Surface Tension — Formula, Capillarity & Applications

Surface tension is the property that makes a liquid surface resist expansion and pull toward the smallest area it can. In physics, it is usually written as $\gamma$ and measured in $\mathrm{N/m}$.

The quick idea is this: a molecule inside the liquid is surrounded by neighbors, but a molecule at the surface is not. That imbalance changes the energy of the surface, so the liquid tends to reduce surface area when it can.

Three formulas appear again and again:

$$\gamma = \frac{F}{L}$$ $$\Delta p = \frac{2\gamma}{r}$$ $$h = \frac{2\gamma \cos\theta}{\rho g r}$$

The first gives force per unit length along a liquid surface. The second is the pressure difference for a spherical liquid droplet of radius $r$. The third is the capillary-rise formula for a narrow cylindrical tube at equilibrium. For a soap bubble, which has two liquid surfaces, the pressure difference is

$$\Delta p = \frac{4\gamma}{r}$$

Use each formula only with its condition. The pressure formulas above are for spherical shapes, and the capillary formula is for a narrow cylindrical tube at equilibrium.

What surface tension means physically

Surface tension is not a literal skin floating on top of the liquid. It is the result of intermolecular forces making the surface behave differently from the bulk.

That is why small droplets tend to become nearly spherical. For a given volume, a sphere has the smallest surface area, so this shape is favored when surface tension matters more than gravity.

People often say the surface "acts like a stretched film." That picture is useful, but it is still only an analogy. The cause is molecular interaction, not an actual elastic sheet.

Surface tension formula and units

In the simplest mechanical definition,

$$\gamma = \frac{F}{L}$$

where $F$ is the force acting tangentially along the surface and $L$ is the length over which that force acts.

This is the clearest way to understand the unit. If a frame or strip pulls on a liquid surface, $\gamma$ tells you the force per unit length along that surface.

You may also see surface tension described as energy per unit area. That description is consistent in SI units, but for most introductory problems, the force-per-length view is easier to use.

Why capillary rise happens

Capillarity is the rise or fall of a liquid in a narrow tube. It depends on both surface tension and the contact angle $\theta$ between the liquid and the tube wall.

If the liquid wets the wall, then $\theta < 90^\circ$ and $\cos\theta > 0$, so the liquid rises. Water in clean glass is the standard example.

If the liquid does not wet the wall, then $\theta > 90^\circ$ and $\cos\theta < 0$, so the liquid level is depressed. Mercury in glass is the standard example.

For a narrow cylindrical tube of radius $r$, the equilibrium capillary height is

$$h = \frac{2\gamma \cos\theta}{\rho g r}$$

where $\rho$ is liquid density and $g$ is gravitational acceleration.

This is an equilibrium formula. It gives the final height difference after the vertical effect of surface tension balances the weight of the liquid column.

Worked example: capillary rise in water

Suppose water rises in a clean glass capillary tube of radius

$$r = 0.50\ \mathrm{mm} = 5.0 \times 10^{-4}\ \mathrm{m}$$

Take surface tension

$$\gamma = 0.072\ \mathrm{N/m}$$

density

$$\rho = 1000\ \mathrm{kg/m^3}$$

and gravitational acceleration

$$g = 9.8\ \mathrm{m/s^2}$$

If the water wets the glass well, then $\theta \approx 0^\circ$ and $\cos\theta \approx 1$. Use the capillary-rise formula:

$$h = \frac{2\gamma \cos\theta}{\rho g r}$$ $$h = \frac{2(0.072)(1)}{(1000)(9.8)(5.0 \times 10^{-4})}$$ $$h = \frac{0.144}{4.9} \approx 0.029\ \mathrm{m}$$

So the water rises by about

$$h \approx 2.9\ \mathrm{cm}$$

The important trend is that a smaller tube gives a larger rise because $h \propto 1/r$ when the other quantities stay the same.

Pressure difference in droplets and soap bubbles

Curved liquid surfaces create a pressure jump.

For a spherical liquid droplet,

$$\Delta p = \frac{2\gamma}{r}$$

For a soap bubble,

$$\Delta p = \frac{4\gamma}{r}$$

The extra factor of $2$ for a bubble appears because a soap bubble has two liquid surfaces, one on the inside and one on the outside. A simple liquid droplet has only one liquid surface of this kind.

This matters most at small scales, because the pressure difference increases as the radius decreases.

Common mistakes with surface tension formulas

Confusing surface tension with viscosity

Surface tension is about the liquid surface. Viscosity is about resistance to flow within the liquid.

Dropping the contact angle without saying so

If you silently replace $\cos\theta$ by $1$, you are assuming complete wetting. That can be a reasonable approximation for some water-glass problems, but it is not always true.

Using the droplet formula for a soap bubble

Use $\Delta p = 2\gamma / r$ for a spherical droplet and $\Delta p = 4\gamma / r$ for a soap bubble.

Forgetting that tube radius changes the answer

The formula shows the opposite: smaller tube radius means larger magnitude of rise or depression.

Where surface tension is used

Surface tension matters in droplets, bubbles, wetting and coating, capillary action in thin tubes, detergents, inkjet printing, and microfluidic devices.

In many of those cases, the main competition is between surface effects and gravity or pressure effects. That is why surface tension becomes especially important at small length scales.

Try a similar problem

Change the worked example by doubling the tube radius while keeping the same liquid and contact angle. Predict the new height before calculating it. Then try your own version with a different liquid or a different contact angle.