Monodisperse micelles composed of poly(ethylene glycol) attached surfactants: platonic nature in a macromolecular aggregate

Abstract

Among the many studies on micelles, dating back more than 100 years, we first found a series of monodisperse micelles: spherical micelles made from calix[4]arene surfactants exhibited monodispersity in aggregation number (N_agg) with values of 4, 6, 8, 12, and 20. We named these Platonic micelles because these values coincided with the face numbers of the Platonic solids. The preferred N_agg values were explained in relation to the mathematical Tammes problem: how to obtain the best coverage of a sphere surface with multiple identical circles. In this paper, we synthesized poly(ethylene glycol)-attached surfactants and carried out small-angle X-ray scattering (SAXS) and analytical ultracentrifugation (AUC) to determine the N_agg. We found that these polymeric surfactants also formed monodispersed micelles and N_agg discontinuously increased from 20 to 24, and then 32 with increasing the alkyl carbon numbers from 9 to 11 continuously. The determined N_agg was greater than 20 and the Platonic solid numbers. We assumed that the preferred N_agg values could be explained in relation to the Tammes problem as well.