A Philomorph Looks at Foam1 - ANU

A Philomorph Looks at Foam1 DENIS WEAIRE Erasmus Smith Professor of Natural and Experimental Philosophy, Trinity College Dublin

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HE WORD PHILOMORPH was coined in the 1960s by a group of people who met from time to time in Cambridge, Massachusetts, to discuss form in nature, science, and art. One of the leaders of that group was Cyril Stanley Smith. Some of my own instincts and principles in research derive directly from Smith, so I have called myself a philomorph on this occasion. Smith reacted against the direction taken by physics—particularly solid state physics—in the twentieth century. He said that its “logical, analytical, reductionist” approach was far too narrow. Its paragon was the single crystal. Only very well defined and controlled local departures from its perfect regularity were considered to be respectable subjects for research. Yet Smith saw that his own subject of metallurgy threw up a vast range of fascinating complex forms, mostly on the mesoscopic length scale—micrometres or greater—rather than the atomic scale of nanometres, which was the solid state physicist’s preoccupation. The phenomena that he admired, such as the patterned steel of a Damascus sword—a product of art, artifice, and nature—were of great utility and aesthetic value. As topics for research, these things had all the attributes that were abhorrent to many of his colleagues. They were often accidental, incidental, capricious. They were history-dependent (for which he coined the word funeous, after a character in a Borges story who tragically lost the capacity to forget). They were metastable, rather than representing true thermodynamic equilibrium. And they were not directly related to the primary physical laws such as Schrödinger’s equation. Furthermore his method was playful and eclectic rather than planned and focussed, and laid great emphasis on the power of analogy. He even called it “holistic,” but this may be misleading today. This was no New Age outcast from modern science: he wanted to broaden it, rather than to supplant it. 1 Read

27 April 2001.

PROCEEDINGS OF THE AMERICAN PHILOSOPHICAL SOCIETY

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VOL. 145, NO. 4, DECEMBER 2001

a philomorph looks at foam

Figure 1. Cyril Stanley Smith, Philomorph

Figure 2. Sir William Thomson (Lord Kelvin)

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Smith cried out as a voice in the wilderness (although Cambridge was not quite a wilderness). And it came to pass that his physicist colleagues did eventually see the point. As Smith said in 1977, quoting his wife, “things on man’s own scale are returning as a topic central to scientific (I think she really meant physical) enquiry.” This change is partly reflected in the change of the names of many journals and research groups from “solid state physics” to “condensed matter and materials science.” One branch of the new, wider subject is concerned with soft matter. I want to tell you a little about two examples of soft matter that are familiar from everyday life, particularly at the seaside—sand and foam. It happens that I can associate the names of two famous Irishmen with these: Sir Osborne Reynolds and Sir William Thomson, later Lord Kelvin. Sand Sand can stand for that class of materials—granular materials—that consist of hard particles, including all sorts of powders, cereal grains, and the like. The physicist may idealise it as a packing of identical hard spheres. So in one corner of the subject there is the Kepler problem—to find the densest packing of identical spheres. This seems to have been finally laid to rest by Thomas Hales in recent years. The greengrocer’s stack of oranges now has the official mathematical seal of approval as the densest possible. But sand, like most granular material, is random, and its random structure has strange properties. Its density depends on how you prepare it, which is why the Bible gives instructions on the meaning of good measure. Once it has been induced into settling into a dense (although still random) structure, it cannot be induced to flow or deform without a small expansion. It’s much the same when people want to get out of a tube train in London or Tokyo: some obliging passengers have to step off, to lower the density and allow others to go with the flow. Osborne Reynolds was the first to recognise this property, while walking on the seashore in the 1870s. So sand has properties intermediate between those of a solid and a liquid. Even the way that powder pours out of hoppers, or out of the top of an egg timer, is not what you would expect from elementary physics. Such processes do not obey the ordinary laws of liquids, and people are still struggling to find new equations for them. And why do Brazil nuts always come to the top in mixtures, when you’d expect them to sink? Reynolds had a great love of the physics of everyday and outdoor things, but he was not motivated by this alone—he also had an alto-


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gether grander goal. He was at that time in pursuit of the nineteenth century’s Holy Grail of physics. (Since everyone involved seemed to be a knight errant, this is an appealing metaphor.) The object of their quest was the all-pervading ether of space, the mysterious substance that was supposed to carry light waves as its vibrations. The trouble with the ether was that it seemed to have inconsistent properties: you could move through it, so it had to be a fluid. But it carried light as transverse vibrations, so it had to be a solid. Reynolds supposed that might be a granular material. J. J. Thomson, who attended Reynolds’s lectures in Manchester, said in his autobiography that the professor’s mind was weakening at this point. Indeed, his quest for the ether was quixotic to the point of dottiness, but it was typical of the times. Indeed, my next character tilted at the same windmill, just a decade of so later. He was lying in his bed in the splendid country chateau he had built in Ayrshire on the proceeds of his work on telegraphy—it was his mirror galvanometer that made the transatlantic cable of 1858 a success, and brought two continents into close communion. Waking at 7:15 in the morning, Sir William Thomson (later Lord Kelvin) decided that the ether of space must be a foam. Foam Kelvin’s flirtation with foam is just one short episode in an unparalleled career that spanned many decades and many subjects, but at the time it excited him greatly. He was undeterred by sceptical comments of colleagues such as George Darwin, who declared the idea to be “utterly frothy.” He applied himself to it with characteristic energy. In due course, even his wife was pressed into service to make a pin-cushion that would illustrate the ideal structure that he had conceived for the ether. Just as sand can stand for any granular material, made up of hard grains, so foam can stand for any cellular material, made up of soft bubbles. Foam structures occur, or are conjectured to do so, on every length scale, from the Planck scale (10 to the power of minus 35 metres) to that of the large-scale structure of the universe. On the everyday length scale of your favorite pub, foam is what we get when bubbles rise out of a liquid, such as beer. They pack together as polyhedra, whose curved faces are liquid films. The individual bubble has an eventful life. Born at a small pit in the glass, it rises and joins its colleagues in a wet foam at the surface of the liquid. Continuing to rise with its neighbours, the bubble is surrounded by less and less liquid, as the pull of gravity drains that liquid away. The bubble changes its shape from time to time in sudden topological re-

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arrangements, and it grows or shrinks as gas diffuses between it and its neighbours. Given enough time, it may even shrink to the point of vanishing entirely, having been consumed by its neighbours. If it survives and grows instead, its eventual demise will be caused by bursting at the surface. It is a moral tale, not overlooked by poets and philosophers. This life story fascinated Smith, who was drawn to it by analogy with the grain structures of metals. He tried to make sense of that restless random structure. Others have been more concerned with the mechanical properties of a foam, as Kelvin was for his particular purpose. Like sand, foam can be solid or liquid depending on circumstances. When you shave, it is a solid while it sticks to your skin, but it is easily removed because it is a fluid when subject to sufficient force. One may drink Guinness without much effort, yet a skilled barman can sketch a shamrock in its stiff foamy surface. This dual nature must have appealed to Kelvin—he corresponded a lot with his lifelong friend and fellow Irishman George Gabriel Stokes on the search for what we would today call a complex fluid, as a model for the ether. And there were further attractions in the details of the details of the elastic properties of foam, which he thought particularly appropriate for the vibrations that were supposed to be light waves. Kelvin supposed that all bubbles were of equal volume in his ideal ether foam, and formulated the following question, as his starting point: What foam structure will minimise energy, which is just the total surface area of all of the films? This is the Kelvin problem, rather like that of Kepler for hard spheres. In two dimensions its solution is well known to be the obvious one—that of the honeybee’s comb. This has nevertheless only been proven recently—again by Thomas Hales. Kelvin made a brilliant and immediate analysis of this problem and published it in the Philosophical Magazine of London, Edinburgh and Dublin. He did so rather more expeditiously than one could achieve with most journals today. Admittedly, he was the editor. He used principles of local equilibrium for soap films, enunciated by the blind Belgian physicist Joseph Plateau in 1873. His niece recorded that he played at home with the wire frames that Plateau had invented, to demonstrate his laws. This is typical of his hands-on approach to physics, and also that of Smith, described in our introduction. Indeed, Kelvin might be said to have founded such a tradition in British crystallography, to be contrasted with the abstractions of the French school. If so, its finest flowering was surely the DNA structure, assembled by Crick and Watson, using their hands and eyes as well as their brains. What Kelvin came up with in this way is what we would today call


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Figure 3. Joseph Plateau

Figure 4. A page from Kelvin’s notebook (from the Kelvin Collection; courtesy of Cambridge University Library)

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Figure 5. Two Kelvin cells

Figure 6. “Kelvin’s bedspring” (courtesy of the University of Glasgow)


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Figure 7. Whereas close packing of spheres (the Kepler problem) is achieved by face-centred cubic packing, Kelvin’s conjecture for foam was related to the bodycentred cubic packing.

Figure 8. Denis Weaire and Robert Phelan

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the body-centred cubic structure, of which he soon had a wire model constructed. In this, all bubbles have an identical and quite subtle shape, which he proceeded to analyse and describe in detail. He called it the tetrakaidecahedron, but let us call it the Kelvin cell. The Kelvin cell has survived, while the ether is forgotten. Kelvin’s structure conforms to Plateau’s rules, but is it really the best possible, in the sense defined above? This question eventually took on a life of its own in the minds of mathematicians and others, but proved intractable to both mathematical logic and computation, and indeed also to experiment. Let us fast-forward for more than a century, to 1994, when we succeeded in Dublin in computing an alternative structure with a lower energy, and produced some fragmentary experimental evidence for it. Our method was based in part on analogy with structures found in chemistry. Our structure is more complicated than that of Kelvin, and has two different kinds of cells. This is a case in which nature departs from the principle once stated by the mathematician McLaurin that “the perfection of mathematical beauty is such that whatsoever is most beautiful and regular is also found to be the most useful and excellent.” Not quite, in this case. For the physicist, the Kelvin problem may well be regarded as resolved, since it now seems very unlikely that this structure will ever be surpassed. Substantial amounts of computer time have been devoted to trying to do so. The mathematicians will have to labour on, I would guess for a decade or two, since there remains no rigorous proof that the new structure is best. This should keep Thomas Hales quiet for quite some time. As for the properties of real foams, we now have a healthy mix of physicists who can simulate structures, chemists who understand liquid films, and chemical engineers who can give us good economic reasons to pursue this research. They work on three different length scales. For example, several groups, such as that of Doug Durian at UCLA, study and use the scattering of light in a foam, which makes it white. This topic goes back to the work of yet another Irishman—and a founder of the Royal Society—Robert Boyle. Boyle discussed this property of whiteness, after doing some foam experiments with his own urine, egg white, and other liquids, and he gave a very correct account of it in terms of light scattering. Another topic of current research is the drainage of liquid from the foam. We have an elegant mathematical theory of this, based on a nonlinear differential equation that exhibits—among other things—solitary waves, analogous to the bore on a river or those mysterious traffic jams on freeways. This research ran into some controversy two years ago.


Figure 9. An extended view of the Weaire-Phelan structure

Figure 10. A typical foam (photograph by M. Boran)

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The formulation of the theory and its experimental comparison had been carried out by my group, in association with Shell Research Laboratories. Some of what we did echoed Russian work that had never reached us from the East, but the difficulties of which I speak came rather from the West—from Cambridge, Massachusetts, where we began. Howard Stone’s group at Harvard developed a very much improved experimental technique for drainage experiments. Their detailed data did not agree with the conventional wisdom established by us. By changing the physical assumptions of our model, they could arrive at a slightly different one that fitted their new data very well. Had we been misleading ourselves? After a little while, this embarrassment was resolved, as follows. Both groups, working in the spirit of Smith, had used what first came to hand to make stable foams. We used the European brand, Fairy Liquid. They used the American one, Dawn. We now realise that they have different properties: all dishwashing detergents are not created equal. The nature of the local flow of liquid between the bubbles is different in these two cases. Both groups, both experiments, both theories would appear to be at least roughly correct. If only all transatlantic differences could be so readily reconciled. Bibliography Aste, T., and D. Weaire. The Pursuit of Perfect Packing. Bristol: IOPP, 2000. Ball, P. The Self-made Tapestry. Oxford: Oxford University Press, 1999. DeGennes, P.-G., and J. Badoz. Fragile Objects. New York: Springer-Verlag, 1996. Smith, C. S. A Search for Structure. Cambridge, Mass.: MIT Press, 1981. Weaire, D., and S. Hutzler. The Physics of Foams. Oxford: Oxford University Press, 1999. Weaire, D., ed. The Kelvin Problem. London: Taylor and Francis, 1987.