The Entropy of Mixing in Self-Assembly and the Role of Surface Tension in Modeling the Critical Micelle Concentration

A theory for the micelle formation of nonionic head-tail amphiphiles (detergents) in aqueous solutions is derived based on the traditional molecular thermodynamic modeling approach and a variant of the Flory-Huggins theory that goes beyond lattice models. The theory is used to analyze experimental values for the critical micelle concentration of n-alkyl-beta-D-maltosides within a mass action model. To correlate those parts of the micellization free energy, which depend on the transfer of hydrophobic molecule parts into the aqueous phase, with molecular surfaces, known data for the solubility of alkanes in water are reanalyzed. The correct surface tension to be used in connection with the solvent-excluded surface of the alky tail is similar to 30 mN/m. This value is smaller than the measured surface tension of a macroscopic alkane-water interface, because the transfer free energy contains a contribution from the incorporation of the alkane or alkyl chain into water, representing the change in free volume in the aqueous phase. The Flory-Huggins theory works well, if one takes into account the difference in liberation free energy between micelles and monomers, which can be described in terms of the aggregation number as well as the thermal de Broglie wavelength and the free volume of the detergent monomer.