Living networks: The role of cross-links in entangled surfactant solutions

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Abstract
We study theoretically the effects of cross‐link formation in the entangled micellar phase of viscoelastic surfactant solutions. We consider equilibrium cross‐links, of energy E≫kBT, that can arise through local fusion of the micelles. For simplicity, we assume that the end‐cap energy is much larger than E, so that the system forms a network in which free ends can be neglected. We consider cases when the material between cross‐links is (i) a rigid rod, (ii) a Gaussian chain, and (iii) an excluded‐volume chain. In each case, we define a ‘‘saturated’’ network as one for which L≂Lb, where L is the arc length between cross‐links and Lb that of a geometrical subunit or ‘‘blob’’ of the network. (An ‘‘unsaturated’’ network is, in contrast, one for which L≫Lb. ) In the rigid‐rod case we predict an unsaturated network at all concentrations. For flexible chains we predict that the network is unsaturated at high concentrations, but as the concentration is decreased, these systems should reach saturation, followed at lower φ by a phase separation between a dilute saturated network and excess solvent. Surprisingly, this phase separation should occur even for the case of a very good solvent: there is an entropy gain associated with creating threefold junction points which increases with dilution, and asymptotically overcomes any finite cross‐link energy E. In practice, however, this may often be unobservable due to the very small concentrations required, and/or the effects of free ends. We briefly discuss the possible implications of our results for the viscoelastic properties of surfactant solutions.