
et al. [16]
tert-
butanol
0.12 M
Xe 20 C
2
Swan
(d
3
P
g
–a
3
P
u
)
14,000 ±1000 2500 ±1000 Pflieger
et al. [16]
Ammonia
0.1 M
Ar 20 OH (A
2
R
+
–
X
2
P)
9000 ±1000 5000 ±500 Pflieger
et al. [17]
NH (A
3
P–
X
3
R
−
)
7000 ±1000 4000 ±500
Ammonia
0.1 M
Ar 359 OH (A
2
R
+
–
X
2
P)
13,000 ±2000 6000 ±1000 Pflieger
et al. [17]
NH (A
3
P–
X
3
R
−
)
10,000 ±1000 2200 ±500
46 2 Sonoluminescence

an optically opaque plasma core, whereby measured emissions would occur from the
outer “cooler”part of the bubbles.
Spectroscopy belongs to the very few experimental methods to estimate tem-
peratures. Another one is a chemical determination, where a temperature is esti-
mated by comparing the yields of two reactions, the reaction rates of which are
known as a function of temperature. For instance, Ciawi et al. [19,20] studied the
kinetics of recombination of methyl radicals in the sonolysis of aqueous tert-
butanol solutions under Ar at different US frequencies. The chemical temperatures
they obtained were 3400 ±200 K for 20 kHz, 4300 ±200 K for 355 kHz, and
3700 ±200 K for 1056 kHz. Obviously, chemical temperatures are significantly
lower than vibrational temperatures and therefore than electron temperatures. This
difference can be explained by the fact that the chemical temperature is a mean
temperature that is not spatially and temporally well defined, whereas SL occurs at
the hottest point in the bubble life. What is more, this chemical model assumes
thermal equilibrium, and parallel reaction pathways are neglected. Despite every-
thing, it is noteworthy that the same trend of higher temperatures at high frequency
is observed by both methods.
2.3.2 Current Limitations of the Fitting of SL Emissions
Molecular emissions in SL present severe drawbacks when it comes to fitting them,
especially in aqueous solutions. First of all, they lie on top of an intense continuum.
Second, they are dim and very broad. Despite these limitations, in many cases it
appears possible to fit them, as seen above. Yet, one can find several examples in
the literature where fitting was not feasible. For instance, in Fig. 2.5 simulations of
OH (A
2
R
+
–X
2
P) and NH (A
3
P–X
3
R
−
) come very close to the experimental
spectra at 20 kHz but do not manage to reproduce them properly at 359 kHz. This
impossibility to fit properly these emissions at high frequency (and at any frequency
in the presence of Xe) was attributed by Ndiaye et al. [21] to a deviation from
Boltzmann law of the populations of excited vibrational levels of OH (A
2
R
+
),
whereby higher excited levels would be overpopulated. Flannigan and Suslick [22]
also reported a non-Boltzmann distribution, namely of the vibrational levels of SO,
in SBSL spectra from 80 wt% aqueous sulfuric acid solutions containing dissolved
neon, while it was Boltzmann for a 65 wt% sulfuric acid solution.
Another factor was recently [17] underlined that would need further develop-
ments to enable huge improvement in the fitting of SL spectra: the taking into
consideration of Stark effects in spectroscopy softwares. Indeed, in a dense plasma
like the one formed in cavitation bubbles, the main broadening mechanism is the
collisional one that consists in collisions with neutral species (the pressure broad-
ening) but also with charged species (Stark effects). The perturbation of the energy
levels by the electric field of charged species leads to two effects: a broadening and
a shift in emission wavelengths. Unfortunately, these Stark effects are not easy to
quantify for molecular species and the corresponding equations are pretty much
2.3 Plasma Diagnostics to Derive Information on the Plasma …47

species dependent [23]. These effects being negligible in low and atmospheric
pressure plasmas, they are at the moment not taken into account in existing spec-
troscopy softwares. Devoted developments would be needed to enable SL fitting
especially at high frequency and in the presence of Xe [17]. Indeed, in these cases,
the higher ionization degree of the intrabubble plasma means stronger Stark effects
and much more broadened emissions. Interestingly, the strong broadening of
emissions was not only observed in MBSL in aqueous solutions, but also in the
spectra of a moving single-bubble in phosphoric acid [15], in the presence of Kr and
Xe (i.e., the rare gases with lower ionization potentials), rendering the spectral
fitting impossible.
2.3.3 Pressure/Density Determination
The following section is adapted from the discussion on the pressure determination
in [17]. It is generally reported that a pressure of several hundred bars is reached
during acoustic cavitation, though direct measurements of it are lacking. The usual
approach [1] leading to a rough estimate of the maximum reached pressure is to
consider adiabatic compression of a bubble filled with a pure ideal gas. This
approach strongly relies on the value of the maximum temperature and on the
assumption of the unicity of this temperature—which, as seen above, is generally
not supported by SL spectra, especially in aqueous solutions. To take an example,
considering a pure Ar bubble in the adiabatic compression model, a peak pressure
of 146 atm is estimated for a maximum temperature of 2200 K, 1130 atm for
5000 K, and 6350 atm for 10,000 K. This simple approach can thus serve to
roughly estimate the conditions at collapse, keeping in mind the underlying
hypotheses and the problematic of the maximum temperature.
Spectral analysis (measurements of line widths or shifts) was used to experi-
mentally determine the maximum pressure. Indeed, an increase in pressure, i.e., in
the intrabubble density, leads to disturbance of the emitting species, of its lifetime
and of the energies of its emitting levels, effects that result in line asymmetry, shift
and broadening [24]. Quantification of these effects allows to estimate the relative
gas density inside the bubble at the time of emission. Since the value that speaks to
most people is the pressure, the obtained gas density is then often converted into a
pressure by assuming a certain temperature value.
In the first method, the pressure is derived from line shifts. Lepoint-Mullie et al.
[25] measured Rb resonance line shifts during the 20-kHz sonication of RbCl
solutions (aqueous solutions and in 1-octanol) under Ar or Kr and compared these
shifts, in the range 0.4–0.7 nm, with tabulated ones as a function of relative density.
A relative density of 18 ±2 was deduced therefrom. The same approach was used
by McNamara et al. [12] on chromium carbonyl solutions in silicon oil sonicated at
20 kHz under Ar. Comparison of the experimental shifts of the excited Cr*
emission with reference shifts from ballistic compression data gave a relative
density of 19 ±2, showing a perfect agreement with the value of Lepoint-Mullie
48 2 Sonoluminescence

et al. [25] From this relative density and taking into account a temperature of
4700 K, they estimated a pressure of approximately 300 bar.
The second used approach tries to correlate the line broadening to the relative
gas density. The underlying assumption is that the main sources of line broadening
are instrumental broadening and collisional broadening and that the others can be
neglected. Sodium showing a strong emission in SL, it was naturally most often
taken as a probe. Sehgal et al. [26] calculated a relative density of 36–50 in aqueous
alkali metal salt solutions sonicated at 460 kHz under Ar. In a similar system (NaCl
solutions sonicated at 138 kHz under Ar), Choi et al. [27] estimated the maximum
relative density around 59.5. Using perfect gas law and 4300 K for the maximum
temperature, they translated this density into a pressure of 873 atm. In these two
works, the pressure effect on the emission line broadening was calculated in the
model of collisional broadening. A similar but more empirical approach was used at
lower US frequencies (22 and 44 kHz) by Kazachek and Gordeychuk [28] who
studied the SL of NaCl aqueous solutions under Ar and estimated pressures of 800–
1200 bar, thus similar. Also, the pressure values derived from a spectral fit with
Specair [9] software of the emissions of NH (A
3
P–X
3
R
−
) and OH (A
2
R
+
–X
2
P)in
SL spectra of aqueous ammonia solutions under Ar [17] were of the same order of
magnitude: The pressure estimated from NH (A
3
P–X
3
R
−
) emission was 500 bar at
20 kHz and 1200 bar at 359 kHz, that estimated from OH (A
2
R
+
–X
2
P) emission
1400 bar at 20 kHz and 2000 bar at 359 kHz. All these values of relative gas
density estimated from line broadening under the assumption that the latter would
result from a combination of instrumental and collisional broadenings are clearly
much higher than those derived from line shifts. This apparent discrepancy confirms
the occurrence of strong Stark effects and that their contribution to line broadening
cannot be neglected. Stark broadening is highly dependent on the nature of the
emitting species [23], which explains the very different “pressure”values derived
from the emissions of OH and NH in [17].
Interestingly, in one SBSL study, [29] in sulfuric acid partially regassed with Ar,
the Stark effects were taken into account. In this work, three contributions to Ar line
broadening were considered: the instrumental one, the pressure broadening, and the
Stark broadening. The three contributions were decoupled and the reached density
was derived and converted into a pressure of 1400 bar. Though conditions achieved
in a single bubble in sulfuric acid may be far from those reached in MBSL bubbles
in aqueous solutions, this study exemplifies the possibility to separate the various
broadening contributions to derive information on the plasma, here in the relatively
simple Ar case.
2.3.4 Electron Density and Electron Temperature
In more conventional plasmas, electron properties are usually measured by an
electrostatic probe such as a Langmuir probe [30]. Due to the peculiarities of the
sonochemical plasma (in particular, its very small size), this technique is not
2.3 Plasma Diagnostics to Derive Information on the Plasma …49

available to sonochemists. Other techniques involve emission spectroscopy and
some of them may be used with SL spectra. In usual plasmas, the electron tem-
perature has been obtained using emission intensities from two different electronic
levels of a rare gas [31–33], though the latter was contained in a small quantity in
the plasma, contrary to usual SL cases. The spectral line broadening caused by the
Stark effect is also the basis of a very efficient plasma diagnostics method that can
be extrapolated to the sonochemical plasma. Indeed, for any emitter, the broadening
and shift in wavelength will depend on electron temperature and electron density
[23,34]. This dependency is complex and species dependent, but the method is
promising.
Measurement-based estimations of the electron density and temperature in SL
are very scarce. In water, it was long hindered by the lack of molecular emissions in
the spectra. First attempts thus relied on the light continuum: Lepoint et al. [35]
estimated electronic temperature (T
e
20,000 K) and density (N
e
=10
25
m
−3
)
from the SBSL continuum of an Ar bubble in water, considering its origin as being
radiative recombination and bremsstrahlung. This very first estimation was based
on hypotheses that condition the obtained values, namely that the bubble size at the
moment of emission would be 2 µm and the light pulse duration 50 ps. The same
group also estimated the electron temperature T
e
from MBSL spectra of water
saturated with Ar, by comparing the intensities of a line to that of the adjacent
continuum [36]. They observed an increase in this temperature with the US fre-
quency, but unfortunately data were not published and only a conference abstract
can be found.
In 2010, Flannigan and Suslick [37] analyzed the emission lines from elec-
tronically excited Ar atoms in SBSL spectra of concentrated sulfuric acid con-
taining Ar at 5% of saturation, and determined the plasma electron density as a
function of the acoustic pressure from the shape of the emission and in particular
from its deviation from a Lorentzian shape (Fig. 2.6). They also derived a
Fig. 2.6 Left: Ar emission line profiles as a function of the acoustic driving pressure, p
a
, in SBSL
spectra of concentrated sulfuric acid containing Ar at 5% of saturation; Right: comparison of one
SBSL Ar line profile with a Lorentzian curve. Figures reprinted with permission from [37]; ©
Springer Nature
50 2 Sonoluminescence

temperature from the fitting of Ar emission. In this particular case, fitting with one
single temperature was possible. Table 2.2 presents the values they obtained as a
function of the applied acoustic pressure (p
a
): both temperature and electron density
strongly increase with p
a
.
Apart from the works by Lepoint et al. [35,36], the determination of electron
density and electron temperature in MBSL in aqueous solutions remains for the
moment an almost blank territory. Due to the limitations discussed above, classical
plasma diagnostics methods could not be applied until now. Nevertheless, several
hints can be found in the literature that the electron energy can be quite high in
cavitation bubbles in water, especially at high US frequency. For instance, it was
shown that a significant number of O
2
molecules can be dissociated in cavitation
bubbles during water sonolysis at high frequency under Ar [38]. The dissociation
energy of O
2
being 5.2 eV, this means that a significant number of electrons have
an energy 5.2 eV. The same phenomenon was observed for N
2
molecules: Water
sonolysis in the presence of Ar and N
2
leads to NH (A
3
P–X
3
R
−
) emission in
MBSL spectra at high frequency but not at 20 kHz [39], whereby the most probable
formation mechanism of NH requires N
2
dissociation and an energy of 9.8 eV. The
same gas nature effect was observed in this study as in [5,15,16,21], namely that
changing Ar for Xe increases the electron energy: While under Ar–N
2
NH (A
3
P–
X
3
R
−
) emission was not observed at 20 kHz, it was clearly present when sonication
was performed under Xe–N
2
mixture.
2.4 Emission of Non-volatile Solutes in SL
It has been known since 1970 that sonication can excite non-volatile solutes and
that corresponding light emission can be observed on SL spectra [40]. Several
studies thus focused on SL spectra of salt solutions to try to derive information
either on the conditions reached in the cavitation bubbles at collapse, or on the
mechanism of formation of the excited species that emit light. They mostly focused
on two ion families: alkali metal ions and lanthanide ions.
Table 2.2 Temperatures and electron densities derived from Ar emission in SBSL spectra of
concentrated sulfuric acid containing Ar at 5% of saturation [37], as a function of the applied
acoustic pressure
p
a
, bar T
Ar
,K N
e
,cm
−3
2.7 7000 4.10
17
3.0 10,000 1.10
18
3.3 13,000 2.10
19
3.6 15,000 5.10
20
3.8 16,000 4.10
21
2.3 Plasma Diagnostics to Derive Information on the Plasma …51

2.4.1 Alkali Metals
In air-saturated aqueous solutions of salts [41], the SL spectra consist of a fea-
tureless continuum. On the contrary, in the presence of a rare gas, emissions arising
from electronic transitions appear in the SL spectra: OH (A
2
R
+
–X
2
P) that is typical
of aqueous solutions and emission from excited metal (e.g., Na, K, and Rb) atoms
accompanied by a blue satellite (Fig. 2.7). The latter corresponds to the transition of
an alkali metal–Ar exciplex, a van der Waals molecule formed within the cavitation
bubbles [25].
The mechanism of Na
*
(or other electronically excited alkali metals) formation
was long a subject of debate, whether inside the bubbles or at their interface. The
nowadays accepted mechanism is summarized in Fig. 2.8: Some volume of solution
is mechanically added to a collapsing bubble (by droplet injection); therein, the salt
molecules are released in the plasma phase and homolytically cleaved, producing
Na and Cl. The metal atoms are then electronically excited by three-body reactions,
leading to Na
*
. As for the electronically excited Na–Ar* exciplex, it would be
formed following a three-body collision with two rare gas atoms. It is to be noted
that two different excited states of the exciplex are populated, leading to the blue
satellite observed around 557 nm (B–X transition) and to the apparent line dis-
tortion of Na
*
peak toward higher wavelengths (A–X transition).
Emission spectra of electronically excited alkali metals were used to derive
information on the conditions reached at collapse, since their emission occurs inside
cavitation bubbles. In particular, several attempts aimed at estimating the intra-
bubble density, with the two methods presented above. The first one relies upon the
shift in wavelength of emission: Using it, Lepoint-Mullie et al. [25] determined the
intrabubble density in aqueous solutions sonicated at 20 kHz under Ar to be
18 ±2. The second method is based on the assumption that the pressure would be
Fig. 2.7 SL spectra of NaCl (a) and RbCl (b) solutions sonicated at 20 kHz under Ar. Adapted
with permission from [25]; ©Elsevier
52 2 Sonoluminescence

the main source of peak broadening and led to much higher relative densities, of the
order of 36–80 [26–28]. As discussed above, the latter values are overestimated
because this method neglects the strong broadening due to the presence of charged
species.
Recent spectroscopic studies of alkali metal solutions revealed that the global
emission of the metal and of its exciplex is in fact composed by the superposition of
two components that do not emit at the same time [42]: a thin, not-shifted emission
of the excited alkali metal atom and a broadened red-shifted emission of the
exciplex. Not only is the time of emission different, but also the spatial distribution,
suggesting that both emissions would arise from different bubble populations.
Interestingly, the broadened component was found to be prominent under He and
Ne, while the narrow one was enhanced under Ar, Kr, or Xe. Further research is
needed to clarify this observation.
Fig. 2.8 Scheme of the reaction pathways leading to Na
*
and Na–Ar* formations as proposed by
Lepoint-Mullie et al. [25]. Reprinted with permission from Lepoint-Mullie et al. [25]; ©Elsevier
2.4 Emission of Non-volatile Solutes in SL 53

It was also shown in recent literature that different bubble populations were
responsible for the emissions of Na
*
and of the SL continuum. Abe and Choi [43]
reported a difference in the timing distributions of both SL emissions during the
sonication at 137 kHz of a NaCl 2M aqueous solution saturated with Ar.
Spectroscopy of SL indicated that the evolution of sonoluminescing bubble pop-
ulation and of Na-emitting bubbles with gas flow, power, and frequency were
different [44]. Moreover, visual observations of SL from both concentrated sulfuric
acid solutions [45,46], phosphoric acid [47], and in aqueous solutions [43,48]
evidenced a spatial separation of continuum emission and alkali metal emission
(blue continuum and orange Na
*
emission). The former was attributed to
higher-temperature bubbles and the latter to lower-temperature bubbles [43]. It was
also suggested based on a bubble radius simulation that continuum emission would
result from smaller bubbles than Na-atom emission. The latter assumption was
confirmed by high-speed imaging in sulfuric acid at 23 kHz under Xe: [47]
Different cavitation bubble populations were observed in the zones of different SL
colors, namely slow and spherically collapsing bubbles in the blue-emitting zones,
fast bubbles subject to liquid jetting during their collapse in the red-emitting zone.
However, while it is clear that Na
*
emission arises only from jetting bubbles, it has
also been reported in Xe saturated phosphoric acid that SL continuum emission can
also be observed from large bubbles having a translational motion and therefore
showing jetting [49].
2.4.2 Lanthanides and Uranyl
A second chemical family has been a topic of interest in SL studies, namely the
trivalent lanthanide ions, and the similarly behaving uranyl ion (UO
2
2+
). Indeed,
these ions can be excited by two different mechanisms: either by photon absorption
or by collisions with highly energetic species (e.g., in radiolysis), with excitation
yields that depend on the ion nature. The emission spectrum is the same in both
cases. In a solution submitted to ultrasonic irradiation, the two mechanisms of
excitation are a priori possible.
The feasibility of the excitation of lanthanide ions under ultrasound was shown
by Sharipov et al. [50] for single-bubble and multibubble configurations at 20 kHz.
Their work was confirmed and extended to high frequencies by Pflieger et al. [51].
In SBSL, the sole excitation mechanism is by absorption of the photons of sono-
luminescence. In MBSL, photoexcitation is prominent for Ce
3+
, while collisions are
more efficient to excite Tb
3+
and Eu
3+
. Sharipov et al. explained this difference by
collisional excitation occurring inside bubbles and made possible by droplet
injections. As for uranyl ions, they are mainly excited by photons at low uranyl
concentrations ( 10
−2
M) while the collisional mechanism becomes important
above 0.03 M [52].
A major difference between sonoexcitation and classical photoexcitation was
brought to light [51,52] namely the presence of an extensive quenching of excited
54 2 Sonoluminescence

species. This non-radiative de-excitation was attributed to the enhanced collisional
(Stern–Volmer) quenching in the overheated zone around cavitation bubbles, by
chemical reactions with sonolytical products and by non-radiative relaxation via
coupling with OH vibrations of the solvent. This quenching leads to a decrease of
the luminescence apparent yield. Its extent can be reduced by complexation of the
lanthanide ions by citrate ions (respectively of uranyl by phosphate ions).
The ultrasonic frequency was shown to play an important role on the observed
yield of emission [52]. In the more simple case of Ce
3+
(where excitation occurs by
photon absorption with a yield of 1), for instance, the SL yield, defined as the ratio
of the number of emitted photons to that of absorbed photons, was 0.9 at 204 kHz
versus 0.4 at 20 kHz, indicating a much more intense quenching at low frequency,
possibly due to the larger size of the bubbles leading to higher quenching. On the
contrary, in the case of terbium, whose excitation mechanism is mainly by colli-
sions, the apparent SL yield decreased with the US frequency, suggesting a
decrease in the excitation efficiency.
2.5 Bubble Size Estimation by SL Intensity Measurement
Under Pulsed Ultrasound
The emission of SL by cavitation bubbles at collapse was also taken advantage of to
estimate bubble sizes and bubble size distributions, in a method developed at the
University of Melbourne [53]. This method is based on the dissolution of bubbles in
a pulsed acoustic field and measurement of the SL light intensity. The principle is
depicted in Fig. 2.9.
During the pulse on time (t
on
), bubbles form and grow. Its value is chosen to
allow the formation of a population of active bubbles while avoiding non-desired
interactions between them. It is followed by a pulse off time, t
off
, during which
bubbles dissolve. Some bubbles can also coalesce, which would impact the bubble
size distribution, but this effect is generally neglected [53–56]. The size of the
Fig. 2.9 Evolution of the
bubble population under
pulsed ultrasound. Reprinted
with permission from [53]; ©
American Chemical Society
2.4 Emission of Non-volatile Solutes in SL 55

bubbles at the end of t
on
determines whether they fully dissolve during t
off
or only
partly, turning into bubble nuclei for the subsequent t
on
.Ift
off
is very short, the
dissolution underwent by bubbles (whatever their size) is only partial and they all
remain as nuclei for the next t
on
.Ast
off
is increased, more and more bubbles
dissolve below a critical size range. Thus, the number of nuclei present at the
beginning of t
on
decreases, and so does the number of active bubbles, which
illustrates in a decrease of the SL (or SCL) intensity. Figure 2.10 presents typical
SL versus “off-time”data observed for a 0.5 M NaCl solution under Ar at 355 kHz.
Around 700 ms, the curve shows a second inflexion point, after which the decrease
in SL intensity almost stops: An off time of 700 ms is sufficient to decrease the size
of the maximum number of active cavitation bubbles below a critical size range
allowing them to act as nuclei in the subsequent on-pulse. In the range of t
off
between the two inflexion points, each increment in t
off
leads to the dissolution of
bubbles with a corresponding size.
The bubble size and bubble size distribution are then derived from the SL
intensity evolution with t
off
, using the dissolution equation of a single stationary
bubble [57]:
DCs
qgR2
0
!
t¼1
3
RTqgR0
2Mcþ1
 ð2:4Þ
In this equation, Dis the gas diffusion coefficient, C
s
is the dissolved gas
concentration, q
g
is the gas density inside the bubble, R
0
is the radius of the bubble
before it starts to dissolve (it corresponds to ambient radius, i.e., the radius of the
bubble when the acoustic pressure is zero), t is the total dissolution time (t
off
), Mis
the molecular weight of the gas, Ris the universal gas constant, Tis the temperature
of the liquid, and cis the surface tension.
Measurements of bubble size distributions in water and 1.5 mM SDS at 515 kHz
under air [53] pointed out that the addition of SDS lead to a decrease in bubble size
0 200 400 600 800 1000
0.0
0.2
0.4
0.6
0.8
1.0
normalised SL intensity
off-time, ms
Fig. 2.10 Typical SL
intensity versus pulse off-time
curve; 0.5 M NaCl solution,
Ar bubbling, 355 kHz.
Reprinted with permission
from [56]; ©American
Chemical Society
56 2 Sonoluminescence

and in the width of the bubble size distribution. It was also shown [55], in
air-saturated solutions sonicated at 575 kHz, that sonochemically (SC) active
bubbles (measured in luminol solutions) were smaller than sonoluminescence
(SL) bubbles (in water). Another important result of these studies on bubble size
and bubble size distribution was the experimental confirmation that the radius of SC
bubbles decreased with an increase in ultrasonic frequency, accompanied by a
narrowing of the size distribution (Fig. 2.11)[55]. It is remarkable that despite the
assumptions made in this technique of bubble size determination (in particular, the
funding one, that nothing else but dissolution happens during the pulse off time),
the obtained sizes, at least in this particular case, are in pretty good agreement with
the calculated ones in Fig. 1.11.
Other experimental parameters that do affect the bubble size are the dissolved
gas nature and the presence of salts in solution. Measurements at 515 kHz [54]
indicated an increase in the bubble size when the gas was changed from helium to
air to argon. The same study showed that an increase in the salt (NaCl, KCl, and
NaNO
3
) concentration led to smaller bubbles. Brotchie et al. [54] explained this
effect arguing that the presence of salts in solution decreased the gas solubility.
These salts were also shown to reduce the extent of bubble coalescence, confirming
the importance of this mechanism in the bubble growth.
The effect of dissolved gas was further investigated [56] in aqueous NaCl
solutions sonicated at 355 kHz and submitted to continuous Ar or He gas flow
(whereas previous studies considered pre-saturated solutions). Similarly to previous
Fig. 2.11 Bubble size distributions for 213, 355, 647, 875, 1056, and 1136 kHz (the data for 875,
1056, and 1136 kHz have been scaled down by a factor of 4) in an air-saturated luminol solution.
Reprinted with permission from [55]; ©American Physical Society
2.5 Bubble Size Estimation by SL Intensity Measurement Under Pulsed Ultrasound 57

results on Ar or He pre-saturated solutions, the bubble size was reported to decrease
with increasing NaCl concentration. The continuous gas flow strongly enhanced
this decrease. Thus, it is the combination of several parameters that determines the
bubble size, including the gas concentration, controlled by the salt concentration,
and the number of cavitation nuclei, that are introduced, e.g., by a continuous gas
bubbling. Besides, the gas diffusion coefficient also appears to play a role in
defining the bubble size: The combination of high gas solubility and high gas
diffusion coefficient allows a faster bubble growth in each expansion cycle and
subsequent bigger bubble sizes.
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60 2 Sonoluminescence

Chapter 3
Sonochemistry
3.1 Introduction
The chemical effects of ultrasound were reported for the first time by Richards and
Loomis in 1927 for the processes of dimethyl sulfate hydrolysis and the reduction
of potassium iodate by sulfurous acid known as iodine “clock”reaction [1].
However, the most famous sonochemical reaction of water molecule splitting was
discovered two years later by Schmitt et al. [2]. This reaction has attracted a lot of
attention of researches for several reasons. First, hydroxyl radicals and hydrogen
peroxide formed during water sonolysis are widely used for sonochemical oxidation
of organic pollutants in aqueous solutions [3]. In addition, hydrogen produced
simultaneously with hydrogen peroxide was found to be effective for sonochemical
reduction of noble metal ions resulting in highly monodispersed metal nanoparticles
without addition of any side reagents [4–6]. Finally, the formation of hydroxyl
radicals and hydrogen peroxide is often deployed as a chemical dosimeter to
measure the specific acoustic power absorbed by solution submitted to power
ultrasound [7,8]. In fact, the dissociation of water molecule is a strongly
endothermic process (ΔG= 113 kcal mol
−1
). Therefore, appearance of sonolytical
products during ultrasonic treatment of water clearly indicates transient cavitation
which provides drastic conditions inside the imploding bubble required for the
rupture of O–H bond. Spectroscopic studies of multibubble sonoluminescence
described in the previous chapter revealed that sonochemical splitting of water
molecule occurs via electronic excitation mechanism. Water molecule can be
excited to A
1
B
1
,B
1
A
1
, and C
1
B
1
states [9]. The A
1
B
1
and B
1
A
1
states are repul-
sive, and excited H
2
O(A
1
B
1
) and H
2
O(B
1
A
1
) molecules dissociate yielding OH
(X
2
P) and OH(A
2
R
+
) radicals, respectively. By contrast, H
2
O(C
1
B
1
) molecules
relax through A
1
B
1
or B
1
A
1
states. At high ultrasonic frequency and in the presence
of easily ionized noble gases, such as Kr and Xe, highly energetic OH(C
2
R
+
) state
is also formed due to the electron impact in non-equilibrium plasma produced
during bubble collapse at these conditions [10]. Hydroxyl radicals in the ground
©The Author(s), under exclusive licence to Springer Nature Switzerland AG 2019
R. Pflieger et al., Characterization of Cavitation Bubbles and Sonoluminescence,
Ultrasound and Sonochemistry, https://doi.org/10.1007/978-3-030-11717-7_3
61

state as well as in the excited states enable various sonochemical reactions inside
the bubble and at the bubble/solution interface. Therefore, kinetics of OH∙radicals
or H
2
O
2
molecules’formation can be used for quantification of acoustic power
delivered to the system. This chapter will focus on the influence of several fun-
damental parameters, such as ultrasonic frequency, saturating gas, and some soluble
nitrogen compounds on chemical reactivity of multibubble cavitation in homoge-
neous aqueous media in connection with the recent data on multibubble sonolu-
minescence overviewed in Chap. 2.
3.2 Sonochemical Dosimetry
Sonochemical water splitting was observed for the first time due to the oxidation of
iodide ion in sonicated aqueous solutions [2]. Since that time, iodometric method is
often used to quantify the sonochemical activity [7,11]. In general, it is suggested
that iodide ion is oxidized by hydrogen peroxide or by hydroxyl radicals formed
during sonochemical splitting of water molecule:
H2OÞÞÞ ! HþOH ð3:1Þ
2H!H2ð3:2Þ
2OH !H2O2ð3:3Þ
H2O2þ2I
þ2Hþ!I2þ2H2Oð3:4Þ
OH þI!OHþIð3:5Þ
IþI!I
2ð3:6Þ
2I

2!I2þ2Ið3:7Þ
I2þI!I
3ð3:8Þ
This reaction scheme is true for water saturated with noble gases. However,
iodometric method is hardly applicable to aerated aqueous solutions submitted to
power ultrasound. Sonolysis of water in the presence of nitrogen yields nitrous acid
[12] which readily oxidizes iodide ion [13]. Moreover, this reaction is catalyzed by
dissolved oxygen [14]. Consequently, the quantification of kinetic data for this
system becomes uncertain. Obviously, the Fricke sonochemical dosimeter based on
Fe(II) oxidation [7] suffers from a similar drawback.
In terephthalic acid (TA) dosimetry, TA solution reacts specifically with a
hydroxyl radical yielding 2-hydroxyterephthalic acid which can be detected using
fluorescence spectroscopy [15]. It is noteworthy that only part of OH∙radicals
produced by cavitation bubble and reaching bubble interface reacts with TA,
62 3 Sonochemistry

whereas a significant fraction gives H
2
O
2
after OH∙radical recombination. The
kinetics of OH∙radical formation during multibubble cavitation in water also can be
measured using salicylic acid (SA) as radical trapping reagent [16,17]. The
products of SA sonochemical oxidation with OH∙radicals, such as
2,3-dihydroxybenzoic acid and 2,5-dihydroxybenzoic acid, can be measured by
HPLC technique.
Both TA and SA dosimetric systems are highly sensitive to OH∙radicals.
However, they require quite complex analytical equipment. Therefore, for many
practical uses the simple spectrophotometric method of H
2
O
2
concentration mea-
surement with Ti(IV) was used as sonochemical dosimeter [18,19]. In acidic
medium, Ti(IV) ions form stable yellow-colored peroxide complexes allowing
quantitative analysis of hydrogen peroxide formed during water sonolysis. It should
be mentioned that in the absence of OH∙or H
2
O
2
scavengers, the rate of H
2
O
2
sonochemical formation follows zero-order kinetic law in a wide range of experi-
mental conditions which is convenient for chemical dosimetry [18].
Whatever chemical dosimetric system, the amount of formed sonochemical
products should be normalized to specific acoustic power (P
ac
) in order to compare
different sonochemical conditions. Usually, P
ac
is measured by thermal probe
method presuming that most of acoustic power delivered to solution is transformed
to heat: [20]
Pac ¼qCp
DT
Dsð3:9Þ
where q(g mL
−1
) is the density of the sonicated liquid, C
p
(J g
−1
K
−1
) is the heat
capacity of the liquid, and DT
Ds(K s
−1
) is the initial heating rate of sonicated liquid
measured under quasi-adiabatic conditions when T increases linearly with time of
ultrasonic treatment. Using kinetic data and P
ac
, it is possible to calculate the yield
of the sonochemical reactions (G, lmol kJ
−1
) useful for comparison of sono-
chemical efficiency at different experimental conditions.
3.3 Effect of Ultrasonic Frequency
Since a long time, it is recognized that the increase of ultrasonic frequency leads to
more efficient OH∙radical production during water sonolysis [21]. However, the
quantification of frequency effect is still challenging because of the complexity of
phenomena occurring in solution when the ultrasonic frequency is increased:
diminishing of the resonance bubble size, acceleration of the bubble implosion,
modification of bubble size distribution as well as the geometry of the bubble cloud,
and increase of the rovibronic temperatures of intrabubble plasma [8,22–24]. In
terms of sonochemical activity, the most suitable approach to compare different
frequencies appears to be the use of sonochemical yield, G, defined in Sect. 3.2.
Several studies revealed that the yield of hydrogen peroxide during sonolysis of
3.2 Sonochemical Dosimetry 63

argon-saturated water reached its maximum between 200 and 400 kHz and then
progressively diminished [19,22,25]. The optimal value of ultrasonic frequency
can potentially be related to a significant decrease of the bubble volume with
ultrasonic frequency. For example, the linear resonance radius of the cavitation
bubble in water is equal to 10.0 lm at 358 kHz and to only 3.3 lm at 1071 kHz
[20]. Taken these values as the average bubble size, the bubble volume at 1071 kHz
would be almost 30 times smaller than that at 358 kHz, leading to a much smaller
relative active volume producing primary products of sonolysis. From Chap. 1,we
have seen, however, that the bubbles tend to shift their sizes toward the Blake
threshold (depending on the applied driving pressure), which would result in
somehow smaller values than the linear resonance radii. This view is corroborated
by the bubble size measurements reported at the end of Chap. 2where SL in pulsed
fields had been used. Thus, one might employ the same argument, but for the
measured radii (Fig. 2.11) of about 3.2 µm at 358 kHz and 1.5–2µm at 1071 kHz.
Still, the volume change would amount a factor of up to 10. Of course, the absolute
active bubble number is assumed similar to this comparison, which might not be
precise. In conclusion, the sonochemical activity at different frequencies derives
from the superposition of several, partly counteracting phenomena and is difficult to
predict. A smooth variation of yield with the frequency, including some optimum,
is to be expected if the comparisons are performed in a “fair”fashion, i.e., with
comparable powers. However, this problem touches the notoriously difficult issue
of scaling acoustic cavitation and its effects, and in many cases experimental
parameter tests cannot be fully avoided for optimization.
3.4 Effect of Oxygen
Several research groups reported a significant increase of H
2
O
2
sonochemical yield
in the presence of oxygen [19,26,27]. Maximal yield of H
2
O
2
was observed at
20–30 vol.% of oxygen in O
2
/Ar gas mixtures whatever the ultrasonic frequency.
Similar to neat argon, the optimal frequency for H
2
O
2
formation in O
2
/Ar gas
mixture is at 200–400 kHz as it can be seen from Fig. 3.1. Interesting is that
high-frequency ultrasound is approximately 5 times more efficient for sonochemical
production of hydrogen peroxide compared to 20 kHz ultrasound.
Enhanced sonochemical efficiency of high-frequency ultrasound is related to
intrabubble conditions. Study of multibubble sonoluminescence spectra revealed
more efficient O
2
dissociation at 362 kHz compared to 20 kHz which should lead to
the increase of G(H
2
O
2
)[19]. In general, sonochemical formation of H
2
O
2
in the
presence of oxygen can be described by the following scheme:
H2OÞÞÞ ! HþOH ð3:10Þ
64 3 Sonochemistry

O2ÞÞÞ ! 2O ð3:11Þ
OþH2O!2OH ð3:12Þ
HþO!OH ð3:13Þ
2OH !H2O2ð3:14Þ
HþO2!HO2ð3:15Þ
2HO
2!H2O2þO2ð3:16Þ
Scavenging of H atoms by molecular oxygen (reaction 3.15) is confirmed by
sharp decrease of H
2
formation rate in Ar/20%O
2
compared to pure Ar [19].
Relatively low G(H
2
O
2
) in pure O
2
most probably is related to reported argon
plasma quenching by O
2
molecules [28].
3.5 Effect of Nitrogen and Ammonia
The sonochemistry of nitrogen in aqueous solutions was pioneered in 1936 by
Shultes and Gohr [29]. They reported the formation of HNO
2
and NO
3
−
ions under
the effect of 900 kHz ultrasound in water sparged with air. Later, Miŝik and Riesz
[30] suggested that H
2
O
2
and NO
2
−
were the primary products of water sonolysis in
the presence of air and that NO
3
−
ion resulted from the secondary oxidation of
nitrous acid by hydrogen peroxide. According to Wakeford et al. [31], the highly
reactive oxygen required for nitrite formation from molecular nitrogen would come
from the dissociation of oxygen molecules inside the cavitation bubble:
Ar Ar/20%O2 O2
0.0
0.5
1.0
1.5
2.0
2.5
G(H2O2), µmol/kJ
gas nature
20 kHz
204 kHz
362 kHz
613 kHz
Fig. 3.1 Effect of saturating
gas and ultrasonic frequency
on the sonochemical yield of
hydrogen peroxide. T=
20 °C, gas bubbling at
80 mL min
−1
, no mechanical
stirring. 20 kHz, P
ac
=
33 W; 204 kHz,
P
ac
=41W; 362 kHz, P
ac
=43W; 613 kHz, P
ac
=
43 W. Reproduced from [19]
3.4 Effect of Oxygen 65

O2ÞÞÞ ! 2O ð3:17Þ
N2þO!NO þNð3:18Þ
NþOH !NO þHð3:19Þ
Then, NO is oxidized by OH∙radicals originated from water molecule splitting
or by O
2
molecules:
H2OÞÞÞ ! HþOH ð3:20Þ
NO þOH !HNO2ð3:21Þ
2NOþO2!2NO
2ð3:22Þ
NO2þOH !HNO3ð3:23Þ
In the presence of N
2
/Ar gas mixtures, sonochemical processes in water involve
the formation of intermediate NH radicals [12] are originated from the intrabubble
dissociation of N
2
molecules [32]. H
2
,H
2
O
2
, HNO
2
, and NO
3species were
identified as stable products of sonolysis in studied system [12]. It was shown that
the dissociation of N
2
molecules is more efficient at higher ultrasonic frequency in
agreement with more drastic intrabubble conditions at high frequency revealed by
multibubble sonoluminescence spectroscopy (see Chap. 2).
Formation of NH∙radicals was also observed during sonolysis of ammonia
solutions saturated with noble gases [33]. Ammonia is known to be volatile, and for
that reason its splitting most likely occurred inside the cavitation bubble simulta-
neously with water molecules:
NH3ÞÞÞ ! NH þH2ð3:24Þ
Chemical analysis of ammonia solutions submitted to ultrasound revealed the
formation of hydrazine. This process was attributed to the mutual recombination of
NH
2
∙radicals or to NH∙radicals scavenging with NH
3
molecules probably at the
cavitation bubble/solution interface:
H2OÞÞÞ ! HþOH ð3:25Þ
NH3þOH !NH2þH2Oð3:26Þ
NH2þNH2þM!N2H4þMð3:27Þ
66 3 Sonochemistry

NH þNH3þM!N2H4þMð3:28Þ
NH þOH !NO þH2ð3:29Þ
Similar to other sonochemical systems, yield of hydrazine is larger at
high-frequency ultrasound compared to 20 kHz. However, monitoring of hydrogen
peroxide indicates that, in contrast to pure water, H
2
O
2
does not accumulate during
the sonolysis of ammonia solutions, most probably due to the OH∙radicals scav-
enging by NH∙radicals (reaction 3.29) and/or NH
3
molecules (reaction 3.26)or
because of the rapid reaction of H
2
O
2
with hydrazine:
N2H4þ2H
2O2!N2þ4H
2Oð3:30Þ
The principal gaseous sonolytical product of ammonia solutions is hydrogen gas.
Higher yield of H
2
in ammonia solutions compared to pure water is undoubtedly
related to the sonochemical splitting of NH
3
molecules and secondary reactions
with nitrogen-containing intermediates.
3.6 Effect of CO and CO
2
Sonochemistry of CO and CO
2
in aqueous media is much less studied than that of
O
2
and N
2
. Nikitenko et al. [34] reported that the sonication of water with 20 kHz
ultrasound in the presence of CO/Ar gas mixture causes a drastic decrease in the
H
2
O
2
formation rate relative to that in pure Ar. Furthermore, sonolysis at these
conditions causes water acidification which is not observed in pure argon. More
recently, it was shown that the yield of H
2
during water sonolysis increases dras-
tically in the presence of CO (10 vol.% CO/Ar) [6] indicating that the suppression
of H
2
O
2
and water acidification is related to OH∙radicals scavenging rather than to
the decrease of global sonochemical activity:
CO þOH !CO2þHð3:31Þ
2H!H2ð3:32Þ
CO2þH2OHþþHCO
3ðpK ¼3:60Þð3:33Þ
Interesting is that at higher concentration of CO (20 vol.% CO/Ar) sonolysis
leads to the formation of carbonaceous products which pointed out the CO dis-
proportionation during bubble collapse [34]. It is worth noting that the amounts of
carbonaceous products formed during sonolysis are very sensitive to the experi-
mental conditions. Decreasing the CO concentration to 10% or heating the soni-
cated water to 45 °C resulted in a sharp decrease in their yield. Chemical analysis
revealed that the composition of these products is close to hydrated poly(carbon
3.5 Effect of Nitrogen and Ammonia 67

suboxide). The anhydrous cyclic polymer with a basic formula (C
3
O
2
)
n
is formed in
strongly non-equilibrium plasma [35]. At such conditions, endothermic CO dis-
proportioning (DH= 5.5 eVmol
−1
,E
a
= 6 eVmol
−1
) can be significantly acceler-
ated by the vibrational excitation of CO molecules:
COðm1ÞþCOðm2Þ!CO2þCð3:34Þ
CþCOðm0ÞþAr !CCO þAr ð3:35Þ
CCO þCOðm0ÞþAr !C3O2þAr ð3:36Þ
nC3O2!ðC3O2Þnð3:37Þ
where m
0
and m
1
;m
2
are the ground state and the excited vibrational levels of CO
respectively. The similarity of the solid products for the sonochemical and plasma
chemical reactions allowed to assume that the mechanism of ultrasonically driven
disproportionation of CO is similar to that in non-equilibrium plasma. This
hypothesis was confirmed by inverse kinetic isotope effect observed during sono-
chemical CO disproportionation. In fact, carbonaceous product was found to be
enriched with the heavy
13
C isotope (a=
13
C/
12
C = 1.053 −1.055). According to
semiclassical kinetic isotope effect theory, the reaction products should be enriched
by light isotopes because of their higher zero vibrational-level energy [36]. By
contrast, the vibrational excitation of CO molecules in non-equilibrium plasma
leads to reverse isotope effect since heavy isotopes have higher vibrational tem-
perature. The finding of reverse kinetic isotope effect was the first evidence for
non-equilibrium plasma formation during multibubble cavitation. Further studies of
sonoluminescence spectroscopy considered in Chap. 2confirmed this striking
phenomenon.
It should also be noted that CO can be formed during sonochemical degradation
of organic compounds, such as formic acid [25]. Carbon monoxide molecules
enable to diffuse inside the bubble and influence the overall sonochemical mech-
anism. Thus, C
2
Swan band emission in the sonoluminescence spectra of HCOOH
aqueous solutions observed at high-frequency ultrasound was attributed to CO
disproportionation followed by the formation of excited C
2radical: [37]
2CO !CO2þCð3:38Þ
CþCþM!C
2þMð3:39Þ
Carbon monoxide originated from HCOOH degradation and accumulated inside
the bubble can also prevent H
2
O
2
formation via reaction (3.31).
The sonochemistry of CO
2
was found to be very different from CO.
Sonochemical activity of CO
2
has never been reported at low (20 kHz) ultrasonic
frequency. At high frequency, addition of only 1% CO
2
to Ar causes a dramatic
decrease in total sonoluminescence intensity of water [37,38]. In terms of
non-equilibrium plasma model of cavitation, this effect can be attributed to effective
68 3 Sonochemistry

vibrational excitation of CO
2
molecules by collisions with electrons [35] which
leads to rapid dissipation of electron energy. On the other hand, Henglein [39] and
Harada [40] reported enhanced sonochemical dissociation of CO
2
at its very low
content in argon (0.03–0.04 mol fraction). Carbon monoxide and formic acid were
identified as principal products, but the yield of HCOOH was about 30 times
smaller than the CO yield. Finally, the inhibiting effect on sonochemical oxidation
of I
−
ions in aqueous solutions increases with CO
2
concentration in CO
2
/Ar gas
mixture which most likely is related to the general decrease of sonochemical
activity.
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References 71

Conclusion
Quite a large knowledge has been gathered on cavitation bubbles in the past years,
aiming at characterizing them from a physical and chemical point of view and at
understanding the impact of various experimental parameters on their behavior and
properties. Many works have dealt with bubble dynamics: measurements of the
evolution of the bubble radius and of the bubble shape with time together with the
development of more and more sophisticated models now allow to understand the
concentration of energy at bubble collapse, the emission of acoustic waves, the
formation of microjets, etc. Also the interactions between bubbles have been
studied, interactions that lead to the formation of the observed bubble structures
(clusters, filaments etc.). In parallel to the dynamics study of bubbles on
the microscopic level, two other directions were pursued to characterize acoustic
cavitation bubbles. The emission of light (sonoluminescence) at the bubble
collapse, and chemical reactions (sonochemistry). Sonoluminescence gives an a
priori direct insight into the conditions reached at collapse. Observed spectral
features clearly indicate the formation of plasma that is not at equilibrium. Though
more research is needed to properly characterize the formed plasma and understand
its formation (in particular the electron temperature and density are mainly
unknown), rovibronic temperatures of excited species were measured that indicate
the large impact of, e.g., the ultrasonic frequency or the nature of the dissolved gas.
These effects are in agreement with measurements of the sonochemical activity.
However, connections between the gained knowledge in each field remain poor,
which is partly due to the different scales of study: Bubble observations (dynamics,
shape, etc.) focus on the microscopic scale, while sonoluminescence and sono-
chemistry deal with measurements on a macroscopic scale. Therefore, the latter two
deliver an average picture of the effects of acoustic cavitation: averaged over space
and also over time. A second difference is in the definition of the systems of
interest: While measurements of bubble dynamics deal with a single or a few
bubbles, sonochemical systems are composed of many bubbles. The bubble pop-
ulation is broad and covers bubbles of very different sizes, active or passive,
spherical or strongly deformed, all of them interacting and evolving with time.
©The Author(s), under exclusive licence to Springer Nature Switzerland AG 2019
R. Pflieger et al., Characterization of Cavitation Bubbles and Sonoluminescence,
Ultrasound and Sonochemistry, https://doi.org/10.1007/978-3-030-11717-7
73

Furthermore, even the ultrasonic frequencies investigated can be different: High
frequencies are often more interesting for chemists (due to the higher chemical
activity provided), while measurements of bubble dynamics focus on low
frequencies (favored due to bigger bubble sizes and longer acoustic periods).
Some technical limitations may be overcome in future years, which will help
linking the knowledge gained from the various approaches (sonochemical activity,
sonoluminescence spectroscopy, and bubble dynamics). First, observations of the
bubbles may be extended toward higher frequencies if (even) faster cameras and
special microscopic techniques are developed. Second, there is a need to go beyond
the spatially and temporally averaged measurements of sonoluminescence spectra.
Since the amount of emitted photons of the systems of interest for chemistry cannot
be much increased, the sensitivity of the light detector should be improved to allow
spatial resolution. The objective is to link spectral features, chemical activity, and
the characteristics of each considered emitting bubble. As already observed (see
Sect. 2.3.1) different types of bubbles can emit light, with different corresponding
sonoluminescence spectra. The big challenge is now to get a full picture of which
bubbles emit which spectra (that reflect the formed plasma in their core) and to link
these observations to the chemical activity of each bubble. In a further step, one
might think of a better control of the desired bubble populations and their dynamics.
In conclusion, cavitation and collapsing bubbles form a quite complex tool,
employed since decades and not only for chemical applications. Nowadays, such
systems are much better understood than in the early days, and the picture becomes
more and more complete. Still there is plenty of work ahead, and we are convinced
that the field will remain active and surprising also in the future.
74 Conclusion