A novel approach to the mechanics and thermodynamics of spherical micelles

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Abstract
A detailed mechanical theory for a spherical micelle is presented, based upon the notion that a micelle has the nature of an interface throughout. Accordingly, we presume that the (ensemble average) pressure tensor is generally anisotropic in a micelle except at r= 0 (for symmetry reasons). Following the Gibbs scheme for treating curved interfaces, the surface tension is usually defined with reference to the surface of tension (s.o.t.). In the case of a micelle, however, this method is inconvenient since the location of the s.o.t. is highly variable and difficult to assess a priori. An alternative approach is developed whereby the hydrocarbon core volume is employed to define the hydrophobic dividing surface (h.d.s.) located at r=R, which separates the internal (i) core part of the micelle from the exterior (e) water solution phase. In order to achieve a complete thermodynamic description it is then necessary to introduce an additional intensive variable, the interaction (disjoining) pressure πm. At the h.d.s., the Laplace equation can be written on the modified form pi–pe= 2γe/R–πm where pi is the average tangential pressure within the hydrocarbon core and γe is the surface tension originating outside r=R. πm is normally attractive (< 0) and is due chiefly to the hydrocarbon-chain packing constraints and the effect of curvature on the electrostatic interactions. Surfactant micelles are very small systems for which fluctuations of the aggregation number, N, are important. Hence the surface thermodynamics of micelles is calculated, taking account of the general principles of the thermodynamics of small systems developed by Hill. On this basis we show that, at phase equilibrium, the average surfactant chemical potential in the micellar state, 〈µN〉, (but not each µN separately) is equal to the monomer chemical potential in the intermicellar solution and, moreover, that the average net work expended to form a micelle with surface area A is equal to 〈(γe+πmR)A/3〉. At aggregation equilibrium this work is counterbalanced by free-energy gains of entropic origin caused by the dispersion of the micelles in the solution and their size fluctuations. Our model calculations for SDS micelles show that a consistent, quantitative theory of micelle formation can be obtained by using (i) Tanford's expressions for the solubility of hydrocarbons in water, (ii) the Poisson–Boltzmann/cell-model treatment of the electrostatics of a uniformly charged sphere, (iii) the macroscopic excess free energy, 50 mJ m–2, to account for the hydrocarbon-core/water contact and (iv) the conformation free-energy function for hydrocarbon chains packed in a spherical aggregate computed by Gruen and de Lacey. The large and negative πm components caused by factors (ii) and (iv) strongly promote micelle formation. The average pressure level in the hydrocarbon core is predominantly determined by the Laplace term 2γe/R and may typically amount to several hundred atmospheres, in agreement with experiments on the solubility of gases in micellar solutions.

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