Theory of gelation with conformation change

To study the equilibrium gelation strongly coupled to the polymer conformational change, we consider a polydisperse mixture of primary polymer chains carrying variable numbers of functional groups in a solvent. A molecule is distinguished by the number / of active functional groups it carries, which is a variable that depends on the temperature and polymer concentration, as well as other environmental parameters [57], 

Let Nx be the number of X-molecules, and (px = nNx/ be their volume fraction. The subscript X indicates / = 0. The volume fraction of /x-molecules is then given by 

The number-average functionality / of the /u.-molecules is then given by 

These depend on the temperature and the concentration, and play a central role in the following analysis of the sol-gel transition. 

The free energy change on passing from the reference states to the final solution, at equilibrium with respect to cluster formation, consists of three parts  

---