Phenomenological theory of the dynamics of polymer melts. I. Analytic treatment of self-diffusion

Abstract

In the context of dynamic Monte Carlo (MC) simulations on dense collections of polymer chains confined to a cubic lattice, the nature of the dynamic entanglements giving rise to the degree of polymerization n, dependence of the self-diffusion constant D∼n−2 is examined. Consistent with our previous simulation results, which failed to find evidence for reptation as the dominant mechanism of polymer melt motion [J. Chem. Phys. 86, 1567, 7164, 7174 (1987)], long-lived dynamic entanglement contacts between pairs of segments belonging to different chains are extremely rare and are mobile with respect to the laboratory fixed frame. It is suggested that dynamic entanglements involve the dragging of one chain by another through the melt for times on the order of the terminal relaxation time of the end-to-end vector. Employing the physical description provided by the MC simulation, the general expression of Hess [Macromolecules 19, 1395 (1986)] for the friction constant increment experienced by a polymer due to the other polymers forms the basis of a phenomenological derivation of D∼n−2 for monodisperse melts that does not require the existence of reptation. Rather, such behavior is dependent on the relatively benign assumptions that the long distance global motions of the chains are uncorrelated, that the dynamic contacts can be truncated at the pair level, and that the propagator describing the evolution between dynamic contacts contains a free Rouse chain component. The mean distance between dynamic entanglements is predicted to depend inversely on concentration, in agreement with experiment. Moreover, as the free Rouse component is frozen out, for chains greater than an entanglement length ne, a molecular weight independent glass transition is predicted. Extension to bidisperse melts predicts that the probe diffusion coefficient Dp depends on the matrix degree of polymerization, nm, as n−1m. Finally, comparison is made between the theoretical expressions and MC results for mono- and bidisperse melts.