Proc. Nati. Acad. Sci. USA Vol. 81, pp. 4601-4605, July 1984 Chemistry Statistical thermodynamics of amphiphile chains in micelles (chain conformations/curved micelles and bilayers/maximal entropy/lattice model/order parameters) A. BEN-SHAUL*, I. SZLEIFER*, AND W. M. GELBARTt *Department of Physical Chemistry and The Fritz Haber Research Center for Molecular Dynamics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel; and tDepartment of Chemistry, The University of California, Los Angeles, CA 90024 Communicated by Howard Reiss, February 21, 1984 ABSTRACT The probability distribution of amphiphile droplet" assumption, the statistical-thermodynamic proper- chain conformations in micelles of different geometries is de- ties of the tails are insensitive to the micellar geometry (3-5). rived through maximization of their packing entropy. A lattice Consequently, the geometry dependence of ,u is usually at- model, first suggested by Dill and Flory, is used to represent tributed exclusively to the "opposing forces" that act at the the possible chain conformations in the micellar core. The po- surface of the micelle (3). These are the repulsion between lar heads of the chains are assumed to be anchored to the mi- the hydrophilic heads and the hydrophobic effect, which cellar surface, with the other chain segments occupying all lat- tend, respectively, to maximize and minimize the average tice sites in the interior of the micelle. This "volume-filling" area per head group on the micellar surface. But these sur- requirement, the connectivity of the chains, and the geometry face effects give rise to rather small variations in ,u for dif- of the micelle define constraints on the possible probability dis- ferent micellar geometries (A/uIl c kT), and it is thus not a tributions of chain conformations. The actual distribution is priori reasonable to neglect the effects of the hydrocarbon derived by maximizing the chain's entropy subject to these chains. constraints; "reversals" of the chains back towards the micel- In this paper we present a statistical-thermodynamic the- lar surface are explicitly included. Results are presented for ory for amphiphile packing in different micellar geometries. amphiphiles organized in planar bilayers and in cylindrical A large number of theoretical studies have dealt with the sta- and spherical micelles of different sizes. It is found that, for all tistical thermodynamics of phospholipids in planar mem- three geometries, the bond order parameters decrease as a brane bilayers, particularly within the context of the gel-liq- function of the bond distance from the polar head, in accord- uid crystal phase transition (6). On the other hand, very few ance with recent experimental data. The entropy differences studies (7-13) have addressed the question of chain packing associated with geometrical changes are shown to be signifi- in nonplanar (e.g., spherical and cylindrical) micelles. Most cant, suggesting thereby the need to include curvature (envi- pertinent to our present paper is the theory of Dill and Flory ronmental)-dependent "tail" contributions in statistical ther- (11, 12). modynamic treatments of micellization. In the theory of Dill and Flory, the micellar core is repre- sented by a cubic lattice, appropriately modified for curved Micelles are aggregates of amphiphilic molecules composed surfaces (see Fig. 1). Every chain conformation is regarded of a hydrophilic (polar, ionic, or zwitterionic) "head" and a as a sequence of steps on the lattice, originating at the sur- hydrophobic "tail" that is usually a flexible hydrocarbon face. To conform to the geometry of the cubic lattice, every chain (1, 2). In aqueous solutions the tails form the interior of chain segment is taken to represent -3.5 methylene groups the micelle, while the heads are at the hydrocarbon/water of real alkyl chains. Differences in internal energy of differ- interface. The aggregates exist in a variety of sizes, shapes, ent conformations are disregarded, and the only constraints and phases, depending on their constituent amphiphiles and on a chain conformation are due to the presence of other "external" conditions like concentration, temperature, and chains and the ("volume-filling") requirement that all lattice ionic strength. Some amphiphiles, like NaDodSO4, form sites are occupied. The probabilities of the various chain spherical micelles at low concentrations (just above the cmc, conformations are generated by a stochastic matrix whose the critical micelle concentration) that grow into rod-shaped elements describe single-step probabilities on the lattice. micelles as the concentration increases. Further increase in The matrix elements are evaluated algebraically through concentration results in a phase transition from an isotropic equations representing the volume-filling condition. A key solution to an ordered, hexagonal phase of long rods. Most assumption that facilitates the formulation and solution of phospholipids, on the other hand, aggregate spontaneously these equations is the neglect of chain "reversals," (i.e., into large, nearly planar bilayers arranged as vesicles or la- "backward" steps towards the surface; see Fig. 1). The mellae. bond-order parameters predicted by the Dill and Flory mod- Aggregation occurs because /4 - A4 < 0, where /4 is the el for planar surfaces are in rather good agreement with ex- standard chemical potential of a single (monomeric) mole- perimental results, but their predictions for spherical and cy- cule in solution, while /4 is the standard chemical potential lindrical micelles differ qualitatively from recent reported of a molecule in a micelle (3-5). (In NaDodSO4, for instance, data for these systems (7, 14, 15). ,A - =4 -lOkT.) The value of /4 for a given amphiphile The theory presented below uses, partly for the sake of depends on its position in the aggregate (i.e., on the local comparison, the cubic lattice of Dill and Flory. We also im- "geometry" of the micelle). A common assumption in the pose the volume-filling condition and disregard chain confor- existing models of amphiphile self-assembly is that the hy- mation energy. The two major differences between the two drophobic tails forming the micellar core behave like in the theories are: (i) our expressions for chain conformation corresponding liquid hydrocarbon (e.g., dodecane in the probabilities are different and are derived by using the maxi- case of NaDodSO4). According to this "liquid hydrocarbon mal entropy principle (the information-theory approach) (16, 17), and (ii) we allow explicitly for chain reversals. We show The publication costs of this article were defrayed in part by page charge that incomplete optimization of the conformational entropy payment. This article must therefore be hereby marked "advertisement" can lead to incorrect predictions concerning the chain statis- in accordance with 18 U.S.C. §1734 solely to indicate this fact. tics. In particular, a simultaneous maximization of the entro- 4601 Downloaded by guest on September 23, 2021 4602 Chemistry: Ben-Shaul et al. Proc. NatL Acad Sci. USA 81 (1984) 9 9 9 0--, 9 r--o I Y 40 9 4- i-= 1 I r- ' 4 i=2 -II I L r-i -- ri- - I 01- ]---L 4- i=L FIG. 1. A two-dimensional representation of the Dill and Flory lattice models (11, 12) describing cylindrical and spherical micelles (Right) and one-half of a planar bilayer (Left). The circles in the surface layer designate the polar heads (or the first chain segment). The arrows point out chains with reversals. py throughout all layers of the micelle provides bond order- density of head groups). Using the appropriate expressions parameter profiles in agreement with experiment; a sequen- for M (=Nn) and M1, we find tial scheme applied successively to one layer at a time, on the results of Dill and Flory, according n the other hand, gives Ml = - plane [2a] to which bond alignment increases from "head" to "tail" for L curved aggregates. cylinder [2b] THEORY (Lj)( 2L the hy- Lattice Representation. The lattices representing = ( - L drophobic cores of planar bilayers, cylindrical and spherical Ml +i-7 sphere [2c] micelles (11, 12), and some chain conformations are shown two-dimensionally in Fig. 1. (The theory is for three dimen- Eq. 2 shows that for the same L, the average head group sions.) The cores are divided into L equally thick layers areas corresponding to the planar, cylindrical, and spherical (shells), and every layer i is divided into Mi cells of equal geometries relate, approximately, as 1:2:3; these ratios are volume. The small circles in the outermost layer represent fundamental in the theories of micelle formation (3-5). the hydrophobic heads or the first chain segments connected The conformations of n-segment chains can be classified to the heads ifthe heads are large or assumed to be surround- into ordered sequences of n - 1 numbers, a = i2, i3, *- i.... ed by water. In all geometries the surface layer is denoted as in, where ik denotes the layer in which the kth segment is i = 1 and the innermost as i = L. The length unit in all calcu- located (recall il 1). Thus, for instance, a = 2, 3, 4, 4, 3 lations will be the layer thickness, and the volume unit will describes a six-membered chain making three consecutive be the volume of a cell. Thus, L is the thickness or the radius ("radial") steps towards the center of the core, then a lateral of the micelle. step in layer 4 and finally a reversal towards the surface (see Experiments show that the density inside the core is simi- also Fig. 1). The assignment a = i2, ..., in does not fully spec- lar to the liquid hydrocarbon density (2, 3) and that water ify the conformation. For instance, a = 1, 1, 1 includes 36 penetration is negligible (15). Accordingly it is assumed here, conformations all confined to the first layer, of which four as in refs. 11 and 12, that all lattice sites are occupied by describe straight chains while the others involve one, two, or chain segments.