A frugal way to study complex systems and materials

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To celebrate the 60th birthday of King Oscar II of Sweden and Norway in 1889, the newspaper Mathematical Act offered a prize for manuscripts that could help solve the following question, commonly referred to as the 3-body problem: Can we predict the orbits of planets, moons, and other celestial bodies over time? Although the mathematician Henri Poincaré received the gold medal and the 2,500 Swedish kronor prize for his submission (it was later found to be in error), the general analytical solution to the problem of “n bodies »Remained difficult to follow. Beyond celestial mechanics, “n-body” issues arise in everything from how proteins fold to understanding complex materials.

Illustration of droplet networks that can be rapidly prototyped to study the role of geometry in macroscopic analogues of complex materials. (Image credit: Rebecca Konte, Artist-in-Residence, Prakash Lab)

Although no analytical insight has solved these complex problems, various macroscopic analogies that mimic the multi-body interaction and the specific geometry of the problem are extremely insightful. Among these, a simple table-top experimental method developed by engineers at Stanford University. All you need to get started is a slippery surface (say a glass slide), a permanent marker, and a mixture of water and propylene glycol, a common ingredient in food coloring.

With these supplies, the researchers invented a new way to rapidly prototype complex geometries reflecting the symmetries present in the problems of interest. Instead of planets scattered throughout the solar system, many tiny droplets interact from a distance and the observer can directly observe and manipulate the evolution of the system over time. The researchers detailed their new method in an article published Aug. 24 in Proceedings of the National Academy of Sciences.

The simple – and inexpensive – technique could be applied to many different questions in a myriad of fields. Other methods for studying these problems tend to be purely theoretical or require expensive and scarce nanoscale equipment. For their part, researchers are particularly interested in the physics of exotic materials, including new magnetic materials that are unlike anything found in nature.

“People are starting to be able to make almost any material in any geometry they want – but these systems are so complex that people don’t have the ability to really understand them,” Anton said. Molina, lead author of the article and a graduate student in the laboratory of Manu Prakash, associate professor of bioengineering. “So we’re excited to be able to use this fast and economical tool to quickly explore many possible configurations. “

Feedback from envelope experiences

The idea to explore the role of geometry in the self-interactive system arose many years ago when the group released a new class of active ingredient system called “dancing droplets”- where a complex interplay of interactions emerges in the evaporation of droplets that can detect and move autonomously, almost like crawling cells. The next big challenge in taming this system was to incorporate defined interactions.

A network of gold droplets with “dancing droplets” of propylene glycol. (Image credit: Prakash Lab)

“What we wanted was the simplest possible system in which the geometry of the interaction is completely programmable and tunable, while behaving like a complex system and producing things that we could not predict,” said Prakash, who is a senior. author of the article.

Based on this dancing droplet work, the researchers knew that this food additive was able to mimic interactions between different “bodies” in multi-body systems, such as gravitational forces between celestial objects or electrostatic forces between them. atoms. Next, to introduce complex geometry essential for emulating many complex systems, the researchers created arrays of droplets either neatly printed in gold or literally drawn with a permanent marker.

“We decided to use lithographic printing for more precision, but we also did a lot of the prototyping using permanent markers,” said Molina. The straightforward process meant researchers could go from sketching out a design while relaxing in the yard outside their lab to experimenting with testing that design in a matter of hours. Prakash likened the process to “back-of-the-envelope calculations” for the experiments. This rapid exploration of the role of geometry in dynamical systems makes it possible to understand exotic configurations and test new ideas at a rapid pace.

The researchers initially focused on hexagonal lattices made up of smaller hexagons, as this is the simplest structure that leads to a non-trivial evolution of the dynamics in these droplet lattices. Motivated to achieve the lowest possible energy state for the system as a whole, the droplets form clumps of three and, as a whole, move towards the center of the network. Their first movements occur shortly after they are put into the array, but the individual changes – which then trigger changes among the other droplets – continue for several minutes. The multi-stage organization of the system is a universal feature of systems with long-range interactions; where all the droplets communicate and pull and push each other simultaneously via an invisible vapor phase.

“Over time, everything evaporates and the water vapor leaves the system. Locally, these triplets preserve the lifespan of the gout the longest, ”explained Molina. “Watching the system gives you an answer as to how these droplets actually do it, and yet it is still confusing to understand the individual movement of a given droplet. “

An invitation to discover

In addition to their hexagons, the researchers created various square arrays that correspond to common patterns in physics, computer science, and materials science. A better understanding of these systems could help inform the design of next-generation computing substrates that might deviate from the conventional architectures we are used to seeing in today’s microchips.

However, as often happens with the work of the Prakash Lab, this experimental tool is not just intended for academics and experts. A marker and food coloring could help solve Mathematical Actquestion of price, or it could be a way to explain the fundamental role of geometry in materials or complex energy states to school children.

“You might say, ‘Oh, to do basic science and discover new rules of nature, I need this and that. “But even for the experiments, that’s not true,” Prakash said. “So I hope people take this as an open invitation to explore because we took these experiences that were incredibly difficult and made them really, really simple. With creativity, there are ways to ask really basic questions.

Other Stanford co-authors include postdoctoral researcher Shailabh Kumar and former postdoctoral researcher Stefan Karpitschka (now at the Max Planck Institute for Dynamics and Self-Organization). Prakash is also a principal investigator at Stanford Woods Institute for the Environment; a member of Bio-X, the Maternal and Child Health Research Institute and the Wu Tsai Institute of Neuroscience; faculty member, Howard Hughes Medical Institute; and an investigator from Chan Zuckerberg Biohub.

This research was funded by the National Science Foundation, the Keck Foundation, an HHMI-Gates Faculty Scholar Award, and a CZI Biohub Investigator Award.

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