The coherent scattering function of the reptation model: simulations compared to theory
We present results of Monte Carlo simulations measuring the coherent structure function of a chain moving through an ordered lattice of fixed topological obstacles. Our computer experiments use chains up to 320 beads and cover a large range of wave vectors and a time range exceeding the reptation time. For additional information we also measured the coherent structure function of internal pieces of the chain. We compare our results (i) to the predictions of the primitive chain model, (ii) to an approximate form resulting from Rouse motion in a coiled tube, and (iii) to our recent evaluation of the full reptation model. (i) The primitive chain model can fit the data for times t >? 20 T2, where T2 is the Rouse time of the chain. Besides some phenomenological amplitude factor this fit involves the reptation time T3 as a second fit parameter. For the chain lengths measured, the asymptotic behavior T3 ? N3 is not attained. (ii) The model of Rouse motion in a tube, which we have criticized before on theoretical grounds, is shown to fail also on the purely phenomenological level. (iii) Our evaluation of the full reptation model yields an excellent fit to the data for both total chains and internal pieces and for all wave vectors and all times, provided specific micro-structure effects of the MC-dynamics are negligible. Such micro-structure effects show up for wave vectors of the order of the inverse segment size and enforce the introduction of some phenomenological, wave vector dependent prefactor. For the dynamics of the total chain our data analysis based on the full reptation model shows the importance of tube length fluctuations. Universal (Rouse-type) internal relaxation, however, is unimportant. It can be observed only in the form of the diffusive motion of a short central subchain in the tube. Finally we present a fit formula which in a large range of wave vectors and chain lengths reproduces the numerical results of our theory for the scattering from the total chain.