Rack and Pinion a la Casimir

Noncontact Rack and Pinion Powered by the Lateral Casimir Force

Arahs Ashourvan, MirFaez Miri, and Ramin Golestanian

PRL 98, 140801 (2007)

URL: http://link.aps.org/abstract/PRL/v98/e140801

The authors propose a nanoscale rack and pinion where the Casimir force (rather than contact between the cogs) allows the pinion to move the rack. They find two regimes. There is a contact regime, where the gears move as if they were in direct contact with one another, and a skipping regime where the teeth can slip by one another. The interesting aspect of this device is that the cogs are never in physical contact with one another.

The equation of motion for the system presented in Eq. (1) is rather simple. You could write it down without knowing much about racks, pinions, or vacuum fluctuations. It simply describes the motion of a pendulum with damping and a driving torque. It's a nonlinear equation that cannot be solved analytically, but it is a classical system.

The authors go on to study the system in four perturbative regimes. First, they study the case of no damping or torque. This system can be integrated, and shows crossover behavior between the contact and skipping regimes mentioned earlier. Depending on the velocity of the rack, the pinion can rotate in either direction. When a torque is applied, the same general behavior is observed. The main differenc is that the boundary between the two regions depends on the applied torque.

The case with dissipation cannot be solved exactly. In the case of weak dissipation, the authors treat the damping as a small perturbation using something called the Melnikov Method. This introduces an interesting regime where the pinion velocity is independent of the torque, and a load at which the velocity drops to zero --- the stall force.

In the case of strong dissipation, the authors discard the acceleration term (like a Langevin equation) and integrate the approximate equation of motion. There is again a stall force, and the behavior is similar to case of weak damping above the skipping velocity.

Finally, the authors investigate the actual form of the Casimir force in their system. As I said earlier, the above analysis is for a damped pendulum under constant torque. To say anything meaningful about the role of the Casimir force, the authors have to introduce it into this phenomenological model. The force decays exponentially with increasing separation between the rack and pinion. The skipping velocity is a power law at small separations and decays exponentially at large separations (like a gamma distribution, I suppose). This is quite useful in applications.

I kept this article in my files because it seems like the type of fundamental nanotechnology research needed to move the field forward. We can't just build little versions of big machines because effects like thermal fluctuations and friction can ruin everything on tiny scales. These authors have shown how you can take advantage of an effect that only happens at these small scales. The rack and pinion steering of an automobile will never utilize the Casimir force, but if you want to build a tiny ratchet out of nanotubes and buckyballs, then you might not have any other option. Well done guys.