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The Journal of Physical Chemistry, Vol. 82, No. 7, 1978
A. Wulf
Statistical Mechanical Theory of Nonionic Micelles A. Wulft Department of Applied Mathematics, Research School of Physical Sciences, The Australian National University, Canberra, A.C. T. 2600, Australia (Received September 12, 1977) Publication costs assisted by the Instituto Venezolano de Investigaciones Cientificas
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. Introduction A 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 micelles, vesicles, and bilayers in one t h e ~ r yand , ~ may be important in the organization of biological membranes, as suggested by Israelachvili.8 Interactions between micelles can be included in the theory by an expansion in powers of micelle molalities which is formally similar to the virial expansion of a multicomponent imperfect gas. For simplicity, the expansion is carried out here only to second order; the expansion coefficients obtained are similar to second virial coefficients, except that potentials of mean force, rather than the direct interaction potentials between particles, t Visiting Fellow. Permanent Address: Centro de Fisica, Instituto Venezolano de Investigaciones Cientificas (IVIC), Apartado 1827, Caracas 101, Venezuela.
0022-3654/78/2082-0804$0 1.OO/O
must be used. If it is assumed that these expansion coefficients are determined mainly by the spatial dimensions of the micelles, one estimates ctot for the total amphiphile molality ctot (in moles of amphiphile per mole of water solvent) a t which interactions between micelles become important for the micelle structures. (For very long cylindrical micelles, ctot might be smaller, see end of section VI.) This is much larger than typical cmc’s in aqueous solutions which have molalities of order Statistical theories of micelles have been previously published by Hoeve and Bensong and by Aranow.lo The work of Hoeve and Benson is based on the canonical ensemble; they did not consider interactions between micelles or derive any thermodynamics. Aranow starts with the same ensemble used here, but then assumes that the solution is incompressible and bases his subsequent development on Hill’s physical cluster the0ry.l’ As a consequence, Aranow’s theory centers around the micelle size distribution, and thermodynamic quantities or the micelle structures are hardly considered. (On the other hand, Aranow presents an approximate treatment of ionic micelles which we do not consider.) In sections 11-IV we develop the general theory, and in sections V and VI we consider briefly applications to micelle thermodynamics and micelle structure, respectively.
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11. Micelles in Water as a Multicomponent Solution In this section we derive an expression for the Gibbs thermodynamic potential G(M,, N,, p , r ) of a dilute solution of N,micelles of aggregation numbers n, n = 1,2, 3, ..., in Mo molecules of solvent (water) and Mi, i = 1, 2, ..., r ions of various salts. We assume here that the salts in solution are fully dissociated. [If the salts are not perfect electrolytes, one should consider partial dissociation. The degree of dissociation depends on p , T , the total salt concentrations, and the molar ratios c, = Nn/Mo. For small e,, the shift of the dissociation equilibrium away from the unperturbed value represented by M , = MLo(neutral salt molecules and ions now), that obtains for the solution of water and salts only, is small. It is not difficult to see that in first order this shift is accounted for by including 0 1978 American
Chemical Society
The Journal of Physical Chemistry, Vol. 82, No. 7, 1978 805
Statistical Mechanical Theory of Nonionic Micelles
an additional term quadratic in the c, in G(M,,N,, p , T ) , eq 18, and putting M , = M,O. For strong electrolyte salts the correction term should be unimportant.] We also assume that M,
