## Soap-bubbles, surface tension, minimal shapes, soap bubble colours and formation.

Soap and soap films exhibit beautiful interference colours. They also demonstrate phenomena associated with surface tension and surface free energy, and thus can demonstrate minimal surfaces constrained by different boundaries. Explanations of how interference produces these colours are given on a separate page: Thin film interference and reflections.

This page supports the multimedia tutorial Interference but has some film clips (and links to longer ones) that were too long for the introductory presentation.

A thinning soap film.

### Thinning soap films

On the macroscopic scale, soap bubbles are thin and fragile. Those shown here are made from water, detergent, which reduces its surface tension, and glycerol, which increases its viscosiy. A closed circular frame is used at left to make a roughly spherical bubble. This gives a fleeting glimpse of interference colours in patches that have appropriate orientation to the light and the observer.The clip at right is included mainly because it is pretty: the bubble drains in a stationary frame but complicated convection effects produce changing patterns of colours, while air currents in the lab change the shape of the film itself.

 Forming a bubble with a closed wire frame (left) and a plane film thinning as solution drains (right)

The draining in the film above is rather complicated, in part because of its initial motion and convection. If the frame is held stationary, one can observe draining under gravity: the solution gradually flows downwards due to its own weight, so the film is thinnest at the top. We discuss the physics involved in Thin film interference and reflections.

Three soap films at different stages of draining. All have reached the stage, however, where a section at the top is much thinner than a wavelength of light, so that the destructive interference for all colours produces a black film.

### Surface tension

In this photo, I made a thin film on a closed circular frame across which was tied a cotton thread. Then I burst the film on the lower side. The tension in (both sides of) the upper film supports the weight of the thread, and also imposes g/cma small tension on it.

The weight of the thread is about 10 mN.m−1, so the surface tension of each side must be greater than about 5 mN.m−1. My guess is that it is about 20 mN.m−1 on each side, a few times smaller than that of pure water (74 mN.m−1).

### Minimal surfaces

The detergent in these films reduces the surface tension well below that of pure water, but it is still large enough to pull the surface into the shape that approximates the surface with the smallest surface area, subject to the external constraints, including those imposed by the supporting frame. In the two top films, air is captured near the centre of the cube, which prohibits contraction of the central closed volume, which is bounded by six spherical segments.
 Film on a cubic frame #1 Film on a cubic frame #2 Film on a helical frame Film on a tetrahedral frame

### Soap bubbles colours

This photo, repeated from above, is a soap bubble made from water, detergent and glycerol. The glycerol makes the solution more viscous and makes the film more stable. Soaps and detergents are surfactants or surface active agents: so called they accumulate at the water surface. Once the bubble is formed on the vertical frame, the water-glycerol mix drains downwards under gravity – relatively slowly because of its viscosity. This makes the bubble thinner at the top. Let's now explain why the bubble is black, then white, then yellow etc. Now refer to the inteference diagram.

Light is reflected when it strikes the first surface and, from air (low n) towards water (higher n), the reflection has a phase change of π. Some of the light is transmitted (with no phase change) and reflected on the more distant interface. At this reflection, water towards air, the phase change is zero.

Now consider a region of the bubble where its thickness t is very much less than a wavelength λ. The difference in path length of the two rays contributes negligible phase change, so the phase difference between the two reflected rays is π, or half a cycle. This gives destructive interference, for all wavelengths.

In the top third of the bubble in the photo, the film has reached the state where t << λ. The destructive interference is almost complete: very little reflection for any wavelength, so the film is nearly as black as the background.

Next, consider a region of the bubble where its thickness t is half the wavelength of blue light in water, i.e.

t  =  λblue/2n
where n is the refractive index of water. In this region, there would be a phase difference of 2π due to the path difference and π due to the different reflection conditions, so, for blue light, the rays would be 1.5 cycles out of phase: very little blue would be reflected.

At this thickness, the interference is not destructive for the other wavelengths so, with blue missing, we should still see red and green, which would give yellow. So we deduce that the yellow band about half way up the frame is where the film thickness is λblue/2n, or about 150 nm.

A little further down the film, where its thickness isλgreen/2n, or about 200 nm, green light is missing, so we see red + blue = magenta. Further down still, where the thickness is λred/2n, or about 250 nm, red light is missing, so we see green + blue = cyan.

Further down, there is another set of coloured bands: where the film thickness is 2λblue/2n, or about 300 nm, there is destructive interference for blue again, so again the colour is yellow.

What about the region between the black and the first yellow band? Here, the thickness is less than λblue/2n, so there is no destructive interference for any visible light, so all colours are reflected, and the film is white.

The abrupt border between white and black may seem surprising. This is due to van der Waals attraction between the surfactant layer, which squeezes out the water between them until their separation is very small. The thickness between the water soluble parts of the surfactants is not quite zero however. There are repulsive forces of two sorts between the surfactant layers: electric repulsion between charges and/or dipoles in the water-soluble parts, and hydration force repulsion, because water is strongly attracted to these parts. Which brings us to:

### Soap bubble formation

Bubbles are not stable in pure (surfactant-free) water, because of the surface tension. Water molecules are strongly attracted to other water molecules, because of dipolar electrical forces. Consequently, it takes energy to make a water-air interface, because to do so all the water molecules in the new surface must be separated from other water molecules. Let's call the work per unit surface area g. Because of this energy, water-air interfaces spontaneously contract. This is why a rain drop is spherical: a sphere is the shape with the smallest ratio of surface area to volume, so the rain drop surface contracts until it reaches this shape. A bubble of air in pure water either floats to the surface and fuses with it, thereby losing its area, or else contracts as the air is dissolved into the water. So splashing around in pure water doesn't create bubbles.

If we deform an air-water surface to increase its area, we have to apply a force along the perimeter that is being moved. If the perimeter has length L and is moved a distance dx, then the area is Ldx and the work done Fdx = γdA = γLdx, where γ is the surface tension. So the force per unit area is also equal to γ, and so may be expressed in either J.m−2 or N.m−1. The surface tension or interfacial specific energy of the water-air interface is about γ = 74 N.m−1.

When surfactants are present, however, they are concentrated at the surface. At low concentration, they make a two dimensional gas and exert a lateral pressure in the interface, and so reduce the surface tension or the surface energy. Consequently, it is relatively easy to produce bubbles in soapy water. In our experiment, a high concentration of detergent was used, but we can see that the surface tension is still positive, because the bubbles on the frame contract (reduce their area) to form a plane.

A common detergent is SodiumDodecylSulphate (SDS). We could sketch the molecule like this

where the kinked line represents the 12 carbon hydrocarbon change (dodecane). Looking at this sketch, we can see that the NaSO4 part looks like a salt and so should be water soluble, while the hydrocarbon looks like an oil and will not be soluble in water. Which explains why these molecules collect at the surface, with the salty part in the water and the hydrocarbon facing the air.