# 17.43 Walking on water – surface tension The insect in the picture is walking on water. You can see that it isn’t simply floating. To float, part of it would have to be submerged so that the upward force that previously acted on the displaced water balanced the downward force (weight) exerted by gravity on the insect (post 17.6). Instead, it is as if the water were covered by a skin or membrane that it strong enough to support the insect but not something with a higher mass (like a frog). We can even see the little dents in this apparent skin, where it is pushed downwards by the insect’s feet.

The phenomenon that causes a liquid surface to behave in this way is called surface tension.

To think about the skin-like behaviour of the surface a little more, let’s think about the behaviour of a balloon filled with water. If we place a small object on the balloon it will be supported. The water is contained by the balloon and so cannot flow away. It then exerts an equal and opposite force to oppose the gravitational force (post 16.16) exerted by the object, following Newton’s third law of motion (post 17.26). The object will create a dent in the balloon in the same way that the insect’s feet create dents in the water surface. But a heavy object will burst the balloon – so it’s not supported.

What causes surface tension? There must be an attractive force between the molecules in any liquid, otherwise they would have complete freedom of movement and form a gas (post 16.37). However, the surface layer only has water molecules below it and so there is a resultant force pulling it down towards the bulk of the liquid. This is what a balloon does when it is full of water – inflating the balloon with a liquid stretches it. The stretched balloon is rather like a stretched spring (post 16.49) that stores energy that it would use to recoil to its original dimensions if it weren’t full of water. The force that would drive the recoil now pushes the water together. So surface water molecules are pulled into the bulk by the balloon just as the water molecules in a free surface are pulled into the bulk by forces between molecules. Think of the balloon as a skin or a membrane and we can see why molecules in a free surface of water act as if they were covered by a skin.

These arguments would apply to any liquid, so we can see that surface tension is a property of all liquids. Incidentally, the analogy between a balloon and a spring is not exact because the rubber of the balloon does not store all the energy used to deform it – it’s behaviour is part way between the elastic behaviour of a spring (post 16.49) and the viscous behaviour of a fluid (post 17.17) – we say that it is viscoelastic. Now let’s return to our insect. If it has a mass m, the gravitational force acting on it has a magnitude mg where g is the magnitude of the gravitational field (posts 16.16 and 17.24). Let’s suppose that the dents under its feet are tiny hemispheres each of radius r. Now the dent is surrounded by a circle of radius r on the surface; the circumference of this circle is 2πr. So we can think of the distorted surface as generating an upward force of 2πrT, on each foot, where T is the force exerted by a unit length of surface. The value of T is the numerical value of surface tension and it is measured in the units of force (N) divided by the units of length (m) (post 16.13) or N.m-1.

The insect is supported when

mg = 2πnrT

where there are n legs in contact with the liquid. For an insect n is usually equal to 6. Then our simple theory predicts that a dent will have a radius of

r = mg/(12πT).

A pondskater (an insect that walks on water) has a mass m = 2.5 ×10-5 kg, g = 9.8 m.s-2 (post 16.16) and for water T = 7.3 × 10-2 N.m-1. The equation above then predicts that r has a value of about 0.1 mm. The hemisphere is the least shallow dent that the insect can make. The picture above shows that, for a more shallow dent,

r = mg/(12πTcosϑ)

where ϑ characterises the shallowness of the dent. Since cosϑ has a maximum value of 1, 0.1 mm is the predicted minimum radius for a dent.

The first picture shows dents that appear to be more shallow than a hemisphere, so it seems likely that the radius of a dent will be around several tenths of a millimetre.

If we think of the size of a pondskater (length a few millimetres), our theory seems provide a reasonable prediction for the size of the dents in the first picture. The table above gives the values of surface tension for a selection of liquids. You can see that water has a high value. This isn’t surprising because of the strong attractive forces between water molecules caused by hydrogen bonds (post 16.45).

In post 16.45, we found that hydrogen bonds help keep fish alive in winter; now we find that they help pondskaters walk on water!

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