Physics · Ch 9 — Mechanical Properties of Fluids

Surface Tension

9.6

Surface Tension

9.6 Surface Tension

The behaviour of a liquid at its free surface is strikingly different from its behaviour in the bulk. A liquid surface behaves as if it were a stretched elastic membrane under tension. This property is called surface tension.

What Causes Surface Tension?

Inside a liquid, a molecule experiences equal attractive forces from all directions — the net force on it is zero. But a molecule at the surface has neighbours only below and to the sides; there are no liquid molecules above it. The unbalanced inward pull pulls surface molecules into the bulk, making the surface contract to the smallest possible area. The surface is therefore in a state of tension, like a stretched film.

Note

Surface tension is a molecular phenomenon. It arises from the cohesive forces between liquid molecules. The stronger the cohesive force, the higher the surface tension.

Definition of Surface Tension

Consider a liquid film stretched on a wire frame with a movable slider of length $L$ . To keep the film from contracting, you must apply a force $F$ to the slider. The film has two surfaces (front and back), so the total length of the surface edge being pulled is $2L$ .

Surface tension $\sigma$ (or $S$ ) is defined as the force per unit length acting along the surface, perpendicular to any line drawn on the surface:

\sigma = \frac{F}{2L}

More generally, if a force $F$ acts along a line of length $l$ on the surface, the surface tension is:

\sigma = \frac{F}{l}

The SI unit of surface tension is $\text{N m}^{-1}$ (newton per metre). Its dimensional formula is $[M^1 L^0 T^{-2}]$ .

\sigma = \frac{F}{l}

Surface Energy

Work must be done to increase the surface area of a liquid because molecules must be brought from the interior to the surface against the inward pull. This work is stored as surface energy.

Consider the same wire frame with a slider. If you pull the slider by a distance $\Delta x$ , the work done is:

W = F \cdot \Delta x = (\sigma \cdot 2L) \cdot \Delta x = \sigma \cdot (2L \Delta x)

The increase in total surface area of the film (two surfaces) is $\Delta A = 2L \Delta x$ . Therefore:

W = \sigma \cdot \Delta A

This work equals the increase in surface energy $\Delta U$ . Hence:

\sigma = \frac{\Delta U}{\Delta A}

Surface tension is numerically equal to the surface energy per unit area.

Important

Surface tension can be defined in two equivalent ways: as force per unit length ( $\text{N m}^{-1}$ ) or as surface energy per unit area ( $\text{J m}^{-2}$ ). Both have the same dimensions.

Angle of Contact

When a liquid meets a solid surface, the liquid surface curves near the wall. The angle of contact $\theta$ is the angle between the tangent to the liquid surface at the point of contact and the solid surface, measured inside the liquid.

For water on clean glass: $\theta \approx 0^\circ$ (water wets glass completely)
For water on waxed surface: $\theta \approx 107^\circ$ (water does not wet)
For mercury on glass: $\theta \approx 140^\circ$ (mercury does not wet glass)

Watch out

The angle of contact is always measured inside the liquid, not outside. A common mistake is to measure it the other way.

Capillary Rise

When a narrow tube (capillary) is dipped into a liquid, the liquid either rises or falls in the tube relative to the surrounding liquid level. This phenomenon is called capillarity.

Derivation of Capillary Rise Height

Consider a capillary tube of radius $r$ dipped into a liquid of density $\rho$ . The liquid rises to a height $h$ in the tube. The meniscus is approximately hemispherical with radius of curvature $R = r / \cos\theta$ , where $\theta$ is the angle of contact.

The pressure difference across a curved liquid surface is given by the Young-Laplace equation:

\Delta p = \frac{2\sigma}{R}

For a spherical meniscus of radius $R$ , this becomes:

\Delta p = \frac{2\sigma}{R} = \frac{2\sigma \cos\theta}{r}

This pressure difference supports the weight of the liquid column of height $h$ :

\Delta p = \rho g h

Equating the two expressions:

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

Therefore:

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

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

Key Observations from the Formula

For wetting liquids ( $\theta < 90^\circ$ ): $\cos\theta > 0$ , so $h > 0$ — the liquid rises in the capillary.
For non-wetting liquids ( $\theta > 90^\circ$ ): $\cos\theta < 0$ , so $h < 0$ — the liquid is depressed in the capillary.
Height is inversely proportional to tube radius: narrower tubes give greater rise (or depression).
Height is directly proportional to surface tension: higher surface tension gives greater rise.

Tip

For water in a clean glass tube, $\theta \approx 0^\circ$ , so $\cos\theta \approx 1$ and the formula simplifies to $h = \frac{2\sigma}{\rho g r}$ .

Excess Pressure Across a Curved Surface

A curved liquid surface always has a pressure difference across it. The pressure on the concave side is greater than that on the convex side.

For a Spherical Drop

A liquid drop has only one surface. The excess pressure inside a spherical drop of radius $R$ is:

p_i - p_o = \frac{2\sigma}{R}

For a Spherical Bubble

A soap bubble has two surfaces (inner and outer). The excess pressure inside a soap bubble of radius $R$ is:

p_i - p_o = \frac{4\sigma}{R}

Watch out

Do not confuse the drop and bubble formulas. A drop has one surface; a bubble has two. The factor of 2 difference is critical in exams.

For a Cylindrical Surface

For a cylindrical liquid surface of radius $R$ , the excess pressure is:

p_i - p_o = \frac{\sigma}{R}

Factors Affecting Surface Tension

Temperature: Surface tension decreases with increasing temperature. At the critical temperature, surface tension becomes zero. …

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