Mathematicians and engineers from the University of Utah have teamed up to show how ultrasonic waves can organize carbon particles in water into something like a model that never repeats itself. According to them, the results can lead to materials called “quasicrystals”

The study was published in *Physical examination letters*.

“Quasicrystals are interesting to study because they have properties that crystals don’t have,” said Fernando Guevara Vazquez, an associate professor of mathematics. “They have been shown to be harder than similar periodic or disordered materials. They can also conduct electricity or scatter waves in ways other than crystals.”

**Models without patterns **

Imagine a chessboard. You can take a square two by two of two black tiles and two white (or red) tiles and copy and paste to get the whole chessboard. Such “periodic” structures, with patterns that *do* repeat, naturally occurring in crystals. Take, for example, a grain of salt. At the atomic level, it is a lattice lattice of sodium and chloride atoms. You can copy and paste the grid from one part of the crystal and find a match in any other part.

But the quasi-periodic structure is misleading. One example is a model called a Penrose lining. At first glance, the diamond-shaped geometric tiles look in the right pattern. But you can’t copy and paste this template. Will not be repeated.

The discovery of quasi-periodic structures in some metal alloys by materials scientist Dan Schechtman won the 2011 Nobel Prize in Chemistry and opened the study of quasicrystals.

Since 2012, Guevara and Bart Raymakers, an associate professor of mechanical engineering, have been collaborating on the design of materials with specially designed structures on a micro scale. Initially, they did not seek to create quasi-periodic materials – in fact, their first theoretical experiments, led by PhD Mauk, a doctoral student in mathematics, focused on periodic materials and what particle patterns could be achieved with ultrasonic waves. In each dimensional plane, they found that two pairs of parallel ultrasonic transducers were sufficient to arrange the particles in a periodic structure.

But what would happen if they had another pair of converters? To find out, Raeymaekers and student Milo Prisbray (now at the National Laboratory in Los Alamos) provide the experimental tools, and mathematics professor Elena Cherkaev provides experience in the mathematical theory of quasicrystals. Guevara and Mock performed theoretical calculations to predict the models that the ultrasound transducers would create.

**Creating quasi-periodic models**

Cherkaev says quasi-periodic models can be seen as using a “cutting and designing” technique instead of a cutting and placing approach.

If you use cropping and design to design quasi-periodic line patterns, you start with a square grid on a plane. Then draw or cut a line so that it passes through only one node of the grid. This can be done by drawing the line at an irrational angle, using an irrational number like pi, an infinite series of numbers that is never repeated. You can then design the nearest network nodes on the line and make sure that the patterns of the distances between the points on the line are never repeated. They are quasi-periodic.

The approach is similar in the two-dimensional plane. “We start with a grid or periodic function in a larger space,” says Cherkaev. “We cut a plane through this space and follow a similar procedure to limit the periodic function to an irrational 2-D cut.” When ultrasonic transducers are used, as in this study, the transducers generate periodic signals in this larger space.

The researchers placed four pairs of ultrasonic transducers in an octagonal stop device. “We knew it would be the simplest setting where we could demonstrate quasi-periodic particle arrangements,” says Guevara. “We also had limited control over what signals to use to drive the ultrasonic transducers; in essence, we could only use the signal or its negative.”

In this octagonal setting, the team placed small carbon nanoparticles suspended in water. Once the transducers are turned on, the ultrasonic waves direct the carbon particles into place, creating a quasi-periodic pattern similar to a Penrose lining.

“Once the experiments were done, we compared the results with the theoretical predictions and got very good agreement,” Guevara said.

**Custom materials**

The next step would actually be to make a material with a quasi-periodic arrangement of the pattern. This would not be difficult, Guevara says, if the particles were suspended in a polymer instead of water, which could be cured or hardened once the particles were in place.

“The most important thing is that with this method we can create quasi-periodic materials, which are either 2-D or 3-D and which can have essentially any of the common quasi-periodic symmetries, by choosing how we arrange the ultrasonic transducers and how we drive them. “, says Guevara.

It remains to be seen what these materials can do, but a possible application could be the creation of materials that can manipulate electromagnetic waves, such as those used by 5G cell technology today. Other already known applications of quasi-periodic materials include non-stick coatings due to their low friction coefficient and heat-insulating coatings, Cherkaev said.

Another example is the hardening of stainless steel by incorporating small quasicrystalline particles. The press release for the 2011 Nobel Prize in Chemistry mentions that quasicrystals can “strengthen the material like armor.”

So, researchers say, we can hope for many new exciting applications of these new quasi-periodic structures created by assembling ultrasonic particles.

The cascade to criticality

**More information:**

Elena Cherkaev et al., Wave-driven assembly of quasi-periodic particle models,

*Physical examination letters*(2021). DOI: 10.1103 / PhysRevLett.126.145501

Provided by the University of Utah

**Quote**: Using sound waves to make models that never repeat (2021, April 14), extracted on April 14, 2021 from https://phys.org/news/2021-04-patterns.html

This document is subject to copyright. Except for any fair transaction for the purpose of private examination or research, no part may be reproduced without written permission. The content is provided for informational purposes only.