CMC of the surfactant molecules

It is known that surfactant molecules form micelles above a certain concentration (see Section 6.7). The concentration at which micelles appear is referred to as the critical micelle concentration (CMC). For mixed solutions of associating polymers and surfactants, the concentration at which mixed micelles starts to form is referred to as the critical a regation concentration (CAC). To see how the CMC of the surfactant is affected by the presence of polymers, let us consider the siufactant molecules that are 

One conventional criterion for CMC is to find the concentration at which the osmotic pressure changes its slope most rapidly [28]. The contribution to the osmotic pressure from the surfactant molecules that are not connected to the polymers is proportional to 

To study the relative positions of CPMC and SMG concentration, we consider a fixed multiplicity model in which the multiplicity is fixed at a single value ko = km = k. The function m(x) in this fixed multiplicify model lakes the form m(x) = 1 +x , and leads to a set of equations 

There is an intersection between the solid broken lines for each multiplicity k. This is a special point where the sol-gel transition and the CPMC take place simultaneously. We call this special point the micellization gelation point (MGP). For instance, for A = 8 this point is located at a surfactant concentration that is nearly twice as large as the SMG concentration. 

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