SONOLUMINESCENCE AND BUBBLE STABILITY CRITERION

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SONOLUMINESCENCE AND BUBBLE STABILITY
CRITERION Ryan Pettibone
BUBBLE STABILITY
ABSTRACT
THEORY
CONCLUSIONS

• The antinode of standing acoustic waves is an
  equilibrium for bubbles.
• The bubbles expand and contract according to the
  Rayleigh-Plesset Equation2
• Where R radius of the bubble,
• ? density
• pg gravitational pressure
• P(t) acoustic wave pressure
• ? viscosity
• ? bubble surface tension.
• If the pressure oscillates rapidly enough, the
  compression is adiabatic.
• The temperature increases high enough during
  compression for the bubble to radiate by some
  mechanism (most likely Bremsstrahlung)

The theoretical and experimental aspects of the
sonoluminescence experiment are presented. Then
our experimental results and innovations are
presented, which suggest that the bubbles are
unstable in the flasks acoustic field. To test
this hypothesis, a simulation of a bubble in a
one-dimensional flask was performed. The results
strongly suggest that bubbles are indeed unstable
and will float to the side of the flask unless
certain bubble stability criteria are met.

• As bubbles were extremely difficult to trap (they
  float to the edge of flask), perhaps the
  antinodes are not stable equilibria.
• To test this, a bubble in a one-dimensional
  flask of length L was simulated.
• The standing acoustic pressure wave in the flask
  is represented by
• Where n is the harmonic number and we have
• Newtons second law is
• Where x is the position coordinate of the
  bubble, m is the effective mass of the bubble,
  and K is a constant of proportionality. We can
  define dimensionless variables by

• We found flask resonances, and amplified the
  signal to levels that should be able to produce
  SL.
• However, we could not observe SL because we could
  not trap bubbles they just float to the surface.
• A 1-D flask was simulated with Mathematica to
  examine the stability of bubbles at the
  antinodes.
• Simulations indicate that if the acoustic field
  is too strong, the bubble strays too far from the
  antinode, or the harmonic number is too high,
  antinode is an unstable equilibrium.
• The buoyant force on bubbles probably pushes them
  out of equilibrium.
• Even though the simulation is 1-D, Figure 4
  provides a back of the envelope estimate of
  bubble stability for a 3-D flask.
• To obtain SL, the acoustic wave must be powerful
  enough and the bubble must be stable at the
  antinode. Therefore, the parameter space over
  which SL may occur is smaller than previously
  thought.

• Here F represents the strength of the acoustic
  field. Then Newtons Law becomes
• The coordinate of the first antinode is
• Epsilon represents the displacement from the
  center of the antinode. The solution to the above
  differential equation with the above initial
  conditions was simulated with Mathematica. (See
  Figure 3 for sample solutions).
• If bubbles are displaced too far from the
  antinode, they are unstable (? too big). Also,
  if the acoustic waves are too strong, the bubbles
  are unstable (F too big). For each harmonic a
  plot containing this information was constructed
  by analyzing the data Mathematica yielded (Figure
  4). Data points represent the border between
  regions of stable and unstable bubbles.

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EXPERIMENTAL SETUP

• Fill a clean spherical flask with distilled,
  degassed water.
• Glue two transducers to the side of the flask to
  drive it, and one transducer to the bottom to be
  driven, acting as a microphone.
• Suspend the flask so that it is able to vibrate
  freely.
• Connect the circuit below (the transducers act as
  capacitors in the circuit)
• Set L0 and adjust the frequency of signal
  generator to a resonant frequency ?0 of the flask
  to set up a standing wave in the flask. (As
  measured by the microphone output)
• Adjust the variable inductor so the RLC circuits
  resonant frequency is ?0 (transducers act as
  capacitors), this further boosts the strength of
  the acoustic wave in the flask.
• Turn up the voltage level on the signal
  generator to produce a strong acoustic wave.
• Pull a small amount of water from the flask and
  drip it back in to produce bubbles, which should
  sonoluminesce.

Figure 1. Microphone Output at Resonance
Figure 3. Sample solutions for ?(?)
EXPERIMENTAL RESULTS

• We found powerful flask resonances at 33, 52, and
  79 KHz. The microphone voltage versus frequency
  was measured (Figure 1). This data was analyzed
  to find the quality factor of each resonance
• We constructed our own amplifier using op-amps
  (Figure 2). This is an economical approach as
  op-amps chips cost only pennies apiece, and also
  practical as op-amps can be tailored to the
  specific application.
• With the inductor in place, we were able to
  attain a microphone voltage of up to 80 Vpp. The
  last group achieved SL at a microphone voltage 4
  Vpp.
• Bubbles visibly quivered in the flask, that is,
  the radius of the bubble oscillated. The bubbles
  were easy to produce but difficult to trap in the
  interior of the flask. Conclusion the acoustic
  field strongly affects the bubbles dynamics.
• We could not observe sonoluminescence because we
  could not trap bubbles. Why is it so hard to trap
  bubbles? (Especially at higher harmonics)

Stable Bubble
Unstable Bubble
Quality factor is a measure of the sharpness
of a resonance. Note that microphone voltages
are of the proper strength for SL.
REFERENCES
1 http//www.pas.rochester.edu/advlab/11-Sonol
um/sono1.pdf 2 Brenner et. al., 2002,
Single Bubble Sonoluminescence, Rev. Mod. Phy.
74, 0034-6861
Figure 4. Bubble Stability Plots
Figure 2. The Op-Amp Amplifier
Bubbles Unstable at Antinodes
Bubbles Stable
Please note that the border between stable and
unstable regions is harmonic-number dependent.
Vo/Vi-Rf/Ri