Titration of mixed micelles containing a pH-sensitive surfactant and conventional (pH-Insensitive) surfactants: a regular solution theory modeling approach.
We present a thermodynamic theory to model the hydrogen-ion titration of mixed micelles containing a pH-sensitive surfactant and any number of conventional (pH-insensitive) surfactants. In particular, a simple expression is derived for the pKm, a parameter analogous to the pKa of simple acids, which describes the deprotonation equilibrium of the micellized pH-sensitive surfactant. The pseudophase approximation and regular solution theory (RST) are used to relate the pKm to (1) the pKa of the surfactant monomers, (2) the critical micelle concentrations (cmc's) of the protonated and deprotonated forms of the pH-sensitive surfactant, (3) the composition of the mixed micelle, and (4) parameters characterizing pairwise interactions between the surfactant molecules in the mixed micelle. Micellar titrations can be used to determine the magnitude of these interaction parameters. Conversely, knowledge of the cmc's and the interaction parameters allows the prediction of the pKm, which can then be used to calculate the micelle composition and surface charge as a function of solution pH. In addition, we have found that, in the context of RST, multicomponent surfactant mixtures are equivalent to a binary surfactant mixture of the pH-sensitive surfactant and a single effective surfactant whose interactions with the pH-sensitive surfactant are an average of those in the multicomponent surfactant mixture. We also discuss the experimental uncertainty in the pKm measurements. To account for the increased uncertainty in the pKm data at extreme micelle compositions, a weighted regression is proposed for the analysis of experimental titration data characterized by widely varying uncertainties. The theory presented here is validated using micellar titration data from the literature for several pH-sensitive surfactants in solutions containing 0.1 M salt. In most cases, the parameters extracted from an analysis of the titration data agree with the cmc and interaction parameters obtained by other means. One notable exception is the surfactant tetradecyldimethylamine oxide (C14DAO), which appears to have concentration-dependent interactions due to extensive growth of cylindrical micelles. Micellar titrations were also conducted on binary surfactant mixtures containing the pH-sensitive surfactant dodecyldimethylamine oxide (C12DAO) and either the cationic surfactant dodecyltrimethylammonium bromide (C12TAB) or the nonionic surfactant dodecyl octa(ethylene oxide) (C12E8). The theory provides a reasonable description of the experimental titration data at all surfactant mixing ratios, although a larger discrepancy is found in the C12DAO/C12E8 system, in which C12E8 interacts preferentially with the protonated, cationic form of C12DAO. Interestingly, C12TAB was also observed to interact preferentially with the protonated, cationic form of C12DAO, although the preference is much weaker than that in the C12DAO/C12E8 system.