A topological study of the dynamical effects of entanglements

Abstract

The constraint that extended molecules such as polymers cannot pass through each other is formulated in terms of a topological invariant. A simple model of the motion of a labelled loop entangled in a uniform background of amorphous loops is studied. A dynamical weighting factor is found which effectively removes configurational changes that would lead to the labelled loop passing through the strands of the background material. A perturbation treatment of this weighting factor is presented which describes the effects of the entanglements in terms of random environmental forces possessing considerable space and time correlations. A detailed analytic calculation is presented for the case of a labelled molecule with a rigid-ring configuration. This calculation illustrates the limited validity of perturbation theory in dealing with entanglements and suggests the existence of a transition in time to a random 'tube'-like behaviour, where the centre-of-mass motion of the ring depends on the time as t1/4 instead of the usual diffusive t1/2. The origin of this behaviour can be located in the long-time persistence of the bond vector correlation function of the labelled ring.