The supercooled water that only partly freezes

A bottle of purified water left in a freezer overnight is sometimes still liquid in the morning, though the freezer is at −18 °C. Tap the bottle, or pour the water onto a piece of ice, and crystals shoot through it in a second or two, and what was a clear liquid becomes a white slush. The display is often described as the water “freezing instantly”. Pick up the bottle, though, and it is not a block of ice. It is mostly still water, full of a fine mesh of ice crystals, and it is at exactly 0 °C — eighteen degrees warmer than it was a moment before.

The barrier a new phase has to climb explained why the water was liquid at all: a crystal must start from a tiny nucleus whose surface costs more free energy than its volume gains, and without a speck of dust or a scratch to start on, pure water can be cooled to about −38 °C before a nucleus forms by chance. This essay is about what happens next — why nucleation in a supercooled liquid freezes only part of it, and how much.

The heat that stops the freezing #

When water freezes it releases its latent heat, 333.6 kilojoules for every kilogram, the heat the heat that changes no temperature found hidden in every change of phase. Ordinarily that heat leaves as fast as it is released, through the walls of the ice tray, which is why freezing water sits at 0 °C for so long. In supercooled water the first crystals release their heat into the liquid around them, which is colder than 0 °C and can absorb it. The crystals grow, the liquid warms, and growth continues until the liquid is no longer supercooled — until the whole mixture has reached 0 °C. There it stops, because at 0 °C ice and water can coexist and there is no longer any drive to freeze.

The crystals race through the liquid in a second or so, far faster than any heat can leave through the bottle, so in that instant the bottle and its contents are effectively insulated. The energy is conserved: the latent heat released by the frozen share has to equal the heat needed to warm the whole of the water from its supercooled temperature to 0 °C. For water cooled to $T_s$ below zero, the frozen fraction is

$$f = \frac{1}{L}\int_{T_s}^{0} c_w(T)\,dT,$$

with $c_w$ the heat capacity of liquid water and $L$ its latent heat at 0 °C.

With the heat capacity of water at 0 °C, 4.2 kJ per kilogram per kelvin, every degree of supercooling freezes about one and a quarter per cent, and the curve is a straight line. The real curve bends upward below about −20 °C, because supercooled water’s heat capacity is not constant: it rises steeply as the water is cooled, from 4.2 at 0 °C to over 5 near −38 °C, one of the anomalies that make water’s supercooled state an active research problem. The curve stops at −38 °C because no liquid water in bulk survives below that. At that limit, about half freezes at once.

To freeze it all in one step, water would have to be supercooled by $L/c_w$, about 80 degrees, so that the heat released by freezing the whole of it only just brought it to 0 °C. Nobody has seen bulk liquid water at −80 °C; it crystallises on its own forty degrees before that. A supercooled bottle of water always ends up as slush.

A horizontal line on the enthalpy diagram #

The same result has a geometric form that makes it look inevitable, and it is worth drawing because it shows why the answer does not depend on how the freezing is imagined to happen.

Enthalpy is the quantity conserved when a substance changes at constant pressure and exchanges no heat. Plot it against temperature and the liquid is one line, the ice another, lower by the latent heat, and between them at 0 °C a vertical segment of mixtures, every point on it a different proportion of ice. Supercooled water is a point on the liquid line continued below 0 °C, where liquid is not supposed to be. Nucleated, it cannot exchange heat in the instant of freezing, so it moves horizontally — enthalpy constant — until it reaches the only states at that enthalpy that are not themselves supercooled, which are the mixtures at 0 °C. Where it lands on the vertical segment is the fraction frozen, by the lever rule that why the triple point is a point used for coexisting phases.

The diagram also settles a question that the formula hides. One could imagine the freezing happening differently: a fraction $f$ freezes at −15 °C, releasing the latent heat ice has at −15 °C — which is smaller than at 0 °C, by about 30 kilojoules per kilogram, because ice and water have different heat capacities — and then the mixture warms to 0 °C. Enthalpy is a function of state, so both routes land at the same point, and the figure’s generator checks that they agree. The answer depends only on where the water starts and where it ends.

A raindrop that freezes in two steps #

Supercooled water is not a laboratory curiosity. Most clouds above the freezing level are made of supercooled droplets, which is the starting point of the ice that grows by stealing from the droplets: a few ice crystals among them grow at the droplets’ expense. And a drop falling through cold air goes through the whole sequence on its way down.

The figure follows a drop two millimetres across. It cools smoothly through 0 °C, because nothing in the clean drop starts a crystal, and keeps cooling until, here at −15 °C, a nucleus forms. In that instant a fifth of it freezes and the drop jumps back to 0 °C — the jump is called recalescence, and in metals that are supercooled before solidifying it can be bright enough to see. Then the drop sits at 0 °C for three-quarters of a minute while the air carries off the latent heat of the remaining four-fifths, a slow process set entirely by how fast heat crosses its surface. Only when it is solid does it cool again, and faster than before, because ice stores less heat per degree than water.

The sequence explains freezing rain, one of the most destructive kinds of winter weather. Rain that falls through a layer of air below freezing near the ground arrives supercooled, at a few degrees below zero. When a drop strikes a branch or a power line, it nucleates on the surface, freezes a few per cent at once, and the remaining liquid spreads over the surface before it freezes in turn, as its heat is conducted into the cold branch and the air. The result is not frost or rime but a clear, dense glaze, layer on layer, which can coat a power line in centimetres of ice and bring it down. Had the drops frozen completely on impact they would have stuck as loose white pellets; it is the unfrozen four-fifths that lets them flow into a solid sheath.

The heat a cloud gets when it glaciates #

High in a thunderstorm the same partial freezing happens to a whole cloud. Updrafts carry droplets far above the freezing level, and cumulus towers routinely hold liquid water at −20 °C and below. When those droplets freeze — by contact with ice crystals, by collision with hail, or by reaching the temperature where they nucleate on their own — each gives up its latent heat to the air around it. The cloud that cools more slowly than the air found condensation warming rising air enough to change its buoyancy; freezing adds about another eighth on top of the heat of condensation, released high in the cloud where it most strengthens the updraft. Storm models that leave the freezing out make weaker storms.

The same release is why cloud seeding was once expected to do more than it does. Seeding a supercooled cloud with silver iodide or dry ice supplies nuclei, the droplets freeze, the released heat lifts the cloud, and the ice crystals grow at the droplets’ expense into snow. Whether that reliably increases precipitation, rather than redistributing it, has been tested for seventy years with results that remain arguable; that it changes the cloud is not in doubt, because the latent heat of the supercooled water is real and is released wherever the nuclei are put.

Supercooling also complicates one of the most argued-over claims in kitchen physics, that hot water can freeze faster than cold. The hotter start that cools first found the evidence for water weak, and supercooling is one reason: two samples that cool identically may nucleate at temperatures several degrees apart, by chance, and whichever nucleates first starts its plateau first. A difference in when crystals start is easily mistaken for a difference in how fast heat leaves.

A hand warmer is the same physics above body temperature #

The arithmetic of partial freezing is the working principle of the reusable hand warmer: a soft plastic pouch of clear liquid with a small metal disc inside. Click the disc, and white crystals spread from it through the liquid, and the pouch becomes warm — about fifty degrees — and stays warm for half an hour.

The liquid is sodium acetate trihydrate, a salt whose crystals contain three molecules of water for each formula unit, and which melts — dissolves in its own water of crystallisation — at 58 °C. It supercools easily and deeply: once melted by boiling the pouch, it cools to room temperature and stays liquid for months. Clicking the disc releases a few microscopic seed crystals trapped in cracks in the metal, and crystallisation spreads. As with water, the latent heat released warms everything to the melting point, and crystallisation stops when the pack reaches it. From room temperature, 43 per cent crystallises at once and the pack jumps to 58 °C.

That is the same calculation as the bottle of water, but with a melting point above body temperature, so the plateau is a useful warmth rather than a chill. The pack then sits at its melting point while the rest crystallises, slowly, at the rate the hand and the air take heat from it — exactly the plateau of the falling drop. Nothing in it reacts chemically. Boil it, and the crystals redissolve, the latent heat is stored again, and the pack is ready for another winter’s day. It is a battery for heat, charged at 58 °C and discharged on demand, and the charge is held by a nucleation barrier.

How deep a liquid must be cooled to freeze all at once #

The supercooling at which the latent heat would only just bring the whole liquid back to its melting point — $L/c$ — is a property of each substance, and it varies a great deal.

For water it is 79 kelvin, twice as deep as any bulk water has been cooled. For metals it is larger, because their latent heats are large compared with their heat capacities: hundreds of kelvin. Metals can be supercooled far more than one might expect if they are kept clear of the container walls on which crystals start — by levitating molten drops in a magnetic field or an electrostatic trap, or in the microgravity of space — and nickel has been supercooled by more than three hundred kelvin that way. Below the depth in the figure, a nucleated liquid would solidify completely in one step, with no liquid left to freeze afterwards and no plateau: the hypercooled regime, which materials scientists use to make solids with structures that ordinary, slow freezing cannot produce, since the whole solidification is over before atoms can rearrange into their usual arrangements.

Where the speed of the crystals comes from #

The figures treat the crystallisation as instantaneous, and on the scale of the drop’s cooling it is. But the crystals have a finite speed, and the speed is set by the same balance. The tip of a growing ice crystal releases latent heat, which has to diffuse away into the supercooled liquid ahead of it before the tip can advance; the more supercooled the liquid, the steeper the temperature gradient ahead of the tip and the faster the heat leaves. So the crystals grow faster in more deeply supercooled water — from a fraction of a millimetre per second within a degree of 0 °C to several centimetres per second ten or twenty degrees below — and they grow as dendrites, branched needles, because a sharp tip sheds its heat into a larger volume than a flat front. That is the mesh of needles in the slushy bottle, and the reason it is a mesh and not a block: each needle stops growing as the liquid around it warms to 0 °C.

The same latent heat is why the ice that grows more slowly the thicker it gets grows as it does — its rate set by how fast the heat of freezing can be conducted out through the ice already formed. The supercooled bottle shows what happens when the heat has nowhere to go: the freezing stops of its own accord, at the point where it has undone the supercooling that drove it.

An instant with no heat lost, and a drop of one temperature #

The fraction-frozen figure assumes the crystallisation is adiabatic — that no heat leaves during it — which is excellent for a bottle that slushes in a second and poor for a thin film on a cold metal plate, where heat leaves through the plate as fast as the crystals form and much more of the film freezes at once. The heat capacity of supercooled water is interpolated from measured values, which become uncertain below −30 °C because the water crystallises during the measurement; the share frozen at −38 °C is good to a few per cent. And the bottle is assumed pure: dissolved air and salt lower the melting point slightly and come out of solution as the ice forms, which is why freezer slush is often cloudy and why the last liquid to freeze is the saltiest.

The drop’s history uses a fixed heat-transfer coefficient and treats the drop as uniform in temperature, which holds for millimetre drops falling through air and fails for large ones, where an ice shell can form round a liquid core and crack as the core freezes and expands. The hand warmer’s numbers use the pure trihydrate; commercial packs contain extra water, which lowers the plateau a few degrees and keeps the crystals from setting into a single hard block. The depths figure uses handbook values for the liquids’ heat capacities at their melting points, typed in, not computed.

Still open: what water does below −38 °C #

Between −38 °C and about −120 °C, liquid water crystallises too fast to be studied — the region physicists call no man’s land. Above it, supercooled water’s heat capacity, its compressibility and its expansion coefficient all grow as though heading for a singularity somewhere near −45 °C; below it, glassy water made by very fast cooling behaves as though it came in two forms of different density. One proposal, made in 1992, is that liquid water has a second critical point inside no man’s land, where two distinct liquids — a low-density one and a high-density one — become one, as the gas and liquid become one at the point at which the two become one. Experiments since 2017 using micrometre droplets cooled at a million degrees a second and probed with X-ray pulses have reached into the region and found signs of the predicted behaviour. Whether a second critical point exists, and where, is not yet settled; the rising heat capacity that bends the fraction-frozen curve is one of the clues.

The bottle in the freezer carries the general rule. A supercooled liquid that is nucleated freezes only until the latent heat it releases has warmed it back to its melting point; the share is the stored cooling divided by the latent heat, the rest waits for the surroundings, and a liquid has to be cooled by L/c before it can all freeze at once. Water rarely gets halfway there. A hand warmer, a falling drop and a freezing-rain glaze all live in the partial freezing that comes first.