Lay a dry steel needle gently on the water in a bowl, and it stays on top. Steel is almost eight times as dense as water, so it ought to sink at once. On a village pond in the monsoon, a thin insect called a pond skater does the same trick. It runs across the water on six long legs, and its feet make small dents in the surface without breaking through. Push the needle under, and it goes straight to the bottom.
The pull of the neighbours
The molecules of a liquid attract one another. This attraction between molecules of the same kind is called cohesion. It is strong only over a very short distance, a few molecular diameters, about a nanometre. So each molecule feels the pull of the few molecules close around it, and no others.
Look at molecule A in Fig 1.1, deep inside the liquid. Neighbours surround it on every side, and they pull it equally in all directions. The pulls cancel, and A feels no net force. Now look at molecule B, in the top layer. It has neighbours beside it and below it, but only air above it, and the air has very few molecules. The sideways pulls cancel, but nothing balances the pulls from below. So B feels a net pull into the liquid.
Why a free drop is round
To bring a molecule from inside up to the surface, work must be done against this inward pull. So a molecule at the surface has more potential energy than one inside. Every piece of surface is a store of extra energy. A liquid, like any system left to itself, settles where its energy is least. It therefore makes its surface as small as it can.
For a given volume, the shape with the least surface area is a sphere. That is why a drop of water on a waxed leaf is nearly round. So are a drop of oil floating in water and a raindrop. Gravity flattens a large drop, but for a small drop the surface wins.
The surface of a liquid behaves like a stretched elastic skin. It tends to contract to the smallest area that its surroundings allow.
The skin is only a picture, and it differs from a rubber sheet in one way. Stretch rubber, and its pull grows. Stretch a liquid surface, and new molecules simply come up from inside to fill the new area. The pull per unit length stays the same, however much the surface grows.
Why does a small drop of liquid, falling freely, take the shape of a sphere?
Use the molecular picture to explain why a molecule at the surface of a liquid has more energy than a molecule deep inside.
A molecule deep inside is pulled equally on all sides by its neighbours, so no net force acts on it. A molecule at the surface has neighbours only beside and below it, so it feels a net pull into the liquid. To bring a molecule from inside to the surface, work must be done against this pull. That work is stored as extra potential energy, so a surface molecule has more energy than one inside.
A pull across every line
Imagine a line drawn on the surface of a liquid at rest. The surface on each side of the line pulls on the surface on the other side. The pull acts along the surface, at right angles to the line. It is a pull, never a push. A longer line carries a larger pull, in proportion to its length.
The surface tension \(T\) of a liquid is the force per unit length that acts across any line drawn in its surface. The force lies in the surface and is at right angles to the line.
Surface tension depends on the liquid and on its temperature. Water at room temperature has \(T \approx 0.072\,\text{N/m}\), and this book rounds it to \(0.07\,\text{N/m}\). Soap solution has about \(0.03\,\text{N/m}\). Mercury, whose atoms hold one another very strongly, has about \(0.5\,\text{N/m}\). These are small forces: a line \(1\,\text{m}\) long on water carries only \(0.07\,\text{N}\).
The U-frame and its slider
Bend a wire into a U and lay a light, straight wire across its two arms, so that it can slide along them. Dip the frame in soap solution and lift it out. A film fills the space between the frame and the slider. Let the slider go, and the film pulls it in, shrinking, as in the first chip of Fig 1.2.
The film has two surfaces, a front and a back. Each surface pulls on the slider with \(Tl\), where \(l\) is the length of the slider in contact with the film. So the total pull of the film is \(2Tl\). A weight \(W = 2Tl\) hung from the slider holds it at rest. For soap solution and a slider \(10\,\text{cm}\) long, \(2Tl = 2 \times 0.03 \times 0.10 = 6 \times 10^{-3}\,\text{N}\), the weight of about \(0.6\,\text{g}\).
The factor 2 belongs to films, not to liquids. A soap film, a bubble or the film on a U-frame has two surfaces, and each one pulls. A needle resting on water, a ring lifted from water or the rim of a column in a tube touches one surface.
A film of an unknown liquid on a U-frame holds a slider \(5\,\text{cm}\) long at rest against a weight of \(4 \times 10^{-3}\,\text{N}\). Find the surface tension of the liquid.
The film has two surfaces, so its pull is \(2Tl\). At rest, \(2Tl = W\).
\(T\) \(= \dfrac{W}{2l}\) \(= \dfrac{4 \times 10^{-3}}{2 \times 0.05}\) \(= 0.04\,\text{N/m}\).
A film of water on a U-frame pulls on a slider \(4\,\text{cm}\) long. Take \(T = 0.07\,\text{N/m}\). Find the pull of the film on the slider, in millinewtons.
The needle and the pond skater
Now the needle of the opening. Under the needle the surface dips into a dent, as in Fig 1.3. On each side, the surface meets the needle along its whole length \(l\) and pulls on it with \(Tl\), along the surface. The surface there is tilted at an angle \(\varphi\) to the horizontal, so each pull has an upward part \(Tl\sin\varphi\). Together the two sides lift the needle with \(2Tl\sin\varphi\).
A heavier needle pushes the dent deeper, and \(\varphi\) grows until the lift balances its weight. But \(\sin\varphi\) can never exceed 1. So the largest lift the surface can give is \(2Tl\), reached only if the surface at the needle turned vertical. For a needle \(5\,\text{cm}\) long on water, \(2Tl = 2 \times 0.07 \times 0.05 = 7 \times 10^{-3}\,\text{N}\). A needle of mass \(0.5\,\text{g}\) weighs \(5 \times 10^{-3}\,\text{N}\), and it floats. A needle of \(1\,\text{g}\) weighs \(10^{-2}\,\text{N}\), more than the surface can hold, and it sinks.
A needle rests on water, and the surface on each side makes \(30^\circ\) with the horizontal. A slightly heavier needle of the same length is put in its place. What happens?
A pond skater has a mass of about \(10\,\text{mg}\). Assume that the surface pulls up on each of its feet along both sides, as for the needle, with the surface nearly vertical. What total length of leg must touch the water, in millimetres, to hold it up? Take \(T = 0.07\,\text{N/m}\).
A ring pulled off the surface
Hang a thin wire ring of radius \(r\) from a spring balance so that it lies flat on water. Raise it slowly. The water clings to the ring and is drawn up into a short film. The film touches the ring along its inside edge and its outside edge, each a circle of length \(2\pi r\). So the surface pulls down on the ring along a length \(4\pi r\). Just before the film breaks, the surface is vertical, and the balance reads the weight of the ring plus \(4\pi r T\).
A thin wire ring of radius \(2\,\text{cm}\) hangs from a spring balance and touches a water surface. It is raised slowly. Find the extra force that the balance reads just before the film breaks. Take \(T = 0.07\,\text{N/m}\).
The water clings to the ring on both edges, inside and outside. Just before the film breaks, the surface pulls straight down.
The length in contact is \(2 \times 2\pi r = 4\pi r\), so the extra force is \(F = 4\pi r T\).
\(4\pi r = 4 \times 3.14 \times 0.02 \approx 0.25\,\text{m}\), and \(F = 0.07 \times 0.25 \approx 1.8 \times 10^{-2}\,\text{N}\).
The balance reads about \(1.8 \times 10^{-2}\,\text{N}\) more than the weight of the ring, the weight of about \(1.8\,\text{g}\).
A thin ring of radius \(3\,\text{cm}\) is pulled slowly off a water surface. Take \(T = 0.07\,\text{N/m}\). By how much does the pull needed exceed the weight of the ring, just before the film breaks? Give the answer in newtons.
Warming weakens the skin
Warm a liquid, and its molecules move faster and sit a little farther apart. Each molecule is then held less strongly by its neighbours, so the pull across a line in the surface falls. For water, \(T\) is about \(0.076\,\text{N/m}\) at \(0\,^\circ\text{C}\) and about \(0.059\,\text{N/m}\) at \(100\,^\circ\text{C}\). At the critical temperature of a liquid, the liquid and its vapour can no longer be told apart. There is no surface left, and \(T\) is zero.
Soap and the washing of clothes
Soap and detergent molecules have two ends. One end likes water, and the other avoids it. They crowd to the surface of water, with their water-avoiding ends pointing out, and they weaken the pull between the water molecules there. So soap lowers the surface tension of water, from \(0.07\) to about \(0.03\,\text{N/m}\).
A drop of plain water tends to sit on greasy cloth as a bead, holding itself together. Soapy water pulls itself together less strongly. It spreads into the narrow gaps between the fibres of a shirt and creeps under the grease and dirt. The water-avoiding ends of the soap molecules also grip the grease, which is then rinsed away. Warm soapy water, with a lower \(T\) still, washes best of all.
Oil on water
Put a drop of cooking oil on water in a bowl in the kitchen, and it spreads out into a wide, thin film. Look at the edge of the oil. The clean water surface outside pulls the edge outward with about \(0.07\,\text{N}\) per metre. The oil surface and the oil-water boundary pull it inward, together with less. So the edge is pulled outward, and the oil spreads until the film is very thin.
Which statements about the surface tension of water are true?
Mosquito larvae hang just below the surface of still water. Each one breathes through a small tube whose tip pokes through the surface, held there by surface tension. For many years health workers in India sprayed a thin film of oil on ponds and drains. The oil lowers the surface tension, the larvae can no longer hang from the surface, and they drown.
Agnes Pockels lived in Brunswick, Germany, and was not allowed to study at a university, which then closed its doors to women. Washing dishes at home, she noticed how soap and grease changed the surface of water. She built a long tin trough with a movable strip across it, to sweep the surface clean and to change its area. With a small disc hung from a balance, she measured the pull of the surface. In 1891 Lord Rayleigh sent her letter to the journal Nature, and her trough became a standard tool of surface science.
- Using \(F = Tl\) for a film on a U-frame.A film has two surfaces, so the pull is \(2Tl\).
- Thinking that stretching a liquid surface makes it pull harder.\(T\) stays the same as the surface grows; new molecules come up from inside.
- Taking the lift on a floating needle to be always \(2Tl\).The lift is \(2Tl\sin\varphi\); \(2Tl\) is only the largest possible value.
- Forgetting the inside edge of a ring.The water touches a thin ring on both edges, \(4\pi r\) in all.