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Physics for Civil Engineering

This is an introduction to Electricity, Strength of Materials and Waves.

Lecture 8 (Surface Tension and Surface Energy)

In this lecture:
• surface energy is defined,
• the effects of temperature and contaminants on the surface is discussed,
• methods of measuring surface energy in solids and surface tension in liquids are given,
• the angle of contact between liquids and solids is defined,
• capillary action is seen as a surface tension effect,
• the size of bubbles is seen as a balance between excess pressure and surface tension, and
• Laplace's law, for cylinders of fluid.

Surface Energy

Surfaces have energy associated with them because work is needed to form them.
Surface energy is the work per unit area done by the force that creates the new surface.

Typical surface energies (Source: Dr. B.R.Lawn)


Surface Energy J.m-2







Mica (in Air)


Mica (in vacuum)














Al2O3 (Sapphire)


Al2O3 (Polycrystalline)






C (Diamond)


C (Graphite)




Fe (Cast Iron)


From the table, the surface energy is very large for Cast Iron, which is a brittle material that shatters without much warning. Since brittle fracture creates new surfaces, the surface energy varies inversely with the tendency to brittle failure.

Rough tendencies for surface energy:
Ionic solids < 1 J.m-2
Metals ~ 1 J.m-2.
Covalent solids >1 J.m-2

Example T1
At high temperatures there is a tendency for glasses to change shape into a sphere. The surface energy of a glass at 650°C is 0.3 J.m-2. If the glass changes, from a cylinder of length 100 mm and diameter 20μm, into a sphere, find the energy released.

Answer T1
First find the radius of a sphere with the same volume as the cylinder.

answer T1

Now compare the surface areas of the two shapes.

surface areas

The sphere has a smaller area so surface energy is released.

surface area


Surface Energy and Temperature

In the bulk, atoms are evenly surrounded and the cohesive forces between the atoms tend to balance.

On the surface there are atoms on one side only, so there is a net inward cohesive force. This creates a force on the surface that tries to minimise its area. When considered as a force rather than an energy, the force is called "surface tension".

surface energy and temp

As temperature increases, the atoms in a solid vibrate more, and reduce the cohesive force binding the atoms.
The surface energy depends on the net inward cohesive force and so surface energy decreases with increasing temperature.
The surface energy for many metals (e.g. Ag, Au, and Cu) goes down by about 0.5 mJ.m2.K-1 with increasing temperature.
Water goes down by about 160 mJ.m2.K-1.

Surface Energy and Contamination

Contaminant molecules adhere to the surface ("like" cohere and "unlike" adhere).

The contaminant molecules thus change the balance of forces and reduce the net inward force. Since the net inward force is related to the surface energy, the surface energy is reduced by contaminants.

contaminant molecules

Measuring the surface energy of solids

Fracture method:

A crack is opened up by forces pulling the edges apart.
A "double cantilever" forms. The work done by the applied force is equal to the potential energy of the "leaf springs" and the surface energy. Solving for the surface energy (eventually) gives:


Measuring Young's modulus, E, and the lengths x, y and d, will give T.

surface are of solids

Indentation method:

With small specimens an indentation method is used.

A diamond point is forced into the surface and microcracks appear at the sharp edges.

It can be shown (but not in this course) that the surface energy is given by:


Measuring the lengths, a, and c, and the indenting force, F, will give the surface energy.

indentation method

Example T2
A razor blade inserted into the edge of a thin sheet of mica in high vacuum drives a crack to an equilibrium length along the central cleavage plane parallel to the sheet faces. The surface energy is measured as 5.0 J.m-2. When air is let in, then the crack length increases 1.9 times. Find the surface energy of mica in air.

Answer T2
From the fracture method:

answer T2

Air molecules have adhered to the new surfaces and reduced the net inward force thus reducing the surface energy.

Surface Tension

In dealing with liquids, it is more usual to use the idea of Surface Tension rather than Surface energy, even though they refer to the same dimensional quantity. This is shown in the following dimensional analysis.

surface energy

The net inward force on the surface of a liquid makes the surface act as if it was an elastic skin that constantly tries to decrease its area.

, acts in the surface and normal to an imaginary line in the surface.

Measuring Surface tension

To measure surface tension, the "wire frame" method is often used. A rectangular wire frame is suspended into a liquid and pulled upwards with force, F, to balance the downward force of surface tension, T.

Make the applied upward force, F up, balance the surface tension force, force down from the two surfaces clinging to the top of the frame.

force down
measuring suface tension

Surface tensions for some liquids in contact with air.


Surface Tension

Temperature °C


5.2 mN.m-1



15.7 mN.m-1


Ethyl alcohol

22.3 mN.m-1


Olive Oil

32.0 mN.m-1



58.9 mN.m-1



66.2 mN.m-1



72.8 mN.m-1



75.6 mN.m-1



465. mN.m-1



800. mN.m-1



1.0 N.m-1



1.1 N.m-1


Angle of contact

For a solid/liquid/gas interface, the adhesion between the liquid and the solid will curve the liquid surface to form a meniscus (Greek word for "crescent").

The angle of contact is always measured through the liquid.

angle of contact

The forces act along the interfaces, as shown.

FSG is the upward force between the solid and the gas.
FSL is the downward force between the solid and the liquid.
FLG is the inclined force between the liquid and the gas.

angle of contact

Resolving the vertical forces, with the proviso that the force between solid and gas, FSG is much smaller than the other two forces:


When FSL and FLG are in the same direction:
• cosα is positive i.e. α is less than 90°

• the meniscus is positive, and
• the liquid "wets" the surface.

meniscus positive

When FSL and FLG are in the opposite direction:
• cosα is negative i.e. α is greater than 90°

• the meniscus is negative, and
• the liquid does not "wet" the surface.

meniscus neg

Contact angles for some interfaces

most organic liquids - glass

0° - 10°

mercury - copper

pure water - glass

water - glass


kerosene - glass


water - silver


water - parafin


mercury - glass


Capillary Action

As a result of surface tension acting around the inner circumference of a small-bore tube (or capillary), that is partially immersed in a liquid, there will be a raised or depressed column of liquid inside it.

The case of a raised column is shown on the right.

capillary action

The upward component of the surface tension force will balance the weight of the liquid column.

From this, the height of the column can be calculated.

height of column

The same maths applies if α is greater than 90° but there is a depressed column.

Example T3
A capillary tube with an inside diameter of 250 μm can support a 100mm column of liquid that has a density of 930 kg.m-3. The observed contact angle is 15°. Find the surface tension of the liquid.

Answer T3

gas bubble

Pressure difference for a gas bubble in a liquid

A gas bubble in a liquid has two balancing forces that determine its size.

These are the outward force from internal gas pressure, and the inward force from surface tension trying to reduce the surface area.

Changing to energy, and using
(force)×(distance) = (pressure)×(volume)

gas bubble

The surface energy of the gas bubble is due to the difference between the bubble filled with gas and the bubble filled with liquid.

Divide top and bottom by the radius.

How the volume and surface area change with radius is now calculated.

The final result is that the pressure difference between the inner gas and the outer liquid is directly proportional to the surface tension and inversely proportional to the radius of the bubble.

What happens as a bubble rises and the outer liquid pressure decreases?

Laplace's law (Pressure difference across a tube of liquid)

For a cylinder of radius R and length such as a blood vessel, the wall supplies an inward force and the liquid supplies an outward pressure.

Laplace Law Pressure

The volume and surface area of the cylinder are given by:

volume surface area cylinder

This gives:


There is a greater pressure difference for a smaller radius than a larger one. This inverse relationship is called Laplace's law. Note that if the outside pressure decreases, the inside pressure also decreases so the radius increases as expected.

Example T4
A bubble of air has a diameter of 1mm when it is 0.5m under the surface of water (surface tension 73 mN.m-1). Find the gauge pressure inside the bubble.

Answer T4

answer T4

Example T5
A soap bubble in air (two surfaces) has surface tension 0.03 N.m-1. Find the gauge pressure inside a bubble of diameter 30mm.

Answer T5

answer T5


Surface energy is the work per unit area done by the force that creates the new surface.
Roughly, surface energy varies inversely with the tendency to brittle failure.
The surface energy decreases with increasing temperature.
Contaminants on the surface reduce the net inward force and decrease surface energy.

The surface energy of solids can be measured by fracture and indentation techniques.
The surface tension in liquids can be measured by using wire frames.
The angle of contact is the angle through the liquid to the solid.
Capillary action can support a column of liquid to a height (or depth) given by: capillary action.
The size of bubbles is a balance between excess pressure and surface tension.
For a single spherical surface: single spherical surface

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