Surface Tension

Why surface tension matters

• Sets the shape of droplets and bubbles. A free droplet minimizes surface area at fixed volume, so it adopts a sphere. Gravity flattens it once the Bond number Bo = ρgL²/γ exceeds order unity — for water this happens at L ≈ 2.7 mm. Below this, droplets are spherical; above, they puddle.
• Drives capillary rise into porous media. Jurin's law h = 2γ cos θ / (ρgr) lifts water 14 cm in a 0.1 mm radius glass capillary at 20°C. Plant xylem (10–500 μm radius) lifts sap 1 to 10 m by capillarity alone, with transpiration cohesion-tension extending this to 100 m in tall trees.
• Stabilizes soap films. A flat soap film has two surfaces, each contributing γ ≈ 25 mN/m (lowered by surfactant). The Marangoni effect — flow induced by surface-tension gradients — heals local thinning by drawing surfactant-rich liquid into thin spots, the basis of the soap-film "self-healing" behavior.
• Critical to lung function. Dipalmitoylphosphatidylcholine surfactant lowers alveolar γ from 72 mN/m (pure water) to as little as 1 mN/m at end-expiration, reducing breathing work by ~75% and preventing collapse via Laplace instability between unequal-radius alveoli.
• Sets contact angles for wetting. Young's equation γ_sv = γ_sl + γ_lv cos θ predicts whether a liquid spreads (θ → 0°) or beads (θ → 180°) on a given solid. Hydrophilic glass shows θ ≈ 0° for water; freshly waxed surfaces give θ ≈ 110°; lotus-leaf nanostructure plus wax gives θ > 150° — superhydrophobicity.
• Enables locomotion for small animals. Water striders, mosquito larvae, fishing spiders, and small basilisk lizards exploit surface tension. Force per leg ≈ γ × perimeter × cos θ; a 10 mg insect needs only ~0.2 mm contact length per leg.
• Foundation for industrial coating, printing, and oil recovery. Surface-tension gradients drive Marangoni flows in inkjet printing, modulate spray atomization in fuel injection, and control whether crude oil is mobilized or trapped in reservoir pores during enhanced oil recovery.

Common misconceptions

• Surface tension is a "skin" on the liquid. There is no separate phase or layer — γ is just the excess Helmholtz free energy per unit interfacial area. The interface is a few molecular diameters thick (~0.5 nm) where density transitions from liquid to vapor; surface tension is the integrated cost of that gradient.
• Soap "breaks" the surface tension. Surfactants reduce γ by adsorbing to the interface (their hydrophilic heads in water, hydrophobic tails sticking out), but the surface still has tension — typically 25–35 mN/m for soap solution, less than pure water but nonzero. "Break" is loose talk.
• Capillary rise works against gravity for free. The energy comes from reducing solid–liquid interfacial energy as liquid wets the capillary wall — the system finds a lower total free energy. There is no perpetual motion machine, because to extract energy you would need to drain the rise.
• Mercury's high γ means it spreads. Opposite. High liquid–vapor γ relative to its interfacial energy with most solids gives mercury contact angles of 130°–140° on glass, so it beads up and even floats on its own surface. This is why barometers have a convex meniscus.
• Surface tension and surface energy are different things. For most liquids, γ (force per length, mN/m) and surface energy (energy per area, mJ/m²) are numerically identical. They differ for solids (where the surface is anisotropic and stretches without re-arranging), giving rise to the distinct "surface stress" concept.
• Laplace pressure is always inward. A spherical droplet has internal overpressure (curving outward → ΔP outward toward higher pressure in the liquid). A spherical bubble inside a liquid is the same. But a saddle-shaped surface, or a flat film, can have zero or even negative net mean curvature, with no Laplace pressure jump.

Derivation from molecular cohesion

Inside a bulk liquid, every molecule is surrounded by neighbors and feels attractive interactions in all directions, with no net force. At the surface, molecules have neighbors only on one side; the net inward attraction is what makes growing the surface energetically expensive. To create new surface area dA at constant T, V, μ, you must do work γ dA. Equivalently, the surface bears a tangential force per unit length γ that opposes any attempt to stretch it — like a thin elastic skin, but with energy proportional to area not strain. For pure water at 20°C, lifting one molecule from bulk to surface costs roughly 4 × 10⁻²¹ J (about kT), and the surface number density of water is ~10¹⁹ molecules/m², giving γ ≈ 4 × 10⁻² J/m² = 40 mN/m. The actual value 72.86 mN/m is higher because hydrogen bonds at the surface are partly preserved.

Two foundational equations follow. Young's 1805 equation balances horizontal forces at a three-phase contact line on a smooth solid: γ_sv = γ_sl + γ_lv cos θ. The contact angle θ measured through the liquid is therefore set by the three interfacial energies. Laplace's 1806 equation relates pressure jump across a curved interface to mean curvature: ΔP = γ (1/R₁ + 1/R₂). For a spherical droplet of radius R both radii equal R, so ΔP = 2γ/R; for a soap bubble with two surfaces ΔP = 4γ/R. Combining Young and Laplace inside a capillary tube — meniscus radius r/cos θ, hydrostatic balance ρgh = ΔP — gives Jurin's law h = 2γ cos θ / (ρgr).

Surface tension decreases with temperature because thermal motion weakens cohesion. The Eötvös rule γ V_m^(2/3) = k(T_c − T) holds approximately for non-polar liquids, with γ going to zero at the critical temperature where liquid and vapor become identical. For water, γ drops from 75.6 mN/m at 0°C to 58.9 mN/m at 100°C — a slope of about −0.15 mN/(m·K). For mercury the slope is −0.20 mN/(m·K). The temperature derivative −dγ/dT equals the surface entropy density, giving direct experimental access to entropy per molecule at the surface.

Surface tension γ for common liquids at 20°C

Liquid	γ (mN/m)	Dominant cohesion	Notes
Mercury (Hg)	485	Metallic bonding	Highest of common room-T liquids; convex meniscus on glass
Water (H₂O)	72.86	Hydrogen bonds	Highest among molecular fluids at 20°C
Glycerol (C₃H₈O₃)	64	Multiple H-bonds + viscosity	Used as a Marangoni-stable carrier in cosmetics
Ethylene glycol	48	Two H-bond donors	Antifreeze; γ falls slowly with T
Olive oil	33	Dispersion + weak dipole	Tagged with "oil" for general triglyceride
Acetone (CH₃COCH₃)	23.5	Polar dipole, no H-bond donor	Common solvent
Ethanol (C₂H₅OH)	22.0	Single H-bond donor	Used to demonstrate Marangoni effect (tears of wine)
n-Hexane	17.9	London dispersion only	Reference for non-polar fluid
Liquid helium (4 K)	0.37	Quantum + van der Waals	Lowest known γ of any liquid

Surface tension of water as a function of temperature

T (°C)	γ_water (mN/m)	Comment
0	75.64	Just-frozen water; high cohesion
10	74.23
20	72.86	Standard reference
25	72.01	Common biology / "room temperature"
40	69.55
60	66.18
80	62.61
100	58.85	Boiling at 1 atm
200	37.7	Saturated steam pressure ≈ 1.5 MPa
374 (T_c)	0	Critical point — γ vanishes

Applications

• Lung surfactant therapy. Type II pneumocytes secrete dipalmitoylphosphatidylcholine plus surfactant proteins (SP-A, SP-B, SP-C, SP-D) that lower alveolar γ from 72 mN/m to as little as 1 mN/m. Premature infants lacking it develop neonatal respiratory distress syndrome; exogenous surfactant (Survanta, Curosurf, Infasurf) administered since the 1980s has cut mortality from 50% to under 10%.
• Surface chemistry of detergents and soaps. Sodium dodecyl sulfate (SDS), linear alkylbenzene sulfonates (LAS), and nonionic alcohol ethoxylates lower water γ from 73 mN/m to 25–35 mN/m, allowing water to wet greasy surfaces and lift dirt by emulsification. Critical micelle concentrations are typically 0.1 to 10 mM.
• Water-strider and small-arthropod locomotion. Gerris (water strider), Tetragnatha (long-jawed spider), Anolis basilisk lizards on water — all exploit γ × leg-perimeter × cos θ supporting body weight. Detergent in a stream causes immediate sinking.
• Inkjet printing and microfluidics. Surface-tension-driven Marangoni flows shape the picoliter-scale ink droplets at the print head; capillary action wicks ink into paper fibers; pinning at the contact line determines line width and edge sharpness. Modern photolithography uses immersion lithography where γ at the lens-water-resist interface controls bubble nucleation.
• Enhanced oil recovery. About two-thirds of original oil-in-place is trapped by capillary forces in reservoir pore throats. Injecting CO₂, nitrogen, or surfactant solutions reduces interfacial tension between water and oil from 30 mN/m to under 0.01 mN/m, mobilizing the residual oil and recovering an extra 5–15% of reserves.