Molecular Packing in Cylindrical Micelles

The elongated shape of linear, worm-like, micelles is among the most common aggregation geometries of amphiphilic molecules in aqueous solutions. The cylindrical body of these uni-dimensional micelles can be regarded as an intermediate packing geometry, of a higher growth dimensionality than that of a spherical micelle and lower than that of a planar bilayer. In all these three canonical structures, the hydrophobic tails of the constituent amphiphiles form a compact, liquid-like, hydrocarbon core, with their polar headgroups residing on its surface, thus largely shielding the hydrocarbon tails from direct contact with water. The planar bilayer is a two dimensional (2D) object; it can grow laterally along d = 2 directions, but its third dimension is always microscopic. Namely, its thickness, or, more precisely, the distance (2b) between its two hydrocarbon-water interfaces cannot exceed 2bmax, where bmax is the length of the fully extended hydrocarbon tail. Similarly, the growth dimensionality of a worm-like micelle is d = 1. It can elongate along the cylindrical axis, but its diameter (2b) – and hence the two perpendicular dimensions – cannot exceed 2bmax. The growth dimensionality of a spherical micelle (keeping its spherical symmetry) is, of course, d = 0, since to maintain a compact hydrophobic core all its three dimensions must be smaller than (2bmax. Note that this last statement is only relevant to systems where a spherical micelle is the intrinsically preferred (or, referred ’spontaneous’) packing environment of the amphiphiles. As briefly discussed later in this chapter, many other amphiphiles, in fact most of those that prefer packing into long cylindrical micelles, first assemble – for entropic reasons – into small spherical micelles. However, above the critical micelle concentration (CMC), driven by the lower packing energy in the cylindrical geometry, the added amphiphiles incorporate into the middle of the aggregate, forming a gradually elongating cylindrical midsection, capped by two approximately hem-spherical micellar caps. In these systems the spherical micelle is just the low concentration limit of a linear aggregate. The reversible self-assembly of amphiphiles in aqueous environments is a fundamental processes, omnipresent in living organisms and underlying numerous technological applications.1 Thermodynamically, micellar solutions belong to the wider class of systems known as complex fluids.2,3 One of the special characteristics of these solutions is that the solute particles, unlike in ordinary molecular solutions, can modify their size, shape and even their aggregation geometry. Such changes occur generally in response to varying ambient conditions, like the total concentration, temperature or ionic strength. For instance, upon increasing the amphiphile concentration in a solution of cylindrical (‘rodlike’) micelles, the average micelle size increases monotonically, resulting in substantial inter-micelle interactions. Eventually, these interactions become strong enough to drive a transition from an isotropic to a nematic phase of rodlike aggregates. However, unlike in ordinary liquid crystalline solutions, the micelle size in the nematic phase is much larger than in the coexisting isotropic phase, demonstrating a subtle coupling between the inter-molecular (or, intraaggregate) and inter-micelle interactions.4 This coupling arises because the (non-covalent) forces holding together the amphiphiles in a micelle are relatively weak, typically on the order of kBT per molecule, where