Langmuir 1996, 12, 913-915
913
van der Waals Interactions in Nonionic Micelles R. Ricceri, N. Vila Romeu,† and G. Taddei* Dipartimento di Chimica, Universita` di Firenze, via G. Capponi 9, 50121 Florence, Italy Received July 21, 1995. In Final Form: November 6, 1995X The aggregation energies of prolate and oblate ellipsoidal neutral micelles in a binary oil/water system were calculated by using a semiempirical model. The interactions among the polar heads of the amphiphilic molecules at the micelle interface were evaluated via a central potential of the Lennard-Jones 6-12 type. The interactions among the hydrophobic chains in the micelle bulk were evaluated via a phenomenological model. The results, which refer only to micelles of low eccentricity, show that the prolate and oblate micelles of equal volumes (containing the same number of molecules) have approximately the same aggregation energies.
Introduction The calculation of the aggregation energy of micelles in aqueous media in terms of the interaction energy among the amphiphilic molecules is an open problem. There are several thermodynamical treatments,1 but they do not specify the kind and the role of the intermolecular forces responsible for self-assembly of the amphiphilic molecules. More recently some semiempirical models have been proposed in which the aggregate stability is calculated in terms of the elasticity of the membrane.2-6 However, in these calculations as well the nature of the intermolecular forces interested in self-assembly is completely disregarded. The major difficulty of the molecular approach concerns the actual conformations of the hydrophobic chains in the micelle bulk, conformations which are not known in principle. Various authors treat the bulk of the micelles essentially as liquid hydrocarbons,7 neglecting the anchorage of the chains to their polar heads disposed at the micelle interface. Other authors8,9 assume that the amphiphilic molecules show characteristic shapes and sizes (see the surfactant parameter model) and that the hydrophobic chains assemble in a fencelike close packing. It is certain that the micelles are dynamical aggregates whose structure changes continuously in the course of time, so that both pictures are partially true. In this paper we propose a semiempirical model for the micelle stability where some of the interacting forces among the amphiphilic molecules were considered. In general, the intermolecular forces in micelles are of the van der Waals and Coulombic types. The latter occur if the polar heads contain ionic charges. The basic assumptions of the present work are the following: (i) the polar heads at the micelle interface are electrically neutral and interact with each other only via van der Waals forces * Author to whom correspondence should be addressed. † Present address: Departamento de Fisico-Quimica, Facultad de Farmacia, Universidad de Santiago de Compostela, Avenida de las Ciencias, Santiago de Compostela, Spain. X Abstract published in Advance ACS Abstracts, February 1, 1996. (1) Israelachvili, J. N. Intermolecular and surface forces; Academic Press: New York, 1985. (2) Helfrich, W. Z. Z. Naturforsch., C 1973, 28, 693. (3) Hyde, S. T. J. Phys. 1990, C7, 209. (4) Fogden, A.; Hyde, S. T.; Lundberg, G. J. Chem. Soc., Faraday Trans. 1991, 87, 949. (5) Taddei, G. J. Chem. Soc., Faraday Trans. 1993, 89, 1749. (6) Taddei, G. Colloid Polym. Sci. 1994, 272, 1300. (7) Tanford, C. The Hydrophobic Effect, 2nd ed.; John Wiley: New York, 1980. (8) Israelachvili, J. N.; Mitchell, D. J.; Ninham, B. W. J. Chem. Soc., Faraday Trans. 1976, 72, 1525. (9) Mitchell, D. J.; Ninham, B. W. J. Chem. Soc., Faraday Trans. 1981, 77, 601.
0743-7463/96/2412-0913$12.00/0
which are described by Lennard-Jones 6-12 potentials; (ii) the force field of the polar heads is central, i.e., the polar heads show a spherical symmetry; (iii) the interactions among the hydrophobic chains in the micelle bulk become attractive to the highest degree when the micelle interface area attains its lowest value (for a given micelle volume); (iv) only spherical and ellipsoidal (prolate and oblate) micelle shapes were considered. The phenomenological assumption iii is very approximate. It seems more reliable when the chain-chain repulsive forces are weaker with respect to the attractive forces. This occurs at the usual densities of the bulk of a micelle whose amphiphilic molecules have a surfactant parameter p < 1 (see below). Note that assumption iii is unavoidable in the model presented here. We are in fact unable to adopt the Lennard-Jones model for the chain-chain interactions since the positions of the atoms of the chains in the micelle bulk are in principle unknown. The aim of this paper is to ascertain the aggregation energies of prolate, oblate, and spherical micelles, considering the values that the surfactant parameter of the amphiphilic molecules p ) v/al (where v and l are, respectively, the volume and the length of the hydrophobic chain and a the polar head area) can actually assume. We had considered only spheroids of low eccentricity (with axial ratio in the range 1-1.5) since for higher eccentricities both prolate and oblate ellipsoids show physically meaningless local curvatures (at the poles for the prolate and at the equatorial belt for the oblate).5,6 In other words, low-eccentricity spheroids satisfy the condition that neither voids nor overlappings among the hydrocarbon chains occur inside the micelle. Note that there are uncertain indications about the relative stability of the prolate and oblate forms.10-12 The calculations presented in this paper consider an arbitrary amphiphilic molecule having a characteristic surfactant parameter p0 in the range 1/3-1/2. Theory The basic equation for calculating the energy of the micelle is N
U ) -K As/Ae + u0/2
[(r0/rij)12 - 2(r0/rij)6] ∑ i*j
(1)
where As and Ae are, respectively, the surface area of the sphere and ellipsoid, u0 and r0 are the coordinates of the (10) Tanford, C.; Nozaki, Y.; Rohde, M. F. J. Phys. Chem. 1977, 81, 1555. (11) Leibner, J. E.; Jacobus, J. J. Phys. Chem. 1977, 81, 130. (12) Birdi, K. S. Prog. Colloid Polym. Sci. 1985, 70, 23.
© 1996 American Chemical Society
914
Langmuir, Vol. 12, No. 4, 1996
Ricceri et al.
Figure 1. U/u0 against the micelle aggregation number N of prolate (up) and oblate (down) micelles of the same volume. The data refer to amphiphilic molecules whose hydrophobic chain contains 12 carbon atoms (see ref 14). The symbols indicate the values of the polar head areas (expressed in square angstroms). The numbers indicate the micelle axial ratios.
Figure 2. The same as in Figure 1 with the exception of the number of carbon atoms of the hydrophobic chain length here being 16.
minimum of the Lennard-Jones potential, rij is the headhead distance, and N is the micelle aggregation number. Constant K of eq 1 can be evaluated in terms of u0 according to the equation:
|K| ) (x2l/2r0)N|u0|
(2)
Equation 2 can be easily obtained from the equations of the dispersion energy between two cylinders (i.e., the hydrophobic chains) closely packed, when the polarizability integrals of the chains and polar heads (assumed spherical) are approximately equal.13 The radii of the spherical micelles were calculated according to the assumptions of Tanford.14 The larger and the shorter half-axes of the prolate and oblate ellipsoids were calculated assuming micelles of equal volumes (with respect to the spherical micelles). Polar heads, whose diameter is r0, were distributed at the interface of the micelles along the micelle parallels of latitude in order to avoid overlappings among the heads and to reduce to a minimum the voids between nearestneighbor heads. Results and Discussion Figures 1-3 show the calculated U/u0 of the prolate and oblate micelles against the aggregation number N for different values of the hydrophobic chain length, polar head area, and axial ratio. The points of the figures are a little scattered since the packing of the polar heads at the micelle interface is by nature discontinuous. In other words, some voids between nearest-neighbor polar heads are unavoidable, since the space available to the polar heads on the micelle interface cannot always be filled by an integer number of heads. Inspection of Figures 1-3 shows that in general the aggregation energy becomes more negative with (i) the length of the hydrophobic chain, (ii) the lowering of the polar head area, and (iii) the anisotropy of the micelle. (13) Langebein, D. van der Waals attraction; Springer Verlag: Berlin, 1974. (14) Tanford, C. J. Phys. Chem. 1972, 76, 3020.
Figure 3. The same as in Figure 1 with the exception of the number of carbon atoms of the hydrophobic chain here being 20.
Point i confirms that the hydrophobic effect is very important in self-assembly. Point ii indirectly shows that the aggregation energy becomes more negative, increasing the micelle aggregation number. Point iii together with point ii confirms that spherical micelles must undergo form alterations when the micelles accommodate a large number of amphiphilic molecules.7 During the calculations we always disregarded those cases in which the calculated surfactant parameter p assumes values not compatible with the overall form of the micelle (see refs 7-9).
Nonionic Micelles
Langmuir, Vol. 12, No. 4, 1996 915
The aggregation energy per amphiphilic molecule u ) U/N can be interpolated from the data of Figures 1-3.
u/u0 ) -2.88 - 1.01 × 10-2 N
(3)
u/u0 ) -2.74 - 1.08 × 10-2 N
(4)
Equations 3 and 4 refer, respectively, to the prolate and oblate micelles. These equations are practically the same within the uncertainties due to interpolation and to polar head packing. They show that the aggregation energy per molecule becomes more negative when the aggregation number grows. Since N increases with the hydrophobic chain length (for a given polar head), the dependence of u/u0 on N is due principally to the attractive chain-chain interactions. In this paper the forms of the micellessprolate and oblate ellipsoids of low eccentricityshave been chosen a priori (see the fourth assumption in the Introduction of this paper) on the basis of a suitable molecular packing inside the micelle. In this case, the micelle aggregation energy U is notsin principlesminimizable. The minimization of U (at constant volume and N) would introduce local changes in the curvature of the micelle which are not compatible with the assumed micelle form. In addition, these local changes would produce some voids and/or overlappings among the hydrocarbon chains in the micelle bulk. (For a detailed discussion on this problem, see ref 6.)
The results of this paper show that the aggregation energy of spheroidal micelles of low eccentricity does not allow us to determine the relative stabilities of the prolate and oblate forms. This conclusion is confirmed indirectly by the calculation of the surfactant parameter p averaged over the whole prolate and oblate micelles by the method reported in ref 5. The values of p are 0.37 and 0.34, respectively, when the micelle axial ratio is 1.5. These values show that the amphiphilic molecules, which assemble spontaneously in prolate or oblate micelles, must have very similar shapes in both cases. Probably, for higher eccentricities prolate and oblate forms have different aggregation energies. However, it has been shown5,6 that in these conditions the amphiphilic molecules which are located in particular zones of the micelle interface are submitted to high elastic strains which are physically unacceptable. The micelle forms which accommodate the largest number of amphiphilic molecules are probably not ellipsoids, but rather sphero-cylinders, rods, or other different forms.5-7,11 The calculations reported here are based on certain assumptions. Some of them are inherent to the adopted mathematical model, while others have a larger significance. For instance, among the latter assumptions, we have neglected the entropic contributions which favor micelles of smaller sizes. We have disregarded the elastic contributions of the micelle membrane during the micelle growth. LA950609R
