Angle of contact, drops and bubbles

A liquid sticks to a solid if the solid-liquid surface energy is less than the sum of the solid-air and liquid-air surface energies.

Angle of contact θ: where liquid meets solid, draw the tangent to the liquid surface; θ is the angle from the solid surface to that tangent, measured through the liquid. It depends on the particular liquid-solid pair.

θ decides whether a liquid spreads or forms droplets: water beads up on a lotus leaf but spreads over a clean plastic plate.

Balancing the three interfacial tensions at the line of contact: S_la cos θ + S_sl = S_sa.

θ is obtuse when S_sl > S_la (water on a leaf, water on wax or oil, mercury on any surface): the liquid's molecules attract one another strongly and the solid's weakly, so the liquid does not wet the solid.

θ is acute when S_sl < S_la (water on glass or plastic, kerosene on almost anything): strong liquid-solid attraction lowers S_sl, cos θ grows and the liquid spreads.

Soaps, detergents and dyeing substances are wetting agents that make θ small so they penetrate well. Waterproofing agents make θ large between water and fibres.

For a given volume, the sphere has the least surface area, so free drops and bubbles are spherical when gravity and air resistance can be ignored.

Pressure inside a drop exceeds that outside. Growing a drop of radius r by Δr costs surface energy 8πrΔr S_la and gains work (P_i − P_o)4πr²Δr, so P_i − P_o = 2S_la/r.

Across any curved liquid-gas surface the concave side is at the higher pressure. An air bubble (cavity) inside a liquid has one surface: excess pressure 2S/r.

A soap bubble in air has two surfaces, inner and outer: P_i − P_o = 4S_la/r. That is why blowing a soap bubble needs a little extra pressure, but not too much.

Capillary of diameter 2.00 mm dipped 8.00 cm into water (S = 7.30 × 10⁻² N m⁻¹, g = 9.80 m s⁻², 1 atm = 1.01 × 10⁵ Pa), hemispherical bubble of radius 1.00 mm at its end: P_o = 1.01 × 10⁵ + 0.08 × 1000 × 9.8 = 1.01784 × 10⁵ Pa; excess pressure 2S/r = 146 Pa; P_i = 1.02 × 10⁵ Pa to three significant figures.