Surface tension and capillarity

Young–Laplace, contact angle, wetting, meniscus, surface free energy, capillary rise.

Every interface between two fluids carries energy. A molecule at the surface of a liquid has fewer neighbours than one in the bulk and therefore higher energy; creating new surface costs work; minimising surface area is the equilibrium criterion. This is the operative force behind a bubble’s pressure jump, a drop’s spherical shape, a meniscus in a capillary, and the cavitation-relevant Young–Laplace condition.

Surface tension as a coordination deficit

A molecule deep inside a liquid is surrounded by ~12 neighbours in 3-D close packing — every direction has another molecule pulling on it. A molecule at the surface, by contrast, loses about half of those neighbours on the open side.

Molecules in the bulk are surrounded by ~6 nearest neighbours (in 2-D hex packing; ~12 in 3-D close packing). Molecules at the surface lose ~half of their neighbours on the open side, costing roughly 2-3 missing bonds of energy ε each. Multiplied by the surface number density (one molecule per σ²), this gives the surface tension γ ∼ ε/σ². For water, ε ∼ 0.25 eV (hydrogen bond) and σ ∼ 0.3 nm gives γ ∼ 70 mN/m — matching the measured 72 mN/m.

Each missing bond costs an energy of order $\varepsilon$ (the pair-interaction energy from the intermolecular-forces chapter). Summed over the surface number density $\sim 1/\sigma^2$ , this gives a surface energy per unit area

\gamma \;\sim\; \frac{\varepsilon}{\sigma^2}.

For water with $\varepsilon \approx 0.25\,\text{eV}$ (hydrogen-bond energy) and $\sigma \approx 0.3\,\text{nm}$ , this is $\sim 70\,\text{mN/m}$ — matching the measured 72 mN/m. The thermodynamic definition formalises this: $\gamma = (\partial G/\partial A)_{T, V, N}$ — the reversible work required to create unit area of interface.

Equivalently, $\gamma$ is the in-plane force per unit length acting along any cut in the surface. The two definitions are equivalent because increasing surface area by stretching it against this tension does work proportional to that area. Units: $\text{N/m} = \text{J/m}^2$ .

The Young–Laplace pressure jump

A spherical interface concentrates surface energy. Shrinking a bubble of radius $R$ by $dR$ reduces its area by $8\pi R\, dR$ , freeing energy $-8\pi R \sigma\, dR$ . By work-energy bookkeeping, this energy must come from somewhere — and it comes from the pressure difference doing work on the changing volume $-4\pi R^2\, dR$ .

▶ Deriving Δp = 2σ/R

For a spherical bubble of radius $R$ , expanding by $dR$ adds $dA = 8\pi R\,dR$ of surface and $dV = 4\pi R^2\, dR$ of volume. The free-energy change is

dG \;=\; \sigma\, dA \;-\; \Delta p\, dV

where $\Delta p = p_\text{inside} - p_\text{outside}$ . At equilibrium $dG = 0$ :

\sigma \cdot 8\pi R\, dR \;-\; \Delta p \cdot 4\pi R^2\, dR \;=\; 0 \quad\Longrightarrow\quad \Delta p \;=\; \frac{2\sigma}{R}.

For a general doubly-curved interface with principal radii $R_1$ , $R_2$ :

\Delta p \;=\; \sigma\!\left(\frac{1}{R_1} + \frac{1}{R_2}\right).

For a soap film with two interfaces, both contribute: $\Delta p = 4\sigma/R$ .

Bubble radius R = 1.00 mm Surface tension σ = 72.0 mN/m Soap bubble (two interfaces — uses 4σ/R)

Presets:

The pressure inside a curved interface is always higher than outside by Δp = 2σ/R (or 4σ/R for a thin film with two interfaces). The R⁻¹ scaling makes microbubbles enormously over-pressurised: a 10 μm vapour bubble in water carries 14 kPa of extra pressure inside, while a 1 cm raindrop has only 14 Pa. This is why surfactants — which drop σ from 72 to ~25 mN/m at the alveolar wall — make breathing energetically feasible.

The $1/R$ scaling makes microbubbles enormously over-pressurised. A 10 μm bubble in water at $\sigma = 72\,\text{mN/m}$ carries 14 kPa overpressure; a 1 cm raindrop only 14 Pa.

Contact angle and Young’s equation

At a three-phase contact line where a liquid drop meets a solid surface in air, three interfacial tensions meet: $\sigma_{sg}$ (solid-gas), $\sigma_{sl}$ (solid-liquid), $\sigma_{lg}$ (liquid-gas). Horizontal force balance along the surface gives Young’s equation:

\cos \theta_c \;=\; \frac{\sigma_\text{sg} - \sigma_\text{sl}}{\sigma_\text{lg}},

where $\theta_c$ is the contact angle measured through the liquid.

cos θ_c0.417

θ_c65.4°

σ_sg (solid–gas) = 70.0 mN/m σ_sl (solid–liquid) = 40.0 mN/m σ_lg (liquid–gas) = 72.0 mN/m

At the contact line, three surface tensions pull along their respective interfaces. Horizontal force balance reads σ_sg = σ_sl + σ_lg cos θ_c. Solving for θ_c: when σ_sg > σ_sl, the drop wets and θ < 90° (hydrophilic); when σ_sl > σ_sg, the drop beads and θ > 90° (hydrophobic). If |σ_sg − σ_sl| > σ_lg the equilibrium contact angle vanishes (total wetting or non-wetting).

Three regimes:

$\theta_c < 90°$ — hydrophilic. Drop spreads.
$\theta_c > 90°$ — hydrophobic. Drop beads.
$|\sigma_{sg} - \sigma_{sl}| > \sigma_{lg}$ — no equilibrium angle; total wetting or total non-wetting.

Water on clean glass has $\theta_c \approx 0$ ; water on Teflon has $\theta_c \approx 110°$ . The contact angle is the macroscopic shadow of microscopic adhesive vs. cohesive energetics.

The capillary length

In the presence of gravity, a free liquid surface adopts a curvature that balances the Laplace pressure against the hydrostatic head $\rho g h$ . The crossover length where these two effects are comparable is the capillary length:

\ell_c \;=\; \sqrt{\frac{\sigma}{\rho g}}.

σ72.0 mN/m

ρ1000 kg/m³

ℓ_c = √(σ/ρg)2.71 mm

Surface tension σ = 72.0 mN/m Density ρ = 1000 kg/m³

The capillary length is the crossover scale where surface-tension force (per unit length, σ) balances the hydrostatic pressure from a head of height ℓ (ρg ℓ²). Below ℓ_c, drops are nearly spherical and capillary effects matter; above it, drops flatten and gravity dominates. For water, ℓ_c ≈ 2.7 mm; for mercury, ~1.9 mm. This is the natural scale of every meniscus, raindrop, and small bubble.

For water in air, $\ell_c \approx 2.7\,\text{mm}$ . On scales much smaller than $\ell_c$ — within a drop, inside a capillary tube — surface tension dominates and gravity is negligible. On scales much larger, gravity wins. This is why small raindrops are nearly spherical but large puddles are nearly flat.

Capillary rise (Jurin’s law)

Insert a thin tube of inner radius $r$ vertically into a wetting liquid. The liquid climbs until the weight of the lifted column balances the Laplace pressure across the curved meniscus:

▶ Deriving Jurin's law

The meniscus at the top of the column has radius of curvature $R = r/\cos\theta_c$ (the meniscus is part of a sphere intersecting the tube wall at angle $\theta_c$ ). The Laplace pressure pulling the liquid upward is therefore $2\sigma/R = 2\sigma\cos\theta_c/r$ .

This must equal the hydrostatic pressure of the lifted column: $\rho g h$ . Solving for $h$ ,

h \;=\; \frac{2\sigma\cos\theta_c}{\rho g r}.

r0.50 mm

σ72.0 mN/m

θ_c20°

h = 2σ cos θ / (ρ g r)27.59 mm

Tube radius r = 0.50 mm Surface tension σ = 72.0 mN/m Contact angle θ_c = 20°

Liquid climbs a thin tube against gravity until the weight of the lifted column balances the Laplace pressure across the curved meniscus. The result, *Jurin's law*: h = 2σ cos θ_c / (ρ g r). Halving the tube radius doubles the rise. Hydrophilic surfaces (θ_c < 90°) cause positive rise; hydrophobic surfaces (θ_c > 90°) cause *depression* (h < 0; the mercury-in-glass case).

Halving $r$ doubles $h$ . Hydrophilic surfaces ( $\theta_c < 90°$ ) give positive rise; hydrophobic surfaces ( $\theta_c > 90°$ ) give depression (mercury in glass). The Cavitation book’s tensile-strength chapter builds on this to argue that pure water can in principle sustain tensions of tens of MPa in a sufficiently fine capillary system.

Stability of crevice-trapped gas

A small gas pocket trapped in a crevice on a solid surface is the canonical heterogeneous-nucleation site. Whether the pocket is stable under applied tension depends on the contact-angle / crevice-geometry combination.

contact angle θ_c110°

crevice half-angle β60°

θ + β170°

criterion θ+β > 90°✓ stable

Contact angle θ_c = 110° (hydrophobic) Crevice half-angle β = 60°

A gas pocket inside a crevice is the canonical cavitation-nucleation site. Whether the liquid floods the crevice or the gas pocket remains stable depends on the contact-angle / geometry combination θ_c + β > 90°. Hydrophobic crevices (θ_c large) trap gas trivially; hydrophilic crevices (θ_c small) require a very narrow opening (β large) to hold gas. Real solids have many crevices spanning a range of geometries — and the weakest (most easily flooded, most easily nucleating) determines the practical tensile-strength threshold of the liquid.

For a V-shaped crevice of half-angle $\beta$ , the meniscus is stable when $\theta_c + \beta > 90°$ . Hydrophobic crevices trivially trap gas; hydrophilic crevices require a very narrow geometry to do so. The Cavitation book’s heterogeneous & crevices chapter develops this in full.

Dimensionless numbers — Weber, Capillary, Bond

When surface tension competes with other forces, dimensionless groupings name the regimes:

Weber number $\mathrm{We} = \rho U^2 L/\sigma$ — inertia vs. surface tension. High $\mathrm{We}$ : splash; low $\mathrm{We}$ : coalescence.
Capillary number $\mathrm{Ca} = \mu U/\sigma$ — viscous vs. surface tension. Coating-flow regimes.
Bond number $\mathrm{Bo} = \rho g L^2/\sigma = (L/\ell_c)^2$ — gravity vs. surface tension. Interfacial-shape diagnostic.

All three are catalogued in the scaling chapter.

⏳ The history — From Young's contact angle to the alveolar surfactant problem

Thomas Young in 1805 published two papers introducing what we now call surface tension and the contact-angle relation that bears his name. He computed (without modern thermodynamics) the equilibrium shapes of menisci and droplets. Pierre-Simon Laplace gave the curved-interface pressure jump in 1806 in Mécanique céleste; combining the two results gave nineteenth-century physics its mature theory of capillarity.

The most consequential modern application is medical. Kurt von Neergaard noticed in 1929 that the pressure required to inflate excised lungs with air was several times the pressure required to inflate them with liquid. The discrepancy implied a substantial surface-tension force at the air-liquid interface inside the alveoli. The “something” the body uses to manage it is pulmonary surfactant, a phospholipid–protein mixture secreted by type II alveolar cells that drops the air-liquid surface tension from water’s 72 mN/m down to 1–5 mN/m at small alveolar radii. Without it, the Young–Laplace pressure across a 100 μm alveolus would be ≈ 1.4 kPa — beyond the muscles available to draw breath. Premature infants who have not yet started producing surfactant suffer respiratory distress syndrome; surfactant-replacement therapy, introduced in the 1980s, transformed neonatal medicine.

For the cross-book applications — cavitation crevice nucleation, alveolar surfactant, bubble pressure inside Rayleigh-Plesset — see the key examples sub-page.