Q4(b). Capillary action, contact angle and γ=12hρ gr

Capillary action and the contact angle

1. Capillary action: the rise or fall of a liquid in a narrow tube relative to the outside level. It comes from the competition between adhesion (liquid–glass) and cohesion (liquid–liquid), expressed through the contact angle θ between the liquid surface and the wall, measured through the liquid.
2. θ<90° (adhesion > cohesion; water, alcohol on clean glass): the liquid wets the wall, the meniscus is concave, and the liquid rises. For water on clean glass θ≈0.
3. θ>90° (cohesion > adhesion; mercury on glass): the liquid does not wet, the meniscus is convex, and the liquid is depressed (θ<0, h<0). At θ=90° the surface is flat and there is no change.

Derivation

1. The liquid surface meets the wall along a circle of length 2π r. Surface tension acts along the surface, at angle θ to the wall; its vertical component pulls the column up: Fᵤₚ=2π r γθ
2. The liquid rises until the weight of the raised column (meniscus volume neglected) balances this pull: W=mg=π r²h ρ g
3. At equilibrium: 2π rγθ=π r²hρ g, so γ=(hρ gr)/(2θ)
4. For a liquid that wets the glass completely, θ=0, θ=1: γ=12hρ gr
5. Same result from pressure: the concave surface of radius r has a pressure deficit 2γ/r beneath it (Laplace); the column rises until hρ g=2γ/r.

Example: water (γ=72.75 mN m⁻¹, ρ=998 kg m⁻³) in a tube of radius 0.10 mm rises h=2γ/ρ gr=14.9 cm; h∝1/r (Jurin's law).

Application. the capillary-rise method is a standard way to measure surface tension; the same action draws water up soil pores and plant xylem, ink into a fountain-pen nib and solvent up a TLC plate.