Monodisperse Micelles with Aggregation Numbers Related to Platonic Solids

Recent studies show that calix[4]arene-based micelles are monodisperse with defined N_agg values chosen from 4, 6, 8, 12, 20, and 32. Interestingly, all these numbers coincide with the face numbers of Platonic solids, so they are called "Platonic micelles." As long as a certain geometric condition is fulfilled, any amphiphilic molecule can form a Platonic micelle. The preferred N_agg values are explained in relation to the mathematical Tammes problem, namely, how to obtain the best coverage of a sphere's surface with multiple identical circles. The coverage ratio can be calculated and produces maxima at 4, 6, 12, 20, and 32, coinciding with the observed N_agg values. In this feature article, Platonic micelles as well as their morphological transition by controlling pH, salt, temperature, and electrostatic interactions are summarized.