The wonder of water

The tell-tale sign of the suspension of the laws of physics is someone seen walking on water. The suspension is always temporary – as soon as a second person tries to follow, the laws of physics are re-instated and the person looking for the ‘me too’ experience instead encounters full immersion. But walking on water is in fact quite possible, even within the legal constraints of the natural world. It should be viewed as a seasonal activity. If you live in the right place (Sweden or Alaska comes to mind), in winter you can walk on water without requiring any special ability apart from some warm clothes and non-slip shoes. In Siberia, you can even drive on it. This is due to a strange oversight in those laws of physics. Water has found a loophole.

It so happens that the freezing point of water is within the range of natural weather variability. It also so happens that frozen water has lower density than liquid water. Most substances expand when they melt, so the liquid rises and the frozen phase sinks. It is what makes volcanoes so interesting: when you melt rock, the magma percolates upward. But if you melt ice, the water goes down while the ice stays up. If rock were like water, volcanoes would erupt their lava downward, into the crust, still of interest to a geologist but the viewing experience would be severely limited. Due to this double accident, in winter you may find a sheet of frozen water lying on top of the liquid phase below. It is a remarkable coincidence that the most common liquid on earth has such uncommon properties. If walking on water makes you doubt physics, you should realize that the miracle is not the walking. It is the water itself.

(Be aware that in England, or any other place with too much weather and not enough climate, walking on water is physically impossible because the ice remains too thin to carry your weight: it cracks, breaks, and leaves you just as submerged as in summer but in much colder water. Just a warning – we don’t want to lose any of our valuable readers and commenters to a fatal misunderstanding. Volcanoes are already dangerous enough without water.)

Water is also a brilliant example of the very different behaviour of solids and liquids: one is strong and stable, and the other weak and fickle. But in fact not all frozen water is equal. Try walking on snow and your experience will be very different. It may still be officially a solid, but you do sink – part-way at least. Snow is a solid disappointment.

But ice also does not always behave like a well-trained solid should. The video is an example of ice solidly misbehaving, where a flow of ice moves like a hostile, alien substance from Doctor Who. In the video, it takes a while before the person filming it realizes that this is more than a lecture in the wonders of physics, and that the army may be required. The moment the penny drops is indicated by a sudden R-rated, uncensored word.

(The videos are an important part of this post and I hope you have time to watch them.) Glaciers do much the same thing. Although made of solid ice, glaciers flow under the influence of gravity. The ice making up a glacier is constantly on the move. It does not move as a solid block, but like an exceedingly slow river. On occasion, this can go terribly wrong. If a glacier looses its footing, it can become run-away. A surge can reach speeds of many meters per day, but it can also form an ice debris flow with speeds reaching a hundred meter per second. We have called this a cryoclastic flow. It is very rare, but devastating. The video below gives a good introduction to the moving ice of a more sedate glacier.

And in fact the solid earth does much the same thing, just not on a time scale that we can comprehend. When Italy crashed into Europe and joined the EU, it created a crumple zone where the Alps were pushed up. Solid rock it might be, but mountains were on the move with layers being thrown up and over each other. One big block was pushed horizontally so far that it became a bit of Africa amidst solidly European rocks: the Matterhorn. We can illustrate the process in the lab, using sand rather than rock. Here is your lecture in sandpit geology. (The only thing missing is a volcano.) But if feels strange that solid rock can behave like loose sand.

Solids in motion

The essence of a solid is that it keeps its shape. There may be a bit of deformation: for instance when you sit on a sofa, the surface sinks a bit, thus providing a solid yet flexible anterior foundation. But when you stand up again, the sofa recovers. In its deformation, different parts of the sofa stretch. But the individual molecules are strongly attached to each other and don’t change places. After you stand up, all molecules go back to their original location. In physics, this behaviour is called elastic. A liquid behaves very differently: molecules in a liquid are only loosely attached and easily move past each other. It never retains its configuration: sit on it and stand up again, and even though the water surface will go back to the previous location, the individual molecules never will. The ability of the molecules to move around means that a liquid leaves no empty spaces. A moving molecule in a liquid will experience friction with the neighbouring molecules. A material where this happens is called viscous. In an ideal solid, this does not happen because the molecules all move together, locked in place. In an elastic process, no energy is lost. In a viscous process, energy is lost (or becomes heat). If you drop a ball on the road, it bounces back up, and keeps hold of its energy. If you drop it on mud, it gets stuck with a total loss of its energy. The first behaviour is elastic, the second is viscous. A viscous material gives you a soft landing.

In the first two videos, ice showed a very strange phenomenon. Even though ice is solid, and strong enough to carry your weight, it also flows like a liquid. In mountain building, rock does the same thing. What really happens? How can something act like a solid at one moment, but move like a liquid over much longer time? Politicians understand this very well. For how do you make people accept change? It turns out, people are resistant to change. A culture behaves like a solid: apply a force (i.e. enforce some unpopular law) and you may get a temporary shift in behaviour, but when you remove the force everyone goes back to their previous behaviour. But apply a little force over many years, and people become far more accommodating. They get used to the new situation, and when you remove the force, people’s behaviour and/or opinions have changed permanently. Politicians move in small steps, continuously assuring people that nothing will change and this is just a minor law which will never be used. People are malleable given enough time. Apply a sudden jolt and human culture applies a counter force, just like a solid body, ensuring nothing changes. But apply pressure over a long time, and reactions begin to shift. The glacier of culture is now on the move.

Physics has defined two parameters which are related to this phenomenon. With the risk of turning our readership away in despair, I’ll try to explain! Please bear with me – complaints can be discussed in the VC bar.

The first parameter is the elasticity, and it describes the essence of ‘solid behaviour’. If you stretch a solid material, a counterforce tries to pull the material back to its original shape. This counterforce comes from the strong attachment of molecules to their neighbours. Good examples are seen in springs and in elastic bands. Some quick experiments will show you that (1) not all springs/bands are equal: some solids are much stiffer than others; (2) the force increases with the stretch – stretch it four times as much and the force that pulls back is four times larger; (3) the material returns to its original state if you remove the force; (4) there is a limit, beyond which the material simply breaks. The force that pulls the material back is the elastic force; the fact that the material returns to its original state shows that no energy is lost in the process.

The second parameter is also already mentioned, and is also one you know from experience: it is viscosity, or ‘stickiness’ and it applies to liquids. The word comes from the Latin word for mistletoe. You may now have visions of stealing sticky kisses under the mistletoe, but this would be the wrong connotation: apparently the berries can be used to make a rather sticky substance. Viscosity is related to friction. Friction is a force that acts on things that are moving, and causes them to slow down by dissipating the energy of the motion. The next video shows an example of viscosity: dropping marbles into different kinds of liquids.

Now you know the meaning of the English (actually German) expression ‘blood is thicker than water’. It means that blood is more viscous than water – which it is. (In reality it means that family bonds are strong – so it is about the viscosity of blood relations.)

Now take a river. The water at the surface flows at some speed. The water right at the bottom is stationary, held in place by the rough rock below. But because of the low viscosity of water, this stationary level does not slow down the water above it, and the flow at depth is about as fast as the surface flow. The stationary level is rather thin. Now take a lava river. The flow at the top can be fast, as we saw at the infamous fissure 8. The very bottom again is stationary. The viscosity of lava is much higher than that of water, and so the bottom layer really slows down the lava above it. So only the top layer flows fast – the lava just below the surface goes much slower because of the friction with the layers below.

This behaviour is illustrated in the video below. It shows a fast flowing lava river from Hawai’i’s fissure 8, with bits of solidified crust on the surface, flowing at the same speed as the lava. But at some point in the video a large lava boat appears which moves much slower. It sits much deeper, and its speed shows how fast the lava goes deeper below the surface. And that is not nearly as fast as the top layer. The lava boat is not scraping along the bottom: you can see that because the motion is smooth. It gets its slowness from the river itself.

These two parameters distinguish solid and liquid behaviour. But the distinction between solids and liquids is far less clear than water makes it appear. Solids can behave as if they are very sticky liquids. Playdoh is an example of such a solid. It is in-between, solid but malleable. Its molecules can flow. In fact, this is true for all solids. They have very high viscosity, but it is not infinite. Solids can flow – if you give them long enough. It may take longer than the Universe is old, but anything can move. This is called creep. If you have a synthetic rope which carries a load over a long period of time, the rope slowly lengthens and eventually has to be discarded. Inside the rope, the chains of molecules move past each other and it is is this movement that makes the rope loose strength. The rope never returns to its old state: in the process, energy is lost. It behaves as friction, converting motion to heat.

Viscous physics

Still with me? Let’s now take something solid (such as a piece of rock, a sheet of glass, or a handful of sand) and apply a force – for instance, by standing on it so your weight pushes it down. It deforms a bit: by how much is determined by the elasticity. But this movement leads to internal friction, and therefore the material also has viscosity.

This was first studied by James Maxwell (1831-1879). He was probably the greatest physicist of the 19^th century, best known for the equations of electromagnetism, which explained the speed of light and formed the foundations of Einstein’s work. But he did many more things. He deduced that Saturn’s rings had to be made up of small particles, worked on polarimetry, developed colour photography, initiated the field of control engineering, explained heat, and won prizes (as a child) for poetry. Maxwell died before his 50^th – 20^th-century physics may have been delayed by decades because of his early death. The theory of visco-elasticity must have been child’s play for him: it was something he published when he was 18(!), having already been at university for two years. Can anything good come from Scotland? I think we can answer that question with a resounding ‘yes’.

The way Maxwell envisioned an elastic but viscous material was with a spring and a damper. The schematics is shown in the diagram. If you pull on the spring, it expands. This puts a force on the damper (or perhaps a damper on the force), and the damper in turn slowly expands, like the synthetic rope discussed above. As the damper moves, it suffers internal friction, and the energy stored in the stretch of the spring is eventually lost to heat – the spring now relaxes over time. If the force continues, the result of the lengthening damper is that the spring slowly moves as a whole, without becoming any more stretched, at a rate that is determined by how quickly the damper can damp. The spring is the elastic bit, and the damper represents the viscosity.

The amount of expansion of the spring is set by the constant of elasticity, which is called E in the diagram. If the force is F, the spring extends by an amount x = F/E. Now the damper kicks in. This equation is slightly different, because friction relates a force to a velocity (friction doesn’t dissipate any energy if you are not moving). The extended spring puts a force on the damper, and it begins to move, with a velocity v = F/η, where η is the viscosity constant. As you can see, the larger this constant is, the slower things move. Think treacle (or re-watch the marble video above). The force from the extension is fully dissipated once the damper has moved the full distance x. Remembering the high school equation x = vt, you can now see how long this will take. The system has adjusted after a time t = x/v, when the elastic force has been dissipated. Fill in x and v using the equations above: x = F/E and v = F/η. The force F appears twice and cancels and you are left with the simple equation t = η /E. The values for the two constants can be looked up, and t calculated, for a particular material.

This t is called the Maxwell time. If you apply a force to a solid for a time shorter than t (for instance you hit it with a hammer) it behaves ‘elastically’ which is how a solid should behave. But apply the force over a time longer than t, such as the rope under a long-lasting load, and the solid absorbs the elastic energy – it ‘flows’. In geology, an earthquake counts as a sudden, fast force. Rocks quickly deform and go back to their original shape: this motion gives rise to the seismic wave traveling through the Earth. But apply a force over a very long time, and the rocks flow and a mountain can rise up.

What are typical values for the two constants? For η, this ranges from 0.05 Pa s for fruit juice and motor oil, to 1.4 Pa s for baby food, 70 Pa s for tooth paste and 30,000 Pa s for tar. (The units are Pascal-second.). For solids the values go very much higher: for glass it can be as high as 10²¹ Pa s. A famous experiment measured the viscosity of pitch (bitumen) at room temperature by counting drops dripping from a container of the material. There was a drop about once a decade. After 80 years the scientists found a value of 2.3×10⁸ Pa s. Science gets there in the end. This experiment received the 2005 Ignobel prize.

Typical values for the elasticity constant E are around 5 GPa (giga-pascal) for (cold) asphalt, 10 GPa for wood, 50 GPa for glass and 100 GPa for metals. (This is also called Young’s modulus.) They can change a lot with temperature. That is true for the viscosity as well, so it is important to measure both at the same temperature! The elasticity constant is only available for solids.

Compare the two constants, and you see that η varies all over the place, but E has fairly similar values for different materials. If you have a material with E = 10 GPa and η = 10¹⁸ Pa s, its Maxwell time becomes 10⁸ s, which is about 3 years. If you keep it under a constant force for 3 years or more, this material will begin to flow. Keep the force on for less time, and it behaves elastically, in a manner worthy of a real solid. The most important force that stays on for this long is gravity. In fact, gravity does not have an ‘off’ switch. Under the force of gravity, any solid can begin to flow – eventually.

Creepy glass, un-runny honey, and salt of the earth

The problem is often discussed in connection to window glass. In old (medieval) buildings you often find that a glass pane is thicker at the bottom than it is at the top. Can it be that the glass has slowly crept downwards under the influence of gravity? This is often claimed. But let’s look at the numbers. Using the values listed above, the Maxwell time for glass becomes 5 10¹⁰ s. That is rather a lot – in seconds. A year lasts 3.15 10⁷ seconds, so this becomes 1600 yr. That still seems a bit long, although not that much more than the age of European medieval buildings. On a hot day the glass may become a bit softer, so perhaps the creep happens during the occasional hot day. However, that doesn’t work because it would require that the hot day is as long as the Maxwell time: a couple of hours won’t do. I did use a high value for the viscosity: for some types of glass it can be one or two order of magnitudes less. So for some types of glass, the Maxwell time could be 10 to a 100 times less, which seems more reasonable to support the case of creeping medieval glass.

However, the argument still fails. For even if you wait this 1600 years, the creep will be by about the elastic extension. And this extension is very small. For a glass pane of 1 square meter, I get about an elastic extension under the force of gravity of about 0.4 micrometer. Even if you take the softest type of glass, you would still have to wait a VERY long time before it can creep even a centimeter: about half a million years, in fact. Not even our university buildings are that ancient.

I once had an office in a colonial building which dated from the 1820’s – and it had this kind of glass. The window frame (which was original) fitted the shape of the glass, including the thicker bit at the bottom. So it had already had that shape when the glass was put in. In those days, glass panes were made on a rotating disk. The glass on the disk would become thicker near the centre. The panes were cut from this disks, and would have one side thicker then the other. The installers would always put the thicker, heavier part at the bottom. Physics still works; however the effect came during manufacturing and installation, not during the long years of academic research.

Glass is crystallized sand, and crystals are hard. Adding some crystals can harden any material. (I won’t mention diamond.) The effect can be viewed by using your diamond ring to scratch the window glass. The harder material will scratch the softer one – diamond is even harder than glass. Try the same thing with graphite (also carbon but a non-crystal form) and the glass will laugh at you. However, an easier way to make the case may be hiding in your fridge: it may contain a jar of honey which someone (no names mentioned) has accidentally put there. Bring it out and all attempts to pour the honey are fruitless. But it is not a solid: the freezing temperature of honey is something like -40C. What happened? The cooling honey formed crystals and these turned the watery honey into a far more viscous fluid – one that won’t let itself be poured. And the crystals are not easy to get rid off: putting it back in the cupboard, at its original temperature, does not work: the honey retains its solid refusal to flow. It does still flow, though: put it upside and after some days you find that it has miraculously moved to the new bottom. But to make it fluid enough to pour, you have to heat the honey to well above room temperature. The crystals dissolve, and the honey can now be put back where it belongs, in the cupboard. But even at room temperature, over time the crystals will still grow and during the coming weeks the honey becomes less runny. The viscosity of runny honey is a few Pa s. That of crystallized honey at fridge temperature is around 1000-3000 Pa s.

(You may be interested how honey compares to magma. The viscosity of basaltic magma is between 10 and 100 Pa s, in between runny and non-runny honey. Andesitic and rhyolitic magmas are far more viscous, around 10³ – 10⁵ Pa s. (Rhyolitic is the most viscous.) It is quite dependent on temperature. But it is clear that such magmas do not flow easily. Even the very best rhyolitic magma is only about as runny as the honey in your fridge. )

Another example of a crystallized material is salt. Wherever large bodies of water have dried up, large deposits of salt were left behind. Many places have these deep salt layers underground. But they are not entirely stable. The salt is not very dense compared to the rock, and therefore is buoyant: it tries to rise and flow. In some places, salt domes have formed from this rising. Elsewhere, whole mountains were made by rising salt. A well known example is near Hormuz in Iran. They are not actually mountains: once above ground, salt erodes rather quickly, and the salt sticks out perhaps 20 meters. The salt is rising at velocities up to 7 mm/yr. From the size and velocity, a viscosity has been derived of around 10¹⁹ Pa s.

Making mountains

So salt can flow to make mountains. How about real mountains? How are they made? Rock is strong but it does have limits, and these limits are reached faster if the rock is heated. This happens deep under ground: the rock here is hotter, and under long-lasting pressure of an approaching plate (think India or Italy), it buckles. Later, erosion removes the layers above and the buckled layers become visible at the surface. It is worth searching in mountain regions for places where this is visible, hidden places where the root of the mountain is revealed. The image below is for the Swartberg in South Africa, near Oudtshoorn. If you have time and a fairly decent car (a four-wheel drive is not needed), the drive over the Swartberg pass is spectacular. The road is in good shape but unpaved. Driving towards it from the north, the mountain range rises steep from the plain and it is impossible to see a path the road could take. It just clings to the side. The rock contortions are on the far side where you descend into a deeply eroded valley.

So flowing rock can be used to build mountains. They can also be used to get rid of one, a bit like the lake above. Mountains require strong crust to carry them. Volcanoes, on the other hand, form where the crust is heated, partly liquid (magmatic) and not as strong as it should be. This is not quite the right combination for strong and stable. Take Hekla. It is a young mountain, dating from the holocene, and is rapidly growing. But the growth is being hampered by a small detail: the area around it is sinking. The crust responds to the weight of the new mountain. In itself, this is not abnormal. Mountains don’t do magic and something needs to carry their weight. The rocks below Hekla are not quite up to the task. This is because they are hot and ductile (and partly melted). It is as if Hekla is sitting on very old brie (or camembert if you can stand the smell). It sinks over time. (Actually, it is growing faster than it is sinking, so Hekla is getting higher while going down. But that is beside the point.)

The time it takes Hekla to sink is given by the equation t = η/ρ gλ where η is the viscosity of the ductile rocks, λ is the depth of those rocks, and g is gravity. The sinking should not be overstated – it is a small effect. Measurements have indicated that the relaxation time is 100 yr and the thickness of the upper elastic (i.e. stiff) layer is 3.5 km – the ductile layer is below this. Put those numbers into the equation, and the viscosity of the hot deep rocks can be calculated. Underneath Hekla, a rough estimate gives a value of order 3 × 10¹⁷ Pa s for the ductile layer below 3.5 km. It is a local effect: over most of southeast Iceland, the elastic layer is around 10 km thick, underlain by a 30 km more viscous layer (1 × 10²⁰ Pa s) on top of quite a ductile layer (1 × 10¹⁹ Pa s). The last layer may represent magma accumulation at the Moho.

Fast-sinking mountains are not the only indication of a ductile crust. A more common sign is a lack of earthquakes. Earthquakes happen in brittle rocks, and their absence (as in the Lurking Dead Zone in Iceland) can indicate low viscosity. Of course it may also just be due to a stress-free region where there is nothing to cause shakes!

We have looked at why solids flow, why mountains grow and how volcanoes sink. The next part will look at a bigger scale. Why do continents move, and what makes the mantle convect and plume? Find out in the next instalment of The Wandering Earth.

Albert Zijlstra, June 2019