Quasicrystals Are Nature’s Impossible Matter

What do a frying pan, an LED light, and the most cutting edge camouflage in the world have in common? Well, that largely depends on who you ask. Most people would struggle to find the link, but for University of Michigan chemical engineers Sharon Glotzer and Michael Engel, there is a substantial connection, indeed one that has flipped the world of materials science on its head since its discovery over 30 years ago.

The magic ingredient common to all three items is the quasiperiodic crystal, the “impossible” atomic arrangement discovered by Dan Shechtman in 1982. Basically, a quasicrystal is a crystalline structure that breaks the periodicity (meaning it has translational symmetry, or the ability to shift the crystal one unit cell without changing the pattern) of a normal crystal for an ordered, yet aperiodic arrangement. This means that quasicrystalline patterns will fill all available space, but in such a way that the pattern of its atomic arrangement never repeats. Glotzer and Engel recently managed to simulate the most complex quasicrystal ever, a discovery which may revolutionize the field of crystallography by blowing open the door for a whole host of applications that were previously inconceivable outside of science-fiction, like making yourself invisible or shape-shifting robots.

While most of the current applications of quasicrystals are rather mundane, such as the coating for frying pans or surgical utensils, Glotzer and Engel’s simulation of a self-assembling icosahedral quasicrystal opens up exciting new avenues for research and development, such as improved camouflage.

“Camouflage is all about redirecting light to change the appearance of something,” said Glotzer. “Making camouflage materials or any kind of transformation optics materials is all about controlling the structure of the material, controlling the spacing of the building blocks to control the way light is absorbed and reflected.”

Icosahedral quasicrystals (IQCs) are one of the several unique structures which have something called a photonic band gap, which dictates the range of photon frequencies which are permitted to pass through the material. Photonic band gaps are determined by the spatial arrangement of an atomic lattice. In other words, whether or not a photon becomes “trapped” in the lattice depends on the photonic frequency (measured as a wavelength) in relation to the space between atoms and the way these atoms are arranged (periodically, aperiodically, etc). If the wavelength falls within the range of the photonic band gap for the specific material, then the photons will not be able to propagate through the structure.

Thus, being able to manipulate photonic band gaps means that one can manipulate atomic structures in such a way that the material will only be visible within determined photonic frequencies, a critical advancement for those concerned with making people invisible, which probably at least partly accounts for why the US Department of Defense and the US Army both helped fund Glotzer and Engel’s study.

While the existence of photonic bandgaps is nothing new, being able to manipulate solid-state matter in such a way that allows one to fully exploit these bandgaps has remained elusive. In this sense, Glotzer and Engel’s simulated quasicrystal represents a return to the fundamentals of crystallography, rather than something entirely novel.

According to the team, before their simulation, scientists knew that mixing certain metals in the right thermodynamic conditions (pressure, temperature) would result in the formation of a quasicrystal. They also knew that given the correct environmental conditions, it was possible for quasicrystals to form in nature (two natural quasicrystals have been discovered to date: the first in 2009 and the second was reported on March 13, coming from a 4.5-billion year old meteorite in Russia).

What scientists didn’t understand, said Engel, was what was happening in the reaction to make these quasicrystals form. There was an input and output, but what went on inside the blackbox remained a mystery. Glotzer and Engel’s experiment was a first step in solving this a-list conundrum in materials science.

“For a long time people have looked for methods to actually model [how icosahedral quasicrystals form],” said Engel. “This is more of a fundamental importance, it doesn’t necessarily make [IQCs] have better properties or applications, but it allows us to study how these crystals form.”

Understanding how these quasicrystals form is the first step in manipulating them toward desired ends. While this ability to manipulate quasicrystals is still in a very young phase, increasing technical sophistication could conceivably lead to some pretty wild developments in the future, like Terminator-style shape-shifting robots.

Part of the reason robots modeled after T-1000 don’t roam the Earth already is because our understanding of matter and our ability to find useful applications for the staggering variety of metals found in nature is still relatively rudimentary. Understanding how quasicrystals form will fill in a huge gap in our knowledge of solid-state physics and chemistry. Increasing this knowledge in all of its forms is essential to future physical manipulation, whether or not this manipulation is directly linked to quasicrystals.

“It’s not that the icosahedral quasicrystal itself would necessarily be the structure you would shoot for [in shapeshifting materials], but it represents the kind of complexity and control that one would like to have over the building blocks of matter,” said Glotzer. “If you understand what is required to get a certain structure, than you could imagine that we could change conditions and change the structure that we get. Everything about a material depends on its structure.”

T-1000 Shapeshifter in Terminator

The quasicrystalline structure was discovered by Dan Shechtman, a professor of materials science at the Technion-Israel Institute of Technology, in 1982 while he was observing an alloy of rapidly cooled aluminum and manganese with an electron microscope.

What he saw defied the laws of nature as they were understood at the time.

Rather than finding a random collection of atoms as expected, Shechtman observed a diffraction pattern with ten-fold rotational symmetry, something which was thought to be impossible (subsequent experiments would demonstrate that what Shechtman had discovered was actually five-fold symmetry).

Shechtman’s five-fold symmetry defied the basic definition of a crystal which had stood unchallenged since crystallography’s inauguration as a science some 70 years prior. According to the received wisdom at the time, a crystal was something which by definition was both ordered and periodic, meaning that it exhibited a certain pattern at regular intervals. On this definition crystals were only capable of exhibiting a two, three, four, or six-fold rotational symmetry (the ability to retain symmetry after being rotated so many times along an axis—in other words, after rotating the crystal along an axis so many times, it will look the same as when you started).

Upon his discovery of a diffraction pattern with five-fold symmetry, Shechtman allegedly exclaimed that “there can be no such creature.” His colleagues agreed with him.

Electron diffraction pattern showing five-fold symmetry from an aluminum-copper-iron quasicrystal. Image via

“Since 1912 all crystals that had been studied were periodic—hundreds of thousands of different crystals were studied. People did not believe that there was anything different because so many thousands of excellent scientists developed the field and found only crystals which were periodic,” Shechtman told me over Skype.

Thus, when Shechtman revealed his discovery, which would earn him a Nobel Prize in chemistry in 2011, he was met with not only incredulity, by outright hostility. Upon hearing of Shechtman’s discovery, the head of his laboratory allegedly told him to revisit a textbook covering the basics of x-ray diffraction, so that he might understand why his “discovery” was impossible. When Shechtman informed him that he had no need of the book since his discovery was not included in the material, he was told that he was a disgrace to the team.

He would be discredited by scientists around the world, including heavyweights such as Linus Pauling, the two time Nobel Prize winning chemist who dismissed Shechtman’s results as the product of “twinning,” the fusion of two normal crystals at an angle.

“When I came out with my results, people found it difficult to accept. It was easier to say ‘Don’t they know anything about crystallography at Technion? Don’t they read the books?’ I had to defend it awhile,” said Shechtman. It took two years from the initial discovery for Shechtman to publish his results. After their publication, according to Shechtman, “all hell broke loose.”

“Shortly after the first publication, there was a growing community of avant-garde young scientists from around the world who all supported me and joined the fight, so I was not alone anymore,” he said. “But in the first two years I was alone.”

Dan Shechtman displays a model in his lab in Israel. Image via

Shechtman’s discovery prompted the International Union of Crystallography to redefine just what was meant by a crystal in 1992. The current definition now reads that a crystal is defined by “discrete diffraction patterns,” which accounts for both the periodic structures which traditionally defined a crystal, as well as the aperiodic quasicrystalline structures discovered by Shechtman.

“Quasiperiodic crystals are still crystals—they have nothing to do with amorphous materials,” said Shechtman. “Amorphous materials are non-ordered (like glass), quasicrystals are crystals, but the atomic relation within them is different than periodic crystals. It is perfectly ordered, but not periodic.”

The math underlying Shechtman’s design has a long history, dating back to Leonardo Fibonacci who in 1202 sought to discover how fast rabbits could breed in ideal circumstances (the sequence ‘discovered’ thereby actually long pre-dated Fibonacci in Indian mathematics—Fibonacci is most accurately credited with introducing it to the West).

Fibonacci began his thought experiment by assuming that two rabbits are placed in a field and produce a new pair of rabbits at the end of a month. It takes each new pair one month before they are able to breed another pair. The question Fibonacci sought to answer was how many pairs there would be at the end of one year. The sequence inaugurated by this pattern (1,1,2,3,5,8,13,21,34,55,89,144) is known as the Fibonacci sequence wherein the next number can always be derived by adding the two numbers which precede it in the sequence.

The Fibonacci sequence can be seen as a 1-D analog to Shechtman’s quasicrystal, in which there is order without repetition. The 2-D analog was discovered in 1974 by the famous English mathematician and physicist Roger Penrose.

That’s what order means: there is correlation between how it looks in one place versus another.

In addition to proving that black holes could result from the gravitational collapse of stars, Penrose discovered a method of tiling a plane aperiodically, which became the first demonstration of five-fold rotational symmetry. (Or the ability to rotate 72 degrees without changing the pattern.) In Penrose’s initial iteration, he used four different shapes all related to a pentagon. He would eventually narrow this down to an aperiodic tiling which used only two rhombuses, a “fat” rhombus and a “skinny” rhombus.

“Locally [Penrose tiles] are a very simple structure: there are only two building blocks and the way that they are put together means that it is not perfectly repeating,” said Engel. “But still there’s always a discrete number, a finite number of ways that you can arrange them. The fact that you only have these finite ways of arranging them makes it such that even if you are infinitely far away from where you started building the structure, it’s still in a way predictable. So there is correlation, meaning they are not independent of one another and that’s what order means: there is correlation between how it looks in one place versus another.”

Translate Penrose Tiling to a three-dimensional atomic lattice and you have the essence of a quasicrystal. The important takeaway here, according to Shechtman, is that “there is not a motif of any size that repeats itself. So there is order, and yet there is no periodicity.” The order is derived from the fact that anyone could reconstruct the Fibonacci sequence or Penrose tiles, yet despite this order, if the sequence or tiling is shifted in anyway it is impossible to derive an exact repetition.

Roger Penrose stands on Penrose Tiling, the first instance of Penrose inception in recorded history

In the 30 some years since Shechtman’s discovery, hundreds of quasicrystals have been discovered, many of which are aluminum-based alloys. The first naturally occurring quasicrystal, icosahedrite, was found in Russia in 2009. Quasicrystals, both natural and artificial, are divided into two primary types: polygonal and icosahedral quasicrystals. The former category exhibits periodicity in one direction (perpendicular to the quasiperiodic layers); the latter exhibits no periodicity whatsoever, which is precisely what makes Glotzer and Engel’s simulation such a big deal.

“The icosahedral quasicrystal is the most exotic,” said Engel. “It’s the most spatially or geometrically complex.”

In their experiment, Glotzer and Engel set out to answer one of the fundamental questions dogging the field of crystallography: How can long range order be generated from local interactions which exhibit no periodicity? While most real quasicystals are made of two or more elements, the University of Michigan team ran simulations using only one type of particle, another first in the field.

In essence, the team was attempting to determine what thermodynamic conditions favored the formation of icosahedral quasicrystals given certain initial parameters which determined the force field, or the way the particles would interact with one another. These parameters were designed so that they could be recreated in a laboratory setting. For instance, one parameter dictated that the particles were only allowed to interact with other particles which were within three particle distances of themselves.

“10 fold???”: A page from Shechtman’s notebook the day he discovered the quasicrystal. Image: via

“Basically, we were solving Newton’s equation of motion,” said Glotzer. “What you have is a bunch of particles and they interact according to a certain force field. So that means at a given time, every atom in the system has a force exerted on it by every other atom in the system. You add up all those forces on every atom, and then you solve F=ma. By adding up all those forces you can solve for the acceleration which tells you how to move the particles. Then you do this for all the particles in the system.”

The end results of these calculations tells the team where the particles “want be” under varying thermodynamic conditions, such as pressure and temperature. Given these initial conditions, the crystal “self-assembles” in the simulation.

“All we know are the force fields between the particles and Newton’s equation,” said Glotzer. “We don’t know what will come out when we start—it’s very different from building [icosahedral quasicrystal] by hand.”

As the team discovered, the interaction of their particles in such a way that a quasicrystal was formed was favored by interactions governed by the golden ratio. The golden ratio is an irrational number which starts as 1.61803, and is derived from the ratio of two numbers whose ratio to one another is the same as that between their sum and the larger integer. It is related to the Fibonacci sequence insofar as each digit you progress in the sequence, the ratio between the current digit and the one before it approaches the golden ratio—it is an infinite approximation.

In a talk given by Penrose at the Royal Institution in 2014, the renowned scientist speculated that icosahedral quasicrystals might be governed by quantum mechanical interactions, given that a complex aperiodic structure demonstrated long range order solely from local potentials, or interactions. Glotzer and Engel’s findings suggest this might not be the case.

“Our simulations suggest that maybe quantum mechanics are not even necessary,” said Engel. “Maybe you can get it from classical, non-quantum interactions. How that works, exactly, is a wide open question. Right now we hope to address this question with our model.”

While shape-shifting robots derived from quasicrystalline principles may be a long way off, quasicrystals are already beginning to play a major role in everyday life. They are most commonly found as a reinforced coating (such as on a frying pan or surgical tool) but are increasingly being added in small quantities to normal metal alloys to reinforce them while retaining lightness. They are also becoming very popular in additive manufacturing, otherwise known as 3D printing, due to their low friction and resistance to wear.

Where the future of quasicrystals will take us is relatively uncertain at the moment. What is, known however is that quasicrystals, nature’s “impossible matter,” provide us with a very important missing link in the study of matter, and may very well hold the key to the total manipulation of the solid universe in the future.