Using sound waves to create patterns that never repeat

Mathematicians and engineers at the University of Utah have teamed up to show how ultrasonic waves can organize carbon particles in water into a kind of pattern that never repeats itself. The results, they say, could lead to materials called “quasi-crystals” with customized magnetic or electrical properties.

The research is published in Physical review letters.

“Quasicrystals are interesting to study because they have properties that crystals do not have,” says Fernando Guevara Vasquez, an associate professor of mathematics. “They have been shown to be stiffer than similar periodic or disordered materials. They can also conduct electricity or scatter waves in different ways in crystals.”

Models that are not patterns

Imagine a chessboard. You can take a square two by two of two blackboards and two whiteboards (or redboards) and copy and paste to get the whole chessboard. Such “periodic” structures, with patterns that do repeat, occur naturally in crystals. Take, for example, a grain of salt. At the atomic level, it is a network similar to the network of sodium and chloride atoms. You can copy and paste the grid from one side of the crystal and find a match anywhere else.

But a quasi-periodic structure is misleading. An example is the model called Penrose tile. At first glance, diamond-shaped geometric plates seem to have a common pattern. But you can’t copy and paste this pattern. It will not happen again.

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

Since 2012, Guevara and Bart Raeymaekers, associate professor of mechanical engineering, have been collaborating on the design of materials with customized structures under a microscope. Initially, they did not seek to create quasi-periodic materials – in fact, their first theoretical experiments, led by China Mauck, PhD in mathematics, focused on periodic materials and what patterns of particles could be made using ultrasound waves. In each dimensional plane, they found that two pairs of parallel ultrasonic transducers are sufficient to arrange the particles in a periodic structure.

But what if he had another pair of translators? To find out, Raeymaekers and graduate student Milo Prisbrey (now at Los Alamos National Laboratory) provided the experimental tools, and math teacher Elena Cherkaev provided experience with the mathematical theory of quasi-crystals. Guevara and Mauck performed theoretical calculations to predict the patterns that ultrasonic transducers will create.

Creating quasi-periodic patterns

Cherkaev says that quasi-periodic models can be considered as using, instead of a cutting and gluing approach, a “cutting and designing” technique.

If you use cutting and design to project quasi-periodic patterns on a line, start with a square grid on a plane. Then draw or cut a line so that it passes through a single network node. This can be done by drawing the line at an irrational angle, using an irrational number as pi, an infinite series of numbers that never repeat. Then you can design the closest 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 a two-dimensional plane. “We start with a grid or a periodic function in the larger space,” says Cherkaev. “We cut a plan through this space and follow a similar procedure to restrict periodic function to an irrational 2-D slice.” When using ultrasonic transducers, as in this study, the transducers generate periodic signals in that larger space.

The researchers set up four pairs of ultrasonic transducers in an octagonal arrangement with a stop sign. “I knew this would be the simplest configuration in which we could demonstrate quasi-periodic particle arrangements,” says Guevara. “We also had limited control over the signals we used to drive ultrasonic transducers; in essence, we could only use the signal or its negative.”

In this octagonal configuration, the team placed small carbon nanoparticles suspended in water. Once the transducers started, the ultrasound waves guided the carbon particles into place, creating a quasi-periodic pattern similar to Penrose plates.

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

Custom materials

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

“Crucially, with this method, we can create quasi-periodic materials that are either 2-D or 3-D and that can have essentially any of the common quasi-periodic symmetries, choosing how we arrange the ultrasonic transducers and how we conduct them. “Says Guevara. .

It has not yet been seen what these materials could do, but a possible application could be the creation of materials that can manipulate electromagnetic waves such as those that 5G cellular technology uses today. Other already known applications of quasi-periodic materials include non-stick coatings due to their low coefficient of friction and insulating coatings against heat transfer, says Cherkaev.

Another example is the hardening of stainless steel by incorporating small quasi-crystalline particles. The 2011 press release for the Nobel Prize in Chemistry states that quasi-crystals can “strengthen the material like armor”.

So, say researchers, we can hope for many interesting new applications of these new quasi-periodic structures created by ultrasonic particle assembly.