Statistical mechanical theory of nonionic micelles

Abstract

A statistical mechanical theory of nonionic (single-component) micelles, well suited for applications, is developed by a simple extension of Hill's theory of solutions. An expression for the Gibbs potential of a dilute micellar solution is derived which involves the micelle size distribution and certain effective micelle partition functions. Using this Gibbs potential, both micelle thermodynamics and micelle structure and size distribution can be simply and directly treated. The thermodynamics obtained agrees with Hall and Pethica's small system thermodynamics treatment within the approximations (ideality, monodispersity, neglect of solvent composition changes) made by these authors. The micelle size distribution is found to obey Tanford's law of mass action type relation with a size dependent free energy. An approximate but detailed consideration of the effective micelle partition function leads to the local packing condition for micelles recently introduced by Israelachvili, Mitchell, and Ninham. Interactions between micelles can be included in the present statistical theory by an expansion in powers of micelle molalities; a rough, excluded volume type estimate indicates that these interactions only become important for the micelle properties when the total amphiphile molality (moles of amphiphile per mole of solvent) is greater than about I. IntroductionA statistical mechanical theory of nonionic micelles1 is developed in this paper. The treatment amounts to a straightforward generalization of Hill's theory of solutions2 which is based on a statistical ensemble first extensively used by S t~c k m a y e r .~ An expression for the Gibbs potential of a micellar solution is derived; it involves the distribution of micelle aggregation numbers and also certain effective micelle partition functions. From this Gibbs potential the small system thermodynamics of Hall and Pethica4 for (single-component) nonionic micelles can be derived very simply. We note, however, that except in the monodisperse limit the micellar enthalpy and volume differences defined by Hall and Pethica are not identical with the quantities measured in a dilution experiment.Tanfords has introduced a theory, based on the law of mass action, which relates micelle sizes and the critical micelle concentration to a size dependent free energy of micellization. Israelachvili, Mitchell, and Ninham6i7 modified Tanford's theory by introducing a local packing criterion which imposes certain geometrical constraints on the allowed micelle structures. We indicate how contact between these theories and the present statistical theory is made by means of suitable approximations to evaluate the effective micelle partition functions. In particular, an approximate derivation of the local packing condition of ref 6 mentioned above is given. This packing condition appears to be of considerable importance. It links together

Hydrocarbon Chain Ordering in Liquid Crystals Investigated by Means of Infrared Attenuated Total Reflection (IR-ATR) Spectroscopy

“…4 to get a value for the mean order parameter which is in satisfactory agreement with the experimentally determined S e = 0.11. This result is in agreement with recent theoretical considerations 21 . In addition to the above the coupling constant V rr turned out to be rather unspecific.…”