Dynamics of self-assembling rigid rods

The current understanding of the dynamics of living polymers is based on the pioneering work of Cates and co-workers, who proposed that the scission and recombination kinetics of the polymeric chains can be described by a single time scale $\tau _{break}$, equal to the lifetime of a chain of mean length. For scission-recombination processes that are not exceedingly fast, one theoretically finds that stress relaxation takes place at times $\tau $ = ($\tau _{rep} \quad \tau _{break})^{1/2}$, at least in semi-dilute solution where the regime where reptation is the dominating mode of stress relief, characterized by the relaxation time $\tau _{rep}$. Comparison with experiments on giant surfactant micelles shows that this picture is qualitatively correct on sufficiently long time scales, but that for shorter times deviations do emerge. In order to study the breakdown of the dynamical mean-field approximation, we present a simplified, so-called end-evaporation model for the association kinetics of rod-like equilibrium polymers that takes into account the diffusive motion of the assemblies, and that allows for the re-absorption of monomers that have split off from the assemblies. The model can be solved exactly in the limit of small temperature or T jumps. We find that diffusion modifies the relaxation to the new equilibrium after a T jump significantly. We derive expressions for the time-dependent average chain length and for the size-dependent chain concentrations, and compare these with results of a numerical study.