Externally driven molecular ratchets on a periodic potential surface: a rate equations approach

The long time dynamics of molecular ratchets on a 1D periodic potential energy surface (PES) subjected to an external stimulus is studied using the rate equation method. The PES consisting of repeated waveforms made of two peaks is considered as an example of a spatially symmetric or asymmetric PES. This PES may, for example, correspond to diffusion of a bipedal molecule that moves along an atomic track via an inchworm walk mechanism [Raval et al., Angew. Chem., Int. Ed., 2015, 54, 7101]. Generalisation to a PES consisting of an arbitrary number of peaks of various heights is straightforward. Assuming the validity of the transition state theory (TST) for the calculation of the transition rates between neighbouring potential wells, the probability of occupying each type of potential well on the PES is obtained analytically, and then the net current for the molecules to move preferentially in a particular direction under application of external fields over a long time is derived. Note that different to methods based on solving numerically the corresponding Fokker-Plank equation, our method is entirely analytical in the limit of weak external fields. The results of the analytical calculations are compared with the exact numerical solution of the derived rate equations. The following external stimuli are considered: constant, sinusoidal and shifted sinusoidal fields due to either a spatially uniform thermal gradient or an electrostatic field. The possible applications of the method for extracting energy from the Brownian motion under load and separating molecules of different chiralities on the surface are also discussed.