Models of thermoreversible gelation

Consider the self-assembling of polymer chain R A/ (wa = ) carrying a total of / associative groups A in an inert solvent B (wb = 1). We assume pairwise cross-linking of the associative groups leading to gels with one-component networks [1,2,3]. 

In the last term, is the number of polymer chains that belong to the network. It becomes a macroscopic variable after the gel point is passed. This additional term appears only in the postgel regime. The free energy 5(0) for a chain to be bound to the network depends on the concentration 0. It is negative, and its absolute value increases with the concentration because the network structure becomes tighter and denser with the concentration. 

The chemical potentials of finite clusters composed of / chains, and that of the solvent molecules, are derived by the differentiation of the free energy as 

Similarly, the chemical potential of the polymer chain in the gel network is 

In the postgel regime, the additional equilibrium condition Afu = Afx holds, and hence the relation 

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Flory s treatment gives [-4/(/ - 1)]C instead of C. Again, the sign changes, but the discontinuity remains. Collecting all results, we come to the conclusion that the ideal model of thermoreversible gelation treated here shows a third-order phase transition that is analogous to the Bose-Einstein condensation. 

We can apply the model of thermoreversible gelation of functional molecules whose functionality is not a fixed number but varies depending on the temperature. 

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