The ice that grows by stealing from the droplets

The degree a spoonful of solute buys lowered liquid water’s chemical potential by dissolving something in it, and watched the freezing and boiling points move to wherever the liquid’s potential met the solid’s and the vapour’s. Earlier essays used chemical potential to explain a battery’s voltage, a semiconductor’s junction and a star’s balance of protons and neutrons; in each the rule was that matter moves from where its chemical potential is higher to where it is lower, until the two are equal.

This essay applies the rule to water against itself. Below 0 °C ice has the lower chemical potential and is the stable form, but liquid water can persist far below its freezing point, because freezing has to begin somewhere and the barrier a new phase has to climb can be very high for a small drop. A supercooled droplet is liquid water with a chemical potential higher than ice’s at the same temperature. Put droplets and ice crystals together in the same air and the difference drives water from one to the other, through the vapour, without the two touching. The process was proposed by Alfred Wegener in 1911, worked out by Tor Bergeron in the 1930s and confirmed by Walter Findeisen, and it is the way most precipitation outside the tropics starts.

Why the droplets do not simply freeze #

A cloud droplet at minus fifteen degrees is in a state that cannot last indefinitely and lasts anyway. To freeze, it must first form a small crystal of ice somewhere inside it, and a small crystal is not stable: its surface, between ice and water, costs energy, and below a critical size the cost of the surface outweighs the gain in chemical potential from turning liquid into ice. Embryos of ice form and melt constantly by thermal fluctuation, and freezing begins only when one happens to exceed the critical size, the rare event whose rate the barrier a new phase has to climb computed. The rate depends extraordinarily steeply on the supercooling, because the barrier’s height falls as the square of the chemical-potential difference.

For droplets the size of cloud droplets the rate becomes large only near minus thirty-eight degrees, which is why a cloud colder than that is all ice and a warmer one can be almost all liquid. Between the two, freezing needs help: a foreign surface whose arrangement of atoms resembles ice’s lowers the barrier, and the rarity of good surfaces in the atmosphere is what leaves clouds supercooled. The droplets are not waiting to be cooled further. They are waiting for a template.

Two vapour pressures below freezing #

The vapour pressure of a substance is the pressure of its vapour in equilibrium with it, the pressure at which molecules leave the condensed phase as fast as they return, and a boiling point is a pressure traced how steeply it rises with temperature. A phase with a higher chemical potential loses molecules more readily, so its equilibrium vapour pressure is higher; in fact the ratio of two phases’ vapour pressures at one temperature is the exponential of their chemical-potential difference over RT.

At 0 °C liquid water and ice are in equilibrium with each other, which means equal chemical potentials and equal vapour pressures; the two curves meet at 6.11 hectopascals, the pressure of the triple point. Below it, ice is stable and the supercooled liquid is not, so the liquid’s chemical potential is higher, and its vapour pressure lies above ice’s at every temperature down to where the liquid can exist at all.

The chemical-potential difference is easy to estimate. At the melting point the two phases have equal potentials; cooling below it by ΔT changes each potential by minus its entropy times ΔT, and the liquid’s entropy is higher by the latent heat of melting divided by the melting temperature. So supercooled water exceeds ice by about $L_f\,\Delta T/T_m$ per mole: about 220 joules per mole at minus ten degrees. That is all the driving force the process has, and it is enough.

Saturated for the droplets, supersaturated for the ice #

A cloud is a population of droplets in air that has reached equilibrium with them, so the air’s vapour pressure equals the saturation pressure over liquid water at the cloud’s temperature. Over the droplets themselves it is exactly saturated, and they neither grow nor shrink. To an ice crystal in the same air, the vapour pressure is higher than its own equilibrium value — by ten per cent at minus ten degrees, twenty-two per cent at minus twenty, more than forty at minus thirty-five. The crystal is in strongly supersaturated air.

Vapour therefore condenses onto the crystal as ice, the air near it is depleted, and the depletion spreads outward by diffusion until it reaches the droplets. They now sit in air slightly below their own saturation and evaporate, restoring the vapour, which diffuses to the crystal. The water passes from droplet to crystal through the air, down a gradient of vapour pressure that the chemical-potential difference maintains, and the droplets disappear while the crystals grow.

The process is the same as the one the small bubble blows up the big one followed for two soap bubbles joined by a tube, and as the slow coarsening of any population in which some members have a lower chemical potential than others: the higher-potential members feed the lower until the difference is used up. There it was curvature that made the difference; here it is phase.

The temperature at which it runs fastest #

The supersaturation grows steadily as the temperature falls, but the amount of vapour available does not: cold air holds little water vapour of any kind. The excess of vapour that an ice crystal can draw on, the difference between the two pressures, is a product of a growing fraction and a shrinking whole, and it peaks near minus twelve degrees, at about a quarter of a hectopascal.

That is close to the temperature at which snow crystals grow fastest. Laboratory studies by Ukichiro Nakaya in the 1930s, growing snow crystals on a rabbit’s hair in a cold chamber, mapped how their shapes change with temperature and humidity: plates near minus two, columns near minus five, plates again below minus ten, and the large branched stars, the dendrites of a picture-book snowflake, near minus fifteen, where the vapour excess is greatest and the crystal’s growing tips outrun the rest of it. The shape depends on how fast vapour reaches each face, and the fastest growth happens where the chemical-potential difference pays most.

Outgrowing every droplet #

How fast the crystal grows is set by diffusion, of vapour towards it and of the latent heat it releases away from it. Both obey the equation that only runs forwards, and for a sphere the result is that its radius squared grows in proportion to time, the rate set by the supersaturation and by how fast the surrounding air conducts heat and diffuses vapour. A crystal starting at five micrometres grows to a hundred in about a quarter of an hour at minus twelve degrees, a little slower at minus five, where the supersaturation is smaller, and at minus twenty-five, where there is less vapour.

The droplets cannot keep up because they are not trying: at their own saturation they have nothing to grow on. A cloud of liquid droplets grows its drops only by condensation as it rises and cools, which makes them all much the same size, ten or twenty micrometres, far too small to fall, and by collisions between drops of different sizes, which in a cloud of uniform droplets is slow to start. An ice crystal in the same cloud reaches a hundred micrometres, big enough to fall through the cloud at tens of centimetres a second, in the time a cloud lasts. Once falling, it sweeps up droplets that freeze onto it as rime, grows faster still, and leaves the cloud as snow, or melts on the way down and arrives as rain.

A few thieves take everything #

The process works because ice nuclei are rare. A typical cloud holds a hundred droplets in each cubic centimetre, a hundred thousand in a litre. Pure water droplets that small freeze spontaneously only near minus thirty-eight degrees, the temperature at which the barrier to nucleating ice inside a droplet becomes small enough to be crossed within the droplet’s life; at warmer temperatures freezing needs a foreign particle with the right surface, a mineral dust grain, a fragment of biological material, and such particles are scarce — of order one per litre active at minus fifteen degrees. So a few crystals form among a vast number of droplets, and each crystal can draw on the water of a hundred thousand. Taking all of it, it grows to most of a millimetre, a snowflake, heavy enough to fall.

If ice nuclei were plentiful the process would not work: a cloud with ten thousand crystals per litre shares its water among them all, and each grows to a few tens of micrometres, too small to fall far. The rarity of the starting points is the whole of the mechanism. A cloud has to fail to freeze for it to snow.

Most rain arrives as melted snow #

The consequence for the weather is larger than it sounds. Outside the tropics, the clouds that produce most rain have tops well below freezing, and in them the Bergeron process is the usual way of making particles big enough to fall. The snow falls into warmer air below and melts, and arrives as rain; whether it reaches the ground as rain, sleet or snow depends on the depth of the air above freezing it falls through.

Weather radar sees the melting directly. A falling snowflake reflects radar weakly, because ice scatters microwaves far less than liquid water does. As it begins to melt it becomes a large, flimsy particle coated with water, and it reflects strongly; when it has melted it collapses into a small, fast-falling raindrop and reflects less again. Between the snow above and the rain below, a horizontal layer a few hundred metres thick at the height of the freezing level shines on the radar as a bright band, which forecasters use to locate the melting level and which is visible evidence that the rain beneath it began as ice.

Melting and freezing also move heat. Each gram of supercooled water that freezes onto a growing crystal, and each gram of vapour that deposits on it, releases latent heat into the cloud, as the heat that changes no temperature traced; the snow that melts below the cloud takes heat from the air it falls through and cools it. The freezing adds buoyancy to the upper parts of a storm, on top of the condensation that the cloud that cools more slowly than the air followed, and the melting chills the downdraughts beneath.

The mixture of ice and supercooled water also electrifies storms. Where small ice crystals rising in an updraught collide with soft, rimed pellets of graupel falling through the supercooled cloud, the collisions transfer charge between them, and which way the charge goes depends on the temperature and on how much liquid water is present. The light crystals carried up charge the top of the storm and the heavier graupel the middle, and the separation builds the fields that end in lightning. Storms without a mixed-phase region, warm clouds with no ice, rarely produce lightning at all.

Ice on a wing #

The same supercooled droplets that feed snowflakes are a hazard to aircraft. A droplet that strikes a wing or a propeller finds a surface on which it can freeze at once, and freezes where it hits; a supercooled cloud can build a layer of ice on the leading edge of a wing in minutes, changing its shape and reducing its lift. Small droplets freeze instantly into rough, opaque rime; larger ones spread a little before freezing into smooth, clear glaze ice, which is heavier and harder to see. Aircraft carry heated leading edges or inflatable boots to break the ice off, and forecasts of icing are forecasts of supercooled liquid water, its amount and its droplet size — precisely the quantity the Bergeron process depletes when ice crystals are present, which is why clouds full of ice are less dangerous to fly through than clouds that have failed to freeze.

Seeding a cloud #

The Bergeron process explains why a supercooled cloud can sit for hours without precipitating — it has too few ice nuclei — and it suggests what to do about it. In 1946 Vincent Schaefer, working at the General Electric laboratory, dropped a pellet of dry ice into a chamber holding a supercooled cloud made by breathing into a home freezer, and the chamber filled with sparkling ice crystals. The dry ice had chilled the air below minus forty degrees and frozen droplets directly; a few months later he seeded a real cloud from an aeroplane and watched it turn to snow. His colleague Bernard Vonnegut found that silver iodide, whose crystal structure resembles ice’s closely enough to template it, nucleates ice at about minus five to minus ten degrees, and it became the standard seeding agent.

Whether seeding increases the precipitation from a whole storm, as opposed to changing what a single cloud does, has been contested for seventy years, because natural variation between clouds is large and controlled experiments are hard. Recent experiments with radar tracking of the seeding line have shown that seeding a suitable supercooled cloud does make snow where it is seeded. The physics is not in doubt; the yield is.

Where the picture stops #

The figures use the standard Magnus approximations to the two saturation curves, accurate to a fraction of a per cent over this range, and treat the crystal as a sphere growing in still air with heat and vapour diffusing to it. Real crystals are plates, columns and branched stars, whose growth is better described by treating them as conductors of the shape in question, and which grow faster for their mass than spheres because their tips reach out into fresher vapour. Falling crystals grow faster again as the airflow brings vapour to them. The budget figure assumes the crystals take all the liquid water, which they do only if the cloud lasts long enough; and it ignores the second route by which crystals multiply, the splintering of rime as droplets freeze onto falling ice, which can raise the number of crystals far above the number of nuclei in some temperature ranges.

The domain of the process is mixed-phase cloud, between about 0 and −38 °C, where liquid and ice can coexist. Warmer clouds rain by collisions between droplets, as many tropical clouds do, and colder ones are already ice.

Still open: where the ice in clouds comes from #

The number of ice crystals in natural clouds is often far larger than the number of ice-nucleating particles measured in the air feeding them, sometimes by factors of a thousand. Secondary processes — splintering during riming, droplets shattering as they freeze, fragments broken off colliding crystals — must make most of them, and which dominates where is unsettled. So is the nature of the best natural ice nuclei: certain feldspar minerals, some bacteria and pollen, and particles from the sea surface all nucleate ice at modest supercooling, and how they are distributed through the atmosphere matters for how clouds reflect sunlight and how much they rain. Climate models represent the mixed-phase balance crudely, and the proportion of liquid to ice in clouds at a given temperature is one of the larger uncertainties in how clouds will respond to warming.

The habit worth carrying away is to look for two forms of one substance that are out of equilibrium with each other in the same place. Below 0 °C supercooled water stands $L_f\,\Delta T/T_m$ above ice in chemical potential — 217 J/mol at −10 °C — so air saturated for the droplets is 10.5 per cent supersaturated for the ice, and a crystal grows to 100 μm in a quarter of an hour while the droplets beside it evaporate. One crystal per litre among a hundred thousand droplets takes all the water and falls as snow.