Penetration theory for homologues

The equation of state (2.154) and adsorption isotherm (2.157) correspond to insoluble/soluble mixtures of two homologues, in absence of any 2D aggregation. These equations can be transformed then into 

When the formation of large aggregates of an insoluble surfactant is stimulated by the adsorption of a soluble surfactant, the von Szyszkowski-Langmuir equation of state and adsorption isotherm for the soluble surfaetant read [157] 

As aggregation of the insoluble component occurs only when its surface concentration is sufficiently high, the description of the two components based on Volmer s equation seems to be more appropriate than that based on the Szyszkowski-Langmuir equation. If a first-order phase transition does not occur in the monolayer, i.e. no aggregates are formed, then the simultaneous solution of Volmer s equation (2.159) for the components 1 and 2, and Pethica s equation (2.152) yields the adsorption isotherm for the soluble component 2 (see [156]) 

Assuming that, according to the quasi-chemical aggregation model [97], mixed aggregates are formed, one can transform the generalised Volmer equation (2.159) into 

When mixed aggregates are formed, and the conditions n, 1 and n2 1 hold, Eq. (2.165) yields 

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