2.3 Surface Tension and Capillarity

Surface Tension and Capillarity

Surface tension shapes how liquids behave at interfaces. It's why water beads up on a leaf and insects can walk on ponds. This force arises from molecules at the surface being pulled inward, creating a "skin" effect.

Contact angles and capillary action are the key phenomena that follow from surface tension. They determine how liquids spread on surfaces and rise (or fall) in narrow tubes, with applications ranging from water-repellent coatings to microfluidic devices.

Causes of Surface Tension

Molecules inside a liquid are pulled equally in all directions by their neighbors, so the net force on them is zero. Molecules at the surface, though, have no liquid neighbors above them. They experience a net inward pull, which causes the surface to contract and behave like an elastic membrane.

• Liquids with stronger intermolecular forces have higher surface tension. Water has relatively high surface tension because of hydrogen bonding ($\gamma \approx 0.072 \, N/m$ at 20°C). Mercury's is even higher ($\gamma \approx 0.465 \, N/m$) due to strong metallic bonds.
• Surface tension $\gamma$ is quantified as force per unit length ($N/m$) or equivalently as energy per unit area ($J/m^2$). Both definitions are physically equivalent.
• Surfactants like soap reduce water's surface tension by inserting themselves between water molecules and disrupting hydrogen bonds. That's why soapy water spreads more easily and doesn't bead up the way pure water does.
• Water striders exploit high surface tension: their hydrophobic legs deform the water surface without breaking through, and the resulting upward surface-tension force supports their weight.

Contact Angle and Wettability

The contact angle $\theta$ is the angle formed where a liquid-vapor interface meets a solid surface, measured through the liquid. It tells you how well a liquid wets that surface.

• Wetting ($\theta < 90°$): The liquid spreads over the surface. Adhesive forces between the liquid and solid dominate over cohesive forces within the liquid. Example: water on clean glass.
• Non-wetting ($\theta > 90°$): The liquid pulls away and forms droplets. Cohesive forces dominate. Example: mercury on glass.
• Perfect wetting ($\theta = 0°$): The liquid spreads into a thin film. Example: many oils on clean metal.
• Superhydrophobic ($\theta > 150°$): Extreme non-wetting, as seen on lotus leaves, where micro/nano-scale surface roughness traps air beneath the droplet.

Young's equation relates the contact angle to the three relevant surface tensions at the contact line:

$\gamma_{sv} = \gamma_{sl} + \gamma_{lv} \cos \theta$

where $\gamma_{sv}$, $\gamma_{sl}$, and $\gamma_{lv}$ are the solid-vapor, solid-liquid, and liquid-vapor surface tensions, respectively. This equation is the starting point for predicting how a droplet will sit on a given surface.

Surface Tension in Capillary Action

Capillary action is the rise or depression of a liquid in a narrow tube. It happens because surface tension pulls the liquid along the tube wall (if the liquid wets the wall) or pushes it away (if it doesn't).

For a wetting liquid like water in a glass tube, adhesive forces between the liquid and the wall pull the meniscus upward. Surface tension along the contact line then drags the liquid column up until gravity balances the upward pull. For a non-wetting liquid like mercury in glass, the effect reverses and the liquid is depressed below the outside level.

The capillary rise equation gives the equilibrium height $h$ of the liquid column:

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

• $\gamma$ = surface tension of the liquid
• $\theta$ = contact angle
• $\rho$ = liquid density
• $g$ = gravitational acceleration
• $r$ = tube radius

When $\theta < 90°$, $\cos \theta > 0$ and the liquid rises. When $\theta > 90°$, $\cos \theta < 0$ and the liquid is depressed. Notice that $h$ is inversely proportional to $r$: narrower tubes produce greater rise (or depression).

The Young-Laplace equation gives the pressure difference across a curved interface inside a capillary:

$\Delta P = \frac{2\gamma \cos \theta}{r}$

A smaller radius means a more tightly curved meniscus and a larger pressure jump. This same principle explains why smaller bubbles have higher internal pressure than larger ones.

Calculations for Surface Phenomena

1. Capillary rise height

Given: water in a glass tube with $r = 0.5 \, mm = 5 \times 10^{-4} \, m$, $\gamma = 0.072 \, N/m$, $\rho = 1000 \, kg/m^3$, $\theta = 0°$.

1. Identify that $\cos 0° = 1$.
2. Substitute into the capillary rise equation: $h = \frac{2(0.072)(1)}{(1000)(9.81)(5 \times 10^{-4})}$
3. Calculate: $h = \frac{0.144}{4.905} \approx 0.0294 \, m \approx 2.94 \, cm$.

2. Pressure difference across a curved interface

Given: water-air interface in a capillary with $r = 0.1 \, mm = 1 \times 10^{-4} \, m$, $\gamma = 0.072 \, N/m$, $\theta = 30°$.

1. Calculate $\cos 30° \approx 0.866$.
2. Substitute into the Young-Laplace equation: $\Delta P = \frac{2(0.072)(0.866)}{1 \times 10^{-4}}$
3. Calculate: $\Delta P = \frac{0.1247}{1 \times 10^{-4}} \approx 1247 \, Pa \approx 1.25 \, kPa$.

3. Force to pull a wire loop from a liquid surface

Surface tension acts along both the inner and outer edges of the wire loop. For a circular loop of radius $R$, the total contact length is $2 \times 2\pi R = 4\pi R$ (two sides of the film). The force due to surface tension is:

$F = \gamma \times 4\pi R$

You then add the weight of any liquid film lifted with the loop if the problem requires it. The Wilhelmy plate method works on the same principle: you measure the downward force on a thin plate partially submerged in a liquid, and from the known contact perimeter and contact angle, you back out $\gamma$.