Capillarity & Contact Angle

Have you ever noticed how water seems to climb up a thin glass tube on its own, or how a drop of water beads up on a waxed car but spreads flat on clean glass? These everyday observations are governed by two closely related phenomena: capillarity (or capillary action) and the contact angle. Both arise from the interplay between cohesive forces — water molecules attracting each other — and adhesive forces — water molecules attracting the solid surface beneath them. Capillary action is the ability of a liquid to flow in narrow spaces without the assistance of external forces like gravity. It is why plants can draw water from the soil up through their roots and stems, why ink spreads through paper, and why a paper towel soaks up a spill. The narrower the tube, the higher a liquid can climb — sometimes to surprising heights. This behaviour is quantified by the Jurin's law formula, which links the height of capillary rise to surface tension, contact angle, tube radius, and liquid density. The contact angle θ is the angle measured through the liquid at the point where the liquid–vapour interface meets the solid surface. It determines whether a liquid 'wets' a surface (spreads out, θ < 90°) or 'beads up' (forms a compact droplet, θ > 90°). Water on glass has θ ≈ 0°, giving a concave meniscus and capillary rise, while mercury on glass has θ ≈ 140°, giving a convex meniscus and capillary depression. Understanding these angles is essential in fields ranging from waterproofing fabrics to designing microfluidic lab-on-a-chip devices.

Key takeaways

Surface Tension: The Skin of a Liquid
The Contact Angle and Young's Equation
Capillary Rise and Jurin's Law
Pressure Inside Drops and Bubbles
Angle of Contact in Practice

Surface Tension: The Skin of a Liquid

Surface tension γ (also written T or S) is the energy required to increase the surface area of a liquid by one unit. Molecules in the bulk of a liquid are attracted equally in all directions, but those at the surface have no neighbours above them — they are pulled inward, creating a tendency for the surface to contract like a stretched elastic sheet. Surface tension is measured in N/m (equivalently, J/m²). For water at 20 °C, γ ≈ 0.0728 N/m.

The Contact Angle and Young's Equation

When a droplet sits on a flat solid surface, three interfaces meet at the contact line: solid–liquid (SL), solid–vapour (SV), and liquid–vapour (LV). Thomas Young (1805) showed that equilibrium requires the balance of the three surface energies (surface tensions) along the horizontal: γ_SV = γ_SL + γ_LV cos θ. Rearranging gives cos θ = (γ_SV − γ_SL) / γ_LV. If γ_SV > γ_SL the cosine is positive, so θ < 90° and the liquid wets the surface. If γ_SV < γ_SL the cosine is negative, so θ > 90° and the liquid beads up. This equation is the foundation for all wetting science.

Capillary Rise and Jurin's Law

When a narrow tube of radius r is placed in a liquid that wets its walls (θ < 90°), the curved meniscus creates a pressure difference across the interface — called the Laplace pressure. For a spherical meniscus of radius R = r / cos θ, the Laplace pressure is ΔP = 2γ/R = 2γ cos θ / r. This excess pressure beneath the concave surface drives liquid up until the weight of the liquid column balances it: ΔP = ρgh, giving the capillary height h = 2γ cos θ / (ρgr). This is Jurin's Law. Notice h ∝ 1/r — halving the radius doubles the rise. For a non-wetting liquid (θ > 90°), cos θ < 0, so h is negative, meaning the liquid is depressed below the reservoir level, as seen with mercury in glass.

Pressure Inside Drops and Bubbles

A spherical liquid drop of radius R has one curved interface (liquid–air). The Young–Laplace equation gives the excess pressure inside: ΔP = 2γ/R. A soap bubble, however, has two surfaces (inner and outer walls of the soap film), so the excess pressure inside a bubble is ΔP = 4γ/R. This is why it takes more effort to blow a small bubble than a large one — the smaller the radius, the greater the internal pressure. This principle also explains why small alveoli in the lungs would collapse into larger ones without surfactant to lower γ.

Angle of Contact in Practice

Surfaces can be engineered to control θ. Hydrophobic coatings (like wax or PTFE/Teflon) raise θ above 90°, making water bead up and roll off. Superhydrophobic surfaces (like lotus leaves) achieve θ > 150° using microscale surface roughness combined with low surface energy, causing water droplets to bounce off almost perfectly. Conversely, hydrophilic coatings lower θ toward 0°, promoting spreading and adhesion — useful for paints, adhesives, and biological implants.

The Key Equation

h = \frac{2\gamma \cos\theta}{\rho g r}

h — Capillary rise height (m): Height the liquid rises (or is depressed) above the reservoir level; negative for non-wetting liquids.
\gamma — Surface tension (N/m): Energy per unit area of the liquid surface; for water at 20 °C, γ ≈ 0.0728 N/m.
\theta — Contact angle (degrees (°)): Angle between the liquid–vapour interface and the solid surface, measured through the liquid.
\rho — Liquid density (kg/m³): Mass per unit volume of the liquid; for water, ρ ≈ 1000 kg/m³.
g — Gravitational acceleration (m/s²): Standard value 9.81 m/s².
r — Tube radius (m): Inner radius of the capillary tube; smaller r gives greater rise.

Worked Example

Problem. A clean glass capillary tube of inner radius r = 0.20 mm is dipped vertically into water at 20 °C. Given γ = 0.0728 N/m, contact angle θ = 0°, and ρ = 1000 kg/m³, find: (a) the height to which water rises, and (b) the excess pressure inside a water drop of radius 1.5 mm.

h \approx 7.4\text{ cm};\quad \Delta P \approx 97\text{ Pa}

Answer. Water rises approximately 7.4 cm in the tube; the excess pressure inside the 1.5 mm drop is approximately 97 Pa.

Figure

Diagram

Origins · 1 minThomas Young and how this was discovered

Thomas Young

1773–1829 · British

Thomas Young formulated the equation that bears his name — Young's equation — in 1805, relating the contact angle at a solid–liquid–vapour interface to the three interfacial surface tensions. This provided the first rigorous mathematical description of wetting and laid the foundation for capillarity theory. Young was also responsible for the wave theory of light and the double-slit experiment demonstrating light interference. Young was such a polymath that he has been called 'the last person to know everything.' Beyond physics, he made critical contributions to deciphering Egyptian hieroglyphs on the Rosetta Stone and helped establish the science of Egyptology.

Surface Tension — Surface tension γ is the fundamental quantity driving both capillary rise and the formation of drops and bubbles; capillarity cannot be understood without it.
Viscosity and Fluid Flow — While surface tension drives capillary rise, viscosity determines how quickly the liquid climbs, making both properties essential for real capillary systems.
Pressure in Fluids — The Young–Laplace pressure difference across curved interfaces is the mechanism linking surface tension to capillary rise and the excess pressure in drops and bubbles.