# Research Interests (reptation)

Prof. Dr. Gunter M. Schütz

Homepage G.Schütz | Driven diffusive systems | Quantum spin chains far from equilibrium | Reaction-diffusion systems |

## Reptation of entangled polymers

| Reptation theory describes the snake-like large-scale motion of entangled polymers. One imagines the diffusive motion of polymer segments to be confined to a hypothetical tube which is essentially the sequence of pores of the entanglement network in which the polymer is embedded. This basic picture of reptation is by now experimentally verified and well-established in many contexts. Nevertheless, there is still a number of perplexing open problems including the effects of disorder, finite size and external driving forces which occur e.g. in electrophoresis. With modern experimental techniques it has become possible to directly probe the behaviour of single polymer chains under entanglement conditions. In our theoretical approach we develop lattice gas models for reptation. We investigate analytically and numerically inhomogeneous reptation dynamics, including relaxation phenomena. |

#### Relaxation phenomena

In equilibrium an entangled polymer is not fully stretched but a certain amount of excess mass is stored inside the tube, hence giving rise to a fluctuating equilibrium tube length less than the actual polymer length. Even though this tube is a rather theoretical construct, the implied tube diameter, i.e. the mean entanglement distance, has been measured in neutron scattering experiments [D. Richter et al. Phys. Rev. Lett. **64**, 1389 (1990)]. Indeed, the tube may be seen as a coarse-grained polymer contour and hence be identified with the experimentally observable visual contour of flourescence-marked polymer chains.

The far-from-equilibrium relaxation of an initially stretched, entangled polymer can be studied with an exactly solvable lattice gas model for reptation [44]. Over a significant time range, including an initial universal power law regime, the predicted tube length relaxation is in very good agreement with experimental data for the relaxation of DNA, obtained by Perkins, Smith and Chu [Science **264**, 819 (1994)] using flourescence microscopy. Experimental evidence and theoretical arguments suggest an observed systematic long-time deviation to be due to entanglement fluctuations. This view is confirmed by the analysis of an extension of the lattice gas model which appears to be a reliable foundation for further numerical study of reptation dynamics.

#### Inhomogeneous reptation

If electrophoresis is applied for the separation of long polymers, for instance fragments of DNA, the drift velocity becomes independent of the length. A consequence of this ``band collapse'' is that long polymers cannot be separated efficiently as they will all travel with approximately the same velocity. These experimental results can be predicted from a quite simple one-dimensional lattice gas model known as Rubinstein-Duke model. On the other hand, if the force acts only on a localized section (e.g. at a bead attached to it) a suitable modification of the Rubinstein-Duke model shows that the band collapse disappears.

Other techniques of avoiding band collapse are known, but this unexpected observation demonstrates that the motion of long entangled polymers which are under the influence of an external driving force is still not completely understood, particularly if reptation becomes inhomogeneous either through inhomogeneous forces or through the effects of disorder. In order to investigate finite-size effects and the influence of inhomogeneities we develop one-dimensional lattice gas models which allow for analysis of the tube dynamics and center of mass motion of the polymer.