AkCalculators

Surface tension — fundamentals, measurement and common formulas

Surface tension is a property of the interface between two phases, most commonly liquid–air interfaces. It represents either the energy required to increase the surface area of a liquid by one unit, or equivalently, the force per unit length acting along a line on the surface. Surface tension controls droplet shapes, capillary action, wetting and many small-scale fluid phenomena.

Physical meaning

At the molecular level, molecules in the bulk experience isotropic attractions; molecules at the surface have an unbalanced force pulling them back into the liquid. The system minimizes surface area to reduce the number of high-energy surface molecules, producing an effective tension along the surface. This macroscopic effect is measured as surface tension γ (Greek gamma).

Units and typical values

The SI unit is newton per meter (N/m). In laboratory practice, surface tension for water at 20°C is approximately 72 mN/m (or 72 dyn/cm) — note that 1 mN/m = 1 dyn/cm = 0.001 N/m. Organic liquids often have lower surface tensions (e.g., ethanol ≈ 22 mN/m), while mercury has a very high surface tension (~480 mN/m).

Common relations used to compute surface tension

• Force per unit length: direct measurement methods (force sensors, Wilhelmy plate) give γ = F / L, where F is the measured force and L the wetted perimeter. For a Wilhelmy plate, the wetted perimeter is twice the plate width plus twice thickness in contact; often simplified to plate width × 2 for thin plates.
• Capillary rise: a liquid climbs a thin tube due to surface tension. The equilibrium rise h above the free surface satisfies ρ g h = 2 γ cosθ / r, so rearranged: γ = (ρ g r h) / (2 cosθ), where ρ is liquid density, r the capillary radius, θ the contact angle and g gravitational acceleration.
• Young–Laplace equation: for a curved interface the pressure difference across the surface is ΔP = γ(1/R1 + 1/R2), where R1 and R2 are the principal radii of curvature. For a spherical droplet R1 = R2 = R, giving ΔP = 2γ/R.

Measurement techniques

Several laboratory techniques measure surface tension with different trade-offs:

• Wilhelmy plate — a thin plate is dipped into the liquid and the force is measured. With accurate contact angle and wetting perimeter, this is robust and straightforward.
• du Noüy ring — a ring is pulled from the surface; force is corrected to obtain γ (requires calibration and correction factors).
• Drop/shrinkage methods — analyzing the shape of pendant or sessile drops with axisymmetric profile analysis gives interfacial tension via the Young–Laplace equation.
• Capillary rise — simple, inexpensive, but sensitive to tube cleanliness and accurate contact angle measurement.

Temperature and contamination

Surface tension typically decreases with increasing temperature because thermal agitation reduces cohesive forces. Surfactants and contaminants strongly reduce measured surface tension; even trace surfactants can change measurements by tens of percent. For reproducible lab work, use clean glassware, degassed liquids when required, and controlled temperature.

Worked examples

Example 1 — Capillary rise. A clean glass capillary of radius 0.5 mm shows a water rise of 4.6 cm at 20°C. With ρ = 998 kg/m³, g = 9.80665 m/s² and θ ≈ 0°, compute γ = (ρ g r h)/(2 cosθ). Insert values (r=5e-4 m, h=0.046 m): γ ≈ (998×9.80665×5e-4×0.046)/(2×1) ≈ 0.072 N/m = 72 mN/m, consistent with expected value for water.

Example 2 — Young–Laplace for a droplet. A small spherical drop has internal pressure higher than the outside by ΔP = 2400 Pa, with radius R = 0.0002 m. Then γ = ΔP R / 2 = 2400×0.0002/2 = 0.24 N/m = 240 mN/m — an unusually high value suggesting either the geometry is not spherical or the units need checking.

Practical tips for measurements

1. Always clean the capillary/tube with solvents and flame when possible to remove surfactants and residues.
2. Measure contact angle or use surfaces that are known to be well-wetting (θ≈0) to simplify capillary calculations; otherwise include cosθ in the formula.
3. Maintain constant temperature and avoid drafts that alter evaporation or create convection near the surface.

Limitations

Simple formulas assume equilibrium, no dynamic wetting, and ideal geometry. In practice, dynamic effects (oscillations, vibrating instruments), evaporation, and impurities introduce systematic errors. For high-precision needs use calibrated instruments and recognized protocols (ASTM standards for surface/interfacial tension measurements).

Concluding remarks

Surface tension is a small but powerful quantity controlling many everyday phenomena from droplets to capillary transport. Quick calculations using the relations in this page are excellent for sanity checks and classroom problems; for research or industrial measurements, combine careful technique with full data analysis and uncertainty estimation.