The beauty of soap bubbles and soap films has a timeless appeal to young and old alike. Such prominent painters as Murillo, Chardin, Hamilton, Manet, and Millais have captured it through the ages.

This page contains information regarding soap films, theire history, structure and possible usage.

Historical Review

The scientific study of liquid surface which has led to our present knowledge of soap films and soap bubbles, is thought to date from the time of Leonardo Da Vinci; a man of science and art. Since the fifteenth century researchers have carried out investigations in two distinct camps. In one camp there are the physical, chemical and biological scientists who have studied the macroscopic and molecular properties of surfaces with mutual benefit. The other camp contains mathematicians who have been concerned with problems that require the minimization of the surface area contained by a fixed boundary and related problems. A simple example of such a problem is the minimum area surface contained by a circle of wire. The solution to this problem is well known to be the disc contained by the wire.

In the 19^th century the Belgian physicist Joseph Plateau showed that dipping wire frameworks into a bath of soap solution could produce analogue solutions to the minimization problems. After with- drawing a framework from the bath a soap film is formed in the frame, bounded by the edges of the framework, with a minimum area surface. All the minimum surfaces were found to have some common geometrical properties. This work rapidly attracted the attention of the mathematicians and has resulted in a fruitful interaction between the two camps. These experimental results have inspired mathematicians to look for new analytic methods to enable them to prove the existence of the geometric properties associated with minimum area surfaces and to solve the minimum area problems. However it is only relatively recently that important steps have been made in this direction, particularly the work of Jesse Douglas and his contemporaries in the 1930's, and the recent work of mathematicians in the United States.

In order to make it more clearly for the average web surfer (such as me) to understand soap films one has to be familiar with some basic information regarding the structure and features of soap films.

Surface Tension

This is the force in the surface of a fluid acting on each side of a line of unit length drawn in the surface. Within the water, at least a few molecules away from the surface, every molecule is engaged in a tug of war with its neighbors on every side. For every "up" pull there is a "down" pull, and for every "left" pull there is a "right" pull, and so on, so that any given molecule feels no net force at all. At the surface things are different. There is no up pull for every down pull, since of course there is no liquid above the surface; thus the surface molecules tend to be pulled back into the liquid. It takes work to pull a molecule up to the surface. If the surface is stretched - as when you blow up a bubble - it becomes larger in area, and more molecules are dragged from within the liquid to become part of this increased area. This "stretchy skin" effect is called surface tension. Surface tension plays an important role in the way liquids behave. If you fill a glass with water, you will be able to add water above the rim of the glass because of surface tension.

Soap and Water Solution

Soap decreases the pull of surface tension - typically to about a third that of plain water. The surface tension in plain water is just too strong for bubbles to last for any length of time. One other problem with pure water bubbles is evaporation: the surface quickly becomes thin, causing them to pop.

Soap molecules are composed of long chains of carbon and hydrogen atoms. At one end of the chain is a configuration of atoms, which likes to be in water (hydrophilic). The other end shuns water (hydrophobic) but attaches easily to grease.

In a soap-and-water solution the hydrophobic (greasy) ends of the soap molecule do not want to be in the liquid at all. Those that find their way to the surface squeeze their way between the surface water molecules, pushing their hydrophobic ends out of the water. This separates the water molecules from each other. Since the surface tension forces become smaller as the distance between water molecules increases, the intervening soap molecules decrease the surface tension.

Because the greasy end of the soap molecule sticks out from the surface of the film, the soap film is somewhat protected from evaporation (grease doesn't evaporate), which prolongs the life of the bubble substantially.

Soap Film

A soap film consists of two layers of soap molecules separated by a thin layer of fluid, which may vary in thickness from .

The largest thickness wills occur immediately after the formation of the film.

Once the film is formed it will begin to thin. The surplus water will drain away from the film by various draining processes. The thickness of the film will decrease until a final equilibrium thickness is reached.

In a state of equilibrium the surface tension is the same in all points on the surface.

When the film has reached equilibrium its surface area will have a minimal value, this minimum area property of soap films can be used to solve some mathematical minimization problems.

The Motorway Problem

One of the most interesting minimization problems for which an analytic solution is yet to be found is the problem of linking n point on the same plane in the shortest possible path.

For example let us consider the problem of linking four towns A, B, C and D by a road the towns are situated at the corners of the square of unit length.

What would you think is the shortest path between the four towns?

Lets look at some possible solutions:

In order to reach an analogue solution which will take advantage of the soap film unique feature of reducing surface space to minimum we have to construct two parallel clear perplex plates joined by four pins, perpendicular to the plates, arranged at the corner of a square which will represent the four towns when this arrangements is immersed in a bath of soap solution and withdrawn from the bath a soap film will form between the two plates. It will reach the an equilibrium configuration in which the area, and hence the length, of the film will be a minimum (in the image shown below) the angle between two soap films at an intersection is 120° and its total length is which is about 4% shorter than the crossroad system.

It is a general property of the solutions to these problems that they consist of straight lines forming a number of intersections. These intersections always contain three straight lines with adjacent lines intersecting at 120°.

The number of intersections, in the case of n points, will be in the range zero to a maximum of (n-2).

Minimum Surface in Three Dimensions

A soap film contained by a fixed boundary will acquire a minimum area. Consequently soap films can be used to solve mathematical problems requiring the minimization of a surface area contained by a boundary. In order to obtain analogue solutions we require a frame to form the boundary of a surface. When the frame is withdrawn from a bath of soap solution a soap film will form which will attain its minimum area configuration on reaching to equilibrium.

The best choices of framework for bounding the soap film are perhaps those with the highest symmetry.

The images below show the minimal surface of the following forms: the tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron all of which are platonic figures with regular faces, all congruent, with equal face angle at every vertex and all the angles between adjacent faces are equal.

Joseph Plateau discovered experimentally, over a hundred years ago, that soap films cofttained by a framework always satisfy three geometrical conditions:

1. Three smooth surfaces of a soap film intersect along a line.

2. The angle between any two tangent planes to the intersecting surfaces, at any point along the line of intersection of three surfaces, is 120°.

3. Four of the lines, each formed by the intersection of three surfaces, meet at a point and the angle between any pair of adjacent lines is 109°28'.

The conditions given by Plateau apply to surfaces bounded by any frame. These surfaces do not have to be planar and the lines of soap film do not have to be straight. It is only recently that Frederick J. Almgren Jr. and Jean E. Taylor have shown that these conditions follow from the mathematical analysis of minimum surfaces and surfaces containing bubbles of air or gas at different pressures, both of which can be described by the Laplace- Young differential equation.

There are a few interesting web pages on this subject:

http://www.susqu.edu/FacStaff/b/brakke/evolver/evolver.html

http://hometown.aol.com/Surfacant/index.html

http://hometown.aol.com/Surfacant/index.html

http://www.physics.ohio-state.edu/~maarten/work/soapflow/soapflow.html

http://www-sfb256.iam.uni-bonn.de/grape/EXAMPLES/AMANDUS/bmandus.html

http://www.susqu.edu/facstaff/b/brakke/cones/cones.htm

This web page is only the tip of the iceberg of the science of soap films, if you wish to extend your knowledge you can do so with the help of following books:

1. Boys, C.V. Soap Bubbles and the Forces, Which Mould Them. New York: Doubleday & Company, Inc.,1959. This brief book is a classic of scientific literature. It contains three lectures that Boys delivered before a juvenile audience in 1889 and 1890. Boys describes a number of experiments that anvone can use to demonstrate the effects of surface tension.

2. Isenberg, Cyril. The Science of Soap Film and Soap Bubbles. Somerset, England: Woodspring Press Ltd., 1978 If you are serious about your bubbles, this is a very good college-level book with lots of physics and mathematical exposition on bubbles and minimal surfaces and their applications to the real world.

3. Lovett, David. Demonstrating Science With Soap Films Bristol & Philadelphia: Institute of Physics ISBN 0 7503 0269 0 If you are super-serious about your bubbles, his book will tell you about the chemistry of bubbles, minimal surfaces, the black film. A good history of bubbles is also included.

4. Noddy, Tom. Tom Noddy's Bubble Magic Philadelphia, Pennsylvania: Running Press, 1988 If you are less serious about your bubbles and just want to have fun (and learn some science at the same time) Tom will teach you to do all kinds of cool things with bubbles.

5. Frederick J. Almgren, Jr. and Jean E. Taylor, 'The Geometry of Soap Films and Soap Bubbles', Scientific American, July 1976, 82-93.

6. Jean T. Taylor, 'The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces', Annals of Mathematics 103 (1976), 489-539 See? Real scientists take bubbles seriously too!