teetering on the edge of chaos

PerplexedJenLuc [Originally posted at our new home at Scientific American.]

It's Chemistry Day at the Scientific American blog network, and while casting about for a relevant physics-related topic, I found my inspiration in the 1991 film Terminator 2: Judgment Day. John Connor is in mortal danger again, this time from a new, improved Terminator machine known as the T-1000. The original killing machine (played by Arnold Schwarzenegger) is now the "good guy," having been reprogrammed to act as Connor's protector.

[Note: Spoilers follow. But if you haven't seen the movie yet, what's the matter with you? It's been 20 years!] Unfortunately, the reprogrammed Terminator is Old and Busted, built with outdated technology. The T-1000 is the New Hotness, and well-nigh indestructible thanks to the material from which it is made: a weird kind of metallic substance that shifts back and forth between and liquid and a solid state, enabling it to pass through metal bars and take on the appearance of anyone (or anything) it touches.

T1000 It also heals almost instantaneously from wounds from bullets, rocket launchers, even being cleft in two. Blow its head off? It comes back together. (That link will take you to a clip on YouTube.) Chop off its arm? No worries! The T-1000 will just meld itself back together and keep coming.

How the hell do you destroy such a thing? One of the climactic scenes — which you can watch here) — involves a car chase in which John, his mother, and Ah-nold are being pursued by the T-1000 in a semi carrying liquid nitrogen. Ah-nold crashes the semi, spilling liquid nitrogen everywhere. So when the T-1000 emerges unscathed (as usual) from the wreckage, it freezes into a block of metallic ice. Ah-nold blows the frozen T-1000 to smithereens with the classic line, "Hasta la vista, baby!" You'd think that would be that, but the T-1000 isn't finished yet. See, they crashed into a smelting plant, and sparks are flying, slowly heating the frozen fragments until they melt. And once back in liquid form, it's only a matter of minutes before the fragments all re-meld again:

There's quite a bit of fascinating chemistry and physics behind the T-1000's unique abilities, but I'm going to focus on phase transitions. Most of us were taught in science class that there are three basic states of matter: solid, liquid and gas. (There is also a fourth state, plasma, found mostly in space and in physics labs.) Any substance has a specific moment when the pressure or temperature is just right to cause it to shift from one state to another. Water is the most common example: lower the temperature sufficiently and it will turn into ice; raise the temperature to a boil and it will evaporate into steam. That's a phase transition.

The precise moment when this happens is called the critical point, when the substance is perfectly poised halfway between one phase and the other. That critical point can vary even for the same substance. At sea level, water boils at 212 degrees F (100 degrees C) and freezes at 32 degrees F (0 degrees C). But try to boil water in Denver — the Mile-High City — and water boils at a slightly lower temperature because of the lower atmospheric pressure, with some interesting implications for cooking. Per Wikipedia:

The boiling point of water is typically considered to be 100 °C or 212 °F. Pressure and a change in composition of the liquid may alter the boiling point of the liquid. For this reason, high elevation cooking generally takes longer since boiling point is a function of atmospheric pressure. In Denver, Colorado, which is at an elevation of about one mile, water boils at approximately 95 °C.[2] Depending on the type of food and the elevation, the boiling water may not be hot enough to cook the food properly.

Phase transitions are quite possibly one of the most fascinating areas of physics: different substances behave differently at various temperature and pressure points — sometimes in very remarkable ways. Yet it's easy to lose one's sense of wonder, because we see phase transitions around us every day. Sure, you can produce a phase transition of sorts for just about any common material, but there's still a lot to learn about the underlying physics.

Altered States

Very high temperatures and pressures can give rise to new and intriguing properties for even a common substance like water. For instance, pure water is not a conductor in its normal state. It's the ions in the water (like Na+ and Cl- from dissolved salt) that carry the current. Sea water is approximately 1,000,000 more conductive than ultra-pure water. But what happens when you take water and jack up the temperature and pressure to incredibly high levels?

Metwaterstruct In 2006, an article appeared in Physical Review Letters describing the results of a new computational study by scientists at Sandia National Laboratories indicating that a new conducting form of water — dubbed "metallic water" — could occur at a temperature of 4000 degrees Kelvin and a pressure of 100 gigapascals. And in that phase, water boasts ultra-high energy densities, far beyond those that would occur naturally anywhere on earth. In fact, it would take temperatures and pressures on a par with those believed to exist in the interiors of the gas giant planet Jupiter to produce equivalent energy densities in a lab.

(A side note about units, courtesy of commenter Alison Chaiken back in 2006: One atmosphere is about 100,000 pascals, which in turn is equivalent to about 15 pounds/inch^2. A gigapascal is 10^9 pascals = 10,000 atmospheres. Let's figure that 10,000 atmospheres = 150,000 pounds/inch^2, so you'd get a gigapascal of pressure if you took a 150,000 pound weight and balanced it on a post with the area of a postage stamp.)

Hydrogen as we know it is a gas (and a critical component of water, H20), but on Jupiter, it's believed to exist as a super-hot liquid metal because of the extreme pressures and temperatures that typify that planet. Eugene Wigner predicted back in 1935 that if you squeezed hydrogen gas hard enough, it would eventually metallicize, but the requisite pressure was so intense that physicists weren't able to achieve it for 60 years. In 1996, William Nellis, a scientist at Lawrence Livermore National Laboratory, announced the successful achievement of metallic hydrogen.

In the case of the T-1000 we're talking about some new, as-yet-undiscovered (in the real world, at least) state of matter. Or maybe not. The T-1000 in its amorphous state bears an eerie resemblance to a ferrofluid, a substance that will change its phase in response to a magnetic field. Technically, it's lumped in with "colloidal suspensions," which applies to materials that possess properties of more than one state of matter — solid and liquid, in the case of the T-1000.

A ferrofluid is made of tiny magnetic fragments, usually of iron, suspended in kerosene or another type of oil, and spiked with a touch of surfactant (like oleic acid) to keep it from clumping up — at least, not until you want the fragments to start clumping by applying a carefully controlled magnetic field. This makes them ideal substances to form liquid seals around the rotating drive shafts in hard disks, which are typically surrounded by magnets that hold the ferrofluid in place. The ferrofluid seals ensure that bits of debris don't find their way into your hard drive.

You can buy ferrofluids, but they're crazy expensive, so Popular Science helpfully tells you how to make your own. You can also make some amazing kinetic "sculptures" using ferrofluids, applying magnetic fields strategically to create beautiful constantly changing shapes:

Consider a Spherical Phase Transition

While experimentalists are busy playing around with unusual phases of matter, the theorists have not been idle, no sirree! In fact, phase transitions played a role in the eventual acceptance of "atomism" — the notion that matter is comprised of individual atoms, dating all the way back to Democritus in ancient Greece.

Democritus His peers thought he was a bit loony, insisting upon the existence of tiny things no one had ever seen, but Democritus thought individual atoms acting in concert could explain the well-defined boundaries between different phases of matter, like when water turns to ice, or evaporates into steam. Specifically, he thought that tiny changes at the atomic level could end up impacting what was happening at the larger scale. It just so happens that this property of tiny fluctuations causing large-scale changes in a given system is the essence of modern chaos theory. Democritus was way ahead of his time.

This, mind you, was a very radical idea, and it didn't gain traction for about two thousand years. Chemists were ahead of the physicists on that score: they accepted the idea of atoms by the 19th century, whereas the debate continued among physicists until the early 20th century. But then physicists made up for lost time by inventing quantum mechanics and other revolutionary theories, so that's all right.

But phase transitions continued to be puzzling, theoretically. Sure, there was a general theory for phase transitions in place by 1937, but physicists struggled to come up with a precise theory for what, exactly, happens at the critical point. The equations broke down the closer one got to that point.

It was made more challenging by the fact that there's more than one kind of phase transition. For instance, there are first-order transitions, which occur abruptly, such as boiling water or melting ice. A second order phase transition occurs more smoothly and continuously, such as when ferromagnetism switches to paramagnetism in metals like iron, nickel and cobalt, or when a substance becomes superconductive.

It was a physicist named Ken Wilson who realized that phase transitions are a type of critical phenomena, and as such, are fundamentally different from most other physical phenomena, at least from a theoretical standpoint. And he won the 1982 Nobel Prize in Physics for his work.

Basically, if you're dealing with something "normal" like radio waves or visible light, you're dealing with systems that have a set scale of length associated with the phenomena, which just means that you only need to take into account large-scale fluctuations. But Wilson discovered that at the critical point of a phase transition, you must also factor in many other fluctuations of varying scales of length, all the way down to the atomic level. You have to consider the entire spectrum, not just one piece of it.

Confusing, right? Here's a simpler way to visualize it, courtesy of the Time Lord, who took a few minutes out from grappling with entropy and the origins of the universe to walk me through the basics of something known as the "Ising model" — the quintessential model for the physics of a phase transition. (Physicists do love their spherical cows.)

Imagine a two-dimensional lattice, or grid. Each point on the lattice has a particle at that point with a property called "spin", and it can only be in one of two states: "spin up" or "spin down." It's kind of like the 0's and 1's of a computer. Ideally, spins all like to be aligned with each other. They don't care if they're pointing up or down, so long as they're all pointing the same way. So the over time, and under the right conditions, the spins will order themselves into that kind of perfectly ordered arrangement. Applying a magnetic field can speed up the process by causing all the spins to flip to the up or down position, depending on the orientation of the field.

Isingphases The Ising model starts out in that perfectly ordered state ("infinite order"), but what good is a model if you don't introduce some new variables? In this case, the new variable is temperature. Imagine that we now gradually start to heat up the Ising lattice. What happens then? The spins start to jiggle (because now they have more energy), and some of them start to change states (from up to down, or down to up, depending on your starting spin state). As the temperature gets higher, they jiggle faster and faster, until they reach an incredibly high temperature at which all the original order is gone because the spins are jiggling far too much. Now we have "infinite disorder" (see figure, top left) — a kind of chaos, in the physics/mathematical sense.

Imagine that you track this gradual heating process, taking occasional snapshots at random points and noting how the arrangement of spins changes at each of those points.

Early on, you'll find almost all "spin ups" with a few clumps of "spin downs." Then there will be more and more clumping as you raise the temperature, because of this aforementioned preference for the spins to be aligned in the same direction as their nearest neighbors. You'll soon start to see more small spin-down domains of various sizes in a big sea of spin-ups (see figure, bottom).

You'll know when you reach the critical point — the moment of the actual transition between phases, when the system is perfectly balanced between one phase and the other — because you will have clumps of all sizes (see figure, top right).

That's where physicists find themselves teetering on the edge of chaos. And they're interested in tracking how the various properties of a given phase change as the system approaches that critical point of transition: the speed of sound, the heat capacity, and so forth. They can calculate values for those properties and more often than not, those values go to infinity at the critical point — the equations literally break down. That's a clue for a physicist that something really interesting is going on.

The Ising model is incredibly useful, and has been applied to all kinds of complex systems, from atoms and protein folding, to the dynamics of flocking birds, neural networks, even social behavior, specifically the tendency of human beings to modify their behavior to conform to how others in their social circles behave (i.e., peer pressure).

A Cosmic Phase

There's one more interesting thing about phase transitions — they might be able to shed some light on conditions in the earliest days of our universe. Back in 2006, physicists at the University of Salerno in Italy reported on their efforts to exploit another unusual feature of phase transitions in water. Freezing water very quickly produces not one, but many crystals, separated by so-called "defects" where the crystals meet. (In liquid crystals, this very rapid cooling produces "knots.") The theory — borne out, it seems by experiment — is that cooling water very slowly will create a single smooth crystal.

A similar model could tell us something about our early universe: namely, there is a theory that predicts a precise relationship between cooling rate and how many topological defects developed in our universe. The working theory is that the cosmos cooled so rapidly after the Big Bang, that it passed through a series of phase transitions, only instead of "freezing" into different states of matter, it froze into the states of the fundamental forces: electromagnetism, strong and weak nuclear forces, and gravity). The Salerno researchers performed one of the most precise laboratory tests to date of that theory, super-cooling a pair of niobium rings, with results that matched pretty darn well with those predictions.

And there you have it: a journey from the T-1000 to our early universe, connected by something as simple as how water turns into ice.

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