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THE BASICS OF FOAM
BUBBLES AND GEOMETRY
Take a moment to consider the world around you. Look at its varied colors and textures; touch, smell, and even taste it. No doubt you'll soon notice the remarkable diversity of matter that surrounds us. Now group all that you see into solids, liquids, and gases. Even within these categories, there are subgroups. You may be sitting, for instance, on a soft solid upholstered chair supported by a hard solid floor. As you read this book, you're turning pages that are also solid, yet they can be bent. Without being much aware of it, you are inhaling oxygen, a clear and formless gas, and exhaling carbon dioxide, another gas. You may be sipping coffee, mineral water, or wine.
Why does the world contain solids, liquids, and gases, and how do they differ? What causes the variety even within a given category? Such fundamental questions engaged the earliest thinkers who contemplated the physical world. Now that modern practitioners of physics, chemistry, and allied sciences can explain much about matter in its pure states of solid, liquid, and gas, they are examining it in more complicated forms, including foam.
Unlike the theory of relativity, a lightning-stroke of an idea born in Albert Einstein's unique mind, there is no single theory of foam. Studying foam requires varied scientific tools and encompasses many viewpoints. The basics of foam, however, began to be established in classical times, and were well-known by the end of the nineteenth century.
The Greekthinkers of the fifth and sixth centuries B.C.E. had an idealized view of the world. To them, everything in the universe arose from a single substance, water or perhaps air. In the fifth century B.C.E., the Greek philosopher Empedocles of Sicily proposed instead that the world is made of four elements: earth, air, fire, and water. His genius lay in choosing the four to represent varied physical properties; and in his realization that when properly combined, they could describe even subtle workings of reality. According to Empedocles, the human eye sees as water modifies fire, that is, as the fluid within the eyeball affects rays of light; in his scheme, the bones of the skeleton are made from earth, water, and fire in the ratio 1:1:2. These categories still have meaning; we now associate the elements earth, water, and air with their modern equivalents: solids, liquids, and gases.
As science continued to develop long after Empedocles' time, scientists analyzed the universe by breaking it into progressively smaller bits: first atoms and molecules; then subatomic protons, neutrons, and electrons; and then their constituents, all the way down to quarks. To grasp the world around us, we need to understand how the atomic building blocks relate to the materials around us. The connections are easiest to trace for pure crystalline solids such as salt and diamonds, whose atoms are arranged in specific patterns that repeat over and over, as a huge hotel is built up from the repeating units of its rooms. However, the real world includes clay, wood, and other noncrystalline solids. It also includes liquids, which are harder to understand than solids. And most complicated of all, there are the combinations of solids, liquids, and gases.
Solids, liquids, and gases are combined in foams; in emulsions, where bubbles of one liquid float in another without mixing (milk, which is bubbles of fat suspended in water, is an example); and in colloids, where tiny specks of a solid substance are distributed throughout another material, typically a liquid (gelatin is an example). In each type, and unlike the orderly geometry of a crystal, the' inclusions are arranged randomly in the surrounding medium. That's why these materials tend to have free-form shapes that are easily deformed, so that a bowl of gelatin sets into the shape of its container. For this reason, foam and the other combined systems are called "soft" matter--neither flowing freely like a true liquid nor taking on the hard definite shape of a rigid solid like a diamond.
Empedocles may have been the first to grasp that new properties result when substances are mingled, as his formula for bone suggests. That principle is true for any foam, such as soapsuds. It is neither fully liquid nor completely gaseous; it flows differently from the first and does not dissipate like the second. Its components are stable, yet it lives only a short while. It is made from clear air and water, yet it is opaque. And while neither air nor water sticks to the hand in any great quantity, if you scoop soapsuds onto the palm of your hand, and turn the hand over, the suds remain in place. In most ways, a foam is totally unlike the substances that make it up.
There are varied ways to make foam: by generating gas within a liquid, as when the bubbles are put into champagne; by freeing gas that was held under pressure within a liquid, as when soda spurts from an opened can; or by mixing gas or vapor into a liquid, as when air is beaten into egg whites to make a meringue, or hissing steam froths milk for a cappuccino. In the latter case, bubbles stream from the nozzle and float upward like hot-air balloons, eventually reaching the surface to form densely packed layers on top of the milk. While the foam grows, it also begins to decay, as its oldest bubbles die.
Most foams are short-lived and must be examined on the fly. Even in a long-lived foam, it is difficult to register all the bubbles as they change. And we are ignorant about how a foam flows, which depends on how its bubbles interact; do they distort each other as they move and collide, or do they tumble over each other like rocks rolling downhill? Simple direct observation can reveal several basic facts about foams, including:
· They contain bubbles of gas within a liquid or solid.
· Liquid foams tend to be white, are usually ephemeral, and move differently from a pure gas or pure liquid.
· Foams formed within solids, such as, the bubbles within risen bread, generally begin as liquid foams.
After observation has yielded as much as it can, experiments and analysis are required for deeper understanding. Nowadays we think of science as requiring giant space-going telescopes and enormous particle accelerators costing billions of dollars, but it is possible to study the basics of foam at a much lower cost. In fact, I have constructed a foam laboratory that could not be cheaper--it cost nothing at all--and is remarkably informative.
To show the students in my astronomy class the foamy arrangement of galaxies in the universe, I made a simple model of the cosmos: a small transparent plastic bottle that had once held spring water, half filled with ordinary tap water and a dash of liquid soap. To demonstrate how galaxies are strewn across space, simply shake the bottle. The clear liquid turns into a mass of bubbles, each surrounded by soapy water. Stretch your mind by scaling the foam to cosmic size, so that each bubble becomes a stupendous volume hundreds of millions of light-years across. (A light-year is the distance a ray of light travels in a year, about 6 trillion miles.) Now imagine that the air within each monstrous bubble is replaced by empty space, and that the soapy water between the bubbles is replaced by billions of galaxies, each with its multitude of stars.
This plastic universe works equally well as a bubble laboratory that displays how earthly foam is born, matures, and dies. First, the laboratory demonstrates that foam requires more than pure gas and pure liquid. Omit the soap, and bubbles still form when you shake the bottle; but they die the instant you stop shaking, too unstable to give even a short-lived froth. Only by adding that bit of liquid soap, and shaking well, do you get a lingering foam. As you shake, you see a radical transformation. Transparent air and water are transmuted into an opaque white mass. The slug of liquid you feel sloshing back and forth changes as it becomes well mingled with air, replaced by a mass that hardly moves as you shake and which feels ... well, frothier. It could not be clearer that a foam is utterly different from its constituents.
A close look through a magnifying glass shows that the foam contains bubbles of various sizes, although most are tiny, a small fraction of an inch across. They are piled like cannonballs on a courthouse lawn, but in a disorderly heap rather than a perfect stack. Each is a sphere, surrounded by liquid that isolates it from its neighbors. This is a wet foam, meaning it is more liquid than gas. If you put the bottle down, and wait, the foam changes. Slowly, the proportion of liquid decreases as the water between bubbles drains downward under the pull of gravity. It is easy to see that this is happening, because a layer of clear water appears and gradually deepens beneath the foam.
CASTLES OF AIR
As the drainage continues, the wet foam becomes something more complex: a dry foam, with more gas than liquid. The walls between adjoining bubbles become very thin; instead of each sphere floating serenely in its liquid cocoon, bubbles press against each other. In some places, the wall is breached, and two bubbles join into one. Also, smaller bubbles contain air at higher pressure, which moves through their walls into the larger bubbles. Both processes coarsen the foam, giving it more big bubbles as it ages. The shape of the bubbles also changes. Instead of spheres, adjoining bubbles distort each other into polyhedrons, three-dimensional bodies with flat or gently curved faces where they abut. Each resembles a soccer ball or a geodesic dome, except that its facets are random in size, shape, and orientation. This stage takes hours to achieve, and lasts even longer. Only after several days does the foam finally die, as the last few polyhedral cells vanish, leaving the bottle as it began, half full of soapy water.
The polyhedral stage is worth the wait, for it is an intricate and airy castle of bubbles. They come in various sizes, up to an inch across. Each nestles perfectly among its neighbors, like an irregular piece of fieldstone carefully fitted into a chimney by a master mason. With the water mostly gone, the films defining the bubbles are only micrometers thick (a micrometer is a millionth of a meter) and show rainbows of color that come from the interference of light waves within the films. Unlike the repetitive geometry of a crystalline material such as salt, the foamy structure lacks any obvious regularity. It displays irregular faces with anywhere from three to nine edges. Those that happen to have six edges are reminiscent of the hexagonal cells in a honeycomb, but the perfect symmetry of a true honeycomb is a far cry from the bewildering complexity of the foam.
The first steps toward untangling this singular structure were made by the nineteenth-century Belgian physicist Joseph Antoine Ferdinand Plateau, the most influential early foam scientist. He derived laws for the geometry of a foam that still stand today. Ironically, this student of foam geometry was blind for much of his life, a result of his early research in optics, during which he gazed directly at the Sun. While Plateau could still see, he began to study soap films stretched over wire frames; with the aid of relatives and colleagues, he continued to do so even after losing his vision.
Plateau developed a mixture of soap, water, and glycerin that gave films and bubbles that survived for up to eighteen hours, allowing him to make careful observations. Eventually he evolved a set of universal laws, Plateau's rules, that impose some order on the mass of bubbles in a foam, and that can be confirmed in the plastic bubble laboratory. If you carefully examine the films forming the walls of the bubbles, you can conclude, as Plateau did, that: (1) only three films--no more, no less--ever meet to form the edge of a bubble; (2) any two adjacent films of these three always meet at an angle of 120 degrees; (3) exactly four edges of bubbles--again, no more, no less--ever come together to meet at a point.
Plateau's century-old rules are a triumph of observation and experiment. They are never violated, no matter how intricate the foam. But why are these fundamental rules true, and can we go beyond them to more fully describe a foam and its oddly shaped bubbles?
The remarkable geometry of a foam is sculpted by natural forces. To understand these forces, we must go back to early studies of gases and liquids, air and water; and of their meeting place, the bubble. Water is a good starting point. Compared to other liquids that support foam, such as milk or egg whites, water has a simple microscopic structure. The familiar symbol for its molecule, H2O, represents two atoms of hydrogen and one of oxygen, bound by electrical forces into a boomeranglike shape, with the oxygen at its center and a hydrogen defining each arm.
Water is a collection of such minute boomerangs. Even when water sits motionless in a beaker, bowl, or pond, all of its molecules--an astronomically huge number--are in constant motion. These molecules attract each other (if they did not, water would fly from an open container like a gas), and that fact goes far in explaining how water supports bubbles and foam. They form because of the force called surface tension, which is due to molecular attraction. This molecular understanding is a relatively recent development, but surface tension has been known for centuries. Leonardo da Vinci entered his careful observation of surface tension's effects into his notebook, around 1508. Describing water as it drips slowly from a source, he wrote: "That water may have tenacity and cohesion together is quite clearly shown ... where the drop [of water] before it falls becomes elongated as possible, until the weight of the drop renders the tenacity by which it is suspended so thin that this tenacity, overcome by the excessive weight, suddenly yields and breaks."
The modern picture of molecules reveals why water acts in this way. Think of the many H2O molecules making up the water in a beaker. Forget for the moment that each molecule has a specific boomeranglike shape, and think of it only as a bit of matter attracted by identical bits of matter all around it. Now imagine trying to pluck a single molecule from the beaker with extremely fine tweezers. If your tweezers close around a target deep within the water, that molecule can be easily pulled loose. The reason is that although all the neighboring molecules attract the target, since they surround it in three dimensions--above and below, left and right, front and back--all their pulls average out to zero.
Now clamp your tweezers around a molecule at the top surface of the water. As you try to pluck it loose, the molecules below the target draw it down toward the bulk of the water. With no water above, no molecular force opposes the downward pull, which you feel as resistance to your efforts. This is surface tension--the force that pulls molecules on the open surface of a liquid toward the interior of the liquid, making a drop of water behave as if it were coated with a taut elastic skin. Surface tension appears throughout nature. It plays a role in how water rises from the roots to the top of a plant; it enables the insect called the water strider to literally walk on water, supporting the insect's feet as if they treaded a miniature trampoline. And surface tension is responsible for the formation of drops of water and bubbles of gas within water.
After da Vinci's time, other researchers examined surface tension and bubbles. British physicist Charles Vernon Boys, a particularly deft experimenter, became intrigued by bubbles in the late 1880s. His genius lay in demonstrating their properties so lucidly that schoolchildren could follow his lectures, which were published in 1890 as the classic work Soap Bubbles, Their Colours and the Forces Which Mould Them. Boys employed the most minimal of equipment, such as wire frames, gas flames, and thin fibers made from quartz, a pure form of glass. (Boys first used such fibers to measure minute gravitational forces when he determined the Earth's density. He made them in a typically clever manner: He would dip a crossbow bolt into molten quartz and shoot the bolt into the air, drawing along a liquid tail that solidified into an exquisitely thin thread. Although the thread might break when the bolt landed, this method provided the short bits of fiber that were all he required.)
Boys had a marvelous way of using these simple implements to make abstract properties of bubbles immediately apparent. For instance, he showed that water held within a thin rubber bag behaves like a drop of water under the influence of surface tension, giving convincing concreteness to the effect. And drawing on the late-nineteenth-century knowledge of light and optics, which was well advanced, Boys and his fellow scientists knew that a wet foam looks white because rays of light are deflected or scattered by its bubbles, and that a dry foam shows colors because light waves interfere within its thin films.
THE MINIMIZING PRINCIPLE
Boys and his colleagues also understood the role of energy in bubbles and foam, a powerful insight that explains why isolated bubbles are perfect globes. It had been established much earlier that any physical system is most stable at its lowest energy and acts to reach that equilibrium if at all possible. This behavior is so widespread and well-known in physics that it does not go by a special name, but I'll call it the minimizing principle for easy reference. The principle explains why a ball rolls downhill; why a stretched spring returns to its relaxed state when it is released; and why a laser emits light. In each case, the action happens as the physical system returns to a lower energy after being raised to a higher one. And the minimizing principle is the reason that drops of water and bubbles take on their particular shapes.
It takes energy to maintain a drop of water, and the larger its surface area, the more energy. So the minimizing principle requires that surface tension always acts to minimize area. For a given amount of liquid, the shape with the smallest surface is a sphere, and surface tension tries to pull a drop of water into a perfect globe, although that cannot always be completed. A raindrop forms a bead on a newly waxed automobile, but not a full sphere, since part of the drop clings to the vehicle so strongly that surface tension cannot pull it away. Water dripping slowly from a tap forms a more complete sphere, although elongated by gravity, before it breaks free and falls. Under zero gravity, however, as encountered by astronauts, water forms perfectly round droplets. (Joseph Plateau did not need NASA to demonstrate those ideal spheres. He made bubbles of oil within a mixture of water and alcohol which he had carefully measured out to obtain exactly the same density as the oil; that effectively canceled the effects of gravity, giving beautiful round globes.)
The ideas of surface tension and minimizing principle apply equally well to a gas bubble inside a liquid, which consists of an elastic skin enclosing a volume of gas, like a balloon. Surface tension provides the skin, but something else is needed to make the bubble truly robust. Such a material is called a surfactant (an acronym for surface-active agent), and soap is a prime example. Its molecules, which are released when soap dissolves in water, act to modify surface tension so it can vary across the curvature of a bubble. This allows the bubble to adjust to gravity or other forces that would otherwise destroy it. In addition, the soap molecules interlock with water molecules, thereby strengthening the skin around the bubble; to put it another way, they create a skin from which water drains only slowly, extending the lifetime of the skin and the bubble. Make a large number of such bubbles, with enough surfactant to make them hardy, and you have created a foam.
Soap is not the only surfactant. Seawater contains a rich stew of molecules that play the same role. That is why the sea sustains foam, whereas freshwater does not, unless it contains additional molecules. If these come from polluting compounds, the result may be a foamy scum, the mark of contaminated water. Many edible foams, such as soufflés and whipped cream, are based on egg whites or milk products, whose complex molecules are surfactants. Some surfactants make foams that are resilient, meaning if a rift develops in the film around a bubble, the film flows so as to heal the opening. Others give plastic foams, meaning that liquid and surfactant form especially strong films that allow the bubbles to survive for hours.
No matter what the surfactant, the minimizing principle always applies. This makes it possible to use soap films to answer difficult mathematical questions called minimal area problems. These arise in mechanics, optics, and the theory of relativity, but the simplest of the type has practical meaning. If you want to build a highway linking a certain number of towns, what configuration requires the least length of road (and therefore the lowest cost) while ensuring that one can drive between any pair of towns? For two towns, the answer is obvious: a straight road. But for an arbitrary number in an arbitrary pattern, it is surprisingly elusive. In The Science of Soap Films and Soap Bubbles, British physicist Cyril Isenburg presents a set of striking photographs showing how a soap film automatically indicates the correct route when it clings to a small scale model giving the location of the towns. Other photographs in the series show how a film naturally takes on minimal area under intricate three-dimensional constraints, such as being required to touch every edge of a cube.
The powerful minimizing principle is more than a guide to the shape of an individual soap film; it also requires that the bubbles in an entire foam adopt the most stable shapes. The connection between Joseph Plateau's rules and the minimizing principle was not proved until nearly a century after Plateau's death. That tour de force was carried out at Princeton University in 1973 by American mathematician Jean Taylor, who showed that Plateau's rules could be directly derived from the minimizing principle.
LORD KELVIN'S BEDSPRING
The application of the principle to a foam leads to a classic problem in mathematics. Each bubble within a foam must take a shape that gives minimal surface area and must also be consistent with the constraining presence of its neighbors. If the bubbles are isolated, as in a wet foam, that shape is a sphere. But in a dry foam, where only thin films separate the bubbles, they must fit together to completely fill the volume of the foam, which cannot be done with spheres. A more complex shape is needed, which turns out to be hard to define mathematically even for a foam simplified and idealized, with all bubbles the same size. This particular pattern of nature is a long-standing enigma whose solution is still elusive. Its answer is important in foam science, because without knowing the shape of the bubbles, it is difficult to predict the properties of a foam.
One of the first to consider the problem was the distinguished British physicist William Thomson, Lord Kelvin. He invented the idea of absolute zero, helped develop the second law of thermodynamics, and contributed to the laying of a transatlantic telegraph cable. In 1877, drawing on shapes known for iron crystals, Kelvin theorized that the bubble with minimum area was a strange-looking figure with six square and eight hexagonal faces. He modeled this remarkable shape, called a tetrakaidecahedron ("fourteen faces" in Greek), out of wire, producing what came to be known as Kelvin's bedspring. This was long taken as the definitive minimum-area shape, but had never been tested in an actual foam. Finally, over fifty years after Kelvin's conjecture, the American botanist Edwin Matzke constructed, by hand, a foam of thousands of identical bubbles to discover how cells pack together inside living organisms. However, he failed to find even one of these Kelvin shapes within the mass of bubbles. Although Matzke's experiment was inconclusive, it did cast doubt on whether Kelvin's shape is truly optimum. Even now, the optimum shape of bubbles in a foam is not definitively known.
Early foam scientists used experiments coupled with ideas such as surface tension and the minimizing principle to establish basic rules about bubbles and foam. These rules do not explain everything, such as the shape of the bubbles in foam, or the extraordinary behavior of a foam under stress, that is, when it is pushed or pulled. Under gentle force, a foam holds its shape like a solid, whereas under greater stress, it flows like a liquid. That is why whipped cream stands firm under its own slight weight but spreads easily when urged by a spatula. These and other characteristics of foam, such as its lifetime, raise scientific questions and are important for human needs. Longevity, for instance, may be desirable in whipped cream but not in soapsuds polluting a natural waterway. And so after the understanding that had developed through the nineteenth century, twentieth-century scientists set about finding new techniques to further enlighten us about foam.
|one THE BASICS OF FOAM Bubbles and Geometry||6|
|two EXAMINING FOAM Imagers, Lasers, and Computers||21|
|three EDIBLE FOAM Bread, Beer, and Cappuccino||35|
|four PRACTICAL FOAM Cork, Aerogel, and Shaving Cream||61|
|five LIVING FOAM Cells, Viruses, and Medicinal Bubbles||94|
|six EARTHLY FOAM Volcanoes, Oceans, and Climate||121|
|seven COSMIC FOAM Quanta, Comets, and Galaxies||146|
In my other life as a physicist, I write research articles, where the style is to fill in the blanks in a linear journey from "Introduction" through "Procedures" to "Conclusions." Telling the science story to general readers is a different and engaging experience, because it is more open. In my books and magazine pieces, I use varied kinds of writing to hold the reader's interest while getting the science right -- direct and concise, yes, but also journalistic or lyrical; personal or anecdotal; and biographical or historical. Metaphors and imagery help me convey complex ideas; and unlike the straight road of a research paper, I follow side paths that connect the science to daily life and to general culture.
The first of my popular books, Empire of Light, combines the science of light (from its birth in the Big Bang to the fiber-optic backbone of the Internet) with the artistic use of light by Vincent van Gogh, Edward Hopper, and others. The book does seem to excite the mind's eye, for it has been translated into Braille (as well as German and Chinese, with a TV production, too). Empire of Light has connected me with artists, Internet entrepreneurs, and one very special reader who wrote to tell me that this secular book stirred his religious feelings.
Universal Foam: From Cappuccino to the Cosmos is full of surprises. It covers the science of foam, from the quantum universe to the farthest reaches of the cosmos, and it also displays the amazingly widespread appearances of foam and bubbles in our world -- in Greek mythology, in works of art, in technology, and in the kitchen (think beer, champagne, soufflés, meringue, bread, and more). Between those I interviewed for the book and responsive readers, Universal Foam has introduced me to eminent scientists, world-famous chefs, and an Australian sheepherder who wants to use foam to help shear his flock.
I can't imagine a better job than making these connections, which I hope expand the meaning of science and make it more accessible to people. And that's where science writing takes on the nature of a calling. Scientists in general have not done the best possible job of telling nonscientists what they do. If good writing helps people understand and feel comfortable with the science and technology that shape their lives, if it makes them better-informed citizens as they consider how science should fit into society, then it is doing something truly essential.
When I write science, I'm combining pure pleasure with social value. For me, it doesn't get any better than that. (Sidney Perkowitz)