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Investigation of the Microstructure of Micelles Formed by Hard-Sphere Chains Interacting via Size Nonadditivity by Discontinuous Molecular Dynamics Simulation Basel F. Abu-Sharkh* and Esam Z. Hamad Department of Chemical Engineering, KFUPM, Dhahran 31261, Saudi Arabia Received August 9, 2003. In Final Form: October 29, 2003
Micelle formation by short nonadditive hard surfactant chains was investigated at different size ratios, reduced densities, and nonadditivity parameters using molecular dynamics simulation. It was found that spherical, cylindrical, lamellar, and reverse micelles can form in systems with different head, tail, and solvent characteristics. Hard-core surfactant chains composed of a head segment and three tail segments were simulated in a solvent of hard spheres. The formation of micelles was found to be a strong function of the packing fraction and nonadditivity parameter. Micelles were more stable at higher densities and larger nonadditivity parameters. At lower densities, micelles tended to break into small, dynamic globules.
Introduction The data of molecular physics have provided a good reason to believe that many of the equilibrium and structural properties of fluids can be satisfactorily treated within the framework of models that take into account only the repulsive portion of molecular interactions.1 Real molecules can, thus, be modeled by hard convex bodies, the simplest of which are hard spheres. In the theory of liquids, this concept plays the same role played by the ideal gas in the theory of gases. The attractive forces are less important especially at high temperatures than the repulsive forces and can be accounted for by perturbations. In other words, the proper volume of molecules is a prime factor in interpretation of the behavior of low-molecularweight fluids. The freely jointed hard-sphere model provides a useful extension of the advantages of the hard-sphere model to the study of the behavior of real chain molecular fluids because it is a simple model that incorporates chain connectivity, flexibility, and other essential features of chain molecules. This model is as important for chain fluids as the hard-sphere model for simple fluids. Consequently, considerable attention has been devoted recently to the development of theories and simulation of this model, and several equations of state have appeared.2-8 Development of the theory for the hard-chain model is easier than for more realistic chains, and the hard-chain fluid can act as a reference system for perturbation theories of more complex chain fluids. Although additive hardchain models have made a significant contribution to our understanding of polymers, they cannot account for some important physical phenomena, for example, fluid phase separation. * Author to whom correspondence should be addressed. E-mail:
[email protected]. (1) Malakhov, A. O.; Brun, E. B. Macromolecules 1992, 25, 6262. (2) Dickman, R.; Hall, C. K. J. Chem. Phys. 1986, 85, 4108. (3) Honnell, K. G.; Hall, C. K. J. Chem. Phys. 1989, 90, 1841. (4) Wertheim, M. S. J. Stat. Phys. 1984, 35, 19. Wertheim, M. S. J. Stat. Phys. 1986, 42, 477. (5) Chapman, G.; Gubbins, K. E. Mol. Phys. 1988, 65, 1057. (6) Chiew, Y. C. Mol. Phys. 1990, 70, 129. (7) Chang, I.; Sandler, S. S. Chem. Eng. Sci. 1994, 49, 2777. (8) Gazzillo, D.; Pastore, F. R.; Enzo, S. J. Phys.: Condens. Matter 1989, 1, 3469. Gazzillo, D.; Pastore, F. R. J. Phys.: Condens. Matter 1990, 2, 8463.
Chains composed of nonadditive hard spheres can act as simple models that can be used to study phase separation and self-assembly in molecular systems while at the same time maintain all the benefits of the hardchain model. Nonadditive hard-sphere (NAHS) mixtures perhaps represent the simplest model that can exhibit material instability and fluid-fluid phase separation at high densities and for all size ratios.8 Spherical nonadditive size interactions between spheres of radii σii and σjj are characterized by a cross collision diameter σij of the form
σii + σjj 2
σij ) (1 + ∆)
∆ g -1
(1)
where ∆ is the nonadditivity parameter. Because the interactions are purely repulsive, demixing of NAHSs is purely entropic in nature (excluded volume effect). Entropically driven phase separations have received much attention recently because of their relevance to the observed phase behavior of colloidal suspensions.8,9 Nonadditivity can be useful for interpreting several experimental features of real systems. In addition, NAHS mixtures can be used as a reference for perturbation theories that aim at describing the thermodynamics of realistic mixtures with nonadditive interactions such as various water and gas mixtures and alloys.8,10 Binary mixtures of NAHSs have been studied since the early 1970s.10 The results of these studies indicate that mixtures with positive nonadditivities may demix at sufficiently high pressures as a result of increased repulsions between unlike particles. This type of aggregation (homoassociation) resembles phase segregation observed in liquid alloys and supercritical mixtures.11 For mixtures with negative nonadditivities, the unlike components tend to associate but no phase separation has been found resembling the heterocoordination behavior observed in compound-forming alloys and aqueous electrolyte solutions.11,12 (9) Frenkel, D. Physica A 1999, 263, 26. (10) Schaink, H.; Schaink, M.; Hoheisel, C. J. Chem. Phys. 1992, 97, 8561. (11) Jung, J.; Jhon, M. S.; Ree, F. H. J. Chem. Phys. 1994, 100, 9064. (12) Alblas, P.; Van der Marel, C.; Geertsman, W.; Meijer, J. A.; van Osten, A. B.; Dijkstra, J.; Stein, P. C.; Van der Lugt, W. J. Non-Cryst. Solids 1984, 61/62, 201S.
10.1021/la035460v CCC: $27.50 © 2004 American Chemical Society Published on Web 12/03/2003
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Nonadditive interactions have been used to describe polymer-protein interactions and colloid-polymer interactions in which the polymer coils can interpenetrate.13 In addition, they have been used to describe liquid metalprotein interactions and fluid-fluid phase separation at very high pressures where repulsive interactions dominate.14-17 There are very few studies in the literature that address nonadditive size interactions in hard chains. In a recent study, a general method of representing chain stiffness, segment fusion, ring rigidity, and specific forces using nonadditive size interactions was suggested.18 Assembly in block copolymers and chain mixtures composed of segments with nonadditive size interactions have also been investigated.19 In this work, we use computer simulation to study micelle formation in solutions of hard surfactant chains in a hard-sphere solvent under the influence of positive size nonadditivity between the head, tail, and solvent segments. The method uses nonadditive size interactions to model attractive and repulsive forces in hard-chain fluids. This method has many advantages over other simplistic simulations, for example, modeling tails as needles or nonadditive spheriocylinders, in that it is more realistic.20 Consequently, the range of morphologies that can be generated is larger. Compared to other short-ranged simulations, for example, using soft repulsive interactions such as the Weeks-Chandler-Anderson potential, or short-ranged attractive potentials such as square well potentials, this method is easier to implement via molecular dynamics and has the potential for comparison with equation of state theories. The modeling of hard chains can easily be accomplished by combining polymerization theories such as the thermodynamic perturbation theory4 with analytical expressions for the contact pair correlation functions for nonadditive spheres.21-24 Finally, this method utilizes the advantages of a discontinuous molecular dynamics simulation compared to a continuous molecular dynamics simulation in especially, for example, efficiency when scaled to large system sizes [CPU time scales as N ln(N) where N represents the number of atoms explicitly represented] and realistic densities, in addition to code simplicity and transferability. The modeling of micelles is complex because it involves a balance among the different interactions that exist between the solvent and hydrophilic and hydrophobic components of the amphiphilic molecules, and simplifying assumptions have to be made to obtain specific structural information. Three general approaches have been used in the literature for modeling surfactant systems.25 The first and most simple approach is using lattice models that allow a rigorous theoretical treatment as well as an extensive analysis by simulation.25,26 This approach has contributed significantly to the qualitative understanding (13) Dijkstra, M. Phys. Rev. E 1998, 58, 7523. (14) Haynes, C. A.; Benitez, F. J.; Blanch, H. W.; Prausnitz, J. M. AIChE J. 1993, 39, 1539. (15) Schouten, J. A.; van den Bergh, L. C.; Trappeniers, N. J. Chem. Phys. Lett. 1985, 114, 40. (16) Costantino, M.; Rice, S. F. J. Phys. Chem. 1991, 95, 9034. (17) van Hinsberg, M. G. E.; Verbrugge, R.; Schouten, J. A. Fluid Phase Equilib. 1993, 88, 115. (18) Hamad, E. Z. J. Chem. Phys. 1999, 111, 5599. (19) Abu-Sharkh, B. F.; Hamad, E. Z. Macromolecules 2000, 33, 1345. (20) Schmidt, M.; von Ferber, C. Phys. Rev. E 2001, 64, 051115. Bolhuis, P. G.; Frenkel, D. Physica A 1997, 244, 45. (21) Hamad, E. Z. Mol. Phys. 1997, 91, 371. (22) Hamad, E. Z. J. Chem. Phys. 1996, 105, 3222. (23) Hamad, E. Z. J. Chem. Phys. 1996, 105, 3229. (24) Hamad, E. Z. J. Chem. Phys. 1997, 106, 6116. (25) Gunn, J. R.; Dawson, K. A. J. Chem. Phys. 1989, 91, 6393. (26) Rector, F.; van Swol, J.; Henderson, R. Mol. Phys. 1994, 82, 1009.
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of amphiphilic systems.27-34 The second approach utilizes detailed models that are intended to mimic experimental conditions through application of atomistic potentials in computer simulations. Computer simulation can provide detailed information on the structure and thermodynamics of micelles knowing only the intermolecular and intramolecular force field parameters.35,36 Many of the detailed structural features can be elucidated using this method, and good agreements have been found between simulation results and neutron scattering data.35-37 The fast development of computational power allows the simulation of larger and more complex amphiphilic systems that include large numbers of components, and this type of simulation is becoming more practical with time.38-44 The third approach is intermediate to the above two and utilizes idealized off-lattice potentials. Using this approach, the self-assembly of larger systems over long time scales can be investigated in a more realistic manner. This approach, which we adopt in this study, has been applied by many researchers to study surfactant-solution systems.26-28,45-51 In these studies, the molecular simulation of micelles in solution is usually performed using continuous potentials with attractive and repulsive components. Hard-sphere models represent a very simple approach to understanding the physical behavior of liquids. Hard chains are easy to model, efficient to simulate, and display the most essential features of real systems. An interesting question is whether micellar formation will take place in hard-chain solutions when nonadditive size interactions are a part of a chain molecule, where only some of the segments interact by size nonadditivity. One would expect that the microphase separation becomes a possibility, depending on the magnitude and sign of the nonadditive interactions and on the composition, structure, and distribution of segments interacting with nonadditive interactions. The ability of nonadditive interactions to represent attractive and repulsive forces can be illustrated using Figure 1. The touching solid circles represent bonded segments. The dashed line represents σij of head-tail, head-solvent, and tail-solvent groups. Positive ∆ht leads (27) Dawson, K. A.; Lipkin, M. D.; Widom, B. J. Chem. Phys. 1988, 88, 5149. (28) Larson, R. G. J. Chem. Phys. 1989, 91, 2479. (29) Wheeler, J. C.; Widom, B. J. Am. Chem. Soc. 1968, 90, 3064. (30) Widom, B. J. Phys. Chem. 1984, 88, 6508. (31) Widom, B. J. Chem. Phys. 1986, 84, 6943. (32) Schick, M.; Shih, W. H. Phys. Rev. Lett. 1987, 59, 1205. (33) Halley, J. W.; Kolan, A. J. J. Chem. Phys. 1988, 88, 3313. (34) Gompper, G.; Shick, M. Phys. Rev. Lett. 1989, 62, 1647. (35) Egbert, E.; Berendsen, H. J. C. J. Chem. Phys. 1988, 89, 3718. (36) Van der Ploeg, P.; Berendsen, H. J. C. J. Chem. Phys. 1982, 76, 3271. (37) Karaborni, S.; O’Connell, J. P. J. Phys. Chem. 1990, 94, 2624. (38) Haile, J. M.; O’Connell, J. P. J. Chem. Phys. 1984, 88, 6363. (39) Woods, M. C.; Haile, J. M.; O’Connell, J. P. J. Phys. Chem. 1986, 90, 1857. (40) Watanabe, K.; Ferrario, M.; Klein, M. L. J. Phys. Chem. 1988, 92, 819. (41) Watanabe, K.; Klein, M. L. J. Phys. Chem. 1989, 93, 6897. (42) Karaborni, S.; O’Connell, J. P. Langmuir 1990, 6, 905. (43) Jonsson, B.; Edholm, O.; Teleman, O. J. Chem. Phys. 1986, 85, 2259. (44) Bast, T.; Hentschke, R. J. Phys. Chem. 1996, 100, 12162. (45) Smit, B. Phys. Rev. A 1988, 37, 3431. (46) Smit, B.; Hilbers, P. A.; Esselink, K.; Pupert, L. A. M.; van Os, N. M.; Schluper, A. G. Nature 1990, 348, 624. (47) Smit, B.; Hilbers, P. A. Esselink, K.; Pupert, L. A. M.; van Os, N. M.; Schluper, A. G. J. Phys. Chem. 1991, 95, 6361. (48) Telo da Gama, M. M.; Thurtell, J. H. J. Chem. Soc., Faraday Trans. 2 1986, 82, 1721. (49) Telo da Gama, M. M.; Gubbins, K. E. Mol. Phys. 1986, 59, 227. (50) Drouffe, J. M.; Maggs, A. C.; Leibler, S. Science 1991, 254, 1353. (51) Wilson, M. R.; Allen, M. P. Mol. Phys. 1993, 80, 277.
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Figure 1. Representation of nonbonded head-tail, headsolvent, and tail-solvent nonadditive size interactions: (a) positive head-tail interaction (∆ht > 0), (b) negative headsolvent interaction (∆hs < 0), and (c) positive tail-solvent interaction (∆ts > 0). The gray, white, and dotted circles represent the head, tail, and solvent segments, respectively. The dashed lines represent the nonbonded nonadditive size interactions, while the touching solid circles represent bonded segments. Only positive nonadditive interactions are used in this study.
to a net repulsive interaction, while negative ∆ht leads to a net attractive interaction. It is the objective of this paper to present a simple and highly efficient method for the simulation of the selfassembly of surfactants in solution using only hard-sphere potentials and positive nonadditive interactions. We apply the proposed simulation methodology to generate some of the common micellar structures of surfactants in solvents under different conditions. To accomplish this objective, a system composed of short surfactant molecules (four segments) dissolved in a solvent composed of hard spheres is simulated. The surfactant molecules consist of a large headgroup and three tail groups. The solvent is composed of single hard spheres with characteristics equivalent to those of the head or tail groups. To simplify the notation, only two types of hard-sphere segments are considered, namely, large and small groups. Consequently, two of the head, tail, and solvent segments are always considered equivalent in size and magnitude of nonadditivity with the third type of segment. We investigate the influence of the size ratio and magnitude of nonadditivity on the self-assembly of the surfactant molecules. Simulation Details We consider a system composed of surfactant molecules consisting of a head and three tail hard-spherical segments and solvent molecules composed of single hard spheres in a volume V. The cross interaction between segments is governed by eq 1, and the packing fraction of the chains (y) is given by
y)
πN 6V
∑i xiσii3
(2)
where xi is the mole fraction of spheres of type i. This definition of the packing fraction is exact only for additive systems but is used here as a simple approximation. The positively nonadditive systems have a higher actual packing fraction than the one calculated by eq 2. The total number of spheres in the system, N, is fixed at 512 with 272 solvent segments and 60 surfactant molecules. The surfactant model consists of a large headgroup and three smaller tail groups. In the hard-chain system under study, an infinite potential well of width 2σδ exists between bonded segments, where σ is the distance between the spheres and the minimum allowable distance between the bonded spheres is (1 - δ)σ, while the maximum allowed extension is (1 + δ)σ. The algorithm we use in our
simulations is the discontinuous potential molecular dynamics (DMD), which is widely used for studying the dynamics of hard-chain systems.52 The DMD algorithm is based on the hard-sphere algorithm by Alder and Wainwright,53 which was later extended by Rapaport54,55 and then by Bellemans et al.56 to simulate hard chains by imposing bond constraints between bonded segments. In Belleman’s model, which mimics the tangent hard-chain model, segments of diameter σ belonging to the same chain are allowed to penetrate and get away from each other in such a way that their average distance over time is σ. The minimum allowable distance between bonded segments is (1 - δ)σ, while the maximum allowed extension is (1 + δ)σ, where the extension parameter δ is small compared to 1. As in the original hard-sphere model, segments interact with each other through excluded-volume collisions. The time between collisions is calculated, and the trajectories in phase space are generated by advancing the system collision by collision. After each collision, the new velocities of the colliding pair are calculated from the equations of motion, keeping the total energy and the total momentum conserved. In addition to hard-sphere-like collisions between nonbonded segments, hard-core collisions take place whenever the distance between bonded spheres reaches the minimum penetration (1 - δ)σ or the maximum extension (1 + δ)σ. In our model, nonadditivity ∆ between unlike spheres belonging to different chains is added as in eq 1. In this case, nonadditive, nonbonded segments, instead of colliding at a distance of σ, would collide at a distance given by eq 1. The simulation is performed in the microcanonical ensemble. We use cubic and rectangular boxes and apply periodic boundary conditions in all directions. In the initial configuration, we place the chains in a lattice and assign them initial velocities that are generated according to a Gaussian distribution. We then allow the chains to equilibrate by monitoring the decay of the end-to-end distance of the surfactant molecule. Next, we start the simulation phase. Depending on the density and the value of ∆, the simulation phase takes from 10 million to 50 million collisions. In this study, we have used N ) 512 spheres in most of our simulations. The simulations were conducted at different densities and values of ∆. Results and Discussion To facilitate the visual identification of micelle formation in the equilibrium configurations of the simulated systems, the head, solvent, and tail groups are shown as gray, black, and white spheres, respectively. Characteristics of the different simulated systems are displayed in Table 1. Also shown in Table 1 is the ratio v/al, which is calculated on the basis of interactions between the first tail segment and neighboring head segments. l is calculated to be equal to the sum of the diameters of the tail segments plus the radius of the headgroup. v is volume of the cone (or cylinder) formed by the head and nonadditive size of the tail segment connected to the headgroup. The shape of the cone is shown by the lines displayed in the last column of Table 1. This convention in calculating v is adapted because the interaction between the head and the first tail segments has the greatest influence in determining (52) Smith, S. W.; Hall, C. K.; Freeman, B. D. J. Chem. Phys. 1996, 104, 5616. (53) Alder, B. J.; Wainwright, T. E. J. Chem. Phys. 1959, 31, 459. (54) Rapaport, D. C. J. Phys. A: Math. Gen. 1987, 11, L213. (55) Rapaport, D. C. J. Chem. Phys. 1979, 71, 3299. (56) Bellemans, A.; Orban, J.; Belle, D. V. Mol. Phys. 1980, 39, 781.
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Table 1. Characteristics of the Different Head, Tail, and Solvent Groups and Aggregate Structures
the characteristics of the micelle. This is because the second and third tail segments are placed far enough from the head that they do not feel its nonadditive repulsive effect. Interactions between these two segments have the strongest influence on the way the molecules pack in a micelle because tail groups placed further from the head do not feel the influence of the headgroup once a micelle is formed. The circle surrounding the first tail segment indicates excluded volume to nonbonded head segments. System 1 is depicted in Figure 2. It consists of equally sized large head and solvent groups and small tail groups. The diameter of the tail group is half that of the head and solvent groups. The tail segments interact with both the head and the solvent segments with a positive nonadditivity parameter equal to 0.333. The packing fraction of the system is 0.45. The configuration of the head (gray), tail (white), and solvent (black) groups in the system after 2 × 107 collisions is shown in Figure 2. The cylindrical micelle formation could be identified in which the core is composed of the tail segments surrounded by the head and the solvent segments. The cylindrical micelle structure is maintained even when the packing fraction is lowered to 0.4. Lowering the density to 0.35 breaks up the cylindrical micelles into smaller globules. The number of segments associated with a globule decrease with decreasing the packing fraction until a density of 0.2.
Figure 2. Configuration of tail segments, head segments, and solvent segments of system 1; F ) 0.45, ∆ht ) ∆st ) 0.333.
However, even at the low density of 0.2, association between small aggregates of segments can still be observed. This can be attributed to the relatively large nonadditivity parameter. To study the influence of the size ratio of the head to tail groups on the shapes of the micelles, system 2 is considered for which the head-tail and solvent-tail size
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Figure 5. Configuration of system 4; F ) 0.45, ∆ht ) ∆st ) 0.333. Figure 3. Configuration of tail segments in system 2; F ) 0.45, ∆ht ) ∆st ) 0.5.
Figure 4. Configuration of the head, tail, and solvent segments of system 3; F ) 0.45, ∆ht ) ∆st ) 0.5.
ratios are increased to 3. It can be observed in Figure 3 that, at a packing fraction of 0.45, small spherical micelles form after equilibration. The core of the micelles is composed of tail groups and the shell of the headgroups. The spherical micelle shape and size is maintained at the lower packing fraction of 0.4. However, reducing the density further to 0.35 reduces the size of the micelles. The number of aggregated molecules continues to decrease with decreasing the density to 0.2 when aggregation nearly disappears. The small size of the tail relative to the head forces micelles to be small in size (about 20 segments) because only a few of the bulky headgroups can pack around the short tails. Reducing the size ratio of the head and solvent segments to the tail segment to 1 results in formation of a bilayer configuration. The configuration of the tail and headgroups of system 3 is depicted in Figure 4. The tail segments interact with both head and solvent segments via the positive nonadditivity parameter ∆ht ) ∆ts ) 0.5. An extended lamellar structure is formed with the tail groups occupying the center surrounded by the head and solvent segments. The case in which the tail and solvent segments have similar characteristics is subsequently investigated. System 4 is composed of large head segments and equally sized tail and solvent segments. The headgroups interact with positive nonadditivity with the tail and solvent
Figure 6. Radial distribution function of the head segments at contact in system 4; F ) 0.45.
groups. The value of the nonadditivity parameter is ∆ht ) ∆hs ) 0.333. It can be observed in Figure 5 that with simulation progress, the headgroups start assembling to form a reverse lamellar micellar structure with the headgroups assembled at the center surrounded by the tail and solvent groups. The large value of the nonadditivity parameter (∆) forces the system to assume this micellar structure to minimize the interaction between the head and other groups. The influence of the nonadditivity parameter (∆) on aggregation of the head segments is monitored using the first peak of the radial distribution function of the headgroups. It can be observed in Figure 6 that increasing the value of ∆ from 0 to 0.333 results in a sharp increase in the value of the radial distribution function at contact. Increasing the size ratio of the head to tail and solvent groups to 3 does not change the layered structure of the micelles, as shown in Figure 7 (system 5). However, it causes the lamella to extend diagonally in the box. This diagonal direction is a result of the periodic boundary conditions and the large size of the headgroups. The larger diagonal section of the simulation box compared to a horizontal or vertical section can accommodate more of the headgroups. However, reducing the value of ∆ to 0.067 results in formation of a spherical reverse micellar structure in which the headgroups aggregate at the center
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Figure 7. Configuration of the head, tail, and solvent segments in system 5; F ) 0.45, ∆ht ) ∆hs ) 0.25.
Figure 9. Configuration of the head segments of system 6 as a function of ∆.
Figure 8. Configuration of the head segments of system 6; F ) 0.45, σhh ) 2, σtt ) σss ) 1, ∆ht ) ∆hs ) 0.067.
some of the commonly observed micellar shapes demonstrates the validity and convenience of the method for isolating the size effects on the shapes of the micelles from other factors. The detailed relationships between density, the critical packing parameter, and the nonadditivity parameter are still to be investigated using larger systems. Conclusion
of the micelle and are surrounded by the tail and solvent groups (system 6), as shown in Figure 8. Snapshots of the equilibrium configuration of the headgroups as a function of ∆ are shown in Figure 9 for the packing fraction 0.45. The transition from a random distribution to spherical and lamellar structures can be observed upon increasing the value of ∆ from 0.0 to 0.333. The shapes of the micellar phases are consistent with the available experimental data on the structure of micelles with different head, tail, and solvent characteristics. Table 1 shows the critical packing parameter v/al for the different systems used. The structures of the micelles at different packing fractions correspond with some of the experimentally observed micellar structures; however, the small number of surfactant molecules (60) does not allow a full comparison with the experimental data. However, the ability of the simulation to reproduce
Introducing nonadditive size interaction in hard-chain systems allows the study of self-assembly of systems that could only be investigated using long-range attractive potentials. Common surfactant assemblies, for example, micelle, reverse micelle, and double layer structures, could be generated by simulating short surfactant chains composed of four hard-sphere segments interacting via size nonadditivity between head, tail, and solvent segments. The shapes of assemblies are found to be dependent on the size ratio of the head and tail segments as well as the strength of interaction described by the magnitude of positive nonadditivity. Acknowledgment. The support of King Fahd University of Petroleum and Minerals through research grant CHE/simulation/228 is greatly appreciated. LA035460V
