In 1940, Hendrik Kramers (left) predicted theoretically that in a double-well system (center bottom), transitions between the stable states happen most frequently at intermediate friction (upper right). The background shows a detail of the laser system used to confirm Kramers' prediction experimentally. Source: Jan Gieseler; Image of H. Kramers courtesy of AIP Emilio Segrè Visual Archives, Goudsmit CollectionIn 1940, Hendrik Kramers (left) predicted theoretically that in a double-well system (center bottom), transitions between the stable states happen most frequently at intermediate friction (upper right). The background shows a detail of the laser system used to confirm Kramers' prediction experimentally. Source: Jan Gieseler; Image of H. Kramers courtesy of AIP Emilio Segrè Visual Archives, Goudsmit Collection

In 1827, Robert Brown, an English botanist, made an observation of what seemed like little importance. It turns out this observation played a central role in the development of the atomic theory of matter. Looking through the objective of a microscope, Brown noticed that pollen grains floating in water were constantly jiggling around as if driven by an invisible force. This phenomenon is now known as Brownian motion.

It was later understood that the irregular motion of the pollen particle is caused by the incessant buffeting of the water molecules surrounding the pollen particle. Albert Einstein’s theoretical analysis of this phenomenon provided evidence for the existence of atoms. The collisions of the pollen grain with water molecules have two important effects on the motion of the grain. They generate friction that slows the particle down and their thermal agitation keeps the particle moving at the same time. Brownian motion results from the balance of these competing forces.

Friction and thermal motion caused by the environment deeply affect transitions between long-lived states; for example, phase transitions like freezing or melting. The long-lived states are separated by a high energy barrier as depicted schematically in the illustration. The barrier between the wells prevents the physical system from rapidly interconverting between the two states. As a consequence, the system spends most of the time rattling around in one of the wells and doesn’t often jump from one well to another. These transitions are important for many processes in nature and technology ranging from phase transition to chemical reactions and folding patterns.

How often do these rare barrier-crossing events occur? In 1940, using a simple model system, Dutch physicist Hendrik Kramers showed mathematically that the rate at which transitions occur quickly decreases the growing barrier height. Kramers predicted that the transition rate depends on the friction in an interesting way: For strong friction, the system moved sluggishly, leading to a small transition rate. As the friction decreased, the system moved more freely and the transition rate grew. At sufficiently low friction, the transition rate started to decrease again because it took a long time for the system to acquire sufficient energy from the environment to overcome the barrier. The resulting maximum of the transition rate at intermediate friction is called the Kramers turnover.

So what does this have to do with technology and matter today? In a joint effort, scientists from the ETH Zurich, ICFO in Barcelona and University of Vienna have now succeeded in directly observing the Kramers turnover for a levitated nanoparticle. In their experiment, a nanoparticle is held in a laser trap with two wells that were separated by an energy barrier. Like the pollen grain that was observed by Brown, the nanoparticle constantly collides with the molecules surrounding it and these random interactions occasionally push the nanoparticle over the barrier.

By monitoring the motion of the nanoparticle over time, the scientists determined the rate at which the nanoparticle hops between the wells for a wide range of friction that can be accurately tuned by adjusting the pressure of the gas around the nanoparticle.

The rate obtained from this experiment clearly confirms the turnover predicted by Kramers almost 80 years ago.

"These results improve our understanding of friction and thermal motion at the nanoscale and will be helpful in the design and construction of future nanodevices," says Christoph Dellago, one of the authors of the study.

A paper on this study was published in Nature Nanotechnology.