March 16, 2015

Fig. 1: Single-bubble sonoluminesce setup. The
luminescing bubble is the blue dot at the center of the
flask. (Source:
Wikimedia
Commons) |

Sonoluminescence is a mysterious phenomenon whereby sound is turned into heat and light inside of a bubble. A typical setup for a sonoluminescence experiment (shown in Fig. 1) consists of a small flask of water and two ultrasonic piezoelectric speakers. The speakers drive a high frequency sinusoidal signal that is tuned to the resonant frequency of the water-filled flask. This is about 25 kilohertz but will vary based on the shape and material of the flask. [1] When the pressure of the sound wave is greater than atmospheric pressure, negative pressure in the flask causes bubbles of dissolved gas such as air to form. This process is known as cavitation. As the pressure in the flask becomes positive, the bubble collapses quickly and violently and then oscillates before expanding again. [1] The process then repeats itself. As the bubble collapses, the gas inside heats up and a flash of light is emitted. Some scientists say that it heats up enough to cause nuclear fusion but others disagree.

Nuclear fusion is the process by which two light
atomic nuclei (usually hydrogen) combine to form a heavier atom
(helium). [2] Enormous energy can be released in the form of heat and
scattered particles such as neutrons. The primary obstacle to obtaining
a fusion reaction is the coulomb barrier which is the force of repulsion
felt by the atoms as they get closer in distance. [2] For fusion to
occur, enough energy must be supplied to the atoms to overcome this
force of repulsion. This is usually done by heating up a hydrogen
plasma. The average kinetic energy of the atoms increases as the
temperature increases. Statistically speaking, nuclear fusion can happen
at any temperature. The Maxwell-boltzmann distribution predicts a
non-zero but very low probability for atoms to have a very high kinetic
energy at a low temperature. However, to get a significant amount of
nuclear fusion reactions to occur, the temperature must be large enough
that there is statistically significant probability for atoms to have a
kinetic energy above the coulomb barrier. Unfortunately, the required
temperature is huge. For the easiest form of fusion (deuterium-tritium)
the temperature required for sustained nuclear fusion (ignition) is
about 45 million kelvin. [2] Another obstacle to achieving fusion is
called the Lawson Criterion. A nuclear fusion reaction must satisfy the
Lawson Criterion in order to be self-sustaining. For the criterion to be
satisfied, the density of the fission reactants multiplied by how long
the reaction lasts must be greater than 10^{14}. [2]

A simplified analysis of the bubble dynamics will be
conducted under the assumptions of ideal gas and isentropic compression.
I will assume that the bubble is comprised of air and that the air is
only nitrogen. A good starting radius (R0) for the bubble is 5.75
micrometers. The starting temperature (T0) is 283.15 kelvin and the
initial pressure (P0) is 1 atmosphere or 101,325 Pascals. The amplitude
of the acoustic wave is 1.58 atmospheres and the frequency is 17
kilohertz. The Rayleigh-Plesset Equation is used to model the collapse
of the bubble. [1] First, let's calculate the total energy in the
bubble. The maximum energy stored in the bubble is given by E = (4/3)
π R_{max}^{3} P_{0}. [1] The maximum radius
is Rmax = 10 R0 = 57.5 micrometers. [1] The energy in the bubble is then
8.07 × 10^{-8} Joules. Using initial data, the number of
nitrogen atoms in the bubble is 2.14 × 10^{10}. This gives
an average potential energy per atom of 23.53 eV. The average maximum
collision energy is twice that which would be 47.07 eV. The required
collision energy for fusion is on the order of tens of keV. From this,
we can already see that it is unlikely that enough fusion reactions will
occur to cause ignition assuming an isentropic collapse. Now we can turn
our attention to how efficient the collapse is at turning the potential
energy into kinetic energy. The minimum radius that the bubble collapses
to is about 1 micrometer. [1] Using the isentropic ideal gas relations
plus a correction due to the van der walls hard core, we get a maximum
bubble temperature of 2678.97 Kelvin. [1] This gives an average kinetic
energy of 0.346 eV per atom or 0.693 eV per collision. Thus, only 1.47%
of the total average potential energy has been converted to kinetic
energy. From this, we see that purely isentropic compression cannot
cause the bubble to reach temperatures high enough for fusion.

There is one gaping hole in the simplified analysis
above. I assumed the collapse is isentropic, however, the bubble
collapse reaches supersonic speeds of greater than Mach 4. [1] In the
supersonic regime, a shockwave would form inside the bubble. This
shockwave is non-isentropic and could potentially help to focus the
energy of the collapsing bubble. A spherical shockwave could tightly
compress a small amount of the gas at the very core of the bubble,
possibly increasing its temperature enough to cause fusion. The
resulting heat energy would diffuse throughout the bubble and, if large
enough, can cause ignition throughout the bubble's volume. Shockwave
analysis is a very complicated subject and will not be attempted here.
Lahey Jr *et al.* conducted their own shockwave analysis and their
result is promising. They predicted that interacting shockwaves at a
point close to the core could reach about 100 million kelvin which is
enough to cause deuterium-tritium fusion. However, this only lasts for
about 0.1 to 1 picosecond. [3] This presents a problem because it does
not satisfy the Lawson Criterion. Lahey Jr. *et al.* address this
in their paper. For the Lawson Criterion to be satisfied for their
simulation, it must last about 10000 picoseconds. Thus, while their
simulation gives fusion reactions, it does not last long enough to
sustain itself. They call their results "fusion sparks". [3]

Probably the main problem with sonoluminescence is
the Lawson Criterion. For a portion of the bubble to heat up to
fusion-level temperatures, there absolutely needs to be a very powerful
shock to focus the energy because the energy gained from isentropic
collapse is very negligible (as was shown in the Simplified Analysis
section). For such a shock to form, the bubble must collapse extremely
rapidly which naturally leads to a time scale which is much smaller than
the Lawson Criterion. In the case of Lahey Jr. *et al.*, the time
scale was four to five orders of magnitude less than the Lawson
Criterion. [3] One possible way to overcome this is simply to optimize
the process. Have a rapid enough collapse to heat the bubble up, but
have it be slow enough to meet the Lawson criterion. This, however could
be impossible. Another way would be to somehow (probably using the
acoustic drive signal) hold the bubble at its minimum radius long enough
to satisfy the Lawson Criterion. This is also much easier said than done
because sonoluminescence is very sensitive to the drive signal. Doing
this may just cause unwanted instability rather than fusion ignition.
Another idea could be to stimulate the bubble's contents using something
other than an acoustic drive to amplify the effect of a smaller
shockwave.

Another problem with sonoluminescene is that is very sensitive to the input parameters such as the gas used in the bubble, the drive signal amplitude, and initial temperature and pressure. [1] The most worrying effect is that the bubble becomes unstable above a certain acoustic drive amplitude. This is likely due to distortions in the bubble during its collapse. [1] This is problematic because the drive amplitude affects how fast the bubble collapses, its minimum radius, and ultimately the maximum temperature. Theoretically, one could just increase the amplitude and easily get temperatures hot enough for fusion if the bubble did not become unstable. One possible solution could be to experiment with different drive signals, however once again, this is a parameter that the bubble is sensitive to.

Lastly, another problem is how to extract the energy
released in the bubble if ignition is achieved. I would assume that the
bubble would burst, releasing all the energetic gas molecules into the
surrounding water and heating the water enough to produce steam to drive
a turbine. Let's take a look at the energy associated with this. Assume
the same sonoluminescence situation as in the simplified analysis but
with hydrogen instead of nitrogen. I will be using pure hydrogen instead
of deuterium-tritium to simplify things but I will assume the energetics
of a deuterium-tritium reaction. The total mass of hydrogen in the
bubble is 5.57 × 10^{-11} grams. Since each fusion
reaction takes two hydrogen atoms, that leaves 2.79 ×
10^{-11} grams as the reactive mass. A deuterium-tritium
reaction releases 331 megajoules per gram of reactive mass. [2] That
means that if all the gas in the bubble undergoes sustained nuclear
fusion, only 9.23 × 10^{-3} Joules of energy will be
released. In a 100 milliliter flask typically used in sonoluminescence
experiments, that's not even enough energy to raise the temperature by 1
degree Celsius! One solution is just to keep trying to cavitate more
bubbles of hydrogen gas to fuse. The problem with this is that it will
heat up the water gradually and sonoluminescence is sensitive to the
water temperature so you may reach a maximum temperature at which
bubbles no longer form. Another solution is to do multiple bubble
sonoluminescence whereby multiple bubbles form and cavitate at once,
however not as much fusion-level research has been done in this
area.

Unfortunately, it seems fairly likely that sonoluminescence will be a dead end for nuclear fusion energy. However, there have been some promising developments that show that more research must be done. More powerful and elaborate numerical simulations, such as direct numerical simulation, can be used to probe the more complex effects of the bubble's collapse. More carefully controlled experiments and documentation and explanation of the effects of changing different parameters to control the process can also be done.

© Mario Chapa. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.

[1] S. J. Putterman and K. R. Weninger,
"Sonoluminescence: How Bubbles Turn Sound into Light, "Annu. Rev. Fluid
Mech. **32**, 445 (2000).

[2] L. R. Radovic and H. H. Schobert. *Energy and
Fuels in Society* (McGraw-Hill, 1992).

[3] R. T. Lahey, Jr., R. P. Taleyarkhan, and R. I. Nigmatulin, "Sonofusion - Fact or Fiction?" Rensselaer Polytechnic Institute, 2 Oct 05.