Reptation of a Polymer Chain in the Presence of Fixed Obstacles

Abstract

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 ${\mathrm{(T}}_d)$ is the equilibration time for the defect concentration, and is proportional to $M^2$ . The other time ${\mathrm{(T}}_r)$ is the time required for complete renewal of the chain conformation, and is proportional to $M^3$ . (b) The over‐all mobility and diffusion coefficients of the chain P are proportional to $M^{\text{−2}}$ . (c) At times ${\mathrm{t\hspace{0.17em}<\hspace{0.17em}T}}_r$ the mean square displacement of one monomer of P increases only like ${\mathrm{〈(r}}_t{\mathrm{\hspace{0.17em}-\hspace{0.17em}r}}_0{)}^2\mathrm{〉\hspace{0.17em}=\hspace{0.17em}}\mathrm{const}{t}^{\text{1/4}}$ . These results may also turn out to be useful for the (more difficult) problem of entanglement effects in unlinked molten polymers.