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 Heating quantum matter: A novel view on topology
22.08.2017 | Université libre de Bruxelles

nachricht Engineering team images tiny quasicrystals as they form
18.08.2017 | Cornell 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: Fizzy soda water could be key to clean manufacture of flat wonder material: Graphene

Whether you call it effervescent, fizzy, or sparkling, carbonated water is making a comeback as a beverage. Aside from quenching thirst, researchers at the University of Illinois at Urbana-Champaign have discovered a new use for these "bubbly" concoctions that will have major impact on the manufacturer of the world's thinnest, flattest, and one most useful materials -- graphene.

As graphene's popularity grows as an advanced "wonder" material, the speed and quality at which it can be manufactured will be paramount. With that in mind,...

Im Focus: Exotic quantum states made from light: Physicists create optical “wells” for a super-photon

Physicists at the University of Bonn have managed to create optical hollows and more complex patterns into which the light of a Bose-Einstein condensate flows. The creation of such highly low-loss structures for light is a prerequisite for complex light circuits, such as for quantum information processing for a new generation of computers. The researchers are now presenting their results in the journal Nature Photonics.

Light particles (photons) occur as tiny, indivisible portions. Many thousands of these light portions can be merged to form a single super-photon if they are...

Im Focus: Circular RNA linked to brain function

For the first time, scientists have shown that circular RNA is linked to brain function. When a RNA molecule called Cdr1as was deleted from the genome of mice, the animals had problems filtering out unnecessary information – like patients suffering from neuropsychiatric disorders.

While hundreds of circular RNAs (circRNAs) are abundant in mammalian brains, one big question has remained unanswered: What are they actually good for? In the...

Im Focus: RAVAN CubeSat measures Earth's outgoing energy

An experimental small satellite has successfully collected and delivered data on a key measurement for predicting changes in Earth's climate.

The Radiometer Assessment using Vertically Aligned Nanotubes (RAVAN) CubeSat was launched into low-Earth orbit on Nov. 11, 2016, in order to test new...

Im Focus: Scientists shine new light on the “other high temperature superconductor”

A study led by scientists of the Max Planck Institute for the Structure and Dynamics of Matter (MPSD) at the Center for Free-Electron Laser Science in Hamburg presents evidence of the coexistence of superconductivity and “charge-density-waves” in compounds of the poorly-studied family of bismuthates. This observation opens up new perspectives for a deeper understanding of the phenomenon of high-temperature superconductivity, a topic which is at the core of condensed matter research since more than 30 years. The paper by Nicoletti et al has been published in the PNAS.

Since the beginning of the 20th century, superconductivity had been observed in some metals at temperatures only a few degrees above the absolute zero (minus...

All Focus news of the innovation-report >>>



Event News

Call for Papers – ICNFT 2018, 5th International Conference on New Forming Technology

16.08.2017 | Event News

Sustainability is the business model of tomorrow

04.08.2017 | Event News

Clash of Realities 2017: Registration now open. International Conference at TH Köln

26.07.2017 | Event News

Latest News

Cholesterol-lowering drugs may fight infectious disease

22.08.2017 | Health and Medicine

Meter-sized single-crystal graphene growth becomes possible

22.08.2017 | Materials Sciences

Repairing damaged hearts with self-healing heart cells

22.08.2017 | Life Sciences

More VideoLinks >>>