Is Structural Precision of Platonic Micelles Controlled Solely by Close-Packed Head Groups or Also by Hydrophobic...

Platonic micelles have been defined as structurally precise amphiphilic aggregates with discrete aggregation numbers corresponding to the close packing of spherical caps (representing head groups) on a sphere (representing hydrophobic core), analogous to the Tammes problem in geometry. Here, we use DFT to explore how an actual molecule behaves compared to the idealized picture based on the Tammes problem by also considering the packing of the tails. We modeled micelles of aggregation numbers 4 to 8 generated from the calix[4]arene amphiphile, PACaL3, with the tails forming a close-packed configuration while the headgroups are arranged as in Platonic solids. The DFT calculations reveal that tail packing overwhelmingly influences the equilibrium aggregation number. While the DFT prediction of a PACaL3 micelle of aggregation number 6 agrees with the scattering experiments of the Sakurai group, DFT calculations also suggest small concentrations of micelles of aggregation number 7. More interestingly, DFT calculations reveal that PACaL3 micelle formation occurs even though less than 20% of the hydrophobic tail surface is removed from contact with water, in contrast to the roughly 80% removal observed for classical surfactant micelles. While the close-packed head groups model predicts higher coverage of the hydrophobic surface for aggregation numbers 4 and 6 compared to 5 and 7, the DFT calculations also accounting for tail packing show that the surface coverage for aggregation numbers 5 and 7 is practically no different than that for aggregation number 4. Finally, although both the close-packed head groups model and the DFT calculations agree that the exposed hydrophobic surface area controls the equilibrium micelle aggregation number, the DFT calculations demonstrate how this exposed hydrophobic area is overwhelmingly determined by the tail group packing and not just by the close packing of head groups.