Langmuir 2000, 16, 10547-10552
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C12E2 Reverse Micelle: A Molecular Dynamics Study Rosalind Allen, Sanjoy Bandyopadhyay,* and Michael L. Klein Center for Molecular Modeling, Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6323 Received August 17, 2000. In Final Form: October 16, 2000 Molecular dynamics (MD) simulations have been carried out to study the microscopic properties of a reverse micelle of the poly(oxyethylene) surfactant C12E2 in nonpolar environments (decane and vacuum). The simulations reveal that the core water molecules and the oxyethylene headgroups of the surfactants behave similarly in decane and vacuum simulations. However, the surfactant tails of the micelle behave differently in a vacuum. In the absence of the solvent the hydrocarbon tails collapse onto the surface of the core. The core water molecules were observed to be less mobile and interact strongly with the oxyethylene headgroups and form hydrogen-bonded bridged structures with the oxygen atoms of the same surfactant chain. This leads to a very strong preference for gauche C-C bonds in the oxyethylene headgroups of the surfactants.
I. Introduction The monoalkyl ethers of poly(oxyethylene) glycols, CmEn (CmH2m+1(OC2H4)nOH), form an important class of nonionic surfactants with a wide range of technological applications, such as detergency, cosmetics, pharmacy, and so forth.1 These surfactants exhibit a rich variety of phases in aqueous solution. Depending on the length of the alkyl and poly(oxyethylene) chains (m,n), temperature, and concentration, they form micellar, hexagonal, lamellar, and cubic phases.1-6 Micelles are aggregates of surfactant molecules in aqueous as well as nonpolar solvents and are widely studied both experimentally and theoretically. In aqueous solution, above the critical micelle concentration (cmc), surfactants form quasi-spherical “regular” micelles, where the headgroups or the hydrophilic parts of the molecules are arranged on the surface of the sphere and come in contact with the aqueous solvent, while the hydrophobic parts form the core of the micelle. In nonpolar solvents, surfactants form “reverse” micelles, where the hydrophobic and hydrophilic parts of the molecules reverse their arrangement compared to regular micelles. Both regular and reverse micelles occupy significant portions of the surfactant/oil/water phase diagram.7 Regular micelles act as detergents for oils in aqueous solution, whereas the reverse micelles act as detergents for polar components in a nonpolar medium. The study of the properties of micelles is important from a fundamental point of view in understanding the phenomenon of molecular self-assembly. Relatively few studies have been made of the structure of the reverse micelles despite their potential importance in a wide range of applications such as oil recovery7 and protein extraction from aqueous solution8 and as media * To whom correspondence should be addressed. Telephone: (215) 573-4773.Fax: (215)573-6233.E-mail:
[email protected]. (1) Laughlin, R. G. The Aqueous Phase Behavior of Surfactants; Academic Press: New York, 1994. (2) Mitchell, D. J.; Tiddy, G. J. T.; Waring, L.; Bostock, T.; McDonald, M. P. J. Chem. Soc., Faraday Trans. 1 1983, 79, 975-1000. (3) Roux, D.; Coulon, C.; Cates, M. E. J. Phys. Chem. 1992, 96, 41744187. (4) Fairhurst, C. E.; Holmes, M. C.; Leaver, M. S. Langmuir 1997, 13, 4964-4975. (5) Funari, S. S.; Holmes, M. C.; Tiddy, G. J. T. J. Phys. Chem. 1994, 98, 3015-3023. (6) Funari, S. S.; Rapp, G. J. Phys. Chem. B 1997, 101, 732-739. (7) Luisi, P. L., Straub, B. E., Eds. Reverse Micelles; Plenum Press: New York, 1984.
for catalytic reactions.9 Reverse micellar solutions of the more well-known, ionic, surfactant AOT (sodium di-2ethylhexylsulfosuccinate) have been characterized by small-angle neutron scattering,10,11 fluorescence spectroscopy,12,13 NMR spectroscopy,14 and ESR spectroscopy.15 ESR15 and acoustic and densimetric16 studies on reverse micelles of the nonionic surfactant C12E4 have provided some insights into the degree of hydration of the headgroups and the penetrability of the micelle to solvent. Since hydration of the headgroups is believed to be very important in dictating the phase behavior of the surfactant, information on the structure of the water in the micellar core region is essential. There have as yet been few simulation studies of nonionic reverse micellar systems. Tobias and Klein17 simulated a calcium carbonate/calcium sulfonate ionic reverse micelle in atomistic detail in various solvents as well as in a vacuum, while Watanabe and Klein18 studied shape fluctuations in a sodium octanoate regular micelle. Other simulation studies of reverse micelles19,20 have also focused on ionic systems. Recently, Cummings and coworkers21 reported MD studies of the formation of reverse micelles of surfactants with two different tails (hybrid surfactants) in supercritical carbon dioxide as the nonpolar solvent. They predicted that reverse micellization in CO2 occurs on a much faster time scale than that for micelles in liquid solvents. In this paper we will focus our attention on reverse or inverted micelles formed by the surfactant C12E2 in (8) Luisi, P. L. Angew. Chem., Int. Ed. Engl. 1985, 24, 439-450. (9) Leong, Y. S.; Candau, F. J. Phys. Chem. 1982, 86, 2269-2271. (10) Chen, S. H. Annu. Rev. Phys. Chem. 1986, 37, 351-399. (11) Kotlarchyk, M.; Huang, J. S. J. Phys. Chem 1985, 89, 43824386. (12) Zhang, J.; Bright, F. V. J. Phys. Chem. 1991, 95, 7900-7907. (13) Karukstis, K. K.; Frazier, A. A.; Martula, D. S.; Whiles, J. A. J. Phys. Chem. 1996, 100, 11133-11138. (14) Schwartz, L. J.; DeCiantis, C. L.; Chapman, S.; Kelley, B. K.; Hornak, J. P. Langmuir 1999, 15, 5461-5466. (15) Caldararu, H. Spectrochim. Acta A 1998, 54, 2309-2336. (16) Amararene, A.; Gindre, M.; Hue´rou, J.-Y.; Nicot, C.; Urbach, W.; Waks, M. J. Phys. Chem. B 1997, 101, 10751-10756. (17) Tobias, D. J.; Klein, M. L. J. Phys. Chem. 1996, 100, 6637-6648. (18) Watanabe, K.; Klein, M. L. J. Phys. Chem. 1989, 93, 68976901. (19) Linse, P. J. Chem. Phys. 1989, 90, 4992-5004. (20) Brown, D.; Clark, J. H. R. J. Phys. Chem. 1988, 92, 2881-2888. (21) Salaniwal, S.; Cui, S. T.; Cummings, P. T.; Cochran, H. D. Langmuir 1999, 15, 5188-5192.
10.1021/la001182d CCC: $19.00 © 2000 American Chemical Society Published on Web 11/29/2000
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nonpolar solvents. It has been observed by Funari and Rapp6 using time-resolved X-ray diffraction and more recently by Lynch et al.22 using the diffusive interfacial transport-near-infrared method (DIT-NIR) that, depending on the temperature and composition, C12E2 forms different phases in aqueous solution. These phases include isotropic solution (L1), sponge phase (L3), Lamellar phase (D), concentrated liquid (L2), and two different bicontinuous cubic phases (V2(1), V2(2)).22 At high concentrations of the surfactant the “L2” phase, which is optically isotropic and of low viscosity, forms. Although the detailed nature of the structure of this “L2” phase is not clear, it has been interpreted as consisting of surfactant aggregates with their hydrophobic tails pointing outward and their headgroups solvated by a water core, which resembles a reverse micelle type of structure. We recently reported a simulation study of the lamellar (D) phase of C12E2 in aqueous solution.23 It was observed that there is a strong tendency for water molecules to form H-bonded “bridges” between adjacent oxygen atoms of the same surfactant chain, which leads to a dominance of the gauche configuration over the trans about the C-C bond between the two oxygens. Similar water-bridged structures of C12E2 surfactant were also observed by Kong et al.24,25 In this article we report an atomistic molecular dynamics (MD) simulation of a reverse micelle of C12E2 in the nonpolar solvent decane. Since reverse micelles are normally formed and function in oils that are complex hydrocarbons, it will be interesting to see how the presence of decane affects the structure of the micelle. To facilitate our understanding, we have also carried out a simulation of an isolated micelle (i.e. in a vacuum) and compared its structure with that of the micelle in decane. In the next section we discuss the potential model, the system setup, and the simulation details. This will be followed by the results obtained from our simulations and their interpretation. II. System Setup and Simulation Details The initial configuration of the micelle was generated by arranging 50 surfactant molecules in their extended chain (all trans) configuration, pointing radially outward and placed randomly with the head OH groups of all the monomers constrained to lie on the surface of a sphere of radius 10 Å. The core of the micelle was filled with a sphere of water (containing 119 molecules) obtained from a wellequilibrated configuration of bulk water. This corresponds to a mass fraction of 86.4% aqueous solution, which at a temperature of 300 K corresponds to the “L2” phase of the surfactant.6,22 A short MD run of 2 ps was first performed at 400 K, keeping the surfactant headgroups and the core water molecules fixed, to randomize the hydrocarbon chain conformations. The micelle was then inserted into a large cubic box of equilibrated decane. The dimension of the box was chosen to be 100 Å, large enough to have sufficient bulk hydrocarbon solvent between the micelle and its periodic images. A further short MD run of 16 ps was performed at constant volume and at a high temperature of 1000 K (NVT), keeping the headgroups and the core water molecules frozen, to randomize the hydrocarbon chains of the surfactants and the decane molecules. At (22) Lynch, M. L.; Kochvar, K. A.; Burns, J. L.; Laughlin, R. G. Langmuir 2000, 16, 3537-3542. (23) Bandyopadhyay, S.; Tarek, M.; Lynch, M. L.; Klein, M. L. Langmuir 2000, 16, 942-946. (24) Kong, Y. C.; Nicholson, D.; Parsonage, N. G.; Thompson, L. J. Chem. Soc., Faraday Trans. 1994, 90, 2375-2380. (25) Kong, Y. C.; Nicholson, D.; Parsonage, N. G. Mol. Phys. 1996, 89, 835-865.
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this point the headgroups and the core water molecules were unfrozen and the temperature of the system was lowered to 20 K. The temperature of the system was slowly increased to 300 K over the next 50 ps. The resulting configuration was then equilibrated at constant temperature (T ) 300 K) and pressure (P ) 0) for about 470 ps. This equilibration period was followed by an NPT production run of 1.74 ns in a flexible simulation box with an orthorhombic angular constraint.26 A well equilibrated configuration of the micelle in decane was taken to initiate the vacuum simulation of the micelle. This simulation was carried out at constant volume and temperature (NVT) for approximately 300 ps (including a 100 ps equilibration period). The simulations utilized the Nose´-Hoover chain thermostat extended system method26 as implemented in the PINY-MD computational package.27 A recently developed reversible multiple time step algorithm26 allowed us to employ a 6 fs MD time step. This was achieved using a three-stage force decomposition into intramolecular forces (torsion/bend-bond), short-range intermolecular forces (a 7.0 Å RESPA cutoff distance), and long-range intermolecular forces. Electrostatic interactions were calculated by using the particle-mesh Ewald (PME) method.28 The minimum image convention29 was employed to calculate the Lennard-Jones interactions and the real-space part of the Ewald sum using a spherical truncation of 7.0 and 10 Å, respectively, for the short- and the long-range parts of the RESPA decomposition. “SHAKE/ROLL” and “RATTLE/ROLL” methods26 were implemented to constrain all bonds involving H atoms to their equilibrium values. The intermolecular potential model was based on pairwise additive site-site electrostatic and LennardJones contributions. The rigid three-site SPC/E model30 was employed for water. The CH3 and CH2 groups of the surfactants were treated as united atoms (i.e. these groups were represented by single interaction sites). The potential parameters for the alkane chain groups were taken from the work of Martin and Siepmann,31 while the EO groups were modeled using the OPLS parameters.32,33 The surfactant chains were made flexible by including bond stretching, bending, and torsion interactions. III. Results and Discussion A. Structural Properties. In Figure 1 we show snapshots of the configurations of the micelle near the beginning (a) and at the end of the simulation in decane (b) and in a vacuum (c). The initial configuration (Figure 1a) shows the ordered arrangement of the core as well as the surfactant tails, which is due to the way the micelle was constructed. The micelle lost the initial ordering during the simulation, as is evident from the snapshot taken at the end of the run (Figure 1b). It is also apparent from the figure that in the course of the simulation the structure of the micelle has undergone some distortions (26) Martyna, G. J.; Tuckerman, M. E.; Tobias, D. J.; Klein, M. L. Mol. Phys. 1996, 87, 1117-1157. (27) Tuckerman, M. E.; Yarne D. A.; Samuelson, S. O.; Hughes, A. L.; Martyna, G. J. Comput. Phys. Commun. 2000, 128, 333-376. (28) Darden, T.; York, D.; Pedersen, L. J. Chem. Phys. 1993, 98, 10089-10092. (29) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Clarendon: Oxford, 1987. (30) Berendsen, H. J. C.; Grigera, J. R.; Straatsma, T. P. J. Phys. Chem. 1987, 91, 6269. (31) Martin, M. G.; Siepmann, J. I. J. Phys. Chem. B 1998, 102, 2569-2577. (32) Jorgensen, W. L. J. Phys. Chem. 1986, 90, 1276-1284. (33) Briggs, J. M.; Matsui, T.; Jorgensen, W. L. J. Comput. Chem. 1990, 11, 958-971.
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Figure 2. Number density profiles of different components of the micelle, measured with respect to the center of mass of the core: (a) decane simulation; (b) vacuum simulation. The density profiles from our bilayer simulation23 are shown in part c, where the densities are measured from the center of the bilayer in the direction normal to the bilayer plane (z).
Figure 1. Snapshots of the configuration of the system near the beginning (a) and at the end (b) of the simulations in decane and in a vacuum (c). The solvent decane molecules are not shown for visual clarity. The atom coloring scheme is as follows: CH2/ CH3, gray; O, red; H, dark green.
from its initial ordered structure, with significant penetration of the hydrocarbon solvent (decane) into the hydrophobic surfactant tail region (decane is not shown in the figure for clarity). We will discuss the shape fluctuations of the micelle in detail later. In a vacuum the shape of the micelle looks more compact, as the hydro-
carbon chains of the surfactants have collapsed onto the surface of the core (see Figure 1c), thereby shielding its exposure. The structural properties of the micelle have been characterized using various distribution functions. Figure 2a shows the average number density profiles of the terminal hydroxyl groups (OH), oxyethylene groups (EO), hydrocarbon chain atoms, the core water molecules, and decane solvent carbon atoms, measured with respect to the center of mass of the micelle. This distribution shows the roughness of the core structure with significant penetration of the surfactant headgroups toward the center of the core. On the other hand, the hydrocarbon tails of the surfactants are predominantly directed away
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from the micelle core and mostly extend into the decane solvent. Figure 2b shows the density profiles of the different components of the micelle from the vacuum simulation. The density profiles of the core water molecules and the surfactant headgroups are qualitatively similar in both decane and vacuum simulations. This suggests that the presence of a nonpolar solvent does not have any significant influence on the structure of the polar core and the surfactant headgroups of the micelle. In contrast, the density profile of the surfactant tail hydrocarbons obtained from the vacuum simulation is significantly different from that obtained from the decane simulation. In the absence of the solvent, the peak density of the tail carbon atoms has shifted inward toward the micelle core, which was also clearly evident in Figure 1c. Therefore, Figure 2 shows that although the vacuum simulation can accurately reproduce the solution structure of the micelle core and the surfactant headgroups, the presence of an explicit nonpolar solvent is required to obtain a realistic description of the hydrophobic micelle exterior. Similar behavior was observed by Tobias and Klein17 in their simulation of a calcium carbonate/calcium sulfonate ionic reverse micelle. For comparison, in Figure 2c we show the density profiles of different components from our bilayer simulation.23 It is clear from these figures that the interfacial structures are quite similar in lamellar (D) and reverse micellar (L2) phases. Further insight into the effect of the solvent on the hydrophobic micelle exterior can be obtained from the probability radial distribution function of the surfactant hydrocarbon tail carbons with respect to the center of mass of the micelle. Figure 3 shows these distributions for Ct1, the closest carbon of the tail to the EO headgroups, Ct6, the midchain carbon, and Ct12, the terminal carbon atom. It is clear from the figure that the distribution of Ct1 is quite similar in both the simulations but there is a significant difference as one proceeds toward the end of the hydrocarbon tail. The distributions of Ct6 and Ct12 are much broader in the presence of the solvent (Figure 3a), which shows the influence of a nonpolar solvent in delocalizing the hydrocarbon exterior of the micelle. B. Micelle Shape. It is clear from Figures 1 and 2 that the structure and shape of the micelle have undergone deformations during the course of the simulations, as compared to the initial configuration. To quantify the extent of the deviation of the micelle from sphericity, we measured the time evolution of the eccentricity, η, defined as
η)1-
Imin Iavg
Figure 3. Probability distribution functions of the first C atom (Ct1), the midchain C atom (Ct6), and the terminal carbon atom (Ct12), measured with respect to the center of mass of the core: (a) decane simulation; (b) vacuum simulation.
(1)
where Imin is the smallest and Iavg is the mean of the principal moments of inertia. Therefore, η f 0 corresponds to spherical objects. The time evolution of the eccentricity of the micelle in decane is shown in Figure 4. It is evident from the figure that both the micelle and the core are significantly deformed from their initial spherical shape. We observed that the fluctuations in decane have longer wavelengths than those in a vacuum, suggesting the slow rearrangement of the micelle in the presence of a solvent. This behavior is similar to that observed previously for a calciun carbonate/calcium sulfonate reverse micelle.17 The relaxation time scale for the shape fluctuations of the micelle, as calculated from the time correlation function 〈η(t)η(0)〉, was found to be of the order of 150 ps, showing
Figure 4. Time evolution of the micelle (solid line) and core (dotted line) eccentricities in decane.
much slower motion of reverse micelles in nonpolar solvents than of regular micelles in aqueous solutions.18,34,35 C. Surfactant Conformations and SurfactantWater Interactions. To obtain a more complete micro(34) Shelley, J. C.; Sprik, M.; Klein, M. L. Langmuir 1993, 9, 916926. (35) Bandyopadhyay, S.; Klein, M. L.; Martyna, G. J.; Tarek, M. Mol. Phys. 1998, 95, 377-384.
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Figure 5. Representative snapshot of a water molecule forming a bridged H-bonded structure with the oxygen atoms of the oxyethylene groups of a surfactant chain. The hydrocarbon chain atoms, oxygens, and hydrogen atoms are drawn as gray, red, and dark green spheres while the H-bonds are shown as green lines. Note that the headgroup atoms of the chain are labeled for clarity. Table 1. Percentage Population of Gauche Bonds in the Oxyethylene Part of the Surfactant Headgroup bond
% gauche (decane)
% gauche (vacuum)
C1-C2 C2-O2 O2-C3 C3-C4 C4-O3 O3-C5
12 48 20 99 24 14
14 49 20 99 28 12
scopic understanding of the micelle structure, it is crucial to investigate in detail the influence of the core water molecules on the conformational properties of the surfactant headgroups. In Figure 5 we show a typical configuration of a surfactant chain taken from our simulation. We measured the relative conformational population of the torsional angles around the individual bonds in the oxyethylene part of the surfactant chain. These are listed in Table 1. It is clear from the table that there is an exclusive preference for the C3-C4 bond to remain in the gauche form. In addition, the two bonds adjacent to the C3-C4 bond, namely O2-C3 and C4-O3 bonds, have a strong preference for the trans conformation. About 75-80% of these two bonds were found to be trans. We observed, as shown in Figure 5, that there is a strong tendency for the core water molecules to form hydrogen bonds bridging between the two adjacent oxygen atoms of the oxyethylene groups of a surfactant chain. Such a tendency to form the hydrogen-bonded bridged structure is favored only if the C3-C4 bond assumes a gauche conformation and the adjacent C3-O2 and C4-O3 bonds become trans. Table 1 clearly suggests that this is the case in our simulations. It is also evident from the table that the conformational behavior of the oxyethylene group in a vacuum is similar to that in decane, which once again shows that there is a minimum influence of the solvent on the structure of the micelle core. We recently observed similar conformational properties of the oxyethylene headgroups of C12E2 and the existence of hydrogen-bonded water-bridged structures in its liquid crystalline lamellar phase.23 Therefore, the present work, in conjunction with our previous studies, suggests that the CmEn surfactants exhibit such characteristic conformational properties of the oxyethylene headgroups. We believe that this is a fundamental property of this class of surfactants and is independent of the nature of the phase or the concentration of the surfactant. Similar conformational behaviors and hydrogen-bonded water-bridged structures were also observed in poly(ethylene glycol) polymers (PEG)36 and poly(ethylene oxide) (PEO) solutions,37 as well as in charged surfactant monolayers at an interface.38,39 (36) Heymann, B.; Grubmu¨ller, H. Chem. Phys. Lett. 1999, 307, 425432. (37) Karlstro¨m, G. J. Phys. Chem. 1985, 89, 4962-4964.
Figure 6. Radial distribution functions of the headgroup atoms of the surfactant with water oxygen atoms in decane. The surfactant atoms are labeled as shown in Figure 5.
To obtain more insight into the structure of core water around the surfactant headgroups, radial distribution functions, g(r), of different headgroup atoms with water were calculated for the micelle in decane and displayed in Figure 6. A strong first peak is observed around 2.8 Å for the hydroxyl group oxygen atoms (O1), which corresponds to the nearest neighbor distance between oxygen atoms. The intensity of this peak decreases for the oxyethylene group oxygen atoms (O2 and O3), as they are further from the core water. There is a broad second peak with sufficient intensity for O2, which accounts for the water molecules hydrogen bonded to O1. We estimated the number of water molecules that are nearest neighbors to O1, O2, and O3 by integrating the corresponding g(r) distribution up to ∼3.5 Å. The g(r) curves in a vacuum (not shown here) were found to be identical to those in solvent, as the structure of the micelle core remains mainly unaffected by the solvent. It was found that, on average, there are about 1.7 water molecules per oxygen atom of the hydroxyl group (O1) and 0.4 per oxygen atom of the oxyethylene groups (O2 and O3). These results are similar to those observed for C12E2 in the lamellar phase.23 D. Dynamics of Core Water. To examine the dynamical behavior of the water molecules present in the core of the micelle, we evaluated their average meansquared displacements (〈u2(t)〉), which are displayed in Figure 7. The diffusion coefficient (D) of the water molecules was obtained from the slope of the mean-squared displacement using the well-known Einstein’s relation29
D)
〈u2(t)〉 2dt
(2)
where d is the dimensionality of the system (in this case d ) 3). The calculated value of the diffusion coefficient (D) was found to be 5.4 × 10-10 m2 s-1, which is about five times smaller than the value for pure SPC/E water (2.5 × 10-9 m2 s-1 at 300 K30). This suggests that the core water molecules mostly exhibit local motion that is significantly constrained compared to that for bulk water. Such a reduction in mobility of the water molecules arises due to their strong interactions with the surfactant headgroups, as discussed in the previous section. A rough (38) Schweighofer, K. J.; Essmann, U.; Berkowitz, M. J. Phys. Chem. B 1997, 101, 3793-3799. (39) Schweighofer, K. J.; Essmann, U.; Berkowitz, M. J. Phys. Chem. B 1997, 101, 10775-10780.
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Figure 7. Time evolution of the mean-squared displacements of the core water molecules. The diffusion coefficient (D) value was estimated from a linear fit between 100 and 300 ps.
estimation showed the average residence time for a water molecule bound to a surfactant headgroup oxygen atom to be about 50 ps. IV. Conclusion In this paper we presented the microscopic properties of a reverse micelle of the nonionic surfactant C12E2 in a
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nonpolar solvent, namely decane, using MD simulations. To investigate the effect of the solvent on the structure of the micelle, we also carried out another simulation in a vacuum. It is observed that the overall structure of the micelle core in a vacuum is similar to that in decane. However, the surfactant tails at the exterior of the micelle exhibit different behavior in a vacuum. In the absence of a solvent, the tails collapse on the surface of the core, thus forming a more compact micelle. The shape of the micelle was found to undergo shape fluctuations with a characteristic relaxation time scale of the order of 150 ps. The conformational properties of the surfactant headgroups and the arrangement of water molecules around them were also investigated in detail. We observed that there is a strong tendency for the core water molecules to form hydrogen-bonded bridged structures between the two adjacent oxyethylene group oxygen atoms. This leads to an exclusive preference for headgroup conformations with gauche C-C bonds. These findings coupled with our previous studies on the liquid crystalline lamellar phase of C12E2 show that such conformational behaviors are important characteristics of this CmEn class of surfactants and possibly other related compounds. Acknowledgment. This work was supported by generous grants from the Procter & Gamble Company and the National Science Foundation. We thank Matt Lynch for useful discussions. LA001182D
