PRINCETON, N.J. -- Princeton scientists have toppled the standard way of thinking about a problem that has engrossed mathematicians since biblical times: how spheres, whether oranges or molecules, stack up when poured randomly into a vessel.
The scientists believe that their proposed alternative solution, in addition to fundamentally shifting the theoretical landscape in the field, may one day have important implications in the design and testing of new chemicals and materials.
Scientists in the field have long accepted the notion that, given enough stirring and shaking, a random collection of particles will always settle to a maximum density, a state known as random close packing. Particles in that state were believed always to have the same density -- about 64 percent. For example, if you poured marbles into a box, they would always pack so that they occupy 64 percent of the space and the rest is open air.
In the March 6 issue of Physical Review Letters, Professor of Chemistry Sal Torquato argues that the concept of random close packing is deeply flawed and that the 64 percent figure is not at all universal. Torquato co-authored the paper in collaboration with professor of chemical engineering Pablo Debenedetti and graduate student Tom Truskett.
Using a combination of computer simulations and theoretical observations, the scientists showed that the density of a randomly packed structure depends in large part on how the shaking, stirring and pouring is done. That is, the marbles may fit perfectly into a container if they are poured one way, but may not fit if they are poured another way.
That observation, says Torquato, negates the idea that there is a fixed concept called random close packing. "People have been trying for years to theoretically predict what the percentage would be - to no avail," said Torquato, "And it's because the concept was totally ill-defined."
"It is a big conceptual change," said Professor of Chemistry Roberto Car in reference to Torquato's work. "They proved that a concept that has been around for a long time is not well defined and is incorrect. That is a significant achievement."
Scholars have been interested in how particles pack together since antiquity, when a fair system of taxation or commerce depended on knowing how much material could be expected to fill a particular container. (In Luke 6:38, Jesus refers to "good measure, pressed down and shaken together.") In modern times, the question has become central to much of chemistry, because it defines how molecules order themselves within materials.
Scientists have long known the most efficient way of putting spherical particles together when the packing is done in an ordered, non-random way. The mathematician and astronomer Johannes Kepler first predicted that the best packing strategy would yield a density of about 74 percent. That figure, which was only recently proved in a formal way, corresponds to the so-called "face-centered cubic (fcc)" array, which is a ubiquitous concept in freshman chemistry and materials science.
The understanding of randomly packed particles has been more elusive. A variety of experiments have yielded results between 60 and 68 percent. Torquato and colleagues report that the same "random" method of packing can result in a range of densities all the way from 64 to 74 percent. The problem, said Torquato, is that the concept of randomness is too vague.
"When people called it 'random close packing,' they had no idea what they meant by randomness," said Torquato. He added "the notion of 'randomness' conflicts with that of 'close-packing,' which implies the most ordered structure."
To resolve the problem, the Princeton team proposed a formal method for measuring randomness, a mathematical expression for how strongly the structure varies from the most ordered fcc array. They then propose an entirely different concept, called the maximally random jammed state. The idea is to look for any combination of particles that is so tightly packed that none of the particles can move -- a jammed state -- and then measure how much randomness the structure contains.
In their computer simulations, the most disordered structure wound up having a density of 64 percent. He noted, however, that scientists now need to come up with a systematic way of sorting through jammed states and measuring their degrees of order. Better measures of randomness may be needed too, he said. Another step will be to extend the ideas to systems that are more complex than identical spheres. Understanding the nature of randomness in complex materials is one of the "holy grails," Torquato said.
"There still are some problems with the new definition, but it's a new way of looking at it, and that is important," said Professor of Physics Paul Chaikin.
Torquato, a faculty member of the Princeton Materials Institute, said this new approach could have important implications for the design of materials. For example, scientists may find it helpful to analyze the degree of randomness in materials during their processing history. "This knowledge may aid us to better tailor the resulting material properties," said Torquato.
NOTE: A photo and graphic are available at http://www.princeton.edu/pr/pictures/s-z/torquato/
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