FOAMS and BUBBLES: GEOMETRY and SIMULATION 1. Introduction 2. CMC Surfaces and Forces

International Journal of Shap e Mo deling, Vol. 0, No. 0 1998 000{000

cWorld Scienti c Publishing Company

FOAMS AND BUBBLES:

GEOMETRY AND SIMULATION

JOHN M. SULLIVAN

Department of Mathematics, University of Il linois

Urbana, IL, USA 61801-2975

Received July 1998

Revised April 1999

Communicatedby Michele Emmer

The geometry of soap lms, bubble clusters and foams has a nice mathematical descrip-

tion, though many op en problems remain. Recently, computer simulations, and their

graphical output, have led to some new insights.

Keywords :Foams, soap lms, minimal surfaces, computer simulation.

1. Intro duction

Soap lms, bubble clusters, and foams and froths can b e mo deled mathematically

as collections of surfaces which minimize their surface area sub ject to volume con-

straints. Each surface will have constant mean curvature, and is thus called a cmc

surface. Section 2 intro duces cmc surfaces, and describ es the balancing of forces

within them. Section 3 lo oks at existence results for bubble clusters, and at the

singularities in such clusters and in foams. These are describ ed by Plateau's rules,

which imply that a foam is combinatorially dual to some triangulation of space.

Among the most interesting triangulations, from the p oint of view of equal-volume

foams, are those of the tetrahedrally close-packed tcp structures from chemistry,

describ ed in Section 4. Finally, Section 5 discusses the application of these ideas to

Kelvin's problem of partitioning space.

2. CMC Surfaces and Forces

Wewant to examine the geometry of a surface like one in a foam which minimizes

its surface area, given constraints on the enclosed volume. Rememb er that a surface

in space has at each p oint two principal curvatures k and k ; these are the

1 2

minimum and maximum over di erent tangent directions of the normal curvature.

The mean curvature is their sum H = k + k twice the average normal curvature.

1 2

A p erturbation of a surface M is given byavector-valued function ~v on M ,

telling how each p ointistomove. Let us consider the rst variations of area A and

volume V under this p erturbation. If ~n is the unit normal to M and H = H~n is 1

2 John M. Sul livan

the mean curvature vector, then

Z Z

V = ~v ~n dA; A = ~v H dA;

M M

so wesay that ~n is the gradientofvolume and H is the gradient of area. If we write

H = A= V ,we obtain a second, variational, de nition of mean curvature.

These Euler-Lagrange equations show that a surface is critical for the area func-

tional exactly if H 0. Such a surface is called a minimal surface, and indeed any

small patch of it is then area-minimizing for its b oundary.Any minimal surface

is saddle shap ed, as in Fig. 1. Therefore, a plane broughttowards a patchofa

minimal surface must touch rst at the b oundary of the patch, not in the interior.

More generally, the maximum principle says that the same happ ens whenever any

two minimal surfaces are brought together.

Fig. 1. On the left, the least-area surface spanning xed edges of a cub e. As for any minimal

surface, near each p oint it is saddle shap ed, with equal and opp osite principal curvatures. On the

right, three minimal surfaces meet along a Plateau b order at 120 . This surface would b e achieved

by soap lm spanning three wire Vs; when extended bytwo-fold rotation ab out these b oundary

lines, it would form the Kelvin foam.

A soap lm spanning a wire b oundary, with no pressure di erence across it,

will b e a minimal surface, but in general we need to consider the e ect of a volume

constraint. In this case, we nd that the surface has constant mean curvature H c,

so it is called a cmc surface. Here the constant c is the Lagrange multiplier for the

volume constraint, which is exactly the pressure di erence across the surface; this

relation H =p is known as Laplace's law.

In physical bubble clusters or foams, each cell contains a certain xed amount

of air or other gas neglecting slow di usion e ects. Of course, air is not an

incompressible uid. But for our mathematical theory,we will usually pretend it

is, xing a volume rather than a mass of gas in each cell. This approximation is

no problem: no matter what the relation b etween pressure and volume is for the

real gas, the equilibrium structures will still b e cmc surfaces.

Foams and Bubbles: Geometry and Simulation 3

For instance, a single bubble will adopt the form of a round sphere. This isoperi-

metric theorem, that a sphere minimizes surface area for its enclosed volume, was

known to the ancient Greeks. Indeed, a famous story credits Queen Dido with

making use of the two-dimensional version when founding the city of Carthage.

But a prop er mathematical pro of whichmust demonstrate the existence of a mini-

mizer rather than merely describing prop erties of one if it do es exist was only given

in the last century bySchwarz [1], using a symmetrization argument.

One of the most imp ortant ideas in the mathematical study of cmc surfaces

is that of forcebalancing, as describ ed by Korevaar, Kusner and Solomon [2]. In

general, No ether's theorem says that symmetries of an equation lead to conserved

quantities. The equation for a cmc surface is invariant under Euclidean motions;

the corresp onding conserved quantities can b e interpreted as forces and torques.

The physical interpretation is as follows. Start with a complete emb edded cmc

surface M , and cut it along a closed curve . The two halves of M pull on each other

with a force due to surface tension and to pressure exerted across an arbitrary disk

D spanning . If ~n is the normal to D , and ~ is the conormal to in M that is,

the unit vector tangenttoM but p erp endicular to , the force can b e written as

Z Z

F = ~ H ~n:

Here we think of pressure p = 0 outside and p = H inside. Torques around any

origin can b e de ned in a similar fashion.

The fact that M is in equilibrium, with surface tension balancing pressure dif-

ferences, means that no compact piece feels any net force or torque. If and are

homologous curves on M , their di erence b ounds such a compact piece, so their

forces must b e equal. That is, F actually dep ends only on the homology class

of .

The cmc surfaces of revolution, called unduloids,were rst studied by Delau-

nay [3] in the last century. An unduloid is a wavy cylinder as in Fig. 2; the force

vector, for a curve lo oping around the cylinder, necessarily p oints along the axis

of revolution. The family of unduloids for given H can b e parameterized by the

strength of this force, which is maximum for a straight cylinder. In fact, Delaunay

Fig. 2. Left, a typical unduloid a cmc surface of revolution, whose force vector p oints along its

axis. Right, a typical triunduloid emb edded cmc surface with three ends. Each end is asymptotic

to an unduloid; the three force vectors sum to zero.

4 John M. Sul livan

describ ed the generating curves for unduloids as solutions of a second-order ODE,

and the force is a rst integral for this equation. As the force decreases from that of

a cylinder, we see alternating necks and bulges. The necks get smaller and smaller,

like shrinking catenoids, and in the limit of zero force, the unduloid degenerates

into a chain of spheres, as in a b eaded necklace.

The idea of force balancing has b een very useful in the classi cation of complete

emb edded cmc surfaces. The only compact example is the round sphere [4], with

all forces equal to zero. In general, each end of such a surface is asymptotic to an

unduloid [2]. Thus it has a force vector asso ciated to it, directed along the axis of the

unduloid, pulling the \center" of the surface out towards in nity. These force vectors

on the ends must sum to zero. This balancing condition is the essential ingredient

in classifying such cmc surfaces, and combined with some spherical trigonometry

can lead to a complete classi cation of the three-ended surfaces [5, 6].

3. Bubble Clusters and Singularities in Soap Films

So far, wehave b een rather cavalier ab out assuming that solutions to our problem

of minimizing surface area with constraints exist and are smo oth surfaces hence

of constant mean curvature. Of course, the surfaces in a foam or bubble cluster

are only piecewise smo oth, and the pieces meet in the singularities observed by

Plateau [7].

Almgren formulated the bubble cluster problem as follows: given volumes V ,

.. ., V , enclose and separate regions in space with these volumes, using the least

total area. He then proved, using techniques of geometric measure theory, that a

minimizer exists and is a smo oth surface almost everywhere [8] see also [9].

The pro of do es not giveany control on the top ology of the regions; in fact we

are not allowed even to assume that each region is connected. For all we know,

it may b e that the b est waytomake a cluster with some four xed volumes is to

split one of them into two pieces with the correct total volume, and make what

lo oks like a cluster of ve bubbles. A reasonable conjecture, of course, is that in

any minimizer, each region is connected.

Almgren's result do es, however, say something ab out the shap e of the mini-

mizers: they are so-called M ; "; -minimizing sets. In the plane, an M ; "; -

minimizing set is a nite network of smo oth curves meeting in threes at equal

120 angles. Note that the deformation shown in Fig. 3 of an X into a double Y

which is the Steiner tree on the four vertices of a square shows that the X is not

minimizing|it can b e improved at any small scale.

Of course, the mo dels we use here are meant to describ e dry foams; in a wet foam

with p ositive liquid fraction '>0, there are Plateau b orders near each singularity.

In this case one mightwonder if additional singularities might b e p ossible. In fact,

awet X is stable for large ', though Brakke and Morgan [10] have shown that it

is unstable for small p ositive '. The analogous question is three dimensions is still

op en.

In three-space, Taylor was able to show [11] that the singularities observed

Foams and Bubbles: Geometry and Simulation 5

Fig. 3. The X junction at the left is not an M ; "; -minimizing set in the plane b ecause it can

deform at an arbitrarily small scale to the double Y at the right. In the plane, the Y is the only

singularity allowed.

empirically in soap lms by Plateau are the only ones p ossible in an M ; "; -

minimizing set. That is, such a minimizer consists of a nite union of smo oth

surfaces, meeting in threes at 120 dihedral angles along a nite numb er of smo oth

curves; these curves in turn meet at nitely many tetrahedral corners whichlook

like the soap lm obtained when dipping a tetrahedral frame into soapywater: six

sheets come together along four triple curves into the central singularity.

In addition to proving these Plateau rules,Taylor showed each smo oth piece

has constant mean curvature, and the mean curvatures satisfy a cocycle condition:

numb ers the pressures can b e assigned to each comp onent of the complement

such that the mean curvature of eachinterface is the di erence of these pressures.

Although we cannot guarantee that each region of xed volume is a single comp o-

nent, the di erent comp onents making up a single region will necessarily have the

same pressure.

Of course a minimizing cluster of just one region will b e a round sphere. It is

surprising, though, how little is known ab out other small clusters. The case of two

regions, a double bubble, is one of the geometrical problems where \many mathe-

maticians b elieve, and all physicists know," that the obvious solution is correct.

The b est double bubble should b e the one seen when playing with soap bubbles,

with three spherical caps meeting along a single circular triple junction. Surpris-

ingly, this is not yet a theorem. However, Hutchings showed following a suggestion

of White that the minimizer must have rotational symmetry, and gave further con-

ditions, which, in the case of equal volumes left only one family of comp etitors, as

in Fig. 4 right. Hass and Schla y were able to rule these out with a computer

search [14] see also [15], thus solving the equal-volume case.

We can de ne a foam mathematically as a lo cally nite collection of cmc

surfaces, meeting according to Plateau's rules, and satisfying the co cycle condition.

We will call the comp onents of the complement the cel ls of the foam, call the

interfaces b etween them simply the faces, call the triple junction lines where they

meet the Plateau borders, and call the tetrahedral singularities the corners.We

The quote is from Rogers [12], describing Kepler's sphere packing problem, which has recently

b een rigorously solved by Hales [13].

6 John M. Sul livan

Fig. 4. At the left we see the standard cluster of three soap bubbles, with all interfaces spherical

and singularities following the Plateau rules. Presumably it minimizes total area for its enclosed

volumes. At the rightwe see a p ossible comp etitor for the least-area double bubble, where one

region is a toroidal b elt around the other.

exp ect this de nition to describ e all physical foams whose shap e is determined

by surface tension, though there are not mathematical results like Almgren's that

would apply to the case of in nite clusters.

For many purp oses, we can ignore the geometric parts of Plateau's rules, and

only pay attention to the combinatorial asp ects of the foam's cell complex. The

combinatorial rules mean that the dual cel l complex to a foam is in fact a simplicial

complex, that is, a triangulation of space. To construct this dual, we put a vertex

in each cell of the foam. Vertices in a pair of adjacent cells are connected byan

edge, which is dual to a face of the foam. Where three faces come together along

a b order, we span the three corresp onding edges with a triangle; and where four

b orders come together at a corner, we ll in a tetrahedron.

Combinatorially, the Plateau rules mean that a foam and its dual triangulation

are likeaVoronoi decomp osition of space and the dual Delone triangulation. Given

a set of sites in space, the Voronoi cell [16, 17] for each site is the convex p olyhedron

consisting of p oints in space closer to that site than to any other. The dual Delone

complex, whose vertices are the original sites, has the prop erty that each tetrahedron

has no other sites inside its circumsphere. An example in the plane is shown in

Fig. 5.

This similarity suggests that that we might lo ok for foams as relaxations of

Voronoi decomp ositions. For instance, we can give a mo dern interpretation to

Kelvin's construction [18] of his candidate for a least-area partition of space into

equal volume cells as follows. Start with sites in the b o dy-centered cubic bcc

lattice. Their Voronoi cells are truncated o ctahedra, packed to ll space. If we let

the lms in this packing relax until the geometric parts as well as the combinatorial

Foams and Bubbles: Geometry and Simulation 7

Fig. 5. The heavy lines show a sample Voronoi partition in the plane, for the sites marked with

dots. The thin lines are the edges of the dual Delone triangulation, which connect sites with

adjacentVoronoi cells.

parts of Plateau's rules are satis ed we should get a p erio dic foam, the one prop osed

by Kelvin.

Starting from other p erio dic arrays of sites, we can carry out the same pro cedure

to generate a foam. Mathematically, there is no theory for the relaxation step,

whichwould follow a mean-curvature ow for surfaces with triple junctions and

volume constraints. But in practice this pro cedure is a useful way to compute

foam geometries. The author's vcs software for computing Voronoi diagrams [19]

in three dimensions can send its output to Brakke's Surface Evolver [20]. This latter

program can interactively manipulate, re ne and evolve triangulated surfaces into

energy-minimizing con gurations, mo deling the physical e ects of surface tension

and pressure. New symmetry features in the Evolver [21] allow computation of

just one fundamental domain in a foam of arbitrary symmetry,thus giving greater

accuracy for given computational e ort.

4. TCP structures

The tetrahedral angle arccos 1=3 109:47 , found at all foam corners, is b etween

the average angle of a p entagon 108 and that of a hexagon 120 . Thus foams

with fairly straight b orders are likely to have mostly p entagonal and hexagonal

faces.

Chemists have studied transition metal alloys in which the atoms are packed in

nearly regular tetrahedra. These tetrahedrally close-packed tcp structures were

rst describ ed byFrank and Kasp er [22, 23], and have b een studied extensively by

the Sho emakers [24] among others. In all of these structures, the Voronoi cell of

each atom has one of four combinatorial typ es; these are exactly the four p olyhedra

whichhave only p entagonal and hexagonal faces, with no adjacent hexagons. Math-

ematically,we can de ne of tcp structures in this way [25]: triangulations whose

8 John M. Sul livan

combinatorial duals called tcp foams have only these four typ es of cells. Since

this de nition deals only with the combinatorics of the triangulation, presumably it

includes some examples that havetoomuch geometric distortion to work chemically

as tcp structures.

Fig. 6. There are four typ es of cells found in tcp foams, each with 12 p entagonal faces like the

do decahedron at the left. The remaining three have in addition two, three or four hexagonal faces,

arranged antip o dally, equatorially, and tetrahedrally.

Each of the four typ es of cells found in tcp foams has 12 p entagonal faces, as

seen in Fig. 6. One typ e is the p entagonal do decahedron. Cells of the other three

typ es have additionally two, three, or four hexagonal faces, which are arranged

antip o dally, equatorially, or tetrahedrally resp ectively. The 14-hedron can b e

viewed as a \ -fold unwrapping" of a do decahedron, since it has six-fold symmetry

through the centers of the opp osite hexagons.

It is an interesting op en question just what foams are p ossible with these four

typ es of cells. Dually,we are asking for triangulations of space in which each

edge has valence ve or six, and no triangle has two edges of valence six. We

knowwe cannot use only do decahedra, since a foam made of these alone meeting

tetrahedrally will ll a spherical space, not Euclidean space.

Fig. 7. The three basic tcp structures. A15, left, has two 12-hedra one upp er-left and six

14-hedra in colums likelower right in a cubic cell. Z, center, has three 12s top, two 14s in

vertical columns but separated in the gure, with unequal horizontal hexagons, and three 15s

with vertical hexagons, one lower front. C15, right, has four 12s and four 16s with hexagons

darkened in a cubic cell.

Chemists have describ ed three basic p erio dic tcp structures, shown in Fig. 7.

A15, observed for instance in Cr Si, has one 12-hedron and three 14-hedra in a 3

Foams and Bubbles: Geometry and Simulation 9

fundamental domain. Z, observed in Zr Al , has three 12-hedra, two 14-hedra, and

4 3

two 15-hedra. And C15, observed byFriauf and Laves in MgCu , has two 12-hedra

and one 16-hedron.

If we describ e p otential tcp structures by the ratio of cells they have of the four

typ es, we could plot them all within an abstract tetrahedron, as convex combina-

tions of the four vertices. But the observation of Yarmolyuk and Kripyakevich [26]

is that all known tcp structures are, in fact, convex combinations of the three ba-

sic ones just mentioned, so they all get plotted within the triangle of Fig. 8. The

F M

C15

A15 C14 SM

Fig. 8. The basic tcp structures A15, Z, and C15 lie at the corners of this triangle, where the

1 1

to 13 and the horizontal axis plots the average numb er of faces p er cell ranging from 13

3 2

vertical axis plots the fraction of cells which are 15-hedra ranging from 0 to . Every known

tcp structure, when describ ed by its numb ers of 12-, 14-, 15-, and 16-sided cells, is a convex

combination of these three. Distinct structures may app ear at the same p oint, but need not then

have the same prop erties.

Sho emakers [24] attempt to explain this observation by noting that for a tiling by

tetrahedra only slightly distorted from regular, we exp ect to have z 13:397. Cer-

3 1

for A15, z =13 for Z, and tainly the three basic tcp structures, with z =13

2 7

z =13 for C15, are close to this value.

It is interesting to note that there are other chemical structures which exhibit

the structure of the tcp Voronoi cells that is, the dual tcp foams more explicitly.

These are describ ed in detail by O'Kee e [27], but we outline the ideas here. In

clathrates, large gas molecules are trapp ed at the lo cation of the tcp sites inside

water cages: oxygen atoms sit at the Voronoi corners, b onded byhydrogen along

the b orders Voronoi edges. For instance, chlorine hydrate Cl 46H O, has the

2 6 2

structure of Voronoi cells for A15, called a Typ e I clathrate. Many salt hydrates

have instead the structure of the Z foam called Typ e I I I, while others use non-tcp

structures like the bcc cells of the Kelvin foam. Some zeolites have similar struc-

tures, with Si and p ossibly some Al atoms at the corners, b onded tetrahedrally to

oxygen atoms along the b orders. For instance so dalite Na Al Si O Cl has the bcc

4 3 3 12

foam structure, with Cl atoms at the Voronoi sites, and Na in the centers of the

hexagonal faces. So dium silicide, when heated prop erly, can generate silicon cages

in the A15 or C15 foam pattern Typ e I or Typ e I I clathrates with so dium trapp ed

10 John M. Sul livan

inside. It should b e noted, however, that in other zeolites the 4-connected silicate

nets are less dense and do not close up into cages. See [28] for more information

on manyinteresting crystal structures, and [29] for instance for the structures of

zeolites.

5. Kelvin's problem

Over one hundred years ago, Kelvin prop osed the problem of partitioning space

into equal-volume cells using the least interface area p er cell [18]. He suggested

that the solution might b e what wehave b een calling the Kelvin foam, a relaxation

of the Voronoi diagram for the bcc lattice. Although Weaire and Phelan nowhave

a b etter partition, Kelvin's is still conjectured [30] to b e the b est if the cells are

required to b e congruentortohave equal pressure. Mathematically, there is no

theory to suggest that such a b est in nite cluster should exist, but we exp ect that

it will, and will have the structure of a foam.

Weaire and Phelan [31] were the rst to consider using tcp structures as foams.

Although they initially thought of these as mo dels for wet foams, they quickly

discovered that the dry A15 foam was a more ecient partition of space than

Kelvin's candidate. Togobeyond their go o d numerical evidence from Brakke's

Evolver and give a rigorous pro of that their foam is b etter than Kelvin's, we need

to give a b ound on howmuch the Kelvin foam can relax.

Although, as wehave mentioned, there is no general mathematical theory for

the relaxation step in constructing foams from Voronoi cells, for Kelvin's foam

there is enough symmetry that we need only consider mean-curvature owona

single surface. In Fig. 1, right, wesaw a symmetric unit of Kelvin's foam, b ounded

by lines of rotational symmetry. In that picture, the vertical sheet is in a mirror

plane, so we need only solve for one of the other sheets, a minimal surface with two

xed b oundary lines and one free b oundary with 120 contact angle. We can use

this analysis to show that a unique foam exists in Kelvin's pattern, and a slicing

argument then givesalower b ound on its area [25]. This b ound, it turns out, suces

to prove that even the unrelaxed Weaire-Phelan A15 foam b eats Kelvin. The Color

Plate shows a view of the A15 foam from the inside, using the author's computer

graphics rendering of realistic soap lms [32].

Given this example, it is natural to lo ok for go o d equal-volume partitions among

the other tcp foams. Rivier [33] prop osed that since C15 has among the known tcp

structures the lowest z and thus the highest prop ortion of p entagons, it might give

an even b etter partition. Instead, computer exp eriments [34] suggest that among

all tcp foams, A15 is the most ecient, and C15 among the least. Here, we are

lo oking at equal-volume foams. For each pattern like Z without cubic symmetry,

we adjust the lattice parameters to get rid of any stress tensor; this ensures that we

have the most ecient partition in that pattern. Probably the 12-hedral and 16-

hedral cells of C15 naturally have such di erent sizes that distorting them to make

the volumes equal ruins whatever advantage p entagons give. If we considered a

mo di ed Kelvin problem where one-third of the cells were to have somewhat larger

Foams and Bubbles: Geometry and Simulation 11

volume, then presumably the C15 structure would do very well. The author has

also made computer exp eriments with equal-volume foams generated from other

chemical structures; the foam from -brass, for instance, is b etter than that from

C15, despite having even some triangular faces. Perhaps p entagons are not as

desirable in foams as has b een assumed.

Acknowledgements

The author is partially supp orted by NSF grant DMS-9727859 and NASA grant

NAG3-2122. He would like to thank Hassan Aref, Karsten Groe-Brauckmann,

Andy Kraynik, Rob Kusner, Frank Morgan, Mike O'Kee e and Nick Rivier for

helpful conversations on foams and related structures. Much of this pap er is based

on the author's rep ort [35] on lectures he delivered at the NATO scho ol on foams

in Cargese, and some of the gures originally app eared in [25].

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oams and Bubbles: Geometry and Simulation 13

Color Plate John Sullivan: This computer-graphics image shows the Weaire-Phelan foam, ob-

tained by relaxing the tcp structure A15. This foam is, as far as we know, the b est partition of

space into equal-volume cells. It is rendered realistically as soap lms.