Microscopic Theory of Quiescent and Deformed Topologically Entangled Rod Solutions: General Formulation and Relaxation after Nonlinear Step Strain

We describe in detail a new self-consistent microscopic theory for the transverse tube confinement potential and the slow dynamics of topologically entangled rod polymer solutions under quiescent (equilibrium) and stressed conditions. Building on this advance, a theory is formulated for the nonlinear relaxation of stress and orientational order after an instantaneous step-strain deformation. The theory accounts for a time- and deformation-dependent (dilated) tube diameter, a maximum confinement force, and a transverse entropic barrier. These aspects result in competing (and coupled) time-dependent relaxation mechanisms under stress: reptative rotational motion and activated transverse barrier hopping. We predict an acceleration of the terminal rotational relaxation due to tube dilation and at sufficiently high deformations the emergence of a competitive transverse hopping (tube breaking) relaxation process. Both of these effects are a consequence of the predicted anharmonic form of the dynamic tube confinement potential and the finite strength of the entanglement force experienced by a tagged rod. After a large enough strain the entanglement network is predicted to break down, and the tube is destroyed in a manner akin to a microscopic yielding event. This results in a regime of ultrafast relaxation of stress and orientation on a time scale much shorter than the terminal relaxation time, and additional dynamical features at intermediate times not present under equilibrium conditions. As stress relaxes, the entanglement constraints and confining tube re-emerge, resulting in a return to an unperturbed reptative relaxation process at long times. The approach to this state is characterized by apparent power-law healing kinetics of the tube diameter, maximum confinement force, and rotational relaxation time. An important consequence of the dynamical evolution of the transverse confinement field is the prediction of a different damping function compared to the classic tube model.