Genesis of the surfactant parameter

An outline of the ideas is as follows [62-65]. It suffices to assert that a dilute solution of surfactant molecules can be considered to consist of water plus monomers, dimers, trimers, and larger allowed aggregates (micelles, vesicles, liposomes,. ..). The concentration is assumed to be so low that aggregates can be considered to be non-interacting. The probability distribution of aggregates, is then determined from the law of mass action 

All aggregates are allowed to occur at any concentration, albeit with infinitesimal probability, even below the cmc. Above the erne, defined by 

Xi increases slowly with concentration. It can be shown that if the chemical potential of a monomer in an aggregate is sharply distributed about some N, then the distribution of aggregates peaks at a value of N just less than the N with minimum fifp and is also sharply distributed. Otherwise, pronoimced polydispersity may occur e.g. for long cylindrical micelles). Thus reduced to bare bones it can be seen that the use of the word theory is dubious. We have simply characterised the observation of micelles, and claim that it can be shown that the law of mass action is an appropriate vehicle for this characterisation. 

The entropic term (kT/N) In (Xjsj/N) has considerable nuisance value. In the pseudophase approximation - valid for finite disconnected aggregates - it can be dropped. Then in this approximation no micelles occur below the cmc. This is now the value Xj of monomer concentration for which + 

We can write for the free energy difference f between a monomer in a micelle and in solution 

---