DEEP beneath Arizona鈥檚 Sonoran desert, below the dusty surface scrawled with cacti and saltbush, is a chamber filled with nature鈥檚 finest artwork.
Known as the Throne Room, the chamber in Kartchner Caverns is bedecked with what cavers call pretties. Along with countless glistening stalactites and stalagmites, sheets of limestone known as draperies resemble hanging fabric, stained with the reds of iron oxides, the tans of acidic soil and the blue-blacks of carbon and manganese oxides. Milky white deposits of calcium carbonate that have hardened around clumps of yellow clay look like fried eggs. Limestone spheres called pearls cling to flecks of debris floating in shallow pools. Stalactites that bulge in the middle hang from the ceiling, giving cavers a mole鈥檚-eye view of a turnip.
鈥淭his is a world that artists and scientists have been astounded by since, well, since there have been artists and scientists,鈥� says Raymond Goldstein, a physicist at the University of Arizona in Tucson. 鈥淣ow, after thousands of years, we finally have the right kind of tools 鈥� the computer power and the mathematics 鈥� to start understanding how stalactites grow.鈥�
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Goldstein has spent the past four years trying to decipher the rules that govern how nature uses time and chemistry to cast some of its greatest works. In that time, he and his colleagues have discovered that the way in which mineral deposition paints the masterpieces one thin film at a time is dictated by the mathematics of chaos theory.
We have known for about a century that cave formations arise when groundwater laden with dissolved limestone slowly leaves a trail of minerals as it leaches through cave walls and ceilings. Yet no one knew in detail how this process led to the distinctive shapes seen in caves all over the world. A stalactite is far more than just a spike hanging from the ceiling, says Rickard Toomey, the scientific curator of Kartchner Caverns who helped Goldstein鈥檚 group before moving to Western Kentucky University. 鈥淏ut no one really understood their distinctive carrot shape.鈥�
It鈥檚 something no one had thought to study, says Goldstein, who models pattern formation in natural systems. He hadn鈥檛 considered it either until 1998, when sculptor David Stone turned up with the results of an experiment that had gone wrong.
Stone usually worked with bronze, but having been a professional sculptor for 10 years, he decided to broaden his interests and started taking classes at the University of Arizona. One of his first classes was on the art of scientific discovery. For his final project, Stone tried to grow structures by using an electric current to deposit thin layers of a mineral or metal dissolved in solution onto metal. The technique is commonly used to electroplate protective surfaces onto metals.
鈥淭he idea was to use the physical forces of nature to self-assemble sculptures, instead of forcing yourself upon a medium,鈥� Stone says. When he tried it with a solution of iron, ammonia and sulphate, however, he didn鈥檛 get quite what he expected. 鈥淚 gummed the mixture up. It oxidised and got thick and cloudy with iron hydroxide precipitate. So I counted it as a failure, shut it down and went to throw it away.鈥�
From rust to rock
Before dumping the solution, Stone noticed tubes of rust, only millimetres long, had grown on the negatively charged electrode, the cathode. Neither he nor his supervisor were able to explain the bizarre structures. Stone began questioning every major electroplating and corrosion specialist without success, and then began visiting the offices of the university鈥檚 professors, until he finally arrived at Goldstein鈥檚 door. 鈥淗is eyes got wide, and he immediately led me to a table in his lab and told me to set up there,鈥� Stone recalls.
Goldstein recognised the tubes as a complex self-assembling system, and provided Stone with a camera capable of filming tube growth on a microscopic scale. With this, they were able to see that the electric current through the cathode was producing small bubbles slicked with a thin film of rust that was forming out of the chemical solution. As the bubbles lifted away into solution, they left a ring of rust behind. The next bubble then had a template to hold on to, so it built upon this ring, leaving its own layer of rust. Eventually the layers stacked up to form a rust tube.
While the basic principle was clear, finding a way to describe the growth proved much harder. The computer model that Goldstein eventually came up with required a mathematical language known as non-linear dynamics. Simply put, non-linear dynamics allows one side of an equation to grow out of step with the other. What鈥檚 more, the growth itself is governed by another set of complex equations.
This makes systems described by non-linear dynamics very difficult to predict; small changes in the values you plug into the equations can lead to wildly different results. Such systems are called chaotic, because the equations are capable of swinging so wildly that they seem virtually impossible to predict. For example, in Stone鈥檚 experiment a milligram less iron could mean that the tubes completely fail to form, while a milligram more produces tubes twice as wide.
Two years later, after listening to a presentation of Goldstein and Stone鈥檚 findings, graduate student Chad Jarvis expanded the scope of the investigation dramatically. He suggested that the tubes looked almost identical to a group of cave pretties known as soda straws, centimetre-wide limestone tubes that grow downwards from cave ceilings and can eventually turn into stalactites. 鈥淭hat鈥檚 when we realised that caves were an entirely new realm for modern physics to explore,鈥� Goldstein says. In 2002, he and Stone obtained funding for a small group to enter the nearby Kartchner Caverns and measure the typical shape of soda straws and stalactites.
The life of a stalactite in Kartchner Caverns begins as rain falls on the parched fields of the Sonoran desert (see Diagram). As the water soaks into the soil, carbon dioxide dissolves in it, which makes it very acidic. The water continues to leach downwards until it reaches the bedrock of the caverns, where the water鈥檚 acidity allows it to dissolve calcium carbonate, the main component of limestone. Laden with minerals, the water finally penetrates the cave wall and, when it comes into contact with air, begins to release its dissolved stash of CO2. Without its acidity, the solution can no longer carry its burden of minerals, so it leaves behind a slick of calcium carbonate over every surface it skims.
A soda straw begins to form when a drop of this water falls from the cave ceiling, leaving behind a ring of calcium carbonate. Drops that follow build a tube, just as they did in Stone鈥檚 beaker, but these tubes can reach remarkable lengths. One soda straw in the Throne Room is nearly 6.5 metres long. More commonly however, mineral deposits collect inside the tube and plug it.
Unable to pass down the inside of the tube, water begins to pool around the base of the soda straw, where it attaches to the cave ceiling. From there, the water begins to stream down the outside of the tube in thin films a few micrometres thick. Slowly, the build-up of limestone forms a carrot-shaped structure.
After collecting hundreds of exquisitely detailed pictures and measurements using strobe lights, high-resolution digital cameras and laser calipers, Goldstein returned to his lab and enlisted the help of fellow physicist and cave specialist Warren Beck, also at the University of Arizona in Tucson. Beck鈥檚 main line of research is mapping the variation in concentration of mineral ions in cave formations to help determine what they reveal about past environmental conditions.
Goldstein鈥檚 graduate student Martin Short weaved together the mathematics of stalactite growth: the fluid dynamics of the running drop; the chemistry of CO2 and the resultant release of minerals; and the geometry of the initial soda-straw nub upon which the stalactite forms. What he found was surprising: the non-linear equation that describes the shape of most stalactites depends almost entirely on the initial geometry of the soda straw, and not on chemistry.
As the film of mineral-laden water runs down the side of a stalactite, the amount of calcium carbonate left behind depends on the angle of the stalactite鈥檚 pre-existing form relative to the cave ceiling. The more vertical a straw is, the faster water travels down it. Since the calcium carbonate precipitates out of the water at an almost constant rate, the faster flow means less deposition. Thus, initial skinniness feeds further skinniness.
Some soda straws, however, somehow develop nubs with a wider base of calcium carbonate. These provide a larger surface area for water to collect on before it starts running down the sides of the tube and so lead to fatter stalactites. However simple this concept sounds, the process makes a stalactite an extraordinarily dynamic structure. 鈥淏efore, we really only had a cartoon notion of stalactites,鈥� says Goldstein. 鈥淣ow we are starting to see some of the reality.鈥�
The research could apply to structures well beyond caves. Non-linear mathematics might describe how the delicate ridges of hydrothermal vents and icicles are formed (see 鈥淩iddle of the ripples鈥�). That鈥檚 not to say Goldstein鈥檚 work in caves is done. Indeed, he feels that he has barely scratched the surface. 鈥淪talactites are pretty much the simplest objects we could have tackled, so there鈥檚 still a long way to go,鈥� he says.
For now though, he and his team are savouring the success of their equations. 鈥淲hen we finally returned to the caves after constructing the models,鈥� he says, 鈥渨e were awestruck to see all these different patterns that we had predicted.鈥�
Riddle of the ripples
One of the lingering mysteries of stalactite formation is the 鈥淢ichelin Man鈥� ridges on the outside of older stalactites. Their development remains elusive because they take millennia to form, making them hard to study. So researchers have turned to similar structures that take shape far faster: icicles.
Researchers led by Kazuto Ueno at Kyushu University in Fukuoka, Japan, looked at why large icicles always develop the same type of periodic ridges that appear in stalactites. The two structures form in very similar ways. In stalactite formation, mineral layers are deposited when carbon dioxide is released from a thin film of running water. Meanwhile icicles form when water releases heat as it freezes to the icicle鈥檚 body.
Ueno鈥檚 theory is that a single bump or non-uniformity is enough to give rise to the odd patterns. A film of water travelling over a bump on an icicle鈥檚 surface releases more heat because it has to spread over a wider area. As a result, ice forms more quickly on the bump, which therefore swells faster than the rest of the icicle.
Eventually, bumps form over the entire structure and give rise to ridges of roughly the same thickness, though no one is certain about the details of the process. 鈥淭his is still an open question,鈥� says Stephen Morris, a pattern-formation researcher at the University of Toronto in Canada. One theory is that certain bump sizes are more efficient at releasing heat than others, depending on the flow rate of the water. So even though there are a lot of non-uniformities in icicles, only some eventually form ridges.
To find out more, Morris is gearing up to study the formation of these icicle ridges using a giant cold room at the University of Cambridge. 鈥淭he current theory is compelling,鈥� he says, though he still has nagging doubts about it. 鈥淭here could be a simpler, more elegant solution out there waiting to be found.鈥�
