The melting curve that leans the wrong way

Every boundary on a phase diagram has a slope, and Clausius and Clapeyron give it from two measured quantities and nothing else:

$$\frac{\mathrm{d}P}{\mathrm{d}T} = \frac{L}{T\,\Delta V}.$$

The latent heat $L$ is the heat absorbed on crossing the line, and it is positive for every transition from a more ordered phase to a less ordered one — that is what makes it a transition in that direction. The temperature is positive. So the sign of the slope is the sign of $\Delta V$, and nothing else in the problem can change it.

Boiling lines never do this, because a gas is always less dense than the liquid it came from: the volume change on boiling is large and positive without exception, so a boiling point always rises with pressure. Melting lines are a different matter, because the volume change on melting is small and can be of either sign.

What the sign is a fact about #

The slope’s sign is a piece of thermodynamics reporting a piece of structure, and the structure is worth stating.

Most solids are denser than their melts because a crystal packs its molecules more efficiently than a liquid does. Ice is not, and the reason is the hydrogen bond: each water molecule in ice is held at exactly four neighbours in a tetrahedral arrangement, which is an open structure with a great deal of empty space in it. Melting breaks some of those bonds and lets the molecules fall into the gaps, so liquid water is denser than ice — by nine per cent.

That is the anomaly, and it is a chemical fact rather than a thermodynamic one. Everything that follows from it — floating ice, the backwards melting curve, the density maximum at four degrees, the fact that lakes freeze from the top down — is that one structural feature read through relations that apply to everything.

A small number of other substances do the same, for related reasons: bismuth, gallium, antimony, silicon and germanium all expand on freezing, all have directional bonding that produces an open solid, and all have backwards-sloping melting lines. Water is not unique; it is merely the one everybody meets.

How much pressure actually buys #

The number is the useful part, and it settles a claim that appears in a great many textbooks.

The claim is that a skate works because the pressure under the blade melts a film of water that lubricates it. The arithmetic is available: a seventy-kilogram skater on a blade a millimetre wide and twenty centimetres long puts about thirty-five atmospheres on the ice, which depresses the melting point by a quarter of a kelvin.

Nobody skates at a quarter of a degree below zero. Ice rinks are held at five to ten below, and outdoor skating happens far colder than that; ice remains slippery at forty below, where pressure melting would need thousands of atmospheres. Even at zero the number is too small, since a blade that concentrated its load enough to matter would sink into the ice rather than glide.

What is actually happening is that the surface of an ice crystal is disordered — a layer a few molecules thick, liquid-like, present whether or not anything is pressing on it. Faraday proposed something like it in 1859, on the evidence that two pieces of ice pressed together freeze into one; the layer was confirmed directly in the twentieth century and thins as the temperature falls, which is why ice does eventually become less slippery when cold enough. Friction from the blade’s own motion melts more, and that contributes too.

The interesting thing about the wrong explanation is that it is not wrong about the physics — pressure genuinely does depress the melting point of ice, uniquely among common substances, by exactly the amount computed. It is wrong about the size, and about which of several effects dominates. That is a common shape of error and the remedy is the arithmetic rather than the argument.

Where the effect does matter #

Two atmospheres of pressure buys fifteen millikelvin, and a glacier three kilometres deep provides two hundred and seventy.

The base of a large ice sheet therefore sits at its pressure melting point — around two kelvin below zero — and is wet. That single fact governs how glaciers move: a glacier frozen to its bed creeps by deformation of the ice, slowly; one resting on water slides, and slides much faster. Whether the bed is at the melting point is decided by the pressure, by the geothermal heat arriving from below, and by the heat generated by the sliding itself, and the last makes the system unstable — sliding produces heat, which produces more water, which produces more sliding.

That instability is why ice streams exist: fast rivers of ice within a slow sheet, with no channel to guide them, sustained by the water they generate. It is also why predicting how quickly an ice sheet will lose mass is hard, since the answer depends on a phase boundary two kilometres down.

The same regelation effect gives Faraday’s demonstration and one more that is worth having. A wire loaded with weights over a block of ice cuts through it and leaves the block whole: the ice melts under the wire where the pressure is, the water flows round to the top where the pressure is not, and refreezes — releasing the latent heat, which conducts back through the wire to supply the melting below. The rate is limited by how fast the wire conducts heat, so a copper wire cuts through in an hour and a nylon one of the same load does not move at all.

Seeing the melting line in its diagram rather than on its own settles how small the anomaly is. The backwards lean is a few degrees off vertical on any scale that shows the whole diagram, and every other feature is entirely ordinary.

That proportion is worth keeping. The consequences of the lean — floating ice, wet glacier beds, the whole habitability of a planet whose lakes freeze from the top — are enormous, and the thermodynamic peculiarity producing them is a small angle on a line that is nearly vertical anyway. Large consequences from small anomalies is the usual arrangement rather than a surprise, because the anomaly is a sign rather than a size.

Where the relation comes from #

The derivation is worth having because it explains why only two quantities appear.

Along a coexistence line the two phases have equal Gibbs free energy — that is what coexistence means. Move along the line by a small step in temperature and pressure and both free energies change, by $-S\,\mathrm{d}T + V\,\mathrm{d}P$ each with their own entropy and volume. For the two to stay equal, the changes must be equal, and rearranging gives

$$\frac{\mathrm{d}P}{\mathrm{d}T} = \frac{\Delta S}{\Delta V} = \frac{L}{T\Delta V},$$

using $L = T\Delta S$, which is what latent heat is.

Nothing about either phase enters beyond its entropy and its volume. No model of the liquid, no theory of the crystal, no interatomic potential — just two measured differences. That is what makes it one of the most reliable relations in the subject and one of the most useful: it converts a measurement of a latent heat and two densities into a prediction about how a boundary moves, and the prediction is exact.

It also explains why the boiling line is so much shallower than the melting line. The volume change on boiling is a thousand times the volume change on melting, so the slope is a thousand times smaller, and a boiling point moves by tens of kelvin over a range of pressure that moves a melting point by fractions.

The other slopes, and what they report #

Applying the same relation to the other boundaries is quick and each answer is a fact about a substance.

The sublimation line always slopes forward and always more steeply than the boiling line at the same temperature, because the latent heat of sublimation is the sum of fusion and vaporisation while the volume change is nearly the same — so the slope is larger by the ratio of the two latent heats. That is why the three lines meet at an angle at the triple point rather than passing smoothly, and it is a constraint on the diagram that any tabulation must satisfy.

The boiling line’s slope, at pressures well below critical, is dominated by the gas volume, which is $RT/P$ — and putting that into the relation gives $\mathrm{d}\ln P/\mathrm{d}T = L/RT^2$, whose integral is the exponential vapour-pressure curve every substance follows. That derivation is the same one line, applied with one approximation, and it produces the shape of every boiling curve there is.

And near the critical point every slope becomes ambiguous, because the latent heat and the volume change both go to zero there. The ratio remains finite — the coexistence line has a definite slope right up to the end — but the relation is now zero over zero, and getting the limit right needs the critical exponents rather than the relation.

Where the model stops #

The relation is exact and the straight lines are not. The slope is a derivative, so a coexistence line is straight only where the latent heat and the volume change are constant. Over the first few hundred atmospheres for water that is a fair approximation; over thousands it is not, and the melting line eventually turns round entirely — above about two thousand atmospheres, ice takes on denser crystal structures and the line slopes forwards again like everything else’s.

Ice has at least nineteen known crystalline phases and the essay describes one. Ordinary ice is the low-pressure form, and the backwards slope belongs to it alone. The phase diagram at high pressure is a thicket of triple points between structures that are all denser than water, and none of them floats.

The transition is assumed to be at equilibrium. Water is famous for not being: it supercools readily, to minus forty in small clean droplets, and a supercooled liquid is on the wrong side of a line the relation describes. Nothing in Clausius–Clapeyron says how quickly a phase change happens or whether it happens at all, which is the subject of nucleation.

And the surface layer is not thermodynamics of this kind. A disordered surface on a crystal below its melting point is a surface phenomenon governed by the balance of surface energies, and it is a different calculation entirely — one in which the bulk phase diagram plays almost no part.

Two substances, and which one is the exception #

Putting several substances side by side is what makes “anomalous” a meaningful word rather than a habit.

Carbon dioxide’s triple point is above one atmosphere, which is why solid carbon dioxide sublimes rather than melting in an open room and why it is called dry ice. Nitrogen’s critical point is at 126 kelvin. Water’s is at 647. Those are enormous differences and none of them is an anomaly: they are what different intermolecular forces give, and the shapes of the three curves are the same.

Against that, water’s melting line is a genuine oddity — a sign rather than a scale — and it is the only one in the set. That is worth insisting on, because water is frequently described as having a long list of anomalies, and most of the list is a consequence of the same hydrogen bonding rather than a set of independent surprises. The density maximum at four degrees, the high heat capacity, the high surface tension, the high boiling point for its molar mass and the backwards melting line are one structural fact seen five ways.

Why a positive latent heat is not a convention #

One step in the argument deserves defending, because it is the step that makes the sign rule work at all.

Latent heat is positive going from the more ordered phase to the less ordered one, and that is not a sign convention: it is a consequence of the second law. Melting increases entropy, because a liquid has more accessible arrangements than a crystal; and $L = T\Delta S$, so a positive entropy change is a positive latent heat. There is no substance for which melting is exothermic, and there could not be, because a phase with lower entropy and lower energy would simply be the stable phase at every temperature and there would be no transition to discuss.

That is why the sign rule is clean. One factor in the relation is fixed by thermodynamics and cannot vary; the other is a structural fact about packing and can be of either sign; so the slope’s sign is determined entirely by the second, and asking about it is asking about density rather than about heat.

The same argument fails for a transition between two solid phases, where the entropy change can be small and either sign, and where a coexistence line can accordingly do things a melting line cannot. Several of ice’s high-pressure boundaries do exactly that.

What the pictures cannot show #

The melting lines are drawn over a few kelvin and a few hundred bars, which is a tiny corner of a phase diagram, and the straightness of the lines is an artefact of that. Zoom out and every one of them curves; zoom out far enough and water’s turns round. A figure drawn to make a sign visible cannot also be honest about the shape.

Nor does anything here show what the two phases are. The whole relation runs on differences of entropy and volume between two states, and it is indifferent to what those states are made of — which is its strength and the reason it says nothing about why ice is less dense. That explanation is a structural one and lives outside the diagram entirely.

What the relation is worth as an instrument #

Because the relation contains only measured quantities, it can be run in whichever direction is convenient, and each direction is a technique.

Measure a coexistence line and get a latent heat. That is the older use and it is still the best way to obtain the latent heat of a transition that is hard to calorimeter — a solid-to-solid transition inside a diamond anvil, or a transition in a sample of a few milligrams. The slope is measurable by watching where the transition happens as the pressure is changed, and the volume change from the two densities.

Measure a latent heat and a density change and predict the line. That is how phase diagrams are extrapolated into regions nobody has reached, and it is why a diagram drawn to a hundred gigapascals is largely a computation rather than a set of observations.

And use the relation as a consistency check on data that came from somewhere else. Three quantities that must satisfy an exact relation are two quantities and a test, and published tabulations of latent heats and densities are routinely checked against measured slopes. A disagreement means one of the three is wrong, which is more information than any of them alone provides.

That last use is the one worth generalising. An exact relation among three measurable quantities is a measurement of a fourth thing: whether the other three were measured correctly.

Where the ladder goes next #

The phase-change ladder began with the heat that changes no temperature, went through a boiling point being a pressure, the part of the curve no fluid follows, the point at which the two become one, the barrier a new phase has to climb, the gas that cools by being let go, the transition with nothing to order and why the triple point is a point. This rung asks what fixes the slope of a boundary and what one sign of it means. The rungs after it: the high-pressure ice phases, where the same relation runs over a diagram with a dozen triple points; the Clausius–Clapeyron relation applied to a second-order transition, where both differences vanish and the relation has to be replaced; and the melting of a small crystal, where the surface energy shifts the melting point by an amount that depends on the size.

The habit worth carrying away is that a thermodynamic relation reports rather than explains. The backwards slope of water’s melting line is not a thermodynamic oddity; it is a structural fact seen through an entirely ordinary relation — and separating the two is usually the first step in understanding why anything behaves unusually.