Flow-induced gelation of living (micellar) polymers

We consider the effect of shear velocity gradients on the size (L) of rodlike micelles in dilute and semidilute solution. A kinetic equation is introduced for the time-dependent concentration of aggregates of length L, consisting of "bimolecular" combination processes L + L′ →(L + L′) and "unimolecular" fragmentations L→L′ + (L - L′). The former are described by a generalization (from spheres to rods) of the Smoluchowski mechanism for shear-induced coalesence of emulsions, and the latter by incorporating the tension-deformation effects due to flow. Steady-state solutions to the kinetic equation are obtained, with the corresponding mean micellar size (L̄) evaluated as a function of the Peclet number P, i.e., the dimensionless ratio of flow rate γ̇ and rotational diffusion coefficient D_r. For sufficiently dilute solutions, we find only a weak dependence of L̄ on P. In the semidilute regime, however, an apparent divergence in L̄ at P≃1 suggests a flow-induced first-order gelation phenomenon.