Ultrasound Waves Create Never-Repeating Pattern in Carbon

Using soundwaves, a team of University of Utah engineers and mathematicians demonstrated how to arrange carbon nanoparticles in water in a never-repeating pattern. The new discovery, according to the team, could lead to a new class of materials called "quasicrystals" - having customizable magnetic and electrical properties.

The University of Utah researchers presented their new study in the article "Wave-Driven Assembly of Quasiperiodic Patterns of Particles," published in the latest Physical Review Letters, April 8. In the report, researchers theoretically show that the superposition of plane waves could create small particles to disperse in a fluid, assembling quasiperiodic 2D or 3D patterns.

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The Never-Repeating Pattern and Quasicrystals

Imagine the graphing paper or the mathematical notebook used in schools. These are great examples of a "periodic tiling of space," which in mathematics refer to a space covered by shape or a pattern without gaps or overlapping. Additionally, each of the basic shapes - in this case, a square - creates a pattern that can be moved by one unit, or rotated exactly 90 degrees, and will reveal the same pattern.

A similar pattern occurs in the atomic level, naturally taking place in crystals and crystalline structures like salt - which follows an orderly lattice of sodium (Na) and chloride (Cl) atoms. Any section of the crystal lattice is essentially similar anywhere in a grain of salt.

However, in the 1970s, British mathematician and Physics laureate Sir Roger Penrose discovered that a pattern could be created from two different shapes, using the angles and sides of a pentagon. Now known as Penrose tilings, these two basic tiles can only tile a plan nonperiodically, presenting it first using a pentagon and resulting in fivefold rotational symmetry. This never-repeating pattern is found in quasiperiodic structures - while they look regular and organized, any section of the pattern never repeats anywhere else.

A study of this structure led materials scientist Dan Schechtman to identify them in metal alloys, earning the 2011 Nobel Prize in Chemistry and opening up the study of quasicrystals. It led University of Utah associate professors Fernando Guevara Vasquez (mathematics) and Bart Raemaekers (mechanical engineering) to experiment with designs at the microscale.

Their original work, including math professor Elena Cherkaev, originally focused on periodic materials and their response to ultrasound waves. The team was able to generate theoretical models capable of predicting patterns created by ultrasound waves on quasicrystals.

"Quasicrystals are interesting to study because they have properties that crystals do not have," Vasquez said in a university press release. He also notes that these materials in a never-repeating pattern are commonly stiffer compared to periodic materials, as well as exhibit electrical and optical properties different from crystals.

Using Ultrasound Waves to Create a Never-Repeating Pattern

As they found that a pair of parallel ultrasound waves directed towards particles create a periodic structure, they tried adding another pair of ultrasound transducers. Researchers explain that the quasiperiodic, never-repeating, pattern is created in a "cut-and-project" technique instead of a cut-and-paste one.

To explain, start with a square grid on a plane. Then draw a line at an irrational angle - like the never-ending, never-repeating irrational number pi - so that it would pass only through one node of the grid. Then, project the nearest grid nodes on the line, creating a never-repeating pattern.

Cherkaev explains that the same approach is done in a 2D plane, starting with a periodic function or a grid from higher-dimensional space, cutting a plane, and keeping the irrational cut to a 2D slice.

They implemented the setup using four transducers of ultrasound waves arranged in an octagonal stop sign pattern since they believed that this is the "simplest setup" where quasiperiodic arrangements were possible.

They then placed carbon nanoparticles suspended in water in the middle. As they turned the transducers on, the carbon nanoparticles arranged themselves in a pattern similar to a Penrose tiling quasiperiodic one.