Equilibrium surface properties of micellar solutions

Measurements of the equilibrium surface tension in a broad concentration range above and below the CMC is one of the most conventional methods in the studies of surfactant solutions. The first derivative of the concentration dependence of the surface tension jumps at a certain 

A simple attempt to estimate the composition in mixed micelles from surface tension data at the CMC was made by Funasaki and Hada using a mass balance condition [57]. Indeed the composition of micelles and the bulk solution containing two surfactants can be related to the surfaee tension values at the CMC. The material balance for this system takes the form 

Thus from the break point of the surface tension dependence on the total surfactant concentration one can determine ccmc and X2 (equal to Xtou at the CMC) as a function of the surface tension. Furthermore, measuring the surface tension of aqueous solutions at a given concentration Ctot above the CMC, one can calculate the mole fraction of the surfactant 2 in micelles using Eq. (5.129). To this aim one has to substitute the values of ccmc and X2 that correspond to the measured surface tension. 

Motomura et al. proposed a method of evaluation of various thermodynamic properties of micellar solutions from the surface tension data in the framework of the pseudophase treatment of micellisation [50, 52-55]. According to these authors, the micellar composition at the CMC can be found from the functional dependence of the CMC on the overall surfactant mole fraction using an analogy to the method proposed by Nguyen et al. [49], The approach of Motomura et al. [50] gives also a possibility to determine the relation between the composition of the surface layer and the micelles. Application of the Gibbs-Duhem equation to the whole 

The superscript CMC designates thermodynamic quantities at the CMC. This relationship between the compositions of the mixed micelles and the adsorption layer close to the CMC was successfully employed in a series of papers [58-61]. 

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