Segment Motion in the Reptation Model of Polymer Dynamics. I. Analytical Investigation

TL;DR

This paper provides an analytical study of bead motion in the reptation model of polymer dynamics, revealing microstructure effects and crossover behaviors that influence experimental observations and challenge traditional asymptotic power laws.

Contribution

It offers a rigorous analytical approach to polymer bead motion in the reptation model, incorporating tube renewal and microstructure effects, with insights into crossover phenomena and experimental observables.

Findings

Microstructure effects persist for large times and long chains.

Asymptotic power laws only hold for extremely long chains.

Crossover behaviors overlap and obscure asymptotic regimes in practical chain lengths.

Abstract

We analyze the motion of individual beads of a polymer chain using a discrete version of De Gennes' reptation model that describes the motion of a polymer through an ordered lattice of obstacles. The motion within the tube can be evaluated rigorously, tube renewal is taken into account in an approximation motivated by random walk theory. We find microstructure effects to be present for remarkably large times and long chains, affecting essentially all present day computer experiments. The various asymptotic power laws, commonly considered as typical for reptation, hold only for extremely long chains. Furthermore, for an arbitrary segment even in a very long chain, we find a rich variety of fairly broad crossovers, which for practicably accessible chain lengths overlap and smear out the asymptotic power laws. Our analysis suggests observables specifically adapted to distinguish reptation from motions dominated by disorder of the environment.