Beneath a suburb of Hamburg in Germany, an underground particle accelerator propels electrons at close to the speed of light through a slalom course of magnets. Racing through the twists and turns, the electrons emit bursts of radiation that produce one of the world’s most powerful X-ray laser beams.
This prized machine, the European X-ray Free Electron Laser Facility (XFEL), has helped researchers to make ultrafast movies of chemical reactions and to map the atomic structures of viruses. Now, they are using it to crack the secrets of a seemingly simple process that has bedevilled scientists for decades: how water and other liquids freeze.
For 150 years, theorists have been trying to explain the process that turns pure liquids into solids. But their models of how quickly this happens are often wildly inaccurate when compared with experiments — results can be off by as much as 20 orders of magnitude.
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It’s hard to resolve this problem, says theorist Michele Parrinello at the University of Italian Switzerland in Lugano, Switzerland. “Experiments are very difficult,” he says. “And theory is difficult, and computer simulations are also difficult.” Tiny errors in modelling or in experiments can lead to huge changes in outcomes.
The mystery behind the freezing of liquids — including molten metals — is not just an esoteric puzzle. More accurate insights into freezing would help in understanding how ice develops in clouds high in Earth’s atmosphere and, in turn, would improve the models that are used to forecast how quickly the world will warm because of greenhouse gases. Better theories would also provide information for geophysicists on how Earth’s solid inner core formed and what happens inside other planets.
At the European XFEL and other laboratories with similar capabilities, researchers are finally starting to make headway in cracking the freezing problem. Using innovative experimental designs, they have captured the first few microseconds of the process. These and other developments are helping them to close the gap in understanding how this happens. And it turns out that disorder plays a bigger part in freezing than scientists had thought.
Pure problems
Modern theories about freezing have their roots in the work of physicists Daniel Fahrenheit in the early eighteenth century and Josiah Willard Gibbs more than a century later. Gibbs used statistical mechanics to describe the freezing process of a pure liquid — that is, one that doesn’t contain any contaminating particles.
In nature, most freezing isn’t so pure. Take what happens when you put a glass of water in a freezer. Water molecules first start to crystallize on the surface of the glass and on impurities in the water. This ‘heterogeneous nucleation’ happens much more readily than the ‘homogeneous nucleation’ that takes place in a pure liquid. In most clouds, for example, water freezes on particles of dust and other contaminants through heterogeneous nucleation. But at high altitudes, homogeneous nucleation often occurs because the air is extremely cold and comparatively free of impurities.
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The case of a pure liquid is what preoccupied Gibbs, and he developed a theory based on the energy changes that happen when a substance is cooled below its freezing point. Gibbs thought of freezing as a competition between two states of molecular organization. On the one hand, molecules in a liquid can settle into a lower energy state if they assume an ordered crystal structure. It is energetically favourable for liquid molecules to crystallize when cooled below a certain point.
On the other hand, there is an energy cost to creating a new interface between the crystal and liquid. For small particles, the surface-to-volume ratio is huge, and so as a tiny crystal develops in liquid, it is inherently unstable and rapidly falls back into the uncrystallized liquid form. The transition becomes permanent only when a crystal cluster reaches a critical radius; that’s when it becomes energetically favourable for the crystal to persist and grow.
In Gibbs’s theory, the rate of crystallization depends on the probability of a rare thermal fluctuation — a large number of molecules falling by random chance into the correctly ordered crystalline structure — to push a cluster over this energy hump.
Later scientists developed Gibbs’s thinking into what is known as classical nucleation theory (CNT), and today this framework remains at the heart of most theoretical approaches to nucleation. To apply it, researchers generally need to make some simplifying assumptions — for example, that the critical nucleus (the small initial nucleus that becomes just large enough to begin growing) is spherical, and that the tiny new crystal has the same properties as a large sample of the solid. With these assumptions, the theory can predict nucleation rates: how many nucleation events should take place per second and per unit of volume.
Cirrus clouds are made of ice crystals and have a warming effect on the climate.Credit: Nick Greaves/Alamy
Unfortunately, the theory gives the rate as a wickedly sensitive exponential that depends on liquid–solid surface tension, the liquid viscosity and other variables. This means that small changes in conditions for a pure liquid can yield huge changes in freezing rate, says physicist Robert Grisenti at the GSI Helmholtz Centre for Heavy Ion Research in Darmstadt, Germany.
In the case of pure liquids, “a small droplet of water held at −20 °C won’t freeze in billions of years,” he says, “but cool it 15 degrees further and it will freeze in a fraction of a second.”
Because even tiny differences in conditions cause such big changes, that makes it difficult for researchers to do reproducible experiments. And real-world uncertainties also mean that theorists who make slightly different assumptions can produce shockingly varied results.
Extraordinary sensitivity
A scientist using CNT to estimate the nucleation rate in a substance such as water needs to make modelling choices. For example, a crucial element in the calculation is the surface tension or assumed energy associated with the interface between the critical crystal seed and the surrounding liquid. Different scientists can justifiably make slightly different assumptions. A survey of theoretical estimates shows that these choices can cause predictions of the nucleation rate to vary by as much as 25 orders of magnitude1.
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This sensitivity shows up in experiments as well. Physical chemist Jonas Sellberg at the KTH Royal Institute of Technology in Stockholm and his colleagues have run experiments to measure the nucleation rates in small droplets of water. When the researchers collated measured nucleation rates from many studies performed in the past three to four decades, they also found an enormous variation of values across more than 20–25 orders of magnitude2.
“This is just water,” he says. “It makes you wonder: how can rates vary that much?”
These experiments include studies done using microdroplets, nanodroplets and with thin films of water deposited on a surface. In one set of experiments, Sellberg says, researchers found nucleation rates varied over six orders of magnitude, even in experiments that tried to reproduce similar conditions2. “These are not random errors,” he says, “it’s just that you get very different results depending on how you prepare these films.”
Liquid jets
To try to bring theory and experiment closer together, researchers have focused on a simpler problem. Water is a complicated substance in part because its molecules have a directional character. They connect through hydrogen bonds, which require close proximity and the correct orientation. Other liquids can be much more straightforward to work with.
In one simple model — the Lennard-Jones (LJ) liquid — molecules are attracted to one another across long distances but repel each other when they get too close. LJ liquids occur in nature in the form of liquefied noble gases such as krypton or argon, which condense when chilled sufficiently. Grisenti and his colleagues have used the XFEL to probe the freezing of these simple liquids.
An X-ray diffraction image of freezing krypton.Credit: Adapted from J. Möller et al. Phys. Rev. Lett. 132, 206102 (2024)
They produced high-velocity jets of liquid krypton and argon and sent them through a vacuum, where the liquid cooled rapidly because of evaporation. The researchers fired X-ray pulses at the jets and analysed the resulting diffraction patterns, which provided information about the liquid’s changing atomic configuration. These showed that the two jets evolved from a liquid state to showing signs of crystal structure over a distance of just a few hundred micrometres, and in a few microseconds3.
When the researchers combined the theory with simulations, the predicted nucleation rate was 100–1,000 times higher than the experimental value. However, compared with previous experiments, this is about 100 times closer to agreement.
The improved agreement arises from two aspects, says Grisenti. Because LJ liquids are so simple, slightly different theories and simulations all give relatively consistent predictions. And the measured nucleation rates are probably more accurate than those from previous experiments.
