Rod/disk coexistence in dilute soap solutions

Abstract

Self-assembly theories for dilute micellar solutions generally assume that the chemical potential per surfactant is dominated by the “surface” terms α a + c/a where α a and c/a represent the interfacial and electrostatic energies associated with the head groups (having area a). That is, they suppress the energy and entropy of packing the hydrophobic chains (having volume v and length l). These “bulk” terms depend on the head group area and also on the elastic properties unique to the surfactant environments (e.g., thickness, curvature, etc.) in question: the chains are not “passive”—we cannot optimize the head group situation without taking into account the free energy “price” “paid” by the tails. In this paper we show that the differences in compressional and splay elasticities among the various environments (e.g., sphere, rod, disk, etc.) can be treated phenomenologically by defining a “relative stability” parameter y: a 0 l v  1 + y, O⩽y⩽2 a 0= c γ . For y ≳ 0, only disks survive (as lamellae or finite micelles); for y ⋍ √2 − 1, rods and disks coexist with relative numbers and sizes determined by overall concentration; and for y ≲ 1, the rods are dominant, with spheres taking over for still larger y. After relating y to physical properties of the surfactant and solvent conditions, we discuss our results in terms of recent theories of micellar shapes.