CMC range determination by the second derivative of the Gibbs adsorption equation

The purpose of this work is to develop a reproducible method to determine the critical micelle concentration (CMC) interval in surfactant solutions, eliminating the need for observer-dependent data selection, which is common in approaches based on surface equations of state such as Langmuir, Langmuir–BET, and the integrated Gibbs equation, among others. The proposed methodology is based on the analysis of the surface-pressure isotherm as a function of the natural logarithm of the surfactant mole fraction, starting from the Gibbs adsorption equation. In particular, the first derivative d π / d ln x and, especially, the second derivative d 2 π / d ( ln x ) 2 are evaluated over the entire composition range. An empirical smoothing function is applied to the experimental surface-tension data, thereby producing continuous derivative profiles that enable identification of the premicellar limit, the CMC, and the postmicellar limit. The method is applied to several surfactant systems and the results are compared with CMC values reported in the literature. The analysis shows that the premicellar and postmicellar regions are characterized by near-zero values of the second derivative, whereas the micellization transition is identified by a pronounced minimum in d 2 π / d ( ln x ) 2 . The estimated CMC values show excellent agreement with literature data. In addition, the width of the premicellar–postmicellar interval is found to depend on surfactant structure and temperature, reflecting variations in hydrophobicity and aggregation tendency. • The second-derivative method uses all experimental points. • It distinguishes the pre-micellar and post-micellar regions.