Holzfuss , Matthias Rüggeberg , Robert Mettin
Institut für Angewandte Physik, TU Darmstadt,
64289 Darmstadt, Germany
Drittes Physikalisches Institut, Bürgerstr.
42-44, 37073 Göttingen, Germany
in: Phys. Rev. Lett. 81 (1998) 1961-1964.
Received: September 26, 1996
sonoluminescence has been experimentally
produced through a novel approach of optimized sound excitation.
A driving consisting of a first and second harmonic with selected
amplitudes and relative phase results in an increase of light emission
compared to sinusoidal driving. We achieved a raise of the maximum
photo current of up to 300% with the two-mode sound signal. Numerical
simulations of multi-mode excitation of a single bubble are compared to
PACS numbers: 78.60.Mq, 43.25.Yw, 42.65.Re, 02.60.Pn
ultrasonic waves of high intensity into a liquid, thousands
of tiny bubbles appear. This process of breakup of the liquid is called
acoustic cavitation. The bubbles begin to form a fractal structure that
is dynamically changing in time. They also emit a loud chaotic sound
because of their forced nonlinear oscillations in the sound field
large mechanical forces on objects brought into contact with the
bubbles enable the usage of cavitation in cleaning, particle
destruction and chemistry. Marinesco and Trillat 
found that a photo plate in water could be fogged
by ultrasound. This multi bubble sonoluminescence (MBSL) has been
analyzed by many researchers, and a great amount of knowledge
has been gained . The discovery
 that it is possible to drive a
single stable bubble in a
regime, where it emits light pulses of picosecond duration
has been encouraging scientists to explore the phenomenon and the
effects with a multitude of experiments, theories and simulations. The
experimental results show picosecond synchronicity
, quasiperiodic and chaotic
variability of inter-pulse
times [8, 9], a black body spectrum 
and mass transport stability . The
theories to explain the
source of SBSL range from hotspot, bremsstrahlung , collision induced radiation  and corona discharges  to non-classical light. Numerical simulations
have been focusing
on the bubble dynamics, behavior of the gas content
properties in magnetic fields  and
stability of the bubble.
the final answer concerning the nature of SBSL still remains open.
energy concentration from low energy acoustic sound waves to 3 eV
raises the question, whether the effect can be up-scaled.
In this paper, we report on experimental enhancement of SBSL light
production by a bimodal excitation of the bubble
oscillation. The experiment follows an idea stated in
. We also present numerical
simulations of multi-mode
sound driving that reveal how multi-harmonic excitation can adapt to
highly nonlinear bubble oscillation in the sense of a strong collapse.
experimental setup is as follows: An air bubble is trapped in a
water filled cell consisting of two piezoceramic cylinders connected
via a glass tube . The levitation
cell (``Crum cell'')
 is standing upright with a glass
covering the lower end of the cell. The upper end remains open. A
video camera pointing from the side allows for online monitoring of the
experiment. The experiments were done with distilled and
at room temperature and an ambient pressure of 1 atm. The bimodal
driving signal is produced by synchronized
sine wave generators that allow
to fix the amplitudes , , and the relative phase .
multi-frequency driving signal however is complicated by two
facts. First the transducers have a complex transfer function and
second the standing wave conditions at each frequency in the cell have
to be obeyed . Therefore,
multi-frequency driving results in space
phases and amplitude relations and thus in an effective sound signal (with cylinder coordinates r,z of the
To measure the amplitudes and relative phase that actually appear at
the bubble position, a small hydrophone is used.
The correct position is adjusted by first focusing the camera on the
bubble and then inserting the hydrophone at the bubble site. The
driving signal is digitally recorded
and phases and amplitudes are recovered via a Fourier transform. The
flashes emitted at collapse are measured with a photomultiplier.
small oscillating bubbles of volume V(t) in non-zero
is possible through the
interaction with the driving sound field which depends on space and time.
The time averaged primary Bjerknes force
overcome the buoyancy force and attract the bubble to a fixed position
in space. Weakly sinusoidally driven bubbles of equilibrium
radius are trapped near a
pressure antinode if they are driven below their linear resonance
(Minnaert) frequency [Hz]
for the experimental conditions used here (with polytropic exponent , ambient pressure , and liquid
. However, the situation is more
complicated for strongly
driven bubbles  and also for
excitation, where the standing wave pattern in the
resonator, the Bjerknes forces and thus the bubble position are changed
by a variation of the sound signal parameters , , and .
oscillation responds to the sound signal at the trapping
experiment, we proceeded in the following way:
For fixed drive amplitudes and , a bubble is injected
into the fluid with a syringe. Once the bubble fixes itself spatially
at a stable position,
where the Bjerknes force equals the buoyancy force, the phase
difference between the locked sine wave
generators, one operating at and
the other at 2f, is sequentially increased while the SL
intensity and the
bubble itself are monitored. Fig. 1
(lower) shows the SL intensity as a
function of the phase difference for bar, bar.
With increased phase difference two maxima appear in the light
The dashed line indicates the maximum achievable SL intensity using
single mode driving.
This value is obtained shortly before and after the 2-mode experiment
direct comparison by keeping all other experimental conditions
unchanged. It is seen, that the 2-mode driving yields
100% more SL intensity than
the maximal single mode driving. By further increase of and at selected phases an intensity
gain of 300%
can be achieved, as shown by the open circles. Beyond that, the bubble
1 (upper) reveals,
that with increased
phase difference the bubble traverses vertically through stable and
(surface) unstable regimes.
Figure 1: Bubble
response for 2-mode driving
as a function of the phase
difference (in degrees) between the driving sinusoidal signal and its
harmonic. Upper: vertical position of the bubble. The thick dotted
lines denote unstable
bubble behaviour. Lower: photo current. The open circles show the
maximum SL intensity achieved.
The dashed line is the maximal photo current for
pure sine wave driving.
simulations have been carried out using the Gilmore model
 which describes the radial motion
of a single
The model includes the usual components of the RPNNP
equation  like surface tension and liquid
viscosity , and
also the compressibility of the liquid to allow damping of the bubble
motion by the shedding of shock waves.
the bubble radius, m its equilibrium radius, C,
, and p are the speed of sound in
the liquid, its density, and the pressure at the
bubble wall, respectively. H is the enthalpy of the
liquid. Parameters were set to =1500 m/s, =998 kg/m , =1 bar, =4/3, =0.0725 N/m, =0.001 Ns/m , n=7, B=3000 bar. a= /8.54 is a hard-core van der
Waals-term . The pressure at
infinity includes the multi-modal driving pressure: , .
calculated the driving sound signal that would lead to the most
violent collapse, indicated by the smallest minimum radius during a
bubble oscillation cycle using the above equation.
The search for suitable pressures and phases was
carried out by a heuristical optimization algorithm
 with the boundary condition of
a constant driving signal power, i.e., . was fixed to
1.3 bar and the driving
frequency was the same as in the experiment. Comparing equal power
signals is convenient, because the power stays constant upon
phase changes, making it possible to compare numerics and experiments.
2 shows the driving pressure and
of different driving signals. A strong increase in the maximum radius
can be seen already by adding just the second harmonic to a sine wave.
The numerically computed optimal phase difference is and
the individual amplitudes are bar and bar. The
radius and the adiabatically calculated temperature around the
are shown in Fig. 3. It is seen, that
the bubble radius at collapse is decreased by a large amount and
is approaching the van der Waals hard core already for the 2-mode
Also the maximum temperatures almost double. The higher mode driving
signals are better adapted to the nonlinear bubble oscillation than
the sine signal: they show a deeper rarefaction phase before
collapse, followed by a more rapid rise to the compression phase
Figure 2: Time
series of the driving (top) and
the radius of calculated
bubble collapses (bottom) for single (dashed) and optimized multi-mode
driving signals (2-mode: line, 8-mode: dotted).
calculations for optimal 8-mode driving exhibit only small
additional gain compared to bimodal driving. Because of the
increased difficulties regarding the spatial stability of bubbles in
the resulting complicated sound field, an 8-mode driving may not be
worth being considered experimentally. Though 2-mode driving is an
early truncation of a series expansion, one sees, that already this
shows a trend for a more intense driving of this nonlinear system.
results are located on a single broad
plateau in parameter space. This is in contrast to the experimental
of two maxima. To understand the reason of this obvious discrepancy,
the bubble model (eq. 2) has been
integrated numerically along with the
primary Bjerknes (eq. 1) and
the buoyancy forces to
examine spatial dependencies. This is also motivated by the
observation, that the vertical position of the bubble is altered when
phase is changed (Fig. 1 upper). The
change of position leads to
different effective excitation amplitudes for f and 2f
and thus to a more
The system of equations is integrated using the spatially dependent
Figure 3: Zoom
into the first collapses of Fig. 2.
are the curves for 1-mode (dashed), optimized 2-mode (line) and
optimized 8-mode (dotted) driving. The time series of the bubble
collapses exhibit a decrease in minimum radius (left)
and increase in the adiabatically calculated temperature (right) as a
function of the number of modes in the driving sound.
The minimum radius comes very close to the van der Waals hard core,
shown by the horizontal line in the left graph. The time axis is
shifted so that the collapses take place at 0, 0.5, and 1 ns,
the acoustic wavenumber , bar, bar, f=23.4
kHz). The spatial modes are approximately the same as the experimental
ones, which have been measured with a needle hydrophone. The points in
vertical z-space where the Bjerknes force vanishes and the
criterion is met represent the position of the bubble. The resulting
minimum radii are shown in Fig. 4.
Comparing this with the experimental results in Fig. 1
shows a very close agreement.
Figure 4: Numerically
calculated vertical bubble
position (upper) and resulting minimum radius (lower) as a function of
the phase difference
for double harmonic driving of a bubble. The dashed line in the lower
plot is the minimum radius for single frequency driving with the same
power. The solid
straight line is the van der Waals hard core.
sinusoidal variation of the position of the bubble gives
rise to two minima of the minimal radius/phase dependence. Each of
these minima is smaller
than the one of the single mode driving. The minima coincide with the
experimental observation of increased SL intensity. The slight
in the experiment can be described by the difference in acoustic
impedance of the glass bottom and the open top of the cell. Also,
imperfect standing waves may lead to small travelling components in the
experimental driving. Changing the
amplitude ratio of the driving signal closer to unity while keeping the
power constant results in a complex scenario of stable bubble positions
effective drivings including hysteretic jumps.
we have shown that a bimodal sound excitation can enhance
light production of SBSL. Though spatial modes play a crucial role in
double harmonic driving, it increased the
photo current to a gain of maximally 300% compared
to sine excitation. We suppose that
multi-frequency driving can shift the bubble oscillation to a regime of
strong stable SBSL which is not reachable by pure harmonic
driving. Numerical simulations of an acoustically driven bubble
including Bjerknes and buoyancy forces show that the increased SBSL
is caused by a larger compression. To give quantitative estimates,
however, elaborate models have to be considered that include gas
dynamic equations for
the interior of the bubble and thus can model the shedding of a shock
wave inside a bubble
have been proposed to increase the violence of
bubble collapses. E.g., calculations for thermonuclear D-D fusion in
D O within this context have been done using a
large pressure pulse superimposed on a sine wave.
However, whether advanced forcing by higher modes is large
enough to achieve a reasonable neutron production rate is an open
question. Apart from sonoluminescence, the increase of cavitation
by means of optimized multi-harmonic sound signals  can also
be of use in the context of sonochemistry 
related areas, where higher reaction rates could be induced.
wish to thank R. G. Holt and W. Lauterborn
for stimulating discussions and
the TU Darmstadt for making the research possible. The work has been
funded through the SFB 185 ``Nichtlineare Dynamik'' of the DFG.
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