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Negative Pressures
and Cavitation in Liquid Helium
Cavitation—the formation of bubbles—is a familiar phenomenon. Whenever a liquid is agitated violently, there is a possibility that cavitation will occur (see, for example, figure 1). In the case of boat propellers or hydraulic machines, cavitation is a problem that engineers try to avoid. In other contexts, however, cavitation can be useful—as, for example, in ultrasonic cleaning devices. Desirable or not, cavitation is a complex phenomenon because inhomogeneities in the liquid—such as walls, dissolved gases, vortices, and impurities—usually play a major role in the nucleation of bubbles. As a consequence, our understanding of cavitation is incomplete. To make some progress, our research groups have examined cavitation in superfluid helium, a simple and pure liquid. It is the coldest liquid in nature and exists only at temperatures near absolute zero, where every other liquid is frozen. It can therefore be filtered very efficiently and prepared without impurities. Furthermore, it has been predicted that, at very low temperatures, the nucleation of bubbles will occur by means of quantum tunneling—a process we believe has now been observed. How
does a liquid break? Why do such things happen? At low pressure P, the equation of state for air (or any other gas) is the well-known ideal-gas law. As P decreases toward zero, the volume V varies as the inverse of the pressure. Consequently, when an air bubble is present in the water, the size of the bubble grows without limit as soon as the weight pulls on the piston. But what happens when there is no bubble and the water is able to stay in a state of tension? This state is only metastable: If a bubble forms, the piston will be able to move, and that movement will clearly lower the potential energy of the system. Before a large bubble can be formed, however, the system has to overcome an energy barrier. The existence of
an energy barrier against nucleation is very general. The barrier arises
because the liquid–gas transition is discontinuous, or “first order.”
Such a barrier exists for any first order transition because the interface
between the two phases has a finite energy per unit area. In our example,
this energy is nothing but the surface tension a
of water. Since a is nonzero, the formation
of a bubble with radius R has an energy cost of 4pR2a.
When such a bubble forms, the energy of the whole system also contains
the work of the negative pressure over the bubble volume, so that the
total energy cost of forming the bubble is
A thermal fluctuation may enable the system to pass over the energy barrier. The probability of such an occurrence is proportional to the factor
where T is
the absolute temperature and k is Boltzmann’s constant. In this
simplified model, it is clear that cavitation should be a random process
that depends on temperature. This discussion of the energy required to
form a bubble is inadequate in one very important respect. Equation 3
says that to maintain a given probability that nucleation will take place,
the ratio of the energy barrier DE to
kT must have a particular value; this requirement would lead to
the conclusion that as the temperature decreases, the pressure at which
nucleation occurs should diverge as T–1/2. But such reasoning
ignores the obvious fact that there is an upper limit to the force one
molecule in a liquid can exert on another, so that for some negative pressure
of finite magnitude, the liquid will stretch without limit (that is, the
compressibility will become infinite). The pressure at which that happens
is called the spinodal limit. When it is reached, the sound velocity becomes
zero and the barrier to nucleation vanishes.
Given these considerations, one can see that if the applied pressure is only slightly negative, the energy barrier is very large and the chance that a bubble will form is very small. Then the liquid can exist in a state of negative pressure for a long time—but only if the liquid is very clean. If the liquid contains dirt or dissolved gas, the formation of bubbles is usually much likelier. Bubbles also tend to form preferentially on the walls of a container. Bubble production associated with walls or impurities is called heterogeneous nucleation—as distinguished from homogeneous nucleation, which takes place within the volume of an ideal bulk liquid and is an intrinsic property of the liquid. Everyday
negative pressures The water pressure
can also become negative in vortices generated by a propeller at the rear
of a boat. The pressure decreases because the local velocity increases
towards the vortex core, in accordance with a general law of hydrodynamics
established by Daniel Bernoulli in the 18th century. But the spinodal
limit of water is far from being reached in such vortices. In a complex
medium such as seawater, bubbles grow from seeds that are already present,
such as microbubbles of dissolved air or various particles floating around.
Achieving large negative pressures in water has actually been a challenge
to scientists for more than a century. Marcellin Berthelot claimed in
1850 that he had reached –50 bars in a glass ampoule completely filled
with pure water.1 In 1967, Edwin Roedder at the US Geological
Survey reached –1000 bars with water inclusions in natural rocks.2
The world’s record now belongs to Austen Angell and his collaborators
at Arizona State University, who in 1991 reported achieving –1400 bars
with a similar technique but synthetic materials.3 Such very
large negative pressures are comparable to theoretical predictions by
Robin Speedy of the University of Wellington (in New Zealand) for the
maximum negative pressure in water.4
Helium
at negative pressures One important question concerns the determination of the pressure at the spinodal. One method of estimating the location of the spinodal relies on an extrapolation of the sound velocity into the negative pressure range—the spinodal is the pressure at which the sound velocity reaches zero. This approach requires making some assumption about the way in which the sound velocity goes to zero at the spinodal, an interesting problem of statistical physics that has not yet been completely solved. Estimates based on this approach5 give a spinodal pressure of between –9 and –10 bars at T = 0 K. At the University of Trento (in Italy), Franco Dalfovo and his coworkers6 have used a density functional theory to describe liquid helium-4 and have obtained about –9.5 bars for the zero-temperature spinodal pressure. The Spanish group of Jordi Boronat at the Polytechnic University of Catalonia in Barcelona found –9.3 bars using a Monte Carlo numerical method.7 Thus, it now seems well established that the extreme limit of metastability of liquid 4He is around –9.5 bars. (For 3He, the corresponding result is approximately –3 bars.) At higher temperatures, the spinodal pressure becomes less negative. Indeed, the spinodal line has to reach the liquid–gas critical point (5.2 K and +2.2 bars in 4He), because there the difference between liquid and gas vanishes. The location of the superfluid transition—the “lambda line”—at negative pressures is even more difficult to estimate; the curve shown in figure 4 is based only on a guess. The lambda line is expected to reach the spinodal at some finite temperature, but again, essentially nothing is known about the behavior of the spinodal or the lambda line in the vicinity of where they meet. In the phase diagram of helium, there is thus a new world at negative pressure that has not yet been very much explored and is not shown in textbooks. It extends from the liquid–gas equilibrium curve at small positive pressures down to the spinodal line a few bars below zero. This regime has been our playground in recent years. To study the liquid in this pressure regime, it is important to use very clean liquid. Robert Finch and his coworkers8 made several studies of cavitation in helium in the 1960s and 1970s using liquid from the main bath of a helium dewar. In those early experiments, cavitation was detected even at very small negative pressures of only a few millibars, presumably because of some form of contamination of the helium. A sample of clean liquid helium can easily be prepared by filling a cell through a fine capillary. This procedure removes any particles of solid air that might be present in a helium dewar. It is also important
that the negative pressure be produced in the interior of the liquid,
far from any wall at which heterogeneous nucleation could occur. We use
a sound wave with a frequency of around 1 MHz generated by a hemispherical
ultrasonic transducer (figure 5). The sound comes to a focus in the interior
of the liquid. This method readily produces a pressure oscillation with
an amplitude of several bars at the focal point. The volume throughout
which the pressure swing is produced is determined by the sound wavelength
and is typically on the order of 10–6 cm3. To determine
whether a cavitation bubble has been produced, we shine a laser beam through
the acoustic focus. Bubbles will scatter the light, which is detected
by a photomultiplier.9
Landmarks
in a quantum sea
The radius R of electron bubbles is determined by minimizing the total energy, given by the sum of the ground state energy of the electron (h2/8me R2, with me being the electron mass), the surface energy of the bubble (4pR2a), and the work done against the liquid’s pressure in forming the bubble (4pR3P/3). If the pressure is zero, a bubble will have a radius of around 19 Å. When the pressure is made negative, the bubble grows, and at a critical pressure of around –2 bars, the electron bubble explodes—it becomes unstable and grows without limit. The critical pressure can be calculated in a reliable way in terms of known quantities; therefore, it provides a milepost in the negative-pressure regime. Lying approximately 20% of the way to the spinodal, this milepost offers a good calibration of our system for the production of negative pressure.10 To study the explosions of electron bubbles, we use a radioactive b source to inject electrons into the liquid. Electrons from the b source enter the liquid with high velocity, lose energy, and then form stable bubbles that wander around in the liquid. If such a bubble wanders into the region of the sound focus, it will expand on the negative part of the pressure oscillation, become unstable, and explode (see figure 6). These explosions
are distinct from the processes that occur in a conventional helium bubble
chamber. In a bubble chamber, an energetic charged particle passes through
a liquid that is already at a negative pressure. In fact, there has been
some uncertainty about the mechanism of bubble formation in bubble chambers.
The traditional view has been that the bubble is formed as a result of
the energy that is deposited by one of the secondary electrons produced
along the track of the fast particle.11 Our studies of electron
explosions have revealed another possibility (for liquids in which electrons
form bubbles).12 When a secondary electron is produced, it
quickly comes to rest in the liquid. It then pushes liquid away to open
up a cavity. The inertia of the liquid surrounding the bubble causes the
radius of the cavity to increase beyond the radius corresponding to the
minimum energy configuration. For example, in liquid helium at a pressure
of –0.3 bars, the cavity will reach a maximum radius of about 28 Å and
then oscillate for a while before finally settling down into a state with
a radius of 20 Å. But if the pressure is more negative, the inertia of
the liquid around the bubble may be sufficient to make the bubble reach
a size beyond the critical radius for bubble nucleation. Then the bubble
will grow without limit. The critical pressure that can be calculated
for this process is in very good agreement with the old measurements for
the threshold pressure for operation of helium and hydrogen bubble chambers.
It may be possible to find other landmarks along the path to the spinodal. One possibility is to use light to raise an electron bubble to an excited state; the bubble should then explode at a negative pressure of smaller magnitude that can also be calculated accurately. A second possibility is to introduce quantized vortices into the liquid. When a vortex is present, each helium atom in the liquid near it will have one unit of angular momentum. Consequently, the liquid will circulate around the vortex with a tangential velocity of h/m4r, where r is the distance from the vortex core and m4 is the mass of a helium atom. This circulation will result in a pressure at the vortex that is more negative than the pressure in the bulk of the liquid. Thus, the vortex core should explode before the spinodal is reached, thereby providing another way in which bubbles can form.13 Thermal
versus quantum cavitation
At first sight, it looked as though it would be very difficult to test these theoretical ideas. Even with the electron explosions as mileposts, the pressure estimation was still not very accurate. Consequently, to determine that the pressure was 0.3 bars from the spinodal seemed to be equivalent to finding a spot 0.3 feet from the edge of a cliff on a very dark night. Fortunately, our recent experiments at the Ecole Normale Supérieure have produced two results that provide indirect but strong evidence that quantum cavitation has indeed been seen.17 When the quantum
regime is entered, the pressure at which cavitation occurs should become
independent of temperature; the experiments found the onset of such behavior
at a temperature of 0.6 K. This temperature was much higher than the Tc
of 0.2 K predicted by theory, but the discrepancy was quickly understood.
In the experiment, 0.6 K was the measured temperature of the liquid before
the pressure was reduced; during the expansion of the liquid that took
place before cavitation, however, the temperature should decrease by about
a factor of three. Thus, the measured critical temperature was in reasonable
agreement with theory. The second piece of evidence for quantum cavitation
concerns the statistics of the cavitation. A series of experiments was
performed in which the pressure swing applied to the liquid was controlled
very precisely. Even though the pressure reached in each sound pulse was
the same, sometimes a bubble was produced and sometimes it wasn’t (figure
7). The observations that the cavitation is statistical, but independent
of temperature, strongly suggest that nucleation is occurring by means
of quantum tunneling.
These observations of quantum cavitation have been performed with liquid 4He in the superfluid state. Can bubbles be produced by quantum tunneling only when there is some sort of quantum coherence in the liquid? We hope to answer that question through measurements on liquid 3He, which is not superfluid except at much lower temperatures. For 3He, the crossover temperature from thermal nucleation to quantum tunneling has been predicted16 to be around 120 mK. In measurements down to 40 mK, our preliminary results show that the cavitation pressure is about three times less negative in 3He than in 4He, as would be expected if cavitation occurs by means of quantum tunneling very close to the spinodal limit for both isotopes in the low-temperature limit. Furthermore, the results show that the pressure at which bubbles form in 3He continues to vary with temperature down to at least 100 mK, a temperature much lower than in helium-4 (again, as predicted). However, the existence of a temperature-independent regime in 3He has not yet been established.18 Much more accurate measurements are now in progress. If a “quantum plateau” below 100 mK is found, it will demonstrate that quantum cavitation can take place without the need for the quantum coherence characteristic of a superfluid. If there is no quantum plateau, it will be necessary to reconsider the theory of quantum cavitation in 3He. For example, since 3He is a Fermi liquid, its compressibility is a function of frequency. This behavior may affect the tunneling process, which occurs on a timescale of around 10–11 s. It is also possible that the spinodal line has an unexpected temperature variation, or that the large viscosity of 3He has to be considered. Answering questions about such issues should lead to further improvements in the understanding of liquids close to a spinodal. We thank many friends and collaborators for their help with this research and for discussions. The work at Brown University has been supported by the National Science Foundation. References
© 1999 American Institute of Physics
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