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

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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 f 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: D 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- 0 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 , 1 .. ., V , enclose and separate regions in space with these volumes, using the least k 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.
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