Angle of contact, drops and capillary rise

Angle of contact

Narrated video. Transcript:

Put a drop of water on a clean glass plate: it spreads out into a thin patch. Put a drop of mercury on the same plate: it stays a round bead. Water on a lotus leaf also forms round beads that roll off.

Zoom in where the liquid surface meets the solid. Near the solid, the surface is curved. Draw the tangent to the liquid surface at the point of contact.

The angle between this tangent and the solid surface, measured inside the liquid, is called the angle of contact, θ. It depends on the liquid and on the solid.

Where does it come from? Three surfaces meet at the edge: liquid–air, solid–liquid and solid–air. Each has its own surface tension, pulling along its surface.

For the edge to stay in balance along the solid surface: S la cos θ plus S sl equals S sa. Here S la is the liquid–air surface tension, S sl the solid–liquid one, and S sa the solid–air one.

If S sa is bigger than S sl, cos θ is positive and θ is acute, less than 90 degrees. The liquid wets the solid and spreads. Water on clean glass has θ close to zero.

If S sl is bigger than S sa, cos θ is negative and θ is obtuse. The liquid does not wet the solid and pulls into drops. Mercury on glass has θ of about 140 degrees; water on a waxy leaf, more than 90.

Soaps and detergents are wetting agents: they make θ small, so water soaks into the cloth. Waterproofing agents on raincoats do the opposite: they make θ large.

Drops, bubbles and capillary rise

Narrated video. Transcript:

Surface tension squeezes a drop, the way a stretched skin would. So the pressure inside a drop, P i, is higher than the pressure outside, P o. How much higher?

Let a drop of radius r grow slightly, to r plus Δr. Its surface area grows by 8 π r Δr, so its surface energy grows by S times 8 π r Δr.

This energy comes from the work done by the extra pressure inside, pushing the surface out: P i minus P o, times the area 4 π r squared, times Δr.

Equate the two: P i minus P o equals 2 S over r. The smaller the drop, the bigger the excess pressure. An air bubble inside a liquid has one surface too, so it is also 2 S over r.

A soap bubble in air is a thin film with two surfaces, outer and inner. So its excess pressure is twice as large: 4 S over r.

Now dip a thin glass tube, a capillary, into water. The water climbs up inside the tube, above the level outside. The narrower the tube, the higher it climbs.

The water wets the glass, so its surface in the tube is curved, concave upwards. Just below this curved surface the pressure is lower than the air above, by 2 S over r.

For a tube of radius a, the curved surface has radius a over cos θ. So the pressure just under it is less than atmospheric by 2 S cos θ over a. Water is pushed up until the column’s pressure, ρ g h, makes up this difference.

So the capillary rise is h equals 2 S cos θ over ρ g a. Worked example: water, θ zero, in a tube of radius 0.5 millimetres: 2 times 0.073, divided by 1000 times 9.8 times 0.0005. That is about 3 centimetres.

For mercury, θ is more than 90 degrees, so cos θ is negative and h is negative: the mercury is pushed down in the tube, with a convex surface. Capillary action also lifts oil up a diya’s wick and water into a towel or blotting paper.

Try it: capillary rise

Interactive simulation. Things to try:

• Make the tube narrower: h = 2S cos θ / ρgr grows as 1/r.
• Pick mercury: θ = 140°, cos θ < 0, so the level in the tube falls below the outside level.
• Drag the angle of contact past 90°: the meniscus flips from concave to convex.
• The close-up shows θ measured inside the liquid, from the wall to the liquid surface.

Try it: drops and bubbles

Interactive simulation. Things to try:

• A drop and an air bubble in a liquid have one surface: excess pressure 2S/r.
• A soap bubble has two surfaces: excess pressure 4S/r.
• Halve the radius: the excess pressure doubles. Small bubbles have the highest pressure inside.

Mercury in a capillary

A glass capillary tube is dipped into mercury. What happens to the mercury level inside the tube, and why?

1. It rises, because mercury has a large surface tension
2. It falls below the outside level, because the angle of contact is more than 90°, so cos θ is negative
3. It stays level, because mercury is too dense to be moved by surface tension
4. It rises, because the angle of contact is more than 90°

How high does water rise?

How high does water (S = 0.073 N/m, angle of contact 0°) rise in a clean glass capillary tube of radius 0.25 mm? (ρ = 1000 kg/m³, g = 9.8 m/s²)

1. 0.6 cm
2. 3.0 cm
3. 6.0 cm
4. 12 cm

Summary

• Angle of contact: Sₗₐ cos θ + Sₛₗ = Sₛₐ
• Drop, or air bubble in a liquid: Pᵢ − Pₒ = 2S / r
• Soap bubble (two surfaces): Pᵢ − Pₒ = 4S / r
• Capillary rise: h = 2S cos θ / ρga

• The angle of contact is measured inside the liquid, between the solid and the tangent to the liquid surface.
• θ &lt; 90°: the liquid wets the solid (water on glass). θ &gt; 90°: it does not (mercury on glass, water on a lotus leaf).
• The pressure is higher on the concave side of a curved liquid surface. Smaller drops and bubbles have a larger excess pressure.
• Liquids that wet the tube rise in a capillary; those that do not are depressed. Narrower tubes give bigger effects.