From Semi-Flexible Polymers to Membranes: Anomalous Diffusion and Reptation

The dynamics of semi-flexible polymers and membranes is discussed. The effect of thermal undulations on both the transversal and longitudinal Mean Square Displacement (MSD) of a tagged monomer is studied in free polymers and membranes. The two MSDs are found to be proportional to one another, and behave as ∼ t 3/4 for polymers and ∼ t 2/3 for membranes on the short time scale. The longitudinal motion is shown to be linked to the dynamics of fluctuations of the projected length (area) of the polymer (membrane). We demonstrate how, at long times, these fluctuations lead to reptation motion of the polymer (membrane) in the longitudinal direction. We generalize this approach to investigate the motion of a membrane between two plates and a polymer in a tube. The latter problem is used as a model for polymer motion in semi-dilute solutions in which the persistence length is longer than the entanglement length. Such systems are not suitable for the classical reptation model of de-Gennes and of Doi and Edwards, which was designed for chains that are flexible on the entanglement distance. The reptation diffusion coefficient and relaxation times that we obtain have the same scaling with chain length L as in the classical reptation model, but differ greatly in factors that are dependent on the ratio of persistence length to entanglement length. We also discuss the diffusion of a tagged monomer under imposed tension and liquid crystalline order.