Physics · Molecular physics and thermodynamics · Grade 9

Surface tension and capillarity

How does a steel needle float on water, and why does water rise by itself in a thin tube?

The situation

A pond skater glides across the surface of a puddle as if on ice, and a steel needle laid down carefully does not sink. A soap bubble forms a sphere by itself, a cotton fibre or a towel draws water into itself, and water climbs all the way up to the branches of a tree through thin vessels. All of this is one phenomenon — the surface of a liquid is stretched tight like an elastic film. How can this tension force be measured, and why does water rise by itself in a thin tube?

Topic

Surface tension and capillarity. An insect walking on water, a soap bubble and water in a cotton fibre

Key ideas

Surface tension
Wetting and meniscus
Capillarity

Formulas

σ = \frac{F}{L} = \frac{Δ E}{Δ S}

Surface tension coefficient: the force acting on a unit length of the boundary, or the work needed to increase the surface by 1 m². Water (20 °C): 0.073 N/m.

F = 2 σ L, A = F Δ x = σ Δ S

A soap film on a frame: two surfaces pull the wire; the work of stretching becomes surface energy.

h = \frac{2 σ \cos θ}{ρ g r}

Jurin's law: the height of rise in a capillary tube ( $θ$ is the contact angle).

Δ p = \frac{2 σ}{R}, Δ p_{bubble} = \frac{4 σ}{R}

Laplace's formula: the excess pressure beneath a curved surface (a bubble has two surfaces).

m g = π d σ, F_{m a x} = 4 π R σ

The drop method (Tate's law) and the ring detachment method.

Theory

A liquid and its surface layer

A liquid keeps its volume but takes its shape from the container: its molecules are densely packed, only short-range order is preserved, and they often change places. So a liquid is almost incompressible (the volume of water decreases by only ≈ 0.5 % at a pressure of 100 atm) and expands slightly when heated; the specific heat capacity of water is large — ≈ 4.19 kJ/(kg·K) (in the “Calorimeter: heat balance” experiment). A molecule inside a liquid is pulled equally in all directions by its neighbours. A molecule in the surface layer has no neighbours above it — the resultant force points inwards (the enlarged picture in the scene). Work is needed to bring a molecule to the surface, so surface molecules have extra surface energy and the liquid tends to reduce its surface: drops and bubbles are spherical. The unit of $σ$ is N/m = J/m².

Surface tension force and surface energy

In a frame with a soap film, the film pulls the movable wire with a force $F = 2 σ L$ (the film has two surfaces). Unlike a spring, this force does not depend on the stretching: when the film is stretched, new molecules come from the inside to the surface — the horizontal line on the graph. The work of stretching, $A = F Δ x = σ Δ S$ , becomes surface energy. $σ$ decreases as temperature rises (water: 0.076 N/m at 0 °C, 0.059 N/m at 100 °C) and falls to zero at the critical temperature. Surfactants (soap) form a one-molecule-thick layer on the surface and reduce $σ$ — for monomolecular films, see the oil film experiment in “Brownian motion and diffusion”.

Why do the needle and the pond skater not sink?

The needle does not sink into the water but bends the surface, forming a dimple. The bent surface pulls from both sides with a force $σ L$ , and the sum of the vertical components, $2 σ L \sin ψ$ , balances the weight (the buoyant force on the water beneath the dimple helps slightly as well). The greatest value of $ψ$ depends on wetting: a slightly greasy needle ( $θ \approx 100^{\circ}$ ) bends the surface a lot, but soapy water wets the needle ( $θ \approx 20^{\circ}$ ) — the supporting force drops sharply and the needle sinks. The tiny hairs on a pond skater's legs are not wetted by water at all; soap wets them too. The dimples in the scene are drawn 4 times deeper.

Wetting, meniscus and excess pressure

If the attraction between the molecules of the liquid and the solid (adhesion) is stronger than the attraction of the liquid molecules to one another (cohesion), the liquid wets the solid: the contact angle is $θ < 90^{\circ}$ and the tube shows a concave meniscus (water–glass, $θ \approx 0^{\circ}$ ). Otherwise it does not wet: $θ > 90^{\circ}$ and the meniscus is convex (mercury–glass, $θ \approx 140^{\circ}$ ). Beneath a curved surface there is Laplace excess pressure: in a drop $Δ p = 2 σ / R$ , in a soap bubble $4 σ / R$ because of the two surfaces. The pressure is higher in a small bubble — if you connect two bubbles, the small one is “blown” into the large one.

Capillarity and its applications

A wetting liquid rises in a thin tube because of the reduced pressure beneath the concave meniscus, until the weight of the column balances it: $h = 2 σ \cos θ / (ρ g r)$ — if the tube is half as wide, the water rises twice as high; on the graph, $h$ against $1 / r$ is a straight line. Mercury, which does not wet the glass, goes down instead. The formula holds when $r$ is much smaller than the capillary length $l_{c} = \sqrt{σ / ρ g} \approx 2.7$ mm. The speed of rise is set by viscosity (Poiseuille flow in the tube): glycerol rises very slowly. For flowing liquids — Bernoulli's equation, laminar flow and Poiseuille's formula — see the “Viscosity and flow” and “Aeroplane wing” experiments. Applications: towels, cotton wool and paper soak up water, moisture in the soil rises through capillaries, and a lamp wick draws up fuel. In the lab, by measuring $h$ you can find $r = 2 σ \cos θ / (ρ g h)$ or $σ = ρ g h r / (2 \cos θ)$ .

Methods of measuring σ

The capillary method — from $h$ and $r$ . The drop method: a drop at the tip breaks off when its weight reaches the surface tension force $π d σ$ holding it; if $N$ drops have volume $V$ , then $σ = ρ g V / (π d N)$ . In precise measurements a correction factor is introduced because part of the drop stays on the tip — the model uses the ideal formula. The ring detachment method: a ring being lifted out of a liquid is pulled by the film on both sides, and the greatest force before detachment is $F_{m a x} = 4 π R σ$ . In all methods temperature and purity matter: even a little soap reduces $σ$ considerably.

Did you know?

Among ordinary liquids at room temperature, the surface tension of water (≈ 0.073 N/m) is very high; higher values occur mainly in mercury (≈ 0.49 N/m) and in molten metals.
A pond skater's legs are covered with very fine hairs and are not wetted by water: according to a study published in the journal Nature in 2004, a single leg can support a force about 15 times the insect's weight.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

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Entropy and the arrow of time — Grade 9
Brownian motion and diffusion — Grade 9
Crystalline and amorphous solids — Grade 9
Phase transitions — Grade 10
Thermal expansion — Grade 10
Refrigerator and heat pump — Grade 10
Real gases — Grade 10