Surfactants micelles, assuming ideal

Advances in the theory of mixed-micelle formation have made it possible to calculate the composition of mixed micelles formed by two or more surfactants. A thermodynamic treatment of micellar solutions of mixed surfactants is usually based on the pseudophase separation theory [61,71-74]. The pseudophase models developed for binary surfactant solutions assume ideal mixing of the surfactants in the micelle. 

For a binary system of surfactants A and B, the mixed micelle formation can be modeled by assuming that the thermodynamics of mixing in the micelle obeys ideal solution theory. When monomer and micelles are in equilibrium in the system, this results in ... 

As for the close packed surface layer it is assumed that the micelles do not contain any water, i.e. x + x = 1. Assuming ideal mixing of the two surfactant types in the mixed micelle, i.e. f. = f2 = 1, Equation 23 shows that the cmc is the arithmetic mean or the two single surfactant erne s with the surfactant composition in the micelle as the variable. 

Assuming ideal mixing, the surfactant composition in the micelles can be calculated from Equation 29, or alternatively from the experi-... 

The phase separation model follows exactly the description of a two-phase equilibrium, i.e., equating the respective chemical potentials of the particular surfactant in both phases (i.e., monomers in the nonpolar solvent and the micelles) at the critical concentration (CMC). Thus, (assuming ideal condition)... 

The thermodynamics of small systems developed by Hill [183] has been applied to non-ionized, non-interacting surfactant systems by Hall and Pethica [184]. In this approach the aggregation number is treated as a thermodynamic variable, thereby enabling variations in the thermodynamic functions of micelle formation with the mean aggregation number n to be examined. The thermodynamic functions of micellization assuming solution ideality are... 

Assuming ideal behavior, the chemical potentials, and ixm, of the surfactant monomers and micelles respectively are... 

For binary surfactant mixtures, one might expect that micelles formed at the CMC of the mixture would be enriched in the less hydrophilic surfactant (i.e., the one with the lower CMC). Analysis based on the phase separation model confirms this expectation. The simplest approach is to assume an ideal mixture in the micellar phase (i.e., activity of each species equal to its mole fraction). With this assumption one obtains the following expression for the ratio x Jx2 of the two surfactants in the micellar phase at the CMC (see Problem 4.3) ... 

We now turn to the thermodynamic description of surfactant solutions. First we note that if the micelles are viewed as a separate phase, then the chemical potential of the surfactant S in the two phases is (assuming dilute ideality of the aqueous solution)... 

Equation (21) describes a highly idealized model for the association of nonionic surfactant molecules or ions. The simple mass action model assumes monodispersity of micelles. Counterions of surfactant ions are not included. In reality, large numbers of counterions are associated with micelles of anionic or cationic surfactants. Hence, if the surfactant dissociates into ions, the counterion and the degree of dissociation have to be considered [24) ... 

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