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 the scattering function S_c(q,t;M,N) for the central piece of length M < N of the chain. Using the full reptation model, we treat global creep, tube length fluctuations, and internal relaxation within a consistent and unified approach. Our theory concentrates on the universal aspects of reptational motion, and our results in all details show excellent agreement with our simulations of the Evans-Edwards model, provided we allow for a phenomenological prefactor which accounts for non-universal effects 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 relaxation can be isolated by studying the limit $N \to \infty$, 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 $(M = N \to \infty)$ and times large compared to the internal relaxation time $(t/N^2 \to \infty)$, 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.