Surface Tension

Why a liquid surface is special

Take any molecule in the bulk of a glass of water. It is surrounded on every side by other water molecules, each pulling on it with hydrogen-bonding and van der Waals forces. By symmetry, the net force is zero — the molecule is happy. Now take a molecule at the surface. It still has water beneath and beside it, but above it sits open air with a thousandth the density. The forces no longer balance. The surface molecule is pulled inward, and creating new surface area requires energy — energy that has to come from somewhere.

That energy cost per unit area is what we call surface tension γ. It can be measured in two completely equivalent ways. As an energy: γ is the work in joules required to enlarge the surface by one square metre, so γ has units J/m². As a force: γ is the inward force per unit length acting along any line drawn on the surface, so γ has units N/m. The numbers come out the same: water at 20°C has γ = 0.0728 N/m = 0.0728 J/m². Both pictures are correct, and which one is more useful depends on whether you are computing forces or energies.

Because surface tension is a force per length rather than per area, its effects scale strongly with size. A 1 mm drop has surface area 4π mm² and weight only a few microgrammes. The surface forces holding it spherical (γ × perimeter ≈ γ × few mm) absolutely dominate gravity. A 10 cm drop, by contrast, has weight in grams and surface forces still only γ × cm — gravity wins easily and the drop spreads. The crossover length is the capillary length ℓ_c = √(γ/ρg), about 2.7 mm for water. Below ℓ_c, surface tension rules; above it, gravity rules.

Surface tension of common liquids

Different liquids have wildly different surface tensions, set by the strength of their intermolecular forces. Water is unusually high for a small molecule because of hydrogen bonding; mercury is extraordinary because of metallic cohesion; non-polar oils and alcohols sit much lower.

Liquid	Temperature	γ (mN/m)	Comment
Mercury	20°C	486	Metallic bonding, beads on glass
Water	20°C	72.8	High due to hydrogen bonding
Water	100°C	58.9	Decreases with temperature
Glycerol	20°C	63.4	Multiple hydrogen bonds
Olive oil	20°C	33	Non-polar, weak cohesion
Ethanol	20°C	22.0	Lower than water — wets more easily
Soapy water	20°C	25–35	Surfactants drop γ dramatically
Liquid helium	2 K	0.37	Minimal cohesion at quantum-fluid scale

Surface tension always decreases with temperature because thermal motion partially undoes the cohesive forces. At the critical temperature of a liquid–vapour system, γ goes smoothly to zero — the interface vanishes and we have a single supercritical phase. For water this happens at 374°C; the linear extrapolation works decently as a back-of-envelope tool.

Worked example: how high does water climb in a xylem capillary?

Plants draw water from roots to leaves through narrow xylem channels. The Jurin formula gives the equilibrium capillary rise:

h = 2γ cos θ / (ρ g r)

γ      = 0.0728 N/m   (water at 20°C)
cos θ  = 1            (water perfectly wets clean cellulose, θ ≈ 0°)
ρ      = 1000 kg/m³
g      = 9.81 m/s²
r      = capillary radius

For a typical xylem vessel of radius r = 25 μm = 2.5×10⁻⁵ m:

h = 2 × 0.0728 × 1 / (1000 × 9.81 × 2.5×10⁻⁵)
  = 0.1456 / 0.245
  = 0.594 m

So a 25 μm xylem capillary supports about 60 cm of water column by surface tension alone. That is enough for grasses but not enough for trees. Plants reach much higher heights — California redwoods exceed 100 m — by using the much smaller pore sizes inside cell walls and tracheid pits, with effective radii down to 5 nm. Re-running the calculation:

r = 5×10⁻⁹ m
h = 2 × 0.0728 / (1000 × 9.81 × 5×10⁻⁹)
  = 0.1456 / 4.9×10⁻⁵
  = 2970 m

Three kilometres in principle. The actual transpiration mechanism uses these tiny menisci as anchors and cohesion within the water column to lift the entire continuous fluid thread under tension. The water in a tall tree's xylem is genuinely under negative absolute pressure — sometimes −2 MPa — and remains a coherent column thanks to hydrogen-bond cohesion plus surface-tension anchoring at the menisci.

The Young–Laplace pressure jump

A curved fluid surface has higher pressure on its concave side than its convex side. The Young–Laplace equation makes this quantitative:

ΔP = γ (1/R₁ + 1/R₂)

For a spherical drop of radius R, both principal radii are R, so ΔP = 2γ/R. For a soap film bounded on both sides by air, the surface counts twice and ΔP = 4γ/R. The implications cascade everywhere through fluid statics:

A 1 mm-radius water drop has internal overpressure ΔP = 2 × 0.0728 / 0.001 = 146 Pa above atmospheric. A 1 μm cloud droplet has 1000× more, 0.15 MPa = 1.5 atm of internal overpressure — which is why small droplets evaporate faster than large ones at the same temperature. This pressure-driven Ostwald ripening drives the growth of large raindrops at the expense of small ones inside any cloud.

Soap films stretched on wire frames settle into shapes with zero mean curvature — minimal surfaces — because any curvature would imply a non-zero pressure jump that the air on both sides cannot supply. The catenoid between two parallel rings, the helicoid, the Schwarz P-surface and other beloved geometric objects of differential geometry are all real-world physics: let a soap film find its equilibrium and you get a minimal surface.

Wetting and contact angles

When a liquid drop sits on a solid, three interfaces meet at the drop edge: liquid–vapour (γ_lv), solid–vapour (γ_sv) and solid–liquid (γ_sl). Force balance along the solid gives Young's equation:

γ_sv = γ_sl + γ_lv cos θ
⇒ cos θ = (γ_sv − γ_sl) / γ_lv

Three regimes follow. If γ_sv is much larger than γ_sl (a clean, high-energy surface like glass), cos θ approaches 1 and the drop spreads — water on glass has θ ≈ 5°. If γ_sl is large (a low-energy surface like wax), cos θ can be negative, the drop beads up, and θ can exceed 100°. If γ_sv − γ_sl exceeds γ_lv, Young's equation has no solution and the drop spreads completely into a film — this is "perfect wetting".

Engineered superhydrophobic surfaces (lotus leaves, rose petals, treated glass) push θ well past 150° by combining low surface energy with micro-textured roughness that traps air pockets under the drop. The effective contact area drops by an order of magnitude and the drop sits on top of the texture, almost a sphere, ready to roll off at the slightest tilt. This is the lotus effect, and it self-cleans by sweeping dust away with rolling drops.

Where surface tension shows up

• Water-walking insects. The water strider supports its 5–10 mg body on legs with thousands of micro-hairs spaced ~2 μm apart, giving an effective contact angle >160°. The combined dimple perimeter (~10 cm) at γ = 73 mN/m yields a supporting force of about 7 mN — over 100× the insect's weight, which is why striders survive even on choppy ponds.
• Soap-film minimal surfaces. Architects use soap-film analogues to design tensile membrane roofs. Frei Otto's Olympic Stadium in Munich (1972) and the Sidney Olympic Aquatic Centre were both prototyped using actual soap films stretched on scale wire frames before being computed digitally.
• Inkjet printing and microfluidics. Drop-on-demand printheads eject droplets 10–50 μm in diameter at 10⁵ drops per second. Drop break-up from the nozzle is governed by the Plateau–Rayleigh instability, which is a direct surface-tension phenomenon — a cylindrical jet is unstable to wavelengths longer than its circumference.
• Lung surfactant. Pulmonary surfactant (a phospholipid mixture from type II alveolar cells) reduces alveolar surface tension from ≈70 mN/m down to ≈5 mN/m at end-expiration. Without it, the smallest alveoli would collapse and feed the larger ones (Young–Laplace ΔP ∝ 1/R favours collapse). Premature infants with insufficient surfactant develop respiratory distress syndrome — the leading historical cause of premature mortality before synthetic surfactant was introduced in the 1980s.
• Capillary rise in plants and soils. A 1 μm soil pore can lift water about 15 m by surface tension alone. This is how moisture moves from groundwater up into the root zone after rain, and how paper towels work — every fibre is a bundle of micron-scale capillaries.

Marangoni convection: surface tension as flow driver

Surface tension γ depends on temperature and on dissolved substances. Any gradient in γ across a fluid surface produces a tangential stress that drives flow from low-γ regions to high-γ regions — the Marangoni effect. The most famous everyday example is "tears of wine": as alcohol evaporates faster at the meniscus of a wineglass, the local γ increases, and water-rich liquid is pulled up the glass, eventually getting heavy enough to drop back down as visible droplets.

Marangoni effects are dominant in low-gravity environments where buoyancy disappears. Spacecraft fluid handling, welding pool dynamics, semiconductor crystal pulling and thin-film coating are all governed by Marangoni rather than buoyant convection. A drop of soap dropped onto a water tank covered with pepper grains demonstrates the effect dramatically: the soap reduces γ locally, and the surrounding higher-γ water yanks the surface (with the pepper) outward at speeds of metres per second.

Surfactants: molecules that gather at the surface

A surfactant is a molecule with a polar head and a non-polar tail (or any analogous amphiphilic structure). Dropped into water, surfactant molecules migrate to the air–water interface and orient with heads in the water, tails in the air, where the energetic cost of having a non-polar group surrounded by water is replaced by the much lower cost of having it in the gas. Their accumulation at the surface lowers γ, often dramatically: 1 mM SDS in water drops γ from 73 mN/m to about 35 mN/m.

This is what makes soap "wash" things. The reduced γ allows water to wet greasy surfaces it otherwise wouldn't, and the surfactant tails solubilise oils into micelles that float away in the rinse. The same chemistry — different molecule — protects the alveoli of your lungs and lets dishwashing detergent work.

Variants and extensions

• Capillary length ℓ_c = √(γ/ρg). The lengthscale below which surface tension dominates over gravity. ≈2.7 mm for water, ≈1.9 mm for mercury (denser but higher γ), <1 mm for low-γ alcohols. Above ℓ_c puddles flatten under gravity; below it they round into spherical caps.
• Bond (Eötvös) number Bo = ρgL²/γ. Dimensionless ratio of gravity to surface tension. Bo < 1 means surface tension dominates (small drops, capillary rise); Bo > 1 means gravity dominates (puddles, large waves).
• Weber number We = ρv²L/γ. Ratio of inertia to surface tension. Controls drop break-up in sprays — at We > 12 a falling drop deforms and shatters; below it stays intact.
• Plateau–Rayleigh instability. A cylindrical fluid column is unstable to wavelengths longer than its circumference because surface tension lowers total surface area by breaking the column into beads. Why a leaky tap drips, why cylindrical jets fragment, why soap films span minimal surfaces.
• Disjoining pressure and DLVO. In thin liquid films (foams, emulsions, soap bubbles), surface tension competes with electrostatic and van der Waals forces between the two interfaces. The Derjaguin–Landau–Verwey–Overbeek theory predicts the equilibrium thickness of stable foam lamellae down to nanometre scales.

Common pitfalls

• Confusing surface tension with surface energy. They have the same numerical value for liquids (because liquids re-equilibrate freely) but differ for solids, where the surface stress and surface free energy are distinct quantities. Soft-matter and solid-mechanics literature uses the distinction; everyday fluid-statics texts elide it.
• Forgetting that γ depends on what's on both sides. Water against air is γ_lv = 73 mN/m. Water against oil is closer to 30 mN/m, and water against another aqueous phase can be milli-newtons. Always specify the second phase when quoting γ.
• Treating a soap film as one surface. A free-standing soap film has two air–liquid interfaces. Pressure jumps and force balances must double-count surface tension: a bubble has ΔP = 4γ/R, not 2γ/R.
• Ignoring contact-angle hysteresis. The advancing and receding contact angles on real surfaces differ — sometimes by tens of degrees. A drop on a tilted surface stays put because the downhill (advancing) edge meets a steeper Young's-law angle than the uphill (receding) edge, balancing gravity. Models that use a single equilibrium θ overpredict drop mobility.
• Assuming temperature dependence is negligible. γ for water drops 20 % between 20°C and 100°C. Microfluidic devices, condensation heat exchangers and phase-change cooling all depend on knowing γ(T) to better than 1 %.