Angle of contact and capillary rise

The angle of contact

Put a drop of water on clean glass and it spreads into a flat cap. Put a drop of mercury on glass and it stays almost round. The angle of contact $\theta$ is the angle between the solid and the liquid's surface, measured through the liquid.

• $\theta < 90^\circ$: the liquid's molecules are pulled more by the solid than by each other. It wets the solid (water on glass, about $0^\circ$).
• $\theta > 90^\circ$: they are pulled more by each other. It does not wet the solid (mercury on glass, about $140^\circ$; water on wax).

Capillary rise

Stand a thin glass tube in water, and the water climbs up inside it. Surface tension pulls along the edge of the curved surface, all round the tube, and lifts a column until its weight balances the pull:

You should find that a tube half as wide lifts water twice as high, and that above $90^\circ$ the liquid is pushed down instead.

Watch out. If the tube is shorter than $h$, the water does not overflow. It reaches the top and its surface flattens, so the pull matches the shorter column.

Where it matters

• A lamp's wick and blotting paper are bundles of fine channels that draw liquid up.
• Water rises through the soil to plant roots.
• Detergents lower $S$ and the angle of contact, so water creeps into the fibres of cloth.
• Waterproofing makes the angle of contact large, so water beads up and runs off.

A worked example

How high does water climb in a glass tube of radius $0.15\ \text{mm}$? And what does mercury do in a tube of radius $0.50\ \text{mm}$ ($S = 0.465\ \text{N/m}$, $\theta = 140^\circ$)?

1. Water. $h = \dfrac{2 \times 0.072\ \text{N/m}}{1000\ \text{kg/m}^3 \times 9.8\ \text{m/s}^2} \times \dfrac{1}{1.5 \times 10^{-4}\ \text{m}} = 0.098\ \text{m}$, about $9.8\ \text{cm}$.
2. Mercury. $\cos 140^\circ = -0.766$, so $h = \dfrac{2 \times 0.465\ \text{N/m} \times (-0.766)}{13\,600\ \text{kg/m}^3 \times 9.8\ \text{m/s}^2} \times \dfrac{1}{5.0 \times 10^{-4}\ \text{m}} = -0.0107\ \text{m}$.
3. Meaning. The mercury stands $1.07\ \text{cm}$ below the level outside.

Practise with new numbers

Try it in Python

The program works out both tubes of the example.

Remember these

• $\theta < 90^\circ$: wets; $\theta > 90^\circ$: does not wet.
• $h = \dfrac{2S\cos\theta}{\rho gr}$: thin tubes lift higher; obtuse angles push down.
• A short tube never overflows.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Why does the oil of a lamp rise in its wick?

2. Water climbs $4.0\ \text{cm}$ in one tube. How high in a tube of half the radius?

3. Why is the mercury level inside a thin glass tube lower than outside?

4. Why do waterproof jackets keep rain out?