Forum for Science, Industry and Business

Sponsored by:     3M 
Search our Site:

 

Engineers discover in nature exotic structures envisioned by mathematicians

25.07.2003


Three years before he received the Nobel Prize in Physics, Eugene Wigner published an article entitled "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" (1960). He marveled at how often physicists develop concepts to describe the "real" world only to discover that mathematicians--heedless of that real world--have already thought up and explored the concepts. His own experience of the uncanny applicability of mathematical insights to the physical reality of quantum mechanics led Wigner to observe "that the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it."



When compressed by a liquid droplet, small groups of colloidal microspheres -- plastic spheres with diameters about one one-hundredth that of a human hair -- pack to form an unusual sequence of structures. At top are packings containing four to eleven spheres, as seen through the scanning electron microscope. At bottom are the polyhedra defined by drawing lines between the centers of touching spheres in each cluster. Some of these polyhedra are familiar structures, such as the tetrahedron (4 spheres) and octahedron (6 spheres), but most of the others -- including the "snub disphenoid" (8 spheres) and the "gyroelongated square dipyramid" (10 spheres) -- are probably unfamiliar, despite their attractive symmetry. Nevertheless, all of these structures obey a single, simple mathematical rule: they all minimize a quantity called the second moment. This is the first observation of this packing motif in nature. [Image credit: V. N. Manoharan]



Doubtless the observation of just such an uncanny correspondence between mathematics and physics prompted the editors of the July 25 issue of "Science" to feature on the cover the colloidal particle clusters that are the subject of research by an engineering professor and his two graduate students at the University of California at Santa Barbara (UCSB). That professor, David Pine, holds a joint appointment in the departments of Chemical Engineering and Materials and chairs the Chemical Engineering Department. The first author of the article, "Dense Packing and Symmetry in Small Clusters of Microspheres," is Vinothan Manoharan; the other author is Mark Elsesser.

Their story begins with the iridescence of opals, which are composed of equal-sized spheres about a micrometer in diameter, or roughly a hundred times smaller than the size of a human hair. The spheres are packed into a structure known as the face-centered cubic (FCC) lattice, which is exactly the same arrangement used by grocers to stack oranges or apples. Because the opal’s constituent spheres are about the size of the wavelength of light, their orderly arrangement diffracts light and causes iridescence.


Pine notes, "Opals have interesting optical properties, but not quite interesting enough. We are trying to improve on this structure to make some useful optical materials."

In principle such materials, known as "photonic crystals," would enable new and inexpensive optical circuits and might also improve the efficiency of devices such as lasers and LEDs. How to make a photonic crystal is not the subject of the "Science" article, but what the researchers discovered in the attempt.

They began by trying to find ways to pack tiny spheres, like the ones that make up an opal, into structures different from the FCC. This is a difficult problem, since, as the mathematician Kepler long ago conjectured, the FCC structure is the densest packing of an infinite number of spheres. In other words, the face-centered cubic structure results whenever a large number of spheres are compressed together. But, the researchers asked, how do a finite number of spheres pack? What structures are formed by a very small number of spheres, say, five or eight?

The experiments which answered that question began with Manoharan taking colloidal microspheres of the common plastic polystyrene and trapping the particles in small droplets of the oily solvent toluene. Then he heated the mixture so that the solvent droplets evaporated, effectively shrink-wrapping the particles into little clusters. Finally, using a centrifuge, he separated the clusters according to the number of particles in each i.e., doublets, triplets, etc.

"The thing that really grabbed our attention," said Manoharan, "was that clusters that contained the same number of particles always had the same configuration." Or, in the language of the "Science" paper, "small numbers (n = 2-15) of hard spheres pack into distinct and identical polyhedra for each value of n." Moreover, when Manoharan examined the clusters under the microscope, he found that many of the structures had beautiful and unexpected symmetry. The seven-sphere cluster, for example, resembles a flower with five petals.

Surprisingly, the symmetry of these configurations has nothing to do with chemical bonds or quantum mechanics. The clusters, it turns out, obey a very simple mathematical principle first explored in 1995 by mathematicians N.J.A. Sloane of AT&T Research, John Conway of Princeton, and colleagues. Sloane and Conway derived the structures of sphere packings that minimize a quantity called the "second moment of the mass distribution."

The structures the mathematicians predicted are the same as those of the colloidal clusters.

"What’s amazing," said Pine, "is that their interests had nothing to do with colloids or emulsions. They were studying a problem in pure mathematics."

What is the "second moment"? Said Pine, "Take one of these clusters and define its center of gravity as the point at which if you hang the cluster by a string it will not rotate. Then you take the distances of each of these spheres from that center of gravity (measuring from the center of the sphere) and square those distances and add the squares together, and that’s the second moment of the mass distribution."

The researchers caution that they do not yet fully understand the physical process that causes the clusters to minimize this quantity. But the mathematical connection has a certain elegance.

"Occasionally," said Pine, "there’s a correspondence between mathematics and the way nature behaves that’s really striking. Most of the time the connection is difficult to visualize, but in this case a layperson can explore it with only a package of ping-pong balls and some glue."

Manoharan points out that their results may be relevant to fields other than colloids, since scientists often model the building blocks of matter as spheres. "These clusters tell us something about matter in general--how symmetry arises from simple packing constraints. That may be important in understanding the atomic-scale structure of liquids, for example. Between the mathematical beauty of the cluster structures and their engineering applications, there is some interesting physics in terms of understanding how geometry affects the basic properties of matter."

The research reported in "Science" is part of Manoharan’s thesis for a Ph.D. from Santa Barbara in chemical engineering. He has accepted a faculty appointment in physics and engineering at Harvard University after a postdoctoral fellowship at the University of Pennsylvania.

Jacquelyn Savani | EurekAlert!
Further information:
http://www.engineering.ucsb.edu

More articles from Physics and Astronomy:

nachricht Convenient location of a near-threshold proton-emitting resonance in 11B
29.05.2020 | The Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of Sciences

nachricht A special elemental magic
28.05.2020 | Kyoto University

All articles from Physics and Astronomy >>>

The most recent press releases about innovation >>>

Die letzten 5 Focus-News des innovations-reports im Überblick:

Im Focus: Biotechnology: Triggered by light, a novel way to switch on an enzyme

In living cells, enzymes drive biochemical metabolic processes enabling reactions to take place efficiently. It is this very ability which allows them to be used as catalysts in biotechnology, for example to create chemical products such as pharmaceutics. Researchers now identified an enzyme that, when illuminated with blue light, becomes catalytically active and initiates a reaction that was previously unknown in enzymatics. The study was published in "Nature Communications".

Enzymes: they are the central drivers for biochemical metabolic processes in every living cell, enabling reactions to take place efficiently. It is this very...

Im Focus: New double-contrast technique picks up small tumors on MRI

Early detection of tumors is extremely important in treating cancer. A new technique developed by researchers at the University of California, Davis offers a significant advance in using magnetic resonance imaging to pick out even very small tumors from normal tissue. The work is published May 25 in the journal Nature Nanotechnology.

researchers at the University of California, Davis offers a significant advance in using magnetic resonance imaging to pick out even very small tumors from...

Im Focus: I-call - When microimplants communicate with each other / Innovation driver digitization - "Smart Health“

Microelectronics as a key technology enables numerous innovations in the field of intelligent medical technology. The Fraunhofer Institute for Biomedical Engineering IBMT coordinates the BMBF cooperative project "I-call" realizing the first electronic system for ultrasound-based, safe and interference-resistant data transmission between implants in the human body.

When microelectronic systems are used for medical applications, they have to meet high requirements in terms of biocompatibility, reliability, energy...

Im Focus: When predictions of theoretical chemists become reality

Thomas Heine, Professor of Theoretical Chemistry at TU Dresden, together with his team, first predicted a topological 2D polymer in 2019. Only one year later, an international team led by Italian researchers was able to synthesize these materials and experimentally prove their topological properties. For the renowned journal Nature Materials, this was the occasion to invite Thomas Heine to a News and Views article, which was published this week. Under the title "Making 2D Topological Polymers a reality" Prof. Heine describes how his theory became a reality.

Ultrathin materials are extremely interesting as building blocks for next generation nano electronic devices, as it is much easier to make circuits and other...

Im Focus: Rolling into the deep

Scientists took a leukocyte as the blueprint and developed a microrobot that has the size, shape and moving capabilities of a white blood cell. Simulating a blood vessel in a laboratory setting, they succeeded in magnetically navigating the ball-shaped microroller through this dynamic and dense environment. The drug-delivery vehicle withstood the simulated blood flow, pushing the developments in targeted drug delivery a step further: inside the body, there is no better access route to all tissues and organs than the circulatory system. A robot that could actually travel through this finely woven web would revolutionize the minimally-invasive treatment of illnesses.

A team of scientists from the Max Planck Institute for Intelligent Systems (MPI-IS) in Stuttgart invented a tiny microrobot that resembles a white blood cell...

All Focus news of the innovation-report >>>

Anzeige

Anzeige

VideoLinks
Industry & Economy
Event News

Dresden Nexus Conference 2020: Same Time, Virtual Format, Registration Opened

19.05.2020 | Event News

Aachen Machine Tool Colloquium AWK'21 will take place on June 10 and 11, 2021

07.04.2020 | Event News

International Coral Reef Symposium in Bremen Postponed by a Year

06.04.2020 | Event News

 
Latest News

Black nitrogen: Bayreuth researchers discover new high-pressure material and solve a puzzle of the periodic table

29.05.2020 | Materials Sciences

Argonne researchers create active material out of microscopic spinning particles

29.05.2020 | Materials Sciences

Smart windows that self-illuminate on rainy days

29.05.2020 | Power and Electrical Engineering

VideoLinks
Science & Research
Overview of more VideoLinks >>>