Surface tension and capillarity

Surface tension

Surface tension ($\sigma$) is the tensile force per unit length acting along the free surface of a liquid, so that the surface behaves like a stretched membrane. Unit: N/m.

• Cause: a molecule inside the liquid is pulled equally in all directions by cohesion, but a molecule at the surface is pulled only inwards. This creates a net inward pull and a "skin" at the surface.
• Examples: water drops are spherical, small insects walk on water, a needle can float.
• Pressure inside a droplet: $p = \dfrac{4\sigma}{d}$; inside a soap bubble (two surfaces): $p = \dfrac{8\sigma}{d}$; inside a liquid jet: $p = \dfrac{2\sigma}{d}$.
• $\sigma$ decreases with increase in temperature. For water at 20 °C, $\sigma \approx 0.073$ N/m.

Capillarity

Capillarity is the rise or fall of a liquid in a small-diameter tube dipped in it, caused by surface tension together with adhesion and cohesion.

• If adhesion > cohesion (water and glass), the liquid wets the tube, the meniscus is concave and the liquid rises.
• If cohesion > adhesion (mercury and glass), the meniscus is convex and the liquid falls (capillary depression).

Equating the upward surface tension force to the weight of the raised column:

$$\sigma \pi d \cos\theta = \rho g \frac{\pi d^2}{4} h \;\Rightarrow\; h = \frac{4\sigma\cos\theta}{\rho g d}$$

where $\theta$ is the contact angle ($\approx 0^\circ$ for water–glass, $\approx 130^\circ$ for mercury–glass). The rise is inversely proportional to tube diameter, so manometer and piezometer tubes are made wider than about 6 mm to keep capillary error small.