Thermoreversible gels with multiple cross-linking

Similar calculation on the basis of Flory s treatment gives the factor —4B instead of B. The sign changes, but the discontinuity remains. 

We next consider the temperature derivatives of the fi ee energy, such as entropy and specific heat. The temperature appears through the interaction parameter x (T) and association constant X(T). Because it is evident that the former does not lead to any singularity, we consider the derivatives with respect to the latter. 

Within Stockmayer s treatment of the postgel regime, after a complex calculation, it turns out that there is no singularity up to the second derivatives, but the third derivative contains a discontinuity 

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. 

4 Thermoreversible gels with multiple cross-linking 

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