Fixing Fick’s Law

Winter 2008

Marc Donohue, the Whiting School’s vice dean of research, places a cup of water on a table in his office. A cherry-flavored Life Saver rests at the bottom. “That’s diffusion,” he says, pointing at the cup.

Donohue has been working to better understand diffusion for nearly a decade. Recently, he and a team of colleagues shed new light on a classic diffusion equation—known as Fick’s Law—that has caused questions among mathematicians and researchers for nearly a century.

“Einstein gave a lot of credence to the idea that Fick’s Law is correct,” says Donohue, a professor in the Department of Chemical and Biomolecular Engineering. “But,” he concludes, “his analysis of it was overly simplistic.”

First described by physiologist Adolf Fick in 1855, Fick’s Law provides a mathematical explanation for diffusion. Specifically, it holds that during diffusion (the process by which molecules randomly and gradually spread from a high concentration to a low), the flux—or rate—at which the molecules spread is proportional to the concentration gradient.
Fick's Law

To illustrate, Donohue uses the submerged Life Saver and the ring of reddish-pink that slowly colors the water at the cup’s bottom. Left to sit for a week, the red coloring slowly creeps toward the top, until finally all the water in the cup is an even shade of pale red.

Classical Fick’s Law holds that the time it takes for the red to reach the top of the cup depends on the concentration gradient. “But, for the past century, everyone has known that concentration is not really the right variable to use in the equation,” says Donohue. “Instead, everyone believed that the flux is proportional to the gradient of the chemical potential.” In other words, the amount by which the free energy of the system (i.e., cup’s water) would change if another particle (i.e., Life Saver) was introduced.

So, Donohue, along with Gregory Aranovich, a fellow professor in the department, and a cadre of graduate students, set out to determine whether the chemical potential is the true driving force for the equation.

They concluded that it isn’t. At the same time, they discovered an even bigger problem. “We knew that concentration is not the driving force for diffusion,” Donohue says. “We soon learned that chemical potential is not, either. But then we determined that the equation itself is wrong.”

To develop a more robust and accurate theory, the team began to look at instances in which Fick’s Law doesn’t work. “One of the things we then predicted is that while diffusion is a dissipative process in which molecules move from high to low concentration, it also includes a wave phenomenon.” Previously, it was thought that during diffusion the random movement of molecules occurs on an infinitesimally small scale; that each molecule moves in one direction a tiny bit and then in another a tiny bit, and so forth. Donohue found that those distances can be large and, in some instances, density waves are created that lead to a layered buildup of molecules.

Profiled across a graph, classical Fick’s Law looks like a loaf of bread: an even distribution of molecules over time. Donohue’s theory, on the same graph, looks like a frontal silhouette of Batman’s head. It’s generally round, but the density waves create two spiky ears. “We call it the Batman Profile,” Donohue explains with a hint of a smile.

The team found that density waves manifest in a variety of conditions. For example, in low pressure gases, or when diffusion occurs on a really small scale, such as in nanoporous materials, or on a really large scale, such as in outer space where molecules travel long distances before colliding with anything. Ultimately, it means that the definition of Fick’s Law as Einstein derived it is flawed, and that there is a better explanation to be found.

“But, this is not going to be solved completely in my lifetime,” Donohue says. “Our biggest contribution is that we’ve shown people that the current theory is limited in its applicability, and we’re pointing them in the direction of a more complete theory.”