The melting line that turns round

The heat that changes no temperature found where the energy goes when a substance changes phase, and a boiling point is a pressure found that a transition temperature is a coordinate on a curve. The part of the curve no fluid follows, the barrier a new phase has to climb and why the triple point is a point followed that curve through its instabilities and intersections. The melting curve that leans the wrong way used Clapeyron’s relation — the slope of a coexistence line is the latent heat over the temperature times the volume change — to explain why water’s melting line runs backwards: ice is less dense than water, the volume change on melting is negative, and so pressure lowers the melting point.

That argument ended by pointing to the rest of water’s phase diagram, where the same relation runs over a dozen triple points. Followed there, it shows that the backwards slope is not a property of water at all. It is a property of one of water’s crystals, and at pressures a few thousand times atmospheric that crystal gives way to others that behave like any ordinary solid.

Five ices on one line #

Ice Ih, the hexagonal form of every snowflake and glacier, is one of about twenty known crystalline phases of water, and five of them are stable in contact with the liquid somewhere.

The figure draws water’s melting line from the international standard equations, which fit a century of measurements with one formula for each ice. Along ordinary ice the melting pressure rises as the temperature falls: the backwards slope. At 251.165 kelvin and 209 megapascals it meets ice III, and the liquid, ice Ih and ice III coexist at a triple point. From there the melting line runs the other way. Ice III gives way to ice V at 350 megapascals, ice V to ice VI at 632, ice VI to ice VII at 2.2 gigapascals, and along each the melting pressure rises with the temperature. Ice VII’s melting line runs out beyond 20 gigapascals and 700 kelvin, the limit of the formula, and experiments follow it further.

The phase rule of why the triple point is a point predicts exactly this geometry: two phases coexist along lines, three at isolated points, and four never. Each triple point on the figure is a place where one ice’s coexistence line with the liquid ends and the next one’s begins, and a third line — the boundary between the two ices — leaves it into the solid region.

What the slope says, in numbers #

Clapeyron’s relation is quantitative, and ordinary ice is the case where every term is known. Freezing a mole of water at 0 °C releases 6.01 kilojoules of latent heat and shrinks the volume by 1.63 cubic centimetres — ice at 19.65 cubic centimetres a mole against the liquid’s 18.02. The slope of the melting line is the temperature times the volume change divided by the latent heat: 273 kelvin times −1.63 cubic centimetres, divided by 6,010 joules, which is −74 kelvin per gigapascal, the number the standard equations give independently at atmospheric pressure. The heat that changes no temperature measured the latent heat; the melting line’s slope is that heat’s ratio to a volume, and either can be found from the other.

Run the same arithmetic backwards along the other ices and the slopes of thirty to sixty kelvin per gigapascal become statements about volumes: melting any of the high-pressure ices expands it, by amounts that are a few per cent of its volume, and the latent heats, measured separately, are of the same order as ordinary ice’s. Nothing about the relation changes from one ice to the next. The sign of a single number, the volume change, carries the whole reversal.

The lowest freezing point #

On linear axes the turn is plain.

Squeezing water lowers its freezing point at first, by 0.074 kelvin for every megapascal, the figure the earlier argument measured. The effect accelerates as the pressure grows, and at 209 megapascals — the pressure about eight kilometres down in the Earth’s crust — liquid water can be held at −22 °C without freezing. That is the lowest temperature at which liquid water is stable in equilibrium anywhere on its phase diagram. Squeezing it harder then raises the freezing point: at 632 megapascals water freezes at 0 °C again, into ice VI rather than ice Ih, with a slope that has the opposite sign.

The minimum is a physically meaningful point, not a feature of the fitting. It is where the solid that forms from the liquid changes from one less dense than the liquid to one denser than it. A melting line can only turn round like this at a triple point, where the solid phase changes, because along any single solid Clapeyron’s relation fixes the sign of the slope by the sign of that solid’s volume change.

The slope, ice by ice #

Clapeyron’s relation can be read backwards: the slope of the melting line gives the ratio of the volume change to the entropy change.

Along ice Ih the slope is negative throughout and grows steeper as the pressure rises, because the liquid itself becomes denser under pressure while ice Ih barely compresses, so the volume the liquid gives up on freezing grows. Along every other ice the slope is positive, between about thirty and sixty kelvin per gigapascal. The entropy of melting is always positive — a liquid is always more disordered than the crystal it melts from — so a positive slope means a positive volume change on melting: the solid is denser than the liquid, as it is for almost every substance. On the figure the sign change is a clean break between the first ice and all the rest.

The sea that stands lower because water gives found that liquid water compresses by 4.7 per cent at the bottom of the ocean; at the pressures here it compresses by twenty or thirty per cent, and its structure changes as it does, the hydrogen-bonded network bending and crowding. The ices that coexist with it are the crystalline forms that the compressed liquid’s structure is closest to at each pressure.

Ice above the boiling point #

The positive slope continues without limit, and it carries the melting temperature to places where ice has no business being.

Above 2.2 gigapascals the stable ice is ice VII, and its melting line climbs steadily: at three gigapascals water freezes at 139 °C, at ten at 341 °C, at twenty at 442 °C. None of this contradicts anything about water at ordinary pressure. At those pressures the liquid is itself dense and hot, far from boiling because the pressure suppresses the vapour entirely, and the ice that forms from it is denser still. A sealed diamond-anvil cell holding a drop of water squeezed to a few gigapascals shows crystals of ice growing in it at temperatures that would scald.

Ice VII is not only a laboratory curiosity. In 2018 Oliver Tschauner and colleagues reported ice VII in tiny inclusions trapped inside diamonds that had formed in the Earth’s mantle, hundreds of kilometres down; the diamond, rigid enough to hold the pressure after it rose to the surface, kept the water inside in its high-pressure structure. It was the first natural ice VII found, and it showed that water-rich fluids exist in the deep mantle. The interiors of the large icy moons of Jupiter and Saturn, and of the water-rich planets found around other stars, are expected to contain the high-pressure ices in layers, from ice Ih at the surface to ice VII or beyond at the base.

What makes one ice float and the others sink #

The single fact behind the whole diagram is density.

The figure lists the ices’ densities. Only ordinary ice is less dense than water; ice III is already sixteen per cent denser than water at atmospheric pressure, and ice VII seventy per cent denser than ice Ih. In ice Ih every molecule forms four hydrogen bonds at nearly tetrahedral angles to its neighbours, and a tetrahedral network is open: it leaves large empty channels through the crystal, which is why it is lighter than the liquid, where the network is partly broken and molecules can pack into the gaps. The high-pressure ices keep the four hydrogen bonds but give up the open arrangement. In ices III and V the bonds bend and the rings of molecules distort; ice VI is two interpenetrating networks that do not bond to each other; ice VII is two interpenetrating copies of a network like ordinary ice’s, each molecule sitting in the empty spaces of the other. The price of the open network is paid in pressure, and above a couple of gigapascals the network doubles up.

So the backwards slope of the ordinary melting line — the fact behind lakes freezing from the top and pipes bursting in winter — is the thermodynamic signature of the tetrahedral hydrogen bond and the empty space it leaves. Squeeze the space out and water freezes like anything else.

The line the supercooled liquid follows #

The melting line also shapes how far water can be cooled without freezing at all. Pure water, free of the dust and surfaces that seed ice, can be supercooled to about −38 °C at atmospheric pressure before ice nucleates in it spontaneously, the limit set by the barrier a new phase has to climb. In 1975 Hiroshi Kanno, Robin Speedy and Austen Angell measured that limit under pressure, in tiny droplets, and found it following the melting line downwards: at 200 megapascals water could be held liquid to about −92 °C, and then, beyond the turn, the limit rose again with the melting line of the denser ices. The deepest supercooling of liquid water is reached just where the melting line bottoms out, because the driving force for freezing depends on how far the liquid is below its melting point, and near 209 megapascals the melting point is at its lowest.

That the limit of supercooling tracks the melting line so closely is itself informative: the ice that nucleates first below the turn is ice Ih, and above it a denser ice, so the liquid’s own structure, and the crystal it most easily becomes, change across the same pressure.

The boundaries between the ices #

Each triple point sends a line into the solid region, the boundary between two ices, and Clapeyron’s relation governs those lines too, with the latent heat and volume change of the transition from one crystal to the other. The boundary between ice Ih and ice III runs down from the first triple point to lower temperatures at nearly constant pressure, because the two crystals’ entropies are close and their volumes differ by a fifth: a large volume change and a small entropy change make a steep line in pressure–temperature coordinates. Between ice VI and ice VII the same happens. Where two ices differ in how ordered their hydrogen atoms are rather than in how their molecules pack, as ice VII and ice VIII do, the volume change is almost nothing and the entropy change is large, and the boundary runs across at almost constant temperature, near 0 °C, over a range of several gigapascals.

The whole diagram is thus drawn by one relation applied to every pair of phases, with the slopes set by the two numbers each pair differs in. The phase rule sets the topology — lines meeting in threes, never in fours — and Clapeyron sets the angles. What neither can do is say which crystals exist, which is a question about how water molecules can be packed while keeping their four hydrogen bonds, and it is answered by experiment and, increasingly, by computation.

Worlds with the other ices inside #

The high-pressure ices matter most where there is a great deal of water under its own weight. Jupiter’s moon Ganymede, the largest in the Solar System, is thought from its gravity and its magnetic field to hold a salty ocean beneath a crust of ordinary ice, and below the ocean, where the pressure exceeds a few hundred megapascals, layers of ices III, V and VI resting on a rocky interior. The ocean is liquid partly because of the turn in the melting line: at the pressures at its base, water can remain liquid down to −20 °C, and the denser ices below sink rather than float, so the ocean is sandwiched between two kinds of ice rather than freezing solid from the top. Models of the water-rich planets found around other stars carry the same layering further, with ice VII and denser forms at the base of oceans thousands of kilometres deep. Which of the ices form, and whether a planet’s ocean touches its rock or is sealed off from it by a layer of high-pressure ice, bears on whether its chemistry could resemble the Earth’s, and it is decided by the melting line drawn above.

Ices that are there and ices that are not #

The figures show only the ices stable in equilibrium with the liquid. Several others exist. Ice II is stable in a region of the diagram below the liquid’s range and never meets the liquid; ice IV and ice XII form only as metastable phases, when the liquid is cooled under pressure in particular ways, and are found because the barrier a new phase has to climb can favour the formation of a phase that is not the most stable one. At lower temperatures, the hydrogen atoms in several of the ices order, giving further phases — ice XI from Ih, ice VIII from VII — that differ in how the hydrogens are arranged rather than in how the oxygens are. More than twenty crystalline ices have been identified, the most recent in the last few years, and the list is not closed.

Where the equations stop #

The figures use the international standard equations for the melting curve, which are fits to experiment accurate to a few per cent in pressure; they are not derived from theory, and their end points, especially at the highest temperatures along ice VII, are limits of the data rather than of the ice. The densities in the last figure are approximate values near each ice’s melting line and vary with pressure and temperature along it. Nothing in the figures shows the boundaries between the ices, where two solids coexist, or the metastable ices, or the transition of ice VII to still denser forms above about 60 gigapascals, where the hydrogen atoms move to the midpoints between oxygens and the molecule ceases to be a molecule.

Still open: what the liquid does at the turn #

The turn in the melting line connects to a question about the liquid that is argued over. Liquid water has several anomalies near and below its freezing point — its density maximum at 4 °C, and compressibility and heat capacity that grow as it is cooled — and one proposed explanation is that supercooled water, below the temperatures at which it normally freezes, has a second critical point: a transition between a low-density liquid, structured like ice Ih, and a high-density liquid, structured more like the high-pressure ices. The proposed point lies near 220 kelvin and 100 megapascals, close to where the melting line turns, in a region where water freezes too fast to be studied in bulk. Experiments on tiny droplets cooled in microseconds, and on water confined in pores, have produced evidence for it; whether it exists, and whether it explains water’s anomalies, is not settled.

The habit worth carrying away is to ask whether an anomaly belongs to a substance or to one of its phases. Clapeyron’s relation gives the sign of a melting line’s slope as the sign of the volume change, so water’s backwards slope is a fact about ice Ih’s open hydrogen-bonded lattice — and past 209 MPa, where the denser ices take over, water’s melting line turns round and climbs to ice that is solid above 400 °C. The rule was never broken; the crystal it was applied to changed.