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 micell es, focusing on the early stages of micellar elongation, i.e., on small and "intermediate-size" micelles, whose endcaps are not yet molded into a fina l shape. The internal free energy of a micelle of given size and shape is e xpressed as an integral over local molecular packing free energies in diffe rent regions of the micelle. The free energy per molecule is expressed as a sum of interfacial ("opposing forces") and chain conformational contributi ons, 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 cyl indrical geometry, we show that the early stages of growth involve an energ etic barrier, resulting in a "gap" in the micellar size distribution. That is, at low total amphiphile concentrations only small (globular) micelles a ppear in solution. Their concentration reaches a well-defined saturation va lue, beyond which, all added amphiphiles are incorporated in long micelles, whose "non-interacting" endcaps are well separated by the cylindrical midd le part. This, "second CMC" behavior is demonstrated by numerical calculati ons of micellar size distributions and average aggregation numbers as a fun ction of the total concentration. The conditions necessary for the appearan ce of a second CMC are analyzed theoretically, with explicit reference to t he 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 intermed iate-size micelles must be considerably lower than the asymptotic (long mic elle) value of this quantity. The diameter of the minimal, spherical micell es, as well as that of the final endcaps, is found to be larger than the di ameter of the cylindrical body of the very long micelles. Our results are i n good qualitative agreement with recent cryo-TEM imaging studies of micell ar shape and growth, as well as with previous (less direct) experiments rev ealing second CMC behavior.