Equilibrium model for estimating micellar aggregation number from surface equation

Abstract

The Aggregation Surface Equation (ASE) is attained by coupling the Gibbs adsorption equation and surface aggregation isotherm. The resulting equation can be used to study fluid-fluid interfaces in connection with the micelle formation process. This new equation contains two boundary conditions in the vicinity of the saturation region: (i) the critical micelle concentration (CMC), which corresponds to the onset of micelle formation; and (ii) the critical spherical micelle concentration (CSC) (i.e. the completion of spherical micellar aggregates). These boundary conditions, which appear within a narrow surfactant concentration range, provide the basis to calculate the average aggregation number under the assumption of constant average aggregation number and spherical micelles geometry. The resulting Surface Equation (SE) contains the maximum surface concentration of the surface monolayer, the critical micelle concentration and the micelle average aggregation number. The mass action law allows extending the ASE as a function of the free monomer, as opposed to the total or nominal surfactant concentration, which was recently performed in the STAND model (Langmuir, 32, (16), (2016), 3917-3925). This paper presents a simple and direct method to calculate the micellar average aggregation number from surface tension data and the proposed surface equation.