Surface Tension

4th February 2001 Back to Water Spider Webs Nature’s Maths

Liquids sometimes form drops, and sometimes spread over a surface and wet it. Why does this happen, and why are raindrops never a metre wide? A clue to the answer to the second question may be found in pictures of astronauts playing with large blobs of water in their space-craft.

It all comes down to the forces between atoms or molecules, and the forces between them. These particles are unimaginably small. In one gram of water the number of molecules is about 3.3 X 10²², or 33000000000000000000000. If the gram of water were in the form of a 1 cm cube, there would be about 23000000 molecules on a side.

The force between two atoms or molecules is generally repulsive if they are pushed too close together. The force increases so strongly as the distance is reduced that they behave almost as if they were hard objects. Try compressing some water or steel. But at larger distances the force are attractive. Try pulling pulling the bung from a tube which contains only water and no air. Or try pulling a piece of piano wire in two.

Let’s look at a graph.

The vertical direction represents energy. The horizontal direction represents the distance between the centres of two molecules. Zero is way off the left of the graph because two molecules can’t overlap.

The green line represents the zero of energy. At the right, when two molecules are far apart, the mutual energy is nearly zero. As we move them together, the energy goes down. This means that they are attracting each other, just as a falling object is moving towards positions of lower energy. But closer than a certain distance, the energy starts to rise more and more sharply, until it is very steep indeed. It becomes unbelievably hard to push two molecules closer than a certain distance. This is why substances such as water and steel are almost incompressible. Gases are compressible because there are gaps between the molecules. The distance between air molecules is very roughly ten times their diameter.

The white dots represent molecules, running around in the stable area between the red lines. The vertical spread represents their distribution of energies. The upper diagram corresponds to a higher temperature. We can see that another increment of energy would see a few molecules going above the green line, and able to go infinitely far to the right. In other words, they can escape. This is called evaporation. Further small increments in temperature will greatly increase the number above the green line, and the evaporation rate will increase dramatically.

This is characteristic of phenomena with a threshold. When you wash something in warm water, the absolute temperature is not raised much, but the effect is great. Another example is the conductance of a semiconductor, which rises with temperature. In fact, in the early days of semiconductors, great care had to be taken to stop them burning themselves out.

The diagram comes with a bonus – as the temperature is raised, the average separation of the molecules moves to the right, that is, the molecules move apart. In other words, the material expands.

You may argue that a few substance contract on heating, like water from 0 C to 4 C. This can be explained by invoking changes of structure. Similarly, a well ordered substance might shrink if the aligned molecules wriggle about more vigorously.

Also, as the liquid starts to lose its grip, the surface tension goes down, and by the time the boiling point is reached, it is considerably less than at room temperature. But it doesn’t disappear completely until the critical temperature is reached. At that point, the interface between liquid and solid vanishes; in fact the distinction between liquid and solid vanishes, leaving no reason to have an interface.

So, between the two regimes of repulsion and attraction, two molecules prefer very strongly to sit at a distance where there is no force between them, where their total energy is a minimum. Of course, they are not stationary – they are jiggling about because of thermal motion. Below a certain temperature, molecules cannot easily get away from their position – in this condition, the substance is a solid. At a higher temperature they can wander around, and we have a liquid. Given a high enough temperature, molecules may may escape – this is evaporation. At higher temperature the liquid turns into a gas. As you might expect, adding pressure changes things.

So in a solid or a liquid, the atoms or molecules are in a state of average equilibrium. On average, each one feels no net force from the ones all around. There will be fluctuations in the force on each one as they jiggle around, and in a liquid the molecules will slowly wander about. Two layers of differently coloured water will gradually diffuse into each other. Even in a solid, there can be very slow diffusion.

What has all this to do with water drops? The answer is already implied in the statements given. Every molecule is on average in a sate of lowest available energy, with no net force on it. If a molecule experienced a net force it would move until it didn’t. Molecules in the body of a liquid are surrounded by neighbours . You might imagine that there could be around twelve packed around it – six in a ring, with three above and three below. Any molecule is, on average, in equilibrium with all of these.

But a molecule at the edge has fewer neighbours, perhaps by about three. What is the consequence? Imagine just two molecules coming together. They are attracted until they reach equilibrium, and so when they are together, they have less energy than when they apart. This is like water in a reservoir – water at the surface has more energy than water at the bottom – that is why it can drive a turbine. Ten molecules together have lost about ten lots of energy in coming together. A molecule in the surface has lost less energy because it has joined together with fewer neighbours.

So a molecule in the surface has higher energy than one inside the liquid. What do things do in a position of high energy? They tend to go to a position of lower energy. In the case of gravity this is called falling. But not all the surface molecules can go inside – whatever happens, there has to be a surface. What actually happens is that the surface becomes very slightly depopulated, until all the molecules are in equilibrium.

So at, and very near, the surface of a water drop, the molecules are a little further apart than the normal equilibrium distance. So the surface behaves as if it is in a state of tension. And because this region has higher energy for a given area, it will tend to behave so as to minimize its area in any situation. For a free blob of liquid, the smallest area is obtained with a sphere. In more complicated cases the shape of the surface reflects the complexity of the situation. Click here

The drops in the middle of the picture show the spherical shape, while those on the wire are influenced by gravity, adhesion to the wire, and surface tension.

Larger drops of water that sit on a non-wetted surface are not spherical – the shape is the result of the combination of the surface effect with the force of gravity. Because some surfaces have an affinity for water, drops can hang from them.

Some liquid and solid combinations have little or no affinity. In these cases a liquid drop sits on the surface. But if there is a degree of affinity, the shape of the drop is modified. With strong affinity, the liquid wets the surface and spreads out as a thin layer. What matters is the relative energy of the three interfaces – air-liquid, liquid-solid, and air-solid. In practice, because of surface variations, impurities and foreign bodies, the situation is complex, and real drops may take shapes that cannot be computed, as you can see in some of the photographs above.

What are the shapes of water-drops in ideal circumstances? To some extent they reflect the symmetry of the situation. Very small drops floating in a space craft are close to being spherical, because there is no preferred direction, but as larger and larger drops are more and more affected by effects that can dominate the short-range inter-molecular forces. The kinetic energy and momentum of different parts of large drops makes them wobble around in a rather unstable manner. If the waves that travel around a drop should add in a suitable manner (Interference) the drop may split.

A similar effect is a part of the explanation of the finite list of chemical elements found on earth, which terminates at number 92, uranium. The particles in an atomic nucleus experience short range forces analogous to those between molecules in a liquid. So larger and larger nuclei can behave rather like liquid drops. Because the number of nucleons is so small, a large proportion of them lie in or near the surface. In uranium, about 40 or so lie in the surface, which is about one sixth of the total. In a drop of water, the number of molecules is unimaginably large, about 1.4 X 10¹⁴, or 140 million million. This is the number in a cube with 52000 molecules on a side. The number in a surface layer about one molecule deep is about 2.7 X 10⁹, which is a very small fraction, 1/52000, of the total.

The diagram below shows how the density of a uranium nucleus varies with the distance from the centre. The horizontal axis is in fermis, which are 10^-15 m.

When the big nucleus begins to divide into two smaller ones, enormous tidal effects are set up. Just as the gravitational attraction of earth and moon sets up tides in both, the electrostatic repulsion of the putative nuclei produces huge deformations. This happens because the force varies so strongly with the distance. If you are stretched out by the gravitational field of a black hole, it isn’t the strength of the field that gets you: it is the variation with position. The water of the oceans forms a prolate ellipsoid, bulging along the line joining earth and moon. Jupiter produces much bigger forces in its moons. The fleeing nuclei will be oblate, squashed along the line. As the stable states are spherical, the oval nuclei have a lot of energy to get rid of. They are neutron rich, and some energy may be shed in the form of spare neutrons.

When we get to the dimensions of boats, the effects of surface tension are obviously negligible. Birds keep the water out because the feathers are composed of very tiny parts, with tiny gaps, and so the effective size is small. By keeping the feathers oiled, they prevent them being wetted by the water, which cannot generate enough pressure to penetrate the gaps, because the required pressure is inversely proportional to the width of the gaps.

Cormorants and kingfishers have wettable feathers, and indeed young kingfishers can drown if they don’t learn how to dive and emerge properly. Cormorants spend a lot of time with outspread wings, drying off. Perhaps after millions of years, some of their descendents may have evolved natural oils.

Other considerations, such as quantum mechanics, modify the behaviour of nuclei, but a large enough nucleus can be quite unstable. Furthermore, some of the particles (protons) in a nucleus carry electric charge. These repel each other, like similar magnetic poles. Because the electric force has long range, each proton feels the effect of all the others, whereas it feels only the nearest neighbours for the nuclear force.

So adding particles to a nucleus doesn’t make it bind more strongly, while the electric repulsion gets bigger. So very large nuclei are very unstable. Large enough nuclei have such short lives that they have not survived the age of the earth, even if they were originally present. Nevertheless, during the last sixty years, scientists have been able to create those bigger nuclei and some that probably never existed on earth. An exciting recent development is the discovery of very heavy ones which have anomalously long lives. On the chart of nuclear types this would appear as an island of stability, which was predicted to exist many years ago. This has no counterpart in drops of water – it is a quantum mechanical effect. See also Blobs

Since it is the repulsion between protons which makes big nuclei unstable, it might be thought possible to make nuclei without them, using only the neutral particles – neutrons. One of the reasons why not is that neutrons are unstable, decaying into protons. They only survive in existing nuclei because their energy is effectively reduced in that environment.

But there is another island of stability, which does indeed comprise only neutrons. In a neutron star the force of gravity generated by a huge number of particles crushes everything together into an enormously dense mass. The electric charges are sent packing in the form of electrons – in effect the protons have decayed into neutrons. Surface tension plays no part here – the surface is a negligible part of the whole. Neutron stars are like no other object. Nature seldom repeats herself exactly.

If the symmetry is broken, drops are no longer spherical. The sticky blobs on a spider web have circular, but not spherical, symmetry because the threads are cylinders (Spider Webs). They also have mirror symmetry because the threads have no preferred direction. For water drops resting on a horizontal surface, gravity breaks the up-down symmetry, although the drops are circular in plan. The pressure in the drop increases slightly with the depth, and so the total radius of curvature decreases from top to bottom.

The next picture show approximate simulations for a range of drop sizes, for the case where liquid has no affinity at all for the solid surface. The tiniest drops are almost spherical, but the larger ones are a little less curved at the top.

If the density of the liquid increases, or the surface tension decreases, or we look at larger drops, the drops will be flatter for a given radius or volume. The second, third and fourth diagrams above show examples. The scales decrease from top to bottom in order to accommodate the greater widths. Eventually the drops do not become higher – they just become wider and flatter.

Here is a plastic barrel of grit for use in icy road conditions. The open end of the barrel has been distorted by the weight of the gravel into a shape which is reminiscent of a water drop. The shape differs for several reasons, for example, the tension is not the same all round the barrel, the system is based on a cylinder and not a sphere, and the shape is distorted by the closed end of the barrel, which retains its circular shape. But the relationship with a water drop is clear.

The photograph above shows very small drops that are nearly spherical, with a few larger ones which are noticeably flattened.

The photograph above shows very large drops tending to constant depth, except that in the photograph the angle of contact is different from the value in the diagram, because the water had some affinity for the surface. This is no problem, because we merely need to remove a slice from the bottom of each of the earlier simulations to see these shapes.

This is an example of the way that large and small things are not comparable. Small insects can ride on the surface tension of a pond, but large ones cannot. Nobody will ever make a boat that floats by surface tension. That doesn’t mean that going to sea in a sieve is impossible – you just need fine enough holes in an unwettable material.

Some insects, such as spring-tails and pond-skaters and whirligig beetles, can live on top of the surface of water. The surface is dimpled by their weight, rather like a trampoline, until it provides an upward force equal to the weight. Some insect larvae which live below the surface can push breathing apparatus through the surface, and cling there while they take in air. The breathing tubes often have feathery tips to increase the length of the perimeter without much increasing the weight. Surface tension, by definition, always produces forces in proportion to the length of any edges, and energies proportional to area.

This picture of a pond skater is poor, because it does not show the dimples in the surface. Ideally, the photograph should be taken from a position that shows the dimples by the way they distort the reflection of skylight. Note how four legs are used for support, with the ends almost horizontal, indeed curved upward at the ends, to get a good length of contact. The two forelegs are used to sense the vibrations made by insects that fall into the water. The skater can then approach them and suck out their fluids. Click on the picture to see a bigger version.

This piece of aluminium foil is about 12 cm wide. It is floating in a tray of water. A piece like this will float even if a part of the edge is pushed under the water.

The reason that surface tension affects only small objects is that it is confined to a layer only a few molecules thick. Its energy is proportional to the surface area. But other effects are often related to the volume. Multiplying the length of a shape by ten increases the area a hundred-fold, but the volume a thousand-fold.

The spider Argyroneta aquatica makes a net under the water, under which it traps air to make a home in which it can live and breed. The net needs only to be fine enough for surface tension to stop the air from getting through any holes. The curvature of a bubble or a drop is proportional to the pressure difference between the inside or the outside. If the hole is small enough, there won’t be enough pressure to make the small radius needed for a bubble to get through the hole. The pressure difference across a water-air surface is proportional to the curvature, that is, inversely proportional to the radius of curvature. So a small drop has a bigger pressure difference than a big one. The same is true of bubbles.

This is also how an umbrella or a tent works – the holes are too small to let the water through. Of course, the material must not be wetted by the water. If you have ever been lucky enough to see an Argyroneta making its bubble then you will probably never forget.

This is another example of nature’s versatility in the use of a simple material, in this case, spider web. Not only are there innumerable types in the air and on the ground, here is one in the water.

If we had a deep container with a bottom made of umbrella material, and we started to fill it, the pressure would eventually force the water through. Why not try it with an inverted umbrella, pouring the water in slowly and carefully.

Let’s see if we can get reasonable numbers for the effect of surface tension on diving birds. The diagram below shows the water pushing through between the barbules of the feathers, or between the threads of an umbrella or tent.

This idealised picture represents the point of no return, because the excess pressure across a curved surface decreases as the radius increases, which is what will happen if the water goes any further. So the calculation below is for the maximum depth of a bird in water.

The excess pressure at depth D in a liquid of density d is dDg, where g is the acceleration due to gravity. If we assume rectangular gaps of width w between the tiny parts of the feathers, then the excess pressure is 2T/w, where T is the surface tension of water. T is 0.070 N/m for pure water: we will use this in the absence of knowledge about water found in rivers, lakes and seas.

So we can set dDg = 2T/w for the maximum depth, giving the formula

Dw = 2T / dg = 2 x 0.07 / (1000 x 9.8),

Dw = 0.0000142.

If we express w in mm instead of metres we get

Dw = 0.0142.

So for a depth D of 1 metre, w = 0.0142 mm. If you look at a feather, do you think this is reasonable? You really need to look at the feather of a diving bird. Perhaps there is another possibility: the oil in the feathers might fill the gaps, and it might have a higher surface tension than that of water. Furthermore, as the bird dives down, the trapped air is compressed to a higher pressure. Do you think this helps to keep out the water?

If a liquid wets a surface, drops can hang, as shown below.

Here are some snail eggs. Between some of them you can see a viscous fluid that holds them together. The shapes are surfaces of minimum energy.

Why do you think raindrops hitting windows behave like those depicted below?

Why do the splashes on windscreen look different when the temperature is low, and the rain is almost sleet?

Do you think that the little spikes below are randomly spaced?

And this below is what randomness really looks like –

The next two diagrams show the distribution of distances between the little radial splashes, for the actual drop, and for a simulated one using random positions. The 37 distances were measured with a precision of one degree of arc, and then put in descending order of size. If the distances were all the same, the graph would be flat. The distribution in the second is an exponential, which is what you get with a random uniform distribution of positions, with statistical fluctuations. The distribution for an actual drop is nearly straight, and lies between a flat distribution and an exponential one.

The third graph is a linear combination of a flat graph and an exponential graph, which does not fit the actual shape very well, and is certainly not the correct way to combine them. But it does show that the actuality is far from random, as the shape is composed of 0.6 of the flat graph and 0.4 of the random one. What allows the behaviour of different parts of the drop to be correlated is the surface tension waves which travel around the surface. This picture was made using a drop of liquid from some pickled beetroot, falling on to paper. Perhaps a smooth surface like glass would have made an even more uniform splash.

You can see the waves on the surface of the blobs of liquid that astronauts make in their space-craft. Extremely tiny versions of the same type of waves run around the surface of very heavy atomic nuclei. In the heaviest nuclei, the repulsion of the electric charges is almost enough to overcome the surface tension. A slight input of energy makes waves which can be big enough to allow the nucleus to break in two, forming two smaller ones. Even the largest nucleus has less than 300 nucleons in it, so a significant proportion are "in the surface" – much more so than in a water drop. Like force between water molecules, the nuclear force has a short range, affecting mainly nearest neighbours.

The ocean is like a large drop of water, though it is almost filled by the earth inside it. It experiences waves, caused by the wind, and sometimes by earthquakes. There is a big rotating wave caused by the moon and the sun. None of these waves are connected with surface tension.

A black hole is a sort of large drop, with a surface which is so strong that nothing can escape. But it is unrelated to surface tension.

The largest known "drop" is the universe. This universe is very peculiar, because if information really cannot travel faster than light, we have problems in explaining how regions that are in space-like connection appear to have the same physical properties in every measurable way. If we make an electron-positron pair here, it will have, as far as we know, exactly the same properties as one made a million light-years away. The same is true of the spectrum lines of hydrogen and helium, which are detectable in the light from distant stars. Indeed, the only differences are the red-shifts which first gave a clue to the expanding universe.