An analysis of approximations for calculating the electrostatic contribution to the Gibbs energy of micelle formation

Approximations used to calculate the electrostatic contribution to the Gibbs energy of micelle formation are discussed as applied to the problem of describing the influence of salt admixtures on the aggregation of ionic surface active substances in aqueous solutions. The known approximate solutions to the Poisson-Boltzmann equation for curved charged surfaces (spheres and cylinders) in the presence of electrolytes are analyzed. It is proposed that the electrostatic Gibbs energy be estimated from the Mitchell-Ninham equation for surface electric potentials. The dependences of Gibbs energy estimates on the radius of the surface and surface charge density are compared with the results of the exact numerical solution to the Poisson-Boltzmann equation for spheres and cylinders over a wide range of background salt concentrations. Recommendations are given for selecting the approximation for estimating the electrostatic contribution taking into account the size and shape of micelles, solution ionic strength, and charge density on the surface of micelles. Two approximations that gave the best description of the exact solution over the whole range of the conditions under consideration were used within the framework of the molecular thermodynamic aggregation model for calculating the critical micelle concentration and the size of aggregates depending on the salt content in aqueous solutions of sodium dodecyl sulfate.