Unified relaxation spectrum for polymers of multiple scales based upon a fuzzy constraint method and non-equilibrium statistical thermodynamics

The molecular relaxation mechanisms of polymers with multi-scale units of motion in glassy, rubbery and melt states are proposed based upon a fuzzy constraint method and non-equilibrium statistical thermodynamics. The entanglement effects due to cohesive force and steric hindrance are expressed quantitatively in terms of a membership function. The micro-Brownian motion of a polymer chain is governed by the Langevin equation, which accounts for viscous force, nonuniform tension, entanglement constraint force and random force. Perturbation solutions have been established for different time and size scales. The solutions account for the effects of both intramolecular and intermolecular interactions in the relaxation process. The unified relaxation spectrum over many orders of time scale is a natural consequence of macromolecular structure, which satisfies the time-temperature equivalence in the form of the Arrhenius equation at low and high temperatures and in the form of the WLF equation near the glass transition temperature. The barrier model, the normal mode theory, the retraction and reptation theories can be taken as special cases corresponding to different scales