Surface tension

Molecules of a liquid attract one another (cohesion); they also attract the molecules of a solid they touch (adhesion). Water clings to glass, so a wet glass stays wet; mercury does not.

Deep inside a liquid, a molecule is pulled equally in all directions. A molecule at the surface has no liquid above it, so it is pulled inwards. The surface therefore behaves like a stretched elastic sheet that tries to shrink to the least possible area. That is why a free drop is a sphere, and why a needle can rest on water.

Surface tension S is the force per unit length along a line drawn in the surface, in N m⁻¹. It is also the extra energy per unit area of surface: to make more surface you must do work S × (increase in area).

A soap film has two surfaces (front and back). So a film on a frame pulls on a slider of length L with 2SL, and blowing a soap bubble from r₁ to r₂ takes work 2 × S × 4π(r₂² – r₁²). With S = 0.03 N m⁻¹, going from 2 cm to 3 cm needs 2 × 0.03 × 4π × (0.0009 – 0.0004) ≈ 3.8 × 10⁻⁴ J.

A curved surface means the pressure is higher on the concave (inner) side:

• A drop, or a gas bubble inside a liquid, has one surface: excess pressure 2S/r.
• A soap bubble has two surfaces: excess pressure 4S/r. A soap bubble of radius 1 cm with S = 0.03 N m⁻¹ has 4 × 0.03/0.01 = 12 Pa more pressure inside.

The angle of contact θ is the angle between the solid and the liquid surface, measured inside the liquid. Water on clean glass has θ about 8° (acute): water wets glass and its surface curves up. Mercury on glass has about 138° (obtuse): it does not wet, and its surface curves down. Soaps and detergents lower θ, so water can soak into cloth.

In a thin tube dipped into water, the water rises: the pressure just under its curved surface is lower, and the column climbs until its weight makes up the difference. The capillary rise is h = 2S cos θ/(ρga), where a is the tube's radius. Example: water (S = 0.073 N m⁻¹, θ ≈ 0) in a tube of radius 0.5 mm rises 2 × 0.073/(1000 × 9.8 × 0.0005) ≈ 3 cm. In mercury, cos θ is negative, so the level is pushed down.

Surface tension falls as the liquid warms. Dirt, oil and soap lower it; that is why a piece of camphor darts about on clean water as it dissolves unevenly.