In the classic fairy tale, "The Emperor's New Clothes,"
Hans Christian Andersen uses the eyes of a child to challenge conventional wisdom
and help others to see more clearly. In similar fashion, researchers at the
University of Illinois have now revealed the naked truth about a classic bell-shaped
curve used to describe the motion of a liquid as it diffuses through another
"The new findings raise fundamental questions concerning the statistical
nature of the diffusion process," says Steve Granick, Founder Professor
of Engineering, and professor of materials science and engineering, of chemistry,
of chemical and biomolecular engineering, and of physics at the U. of I.
Diffusion is critical to processes such as drug delivery, water purification,
and the normal operation of living cells. Key to the diffusion process is the
manner in which the motion of one molecule affects the motion of another.
"In high school science classes, students are often assigned the task
of using a microscope to watch a particle of dust sitting in a drop of water,"
Granick said. "The dust particle seems alive, moving back and forth, never
in the same way. The motion of the dust particle is caused by the random ‘kicks'
of surrounding water molecules."
Called "Brownian motion" (after botanist Robert Brown, who noticed
it in 1828), this phenomenon of fluids was described by Albert Einstein in 1905,
when he published his statistical molecular theory of liquids.
According to Einstein, if the motions of many particles were watched, and the
distance each moved in a certain time were recorded, the distribution would
resemble the familiar Gaussian, bell-shaped curve used to assign grades in a
Einstein had it right – almost.
"Like Einstein, we used to think we could describe Brownian motion with
a standard bell-shaped curve," Granick said. "But now, with the
ability to measure very small distances much more precisely than was possible
100 years ago, we have found that we can have extremes much farther than previously
In a paper to be published in the Proceedings of the National Academy of Sciences
Online Early Edition next week, the U. of I. researchers show that Einstein's
explanation, commonly cited in textbooks, fails in certain important cases.
The experiments were conducted by precisely tracking the motion of 100-nanometer
colloidal beads using fluorescence microscopy.
In one series of experiments, the researchers watched as the beads moved up
and down tiny tubes of lipid molecules by Brownian motion. In a second series
of experiments, the researchers watched as the beads diffused through a porous
membrane of entangled macromolecule filaments, again by Brownian motion.
In both sets of experiments, there were many features in full agreement with
Einstein and the bell-shaped curve; but there were also features in significant
disagreement. In those cases, the beads moved much farther than the common curve
could predict. In those extreme displacements, diffusion behavior was not Gaussian,
the researchers report. The behavior was exponential.
"These large displacements happen less often, but when they do occur,
they are much bigger than we previously thought possible," Granick said.
The new findings "change the rules of the diffusion game," Granick
said. "Like the emperor's new clothes, now that we know the bell-shaped
curve isn't always the right way to think about a particular problem,
process, or operation, we can begin to design around it, and maybe take advantage
of it. And, we can correct the textbooks."
Granick is affiliated with the university's Beckman Institute, the department
of bioengineering, and the Frederick Seitz Materials Research Laboratory.
With Granick, co-authors of the paper are graduate research assistant and lead
author Bo Wang, graduate research assistant Stephen M. Anthony and research
scientist Sung Chul Bae.