What do giraffes, zebrafish, and turtles have in common: These animals display well-arranged patterns on their coat, scales, or shells. In 1952, to explain these geometric figures, the English mathematician Alan Turing developed a theory that combined the phenomena of interaction and diffusion. By choosing the parameters of their equivalents, it is possible to describe all these patterns, whether it is several tens of centimeters on a tiger or a few millimeters on a wasp. Recently, Yuki Fusya, of Chufu University in Japan, and colleagues, notably Kamran Behnia, a CNRS researcher at ESPCI, in Paris, observed these Turing structures at a nanoscale, with bismuth atoms.
The researchers initially wanted to study the topology of a single layer of bismuth in two dimensions, on the support for niobium disilinide (NbSe).2). But in 2018, they noticed that bismuth atoms clump together in clusters, sometimes linking together to form Y-type branches. The setup reminded the team of the patterns of the emperor angelfish, a tropical fish with yellow and blue stripes.
Bismuth is an element capable of organizing itself into many different solid structures, which vary easily with pressure conditions and even more so in such a thin layer. But such an arrangement of the atoms was unusual: niobium disilienide is usually a hexagonal crystal, which would have imposed very different constraints on the shape of bismuth. Are these patterns controlled by Turing’s equations? “The equations that underpin the Turing model have no distinct scale,” says Kamran Penhia. So it is not prohibited to use it on a microscopic scale. No one has come up with the idea to use this approach to explain some of the complex structures at the nanoscale. “
To validate this geometric arrangement of bismuth atoms as a Turing structure, Yuki Foscia and Kamran Penhia and colleagues modeled the system taking into account the angles between the covalent bonds of bismuth atoms to each other and to their supports. The obtained differential equation is related to a specific case of Turing states, and is able to theoretically predict the patterns of bismuth atoms with high accuracy compared to observations.
By playing with the parameters of this model, the researchers noticed that the patterns produced were of the same size whatever the initial conditions of the system: 1.7 nanometers in length, or the width of five atoms of bismuth. In principle, the Y junctions observed between the scratches disappear from the system when their composition reaches an equilibrium state, but the team was able to reproduce them by introducing defects or inclusions into the structure.
Depending on the parameters chosen, the bismuth layer can have hexagons, lines connected to Y, or even all parallel lines. This last arrangement constitutes a state of stable equilibrium. The result: in the event of a disturbance, the entire system “self-repairs” and reconfigures the parallel bands. In addition to the aesthetic aspect, the researchers hope to create, from Turing nanoscale shapes and their ability to self-repair, materials with new properties, such as ultra-thin films without any defects.
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