Featured image: An example of zellij tilework in the Al Attarine Madrasa in Fes, Morocco (2012), with complex geometric patterns on the lower walls and a band of calligraphy above. Caption and credit: just_a_cheeseburger (CC BY)
‘Quasi’ means almost. It’s an unfair name for quasicrystals. These crystals exist in their own right. Their name comes from the internal arrangement of their atoms. A crystal is made up of a repeating group of some atoms arranged in a fixed way. The smallest arrangement that repeats to make up the whole crystal is called the unit cell. In diamond, a convenient unit cell is four carbon atoms bonded to each other in a tetrahedral (pyramid-like) arrangement. Millions of copies of this unit cell together make up a diamond crystal. The unit cell of sodium chloride has a cubical shape: the chloride ions (Cl–) occupy the corners and face centres while the sodium ions (Na+) occupy the middle of the edges and centre of the cube. As this cube repeats itself, you get table salt.
The structure of all crystals thus follows two simple rules: have a unit cell and repeat it. Thus the internal structure of crystals is periodic. For example if a unit cell is 5 nanometres wide, it stands to reason you’ll see the same arrangement of atoms after every 5 nm. And because it’s the same unit cell in all directions and they don’t have any gaps between them, the unit cells fill the space available. It’s thus an exercise in tiling. For example, you can cover a floor of any shape completely with square or triangular tiles (you’ll just need to trim those at the edges). But you can’t do this with pentagonal tiles. If you do, the tiles will have gaps between them that other pentagonal tiles can’t fill.
Quasicrystals buck this pattern in a simple way: their unit cells are like pentagonal tiles. They repeat themselves but the resulting tiling isn’t periodic. There are no gaps in the crystal either because instead of each unit cell just like the one on its left or right, the tiles sometimes slot themselves in by rotating by an angle. Thus rather than the crystal structure following a grid-like pattern, the unit cells seem to be ordered along curves. As a result, even though the structure may have an ordered set of atoms, it’s impossible to find a unit cell that by repeating itself in a straight line gives rise to the overall crystal. In technical parlance, the crystal is said to lack translational symmetry.
Such structures are called quasicrystals. They’re obviously not crystalline, because they lack a periodic arrangement of atoms. They aren’t amorphous either, like the haphazardly arranged atoms of glass. Quasicrystals are somewhere in between: their atoms are arranged in a fixed way, with different combinations of pentagonal, octagonal, and other tile shapes that are disallowed in regular crystals, and with the substance lacking a unit cell. Instead the tiles twist and turn within the structure to form mosaic patterns like the ones featured in Islamic architecture (see image at the top).
The discovery of quasicrystals in the early 1980s was a revolutionary moment in the history of science. It shook up what chemists believed a crystal should look like and what rules the unit cell ought to follow. The first quasicrystals that scientists studied were made in the lab, in particular aluminium-manganese alloys, and there was a sense that these unusual crystals didn’t occur in nature. That changed in the 1990s and 2000s when expeditions to Siberia uncovered natural quasicrystals in meteorites that had smashed into the earth millions of years ago. But even this discovery kept one particular question about quasicrystals alive: why do they exist? Both Al-Mn alloys and the minerals in meteorites form in high temperatures and extreme pressures. The question of their existence, more than just because they can, is a question about whether the atoms involved are forced to adopt a quasicrystal rather than a crystal structure. In other words, it asks if the atoms would rather adopt a crystal structure but don’t because their external conditions force them not to.
This post benefited from feedback from Adhip Agarwala.
Often a good way to understand the effects of extreme conditions on a substance is using the tools of thermodynamics — the science of the conditions in which heat moves from place to another. And in thermodynamics, the existential question can be framed like this, to quote from a June paper in Nature Physics: “Are quasicrystals enthalpy-stabilised or entropy-stabilised?” Enthalpy-stabilised means the atoms of a quasicrystal are arranged in a way where they collectively have the lowest energy possible for that group. It means the atoms aren’t arranged in a less-than-ideal way forced by their external conditions but because the quasicrystal structure in fact is better than a crystal structure. It answers “why do quasicrystals exist?” with “because they want to, not just because they can”. Entropy-stabilised goes the other way. That is: at 0 K (-273.15º C), the atoms would rather come together as a crystal because a crystal structure has lower energy at absolute zero. But as the temperature increases, the energy in the crystal builds up and forces the atoms to adjust where they’re sitting so that they can accommodate new forces. At some higher temperature, the structure becomes entropy-stabilised. That is, there’s enough disorder in the structure — like sound passing through the grid of atoms and atoms momentarily shifting their positions — that allows it to hold the ‘excess’ energy but at the same time deviate from the orderliness of a crystal structure. Entropy stabilisation answers “why do quasicrystals exist?” with “because they’re forced to, not because they want to”.
In materials science, the go-to tool to judge whether a crystal structure is energetically favourable is density functional theory (DFT). It estimates the total energy of a solid and from there scientists can compare competing phases and decide which one is most stable. If four atoms will have less energy arranged as a cuboid than as a pyramid at a certain temperature and pressure, then the cuboidal phase is said to be more favoured. The problem is DFT can’t be directly applied to quasicrystals because the technique assumes that a given mineral has a periodic internal structure. Quasicrystals are aperiodic. But because scientists are already comfortable with using DFT, they have tried to surmount this problem by considering a superunit cell that’s made up of a large number of atoms or by assuming that a quasicrystal’s structure, while being aperiodic in three dimensions, could be periodic in say four dimensions. But the resulting estimates of the solid’s energy have not been very good.
In the new Nature Physics paper, scientists from the University of Michigan, Ann Arbor, have reported a way around the no-unit-cell problem to apply DFT to estimate the energy of two quasicrystals. And they found that these quasicrystals are enthalpy-stabilised. The finding answer is a chemistry breakthrough because it raises the possibility of performing DFT in crystals without translational symmetry. Further, by showing that two real quasicrystals are enthalpy-stabilised, chemists may be forced to rethink why almost every other inorganic material does adopt a repeating structure. Crystals are no longer at the centre of the orderliness universe.
The team started by studying the internal structure of two quasicrystals using X-rays, then ‘scooped’ out five random parts for further analysis. Each of these scoops had 24 to 740 atoms. Second, the team used a modified version of DFT called DFT-FE. The computational cost of running DFT scales increases according to the cube of the number of atoms being studied. If studying four atoms with DFT requires X amount of computing power, 24 atoms would require 8,000 times X and 740 atoms would require 400 million times X. Instead the computational cost of DFT-FE scales as the square of the number of atoms, which makes a big difference. Continuing from the previous example, 24 atoms would require 400 times X and 740 atoms would require half a million times X. But the lower computational cost of DFT-FE is still considerable. The researchers’ solution was to use GPUs — the processors originally developed to run complicated video games and today used to run artificial intelligence (AI) apps like ChatGPT.
The team was able to calculate that the resulting energy estimates for a quasicrystal was off by no more than 0.3 milli-electron-volt (meV) per atom, considered acceptable. They also applied their technique to a known crystal, ScZn6, and confirmed that its estimate of the energy matched the known value (5-9 meV per atom). They were ready to go now.
When they applied DFT-FE to scandium-zinc and ytterbium-cadmium quasicrystals, they found clear evidence that they were enthalpy-stabilised. Each atom in the scandium-zinc quasicrystal had 23 meV less energy than if it had been part of a crystal structure. Similarly atoms in the ytterbium-cadmium quasicrystal had roughly 7 meV less each. The verdict was obvious: translational symmetry is not required for the most stable form of an inorganic solid.
The researchers also explored why the ytterbium-cadmium quasicrystal is so much easier to make than the scandium-zinc quasicrystal. In fact the former was the world’s first two-element quasicrystal to be discovered, 25 years ago this year. The team broke down the total energy as the energy in the bulk plus energy on the surface, and found that the scandium-zinc quasicrystal has high surface energy.
This is important because in thermodynamics, energy is like cost. If you’re hungry and go to a department store, you buy the pack of biscuits that you can afford rather than wait until you have enough money to buy the most expensive one. Similarly, when there’s a hot mass of scandium-zinc as a liquid and scientists are slowly cooling it, the atoms will form the first solid phase they can access rather than wait until they have accumulated enough surface energy to access the quasicrystal phase. And the first phase they can access will be crystalline. On the other hand scientists discovered the ytterbium-cadmium quasicrystal so quickly because it has a modest amount of energy across its surface and thus when cooled from liquid to solid, the first solid phase the atoms can access is also the quasicrystal phase.
This is an important discovery: the researchers found that a phase diagram alone can’t be used to say which phase will actually form. Understanding the surface-energy barrier is also important, and could pave the way to a practical roadmap for scientists trying to grow crystals for specific applications.
The big question now is: what special bonding or electronic effects allow atoms to have order without periodicity? After Israeli scientist Dan Shechtman discovered quasicrystals in 1982, he didn’t publish his findings until two years later, after including some authors on his submission to improve its chances with a journal, because he thought he wouldn’t be taken seriously. This wasn’t a silly concern: Linus Pauling, one of the greatest chemists in the history of subject, dismissed Shechtman’s work and called him a “quasi-scientist”. The blowback was so sharp and swift because chemists like Pauling, who had helped establish the science of crystal structures, were certain there was a way crystals could look and a way they couldn’t — and quasicrystals didn’t have the right look. But now, the new study has found that quasicrystals look perfect. Perhaps it’s crystals that need to explain themselves…
