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The world is full of minimisers: civil servants are economical with
the truth, engineers want to cut down the weight of aircraft, bees use as
little wax as possible. And at low temperatures at least, everything tends
to end up in a state of minimum energy. When it comes to searching for geometrical
forms that minimise energy, or some corresponding geometrical quantity such
as surface area, however, the problem requires a lot more than minimal effort
by mathematicians and computer scientists.

Sometimes the minimal solution is obvious: the shortest distance between
two points, for instance, is a straight line. But things become a good
deal more difficult when you start dealing with three-dimensional shapes.
Perhaps the best place to start is the shape that minimises the surface
area for a given volume – a sphere. Take a large number of spheres and you
end up with a froth made of identically sized bubbles. But these bubbles
become deformed as they crowd together at the top of the liquid. Indeed,
the liquid may drain out of them to the point where there is practically
none left and the bubbles take up many-faceted angular shapes in order
to fit together. The overall energy is still governed by their total area.
But what ideal shapes minimise it?

This is more than an interesting geometrical teaser. It is the starting
point for an attack on a real problem: foams consisting of random mixtures
of bubbles of widely different sizes. Such foams abound in the pub, in the
kitchen and in chemicals plants. They are of intense interest to food technologists
as they bake, brew and extrude foaming liquids. But useful foams remain
poorly understood – there simply is no theory that adequately describes
the head on a pint of beer or the foam on a glass of champagne.

Scattered light

One reason is that observing and measuring the complex structure of
such a liquid, or ‘wet’ foam, is extremely difficult. As you pour yourself
a glass of beer, you can see the bubbles surge upwards and form a head.
The bubbles jostle one another to seek a stable arrangement before they
burst and collapse. But what you see is mainly what happens at the surface
of the foam, close to the glass, and many foams – Guinness, shaving foam
and whipped cream, for example – are frustratingly opaque. Even under a
microscope, structural details beyond the first couple of layers of bubbles
often vanish in a fog of scattered light. To find out what is happening
within the foam, and to gather this information in a convenient statistical
form, some researchers are trying to glean information from this scattered
light. Others are experimenting with nuclear magnetic resonance scanning,
a technique better known for its medical applications.

While such information accumulates, research-ers such as our group at
Trinity College, Dublin are attempting to model foams computationally. But
even the comparatively simple task of modelling an ideal froth of equal-sized
bubbles can still spring a few surprises a century after William Thomson
(later Lord Kelvin) seemed to have solved the problem.

Kelvin’s theoretical solution of 1887 was built on his interest in crystal
structures and a knowledge of the basic rules of bubble structures. These
had been laid down in the previous decade by Joseph Plateau, a Belgian physicist
who wrote the definitive work on soap films, bubbles and foams in 1873.
From his observations of soap films, which he made by dipping wire frames
of various shapes in soap solution, Plateau devised a set of rules that
dictate that only three soap films can meet to form lines and that only
four of these lines can meet at a point. Any structure that breaks these
rules is unstable and will spontaneously rearrange. You can observe these
rules at work in the head on a glass of Pilsner, or any other source of
reasonably large bubbles.

Crystal lattice

Plateau’s rules left Kelvin with hardly any choice among the possible
structures in which all the bubbles have the same shape and orientation.
They rule out the face-centred cubic structure, for example. Kelvin chose
what we would call the body-centred cubic lattice, a common crystal structure
which occurs in metals such as iron. Now all he had to do was fix the precise
shape of the bubbles, as this was not entirely set by the requirement that
they fit into the body-centred cubic lattice.

In principle, Plateau had the answer to this too. The surfaces separating
neighbouring bubbles must meet each other at an angle of 120 degrees, as
in a honeycomb, and if the bubbles are identical they must have zero curvature
overall. This does not mean they have to be flat: they can have equal and
opposite curvatures in two directions at any point, like some potato crisps.

So Kelvin came up with a shape that did the job: a tetrakaidecahedron
– that is, a shape with 14 faces. Of these, the square ones are flat and
the hexagonal ones wavy. He made a large wire model of this geometrical
structure, later nicknamed ‘Kelvin’s bed-spring’, which still exists at
the University of Glasgow. This was typical of Kelvin’s down-to-earth approach.
He once advised aspiring students of crystallography to ‘contract with a
wood turner or a maker of beads for furniture tassels or for rosaries, for
a thousand wooden balls of about half an inch diameter each’. But strangely
for such a practical scientist, Kelvin makes no mention of any attempt to
observe his structure in real foam.

It was D’Arcy Wentworth Thompson who advertised its virtues. His book
of 1917, On Growth and Form, is largely devoted to minimal forms in nature.
In his enthusiasm for Kelvin’s structure, Thompson rashly assumed that a
practical demonstration of it was a trivial matter. And there the matter
rested until the 1940s when Edwin Matzke, an American botanist, decided
to test out Thompson’s claim that Kelvin’s structure could be realised ‘in
the twinkling of an eye’ in a foam.

Matzke went to the trouble of making an ‘ideal’ foam by manually assembling
thousands of large equal bubbles, blown one by one with a syringe. Though
he failed to find a single Kelvin cell in the disordered mass that emerged,
most people were still inclined to accept that Kelvin had arrived at the
right solution for the most stable form of a foam. For one thing, Matzke’s
experimental procedure was suspect, because it took a day or so to complete.
Real foams are ephemeral, as anyone who has lain long enough in a bubble
bath can testify. Even those bubbles that do not burst still disappear as
the gas diffuses between them, causing some of them to shrink and vanish.

And then there were sound physical reasons why Kelvin’s cell might remain
elusive in a real foam, as opposed to an ideal, theoretical one. Atoms and
molecules may have enough thermal energy to search for ordered arrangements
that will lower their energy, by jumping repeatedly between alternative
arrangements – Kelvin talked of their ‘molecular tactics’. But bubbles do
not hop around in this manner because they have too much mass. They are
trapped in whatever stable structure they first fall into. So bubbles cannot
be expected to achieve their ideal any more than a bag of marbles will
form the ordered closely packed array that is the ideal theoretical solution
to the problem of packing spheres with maximum density.

But even if you accept Kelvin’s tentative solution for ‘dry’ foam (one
that contains little liquid), the ideal structure of ‘wet’ foams is something
that researchers are still investigating. Late last year Robert Phelan,
a research student at Trinity College Dublin, was trying to find out what
combination of faces would have the lowest energy for foams containing different
proportions of liquid. To do this he was computing the surface area of possible
structures using software developed by Ken Brakke of Susquehanna University
in Pennsylvania, which represents curved surfaces in terms of a large number
of flat tiles.

Dry foam geometry

Phelan and other researchers in our group were amazed to find that
the first alternative we examined had a surface area 0.4 per cent less
than Kelvin’s solution for an ideal dry foam. This may not sound much,
but the Kelvin structure itself only surpasses the face-centred cubic structure,
which is not even a stable structure for an ideal foam, by 0.8 per cent.

In Kelvin’s structure, each of the bubbles lies at one of the points
of a lattice, so all the bubbles are equivalent. In Phelan’s more complicated
structure, two different cell shapes lock together in a unit of eight cells
within a repeating simple cubic lattice. The centres of the cells lie in
the same positions as the atoms in one form of the element tungsten which,
along with many alloys, can adopt this structure. A quarter of the cells
have 12 slightly curved pentagonal faces. The other cells have 14 faces,
the two extra faces being hexagons.

This gives our new structure an average of 13.5 faces per cell, a figure
that caught the attention of Rob Kusner, a mathematician at the University
of Massachusetts at Amherst. Like others before him, Kusner is fascinated
by the number 13.397 . . ., which is the average number of faces a polyhedron
would have if all its faces were flat, all its angles equal and it could
be packed together to fill space. Such a polyhedron cannot exist, because
a real polyhedron must have an integer number of faces, but it is perhaps
an appropriate average to aim for in a minimal structure. In 1992, Kusner
showed mathematically that the average number of faces in the bubbles of
a foam in which all pressures (rather than volumes) are equal cannot be
less than 13.39.

Face value

Kusner and John Sullivan of the University of Minnesota quickly produced
an analytical proof – using pencil and paper rather than computers – that
our new structure is superior to Kelvin’s. But this does not necessarily
mean that it is the best. Nicholas Rivier, a physicist at the University
of Strasbourg, points out that this structure is a member of the crystallographic
family called the Frank-Kasper phases. He says that some of its cousins
come much closer to the ideal average number of faces. We believe that this
is not enough: the number of faces of each individual cell should come as
close as possible to the ideal.

Others have recognised the structure as an old friend in a new guise.
Two French physicists, Jean-Francois Sadoc of the University of Paris at
Orsay and Jean Charvolin of the Laue-Langevin Institute in Grenoble came
up with it in the late 1980s to explain a mysterious diffraction pattern
observed in liquid crystals. But they dismissed it as a candidate for foams,
attributing its existence to the special forces associated with liquid crystals.
Chemists also know the structure well. It is that of many clathrate compounds,
cage-like structures in which the vertices are atoms, the lines are covalent
bonds and other guest atoms reside comfortably in the ‘cages’ which correspond
to our cells. We knew of such compounds and realised that they offered excellent
candidates for foam structures of low surface area.

Near hit

Pierre Clement Grignon, a French artillery engineer, might almost have
recognised it too. In 1775, he gave a detailed specification of what he
took to be the typical form of a grain of steel. It has 14 sides and is
not far removed in shape from the predominant form that occurs in our ideal
structure.

But could we find our structure in the laboratory? When we made a foam
of equal-sized bubbles by blowing them from a nozzle immersed in a soap
solution, we found a perfect structure of Kelvin cells near the glass walls
of the vessel containing it. This was very surprising as Matzke had looked
specifically at the surface region, and it convinced us that his experiment
had indeed been unreliable. For a soap froth of bubbles about a millimetre
in diameter, it is just about possible to see down to the fourth layer,
and the Kelvin structure extended over only the first two. Beyond that,
it became disordered. Among this disordered mass we managed to photograph
what appeared to be small fragments of our new structure. Further work will
reveal just how common these fragments are and whether they can be made
to grow into larger regions. With the benefit of hindsight we can now see
how Kelvin cells adapt particularly well to fitting against the glass wall
of the container and are favoured there.

The new structure might also turn up in emulsions – which are in effect
liquid foams in which the bubbles are filled with a second liquid instead
of a gas. The essential physics is identical. Jean Bibette of the University
of Bordeaux has developed techniques for making the droplet size uniform
and David Weitz of the petrochemicals company Exxon in New Jersey is measuring
structures and elastic constants of such emulsions. The motivation for such
work is similar to ours: it just depends on whether you prefer beer or mayonnaise.

Soon our group will return to studying the real world of mixtures of
bubbles of different sizes – an enormous computational challenge. Our aim
is to be able some day to capture and simulate the dynamics of champagne
bubbles, and use this picture to produce designer foam. Meanwhile, a toast
to Lord Kelvin . . .

Denis Weaire is professor of natural and experimental philoscophy in
the Department of PUre and Applied Physics at Trinity College, Dublin.