Phase separation in solution of worm-like micelles: a dilute n -> 0 spin-vector model

We show how the dilute n --> 0 spin vector model introduced originally by W heeler and co-workers for describing the polymerization phenomenon in solut ions of liquid sulphur and of living polymers may be conveniently adapted f or studying phase separation in systems containing long flexible micelles. We draw an isomorphism between the coupling constant appearing in the excha nge Hamiltonian and the surfactant energies in the micellar problem. We sol ve this problem within the mean-field approximation and compare the main re sults we have obtained with respect to polymer theory and previous theories of phase separation in micellar solutions. We show that the attractive int eraction term chi between monomers renormalizes the aggregation energy and subsequently the corresponding size distribution. Under these conditions, w e observe that the general aspect of the phase diagram in the (Phi, chi) pl ane (where Phi is the surfactant concentration) is different from previous results. The spinodal line shows a reentrant behaviour and, at low concentr ations, we point out the possibility of specific nucleation phenomena relat ed to the existence of a metastable transition line between a region compos ed of spherical micelles and another one corresponding to a dilute solution of long flexible micelles.