The coherent scattering function in the reptation model: analysis beyond asymptotic limits

We calculate the coherent dynamical scattering function $S_{c}(q,t;N)$ of a flexible chain of length $N$, diffusing through an ordered background of topological obstacles. As an instructive generalization, we also calculate thescattering function $S_{c}(q,t;M,N)$ for the central piece of length $Mle N$ of the chain. Using the full reptation model, we treat global creep, tube length fluctuations, and internal relaxation within a consistent and unifiedapproach. Our theory concentrates on the universal aspects ofreptational motion, and our results in all details show excellent agreementwith our simulations of the Evans-Edwards model, provided we allowfor a phenomenological prefactor which accounts for non-universaleffects of the micro-structure of the Monte Carlo chain, present for short times. Previous approaches to the coherent structure function can be analyzed as special limits of our theory. First, the effects of internal relaxationcan be isolated by studying the limit $N ightarrow infty$, $M$ fixed.The results do not support the model of a `Rouse chain in a tube'.We trace this back to the non-equilibrium initial conditions of the latter model. Second, in the limit of long chains $(M = N ightarrow infty)$and times large compared to the internal relaxation time $(t/N^{2} ightarrow infty)$, our theory reproduces the results of the primitive chain model. This limiting form applies only to extremely long chains, and for chain lengths accessible in practice, effects of, e.g., tube length fluctuations are not negligible.