Multistep relaxation in equilibrium polymer solutions: a minimal model of relaxation in "complex" fluids.

We examine the rheological and dielectric properties of solutions of equilibrium self-assembling particles and molecules that form polydisperse chains whose average length depends on temperature and concentration (free association model). Relaxation of the self-assembling clusters proceeds by motions associated either with cluster rotations, with diffusive internal chain dynamics, or with interchain entanglement interactions. A hierarchy of models is used to emphasize different physical effects: Unentangled rodlike clusters, unentangled flexible polymers, and entangled chains. All models yield a multistep relaxation for low polymer scission rates ("persistent polymers"). The short time relaxation is nearly exponential and is dominated by the monomeric species and solvent, and the long time relaxation is approximately a stretched exponential, exp[-(t/tau)(beta)], a behavior that arises from an averaging over the equilibrium chain length distribution and the internal relaxation modes of the assembled structures. Relaxation functions indicate a bifurcation of the relaxation function into fast and slow contributions upon passing through the polymerization transition. The apparent activation energy for the long time relaxation becomes temperature dependent, while the fast monomeric relaxation process remains Arrhenius. The effective exponent beta(T), describing the long time relaxation process, varies monotonically from near unity above the polymerization temperature to a low temperature limit, beta approximately 13, when the self-assembly process is complete. The variation in the relaxation function with temperature is represented as a function of molecular parameters, such as the average chain length, friction coefficient, solvent viscosity, and the reaction rates for particle association and dissociation.