The wire that goes through the ice and leaves it whole

The melting curve that leans the wrong way explained why pressing on ice lowers its melting point: ice takes up more room than the water it came from, so pressure favours the water. It gave the slope from the Clausius–Clapeyron equation and it noted that the effect is small. A famous demonstration has been shown in classrooms since the middle of the nineteenth century. Take a block of ice, support it at both ends, loop a thin wire over the middle and hang weights on the wire’s two ends. Within an hour or so the wire will have sunk into the ice, and eventually it will have gone clean through it and dropped to the floor, along with its weights. The ice block sits there unbroken, with perhaps a slightly cloudy line where the wire went through.

John Tyndall made the phenomenon famous, and he and James Thomson attached the word regelation to it. Michael Faraday had noted that two pieces of ice would freeze together when pressed. The classroom explanation goes: pressure beneath the wire melts the ice; the water flows above the wire where pressure is lower and freezes again. All of that is correct, as far as it goes. And it leaves out the one thing that decides how fast it happens. It also tends to give people the impression that pressure is doing a great deal. It is doing very little.

How little the pressure does #

The Clausius–Clapeyron slope for ice, using the volumes of ice and water and the latent heat of 334 kJ/kg, is about $-0.0074$ K per bar. That is seven thousandths of a degree for a pressure equal to that of the atmosphere.

Take the numbers for a wire 1 mm across, with 5 kg hanging from each end, over a block 20 cm wide. The total force of about 98 N is shared along the 20 cm of contact. Per unit length that is 490 N/m, and spread across a width of 1 mm, that gives about 4.9 bar. So the wire lowers the melting point by only about 36 thousandths of a degree. A skater’s blade, with far more pressure, moves it by about 0.3 K. That number should by itself have ended the story that skating depends on melting by pressure, as ice at −20 °C would need 2,700 bar, and it is curious how long it survived. The real reason ice is slippery has been debated ever since Reynolds proposed the pressure explanation in 1899 and has moved to frictional heating and a disordered surface layer that exists on ice even without pressure, on which there is still active work. Whatever the correct answer, this slope rules out the textbook one, with no need of the more delicate story in the force that takes what it needs about how ordinary friction arises.

So a very small temperature difference has to do all the work of regelation. Why does the process work at all?

Melting takes heat, and the heat is on the wrong side #

The ice just beneath the wire is at slightly above its melting point for that pressure. Melting begins. But melting requires heat, 334 kJ/kg, a very large amount. As the heat that changes no temperature explained, a phase change is a process that takes in latent heat without a change of temperature. Without a supply of heat from somewhere, the melt would chill the interface until the local temperature had dropped to the new melting point. After that nothing further would happen. The wire would stop moving.

The heat comes from the other side of the wire. Water squeezed out from under the wire passes around to the low-pressure region above it. Here it is at slightly below the melting point for that lower pressure, and it freezes. In doing so it releases exactly as much latent heat as was absorbed below. So there is one amount of heat released on top of the wire and one required beneath it. In between lies the wire.

The process is therefore a closed cycle with a heat flow at its heart. Nothing is consumed, apart from the potential energy of the weights descending, which is a very small amount of energy compared with latent heat. The steady state has the wire sinking at the rate that it takes the heat to get across.

Let $\Delta T$ be the temperature difference between the two sides, about 36 thousandths of a degree here. Then the heat flow per unit length of wire is about $S k \Delta T$ for a conductivity $k$ and a shape factor $S$ of about one. Melting a slab of width $2r$ at a speed $v$ takes $v \cdot 2r \cdot \rho L$ per unit length. So

$$v \approx \frac{S\,(k + k_{\rm ice})\,\Delta T}{2r\,\rho L}.$$

That is simple enough to be quite useful, and the ice’s own conductivity is included in parallel, because some heat can go around through the ice. This form was given by Bottomley in the 1870s in essence, and in the twentieth century Nye, Drake and Shreve treated it with care. Everything hard about the phenomenon has gone into $k$.

A wire chosen by its conductivity #

This is what the common account leaves out.

The ice’s own conductivity, 2.2 W/m/K, gives a very poor path. Copper conducts about 400 W/m/K. So a copper wire at a given load should cut much faster than a nylon fishing line of the same width, by over a hundred times, and the difference has been shown in many classrooms. J. T. Bottomley reported in 1872 that a copper wire went through ice quickly, an iron wire more slowly and a piece of twine, with the same load, barely at all. He understood the reason. He had noticed that the demonstration depends on the thermal path and that the material is what mattered. So the classroom result of “pressure melts ice” is close to the opposite of what the most instructive version of the classroom demonstration shows: the same pressure can produce very different rates.

It is important not to read 17 cm per hour as an accurate prediction. A shape factor of one was assumed, the water film’s thickness and how fast it can get around were ignored, and in fact in a real experiment there is a contact resistance. Water flows through a very thin film under high pressure, and it does so at a limited rate, with a resistance of its own. Experiments often report a threshold load below which the wire moves very slowly, perhaps because of impurity in the ice or because air bubbles accumulate. Reported rates with similar arrangements have been up to several times different from this rough estimate, often lower, and their ratio between materials is less extreme than the estimate would suggest. But the structure holds.

There is a pleasant side-effect of the same formula. Pressure rises with load and falls with width.

A thin wire gets the benefit twice. Half the thickness gives twice the pressure and half the path, so the speed goes as the inverse square of the wire’s radius. It is for this reason that demonstrations always use thin wire, and a thread or a chain would not be nearly as good. There is a limit: as the wire gets thin, the ice’s strength begins to matter, as ice can be crushed under the wire at around 100 bar. And at a load near that the ice cracks, and it stops being regelation at all.

A glacier on its bed #

What makes this a matter of more than classroom interest is that the Earth has a very large version of the same process. A glacier slides on its bed. Glacial ice at its base is often at the pressure melting point. Bumps of bedrock stand in its way. Upstream of a bump, the pressure is high and ice melts; downstream, it is low and water refreezes. The heat released on the downstream side goes through the rock of the bump to the upstream side. This is a wire, with the glacier moving past it, seen from the wire’s frame.

In 1957, Johannes Weertman used this idea to build a first theory of how fast a glacier can slide. He observed that there are two ways for ice to get past a bump. It can regelate past it. For that, the formula above says that speed goes as one over size, since small obstacles have a short heat path. Or the ice can deform plastically around the bump, as ice flows in the sense discussed in the oil that is a glass for a quarter of a millisecond and elsewhere, under stress concentrated by the obstacle. For that, speed goes up in proportion to size, because the stress acts over a greater distance.

The two speeds add, as each mechanism is available at the same time, so the slowest path is at the crossing point. Bumps of around 15 cm are therefore what hold a glacier back. Weertman called it the controlling obstacle size, and estimates in the field agree on order of magnitude. The elegance of the result is that small rocks and large hills turn out to be irrelevant. The small rocks have ice passing them by regelation and the hills by flow. The bed’s roughness at a particular scale, of the order of a kitchen chair, sets the speed of a river of ice. Field observations have been made in tunnels under glaciers, where regelated ice shows up with a clear signature in a layer of debris-rich ice at the base, some millimetres to centimetres thick. In some cases, the same process picks up debris from the bed, and glaciers carry rocks with them.

Real glaciers, however, rarely stop at this picture. Subglacial water at high pressure can lift the ice partly off its bed, reducing the contact area and speeding it up, sometimes greatly. Weertman’s theory left this out. A theory of sliding in the presence of cavities came from Lliboutry, and it is the subject of continuing work. Fast ice streams in Antarctica slide over soft, water-saturated sediment, with no bumps of rock at all, and that is a different problem whose physics is still argued over. In fact, sliding is among the largest uncertainties in estimates of how ice sheets will respond to warming. Regelation is not the main story any more for those, but it was the way the first theory got hold of the question.

The same arithmetic in soil and wire #

The heat flow idea has other applications. A body moving through ice under its own weight, with the temperature gradient driving it, moves by the same process with the source of the difference changed. The frost that lifts by drinking depends on a thin film of water along a particle in frozen ground, in which water moves towards colder regions. A particle in ice, in a temperature gradient, migrates towards the warmer side: this was found in experiments by Römkens and Miller around 1973 and it is called thermal regelation. It is thought to matter for the movement of soil particles, and in the way dust and bubbles migrate in ice sheets, with a very slow drift of order a few metres over millennia. Ice-core scientists have taken note of it.

What they have in common is one fact about the ice surface. Ice in contact with a solid has a thin film of water on it, down to well below the normal melting point, and water moves readily along it. Regelation assumes a thin film, which any real wire has to have if water is to get round at all. The physics of that film near the melting point is a subject in itself, and the idea of “premelting” extends far beyond ice.

Why almost nothing else does this #

The demonstration would fail with almost any other solid, because of a sign. Most materials shrink when they freeze, so pressure raises their melting points, and a wire pressing on them moves them further from melting. A hot, loaded wire can cut through solid wax, but it is then just heat from the wire. To regelate, a material has to have the unusual slope that the melting curve that leans the wrong way described. Water is the common example. Bismuth, antimony, gallium, germanium and silicon also expand on freezing. A wire should be able to pass through solid gallium at just under its melting point of about 30 °C, and that has been a neat demonstration, as has passing a current to speed it up. Silicon is not an obvious candidate, given its melting point at about 1400 °C.

Even for ice the effect has limits. On the ice phase diagram the melting curve runs down to about −22 °C at 2,100 bar, then the melting line turns round and heads upward, as denser phases of ice form at high pressure, and they behave normally. So the same mechanism could not operate above about 2 kilobar.

The figure treats the heat flow as confined to the wire and the ice near it, with no other source. It is perfectly possible to supply the heat another way. A warm wire cuts ice readily without any regelation at all: this is just melting. In a cold environment, where the ice is below freezing, a wire does not regelate at all until the ice has warmed to very nearly zero, as it has to be at the melting point for its pressure, and in that case it is the ice’s own conductivity that matters. Below about −1 °C on these loads, no demonstration will work. Ice from the freezer has to sit around until its surface has warmed, and an impatient demonstrator will often find a wire doing nothing for the first half hour. The heat path has a comparison in the metal that feels colder than the wood, where the same contrast in conductivity is felt by a hand, and in the insulation that makes a wire lose more heat, in which the geometry of a thin wire surprises in a similar way.

What a heat-flow estimate cannot hold #

The heat flow formula uses a shape factor of about one, standing for the actual two-dimensional conduction between two curved surfaces, which has a logarithm in it. It also uses the entire $\Delta T$ between the bulk of the top and bottom interface, and it assumes that the temperature difference is what pressure gives and is uniform over the wire’s surface. A real wire’s load is unevenly distributed, and pressure varies from the bottom of the wire around to its sides. The film’s flow resistance can matter, since the water has to get round under a very small pressure difference, and it is not automatically true that the flow will not be limiting. Some careful experiments in the 1970s found that at very low loads the rate is well below this estimate.

The ice also has to be clean. Air bubbles accumulate in front of a slowly moving wire and may bridge the film. Dissolved salt lowers the freezing point locally and it would upset the relationship by far more than the pressure does, since a salt concentration of one part per thousand lowers the freezing point by more than this wire’s whole temperature difference. A demonstration done with ordinary tap-water ice can behave in curious ways on account of this. Near the end of a demonstration, when the wire has got through most of the block, a ripple of incompletely refrozen ice remains behind. It is not literally whole, in a metallurgical sense.

The Weertman figure is much cruder. It assumes one size of cube-shaped bumps, a simple power law for ice flow with an assumed coefficient, and a certain stress. All of the numbers can move the crossing, by factors of a few. It shows the shape, and it should not be used for an estimate of a sliding speed in any actual glacier.

Still open: how a film sets the speed #

The wire experiment’s fine structure turns out to be unexpectedly hard. How thick is the water film? Direct measurement in the working situation is very difficult, as it is thin and at the bottom of a block of ice. Estimates say about a micrometre or so and it may be much less. Theory suggests that the thickness adjusts so as to carry the required flow and that it depends on the load, and on the ice’s crystal orientation. In a moving contact with the ice’s premelted layer near 0 °C, the difference between a fluid and an ordered interface is unsettled. Molecular simulations have started to reach such length scales, and they find that a layer of a few nanometres behaves as neither liquid nor solid in the usual sense. How that layer connects with the continuum flow at a micrometre is still an open question, and the same layer is involved in why ice is slippery, the question that pressure melting failed to answer. Nearly two centuries after Faraday pressed his two pieces of ice together, and in spite of all the classroom demonstrations, the answer is still not fully known.

A wire’s pressure lowers ice’s melting point by only about 36 thousandths of a degree, and the wire sinks only as fast as heat from the refreezing above can cross to the melting below: about 17 cm/h for copper against 0.1 for nylon at the same load, in a formula that also predicts why a glacier is held up by bed bumps of about 15 cm. The classroom story is about pressure and the physics is about heat. The habit worth carrying is to ask, of a process that seems to need nothing, what must be delivered and from where, because a process limited by delivery is telling something about the pipe.