Reptation of a semiflexible polymer through porous media

We study the motion of a single stiff semiflexible filament of length $S$ through an array of topological obstacles. By means of scaling arguments and two-dimensional computer simulations, we show that the stiff chain kinetics follows the reptation picture, albeit with kinetic exponents (for the central monomer) different from those for flexible chain reptation. At early times when topological constraints are irrelevant, the chain kinetics is the anisotropic dynamics of a free filament. After the entanglement time $τe$ transverse modes are equilibrated under the topological constraints, but the chain is not yet correlated over its whole length. During the relaxation of longitudinal modes, both the longitudinal fluctuation of the central monomer and the longitudinal correlation length grow as $∼t$⁠. After time $τr∼S2$ chain ends are correlated, the chain then diffuses globally along the tube and tube renewal takes place. In the reptation regime, the longitudinal fluctuation of the central monomer grows like $∼t1$⁠. The opening of the intermediate $∼t$ regime, absent for a free filament, is a signature of the reptation process. Although the underlying physics is quite different, the intermediate regime is reminiscent of the internal Rouse mode relaxation found for reptating flexible chains. In most cases asymptotic power laws from scaling could be complemented by prefactors calculated analytically. Our results are supported by two-dimensional Langevin simulations with fixed obstacles via evaluation of the mean squared displacement of the central monomer. The scaling theory can be extended to long semiflexible polymers adopting random-walk equilibrium configurations and should also apply in three dimensions for porous media with pore diameter smaller than the persistence length of the filament.