Optical measurements probe the pressure and density of water under tension

It’s easy to think of water as a typical liquid, but many aspects of its behavior are highly atypical. For example, its freezing and boiling points are high compared with those of liquids with similarly-sized molecules, and it becomes less dense as it freezes. The unusual behavior is presumed to have something to do with the hydrogen bonds that form between the hydrogen and oxygen atoms of different water molecules, but the details of the connection are not fully known.

To better understand water’s anomalous nature, and to test theoretical models that connect its microscopic and macroscopic properties, researchers can study its behavior in metastable liquid phases. Supercooled water is metastable (see the article by Pablo Debenedetti and Eugene Stanley, PHYSICS TODAY, June 2003, page 40 ), as is water under tension, or negative pressure (see the article by Humphrey Maris and Sébastien Balibar, PHYSICS TODAY, February 2000, page 29 ).

Just as a spring or rope under tension is stretched beyond its equilibrium length, water under negative pressure is forced to fill a volume larger than it would occupy at zero pressure. Such stretched water could lower its potential energy by partially evaporating, creating a vapor bubble that would allow the remaining liquid to shrink to a smaller volume. But if no bubbles are present already, an energy barrier bars their formation.

Experimental studies of water under tension present mysteries of their own: Different methods of stretching water achieve vastly different maximum tensions before vapor bubbles form. To try to resolve the discrepancy, Balibar, Kristina Davitt, and colleagues at the École Normale Supérieure in Paris measured the equation of state (EOS, the relationship between temperature, pressure, and density) of stretched water. ¹ The last time the EOS was measured so extensively was in 1911, when Julius Meyer collected data down to −3.4 megapascals (−34 atmospheres). The Paris researchers measured it down to −26 MPa.

Stretching water

Like a rope, water can sustain only so much tension before it breaks. All sorts of things—impurities, rough patches on the walls, even cosmic rays—can nudge the water over the energy barrier and induce so-called heterogeneous cavitation. Even in an experiment carefully designed to minimize those effects, thermal fluctuations eventually become sufficient to surmount the barrier, at a point called the homogeneous cavitation threshold. At even more negative pressure, there’s another threshold, called the spinodal, at which the barrier to cavitation disappears and the stretched water is no longer metastable but unstable. The spinodal can be studied theoretically but can never be accessed experimentally, since thermal fluctuations are always present.

There are many methods of stretching water. In all but one, even careful experiments achieve tensions of at most −30 MPa. The exception is water trapped in micron-sized pockets (or “inclusions”) in natural or synthetic minerals. ² Researchers heat those samples until the water fills the entire inclusion (whatever air is present becomes dissolved in the water), then lower the temperature until gas bubbles reform. From the temperature measurements, they infer that the water reaches pressures as low as −140 MPa.

But inferring the pressure so far into the metastable regime requires an EOS extrapolated from positive pressures, and the extrapolation might not be accurate. That was the Paris researchers’ motivation: If the temperatures and densities measured in the mineral-inclusion experiments actually corresponded to a pressure closer to −30 MPa, there would be no discrepancy.

Experimental EOS data at negative pressure are sparse because the experiments are difficult: It’s necessary to determine the stretched water’s temperature, density, and pressure simultaneously. Temperature measurements are straightforward; in this case, the researchers kept their system at a constant 23.3 °C. Pressure and density measurements are harder, especially if the metastable state is present for only a short time.

The Paris researchers created negative pressure using an acoustic method developed in the 1980s to study cavitation in liquid helium. ³ The apparatus is shown in figure 1. All acoustic waves consist of alternating regions of high and low pressure. A short burst of ultrasound launched from a hemispherical piezoelectric transducer can produce positive and negative pressures of tens of megapascals at the sphere’s center.

By confining the most negative pressure to a small region far from any walls, and by sustaining the tension for a short period of time, the researchers limited the effects of heterogeneous cavitation. Still, they found that bubbles consistently formed at −30 MPa. To be safe, they limited their study to −26 MPa.

Pressure and density

To determine the pressure and density of the water, the researchers used two optical methods. The first, employing a fiber-optic probe hydrophone, measured the density. ⁴ An optical fiber (positioned vertically in figure 1) extends into the water, with its tip at the acoustic wave’s focus. IR light directed down through the fiber is partially reflected when it reaches the tip. The reflected intensity depends on the water’s local refractive index, which is a function of the density.

The second optical measurement, Brillouin scattering, used the horizontal green beam in figure 1. Similar in principle to Raman scattering, in which inelastically scattering photons lose or gain energy as they excite or de-excite molecular vibrations, Brillouin scattering involves the excitation and de-excitation of thermal phonons. Those phonons—which are distinct from the applied ultrasound pulse—have energies related to ∂P/∂ρ, the derivative of pressure with respect to density; the energies are revealed in the frequency shifts of the inelastically scattered photons with respect to the elastically scattered photons.

A Brillouin-scattering spectrum for one value of the negative pressure is shown in figure 2. Producing that spectrum—with 50–60 photons at each of the peaks, just enough to reliably determine their locations—required 15 hours of data collection. Keeping the experiment stable enough over that amount of time was a challenge. To check the stability, the researchers also collected data between acoustic pulses and over the positive-pressure half of the acoustic wave—two parts of the phase diagram where the EOS is well known.

Equation of state

Integrating the Brillouin-scattering measurements of ∂P/∂ρ and combining them with the fiber-optic measurements of ρ gives the pressure–density relationship shown by the purple line at the top right of figure 3. The blue and red curves are the two commonly used model EOSs extended to their respective spinodals. ⁵ It’s no coincidence that both curves flatten out on the low-pressure end. The spinodal is the point at which liquid molecules lose their grip on one another, allowing the liquid to break. At tensions approaching the spinodal, the cohesion of the liquid is starting to break down, so a small change in pressure produces a relatively large change in volume.

The experimental EOS doesn’t flatten out, which means that the acoustic method’s cavitation threshold is nowhere near the spinodal. It also implies that the temperatures and densities measured in the mineral-inclusion experiments do not, in fact, correspond to pressures near −30 MPa: The mineral-inclusion method really does reach more negative pressures than any other technique, and it’s still not clear why.

“We’ve gone as far as we can with the acoustic method,” Davitt explains. “We can’t make measurements beyond the cavitation threshold.” To extend their experimental EOS to even more negative pressures, she and her colleagues are working on ways to make simultaneous thermodynamic measurements on water in mineral inclusions.