Forum for Science, Industry and Business

Sponsored by:     3M 
Search our Site:


Engineers discover in nature exotic structures envisioned by mathematicians


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:

More articles from Physics and Astronomy:

nachricht Enhancing the quantum sensing capabilities of diamond
23.11.2017 | The Hebrew University of Jerusalem

nachricht Quantum optics allows us to abandon expensive lasers in spectroscopy
22.11.2017 | Lomonosov Moscow State 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: Frictional Heat Powers Hydrothermal Activity on Enceladus

Computer simulation shows how the icy moon heats water in a porous rock core

Heat from the friction of rocks caused by tidal forces could be the “engine” for the hydrothermal activity on Saturn's moon Enceladus. This presupposes that...

Im Focus: Nanoparticles help with malaria diagnosis – new rapid test in development

The WHO reports an estimated 429,000 malaria deaths each year. The disease mostly affects tropical and subtropical regions and in particular the African continent. The Fraunhofer Institute for Silicate Research ISC teamed up with the Fraunhofer Institute for Molecular Biology and Applied Ecology IME and the Institute of Tropical Medicine at the University of Tübingen for a new test method to detect malaria parasites in blood. The idea of the research project “NanoFRET” is to develop a highly sensitive and reliable rapid diagnostic test so that patient treatment can begin as early as possible.

Malaria is caused by parasites transmitted by mosquito bite. The most dangerous form of malaria is malaria tropica. Left untreated, it is fatal in most cases....

Im Focus: A “cosmic snake” reveals the structure of remote galaxies

The formation of stars in distant galaxies is still largely unexplored. For the first time, astron-omers at the University of Geneva have now been able to closely observe a star system six billion light-years away. In doing so, they are confirming earlier simulations made by the University of Zurich. One special effect is made possible by the multiple reflections of images that run through the cosmos like a snake.

Today, astronomers have a pretty accurate idea of how stars were formed in the recent cosmic past. But do these laws also apply to older galaxies? For around a...

Im Focus: Visual intelligence is not the same as IQ

Just because someone is smart and well-motivated doesn't mean he or she can learn the visual skills needed to excel at tasks like matching fingerprints, interpreting medical X-rays, keeping track of aircraft on radar displays or forensic face matching.

That is the implication of a new study which shows for the first time that there is a broad range of differences in people's visual ability and that these...

Im Focus: Novel Nano-CT device creates high-resolution 3D-X-rays of tiny velvet worm legs

Computer Tomography (CT) is a standard procedure in hospitals, but so far, the technology has not been suitable for imaging extremely small objects. In PNAS, a team from the Technical University of Munich (TUM) describes a Nano-CT device that creates three-dimensional x-ray images at resolutions up to 100 nanometers. The first test application: Together with colleagues from the University of Kassel and Helmholtz-Zentrum Geesthacht the researchers analyzed the locomotory system of a velvet worm.

During a CT analysis, the object under investigation is x-rayed and a detector measures the respective amount of radiation absorbed from various angles....

All Focus news of the innovation-report >>>



Event News

Ecology Across Borders: International conference brings together 1,500 ecologists

15.11.2017 | Event News

Road into laboratory: Users discuss biaxial fatigue-testing for car and truck wheel

15.11.2017 | Event News

#Berlin5GWeek: The right network for Industry 4.0

30.10.2017 | Event News

Latest News

Desert ants cannot be fooled

23.11.2017 | Life Sciences

By saving cost and energy, the lighting revolution may increase light pollution

23.11.2017 | Earth Sciences

Retreating permafrost coasts threaten the fragile Arctic environment

23.11.2017 | Earth Sciences

More VideoLinks >>>