Mpemba effect

The Mpemba effect is the observation that a hot liquid (such as water) can freeze faster than the same volume of cold liquid, under otherwise similar conditions. The effect is named after Tanzanian Erasto Mpemba, who studied the effect in 1963 as a secondary school student, while freezing ice cream. Observations of the effect date back to ancient times; Aristotle wrote that the effect was common knowledge.^[1]

While initially observed in water and ice cream, it has been studied in other colloids,^[2] in gases, and in quantum systems.^[3][4] The exact definition of the effect, the parameters required to produce it, and its physical mechanisms, remain points of scholarly debate.^[5]

Definition

The phenomenon, when taken to mean "hot water freezes faster than cold", was difficult to study because of the looseness of its definition.^[6] Monwhea Jeng proposed a more precise wording: "There exists a set of initial parameters, and a pair of temperatures, such that given two bodies of water identical in these parameters, and differing only in initial uniform temperatures, the hot one will freeze sooner."^[7]

Even with Jeng's definition, it is not clear whether "freezing" refers to the point at which water forms a visible surface layer of ice, or the point at which the entire volume of water becomes a solid block of ice.^[6] When studying this in water, some researchers have instead studied the time it takes for a fluid to reach 0 °C (32 °F; 273 K), before the onset of freezing.^[8]

Jeng's definition suggests simple ways in which the effect might be observed, such as if a warmer temperature melts the frost on a cooling surface, thereby increasing thermal conductivity between the cooling surface and the water container.^[6] Even among experiments that observe the Mpemba effect for some experimental setups, they often do not observe it for all setups and starting conditions.^[6]

Observations

Historical context

Various effects of heat on the freezing of water were described by ancient scientists, including Aristotle: "The fact that the water has previously been warmed contributes to its freezing quickly: for so it cools sooner. Hence many people, when they want to cool water quickly, begin by putting it in the sun."^[9] Aristotle's explanation involved antiperistasis: "...the supposed increase in the intensity of a quality as a result of being surrounded by its contrary quality."^{[citation needed]}

Francis Bacon noted that "slightly tepid water freezes more easily than that which is utterly cold."^[10] René Descartes wrote in his Discourse on the Method, relating the phenomenon to his vortex theory: "One can see by experience that water that has been kept on a fire for a long time freezes faster than other, the reason being that those of its particles that are least able to stop bending evaporate while the water is being heated."^[11]

Scottish scientist Joseph Black in 1775 investigated a special case of the phenomenon by comparing previously boiled with unboiled water.^[12] He found that the previously boiled water froze more quickly, even when evaporation was controlled for. He discussed the influence of stirring on the results of the experiment, noting that stirring the unboiled water led to it freezing at the same time as the previously boiled water, and also noted that stirring the very-cold unboiled water led to immediate freezing. Joseph Black then discussed Daniel Gabriel Fahrenheit's description of supercooling of water, arguing that the previously boiled water could not be as readily supercooled.^{[citation needed]}

Mpemba's observation

The effect is named after Tanzanian student Mpemba, who described it in 1963 in Form 3 of Magamba Secondary School, Tanganyika; when freezing a hot ice cream mixture in a cookery class, he noticed that it froze before a cold mixture. He later became a student at Mkwawa Secondary (formerly High) School in Iringa. The headmaster invited Dr. Denis Osborne from the University College in Dar es Salaam to give a lecture on physics. After the lecture, Mpemba asked him, "If you take two similar containers with equal volumes of water, one at 35 °C (95 °F) and the other at 100 °C (212 °F), and put them into a freezer, the one that started at 100 °C (212 °F) freezes first. Why?" Mpemba was at first ridiculed by both his classmates and his teacher. However Osborne experimented on the issue back at his workplace and confirmed Mpemba's finding, and they later published the results together in 1969, while Mpemba was studying at the College of African Wildlife Management.^[13]

Mpemba and Osborne described placing 70 ml (2.5 imp fl oz; 2.4 US fl oz) samples of water in 100 ml (3.5 imp fl oz; 3.4 US fl oz) beakers in the icebox of a domestic refrigerator on a sheet of polystyrene foam. They showed the time for freezing to start was longest with an initial temperature of 25 °C (77 °F) and that it was much less at around 90 °C (194 °F). They ruled out loss of liquid volume by evaporation and the effect of dissolved air as significant factors. In their setup, most heat loss was found to be from the liquid surface.^[13]

Modern experimental work

Modern studies using freezers with well-understood properties have observed the Mpemba effect where water supercools before freezing. Water that starts out cooler tends to reach a lower supercooled temperature before freezing. Most studies measure the time it takes for a sample to begin to freeze (the start of recalescence, the moment where the heat of freezing first starts to be released) the time it takes to completely freeze, or the difference: the time from the onset of recalescence to the completion of freezing. Some also measure the time it takes for a sample to reach the freezing point of the fluid, before any freezing has begun.

In 1995, David Auerbach studied glass beakers placed into a liquid cooling bath, where the water supercooled to −6 to −18 °C (21 to 0 °F; 267 to 255 K) before freezing. In some cases water which started off hotter began freezing first.^[14] Considerable random variation was observed in the time required for spontaneous freezing to start, and Auerbach observed the Mpemba effect more frequently when the ambient temperature was between –6 and −12 °C (10 °F). James Brownridge later studied a variety of initial conditions and containers, measuring the time to the onset of recalescence, and found that while hot samples sometimes froze first this was also affected by properties of the container holding the liquid.^[15] In 2021 John Bechhoefer described a way to reliably reproduce the effect in water.^[16]

In 2024, Argelia Ortega, et al. studied the freezing of small (1-20mL) drops in a Peltier cell with a thermographic camera, and found that hot drops consistently froze faster than cold ones, with a more pronounced difference for larger drops. In particular, hot drops finished freezing after the onset of recalescence significantly faster, and experienced less of a temperature spike during the freezing process.^[17] This study was notable because of the care it took in ensuring that the starting conditions were identical save for the initial temperature of the water.^[17]

Writing for New Scientist, Mick O'Hare recommended starting the experiment with containers at 35 and 5 °C (95 and 41 °F; 308 and 278 K), respectively, to maximize the effect.^[18]

Modern criticisms of experimental work

Some researchers have criticized studies of the Mpemba effect for not accounting for dissolved solids and gasses, and other confounding factors.^[19] In 2006, Philip Ball, a reviewer for Physics World wrote: "Even if the Mpemba effect is real — if hot water can sometimes freeze more quickly than cold — it is not clear whether the explanation would be trivial or illuminating."^[6] Ball wrote that investigations of the phenomenon need to control a large number of initial parameters (including type and initial temperature of the water, dissolved gas and other impurities, and size, shape and material of the container, and temperature of the refrigerator) and the need to settle on a particular method of establishing the time of freezing. The multidimensional nature of experiments might explain why the effect was not well understood.^[6]

In 2016, Burridge and Linden studied the time it took water samples to reach 0 °C. They carried out their own experiments, and reviewed previous work by others. Their review noted that the large effects observed in early experiments had not been replicated in other studies of cooling to the freezing point, and that studies showing small effects could be influenced by variations in the positioning of thermometers: "We conclude, somewhat sadly, that there is no evidence to support meaningful observations of the Mpemba effect."^[8]

Theoretical explanations

While the definition of the Mpemba effect used in theoretical studies varies,^[19] several explanations have been offered for its occurrence.

In 2017, two research groups independently and simultaneously found a theoretical Mpemba effect and also predicted a new "inverse" Mpemba effect in which heating a cooled, far-from-equilibrium system takes less time than another system that is initially closer to equilibrium. Zhiyue Lu and Oren Raz yielded a general criterion based on Markovian statistical mechanics, predicting the appearance of the inverse Mpemba effect in the Ising model and diffusion dynamics.^[20] Antonio Lasanta and co-authors also predicted the direct and inverse Mpemba effects for a granular gas in a far-from-equilibrium initial state.^[21] Lasanta's paper also suggested that a very generic mechanism leading to both Mpemba effects is due to a particle velocity distribution function that significantly deviates from the Maxwell–Boltzmann distribution.^[21]

Supercooling is likely involved, particularly the tendency of hot liquids that are cooled rapidly to start freezing at a higher supercool temperature.^[22][15] Several molecular dynamics simulations have also supported that changes in hydrogen bonding during supercooling take a major role in the process.^[23][24] In 2017, Yunwen Tao and co-authors suggested that the vast diversity and peculiar occurrence of different hydrogen bonds could contribute to the effect. They argued that the number of strong hydrogen bonds increases as temperature is elevated, and that the existence of the small strongly bonded clusters facilitates in turn the nucleation of hexagonal ice when warm water is rapidly cooled down. The authors used vibrational spectroscopy and modelling with density functional theory-optimized water clusters.^[5]

The following additional explanations have been proposed:

• Microbubble-induced heat transfer: The process of boiling induced microbubbles in water that remain stably suspended as the water cools, then act by convection to transfer heat more quickly as the water cools.^[25][26]
• Evaporation: The evaporation of the warmer water reduces the mass of the water to be frozen.^[27] Evaporation is endothermic, meaning that the water mass is cooled by vapor carrying away the heat, but this alone probably does not account for the entirety of the effect.^[7]
• Convection, accelerating heat transfers: Reduction of water density below 4 °C (39 °F) tends to suppress the convection currents that cool the lower part of the liquid mass; the lower density of hot water would reduce this effect, perhaps sustaining the more rapid initial cooling. Higher convection in the warmer water may also spread ice crystals around faster.^[28]
• Frost: Frost has insulating effects. The lower temperature water will tend to freeze from the top, reducing further heat loss by radiation and air convection, while the warmer water will tend to freeze from the bottom and sides because of water convection. This is disputed as there are experiments that account for this factor.^[7]
• Solutes: Calcium carbonate, magnesium carbonate, and other mineral salts dissolved in water can precipitate out when water is boiled, leading to an increase in the freezing point compared to non-boiled water that contains all the dissolved minerals.^[29]
• Thermal conductivity: A container of hotter liquid may melt through a layer of frost that is acting as an insulator around a cooler container, allowing the container to work more effectively within the same refrigeration setup.^[15]
• Dissolved gases: Cold water can contain more dissolved gases than hot water, which may somehow change the properties of the water with respect to convection currents, a proposition that has some experimental support but no theoretical explanation.^[7]
• Hydrogen bonding: In warm water, hydrogen bonding is weaker.^[5]
• Crystallization: Another explanation suggests that the relatively higher population of water hexamer states in warm water might be responsible for the faster crystallization.^[23]
• Distribution function: ^{[clarification needed]} Strong deviations from the Maxwell–Boltzmann distribution can result in a Mpemba effect in gases and granular fluids.^[21]

Effects in fluids other than water

Strong Mpemba effect

The possibility of a "strong Mpemba effect" where exponentially faster cooling can occur in a system at particular initial temperatures was predicted in 2019 by Klich, Raz, Hirschberg and Vucelja.^[30]

In 2020 the strong Mpemba effect was demonstrated experimentally by Avinash Kumar and John Boechhoefer in a colloidal system.^[2]

Quantum Mpemba effect

Since 2020, a number of quantum researchers have studied Mpemba effects in quantum systems.

In 2024, a team in John Goold's lab at Trinity College described their quantum-mechanical analysis of an abstract problem wherein "an initially hot system is quenched into a cold bath and reaches equilibrium faster than an initially cooler system."^[3] In addition to their theoretical work, which used non-equilibrium quantum dynamics, their paper includes computational studies of spin systems which exhibit the effect.^[3] They concluded that certain initial conditions of a quantum-dynamical system can lead to a simultaneous increase in the thermalization rate and the free energy.^[4]

See also

• Latent heat: Turning 0 °C (32 °F) ice to 0 °C water takes the same amount of energy as heating water from 0 °C to 80 °C (176 °F).
• Leidenfrost effect: Lower temperature boilers can sometimes vaporize water faster than higher temperature boilers.
• Density of water
• Heat capacity
• Water cluster
• Newton's law of cooling