Reptation of a Polymer Chain in the Presence of Fixed Obstacles

We discuss possible motions for one polymer molecule P (of mass M) performing wormlike displacements inside a strongly cross-linked polymeric gel G. The topological requirement that P cannot intersect any of the chains of G is taken into account by a rigorous procedure: The only motions allowed for the chain are associated with the displacement of certain “defects” along the chain. The main conclusions derived from this model are the following: (a) There are two characteristic times for the chain motion: One of them (Td) is the equilibration time for the defect concentration, and is proportional to M2. The other time (Tr) is the time required for complete renewal of the chain conformation, and is proportional to M3. (b) The over-all mobility and diffusion coefficients of the chain P are proportional to M−2. (c) At times t < Tr the mean square displacement of one monomer of P increases only like 〈(rt − r0)2〉 = const t1/4. These results may also turn out to be useful for the (more difficult) problem of entanglement effects in unlinked molten polymers.