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## Simple analytical theory for micelles with widespread radial distribution of charged heads

Modeling aggregation of ionic micelles in solution is an important task that has been attracting attention for years. The molecular-thermodynamic micellization models describe the electrostatic interactions within the Poisson-Boltzmann theory that does not take into account fluctuations of the surfactant heads in the micelle. In this work we apply the statistical field theory to spherical micelles where the charged heads of surfactant molecules undergo fluctuations in the diffuse coronae. From this theory, a simple equation is derived for the electrostatic contribution to the standard free energy of aggregation. The results from the new theory are compared with the numerical results and analytical approximations of the Poisson-Boltzmann theory for the spheres of varying charge and curvature in the ranges relevant to micelles and over a broad interval of solution salinity. The new electrostatic free energy term is employed in the classical molecular-thermodynamic model to describe spherical micelles in the aqueous solution of ionic surfactant with added salt. We introduce an empirical correction that takes into account the effect of fluctuations on the hydrophobic part of the aggregation free energy. Modeling aggregation equilibrium for the spherical SDS micelles in the aqueous solution of salt, we illustrate the performance of the new theory. We compare our results with the structural MD-data and experimental hydrodynamic radii to show that our model with the fluctuating surfactant heads can reflect a realistic picture of the micellar structure and gives reasonable size-estimates for the spherical ionic micelles if proper charge distributions are considered.