Last January, Duke Physics major Travis Byington ’12 published an article in Physical Review Letters (PRL) with Prof. Josh Socolar. While it’s not unheard of for Duke undergraduates to publish in professional journals, Socolar says, “PRL is a pretty prestigious place to publish your first paper.” The paper, titled “Hierarchical Freezing in a Lattice Model,” describes how Socolar and Byington used computer simulations and analytical methods to model the behavior of atoms in a theoretical material as it transitions from a high-energy, disordered state to a low-energy, ordered state. The theoretical material—whose existence is possible, but not certain—is one whose atoms are arranged in a novel way, suggested by a two-dimensional geometric pattern created by Socolar and Joan Taylor, a non-degreed mathematician from Tasmania. She emailed Socolar with the main idea, which the two of them modified and streamlined together. They published their results in 2011, solving a long-standing question in math—is there one tile shaped such that a group of them can be arranged to completely cover a flat surface only in a non-periodic pattern? In other words, if you shift the whole pattern to the left or right, it won’t stay the same. In contrast, a tiling of squares—a checkerboard—is periodic because the pattern stays the same when shifted in either direction.
The special tile is a hexagon with surface decorations. Using just two rules about how the surface decorations must interact with neighboring and nearby tiles, the tiles can be combined to cover the plane, but only in the pattern shown in the photo behind Socolar and Byington. The pattern seems regular and repetitive, but the decorations form larger and larger triangles. There can never be a largest triangle, so the pattern can never be periodic. Once the hexagon and its resulting pattern were created, the next obvious step (well, obvious to theoretical physicists) was to investigate the behavior of a theoretical structure whose atoms, when arranged in their lowest energy state, would correspond to the decorations on the hexagon. Byington says, “The idea of simulating things on lattices is a very common idea. Instead of just talking about an odd arrangement of atoms, we can say they are on these lattice points. We were asking if we had a particular material that behaved like this tile, what would happen in a more realistic situation with it?” This material would be quite unusual because until fairly recently, scientists believed that the atoms of all crystalline solids (that is, non-glass solids) had to be arranged in a periodic fashion. “This is just really a weird system,” Byington says. “We wondered, how can we reach this weirdly ordered state from a disordered state? Could nature ever reach this state through a freezing process?” A good analogy of a material moving from a disordered to an ordered state as the temperature drops is a magnet. At high temperature, the atomic magnets within the larger magnet are pointing in all different directions, but as the temperature falls, they come into alignment. But they don’t line up gradually—they all align at a specific temperature in a process called phase transition. Byington wrote a computer program to simulate a situation where the hexagonal tiles start out in a random, unordered state, and then gradually fall into their lowest-energy state, represented by the two rules that govern how the hexagons come together. Byington says writing the computer code was “fairly easy.” The hard part was coming up with a mathematical measure of order that would distinguish quantitatively among the patterns produced by different rates of freezing. “We were able to look at how the order parameter varied as we brought system down to low temperature. We did both slow and fast quench. For fast quenches it would freeze in disorder at some levels but at slow quench, it would freeze in order at every level,” Byington says. They also discovered something surprising about the phase transition. “There wasn’t just one phase transition like the magnet,” Socolar says. “There’s an infinite number. Each [size of] triangle has a real temperature-dependent phase transition associated with it. Does a material like this exist in nature? “Nobody has reported discovering a material like this,” Socolar says, “but it’s just barely conceivable that there is one out there that people have looked at and didn’t recognize. More likely is that it can be a possible target for designed materials. As far as we can tell there’s no physical principle that prevents that from happening.” Besides publishing in PRL, Byington also won a couple of departmental awards for his work: his research poster was voted the best among undergraduates this spring, and he won the 2012 Daphne Chang Memorial Award for the best undergraduate research. Byington, who is from Clearwater, Florida, is spending the summer at Duke writing a longer paper about the project for the Physical Review. Socolar says, “Travis was especially good at contributing to the analysis of this problem. He and I really worked on this together. I have a lot more experience with it but he made substantial contributions. I’m really comfortable calling it a collaboration rather than just a directed investigation of my own ideas.” You can read the article on PRL's website by clicking here. Non-PRL subscribers can also read the full paper here.
Mary-Russell Roberson is a freelance science writer who lives in Durham.