Molecular theory of the sphere-to-rod transition and the second CMC in aqueous micellar solutions

We present a molecular-level theory for amphiphile packing in linear micelles, focusing on the early stages of micellar elongation, i.e., on small and "intermediate-size" micelles, whose endcaps are not yet molded into a final shape. The internal free energy of a micelle of given size and shape is expressed as an integral over local molecular packing free energies in different regions of the micelle. The free energy per molecule is expressed as a sum of interfacial ("opposing forces") and chain conformational contributions, both depending on the local geometry. The equilibrium shape and energy of the micelle is determined by functional minimization of the total free energy. For amphiphiles exhibiting strong preference for packing in the cylindrical geometry, we show that the early stages of growth involve an energetic barrier, resulting in a "gap" in the micellar size distribution. That is, at low total amphiphile concentrations only small (globular) micelles appear in solution. Their concentration reaches a well-defined saturation value, beyond which, all added amphiphiles are incorporated in long micelles, whose "non-interacting" endcaps are well separated by the cylindrical middle part. This, "second CMC" behavior is demonstrated by numerical calculations of micellar size distributions and average aggregation numbers as a function of the total concentration. The conditions necessary for the appearance of a second CMC are analyzed theoretically, with explicit reference to the underlying molecular packing characteristics. In particular, it is shown that a necessary condition for the appearance of a sharply defined second CMC is that the endcap energies (of at least some) of the small or intermediate-size micelles must be considerably lower than the asymptotic (long micelle) value of this quantity. The diameter of the minimal, spherical micelles, as well as that of the final endcaps, is found to be larger than the diameter of the cylindrical body of the very long micelles. Our results are in good qualitative agreement with recent cryo-TEM imaging studies of micellar shape and growth, as well as with previous (less direct) experiments revealing second CMC behavior.