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Mpemba effect runs in reverse

9 March 2022

Observations of a confined particle reveal that initially cold systems reach a high final temperature faster than initially warm systems, breaking with intuition but not theory.

Melting ice cream.
Credit: Marstourist/Pixabay

In 1963, a 13-year-old Tanzanian student named Erasto Mpemba and his secondary school classmates were tasked with making ice cream. There was limited room in the freezers, and he found himself falling behind other students. His classmates were boiling milk for the treat, then letting the mixture cool before placing it in the freezer. To stay on track, Mpemba put his hot concoction straight into the freezer. Checking on the dessert some time later, he found it perfectly frozen, while his classmates’ remained liquid.

The idea of water freezing faster when it starts at a higher temperature was christened the Mpemba effect after he published the finding in 1969 with physicist Denis Osborne. Although reproducible results have been difficult to obtain for water, toy-model systems made of a single colloidal particle have undeniably demonstrated the nonintuitive effect.

Now John Bechhoefer of Simon Fraser University and his colleagues have experimentally shown the inverse Mpemba effect, observing that under specific conditions a cold particle heats up faster than a warmer counterpart. The team used optical tweezers to create a tilted double-well potential that confined a colloidal particle (a small glass bead). The researchers then measured the particle’s response as a function of its initial temperature.

The new measurements indicate the inverse Mpemba effect is much weaker than the conventional, forward effect, which explains why it had not been as commonly observed. The work also experimentally corroborates some of the predicted mechanisms behind both the forward and the inverse effects.

A controversial history

Hot water freezing before cold water has been documented throughout history. Aristotle reported that ice fishermen would cut holes in ice, insert their reeds into the water, then pour warm water around the reeds. The warm water would freeze and fix the reeds in place faster than if the fishermen had used cold water.

But by the time Mpemba published his observations, the effect had been somewhat forgotten. After all, Newton’s law of cooling stated that the rate of cooling is proportional to the temperature difference, and who was going to disagree with Newton?

However, Mpemba’s work sparked interest—and controversy—in the scientific community about the mechanisms behind the effect and even its mere existence. At first researchers justified the forward effect as a combination of simple explanations: evaporation reducing mass, dissolved gas changing heat capacity, and convection flows.

Then a decade ago, computational chemists simulated water molecules and observed the Mpemba effect despite the absence of the supposedly necessary mechanisms. Recently, researchers have also observed the effect in other liquids and magnetic alloys, which indicates that causes specific to water, like hydrogen bonds, cannot fully explain the effect. Further complicating the investigation of the Mpemba effect is that many water-based experiments involve a phase transition between liquid and ice, which is dependent on conditions like the container and environment; that makes measurements hard to obtain and extremely difficult to reproduce.

To gain clarity on the Mpemba effect, researchers have recently developed kinetic models with molecules moving and colliding in nonequilibrium. Meanwhile, more abstract research into the relaxation process of particles has predicted not only the forward effect but also its inverse. “There had been a bunch of people who claim to see this in cooling,” says Bechhoefer. “But never had anybody claimed to see it in heating.”

A toy system

Bechhoefer and his team used a simple and unambiguous definition to measure the inverse Mpemba effect: the time it takes a system that starts at one equilibrium temperature to reach another, higher temperature. By using a single colloidal particle, they avoided the unnecessary complications of phase transitions in water and other systems.

In their experiment, optical tweezers create a force and thus a potential in which the particle moves. The potential is a tilted double well, as shown in figure 1 below. The particle can settle into two different local minima, the left or the right valley. The potential qualitatively mimics the states of supercooled water: One local minimum has a slightly higher free energy, representing liquid (left), and the other, representing solid ice (right), has a lower free energy because that state is favored.

The tilted double-well potential has local minima representing two possible states.
Figure 1. The tilted double-well potential has local minima representing two possible states. After release, the colloidal particle falls into one of the temperature valleys. Credit: A. Kumar, R. Chétrite, J. Bechhoefer, Proc. Natl. Acad. Sci. USA 119, e2118484119 (2022)

After a particle is released with a starting position that corresponds to an initial temperature, the system is quenched and relaxes quickly to the final temperature as the particle swings and then settles into the pit of a local minimum. The short relaxation time, about one-tenth of a second, allowed the researchers to perform enough measurements to create a useful statistical picture within minutes.

To get the same quality of results observed for the forward Mpemba effect, the team had to perform five times the number of trials—5000 rather than 1000—and they believe they know why. In the forward effect, particles fall quickly into one of the two potential wells. The fraction in the left and the fraction in the right, in general, differ from the fractions that should probabilistically be in each well in equilibrium, after the system has settled to its final temperature. That difference leads to a second, slower step, in which particles hop the barrier into the other well until the correct fractions are attained. If the barrier is tall, the process can be slow and create a sharp separation in time between the initial drop into the well and the hopping. When the Mpemba effect is working at its strongest, the hopping is minimal and the relaxation time to the final equilibrium temperature is short.

In the inverse effect, the separation in time between the release into the potential and the hopping stage is less extreme—there is less of an observable difference between when the particles hop and when they do not. The barrier is lower at high temperatures compared with the particles’ energy, so they can diffuse between the wells more easily. The less staggering separation in time makes the inverse Mpemba effect weaker and more difficult to observe.

Mechanisms behind Mpemba

As for the mechanisms behind the Mpemba effect, the work of Bechhoefer and his colleagues provides one picture among many. For an optimal initial temperature, the first stage of relaxation, as the particles fall into a well, can result in just the right number of particles in the left and right valleys. Without the second, hopping stage, the system skips right into equilibrium and therefore its final temperature. Bechhoefer calls that scenario the strong Mpemba effect. “So, if things could be tuned just right, you would have that first fast stage of relaxation and then nothing,” he says. “You’d be in.”

Figure 2 shows the time to settle into equilibrium (teq) as a function of the ratio of the initial temperature to the final temperature (T0/Tb). The zone of anomalous heating, sandwiched between two regimes of normal heating, designates the region where the inverse Mpemba effect applies. That Goldilocks region occurs because the tuning of the initial temperature is just right to settle directly into equilibrium—not too far in either the cold or the warm direction.

The time for the colloidal particle to reach the equilibrium temperature.
Figure 2. The time for the colloidal particle to reach the equilibrium temperature as a function of the ratio of the initial temperature to the final temperature. Moving from left to right, the particle at first follows normal heating. When it switches to the regime of anomalous heating, the Mpemba effect takes over and the time increases as the initial temperature increases. The time is longest at the point where anomalous heating changes back to normal heating. Credit: Adapted from A. Kumar, R. Chétrite, J. Bechhoefer, Proc. Natl. Acad. Sci. USA 119, e2118484119 (2022)

The path to local equilibrium can itself travel far from equilibrium. During the relaxation in the Mpemba effect and its inverse, sampling the system does not reveal a Boltzmann distribution, and therefore the system does not have a set temperature. Thus, a hot system can skip temperatures and cool faster than a cold system, and likewise a cold system can skip temperatures and heat faster than a hot one. Consider it like stepping stones: If you have the right starting energy, you can jump straight from the first to the third without ever landing on the second. In some scenarios, it is even faster to take the extra step of heating a system before cooling it.

The future (with phase conditions)

Oren Raz, a senior researcher at the Weizmann Institute of Science, was one of the scientists to predict the existence of the inverse Mpemba effect. “This observation really shows this is not an abstract idea,” he says. “It was really a nice demonstration that this is something real.”

Although Raz believes that better understanding of thermal relaxation might one day be useful for applications like heating and cooling extremely quickly, he emphasizes that thermal relaxation is very complicated. Studies of the Mpemba effect and its inverse still have a long way to go, especially in more complex systems such as water.

“The real mystery happens in water at the freezing point,” says Raz. “All the models and this experiment have no phase transitions at all. We understand phase conditions by now quite well, in equilibrium. Going through a phase transition out of equilibrium is something that—we are not there yet. That is a big step in front of us.”

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