Tag Archives: L. Mahadevan

Physics of a singing saw could lead to applications in sensing, nanoelectronics, photonics, etc.

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I’d forgotten how haunting a musical saw can sound,

An April 22, 2022 news item on Nanowerk announces research into the possibilities of a singing saw,

The eerie, ethereal sound of the singing saw has been a part of folk music traditions around the globe, from China to Appalachia, since the proliferation of cheap, flexible steel in the early 19th century. Made from bending a metal hand saw and bowing it like a cello, the instrument reached its heyday on the vaudeville stages of the early 20th century and has seen a resurgence thanks, in part, to social media.

As it turns out, the unique mathematical physics of the singing saw may hold the key to designing high quality resonators for a range of applications.

In a new paper, a team of researchers from the Harvard John A. Paulson School of Engineering and Applied Sciences (SEAS) and the Department of Physics used the singing saw to demonstrate how the geometry of a curved sheet, like curved metal, could be tuned to create high-quality, long-lasting oscillations for applications in sensing, nanoelectronics, photonics and more.

An April 21, 2022 Harvard University John A. Paulson School of Engineering and Applied Sciences (SEAS) news release by Leah Burrows (also on EurekAlert but published on April 22, 2022) delves further into physics of singing saws,

“Our research offers a robust principle to design high-quality resonators independent of scale and material, from macroscopic musical instruments to nanoscale devices, simply through a combination of geometry and topology,” said L Mahadevan, the Lola England de Valpine Professor of Applied Mathematics, of Organismic and Evolutionary Biology, and of Physics and senior author of the study.

…

While all musical instruments are acoustic resonators of a kind, none work quite like the singing saw.

“How the singing saw sings is based on a surprising effect,” said Petur Bryde, a graduate student at SEAS and co-first author of the paper. “When you strike a flat elastic sheet, such as a sheet of metal, the entire structure vibrates. The energy is quickly lost through the boundary where it is held, resulting in a dull sound that dissipates quickly. The same result is observed if you curve it into a J-shape. But, if you bend the sheet into an S-shape, you can make it vibrate in a very small area, which produces a clear, long-lasting tone.”

The geometry of the curved saw creates what musicians call the sweet spot and what physicists call localized vibrational modes — a confined area on the sheet which resonates without losing energy at the edges.

Importantly, the specific geometry of the S-curve doesn’t matter. It could be an S with a big curve at the top and a small curve at the bottom or visa versa. 

“Musicians and researchers have known about this robust effect of geometry for some time, but the underlying mechanisms have remained a mystery,” said Suraj Shankar, a Harvard Junior Fellow in Physics and SEAS and co-first author of the study.  “We found a mathematical argument that explains how and why this robust effect exists with any shape within this class, so that the details of the shape are unimportant, and the only fact that matters is that there is a reversal of curvature along the saw.”

Shankar, Bryde and Mahadevan found that explanation via an analogy to very different class of physical systems — topological insulators. Most often associated with quantum physics, topological insulators are materials that conduct electricity in their surface or edge but not in the middle and no matter how you cut these materials, they will always conduct on their edges.

“In this work, we drew a mathematical analogy between the acoustics of bent sheets and these quantum and electronic systems,” said Shankar.

By using the mathematics of topological systems, the researchers found that the localized vibrational modes in the sweet spot of singing saw were governed by a topological parameter that can be computed and which relies on nothing more than the existence of two opposite curves in the material. The sweet spot then behaves like an internal “edge” in the saw.

“By using experiments, theoretical and numerical analysis, we showed that the S-curvature in a thin shell can localize topologically-protected modes at the ‘sweet spot’ or inflection line, similar to exotic edge states in topological insulators,” said Bryde. “This phenomenon is material independent, meaning it will appear in steel, glass or even graphene.”

The researchers also found that they could tune the localization of the mode by changing the shape of the S-curve, which is important in applications such as sensing, where you need a resonator that is tuned to very specific frequencies.

Next, the researchers aim to explore localized modes in doubly curved structures, such as bells and other shapes.

…

Here’s a link to and a citation for the paper,

Geometric control of topological dynamics in a singing saw by Suraj Shankar, Petur Bryde, and L. Mahadevan. The Proceedings of the National Academy of Sciences (PNAS) April 21, 2022 | 119 (17) e2117241119 DOI: https://doi.org/10.1073/pnas.2117241119

This paper is open (free) access.

Origami and our pop-up future

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They should have declared Jan. 25, 2016 ‘L. Mahadevan Day’ at Harvard University. The researcher was listed as an author on two major papers. I covered the first piece of research, 4D printed hydrogels, in this Jan. 26, 2016 posting. Now for Mahadevan’s other work, from a Jan. 27, 2016 news item on Nanotechnology Now,

What if you could make any object out of a flat sheet of paper?

That future is on the horizon thanks to new research by L. Mahadevan, the Lola England de Valpine Professor of Applied Mathematics, Organismic and Evolutionary Biology, and Physics at the Harvard John A. Paulson School of Engineering and Applied Sciences (SEAS). He is also a core faculty member of the Wyss Institute for Biologically Inspired Engineering, and member of the Kavli Institute for Bionano Science and Technology, at Harvard University.

Mahadevan and his team have characterized a fundamental origami fold, or tessellation, that could be used as a building block to create almost any three-dimensional shape, from nanostructures to buildings. …

A Jan. 26, 2016 Harvard University news release by Leah Burrows, which originated the news item, provides more detail about the specific fold the team has been investigating,

The folding pattern, known as the Miura-ori, is a periodic way to tile the plane using the simplest mountain-valley fold in origami. It was used as a decorative item in clothing at least as long ago as the 15th century. A folded Miura can be packed into a flat, compact shape and unfolded in one continuous motion, making it ideal for packing rigid structures like solar panels.  It also occurs in nature in a variety of situations, such as in insect wings and certain leaves.

“Could this simple folding pattern serve as a template for more complicated shapes, such as saddles, spheres, cylinders, and helices?” asked Mahadevan.

“We found an incredible amount of flexibility hidden inside the geometry of the Miura-ori,” said Levi Dudte, graduate student in the Mahadevan lab and first author of the paper. “As it turns out, this fold is capable of creating many more shapes than we imagined.”

Think surgical stents that can be packed flat and pop-up into three-dimensional structures once inside the body or dining room tables that can lean flat against the wall until they are ready to be used.

“The collapsibility, transportability and deployability of Miura-ori folded objects makes it a potentially attractive design for everything from space-bound payloads to small-space living to laparoscopic surgery and soft robotics,” said Dudte.

Here’s a .gif demonstrating the fold,

This spiral folds rigidly from flat pattern through the target surface and onto the flat-folded plane (Image courtesy of Mahadevan Lab) Harvard University

This spiral folds rigidly from flat pattern through the target surface and onto the flat-folded plane (Image courtesy of Mahadevan Lab) Harvard University

The news release offers some details about the research,

To explore the potential of the tessellation, the team developed an algorithm that can create certain shapes using the Miura-ori fold, repeated with small variations. Given the specifications of the target shape, the program lays out the folds needed to create the design, which can then be laser printed for folding.

The program takes into account several factors, including the stiffness of the folded material and the trade-off between the accuracy of the pattern and the effort associated with creating finer folds – an important characterization because, as of now, these shapes are all folded by hand.

“Essentially, we would like to be able to tailor any shape by using an appropriate folding pattern,” said Mahadevan. “Starting with the basic mountain-valley fold, our algorithm determines how to vary it by gently tweaking it from one location to the other to make a vase, a hat, a saddle, or to stitch them together to make more and more complex structures.”

“This is a step in the direction of being able to solve the inverse problem – given a functional shape, how can we design the folds on a sheet to achieve it,” Dudte said.

“The really exciting thing about this fold is it is completely scalable,” said Mahadevan. “You can do this with graphene, which is one atom thick, or you can do it on the architectural scale.”

Co-authors on the study include Etienne Vouga, currently at the University of Texas at Austin, and Tomohiro Tachi from the University of Tokyo. …

Here’s a link to and a citation for the paper,

Programming curvature using origami tessellations by Levi H. Dudte, Etienne Vouga, Tomohiro Tachi, & L. Mahadevan. Nature Materials (2016) doi:10.1038/nmat4540 Published online 25 January 2016

This paper is behind a paywall.

4D printing: a hydrogel orchid

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In 2013, the 4th dimension for printing was self-assembly according to a March 1, 2013 article by Tuan Nguyen for ZDNET. A Jan. 25, 2016 Wyss Institute for Biologically Inspired Engineering at Harvard University news release (also on EurekAlert) points to time as the fourth dimension in a description of the Wyss Institute’s latest 4D printed object,

A team of scientists at the Wyss Institute for Biologically Inspired Engineering at Harvard University and the Harvard John A. Paulson School of Engineering and Applied Sciences has evolved their microscale 3D printing technology to the fourth dimension, time. Inspired by natural structures like plants, which respond and change their form over time according to environmental stimuli, the team has unveiled 4D-printed hydrogel composite structures that change shape upon immersion in water.

“This work represents an elegant advance in programmable materials assembly, made possible by a multidisciplinary approach,” said Jennifer Lewis, Sc.D., senior author on the new study. “We have now gone beyond integrating form and function to create transformable architectures.”

…

In nature, flowers and plants have tissue composition and microstructures that result in dynamic morphologies that change according to their environments. Mimicking the variety of shape changes undergone by plant organs such as tendrils, leaves, and flowers in response to environmental stimuli like humidity and/or temperature, the 4D-printed hydrogel composites developed by Lewis and her team are programmed to contain precise, localized swelling behaviors. Importantly, the hydrogel composites contain cellulose fibrils that are derived from wood and are similar to the microstructures that enable shape changes in plants.

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By aligning cellulose fibrils (also known as, cellulose nanofibrils or nanofibrillated cellulose) during printing, the hydrogel composite ink is encoded with anisotropic swelling and stiffness, which can be patterned to produce intricate shape changes. The anisotropic nature of the cellulose fibrils gives rise to varied directional properties that can be predicted and controlled. Just like wood, which can be split easier along the grain rather than across it. Likewise, when immersed in water, the hydrogel-cellulose fibril ink undergoes differential swelling behavior along and orthogonal to the printing path. Combined with a proprietary mathematical model developed by the team that predicts how a 4D object must be printed to achieve prescribed transformable shapes, the new method opens up many new and exciting potential applications for 4D printing technology including smart textiles, soft electronics, biomedical devices, and tissue engineering.

“Using one composite ink printed in a single step, we can achieve shape-changing hydrogel geometries containing more complexity than any other technique, and we can do so simply by modifying the print path,” said Gladman [A. Sydney Gladman, Wyss Institute a graduate research assistant]. “What’s more, we can interchange different materials to tune for properties such as conductivity or biocompatibility.”

The composite ink that the team uses flows like liquid through the printhead, yet rapidly solidifies once printed. A variety of hydrogel materials can be used interchangeably resulting in different stimuli-responsive behavior, while the cellulose fibrils can be replaced with other anisotropic fillers of choice, including conductive fillers.

“Our mathematical model prescribes the printing pathways required to achieve the desired shape-transforming response,” said Matsumoto [Elisabetta Matsumoto, Ph.D., a postdoctoral fellow at the Wyss]. “We can control the curvature both discretely and continuously using our entirely tunable and programmable method.”

Specifically, the mathematical modeling solves the “inverse problem”, which is the challenge of being able to predict what the printing toolpath must be in order to encode swelling behaviors toward achieving a specific desired target shape.

“It is wonderful to be able to design and realize, in an engineered structure, some of nature’s solutions,” said Mahadevan [L. Mahadevan, Ph.D., a Wyss Core Faculty member] , who has studied phenomena such as how botanical tendrils coil, how flowers bloom, and how pine cones open and close. “By solving the inverse problem, we are now able to reverse-engineer the problem and determine how to vary local inhomogeneity, i.e. the spacing between the printed ink filaments, and the anisotropy, i.e. the direction of these filaments, to control the spatiotemporal response of these shapeshifting sheets. ”

“What’s remarkable about this 4D printing advance made by Jennifer and her team is that it enables the design of almost any arbitrary, transformable shape from a wide range of available materials with different properties and potential applications, truly establishing a new platform for printing self-assembling, dynamic microscale structures that could be applied to a broad range of industrial and medical applications,” said Wyss Institute Founding Director Donald Ingber, M.D., Ph.D., who is also the Judah Folkman Professor of Vascular Biology at Harvard Medical School and the Vascular Biology Program at Boston Children’s Hospital and Professor of Bioengineering at Harvard SEAS [School of Engineering and Applied Science’.

Here’s an animation from the Wyss Institute illustrating the process,

And, here’s a link to and a citation for the paper,

Biomimetic 4D printing by A. Sydney Gladman, Elisabetta A. Matsumoto, Ralph G. Nuzzo, L. Mahadevan, & Jennifer A. Lewis. Nature Materials (2016) doi:10.1038/nmat4544 Published online 25 January 2016

This paper is behind a paywall.

“Spring is like a perhaps hand,” E. E. Cummings, Harvard, and nano flowers

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It’s always a treat to read a news/press/media release that starts with poetry. From the May 16, 2013 Harvard University press release,

“Spring is like a perhaps hand,” wrote the poet E. E. Cummings: “carefully / moving a perhaps / fraction of flower here placing / an inch of air there… / without breaking anything.”

This was written to celebrate the publication of a paper by Wim L. Noorduin and others, from the press release (Note: Links have been removed),

By simply manipulating chemical gradients in a beaker of fluid, Wim L. Noorduin, a postdoctoral fellow at the Harvard School of Engineering and Applied Sciences (SEAS) and lead author of a paper appearing on the cover of the May 17 issue of Science, has found that he can control the growth behavior of these crystals to create precisely tailored structures.

“For at least 200 years, people have been intrigued by how complex shapes could have evolved in nature. This work helps to demonstrate what’s possible just through environmental, chemical changes,” says Noorduin.

The precipitation of the crystals depends on a reaction of compounds that are diffusing through a liquid solution. The crystals grow toward or away from certain chemical gradients as the pH of the reaction shifts back and forth. The conditions of the reaction dictate whether the structure resembles broad, radiating leaves, a thin stem, or a rosette of petals.

…

Replicating this type of effect in the laboratory was a matter of identifying a suitable chemical reaction and testing, again and again, how variables like the pH, temperature, and exposure to air might affect the nanoscale structures.

The project fits right in with the work of Joanna Aizenberg, an expert in biologically inspired materials science, biomineralization, and self-assembly, and principal investigator for this research.

Aizenberg is the Amy Smith Berylson Professor of Materials Science at Harvard SEAS, Professor of Chemistry and Chemical Biology in the Harvard Department of Chemistry and Chemical Biology, and a Core Faculty Member of the Wyss Institute for Biologically Inspired Engineering at Harvard.

Here are some details about how the scientists created their ‘flowers, from the press release,

To create the flower structures, Noorduin and his colleagues dissolve barium chloride (a salt) and sodium silicate (also known as waterglass) into a beaker of water. Carbon dioxide from air naturally dissolves in the water, setting off a reaction which precipitates barium carbonate crystals. As a byproduct, it also lowers the pH of the solution immediately surrounding the crystals, which then triggers a reaction with the dissolved waterglass. This second reaction adds a layer of silica to the growing structures, uses up the acid from the solution, and allows the formation of barium carbonate crystals to continue.

“You can really collaborate with the self-assembly process,” says Noorduin. “The precipitation happens spontaneously, but if you want to change something then you can just manipulate the conditions of the reaction and sculpt the forms while they’re growing.”

Increasing the concentration of carbon dioxide, for instance, helps to create ‘broad-leafed’ structures. Reversing the pH gradient at the right moment can create curved, ruffled structures.

Noorduin and his colleagues have grown the crystals on glass slides and metal blades; they’ve even grown a field of flowers in front of President Lincoln’s seat on a one-cent coin.

“When you look through the electron microscope, it really feels a bit like you’re diving in the ocean, seeing huge fields of coral and sponges,” describes Noorduin. “Sometimes I forget to take images because it’s so nice to explore.”

Here’s a link to and a citation for the paper,

Rationally Designed Complex, Hierarchical Microarchitectures by Wim L. Noorduin, Alison Grinthal, L. Mahadevan, and Joanna Aizenberg. Science 17 May 2013: Vol. 340 no. 6134 pp. 832-837 DOI: 10.1126/science.1234621

H/T to the May 17, 2013 news item on Azonano.

Jackson Pollock’s physics

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Take a mathematician (L. Mahadevan), a physicist (Andrzej Herczynski), and an art historian (Claude Cernuschi) and you’re liable to get a different perspective on Jackson Pollock*, a major figure in abstract expressionism, art. (I’m pretty sure there’s a joke in there of the: “There was mathematician and a physicist in a bar when an art historian came in …” ilk. I just can’t come up with it. If you can, please do leave it in the comments.)

Let’s start with a picture (image downloaded from the Wikipedia essay about Jackson Pollock’s No. 5, 1948),

No. 5, 1948 (Jackson Pollock, downloaded from Wikipedia essay about No. 5, 1948)

In a recent paper published in Physics Today (Painting with drops, jets, and sheets, which is behind a paywall), Mahadevan, Herczynski, and Cernuschi speculate about Pollock’s intuitive understanding of physics, in this case, fluid dynamics. From the June 28, 2011 news item on physorg.com,

A quantitative analysis of Pollock’s streams, drips, and coils, by Harvard mathematician L. Mahadevan and collaborators at Boston College, reveals, however, that the artist had to be slow—he had to be deliberate—to exploit fluid dynamics in the way that he did.

The finding, published in Physics Today, represents a rare collision between mathematics, physics, and art history, providing new insight into the artist’s method and techniques—as well as his appreciation for the beauty of natural phenomena.

…

“My own interest,” says Mahadevan, “is in the tension between the medium—the dynamics of the fluid, and the way it is applied (written, brushed, poured…)—and the message. While the latter will eventually transcend the former, the medium can be sometimes limiting and sometimes liberating.”

Pollock’s signature style involved laying a canvas on the floor and pouring paint onto it in continuous, curving streams. Rather than pouring straight from the can, he applied paint from a stick or a trowel, waving his hand back and forth above the canvas and adjusting the height and angle of the trowel to make the stream of paint wider or thinner.

Simultaneously restricted and inspired by the laws of nature, Pollock took on the role of experimentalist, ceding a certain amount of control to physics in order to create new aesthetic effects.

…

The artist, of course, must have discovered the effects he could create through experimentation with various motions and types of paint, and perhaps some intuition and luck. But that, says Mahadevan, is the essence of science: “We are all students of nature, and so was Pollock. Often, artists and artisans are far ahead, as they push boundaries in ways that are quite similar to, and yet different from, how scientists and engineers do the same.”

There’s more about this study on the physorg.com site including a video illustrating fluid dynamics. You can also find a June 29, 2011 news item on Science Daily and a June 29, 2011 article in Harvard Magazine about the study. From the Harvard news article,

MODERN ART WAS NEVER more famously lampooned than when Tom Stoppard [playwright and screenwriter] said, “Skill without imagination is craftsmanship and gives us many useful objects such as wickerwork picnic baskets. Imagination without skill gives us modern art.”

The article by expanding on Mahadevan’s research gives the lie to Stoppard’s quote. (I wonder if Stoppard will write a play about physics and art in the light of this new thinking about Pollock’s work?)

This all brought to mind, Richard Jackson’s work which was featured in 2010 at the Rennie Collection in Vancouver (my most substantive comments about Jackson’s work are in my May 11, 2010 posting). Trained as both an artist and an engineer, he too works with paint and its vicosity. I still remember the piece in the gallery basement that featured three (as I recall) cans of paint apparently caught in the act of being poured. In retrospect, one of the things I liked best about the show is that a lot of Jackson’s work is very much about the physical act of painting and the physicality of the materials.

One final note, the L. in Mahadevan’s name stands for Lakshinarayan.

*’Pollock’s’ corrected to Pollock on April 27, 2017.