Candy Science: M&Ms pack more tightly than spheres

Peter Weiss

Pouring M&Ms into a bowl leads to a marvel of packing efficiency, a team of sweet-toothed scientists reports.

Using bench experiments and computer simulations, the team has found that squashed or stretched versions of spheres snuggle together more tightly than randomly packed spheres do.

This surprising result could help scientists better understand the behavior of disordered materials ranging from powders to glassy solids, says Princeton University chemist Salvatore Torquato. The finding could also lead to denser ceramic materials that might make for improved heat shields for furnaces and reduced-porosity glass with exceptional transparency.

He and his colleagues at Princeton, Cornell University, and North Carolina Central University in Durham detail their results in the Feb. 13 Science.

"This work is really beautiful," comments Sidney R. Nagel of the University of Chicago. "It enhances our understanding of one of the outstanding questions in science"—namely, how densely various types of objects can pack together.

Investigations into arrangements of spheres date back centuries, but research into how efficiently aspherical objects aggregate has received scant attention.

In 1611, Johannes Kepler proposed that identical spheres can crowd together no more tightly than oranges do in a grocer's stack, a formation called face-centered cubic packing. In the 19th century, Carl Friedrich Gauss weighed in with a partial proof of Kepler's conjecture. Finally, in 1998, a mathematician offered a full proof, now widely accepted, that relies heavily on computer calculations (SN: 8/15/98, p. 103: http://www.sciencenews.org/pages/sn_arc98/8_15_98/fob7.htm). The grocery store arrangement fills 74 percent of available volume.

"Since [an] ellipsoid is nothing more than a simple deformation of a sphere, we wanted to see what effect that simple change of shape would have on the packing efficiency," Torquato explains. "The effect was dramatic and very unexpected."

Using either plain, regular-size M&Ms or smaller versions known as minis, the scientists tested how efficiently the candies pack when poured into a square box roughly the size of a coffee mug or into a spherical flask up to 5 liters in volume. To confirm the randomness of the internal packing, the team took magnetic resonance imaging scans of a 5-liter flask filled with about 7,500 regular-size M&Ms.

Torquato and other colleagues had shown previously that random stacking of spheres fills markedly less space—about 64 percent—than does the grocery arrangement (SN: 4/1/00, p. 219: Available to subscribers at http://www.sciencenews.org/articles/20000401/note13.asp). For both actual and simulated ellipsoids, Torquato and his colleagues now find that random packings fill as much as 73.5 percent of the space, just a smidgeon less than hand-stacked spheres do.

Why is random packing denser for ellipsoids than for spheres? The team proposes that the asymmetric ellipsoids can tip and rotate in ways that spheres can't, so an ellipsoid nestles close to more neighbors than a sphere does. Indeed, the team finds that as many as 11 neighbors touch an ellipsoid, whereas each tight-packed sphere typically has only 6 adjacent neighbors.

Theoretical physicist Samuel F. Edwards of Cambridge University in England says, "It's a real advance to do [packing experiments and simulations] on ellipsoids."

For such experiments, Torquato says, M&Ms make "great" test objects because they're inexpensive and uniform in size and shape. What's more, he adds, "you can eat the experiment afterwards."

********

Letters:

In this article, I read that an orb of a given size, when slightly flattened, will pack more densely than when perfectly round. No kidding? Do you suppose if we were to crush cars into little cubes we could hurl more into a landfill than we could just by driving them over a cliff? What am I missing?

Marc Gurevitch
Nicholasville, KY

Don't be misled by how many cars might fill a landfill, says Salvatore Torquato of Princeton University. Rather, consider the pile that either whole or crushed cars would form. The M&M team explored what fraction of such a pile is solid material, not air. When deformed spheres such as M&Ms are randomly piled and jostled, their small variations from sphericity boost that fraction unexpectedly high.—P. Weiss

Your story states that "each tight-packed sphere typically has only 6 adjacent neighbors." However, it is well known that closely packed oranges touch 12 adjacent oranges (cubic closest packing). One is tempted to explain the apparent 2-fold discrepancy by considering that each contact is shared by two particles, but that would be wrong. The number of contacting neighbors actually does decrease to about six in randomly close-packed spheres.

Don Garlick
Humboldt State University
Arcata, CA

References:

Donev, A. … S. Torquato, et al. 2004. Improving the density of jammed disordered packings using ellipsoids. Science 303(Feb. 13):990–993. Abstract available at http://dx.doi.org/10.1126/science.1093010.

Weitz, D.A. 2004. Packing in the spheres. Science 303(Feb. 13):968–969. Summary available at http://dx.doi.org/10.1126/science.1094581.

Further Readings:

Monastersky, R. 1990. Loosely packed spheres. Science News 137(June 16):382.

Peterson, I. 2000. Random packing of spheres. Science News 157(April 1):219. Available to subscribers at http://www.sciencenews.org/articles/20000401/note13.asp.

______. 2000. Packing spheres around a sphere. Science News 157(Feb. 26):141. Available to subscribers at http://www.sciencenews.org/articles/20000226/note15.asp.

______. 1998. Cracking Kepler's sphere-packing problem. Science News 154(Aug. 15):103. Available at http://www.sciencenews.org/pages/sn_arc98/8_15_98/fob7.htm.

Sources:

Samuel F. Edwards
Department of Physics
Polymers and Colloids Group
Cavendish Laboratory
University of Cambridge
Madingley Road
Cambridge CB3 0HE
United Kingdom

Sidney R. Nagel
University of Chicago
Department of Physics
5640 South Ellis Avenue
Chicago, IL 60637

Salvatore Torquato
Department of Chemistry
Princeton Materials Institute
Princeton, NJ 08544

From Science News, Volume 165, No. 7, February 14, 2004, p. 102.