Interplay between gelation and phase separation in tree polymers, and the calculation of macroscopic loop density in the postgel regime

We consider a very general model of equilibrium polycondensation of tree po lymers and elucidate (i) the conditions that must be met for gelation to co mpete with phase separation under a variety of conditions; (ii) how gelatio n is different from a thermal transition; and (iii) how gelation can be ind uced not only by lowering but also by raising the temperature, thereby givi ng rise to the [lower critical solution temperature for gelation] phenomeno n observed recently. We also preset a new and direct scheme to calculate th e contributions to various functional densities from finite (sol) and infin ite (gel) clusters. The scheme presents us with an elegant method to calcul ate the loop density explicitly and helps settle a long-standing controvers y about the presence and the nature of loops in the postgel regime. The loo ps are macroscopic in size. Other important features are also discussed. (C ) 1999 American Institute of Physics. [S0021-9606(99)51241-5].