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.

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__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.

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__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.

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__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.

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__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).

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__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!