Linear Aggregation Revisited: Rods, Rings and Worms
  The problem of ring formation in solutions of cylindrical micelles is reinvestigated theoretically, taking into account a finite bending rigidity of the self-assembled linear objects. Transitions between three regimes are found when the scission energy is sufficiently large. At very low densities only spherical and very short, rod-like micelles form. Beyond a critical density, mainly rings but also rod-like chains appear in (virtually) fixed relative amounts. Above a second transition both the length of the linear chains and the relative amount of material taken up by them increase rapidly with increasing concentration. The mass accumulated into long, semi-flexible worms then overwhelms that in rings. The ring-dominated regime is very narrow for semi-flexible chains, confirming that the presence of rings may be difficult to observe in many micellar systems, and indeed disappears completely for sufficiently low scission energy and/or large persistence length. 
P. van der Schoot, J.P. Wittmer