Surface Tension and Capillarity

1Walking on water

A pond skater stands on the surface of still water without sinking. Look closely and you see that its feet do not break the surface: each one presses a small dimple into it, as if the water were covered by a stretched elastic skin. A steel needle laid gently on water floats the same way, although steel is nearly eight times denser than water. Raindrops and dewdrops pull themselves into spheres, and water creeps up a narrow glass tube or into the pores of soil, against gravity. All of these come from one property of a liquid surface: surface tension.

2Why a liquid surface is under tension

Molecules of a liquid attract each other (cohesion). A molecule inside the liquid is pulled equally in all directions. A molecule at the surface has liquid neighbours only beside and below it, so it feels a net pull inwards. To bring a molecule from the inside to the surface, work must be done against this pull, so the surface has extra energy. A liquid therefore tends to make its surface area as small as possible, which is why a free drop is a sphere, the shape with the least surface for its volume.

Defining surface tension

Make a soap film on a wire frame with a sliding wire of length $LL$. The film pulls the wire inwards, and a force $FF$ is needed to hold it still. Doubling $LL$ doubles $FF$. A soap film has two surfaces (front and back), so

If the wire moves out by $\mathrm{\Delta }x\backslash Delta\; x$, the work done is $F\mathrm{\Delta }x=\gamma (2L\mathrm{\Delta }x)F\backslash Delta\; x\; =\; \backslash gamma(2L\backslash Delta\; x)$: $\gamma \backslash gamma$ times the new surface area created. So surface tension is also the surface energy per unit area, and $\mathrm{N}{\mathrm{m}}^{-1}\backslash mathrm\{N\backslash ,m^\{-1\}\}$ is the same as $\mathrm{J}{\mathrm{m}}^{-2}\backslash mathrm\{J\backslash ,m^\{-2\}\}$.

Surface tension falls as temperature rises, and detergents and soaps lower it sharply. That is why hot, soapy water wets and cleans greasy dishes better than cold water.

3Excess pressure inside drops and bubbles

Because the surface pulls inwards, the pressure inside a curved surface is higher than outside. Cut a spherical drop of radius $rr$ in half and look at one hemisphere. Surface tension pulls on it all round the cut circle of length $2\pi r2\backslash pi\; r$, with total force $2\pi r\gamma 2\backslash pi\; r\backslash gamma$. The excess pressure $\mathrm{\Delta }p\backslash Delta\; p$ pushes on the flat face of area $\pi r^2\backslash pi\; r^2$. For balance:

A soap bubble in air has two surfaces, so the pull doubles: $\mathrm{\Delta }p=\frac{4\gamma }{r}\backslash Delta\; p\; =\; \backslash frac\{4\backslash gamma\}\{r\}$. Smaller drops and bubbles have larger excess pressures. Connect a small soap bubble to a large one through a tube and the small one shrinks, blowing up the large one.

4Contact angle and capillarity

Where a liquid meets a solid, two attractions compete: cohesion (liquid to liquid) and adhesion (liquid to solid). The contact angle $\theta \backslash theta$ is the angle between the solid surface and the liquid surface, measured through the liquid. Water on clean glass has $\theta \approx 0^{\circ }\backslash theta\; \backslash approx\; 0^\backslash circ$: adhesion wins, water wets the glass and its surface curves up at the walls (a concave meniscus). Mercury on glass has $\theta \approx {140}^{\circ }\backslash theta\; \backslash approx\; 140^\backslash circ$: cohesion wins, mercury does not wet glass and its meniscus is convex.

Capillarity is the rise or fall of a liquid in a narrow tube. Around the rim of the meniscus, of length $2\pi r2\backslash pi\; r$, the surface pulls on the liquid with $\gamma \backslash gamma$ per metre at the angle $\theta \backslash theta$ to the wall. In equilibrium the vertical part of this pull supports the weight of the raised column:

This is Jurin's law. The rise is inversely proportional to the radius: halve the bore and the water climbs twice as high. If $\theta >{90}^{\circ }\backslash theta\; >\; 90^\backslash circ$, $\cos \theta \backslash cos\backslash theta$ is negative and the liquid is pushed down, as mercury is. Capillarity draws water up through the fine pores of soil towards plant roots and towards the surface, where it evaporates; it lifts oil up a lamp wick and ink into a paper towel.

5Experiment 4.2: measuring surface tension by capillary rise

Aim: to verify that $h\propto \frac{1}{r}h\; \backslash propto\; \backslash frac\{1\}\{r\}$ and to find the surface tension of water.

Apparatus: clean glass capillary tubes of several known bores, a beaker of clean water, a clamp, a travelling microscope (or a ruler and magnifying glass).

1. Clean the tubes (soap or grease lowers $\gamma \backslash gamma$ and changes the contact angle) and rinse them in the water.
2. Clamp each tube vertically in the water, raise and lower it once to wet the inside, and measure the height $hh$ of the bottom of the meniscus above the outside surface.
3. Plot $hh$ against $\frac{1}{r}\backslash frac\{1\}\{r\}$. With $\theta =0\backslash theta\; =\; 0$, the gradient is $\frac{2\gamma }{\rho g}\backslash frac\{2\backslash gamma\}\{\backslash rho\; g\}$.

Illustrative results:

The best-fit line through these points has gradient $1.45\times {10}^{-5}{\mathrm{m}}^{2}1.45\; \backslash times\; 10^\{-5\}\backslash ,\backslash mathrm\{m^2\}$, so $\gamma =\frac{\rho g\times \text{gradient}}{2}=\frac{1000\times 9.8\times 1.45\times {10}^{-5}}{2}=0.071\mathrm{N}{\mathrm{m}}^{-1}\backslash gamma\; =\; \backslash frac\{\backslash rho\; g\; \backslash times\; \backslash text\{gradient\}\}\{2\}\; =\; \backslash frac\{1000\; \backslash times\; 9.8\; \backslash times\; 1.45\; \backslash times\; 10^\{-5\}\}\{2\}\; =\; 0.071\backslash ,\backslash mathrm\{N\backslash ,m^\{-1\}\}$, close to the accepted 0.073.