Content uploaded by Dario Gaitan
Author content
All content in this area was uploaded by Dario Gaitan on Jun 10, 2016
Content may be subject to copyright.
Sonoluminescence and bubble dynamics for a single, stable,
cavitation bubble
D. Felipe Gaitan, a) Lawrence A. Crum, Charles C. Church, and Ronald A. Roy
National Center for Physical•4coustics, University of Mississippi, Coliseum Drive,
University, Mississippi 38677
(Received 8 July 1991; revised 28 January 1992; accepted 29 January 1992)
High-amplitude radial pulsations of a single gas bubble in several glycerine and water mixtures
have been observed in an acoustic stationary wave system at acoustic pressure amplitudes on
the order of 150 kPa ( 1.5 arm) at 21-25 kHz. Sonoluminescenee (SL), a phenomenon
generally attributed to the high temperatures generated during the collapse of cavitation
bubbles, was observed as short light pulses occurring once every acoustic period. These
emissions can be seen to originate at the geometric center of the bubble when observed through
a microscope. It was observed that the light emissions occurred simultaneously with the bubble
collapse. Using a laser scattering technique, experimental radius-time curves have been
obtained which confirm the absence of surface waves, which are expected at pressure
amplitudes above 100 kPa. IS. Hotsburgh, Ph.D. dissertation, University of Mississippi
(1990) ]. Also from these radius-time curves, measurements of the pulsation amplitude, the
timing of the major bubble collapse, and the number of rebounds were made and compared
with several theories. The implications of this research on the current understanding of
cavitation related phenomena such as rectified diffusion, surface wave excitation, and
sonoluminescence are discussed.
PACS numbers: 43.25.Yw, 43.35.Sx
INTRODUCTION
The subject of this paper is the dynamics of bubbles in
acoustic cavitation fields of moderate intensities. Acoustic
cavitation may be defined as the formation and pulsation of
vapor or gas cavities in a liquid under acoustic stress. A par-
ticularly interesting phenomenon associated with acousti-
cally driven gas bubbles is the weak emission of light called
sonoluminescence (SL). This emission has been attributed
to the high temperatures generated during the rapid com-
pression of the bubbles caused by the action of the sound
field. Despite the extensive amount of research done on both
acoustic cavitation and sonoluminescence, many important
questions relating to the nature and dynamics of these phe-
nomena remain unanswered. Attesting to this uncertainty is
the multiplicity of existing models describing the mecha-
nisms of light production as well as the number of conflicting
views and observations of cavitation-related phenomena
found in the literature.
Until very recently, experiments in SL generally in-
volved "clouds" of (sonoluminescing) gas bubbles genera-
ted by various types of transducers immersed in different
liquids, and driven at high amplitudes.•-3 Due to the high
amplitudes needed to generate light, the bubbles were made
to pulsate nonlinearly and nonradially, causing them to
move about rapidly and erratically, and to break up often
into smaller parts. 4 Obviously, the determination of the pa-
rameters relevant to SL (bubble radius, driving pressure,
internal temperature and pressure, etc.) has been difficult
Present address: Naval Postgraduate School, Physics Department, Mon-
terey, CA 93943.
under these conditions. Researchers therefore have been
forced to describe such cavitation systems in terms of ensem-
ble average quantities, or to assume that SL was produced
only for a small range of parameter values. In this paper, we
report on a recently found region in the driving pressure-
bubble-radius parameter space in which a single, light-emit-
ting bubble can pulsate radially and stably for an indefinite
amount of time. Using this system, the radius of the bubble
has been recorded as a function of time and the timing of the
SL flashes has been measured relative to the sound field and
the bubble radius. The timing of SL has also been used to
observe the change of the bubble's equilibrium radius as a
function of time.
Cavitation has generally been classified in two types:
"transient" and "stable" cavitation. This classification
traces its origin to when the first visual observations of cavi-
tation activity were made, 5'6 and was introduced by Flynn 7
in order to describe these observations. Transient cavitation
was used to describe events that lasted only fractions of a
second, usually occurring at high pressure amplitudes.
These events were attributed to vapor or gas bubbles that
expanded to large sizes during the negative part of the pres-
sure cycle, after which they began to collapse. Because of the
large radius attained during the expansion, their collapse
was very rapid and violent, often resulting in the destruction
of the bubbles. According to Flynn, 7 a bubble may be repre-
sented by a transient cavity "if, on contraction from some
maximum size, its initial motion approximates that of a Ray-
leigh cavity..." or by a stable cavity "if it oscillates nonlinear-
ly about its equilibrium radius." Flynn later s defined the
expansion ratio, R,•ax/Ro, where Rma x is the maximum radi-
3166 J. Acoust. Soc. Am. 91 (6), June 1992 0001-4966/92/063166-18500.80 ¸ 1992 Acoustical Society of America 3166
us attained during a given acoustic period and Ro is the equi-
librium radius, and showed that bubbles pulsating with
Rm•x/Ro above this threshold usually developed into tran-
sient cavities, whereas bubbles below this threshold devel-
oped into stable cavities. Although transient bubbles are not
necessarily unstable, they tend to collapse very rapidly, pro-
moting surface instabilities that often result in breakup.
Thus, when the transient cavitation threshold is exceeded,
bubbles are, in general, not expected to survive for more than
a few cycles.
Sonoluminescence (SL) was first observed by Mar-
inesco et al. in 19339 when photographic plates submerged
in an insonified liquid became exposed. But it was not until
1947 that Paounoff et al. 1ø showed that the exposure of the
plates occurred at the locations of pressure maxima (re-
ferred to in this paper as "pressure antinodes" or simply
"antinodes") of the standing wave field. After several years
of investigation, it became clear to researchers that gas bub-
bles were directly responsible for the generation of light in a
sound field.
During the late 1950s and early 1960s, several experi-
ments were done in an attempt to find the relationship be-
tween the phase of SL emission and the phase of the sound
field. n-j5 The motivation behind these experiments was to
discriminate among the different models based on the con-
flicting predictions concerning the phase of the SL emissionß
For example, The Triboluminescence, Microdischarge and
Mechanochemical models predict the light to be emitted
during bubble growth, whereas the Balloeleetric, the Anion
Discharge, the Hotspot, and the Chemiluminescence models
predict it to occur during the bubble collapse. Although it
was not known then, the phase of SL relative to the sound
field is dependent on the experimental conditions such as the
insonation frequency, the initial bubble radii and the phys-
ical parameters of the host liquid ]6 and, for this reason,
those experiments were partly unsuccessful.
The general outline of the rest of this paper is as follows:
Section I describes briefly three mathematical models used
to describe the motion of acoustically driven bubbles. Some
of the basic ideas of nonlinear bubble dynamics necessary to
understand the experimental results are introduced in this
section. Section II contains a detailed description of the ap-
paratus and the experimental arrangements used to acquire
the data, including the calibration methods. In See. III, the
results of the different experiments are presented and dis-
cussed, including observations on the stabilization process
for a single bubble. Data from the single-bubble experiments
are presented first in order to facilitate the interpretation of
the multibubble experimental results. Single-bubble data in-
clude the radius-versus-time curves and the phase of SL
emission. Then, a study of the phase of SL in multibubble
cavitation fields is presented. Finally, conclusions drawn
from the experiments are summarized in Sec. IV.
I. THEORY OF BUBBLE DYNAMICS
A. Introduction
In this section, we present the mathematical formula-
tions used to describe the motion of a single bubble in a spa-
tially uniform acoustic field, which is assumed to vary sinu-
soldally in time. In the calculations, three different
formulations are used, the results of which are compared
with the measurements. These formulations are the Keller-
Miksis •7 radial equation with a linear polytropic exponent
approximation, the Keller-Miksis radial equation with the
more exact formulation for the internal pressure due to Pro-
speretti et al., • and Flynn's •ø formulation, which also in-
eludes thermal effects inside the bubble. These formulations
will be called KM-polytropic, KM-Prosperetti, and Flynn
formulations, respectively. Because of the large pressure am-
plitudes PA used in the calculations, only formulations that
included correction terms for the compressibility of the liq-
uid were considered. Following the procedure used by Kel-
ler and Miksis,'7 Prosperetti2o has obtained the following
equation:
R dPs (R,t)
+ , (I)
p•c dt
where R is the bubble radius, dots indicate time derivatives,
Pt is the liquid density, c is the speed of sound in the liquid,
P8 (R,t) is the pressure on the liquid side of the bubble inter-
face, andpA (t + R/c) is the time-delayed driving pressure.
Here, p8 (R,t) can be expressed in terms of the internal pres-
sure by the continuity of normal stress across the boundary
p• (R,t) = p• (R,t) -- 2a/R -- 4tz • (•/R ), (2)
wherepg is the gas pressure in the interior of the bubble, •r is
the air-liquid surface tension, and/•t is the shear viscosity of
the liquid. The relationship between the internal pressure
and the bubble radius is determined mainly by the thermo-
dynamic properties of the gas. Although this relation can be
assumed to be isothermal or adiabatic under some circum-
stances, a polytropic relation has often been used 7'•6'•7'•'n
of the form
pg =po(Ro/R) •, (3)
where g is the polytropic exponent, and Po is the internal
pressure of the bubble at equilibrium (R = Re ), defined by
Po =Poe + 2tx/Ro, (4)
where P•o is the liquid pressure at infinity. The value of g
varies between 1 and •, for isothermal and adiabatic motion,
respectively, ?' being the ratio of the specific heat capacities
Cp/C•. This exponent is calculated using a method de-
scribed by Prosperetti? ø
Although the polytropic approximation has been gener-
ally thought to be a fairly accurate model, an experimental
study by Crum and Prosperetti 23 has shown significant dis-
crepancies between theory and experiment for oscillations
near the harmonics of the resonance frequency of the bubbleß
Since compressibility and viscous effects are expected to be
small at the amplitudes used by Crum and Prosperetti, their
results suggest that thermodynamics plays a more important
role in the motion of the bubble than originally expected. As
3167 J. Acoust. Sec. Am., Vol. 91, No. 6, June 1992 Gaitan e! al.: Sonoluminesconco and single bubble dynamics 3167
a result, a more accurate expression for the internal pressure
has been obtained by Prosperetti et al. •8 Dropping the sub-
script, the differential equation for pg is given, according to
Prosperetti, by
(r- Or '
where Tis the temperature, K is the thermal conductivity of
the gas, and r is the radial distance. This equation can be
solved simultaneously with the radial equation ( 1 ) once the
gradient of the temperature field at the bubble wall is found.
For this purpose, Prosperetti et al. used the energy equation
that can be written as
ypR •
where
r = K(O)dO,
Or ) Or D V2r,
Oy •=•Y -•y--DP=R2
(6)
(7)
and T• is the ambient temperature. For the convenience of
having a fixed rather than a moving boundary condition, a
new variable was introduced
y = r/R(t). (8)
The Laplacian operator V 2 is with respect to the variable y,
and
D(p,T) -- K(T) _ 7/-- 1 K(T)T (9)
C,p, (p, T) 7/ p
where p is the gas density, and D the thermal diffusivity for
an ideal gas. The boundary condition for the vector r is
r(y = 1,t) = 0. (10)
The third formulation used to compare to the experi-
mental results is the Flynn formulation •9 in which the ther-
modynamics of the bubble interior are included. Just as in
the KM-Prosperetti formulation, Flynn assumes that the
pressure inside the bubble is uniform. An improvement from
the KM-Prosperetti formulation is that the energy equation
in the liquid surrounding the bubble is included. Preliminary
tests showed that, for the pressure amplitudes considered
here ( -- 1.5 atm), the temperature in a thin shell of liquid
around the bubble is increased significantly during the radial
minima. In this respect, Flynn's model should provide addi-
tional information on the thermodynamics associated with
the bubble motion.
Since the derivation of the Flynn formulation is rather
involved, only the radial equation is given here, using Pro-
speretti's nomenclature, as follows:
•-/ • [pB(R,t) --pA(t) -- P• ]
R ( •)dpB(R,t)
+ 1 - (11)
ptc dt
The major difference between this equation and the Kel-
ler-Miksis equation is the factor ( 1 - k/c) multiplying the
time derivative of the pressure just outside the bubble,
dpa/dt. Comparing the results of the Flynn and KM-Pro-
speretti formulations, it appears that this extra compressibil-
ity term makes a significant contribution when the bubble
wall velocities,/•, are high, reducing the strength of the col-
lapse. For the details of the Flynn formulation, the interested
reader is referred to Ref. 19.
Five major assumptions were made in deriving these
three formulations. Four of them are common to all the
models. These are: (a) the bubble remains spherical; (b) the
bubble contents obey the ideal gas law; (c) the internal pres-
sure remains uniform throughout the bubble; and (d) no
evaporation or condensation occurs inside the bubble. The
fifth assumption pertains to the thermodynamic behavior of
the liquid surrounding the bubble, and is different in each
formulation. These approximations place severe limitations
on the models, especially when the pulsations are large and
the wall velocities reach values comparable to the speed of
sound. For a more detailed consideration of the limitations,
see Reft 24.
The three formulations were solved numerically. The
first two, KM-polytropic and KM-Prosperetti, were solved
using an IMSL 25 integration routine. This routine employed
Gear's 26 "backward differentiation" technique. A spectral
solution scheme developed by Kamath and Prosperetti •*
was used to solve for the vector r in Eq. (6). Flynn's m9 set of
equations was solved using an improved Euler method. Five
DEC MicroVaxes in a cluster configuration were used to do
the computations.
B. Basic concepts of radial bubble motion
This section presents some general characteristics of the
motion of a single gas bubble driven sinusoidally. Unless
otherwise stated, the examples plotted in this section were
calculated with the KM-polytropic formulation, which re-
quired the least amount of computational time. The other
formulations give qualitatively similar results.
In Fig. 1, several examples of bubble response curves, as
predicted by the KM-Prosperetti formulation for water at
f= 21 kHz, have been plotted. Here, the normalized maxi-
mum radius (Rmax/Ro) during one period of the driving
frequency after the solution has reached steady state is plot-
ted as a function of the equilibrium radius (Ro) for several
2.5
Radius, Ro/Rres
FIG. 1. Theoretical bubble response curves calculated using Prosperetti's
formulation for water at PA = 0.143.5 atm, f= 21.0 kHz.
3168 J. Acoust. Soc. Am., Vol. 91, No. 6, June 1992 Gaitan et aL: Sonoluminescence and single bubble dynamics 3168
values of the pressure amplitude, PA, as indicated by the
numbers labeling each curve. Since the steady-state solution
has a period equal to that of the driving frequency, Rma n is
single valued. For PA = 0.1 atm (10 kPa), the maximum
response occurs when the normalized radius (Ro/Rr•) is
nearly equal to 1. Other peaks are seen at or near
Ro/R,• = 1/2, 1/3, 1/4 .... etc., each lower in amplitude
than the previous one. They are known as the harmonics of
the resonance response. These peaks, shown in Fig. 1, have
been labeled with an expression n/m, known as the order of
the resonance according to the notation introduced by Lau-
terborn? Here, n and rn are defined as follows: If T R is the
period of the bubble motion, T/the period of free bubble
oscillations, and T the period of the driving frequency, then
for a given steady state solution, we can express TR =mT
and T R = nTf The case when m = 1 and n = 2,3 .... has al-
ready been mentioned and are the well-known harmonics,
whereas the resonances when n = 1 and m = 2,3 .... are
called subharmonics. Thus the motion of bubbles near the
m = 1 peaks can be characterized by the number of minima
occurring in one acoustic period, T, after the solution has
reached steady state. For example, bubbles of radii Ro/R,•
near the 1/1 peak exhibit one radial minimum, bubbles near
the 2/1 peak exhibit two radial minima, and so on. Examples
of radial pulsations of bubbles near the 1/1 (top) and 2/1
(bottom) peaks are shown in Fig. 2. Since T/is proportional
to the bubble radius R o, the number of minima occurring in
one acoustic period should give some indication of the size of
the bubble, with the smaller bubbles having the larger num-
ber of minima.
At pressure amplitudes above 1 atm ( 100 kPa), the pul-
sations of small bubbles ( < 40pm) develop a characteristic
profile. An example era radius-time curve at large pressure
amplitudes is shown in Fig. 3 (center). This example is for a
21-pm bubble in water at PA = 1.2 atm. In general, the mo-
tion follows a relatively slow expansion during the first half
of the acoustic cycle when the pressure is negative, followed
by a rapid collapse and several rebounds. The word "col-
lapse" here refers to the first bubble contraction after Rma•
has been reached. The time when the radius reaches a mini-
mum value is usually called the collapse phase, •o and is
11.0 I 1.5 12.0 12.5 13.0
5
=-- •-
12.0 12 5 13.0 13.5 14.0
Tim• (acoustic periods)
FIG. 2. Theoretical radius-time curves calculated using the polytropie for-
mulation for water at PA = 0.8 arm, f---- 21.0 kHz, Ro/R,• = 0.7 (top),
and Ro/R,• = 0.39 (bottom).
'lime (acoustic periods)
FIG. 3. Theoretical radius- and internal temperature-time curves calculat-
ed using Prosperetti'.• formulation for water at P• = 1.2 atto,
Ro/R,• = 0.14, f= 21.0 kHz.
measured in degrees from the beginning of the negative pres-
sure half-cycle, where • = 0 ø. The beginning of the compres-
sion cycle corresponds to • = 180 ø. The collapse in Fig. 3, for
example, occurs near 220 deg.
The highest temperatures and pressures are generated in
the interior of the bubble during this most violent collapse.
Figure 3 (bottom) is a plot of the temperature inside the
bubble. The large temperature spikes are generated during
the violent, nearly adiabatic, collapses. It is during these col-
lapses that sonoluminescence is believed to be generated.
Similarly, high pressures are generated in the bubble interior
as the gas is rapidly compressed, as shown in Fig. 4 (bottom)
for the same condittons as Fig. 3.
Thus three quantities can be used to describe the radial
pulsations of bubbles at high pressure amplitudes: the maxi-
mum response or pulsation amplitude (R,•,•), the phase of
the collapse (•) and the number of minima (M), which
includes the first miuimum (i.e., the collapse). Note that the
number of minima increases as the equilibrium radius de-
creases.
Up to this point, all quantities have been given a single
value for each solution if, after reaching steady state, the
solutions have the same period as the driving pressure. But
this is not always the case especially at high pressure ampli-
tudes where nonlinear effects are more prominent. In this
4
9 0 9.5 10 0
Time (acoustic periods)
15000
FIG. 4. Theoretical radius- and internal pressure-time curves calculated
using Prospereni's formulation for the same conditions used in Fig. 3.
3169 J. Acoust. Sec. Am., VoL 91, No. 6, June 1992 Gaitan otal.: Sonolumino•cence and single bubble dynamics 3169
regime and under certain conditions, the bubble may display
subharmonic motion (rn = 2,3 .... ). For example, subhar-
monic motion of period two (rn = 2), also known as period-
two motion, occurs when the period of the radial pulsations
is twice the acoustic period ( TR = 2T). Here, the values of
i i
R .... •o and M •, where i = 1,2, alternate between two dif-
ferent values from one acoustic period to the next. Although
period 3, 4 .... etc., motions are theoretically possible, only
period-one or period-two solutions were found numerically
for the conditions used in the experiments. For the rest of
this paper, the superscript i will be dropped from the quanti-
ties R .... •½, and M, unless confusion is likely to arise.
Examples of period-two motion can be found in the re-
sponse curve shown in Fig. 5 (bottom) calculated for a bub-
ble driven at 1.2 arm ( 120 kPa) in water according to the
KM-Prosperetti formulation. This type of motion is indicat-
ed by a double value of the maximum radius for particular
values ofRo/R .... and is usually called a period-doubling or
pitchfork bifurcation. 2s The peaks observed in Fig. 5 are due
to nonlinear resonances of the radial motion encountered as
the equilibrium bubble radius changes; the height of these
peaks is usually dependent on the amount of damping in-
eluded in the theoretical model.
Similarly, a period-two bifurcation is shown in the
phase-of-collapse curve at Ro/Rr½ , •0.14 (20.3/zm) in Fig.
5 (top). Not only does the bifurcation occur in the same
location, but every change in the curve does as well; i.e., an
increase (decrease) in Rmax results in an increase (decrease)
in •c. This correlation is not surprising since a larger re-
sponse requires the bubble to spend more time during the
growth and collapse phase. Because a larger pulsation ampli-
tude results in a stronger collapse (other parameters remain-
ing constant), a greater phase of collapse corresponds to a
more violent collapse. Thus, after the period-doubling bifur-
cation occurs, the uppermost branch corresponds to the
larger pulsation amplitude and the stronger collapse. A
"stronger" collapse means a smaller collapse ratio
(R•in/Ro), which generates higher gas densities, tempera-
tures, and pressures. This is illustrated in Fig. 6 where the
maximum internal temperature ( Tmax ) and pressure (Pmax)
generated during each period has been plotted on the top and
100
0.09 0,04 0.06 0.08 0.1O 0.12 0,14
Equilibrium Radius, Ro/Rres
FIG. 5. Theoretical bubble response and phase of collapse curves calculated
using Prosperetti's formulation for water at PA = 1.2 atm, f= 21.0 kHz,
and Rr• = 150/•m.
0.02 0.04 0.06 0.08 0.10 0.12 0.14
Equilibrium Radius, Ro/Rres
15000 •
•00o0 •
s0o0 •
FIG. 6. Theoretical internal temperatures and pressures calculated using
Prosperetti's formulation for the same conditions used in Fig. 5.
bottom, respectively, for the same conditions used in Fig. 5.
Here Tm• x has been averaged over the bubble interior.
Again, note that these two quantities are directly correlated
to the pulsation amplitude. The close relationship among the
quantities R .... •c, T•, and Pm• will be useful in the next
section where the experimental results will be discussed.
II. APPARATUS AND EXPERIMENTAL PROCEDURE
The experiment was divided in two parts: ( 1 ) the mea-
surement of the radius-time curve of a single cavitation bub-
ble and (2) the measurement of the phase of the sonolu-
minescence (SL) emitted by single- and multibubble
cavitation fields relative to the phase of the acoustic pressure
and to the bubble pulsations.
In both experiments, an acoustic resonator (levitation
cell) was used to excite a standing-wave sound field in sever-
al glycerine/water mixtures. Using this apparatus, single-
and multibubble cavitation was generated at pressure ampli-
tudes in the range 1.0 < P• < 1.5 atm ( 1 atm = 100 kPa).
In the first part of the experiment, a single bubble was
held in position in the levitation cell at amplitudes large
enough to generate SL. Light from an Ar-ion laser was scat-
tered from the bubble and detected with a photodiode. The
amplitude of the scattered light, modulated by the radial
pulsations, was converted to radius via an experimental
transfer function in which the average bubble radius was
related to the dc component of the scattered intensity. By
this method, experimental radius-time (R-t) curves of bub-
bles pulsating periodically in a steady state were obtained.
The radial pulsation amplitude, the timing of the bubble col-
lapse, and the number of rebounds occurring in each acous-
tic cycle were obtained from the R-t curves.
In the second part of the experiment, SL from a single
bubble was detected with a photomultiplier tube in a light-
tight enclosure. By using an intermediate reference, namely
the pill transducer, the emission of SL was correlated in time
with the bubble radius thus obtaining the phase of SL emis-
sion relative to the bubble pulsations. The phase of SL rela-
tive to the sound field in single- and multibubble cavitation
was measured during thousands of consecutive acoustic cy-
cles in order to study the growth, breakup, and steady-state
3170 J. Acoust. Sec. Am., Vol. 91, No. 6, June 1992 Gaitan eta/.: Sonoluminescence and single bubble dynamics 3170
motion of bubbles in cavitation fields. The phase was mea-
sured with a computer-controlled time-to-amplitude con-
verter.
A. Apparatus
The basic apparatus used in the experiments consists of
a levitation cell plus the driving and controlling electronics.
A schematic diagram of this apparatus is shown in Fig. 7 (a)
(nonshaded blocks). In each experiment, a levitation cell
was filled with one of four different water/glycerine mix-
tures and driven by a 75-W amplifier connected to the cell
via an impedance matching coil with an inductance of ap-
proximately 4 mH. The amplifier was driven by a function
generator that was controlled by a process control and data
acquisition system.
A levitation cell is a container filled with liquid in which
a stationary acoustic wave is excited. The radiation force 29
exerted on the gas bubble by the stationary wave is used to
counteract the hydrostatic or buoyancy force, enabling the
bubble to remain suspended in the liquid indefinitely, effec-
tively removed from all boundaries. One cell was made from
two 3.6-cm-longX 7.5-cm-o.d. concentric piezoelectric cy-
lindrical transducers joined by a glass cylinder of approxi-
mately equal dimensions. The transducers were poled to be
driven primarily in the thickness mode. A thin, roughly 3-
mm-thick (i.e., •,• .... d ) Plexiglas disk was glued to the bot-
tom of the cell, providing a nearly free boundary at the fre-
quencies used in these experiments. A pill-shaped
transducer (0.5 cm diam X 0.3 cm thick) was attached with
epoxy to the outside of the glass to monitor the pressure
amplitude and the phase of the acoustic field in the cell.
Besides the cylindrical cell, a rectangular levitation cell
was used to photograph and observe the levirated bubbles
more clearly. These cells were constructed from one-piece
rectangular Pyrex containers with horizontal cross sections
of 4.5 X 4.5, 5.1 X 5. I, or 6.0 X 6.0 cm 2 and heights of 12.0 cm
with the top end open. A piezoelectric hollow cylindrical
Oriel 7182-1
PHOTO-
DETECTOR
SIDE PILL
TRANSDUCER
IDA?A; I
LEVITATION
CELL
Krohn-Hite
model 7500
MATCHING
COIL Hewlett-Packard [
model 3312A
OSCILLATOR
AM VCO
ISAAC II /APPLE II U ]
DATA ACQUISITION/•- }
{CONTaOL SYSTEM I
STOP [ START
{ AMp•. I
R585/C617 •
PMT / PREAMP
SIDE PILL
TRANSDUCER
CH#2 CH#1
•C•o• •oo [(s•c)
DIGITAL I {
OSCILLOSCOPE {
SYNC Nuke 8600A RMS
DATA I T
LEVITATION
CELL
BUBBLE
MATCHING
COIL
Kmhn-Hite
model 7500
Hewlett-Packard]
model 3312A
OSCILLATOR
DATA ACQUISITION
AND
CONTROL SYSTEM
FIG. 7. (a) Schematic diagram of experimental
apparatus used to record the scattered light in-
tensity as a function of time. (b) Schematic dia-
gram of experimental apparatus used to mea-
sure the phase of sonoluminescence.
3171 J. Acoust. Sec. Am., Vol. 91, No. 6, June 1992 Gaitan eta/.: Sonoluminoscence and single bubble dynamics 3171
transducer 5 cm long X 5 cm o.d. was glued to the outside
bottom of each container. As with the cylindrical cell, a pill
transducer was attached to the outside of a wall and calibra-
ted to provide a noninvasive way to monitor the acoustic
pressure in the liquid. Although the rectangular cell was not
as stable in frequency and amplitude as the cylindrical one, it
provided fiat optical surfaces through which gas bubbles
could be observed with excellent clarity. Schematic draw-
ings of the levitation cells can be found in Ref. 24.
When driven at its fundamental resonance, a typical lev-
itation cell generated up to 4 atmospheres of acoustic pres-
sure amplitude at the antinodes. During the experiments the
cell was always driven at resonance, i.e., between 20 and 25
kHz, depending on the cell and the liquid mixture used. The
particular stationary wave excited was determined by the
driving frequency. In all the experiments (using a cylindrical
cell), an (r,O,z) mode of (1,0,1) was used. The cell in use
was mounted on a 2 degree of freedom translation stage thus
allowing the positioning of the bubble anywhere in the plane
perpendicular to the At-ion laser. The entire apparatus ex-
cept for the laser was mounted on an optical table.
To record the radius of the bubble as a function of time,
linearly polarized light from a laser was scattered from the
bubble using the apparatus shown in Fig. 7(a) (shaded
blocks). The scattered light intensity is modulated by the
bubble pulsations and, if measured at the appropriate angle
( • 80 ø measured from the forward direction), this intensity
can be converted to an absolute bubble radius. This tech-
nique was developed by Hansen 3ø to size air bubbles under
similar conditions and has been used to record small ampli-
tude radial and nonradial bubble oscillations by Holt 3• and
Hotsburgh. 32 A more detailed description of this technique
and of the apparatus can be found in these references.
A 3-W water-cooled Lexel model 95 argon-ion laser op-
erated in the TEMoo mode was used as the light source at
0.8-W power level and at a wavelength of 488 nm. This pow-
er setting was determined empirically by using the maxi-
mum power possible before the energy absorbed by the bub-
ble was large enough to affect its motion. An optical rotating
polarizer provided a 1200:1 linear polarization ratio. The
scattered light was detected by an Oriel 7182-1 photovoltaic
silicon photodiode with integral preamplifier, a detection
area of 100 mm 2 giving a subtended solid angle of 0.04 sr. A
488-nm laser line transmission filter was used to reduce
background noise. The output of this diode was connected to
both a dc voltmeter, which provided information on the tem-
poral average radius of the bubble, and via a Stanford Re-
search Systems model SR560 low-noise preamplifier with
40-dB gain, to an ac-coupled LeCroy 9400 digital oscillo-
scope that recorded the scattered light intensity of the pul-
sating bubble as a function of time. The data were trans-
ferred from the oscilloscope to a Macintosh II computer for
analysis and graphical display.
The experimental procedure was straightforward, once
the bubble was stabilized in the radial mode. After position-
ing the bubble in the center of the laser beam, simultaneous
traces of the pill transducer output and scattered light inten-
sity were digitized by the oscilloscope and transferred to the
computer for permanent storage. The process was repeated
for each mixture at several values of the pressure amplitude,
PA using the same bubble whenever possible. Each trace cor-
responded to about 0.2 ms of real time, i.e., four to five
acoustic periods. For the lowest values of the pressure ampli-
tude when the signal-•o-noise ratio was too small, time aver-
ages over 100 traces were computed on the LeCroy oscillo-
scope. Because the scattered intensity was periodic over
several minutes, no information was lost in the time-averag-
ing procedure.
Figure 7(b) is a schematic diagram of the apparatus
used to measure the onset and the phase of the light emission
relative to the sound field (nonshaded blocks). Because of
the low intensity of the light emissions, an 8-ft 3 light-tight
box was built that allowed the use of a Hamamatsu model
R585/C617 PMT/preamplifier combination designed for
single photon counting applications. The dark current and
gain specifications of this system were 1 count/s and 105,
respectively, while the rise and fall times were 15 ns each.
This system provided a time resolution of roughly 1/1000th
of an acoustic period ( < 1ø).
The phase of SL flashes was measured relative to a refer-
ence signal consisting of a series of fast electronic pulses
(- 10 ns wide) generated synchronously with the sound
field. By measuring the interval between these pulses and SL
events, the phase of SL was determined. A schematic dia-
gram of the circuit used to measure this phase and details of
the method can be found in Ref. 24.
The host computer for the data acquisition and control
was a MAC II using the Labview © software package. The
precision of the data acquisition system was 12 bits, which
corresponded to a precision of A•b c = d-0.1 deg of the
phase. The total experimental error, a mixture of electrical,
acoustic and optical noise, was estimated at d- 1 deg.
The general procedure was as follows: After filling the
cell with the desired glycerine mixture, an amplitude and
phase calibration of the pill transducer was made using the
procedure described below. A bubble was then introduced
into the liquid and made to luminesce at the desired pressure
amplitude. After the light-tight box was closed, the phase of
SL was measured for different pressure amplitudes using the
TAC system and was stored in the computer for later analy-
sis. The system was periodically verified to be at resonance to
maintain the calibration.
Four different glycerine/water mixtures were used in
this experiment to obtain data for liquids with several differ-
ent values of the liquid density, viscosity, surface tension and
speed of sound. These mixtures were pure water, GLY21,
GLY35, and GLY42, where the two digits represent the per-
cent by weight of glycerine. The solutions were prepared
from 99.5% pure glycerine and deionized water. The mix-
tures were periodically vacuum-filtered through a 5-/tm Tef-
lon filter to lower the gas content. This procedure also served
to remove impurities after the liquid had been used for some
time. The physical parameters of these solutions are shown
in Table I.
B. Calibration
Two different methods were used to determine the abso-
lute acoustic pressure in the levitation cells. One method, the
3172 J. Acoust. Soc. Am., Vol. 91, No. 6, June 1992 Gaitan ot a/.: Sonoluminescence and single bubble dynamics 3172
TABLE I. Summary of the physical parameters of the solutions used in the
experiments.
Density Viscosity Surface tension Speed of sound
(g/cm •) (centipoise) (dyn/cm 2) (cm/s)
Water 1.00 1.00 71.5 148100
GLY2 ! 1.05 I. 54 69.4 152900
GLY35 1.08 2.47 68.5 165700
GLY42 1.10 3.30 68.0 169000
levitation technique, 33 gave a direct calibration of the pill
transducer from which a needle hydrophone (NDL1) was
calibrated. A second method was used, the comparison tech-
nique, in which the needle hydrophone was calibrated di-
rectly using a factory-calibrated reference (B&K 8103). A
single calibration constant for NDL1 was obtained from the
average of the two results. The pill transducer was recali-
brated at the beginning of each experiment (liquid mixture)
using NDL 1. During the experiments, the output of the side
pill transducer (instead of NDL1) was used to record the
acoustic pressure amplitude after being calibrated. The pres-
sure calibration was accurate to within + 0.1 atm whereas
the precision of the driving electronics was + 0.01 atm.
To determine the instantaneous bubble radius from the
scattered light intensity, an experimental intensity-radius
transfer curve was needed. Thus bubbles of different sizes
were levitated and the scattered light intensity recorded. The
size of the bubbles was determined by the rise-time method
described in Ref. 24 using the appropriate drag law. The dc
component of the measured scattering intensity for bubbles
between 20 and 80/zm for 42% glycerine mixture is shown in
Fig. 8. The solid line is a second degree polynomial fit to the
data. Curves for pure water, 21%, 35%, and 60% glycerine
can be found in Ref. 24. The experimental error was estimat-
ed to be + 3/tm.
The phase of the bubble pulsations was measured rela-
tive to the output of the calibrated side pill transducer. The
phase calibration was made relative to the oscillations of a
levitated bubble driven at a low pressure amplitude and far
from its resonance frequencies. As illustrated in Fig. 9, the
0.10
BuhMe Radiua (micros)
FIG. 8. Measured scattered light intensity versus equilibrium bubble radius
for 42% glycerine. The solid line corresponds to a second degree polyno-
mial fit.
' 012
i 0'3 a•'2 arm R.,'es=la$ rulers, ns
04 06
Radiu•. Ra/Rres
FIG. 9. Theoretical phase of bubble radius minimum of the steady-state
solution for 42% glyceline at PA = 0. I, 0.2, and 0.3 atto and]'= 23.6 kHz
used in the phase calibration of the light-scattering apparatus.
theoretical phase of the radial minimum in the region
0.6R,• <R o <0.75Rr• is not only nearly independent of R o
but also independent of the pressure amplitude. Since this
region corresponds to bubbles away from their harmonic
resonances, the phase corresponds to the only radial mini-
mum occurring each cycle of the driving pressure. This
phase was also found to be independent of the mathematical
formulation used. Thus, by knowing the phase of the bubble
pulsations relative to the sound field, the phase of the pill
transducer also could be found. Bubbles in this size range
were levitated and the simultaneous oscilloscope traces of
both the pill transducer output and scattered light intensity
were stored in the computer. The two traces were displayed
on the Macintosh II using graphics software. The time dif-
ference between minima was measured and the phase differ-
ence was calculated. The total error of the calibration proce-
dure was estimated to be + 5 deg. This error was mostly due
to the uncertainty in the measurement of the bubble size.
III. RESULTS AND DISCUSSION
A. General observations
The initial goal of this study was to find the range of SL-
active bubble radii in a cavitation field at pressure ampli-
tudes above 1 atto (100 kPa), that is, above the pressure
threshold for detectable light emission. The goal was
achieved by measuring the phase of SL relative to the sound
field and comparing it to the phase of bubble collapse pre-
dicted by the different theories of radial bubble pulsations.
Since the SL flashes are very short, if they are emitted at the
time of the bubble collapse (shown later), one can determine
the phase of collapse by measuring the phase of SL. Accord-
ing to the theories, the phase of bubble collapse varies over a
wide range of values depending on the pressure amplitude
and the bubble size, as shown in Fig. 5. It is thus possible to
estimate the bubble sizes that are active during cavitation by
measuring the phase of SL emission, assuming the theories
are correct and tha! SL is emitted during the collapse of the
bubble.
During the initial experiment, it was found that the scat-
ter in the phase of SL at times decreased until it was less than
3173 J. Acoust. Sec. Am., Vol. 91. No. 6, June 1992 Gaitan otal.: Sonotuminescence and single bubble dynamics 3173
_ 1 deg--i.e., below the noise level--indicating that the
phase was constant in time. It was observed that as the pres-
sure was increased the degassing action of the sound field
was reducing the number of bubbles, causing the cavitation
streamers to become very thin until only a single bubble re-
mained. The remaining bubble was approximately 20/•m in
radius and positioned at the antinode. At this point, the bub-
ble was remarkably stable in position and shape, remained
constant in size, and seemed to be pulsating in a purely radial
mode. With the room lights dimmed, a greenish luminous
spot the size of a pinpoint could be seen with the unaided eye,
near the bubble's position in the liquid. The luminous spot
was then located at the bubble's geometric center when ob-
served through a microscope. Although this luminescence
had been observed before by other researchers, t very sensi-
tive instruments were usually required. Furthermore, it had
never been observed for a single, stable bubble.
A single, radially pulsating bubble was stabilized in
slightly degassed water-glycerine mixtures with glycerine
concentrations less than 60%. A description of the visual
observations of the stabilization process as the acoustic pres-
sure was increased follows: After injecting a gas bubble with
a small syringe at about PA = 0.6 atm (60 kPa), the bubble
exhibited dancing motion indicative of surface waves and
asymmetric collapses? As the pressure amplitude was in-
creased, the dancing motion became more vigorous, causing
the bubble to fragment. A bubble cluster was then formed as
the residual bubbles moved around the parent bubble. The
bubble system looked more like a cloud and very often devel-
oped into what has been termed a "shuttlecock." When ob-
served through a microscope, the shuttlecock appeared to be
a cloud of microbubbles surrounding a larger bubble approx-
imately 50 •m in radius. Through the interaction with the
sound field, this cloud developed a definite pattern of motion
from one side of the bubble to the other. In contrast with
single bubbles, the position of the shuttlecock was not on the
levitation cell's axis but a small distance away from it (0.5-
1.0 cm). Microbubbles appeared to be ejected from the side
of the cloud directly away from the antinode and were imme-
diately attracted toward the cloud by Bjerknes forces, 3• so
that they moved around the cloud toward the antinode. As
the microbubbles reached the opposite end of the bubble
cloud from which they had been ejected, they were pulled in,
possibly by interbubble forces directed toward the center of
the cloud. Thus, a rotational pattern was established. The
shuttlecock was observed to emit a low rate of SL flashes.
It is important to note that a well-defined pressure
threshold, pt sT, was observed at which stability was reached.
In addition, a hysteresis effect was observed, i.e., after reach-
ing stability, the pressure could be decreased below the
thresholdpt sx and stability was maintained. In general, stable
bubbles could be driven at values of the acoustic pressure 0.1
arm below the stability threshold. Furthermore, if driven
above a certain pressure the bubble disappeared, possibly
due to dynamic instabilities. For convenience, we will call
these the lower, p}X, and upper, pS, t'r, stability thresholds, re-
spectively, where the subscripts indicate lower and upper
threshold and the superscript indicates a stability (rather
than instability) threshold. These thresholds are illustrated
in Fig. 10. In water, for example, pt sx was measured to be 1.2
atm,p• • at 1.1 atm, andpS•t x at 1.3 atm. The range of pressures
at which bubbles were stable was 1.1 <PA • 1.5 atm depend-
ing on the liquid mixture. For water, it was 1.1 •PA < 1.3 atm
whereas for a 42%-glycerine solution it was 1.3•PA < 1.5
arm. Thus higher concentrations of glyeerine required
slightly higher pressures to achieve radial stability. It was
also found that these ranges could be enlarged (toward the
lower pressures) by decreasing the dissolved gas content of
the liquid. Also, as the amount of dissolved gas increased,
regions of instability within the stability pressure range be-
gan to appear; however, the upper pressure threshold, pSat T, at
which the bubble disappeared, remained constant. It was
also concluded that rectified diffusion 36 played an impor-
tant role in the stability process.
The observation of an upper pressure amplitude thresh-
ST
oldput, above which pulsating bubbles cannot persist, seems
to confirm the existence of a transient cavitation threshold as
defined by Flynn. 7'8 According to Flynn, 8 the motion of
bubbles pulsating with an expansion ratio (Rm8:/n o ) above
a certain value become inertia controlled. The collapse of
these bubbles tends to be violent, often resulting in the de-
struction of the bubbles. The bubbles studied here (• 20
/zm) fall under the "large" category ( > 5/zm) which, ac-
cording to the Flynn model, should exhibit thermally related
effects (e.g., sonoluminescence) before reaching the tran-
sient cavitation (dynamical) threshold. For 20-/zm bubbles,
Flynn predicted a dynamical threshold of Rmax/Ro •2.2.
These predictions appear to be confirmed by the observa-
tions made here, since $L has been observed from stably
I
Stability Threshold
Lower StabUity Th•oi•nUdS• !
•AVI•ATION
I NON-SPHERICAL
PULSATIONS
LOW
P•TIONS
FIG. 10. Diagram of the observed radial stability thresholds for 15- to 20-
mm bubbles in water/glycerine mixtures in an acoustic levitation system at
f= 21-25 kHz.
3174 J. Acoust. Soc. Am., Vol. 91, No. 6, June 1992 Gaitan etal.: Sonoluminescence and single bubble dynamics 3174
pulsating bubbles. After reaching a value of the expansion
ratio R,•a•/R o •4, the bubbles become unstable, i.e., they
disintegrated. Although the observed values of the dynami-
cal threshold differ somewhat from those predicted by
Flynn, it is generally agreed that these theoretical values
"most likely act as lower limits to experimental thresh-
olds. "8
Although the observed upper threshold instability can
be explained in terms ofFlynn's model, the region of stability
above the previously measured surface wave threshold is a
totally new and unexpected observation for which an ade-
quate explanation in terms of the equations of single radial
bubble pulsations has not been found. Despite the lack of
understanding about the particular mechanisms involved,
the discovery of a single bubble pulsating at large amplitudes
has many important consequences. Among these is the abi-
lity to obtain new and more accurate data about the motion
of bubbles in cavitation fields, such as the radius-time curve
of single bubbles pulsating radially from which we can mea-
sure R .... •c, and M to test the applicability of the theories.
One objective of this study, therefore, is to determine which
value or values ofR o give the best agreement between a par-
ticular theory and experiment for these three independently
measured parameters.
Comparison between the measurements and KM-Pro-
speretti model will be made first followed by a comparison
with the other models with the intent of determining the
most accurate theory. The theoretical values of the internal
temperatures, pressures and relative densities predicted by
the KM-Prosperetti and Flynn formulations will be com-
pared. The theoretical temperature calculated at the experi-
mentally determined threshold for SL will be discussed and
compared with previous measurements of temperature. In
addition, the simultaneity of SL and the collapse of the bub-
ble will be verified. We will present the results of the single
bubble experiments first, along with the theoretical results.
Finally, the observations in multibubble cavitation fields will
be discussed.
B. Light scattering experiments
1. Experimental data versus the KM-Prosperetti theory
The light scattering apparatus described in Sec. II was
used to obtain the radius versus time curves for radial pulsa-
tions of bubbles. The value of the equilibrium radius of the
bubble (R o ) was obtained by the rise-time method, 24 al-
though thermal and acoustic pressure currents in the liquid
sometimes made this measurement both difficult and impre-
cise. From these measurements, it was determined that the
equilibrium bubble sizes were roughly 20 + 5/zm for all the
liquid mixtures.
Figure 11 shows a plot of the bubble radius versus time
(dotted line) at PA = 1.22 arm in GLY21 as measured by
the photodetector. This figure is a single trace obtained from
the LeCroy oscilloscope. The y values have been converted
from intensity to radius using the transfer function obtained
during the calibration procedure. This figure is a typical ex-
ample of a radius-time (R-t) curve obtained from a single
bubble pulsating at large amplitudes with a period equal to
that of the driving pressure. Here, R-t curves for a range of
4O 6O 8O
Time (l.mec)
- o
FIG. 11. Theoretical (solid) and experimental (dotted) radius-time curve
obtained with the light-scattering apparatus in 21% glycerine at PA = 1.22
arm and f= 22.3 kHz.
pressures between 1.1 and 1.5 arm were obtained for all the
liquid mixtures. All the R-t curves obtained at these high
driving pressure amplitudes are characterized by a relatively
slow expansion ( - 15-20/zs), followed by a rapid collapse
( • 5-10/zs) after which several rebounds occur before the
next cycle starts. Curves similar to Fig. 11 were also obtained
for the other mixtures and representative curves can be
found in the Appendix of Ref. 24. The number of radial
minima during one cycle was usually 4 or 5, although three
minima occurred occasionally. The differences in the traces
taken in different liquids were mainly in the amplitude of the
pulsations and the phase of the first radial minimum.
For comparison, a theoretical R-t curve using the same
experimental conditions with Re = 19.3/zm has been plot-
ted in Fig. 11 (solid line). The reasons for choosing this
value of Re will be explained below. The general shape of the
two plots is very similar, including the magnitude of the bub-
ble response and the: existence of rebounds after the collapse.
Discrepancies were found in the number of minima and the
apparent increase in the amplitude of the rebounds in the
experimental R-t curve (dotted line), as opposed to the de-
crease predicted by the theory (solid line). Some of these
discrepancies will be discussed below. Periodic pulsations
such as those seen in Fig. 11 were observed continuously for
thousands of acoustic periods. These R-t curves and the ob-
servation of the spherical outline of the bubble pulsations
through the microscope when bubbles were !evitated in the
rectangular cell are most convincing evidence of the period-
icity, stability, and spherical symmetry of the bubble pulsa-
tions.
Several aspects of these nonlinear bubble pulsations
have been measured using the R-t curves of single bubbles
pulsating at large arnplitudes (Rm• •/R o > 3). In the follow-
ing sections, some of these measurements will be presented
and compared to the theoretical values predicted by the dif-
ferent formulations. Since it was not possible to measure the
equilibrium bubble size precisely, calculations were made
for Re = 15, 20, and 25/•m. Although the mechanism by
which the bubbles are stabilized is not well understood, each
bubble did appear to remain constant in size for a given set of
acoustical parameters (pressure amplitude, gas concentra-
tion, etc.).
3175 J. Acoust. Sec. Am., Vol. 91, No. 6, June 1992 Gaitan eta/.: Sonoluminescence and single bubble dynamics 3175
a. Pulsation amplitude. For each R-t curve obtained, the
pulsation amplitude R•,ax was measured and plotted versus
P• for water and GLY42 as shown in Figs. 12 and 13; data
for GLY21, GLY35, and GLY60 can be found in Reft 24. In
the same figures, values of the theoretical predictions made
by the KM-Prosperetti model for the same experimental
conditions have been plotted for R o = 15, 20, and 25 pm, as
indicated in the legend. Results for the KM-polytropic and
Flynn theories for 15 and 20 pm have also been included and
will be discussed later. The data shown in Figs. 12 and 13 are
mean values taken from stored oscilloscope traces. The
traces were taken using the same bubble and each trace con-
sisted of several acoustic periods for each value ofP•. Usual-
ly, only one trace was stored for each value of P• due to the
large amount of time and storage space that each trace re-
quired. The error bars placed on these data represent the
combined experimental uncertainty estimated to be q-5
pm. Some theoretical curves exhibit abrupt changes caused
by resonances of the radial motion excited at different values
ofP•. Bifurcations into period-2 and period-4 solutions also
occur, especially for the 20- and 25-pm bubbles. These re-
sonances and bifurcations have been briefly discussed in Sec.
I. Although values were calculated in increments of
P• = 0.01 atm for each value of Re, not all the data were
displayed. These gaps in the theoretical curves indicate that
no steady-state solutions were found in a reasonable amount
of time, usually 20 periods of the driving pressure.
In general, comparison of the pulsation amplitude data
indicates better agreement with an equilibrium radius of 15
pm, although this is not always true. Some experimental
data, for example, agrees partially with the theoretical re-
sults of a 20-pm bubble. It should be noted here that it was
probable that the equilibrium bubble radius changed as the
pressure amplitude was increased. In fact, for GLY21, closer
agreement with the theoretical results was found using the
15-/tin bubble at the lower pressures, but with the 20-pm
bubble at the higher pressures. This effective increase in
equilibrium radius with pressure amplitude may have result-
ed from rectified diffusion 36 which, in general, is proportion-
100
OoO•null?
1.1 1.2 1.3 1.4 1.5
Drivin• Pre•ure (arm)
FIG. 13. Measured and theoretical pulsation amplitude (R•) versus
acoustic pressure amplitude in GLY42 using all three formulations at
f= 23.6 kHz.
al to P•. On the other hand, these data may reflect a change
in the applicability of the theories as P• is increased.
Theoretically, resonances and bifurcations occur in dif-
ferent regions of the bubble response curve, the exact loca-
tions being very sensitive to the specific set of parameters
values, even for the same values of P• and R o . Thus it was
not expected to find agreement between theory and experi-
ment in the regions where rapid transitions occurred. It is
noted that although subharmonic motion was predicted of-
ten for the range of parameters considered, it was never ob-
served in these experiments.
b. Phase of collapse. From the same R-t curves consid-
ered in the previous section, the phase of collapse • was
measured. The results are plotted in Figs. 14 and 15 with the
theoretical predictions of the KM-Prosperetti formulation
for 15-, 20-, and 25-/.tm bubbles and the KM~polytropic and
Flynn theories for 15- and 20-btm bubbles. The estimated
total error of the calibration procedure was q- 5 deg and it is
indicated by the error bars.
The results of the phase of collapse measurements were
very similar to the pulsation amplitude measurements in the
100
40
Driving Pressure (atm)
FIG. 12. Measured and theoretical pulsation amplitude (Rma x ) versus
acoustic pressure amplitude in water using all three formulations atf = 21.0
kHz.
FIG. 14. Measured and theoretical phase of bubble collapse (•) versus
acoustic pressure amplitude in water using all three formulations atf = 21.0
kHz.
3176 J. Acoust. Sec. Am., Vol. 91, No. 6, June 1992 Gaitan eta/.: Sonoluminescence and single bubble dynamics 3176
26O
220
200,
Driving Pressure (atto)
FIG. 15. Measured and theoretical phase of bubble collapse (•Pc) versus
acoustic pressure amplitude in GLY42 using all three formulations at
f= 23.6 kHz.
Driving Pressure (arm)
FIG. 16. Measured and theoretical number of minima (M) versus acoustic
pressure amplitude in u•ater using all three formulations at f= 21.0 kHz.
previous section. In summary, these results can be interpret-
ed as follows: In Fig. 14 (water), R o appears to increase
from 10•<Ro•15 pm to 15•<Ro•<20 pm as PA increases,
whereas in Fig. 15 (GLY42), R o • 15 pm at low PA and
Re • 20 pm at high PA The collapse phase data for water
predicted larger radii than the pulsation amplitude data
shown in the previous section, although the discrepancy was
small enough to be within the experimental error.
c. Number of radial minima. A third experimental de-
termination that can be used to test the theory is the number
of radial minima observed during one acoustic period. This
parameter is mainly dependent on the bubble size, although
some dependence on the pressure amplitude was found from
the calculations. Figures 16 and 17 show the number of mini-
ma measured in each liquid mixture and those predicted by
the KM-Prosperetti and the KM-polytropic models for 15,
20, and 25pm and the Flynn theory for 15- and 20-pm bub-
bles. Noninteger values of the measured number of minima
indicate that two different values were observed in the same
radius-time curve; In this instance, the mean value was plot-
ted. The error bars indicate the estimated error in the mea-
surement. Double values in the theoretical data mean that a
different number of minima occurred per period in the
steady-state solution, usually due to subharmonic motion--
i.e., m > 1 (see Sec. I).
The number of minima observed were, in general, fewer
than the number expected from the theoretical predictions
based on the pulsation amplitude and collapse phase mea-
surements. This means that the equilibrium bubble radii pre-
dicted based on the number of minima were larger since, as
shown previously, larger bubbles pulsate with fewer minima.
For instance, Fig. 16 (water) predicts Re •20 pm at low
values of PA and Re • 25 pm at the higher values of PA.
Figure 17 (GLY42) predicts Ro•20 pm remaining con-
stant for the range of P• considered. It is not clear why the
number ofminima observed was fewer than predicted by the
theories. The explanation may be linked to another anoma-
lous observation in the rebounds of the radius-time curves.
As shown in Fig. 11, the bubble rebounds appear to increase
in radius within eac. h period instead of decreasing as predict-
ed by the theoretical R-t curve. The experimental R-t curves
also seem to indicate much larger amplitudes for the re-
bounds. Larger amplitudes would indeed result in fewer re-
bounds because each rebound would require more time. In
this respect, the observed fewer number of minima is consis-
tent with the large rebound amplitudes observed in the scat-
tered-light data. No explanation has been found for this in-
crease in the amplitude of the rebounds.
Additional evidence on the size of the bubbles can be
obtained by considering the periodicity of the solutions. For
example, period-two solutions were often predicted by the
theory for the 20-pm bubble. For the 25-pm bubble, few,
stable, period-one :•olutions were obtained whereas all the
solutions for R o = 15 pm were stable and with the same
period as that of the.. driving pressure. In this respect, the fact
that only bubble pulsations with the same period as that of
the sound field were observed in the laboratory suggests that
the bubble radius was less than 20 pm.
In summary, the data presented in the last three sections
may be interpreted as indicating that the equilibrium radii of
Driving Pressure (atm)
FIG. 17. Measured and •heoretical number of minima (M) versus acoustic
pressure amplitude in G:LY42 using all three formulations at f= 23.6 kHz.
3177 J. Acoust. Sec. Am., Vol. 91, No. 6, June 1992 Gaitan eta/.: Sonoluminescence and single bubble dynamics 3177
stabilized bubbles driven at pressure amplitudes between 1.1
and 1.5 arm in different water/glycerine mixtures were be-
tween 15 and 20/tm. It may also be inferred from some of the
data that the equilibrium bubble radii increased as the driv-
ing pressure was increased. The predicted values of Ro based
on the measurements of Rmax and $c are consistent for three
of the five data sets ( see Ref. 24). Inconsistencies were found
in the predictions for water, GLY35, and GLY42. The mea-
sured number of minima indicated, on the other hand, that
the equilibrium bubble radii were larger--between 20 and 25
/•m. This lack of total agreement means that the KM-Pro-
speretti theory, although applicable, may not give a quanti-
tatively accurate description of a cavitation bubble pulsating
at high amplitudes.
2. The KM-polytropic and Flynn theories
Let us first consider the results from the Flynn model.
For clarity, only values of Ro = 15 and 20/•m have been
plotted using the solid symbols as shown in the legend. Ex-
trapolated values were estimated for the Flynn and the KM-
polytropic model data when necessary. The pulsation ampli-
tude data shown in Fig. 12 (water) indicate that Ro • 15/•m,
increasing as PA increases. The data shown in Fig. 13
(GLY42) indicate that Ro •< 15/tm, with Ro increasing to 15
/•m as PA was increased. However, the phase of the collapse
data shown in Fig. 14 (water) suggests that 10•<Ro •<20•tm
with Ro also increasing as PA increases. The data in Fig. 15
(GLY42) indicate that 15 •<Ro •< 20 •tm, with Ro increasing
with PA. The number ofminima data shown in Fig. 16 (wa-
ter) imply that 20•<Ro •25/zm, with Ro increasing with PA
and those in Fig. 17 (GLY42) indicate that Ro •20/zm also
increasing slightly with PA. Note that, in general, the conclu-
sions agree well with KM-Prosperetti in the previous sec-
tion. Let us now consider the results from the KM-polytropic
model plotted in the same figures using the crossed symbols
as shown in the legend. The pulsation amplitude data shown
in Fig. 12 (water) indicate that 10•R o •20/zm, with R o
increasing as PA increases. The data shown in Fig. 13
(GLY42) indicate that Ro • 15/•m, with Ro increasing to 15
/tm or more as PA increases. The phase of collapse data
shown in Fig. 14 (water) suggest that 10•<Ro •<20 •tm with
Ro also increasing as PA increases. The data in Fig. 15
(GLY42) indicate that 15•<Ro •<20/tm, with Ro increasing
with PA. The number of minima data shown in Fig. 16 (wa-
ter) imply that 22•Ro •25 •tm, with Ro increasing with PA.
The data in Fig. 17 (GLY42) indicate that R o • 20/•m, with
Ro also increasing slightly with PA In general, these values
of R o agree well with those based on the KM-Prosperetti
predictions shown in the previous sections, and even better
with those based on the Flynn predictions.
In summary, two conclusions can be made: (1) The
measurements do not constitute a good test to discriminate
between the theories, since the experimental error is larger
than the difference between the values predicted by the theo-
ries using the same conditions; (2) each of the theories pro-
vide excellent qualitative and reasonable quantitative de-
scriptions of the dynamic behavior of stable cavitation
bubbles driven in a highly nonlinear regime.
3. Temperature, pressure, and relative density
The maximum internal temperatures, pressures and rel-
ative densities have been summarized in Tables II, III, and
IV, respectively. Because of the isothermal bubble motion of
the KM-polytropic formulation, the predicted pressures
were unreasonably high and the temperatures unreasonably
low. We have, therefore, not included them in this analysis.
In order to make a meaningful comparison between KM-
Prosperetti and Flynn formulations the maximum tempera-
tures calculated from KM-Prosperetti have been spatially
averaged over the bubble radius. The relative densities,
which have been normalized with respect to equilibrium
conditions, were calculated frompmax/p = (Ro/Rmin ) 3, i.e.,
assuming no condensation or vaporization occurs. Note that
for water, 1.12•PA•1.25 whereas for GLY42,
1.12•PA•1.47 so that higher values are expected for
GLY42.
The third column of each table contains the maximum
internal temperatures (Tma x ), pressures (Pmax), and rela-
tive densities (Pma•) predicted by the KM-Prosperetti theo-
ry, and those predicted by the Flynn theory are shown in the
fourth column. Note that, in every case, the temperatures
and pressures predicted by the KM-Prosperetti theory are
higher than those predicted by Flynn for the same bubble
size, even for the cases where the pulsation amplitude pre-
dicted by KM-Prosperetti was less. The fifth column of each
table contains the same quantities calculated using the Pro-
speretti formulation with Flynn's radial equation (instead of
Keller-Miksis). These values will be referred to as F-Pro-
speretti. The difference in the predictions shown in columns
four and five is entirely due to the way in which the internal
pressure is obtained by each formulation, i.e., the thermody-
namics in the bubble's interior. Note that the values of Tr•a•,
Pma•, andpm• predicted by the Prosperetti theory are lower
when Flynn's radial equation is used instead of Keller-Mik-
sis, suggesting that the additional compressibility term plays
a significant role during the collapse. The values of Truax and
Pma• predicted by F-Prosperetti, however, are still higher
than those predicted by Flynn by about 10%-20%. The val-
ues Ofpma• are about the same in both cases. These results
indicate that the Prosperetti theory does not allow as much
heat to diffuse out into the liquid resulting in higher tem-
peratures and pressures than the Flynn theory for the range
of parameters considered here.
The phenomenon of sonoluminescence is primarily of
thermal origin, i.e., caused by high temperatures rather than
high pressures or densities? The determination of the mini-
mum temperature required for light emission is, therefore,
important in understanding the mechanisms involved. Since
TABLE II. Summary of maximum theoretical temperatures inside 15-to
20-/•m bubbles for the range of P• used during the experiments.
P• (atm) KM-Prosperetti Flynn F-Prosperetti
Water (1.12-1.25) 2500-7000 K 2500-5000K 2500-6000K
GLY42 (1.12-1.47) 3000-10000 K 2500-7000K 3000-8000K
3178 d. Acoust. Soc. Am., Vol. 91, No. 6, June 1992 Gaitan eta/.: Sonoluminoscence and single bubble dynamics 3178
TABLE IIl. Summary of maximum theoretical pressures inside 15- to 20-/•m bubbles for the range of P• used during the experiments.
PA (arm) KM-Prospereni Flynn F-Prosperetti
Water (1.12-1.25) 1006-12 000 arm 2000-7000 arm 2000-8000 arm
GLY42 (1.12-1.47) 2000-40 000 atto 20(•)-22 000 arm 2000-20 000 arm
the acoustic pressure threshold for detectable light emission
was observed in this study to be around 1.1 atm, this indi-
cates that, according to the theories, the minimum tempera-
ture necessary to generate observable SL is between 2000
and 3000 K, corresponding to relative densities of about
100-200.
The most recent and precise measurement of tempera-
tures inside cavitation bubbles is that by Suslick etal., 2 using
a comparative rate thermometry technique in aqueous solu-
tions at 20 kHz. They measured a temperature of
5200 q- 650 K at acoustic intensities of 24 W/cm 2, which
corresponds to about 8 atm ( 800 kPa) assuming plane waves
and ignoring the effect of the cavities. This temperature falls
in the middle of the range of the theoretical temperatures
calculated in this study, corresponding to PA • 1.3 atm,
much lower than the estimated 8 atm used in Suslick's exper-
iment. In any case, the large number of bubbles present in the
apparatus used by Suslick et al. would most likely result in
the actual acoustic pressure being much less than that calcu-
lated using the plane-wave assumption.
As stated earlier, most experiments in which the tem-
perature of collapsing cavities was measured probably re-
quired high-intensity sound fields that generated many cav-
ities at once. In addition, transient cavitation was most likely
the prevalent type ofcavitation, implying that light emitting
cavities did not last for more than a few acoustic periods. In
the stable cavitation observed in this study, it has been deter-
mined that an increase in the acoustic pressure amplitude
results in higher temperatures, as evidenced by the increase
in the light emitted from the bubble. In a cavitation field
where many bubbles are present, it is not obvious whether an
increase in light emission is due to an increase in the amount
of light emitted by each bubble, or an increase in the number
of bubbles emitting light. Since the light emitted by each
bubble should be proportional to the temperatures reached
in its interior, it would be interesting to measure the depend-
ence of the internal temperature on the acoustic intensity in
experiments such as Susliek's.
The internal temperatures measured in Susliek's experi-
ments, which were done at much higher pressure ampli-
tudes, indicate that present theories of bubble pulsations
may overestimate the internal temperature and possibly the
TABLE IV. Summary of maximum theoretical relative densities inside 15-
to 20-/•m bubbles for the range of PA used during the experiments.
P.• (arm) KM-Prosperetti Flynn F-Prosperctti
Water (1.12-1.25) 150-500 200-400 100-400
GLY42 (1.12-1.47) 200-1300 150-700 200-700
internal pressure too. This overestimation may be explained
by the failure of the assumptions made in the models. Specifi-
cally, the dissocialion of the gas molecules due to the high
temperatures attained would increase the number density
and, therefore, the internal (gas) pressure, arresting the
bubble collapse sooner than predicted. When temperatures
of 5000 K or higher are reached, it is likely that the energy of
the collapse is partitioned into other chemical processes, in-
stead of increasing the temperature of the gas. Some of these
processes include dissociation of the gas molecules and ioni-
zation. Thus, as the strength of the collapse increases, i.e., at
larger values of PA, the internal temperature is expected to
reach a plateau. In addition, the models underestimate the
effects of energy dissipation due to the compressibility of the
liquid. A compressible liquid results in the formation of
shock waves that may carry a significant portion of the col-
lapse energy away t¾om the bubble.
4. Phase of sonoluminescence
Using the light scattering apparatus and the photomul-
tiplier (PMT) SL detection system shown in Fig. 7(a) and
(b), the phase of SL was measured relative to the instanta-
neous radius of the bubble. This is the most direct way of
determining the point during the bubble motion in which the
SL flash is emitted. Since both the photomultiplier tube
(PMT) and the laser could not be engaged simultaneously,
an intermediate phase reference, the side pill transducer, had
to be used.
The procedure was as follows: After positioning a pul-
sating bubble in the center of the laser beam, a scattered-light
intensity versus time trace was obtained simultaneously with
the pill output and subsequently stored in the computer. The
laser and the room lights were then turned off and the PMT
activated. Similarly, simultaneous traces of the PMT output
and the pill transducer output were stored in the computer.
Several traces were obtained within a few seconds of each
other to verify that experimental conditions remained as
identical as possible. The correlation between the two traces
was determined graphically by overlapping the two pill
transducer traces until a best fit was obtained. Figure 18 is a
plot of the experimental results clearly showing that $L is
produced during the collapse of the bubble. The error in this
measurement was estimated to be -I- 5 deg. This result is
significant because it dramatically illustrates the causal rela-
tionship between the bubble collapse and the light emission.
C. $onoluminescence experiments
After the simultaneity of the light emission and the col-
lapse of the bubble was established, the phase of the SL emis-
sion was used to study the behavior and the time evolution of
3179 d. Acoust. Sec. Am., VoL 91, No. 6, June 1992 Gaitan ot aL: Sonoluminescence and single bubble dynamics 3179
• •00 400 •0 800 •000
Time
0.2 '•
OA o •
0.0
FIG. 18. Simultaneous plots of the sound field (top), bubble radius (mid-
dle) and sonoluminescence (bottom) in GLY21 at P.• = i.2 atto and
f= 22.3 kHz.
202
Time (acoustic periods)
FIG. 19. Phase ofsonoluminescence emitted by a single bubble as a function
of time. It was measured in water at Pa=l.I atm and f----21.0 kHz
( T = 47.6/•s) by the time-to-amplitude converter system. The oscillation
in the signal is due primarily to 60-Hz noise.
cavitation bubbles as they interacted with the sound field
and with each other. This was done by measmang the phase
of the light emitted by the bubbles to obtain the phase of the
bubble collapse. In view of the already established coinci-
dence of the light emission and the bubble collapse, these two
terms will be used interchangeably.
The time-to-amplitude converter system provided a
very efficient method of recording the phase of SL for thou-
sands of cycles and in real-time if desired. The phase of SL
was measured for single bubbles using the different liquid
mixtures. Measurements were also made of the phase of SL
during streamer activity when the cavitation field was com-
posed of many bubbles sonoluminescing simultaneously.
The purpose of this latter experiment was to study the behav-
ior of bubbles during intense cavitation activity. To interpret
the observations made in multibubble cavitation fields, the
results for a single bubble will be discussed first.
I. Single bubble cavitation field
Figure 19 shows the phase of SL at approximately 1.1
atto in water, plotted as a function of time in units of acoustic
periods. In this data set, a total 2000 data points were taken
corresponding to approximately 0.1 s. The small oscillations
on these data are due to 60-Hz line noise that could not be
easily filtered. Typical 60-Hz noise levels were + 1 deg. This
is an example of the phase of the light emission for a single
bubble pulsating radially.
If one ignores the 60-Hz noise, these data demonstrate a
great stability of this dynamic system. Even though the bub-
ble grows to several times its equilibrium size and then col-
lapses within microseconds with such a violence that it emits
optical energy, it still repeats this process for thousands of
cycles with a regularity more precise than was possible to
measure. Indeed, Barber and Putterman 38'39 have deter-
mined that the "jitter" in the repeatability of this process is
less that 50 ps. Furthermore, the duration of the SL flash was
so short that we were also unable to resolve its length with
the available instruments. Subsequently, Putterman and his
colleagues have obtained photomultiplier tubes with in-
creasingly fast response times and have demonstrated that
the SL pulse duration is also less that 50 ps. TM These latter
experiments demonstrate the remarkable nature of this phe-
nomenon, suggesting that a major part of it is still not clearly
understood.
To determine if a single bubble was present, the output
of the side pill transducer was monitored. This output would
become noticeably irregular and noisy when streamers were
present. For a single bubble, the signal was symmetrical ex-
cept for a small notch, probably due to the shock wave pro-
duced by the bubble collapse. For the case shown in Fig. 19,
the value of the phase was •201 deg. A range of values was
measured between 190 and 220 deg for pressure amplitudes
between 1.0 and 1.4 atto in water.
At the high end of the pressure amplitude range, the
levitated bubble was occasionally observed to become unsta-
ble for a fraction of a second, after which stability was re-
stored. Following this transition, the bubble's position shift-
ed in the vertical direction. This new position was always
away from the antinode. When the light emission was moni-
tored, it was observed that the intensity of each flash was
sharply decreased after the transition had occurred. The
phase of the emissions was also decreased. These observa-
tions are illustrated in Fig. 20. The phase can be clearly seen
215 f
2O0
195
0 200 400 600 800 1000
Time (acauatle periods)
FIG. 20. Phase ofsonoluminescence emitted by a single bubble as a function
of time. It was measured in water at P• = 1.2 atto and f---- 21.0 kHz
( T= 47.6/.rs) by the time-to-amplitude converter system during a transi-
tion {see text).
3180 J. Acoust. Sec. Am., Vol. 91, No. 6, June 1992 Gaitan etaL: Sonoluminoscence and single bubble dynamics 3180
to decrease sharply during the transition at 600 cycles. One
explanation for this decrease of the phase is that there is a
decrease in the equilibrium bubble radius, i.e., a breakup of
the bubble due to surface instabilities. This type of breakup
usually occurs via the ejection of microbubbles. 4'4ø After re-
ducing its size, the bubble regains its stability due to the
larger surface-tension pressure (2a/Ro). This occasional,
and sometimes periodic, breakup can be understood in terms
of the bubble response curve as a function of equilibrium
radius (see Fig. 5). As the bubble grows by rectified diffu-
sion, resonances are encountered and the bubble response
increases rapidly. If the surface instability threshold hap-
pens to be below one of these peaks, breakup will occur and
the bubble size is quickly reduced. This cycle may repeat
depending on PA and the new value of R o, among other
parameters. In addition, by plotting the pulsation ampli-
tude, R .... and the phase of the collapse as a function ofRo,
a direct correlation between these two quantities can be es-
tablished. When Rma x increases (decreases) the phase also
increases (decreases), as shown in Fig. 5. Note that a change
in the radius R o as small as 1/•m can result in the phase of
collapse changing by as much as 20 deg, depending on the
bubble size. Thus, it does not require a large change in radius
to effect a large change in the phase.
It appears that before the bubble regains stability, its
response amplitude or, equivalently, equilibrium size must
be decreased. This is not surprising since instabilities in the
gas-liquid interface are triggered by large accelerationsil 1
This observation can be explained as follows: After develop-
ing instabilities, the bubble sheds microbubbles, reducing its
size and its response amplitude. Both effects increase the
stability of the bubble and the pulsations become spherically
symmetric again. The bubble then grows slowly by rectified
diffusion, sheds microbubbles, etc. This behavior is com-
monly seen in single bubbles at the higher pressure ampli-
tudes ( -- 1.3-1.4 atm).
At the higher end of the pressure range, instabilities oc-
curred repeatedly with a period of a few seconds. At the
higher end, transitions often occurred rapidly and consecu-
tively as shown in Fig. 21. Here, it may be postulated that the
pressure amplitude was high enough that surface waves were
easily excited, and the bubble was unable to regain its stabil-
ity before disintegrating. If the residual microbubbles are
large enough or close enough to the initiating bubble, coales-
cence may occur before they can dissolve. This process may
explain the rapid bubble growth inferred from the rapid in-
crease in the phase of collapse shown in Fig. 21. Sometimes
the bubble disappeared unexpectedly, as occurred here when
the bubble suddenly stopped emitting light around cycle
1250. This bubble disappearance also can be observed with
the unaided eye or through the microscope. A similar type of
bubble annihilation was reported by Nyborg and Hughes 4
during observation ofcavitation on the surface of a vibrating
bar driven at 20 k}tz. Using high-speed photography, they
reported bubbles that disappeared in less than six acoustic
periods. An explanation similar to ours was given by Nyborg
and Hughes 4 for this phenomenon.
In summary, it: appears that bubbles grow and breakup
in a periodic or quasiperiodic fashion during radial pulsa-
tions. A breakup is usually reflected by a decrease in the
bubble response and the phase of the collapse. Bubble
growth can occur by coalescence or rectified diffusion and
can be detected by an increase of the response and the phase
of the bubble collapse.
2. Multibubble cavitation fields
In this section, the observations made from a cavitation
zone composed of streamers will be discussed. High-speed
photographs taken during the experiments revealed that
these streamers consisted of fast-moving bubbles that were
undergoing rapid fission and fusion. An example of the mea-
surement of the phase of collapse in water at around 1.3 atm
is shown in Fig. 22.. In this data set, the phase of collapse
appears to change irregularly between 185 and 215 deg.
Most of the measurements made of the collapse phase in
multibubble cavitation fields were in this range. This range
coincided with the collapse phases measured for single bub-
bles, which were determined to be between 15 and 20/.tm in
radius. It thus appears that in a cavitation field at 21 kHz,
bubbles that emit SL are in the range 15<Ro <20/tm. This
210
190
0 500 10 0 1500
Time (acoustic periods)
FIG. 21. Phase ofsonoluminescence emitted by a single bubble as a function
of time. It was measured in water at P•=I.3 atm and f= 21.0 kHz
(T = 47.6 its) by the time-to-amplitude converter system during bubble
breakup and disappearance.
215
210
205
200
195
190
' • .... • ' ' ' 20•00 •0 • .... 50•00 '
0 1000 30 0 4000
Time (acoustAc periods)
FIG. 22. Phase of sonolaminescence emitted by a multibubble cavitation
field as a function of time. It was measured in water at P.= 1.3 arm and
f= 21.0 kHz ( T = 47.6/•s) by the time-to-amplitude converter system.
3181 J. Acoust. Soc. Am., Vol. 91, No. 6, June 1992 Gaitan ota/.: Sonoluminescence and single bubble dynamics 3181
range of bubble sizes may be even smaller since some scatter
in the measured values of the phase may be due to fluctu-
ations of the driving pressure. Also, the fact that the range of
values for the phase of collapse of single bubbles and cavita-
tion field bubbles coincide may be an indication that only
bubbles pulsating radially emit SL. Evidence for this hypoth-
esis is given by our observations that when a single bubble
undergoes surface oscillations, the SL output decreases by
an order of magnitude. It was not uncommon to observe a
single bubble pulsating radially and emitting SL flashes ev-
ery cycle for as many as 1000 consecutive cycles. When sur-
face waves were excited, as evidenced by dancing motion, the
average number of SL flashes was about one in ten cycles.
This seems to imply that a spherical collapse is necessary to
generate sufficiently high temperatures and pressures in the
interior of the bubble.
In addition, the data displayed in Fig. 23 reveal the same
type of cyclic behavior observed in a single bubble as de-
scribed in the previous section. This behavior was observed
often in a multibubble cavitation field, although it was not
always repeatable. It was usually observed at the higher pres-
sure amplitudes (1.3-1.5 atm). Nevertheless, these data
show that the same cyclic behavior observed for a single
bubble also occurs in a multibubble cavitation field. Similar-
ly, this behavior can be attributed to the fission of bubbles by
surface instabilities followed by coalescence or rectified dif-
fusion.
In summary, we have observed cyclic behavior in both
single and multibubble cavitation fields. This behavior has
been observed before by other investigators and can be
caused by resonances of the bubble motion encountered as
the bubble grows. These resonances trigger surface instabili-
ties that result in a reduction of the bubble size due to fission.
As the bubbles regrow by rectified diffusion or coalescence,
the process repeats itself. In addition, the coincidence of the
phase of collapse for both single and multibubble cavitation
suggests that spherical collapses are required to generate the
high temperatures and pressures responsible for most cavita-
tion-related effects. It has also been demonstrated that the
phase of $L emission can be used to study the behavior of
210
205 ..' .'
/, • ;'• • •'.'• ::• : / •' :.. Z " • •:.
• •. . :• . ß • • • / .'
•95
O
Time (acoustic pe•ods)
FIG. 23. Phase of sonoluminescence emitted by a multibubble cavitation
field as a function of time. It was measured in water at PA = 1.3 atm and
f= 21.0 kHz ( T = 47.6/•s) by the time-to-amplitude converter system.
bubbles in cavitation fields on a time scale on the order of an
acoustic period.
IV. SUMMARY AND CONCLUSIONS
( 1 ) A previously undiscovered region of stability exists
in the pressure-radius parameter space above the level of
both surface instability and the threshold for rectified diffu-
sion thresholds for radially pulsating bubbles. The mecha-
nisms through which this stability is attained are not yet
understood. The observations reported here indicate the ex-
istence of a transient cavitation threshold, which forms an
upper boundary to this stable region. Above this threshold,
spherical stability was never attained. It is believed that the
rapid bubble collapse promotes the formation and enhances
the magnitude of surface instabilities that cause the bubble to
disintegrate.
The phase of the light emission was observed for thou-
sands of acoustic periods to monitor the long-term behavior
of bubbles driven at moderate intensities in the cavitation
field. From these observations, it was concluded that, except
in the small region of stability, bubbles grow by rectified
diffusion until surface waves cause them to fission. These
two phenomena affect the periodic pulsations of the bubbles
so that they exhibit a "life cycle." This life cycle has a rela-
tively slow growth followed by an abrupt fragmentation and
has a period on the order of 10 ms ( - 200 acoustic periods),
depending on the driving amplitude.
(2) Experimental radius-time curves for single bubbles
pulsating radially at large amplitudes have been obtained.
These bubbles were observed in water/glycerine mixtures at
pressure amplitudes between 1.1 and 1.5 atto at 20-25 kHz.
The pulsation amplitude, the phase of the collapse and the
number of minima have been measured for a range of pres-
sure amplitudes and bubble sizes.
(3) Sonoluminescence from stable cavitation at 20 kHz
is most effectively emitted by bubbles in the size range 15-20
$tm rather than by bubbles resonating at their natural fre-
quency. These bubbles are small enough so that, although
pulsating at large amplitudes, they remain spherically sym-
metric, rapidly compressing the gas and heating it to high
temperatures.
(4) The applicability of three theoretical formulations
of bubble dynamics has been evaluated in a range of acoustic
pressure amplitudes near the threshold for sonolumines-
cence. General agreement was found between the experi-
mental and theoretical values of the pulsation amplitude,
phase of collapse and the number of rebounds. The number
of rebounds was observed to be lower and the amplitude of
each rebound larger in amplitude than expected from the
theories. The data suggest that the physical conditions at-
tained during the collapse may be outside the limits of the
theories.
(5) From the theoretical calculations made for the pa-
rameters measured at the threshold for light emission, it is
estimated that internal temperatures in the range 2000-3000
K are necessary to produce detectable sonoluminescence.
This finding supports the Chemiluminescent model as the
mechanism for light production.
3182 J. Acoust. Sec. Am., Vol. 91, No. 6, June 1992 Gaitan eta/.: Sonoluminescence and single bubble dynamics 3182
ACKNOWLEDGMENTS
The authors wish to acknowledge many helpful discus-
sions with Andrea Prosperetti, S. Putterman, K. Suslick,
and the financial support of the Office of Naval Research.
T. K. Saksena and W. L. Nyborg, "Sonolumineseence from stable cavita-
tion," $. Chem. Phys. 53, 1722 (1970).
K. S. Suslick, D. A. Hammerton, and R. E. Cline, "The sonochemical hot
spot," Am. Chem. Soc. 108, 5641-5642 (1986).
L. A. Crum and G. T. Reynolds, "Sonolumineseenee produced by stable
cavitation," $. Acoust. Soc. Am. 78, 137-139 (1985).
W. Nyborg and D. E. Hughes, "Bubble annihilation in cavitation stream-
ers," J. Acoust. Soc. Am. 42, 891 (1967).
SF. G. Blake, "The onset of cavitation in liquids," Tech. Memo. No. 12,
Harvard Acoustics Research Laboratory (1949).
G. W. Willard, "Ultrasonically induced cavitation in water: a step-by-step
process," $. Acoust. Soc. Am. 23, 669 ( 1953}.
H. G. Flynn, Physical.4coustics, edited by W. P. Mason ( Academic, New
York, 1964), Vol. 1, part B.
H. G. Flynn, "Cavitation dynamics. II. Free pulsations and models for
cavitation bubbles," J. Acoust. Soc. Am. -•8, 1160 (1975).
•N. Marinesco and •. •. Trillat, C. R. Acad. Sci. Paris 196, 858 (1933).
top. Paounoff, C. R. Hebd. Searices Acad. Sci. Paris 44, 261 ( 1947}.
P. Gunther, E. Heim, A. Schmitt, and W. Zeil, Z. Naturforsch. 12A, 521
(1957a).
•2W. U. Wagner, Z. Angew. Phys. 10, 445 (1958).
•3K. Negishi, Acustica 10, 124 (1960).
•4 R. Q. Macleay and L. V. Holroyd, "Space-time analysis of the sonolu-
minescence emitted by cavitated water," $. Appl. Phys. 32, 449-453
(1961).
s3p. Jarman and K. $. Taylor. "The timing of the main and secondary
flashes of sonoluminescence from acoustically cavitated water," Acustica
23, 243 (1970).
•6B. E. Noltingk and E. A. Neppiras. "Cavitation produced by ultrason-
ics," Proc. Phys. Soc. London Sect. B 63, 674 (1950).
•7 j. B. Keller and M. Miksis, "Bubble oscillations of large amplitude," J.
Acoust. Soc. Am. 68, 628 (1980).
•a A. Prosperetti, L. A. Crum, and K. W. Commander, "Nonlinear bubble
dynamics," J. Acoust. Soc. Am. 83, 502 (1986).
•9 H. G. Flynn, "Cavitation dynamics. i. A Mathematical formulation," J.
Acoust. Soc. Am. 57, 1379 (1975).
2o A. Prosperetti, "Acoustic cavitation series part two--Bubble phenomena
in sound fields: part one," Ultrasonics 22, 69 (1984).
A. Prosperetti, "Nonlinear oscillations of gas bubbles in liquids: steady-
state solutions," 3. Acoust. Soc. Am. 56, 878 (1974).
22 W. Lauterborn, "Numerical investigation of nonlinear oscillations of gas
bubbles in liquids," I. Acoust. SOC. Am. 59, 283-293 (1976).
L. A. Crum and A. Prosperetti, "Erratum and comments on 'Nonlinear
oscillations of gas bubbles in liquids: An interpretation of some experi-
mental results'," J. Acoust. Soc. Am. 75, 1910 (1984).
D. F. Gaitan, "An experimental investigation of acoustic cavitation in
gaseous liquids," Ph. D. dissertation, University of Mississippi (1990).
IMSL, Mathematical Software Library ( IMSL, Houston, 1987).
26C. W. Gear, Numerical Initial-Value Problems in Ordinary Differential
Equations (Prentice-Hall, Englewood Cliffs, N J, 1971 }.
V. Kamath and A. Prosperertl, "Numerical integration methods in gas-
bubble dynamics," J. Acoust. Soc. Am. 85, 1538 (1989).
W. Lauterborn and U. Parlitz, "Methods of chaos physics and their appli-
cations to acoustics," •. Acoust. Soc. Am. 84, 1975-1993 (1988).
2•L. A. Crum, "The acoustic radiation pressure on a liquid droplet in a
stationary sound field," Tech. Report, U.S. Naval Academy (1970).
søG. M. Hansen, "Mie scattering as a technique for the sizing of air bub-
bles," Ph. D. dissertation, University of Mississippi (1984).
• R. G. Holt, "Experimental observations of the nonlinear response of sin-
gle bubbles to an applied acoustic field," Ph. D. dissertation, University of
Mississippi (1988).
•2 S. D. Hotsburgh, "Radial instabilities of a pulsating air bubble in water,"
Ph.D. dissertation, University of Mississippi (1990).
s• R. K. Gould, "Simple method for calibrating small omnidirectional hy-
drophone," I. Acoust. Soc. Am. 43, 1185 (1968).
ss W. Lauterborn, "Carlration dynamics--new tools for an intricate prob-
lem," Appl. Sci. Res. 38, 165-178 (1982).
3s L. A. Crum, "Bjerknes forces on bubbles in a stationary sound field," J.
Acoust. Soc. Am. 57, 1363 (1975).
aOA. Eller and G. H. Flynn, "Rectified diffusion during nonlinear pulsa-
tions of cavitation bubbles," J. Acoust. Soc. Am. 37, 493 ( 1965 ).
•=C. Sehgal, R. P. Steer, R. G. Sutherland, and R. E. Verral, "Sonolumines-
cence of argon saturated alkali metal salt solutions as a probe of acoustic
eavitation," J. Chem. Phys. 70, 2242-2248 (1979).
3aB. P. Barber and S. J. Putterman, "Observations of synchronous picose-
cond sonoluminescence," Nature 352, 318 ( 1991 ).
•B. P. Barber, R. Hiller, K. Arisaka, H. Fetterman, and S. J. Putterman,
"Resolving the pieosecond characteristics of synchronous sonolumines-
eence," submitted to J. Acoust. Soc. Am.
•øE. Neppiras and E. E. Fill, "A cyclic cavitation process," •. Acoust. Soc.
Am. 46, 1264 (1969).
4• D. Y. Hsieh, "On the thresholds for surface waves and subharmonics of
an oscillating bubbles," J. Acoust. Soe. Am. 56, 392 ( 1974}.
3183 J. Acoust. Sec. Am., Vol. 91, No. 6, June 1992 Gaitan el al.: Sonoluminescence and single bubble dynamics 3183
