The mechanics and thermodynamics of rod-shaped micelles

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Abstract
In Part I of this paper the detailed mechanics of a (hemi-)spherocylindrical rod-shaped surfactant micelle is derived by means of the pressure tensor. The anisotropy of the pressure tensor, particularly within the hydrocarbon core, is found to be of fundamental importance for the mechanical stability of a rod-shaped micelle, which requires that the Laplace pressure of the cylindrical part, γc/Rc, is equal in magnitude to the associated (negative) disjoining pressure, πc, that is chiefly due to the packing constraints imposed on the hydrocarbon-chain conformations. Our numerical calculations for sodium dodecyl sulphate (SDS) micelles indicate that this condition is actually fulfilled for a cylinder radius, Rc, which is significantly smaller than the extended hydrocarbon chain length. In Part II a thermodynamic analysis is carried out based on surface thermodynamics and the thermodynamics of small systems developed by T. L. Hill. Large length fluctuations arise, i.e. long, rod-shaped micelles can exist in appreciable amounts only when the work of forming the (homogeneous) central part of the micelle out of monomers at the chemical potential µ– 2 in the surfactant solution (βkT per monomer) is close enough to zero (⩽10–2kT). Assuming a spherocylindrical shape it is established that Rc < Rs, which is consistent with the notion of the size distributions of small spherical and large rod-shaped micelles, respectively, being separated by a range of comparatively rare, elongated micelles of intermediate size and, furthermore, that the total free-energy excess of the end-caps, αkT, is larger than the corresponding quantity of the spherical equilibrium micelle. Finally, we show that the coexistence of spherical and rod-shaped micelles is largely geared by the relative proximity of the critical micelle concentration and β= 0 conditions.

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