Microscopic theory of the dynamics of polymeric liquids: General formulation of a mode–mode-coupling approach

A formally exact, nonlinear generalized Langevin equation (GLE) for a flexible probe polymer in a dense melt has been derived using molecular phase space kinetic theory and Mori–Zwanzig projection operator techniques. An approximate, linearized dynamic memory function is developed, and the resulting GLE is specialized to the problem of an overdamped liquid of uncrossable Rouse polymers. An analytically tractable, perturbative/short time evaluation of the projected force time correlation function matrix is proposed which accounts for uncorrelated intermolecular pair interaction effects in the polymer melt. The detailed predictions for transport coefficients and various time correlation functions are determined for linear chains, and compared with recent lattice Monte Carlo simulations. Significant slowing down of all dynamical processes relative to the Rouse behavior is found, but the molecular weight scaling is not correctly described. A nonperturbative approach based on a polymeric generalization of molecular-scale mode–mode coupling theory is formulated which does properly capture the strong caging and viscoelastic effects in dense melts. The phenomenological concepts of topological entanglements, a static tube, and primitive path are not employed, and simplified assumptions about liquid structure and mode of motion are not introduced a priori. The microscopic theory is based on an explicit nonlinear coupling of the collective fluid density fluctuations with the segmental density fields of a probe polymer. Equilibrium structural information is naturally incorporated, and the associated renormalized intermolecular potential, or vertex, is found to be spatially long range due to chain connectivity and correlation hole effects. The projected dynamics describing the time evolution of the mode-coupling part of the memory function matrix is evaluated using the short time/pair interaction theory. The polymeric mode-coupling theory can be employed as a rigorous and unified framework for qualitatively and quantitatively studying transport coefficients, material response functions, crossover phenomena, collective density fluctuation dynamical effects, nonlinear molecular architectures (e.g., ring polymers), tracer diffusion, semidilute solutions and blends.