Contact Angle & Capillary Rise
Water climbs a thin glass tube; mercury sinks in it.
Lesson at a glanceShow the key points
- Contact angle θ: between the solid surface (drawn into the liquid) and the liquid surface (drawn away from the solid) at the line of contact. Water–glass ≈ 0°, mercury–glass ≈ 140°.
- Cohesive force: between like molecules; adhesive: between unlike ones. The liquid surface sets itself perpendicular to the resultant force on the surface layer.
- θ < 90°: liquid rises along the solid (concave meniscus); θ > 90°: it is depressed (convex).
- Capillary rise h = 2S cos θ/(rρg) (14.17); for θ = 0 with the meniscus included, h = 2S/(rρg) − r/3 (14.18).
- A tube shorter than h does not overflow; the contact angle adjusts instead.
Dip the corner of a paper towel in water and the water climbs. Stand a thin glass tube in water and the water rises inside it, the higher the narrower the tube. In mercury, the same tube shows the opposite.
The contact angle
Where a liquid surface meets a solid, it curves. Dip a glass plate in water and the water near it curls upward, as if pulled up; dip it in mercury and the surface curls down.
Liquids that rise along the solid have contact angles below 90°; liquids that are depressed have angles above 90°.
| Pair | Contact angle | Pair | Contact angle |
|---|---|---|---|
| Water–glass | 0° | Water–paraffin wax | 107° |
| Mercury–glass | 140° | Methylene iodide–glass | 29° |
Why the surface bends
A liquid at rest can’t sustain tangential stress, so the resultant force on any small piece of its surface layer must be perpendicular to the surface. Near the solid, a piece of the surface feels:
- the adhesive force Fₛ from the nearby solid molecules (force between unlike molecules), pointing into the solid;
- the cohesive force F_L from the neighbouring liquid molecules (force between like molecules), pointing into the liquid;
- its weight.
The surface settles perpendicular to their resultant. If the resultant points into the solid, the surface is concave upward and the liquid climbs the wall; if it points into the liquid, the surface is convex and the liquid is depressed.
Capillary rise
Dip a narrow (capillary) tube of radius r in a liquid of density ρ, surface tension S and contact angle θ. The surface inside is nearly spherical. Consider the column of liquid raised to a height h:
- Along the contact circle (length 2πr) the tube pulls the liquid surface with force S per unit length along the liquid surface. The vertical components add up to ; horizontal ones cancel.
- Air pressure on top and liquid pressure at the bottom (at the level of the outside surface) are both atmospheric, so they cancel.
- The weight of the raised column (ignoring the meniscus) is .
How high does water rise in a clean glass tube of radius 0.30 mm? (S = 0.075 N/m, θ = 0, g = 10 m/s²)
A glass tube of radius 0.50 mm stands in mercury (S = 0.465 N/m, θ = 140°, ρ = 13 600 kg/m³). Find the change in level inside (g = 10 m/s²).
- 1Contact angle θ < 90°: liquid wets the solid and rises; θ > 90°: it is depressed.
- 2The surface sets perpendicular to the resultant of adhesive, cohesive and weight forces.
- 3Capillary rise ; narrower tubes, higher rise.
- 4A short tube doesn't overflow; the contact angle adjusts.
Check yourself
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