Modification of the gel point by surfactants

For the polymer/surfactant system, the sol-gel transition condition (7.116) is explicitly given by 

Combining this condition with the relation (10.3), we find the sol-gel transition curve on the temperature-concentration plane. 

To describe the observed nonmonotonic behavior of the mechanical modulus and viscosity of the polymer-surfactant mixtures, we introduce a mini-max model of the junctions (7.127), for which 

Small micelles whose aggregation numbers are less than Uq and large micelles with aggregation numbers larger than km are assumed to be unstable and dissociate. In fact, sufactant molecules are known to form micelles of a very narrow size distribution. The upper and lower bounds here are determined by the geometrical suitability of the hydrophobes for spatial packing, the flexibility of the polymer chains, and other factors. 

When the polymer concentration is so low that the number of hydrophobes is not enough to form junctions, the addition of surfactants combines the unassociated hydrophobes until their aggregation number exceeds ko and stabilizes them. The surfactant works as a cross-linker. 

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