Molecular Dynamics Simulations of Model Perhydrogenated and

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Molecular Dynamics Simulations of Model Perhydrogenated and Perfluorinated Alkyl Chains, Droplets, and Micelles Donghai Mei and John P. O’Connell* Department of Chemical Engineering, University of Virginia, Charlottesville, Virginia 22904 Received February 19, 2002. In Final Form: August 21, 2002 Molecular dynamics simulations have been performed to study chain conformation and internal structure for three models of n-alkanes (PHA) and n-perfluoroalkanes (PFA). These systems include single chains of 7-29 identical segments in a solvent of segments, “oil” droplets (aggregates of 24 chains), and micelles (experimental mean-sized aggregates of 24 octyl PHA or 40 perfluoroheptyl PFA chains with “nonionic sulfate head” groups attached to one end). Three different united-atom potential models of PHA chains and five models of PFA chains have been used to study the effect of chain potential. A computationally efficient solvent force field represented segment and headgroup interactions with an aqueous solvent and maintained aggregate sizes and shapes. For single chains, the overall trans bond fraction and distribution were independent of chain length while the scaling exponent for end-to-end distances for longer chains agrees with theoretical scaling laws. Several different statistical analyses showed that the conformational and structural properties for chains in droplets and micelles are similar except for slightly higher trans bond fractions for the PFA chains. The bond orientation parameters of micelles had preferential nearsurface orientations but were random in the core. Aggregate shapes were changed in ellipticity from 0.5 (oblate) to 2.0 (prolate) but the only effect was a small change of bond orientation parameter in the PHA micelles.

1. Introduction The study of structural and thermophysical properties of complex systems consisting of flexible chain molecules is of fundamental and applied interest. Linear perhydrogenated n-alkanes (PHA) and their counterpart perfluorinated n-alkanes (PFA) have received the most attention because they are of practical interest in their own in solutions as well as are the building blocks of functional chain substances such as surfactants, biologicals, and polymers.1-7 With the rapid development of computational hardware and resources available, computer molecular simulation has been becoming a standard research tool in obtaining useful and valuable information related to the relationships between microscopic structure and macroscopic properties. In particular, many molecular simulations of PHAs,8-16 PFAs,17,18 and their mixtures with carbon * To whom correspondence should be addressed. E-mail: jpo2x@ virginia.edu. (1) Mittal, K. L.; Lindman, B. Surfactants in Solution; Plenum: New York, 1984; Vol. 1. (2) Zana, R. Surfactant Solutions; Dekker: New York, 1987. (3) Israelachvili, J. N. Intermolecular and Surface Forces, 2nd ed.; Academic Press: New York, 1991. (4) Kissa, E. Fluorinated Surfactants; Dekker: New York, 1994. (5) Evans, D. F.; Wennerstrom, H. The Colloidal Domain; VCH Publishers: New York, 1994. (6) Hiemenz, P. C.; Rajagopalan, R. Principles of Colloid and Surface Chemistry, 3rd ed.; Dekker: New York, 1997. (7) Shah, D. O. Micelles, Microemulsions and Monolayers; Dekker: New York, 1998. (8) Siepmann, J. I.; Karaborni, S.; Smit, B. Nature 1993, 365, 330. (9) Siepmann, J. I.; Karaborni, S.; Smit, B. J. Am. Chem. Soc. 1993, 115, 6454. (10) Siepmann, J. I.; Martin, M. G.; Mundy, C. J.; Klein, M. L. Mol. Phys. 1997, 90, 687. (11) Martin, M. G.; Siepmann, J. I. J. Am. Chem. Soc. 1997, 119, 8921. (12) Chen, B.; M. G. Martin, M. G.; Siepmann, J. I. J. Phys. Chem. B 1998, 102, 2578. (13) Martin, M. G.; Siepmann, J. I. J. Phys. Chem. B 1998, 102, 2569. (14) Martin, M. G.; Siepmann, J. I. Theor. Chem. Acc. 1998, 99, 347.

dioxide19 have been performed in recent years. These simulations have mainly focused on developing inter- and intramolecular potential models for phase equilibria, critical behavior and thermodynamic properties. The principal methods have been Gibbs-ensemble Monte Carlo (MC) and configurational-bias MC simulation methods though some simulations treat chain flexibility and dynamics.20-22 Toxvaerd23-26 and Ungerer and co-workers27,28 have carried out molecular dynamics (MD) simulations of PHA systems in order to investigate the effects of different chain potential models on chain conformation, fluid structure and thermophysical properties. Some studies of single PHA chains immersed in solvent segments have been appeared in the literature since the first work done by Rapaport.29 Most of these investigate the effects of density and nature of the solvent segments, and of the form of the interaction potentials on static and dynamic conformational properties.30-36 In particular, Luque et al.37 (15) Martin, M. G.; Siepmann, J. I. J. Phys. Chem. 1999, 103, 4508. (16) Delhommelle, J.; Boutin, A.; Tavitian, B.; Mackie, A. D.; Fuchs, A. H. Mol. Phys. 1999, 96, 1517. (17) Siepmann, J. I.; Karaborni, S.; Smit, B.; Klein, M. L. AIChE Spring National Meeting, 1994; Paper 94c. (18) Cui, S. T.; Siepmann, J. I.; Cochran, H. D.; Cummings, P. T. Fluid Phase Equilib. 1998, 146, 1251. (19) Cui, S. T.; Cochran, H. D.; Cummings, P. T. J. Phys. Chem. B 1999, 103, 4485. (20) Szczepanski, R.; Maitland, G. C. Adv. Chem. Ser. 1983, 204, 469. (21) Banon, A.; Adan, F. S.; Santamaria, J. J. Chem. Phys. 1985, 83, 297. (22) Steele, D. J. Chem. Soc., Faraday Trans 2 1985, 81, 1077. (23) Toxvaerd, S. J. Chem. Phys. 1987, 87, 6140. (24) Toxvaerd, S. J. Chem. Phys. 1988, 89, 3808. (25) Padilla, P.; Toxvaerd, S. J. Chem. Phys. 1991, 94, 5650. (26) Toxvaerd, S. J. Chem. Phys. 1997, 107, 5197. (27) Ungerer, P.; Beauvais, C.; Delhommelle, J.; Boutin, A.; Rousseau, B.; Fuchs, A. H. J. Chem. Phys. 2000, 112, 5499. (28) Delhommelle, J.; Tschirwitz, C.; Ungerer, P.; Grannucci, G.; Millie, P.; Pattou, D.; Fuchs, A. H. J. Phys. Chem. B 2000, 104, 4745. (29) Rapaport, D. C. J. Chem. Phys. 1979, 71, 3299. (30) Bigot, B.; Jorgensen, W. L. J. Chem. Phys. 1981, 75, 1944. (31) Bruns, W.; Bansal, R. J. Chem. Phys. 1981, 74, 2064.

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have studied the effect of chain length on chain conformations. The scaling exponent for measures of end-toend distances they obtained from MD simulations of linear chain molecules ranging from 8 to 20 segments were about 0.4, which should be compared to the theoretical value38 of 0.6 for a single chain monomer in good solvent according to scaling laws. Karaborni and O’Connell39 found a linear dependence of PHA chain length on end-to-end distance and radius of gyration for hydrocarbon chain with segments from 7 to 21. To our knowledge, no simulation of chain conformational characteristics and aggregate structure for PFA systems has been reported. Analogous to PHA and PFA chain molecules are surfactant molecules which contain a hydrophilic headgroup and an alkyl or fluoroalkyl tail composed of linked segments. These species can form a wide variety of aggregates in solutions and at interfaces such as micelles, monolayers, bilayers and vesicles. Their amphiphilic nature gives them wide application such as in detergency, pharmaceuticals, enhanced oil recovery, catalysis, and many other technological domains. Considerable efforts in experimental and theoretical modeling have been done for micellar systems in the past.1,4-7 Advanced spectroscopic techniques such as small-angle neutron scattering (SANS), as well as nuclear magnetic resonance (NMR), have been used to study the internal packing structures, shapes, and fluctuations of micelle and the growth of micelles,40-62 Often, the most interesting microscopic (32) Bruns, W.; Bansal, R. J. Chem. Phys. 1981, 75, 5149. (33) Joen, S. H.; Oh, I. J.; Ree, T. J. Phys. Chem. 1983, 87, 2890. (34) Khalatur, P. G.; Papulov, Y. G.; Pavlov, A. S. Mol. Phys. 1986, 58, 887. (35) Smit, B.; Cox, K. R.; Michels, J. P. J. Mol. Phys. 1989, 66, 97. (36) Smit, B.; van der Put, A.; Peters, C. J.; de Swaan Arons, J. J. Chem. Phys. 1988, 88, 3372. (37) Luque, J.; Santamaria, J.; Freire, J. J. J. Chem. Phys. 1989, 91, 584. (38) de Gennes, P. G.Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, NY, 1979. (39) Karaborni, S.; O’Connell, J. P. J. Chem. Phys. 1990, 92, 6190. (40) Hoffmann, H.; Tagesson, B. Z. Phys. Chem. 1978, 110, 113. (41) Wennerstrom, H.; Lindmann, B.; Soderman, O.; Drakenberg, T.; Rosenholm, J. B. J. Am. Chem. Soc. 1979, 101, 6860. (42) Hayter, J. B.; Zemb, T. Chem. Phys. Lett. 1982, 93, 91. (43) Hoffmann, H.; Kalus, J.; Reizlein, K.; Ulbricht, W.; Ibel, K. Colloid Polym. Sci. 1982, 260, 435. (44) Bendedouch, D.; Chen, S.-H.; Koehler, T. H. J. Phys. Chem. 1983, 87, 153. (45) Zemb, T.; Drifford, M.; Hayoun, M.; Jehanno, A. J. Phys. Chem. 1983, 87, 4524. (46) Walderhaug, H.; Soderman, O.; Stilbs, P. J. Phys. Chem. 1984, 88, 1655. (47) Cabane, B.; Duplessix, R.; Zemb, T. J. Phys. (Paris) 1985, 46, 2161. (48) Chevalier, Y.; Chachaty, C. J. Phys. Chem. 1985, 89, 875. (49) Holmes, M. C.; Leaver, M. C.; Smith, A. M. Langmuir 1985, 11, 356. (50) Burkitt, S. J.; Ottewill, R. H.; Hayter, J. B.; Ingram, B. T. Colloid Polym. Sci. 1987, 265, 619, 628. (51) Berr, S. S.; Jones, R. R. M. J. Phys. Chem. 1989, 93, 2555. (52) Caponetti, E.; Causi, S.; De Lisi, R.; Floriano, M. A.; Milioto, S.; Triolo, R. J. Phys. Chem. 1992, 96, 4950. (53) Caponetti, E.; Martino, D. C.; Floriano, M. A.; Triolo, R. Langmuir 1993, 9, 1193. (54) Clapperton, R. M.; Ottewill, R. H.; Ingram, B. T. Langmuir 1994, 10, 51. (55) Hirata, H.; Hattori, N.; Ishida, M.; Okabayashi, H.; Frusaka, M.; Zana, R. J. Phys. Chem. 1995, 99, 17778. (56) Iijima, H.; Kato, T.; Seimiya, T.; Yoshida, H.; Imai, M. Prog. Colloid Polym. Sci. 1997, 106, 61. (57) Almgren, M.; Wang, K.; Asakawa, T. Langmuir 1997, 13, 4535. (58) Ito, A.; Kamogawa, K.; Sakai, H.; Hamano, K.; Kondo, Y.; Yoshino, N.; Abe, M. Langmuir 1997, 13, 2935. (59) Damas, C.; Naejus, R.; Coudert, R.; Frochot, C.; Brembilla, A.; Viriot, M. L. J. Phys. Chem. B 1998, 102, 10917. (60) Iijima, H.; Koyama, S.; Fujio, K.; Uzu, Y. Bull. Chem. Soc. Jpn. 1999, 72, 171. (61) Furo, I.; Sitnikov, R. Langmuir 1999, 15, 2669. (62) Lopez-Fontan, J. L.; Suarez, M. J.; Mosquera, V.; Sarmiento, F. Phys. Chem. Chem. Phys. 1999, 1, 3853.

Mei and O’Connell

information must involve interpretation of models of the experimental measurements resulting in limited, fragmented, and even controversial understanding of structure and dynamics. For example, Berr and Jones51 have determined the charge and aggregation number in the micelle by SANS measurements of aqueous sodium perfluorooctanoate solutions as a function of surfactant concentration and suggested a linear relationship between aggregation number and surfactant concentration above the critical micelle concentration (cmc). They proposed that a spherical micelle with a water-free fluoroalkyl core would be formed in the aqueous surfactant solutions. Meanwhile, Caponetti et al.53 concluded that the aggregation number increases as a one-fourth power of the micellized surfactant concentration and found that the same SANS experimental data are better explained and fitted using a monodisperse ellipsoid model. Thus, important molecular structure aspects of micelles may not unambiguously be obtained by experiment alone. Computer simulation also makes it possible to investigate models of micellar systems at the molecular level. Over the past 2 decades, a number of molecular dynamics simulations have been performed to investigate the structure, shape, and fluctuations of the aggregates, the chain conformations, and dynamical properties, the penetration of water molecules into the hydrophobic core, the coupling of counterions to the micelle exterior, and the nature of hydrophobic interactions.63-92 While simplified models are used87,89 for micelle formation, monomer exchange, and distribution of micelle numbers in real systems, detailed potential models with femtosecond time (63) Haile, J. M.; O’Connell, J. P. J. Phys. Chem. 1984, 88, 6363. (64) Woods, M. C.; Haile, J. M.; O’Connell, J. P. J. Phys. Chem. 1986, 90, 1875. (65) Jonsson, B.; Edholm, O.; Teleman, O. J. Chem. Phys. 1986, 85, 2259. (66) Watanabe, K.; Ferrario, M.; Klein, M. L. J. Phys. Chem. 1988, 92, 819. (67) Watanabe, K.; Klein, M. L. J. Phys. Chem. 1989, 93, 6897. (68) Wendoloski, J. J.; Kimatian, S. J.; Schutt, C. E.; Salemme, F. R. Science 1989, 243, 636. (69) Karaborni, S.; O’Connell, J. P.J. Phys. Chem. 1990, 94, 2624. (70) Karaborni, S.; O’Connell, J. P. Langmuir 1990, 6, 905. (71) Shelley, J.; Watanabe, K.; Klein, M. L. Int. J. Quantum Chem.: Quantum Bio. Symp. 1990, 17, 103. (72) Karaborni, S.; O’Connell, J. P. In Surfactants in Solution; Mittal, K. L., Shah, D. O., Eds.; Plenum: New York, 1991; Vol. 11, p 83. (73) Shelley, J.; Watanabe, K.; Klein, M. L. Electrochim. Acta 1991, 36, 1729. (74) Shelley, J.; Sprik, M.; Klein, M. L. Langmuir 1993, 9, 916. (75) Karaborni, S.; O’Connell, J. P. Tenside Surf. Det. 1993, 30, 235. (76) Bocker, J.; Brickmann, J.; Bopp, P. J. Phys. Chem. 1994, 98, 712. (77) Rusling, J. F.; Kumosinski, T. F. J. Phys. Chem. 1995, 99, 9241. (78) MacKerell, A. D. J. Phys. Chem. 1995, 99, 1846. (79) Bast, T.; Hentschke, R. J. Phys. Chem. 1996, 100, 12162. (80) Kuhn, H.; Rehage, H. Ber. Bunsen-Ges. Phys. Chem. 1997, 101 1493. (81) Kuhn, H.; Rehage, H. Ber. Bunsen-Ges. Phys. Chem. 1997, 101 1485. (82) Kuhn, H.; Breitzke, B.; Rehage, H. Colloid Polym. Sci. 1998, 276, 824. (83) Kuhn, H.; Rehage, H. Prog. Colloid Polym. Sci. 1998, 111, 158. (84) Wymore, T.; Gao, X.-F.; Wong; T. C. J. Mol. Struct. 1999, 485, 195. (85) Floriano, M. A.; Caponetti, E.; Panagiotopoulos, A. Z. Langmuir 1999, 15, 3143. (86) Salaniwal, S.; Cui, S. T.; Cochran, H. D.; Cummings, P. T. Langmuir 1999, 15, 5188. (87) Palmer, B. J.; Liu, J.; Virden, J. Langmuir 1999, 15, 7426. (88) Marrink, S. J.; Tieleman, D. P.; Mark, A. E. J. Phys. Chem. B 2000, 104, 12165. (89) Shelley, J. C.; Shelley, M. Y. Curr. Opin. Colloids Surf. Sci. 2000, 5, 101. (90) Tieleman, D. P.; van der Spoel, D.; Berendsen, H. J. C. J. Phys. Chem. B 2000, 104, 6380. (91) Rajagopalan, R. Curr. Opin. Colloids Surf. Sci. 2001, 6, 357. (92) Salaniwal, S.; Cui, S. T.; Cochran, H. D.; Cummings, P. T. Langmuir 2001, 17, 1773.

Alkyl Chains, Droplets, and Micelles

steps are used to study the average equilibrium structure and dynamics of the monomers of micellar aggregates that are set up and equilibrated in the initialization process. Two types of micelle models have been used in simulations. One is a “full” micelle model, in which all surfactants, counterions and water molecules are considered explicitly. The other uses a “solvent force field” or “simplified shell” model in which interactions of the monomers with counterions and water molecules are represented by an external or shell potential instead of being treated with individual interactions. Comparisons of these methods show that equilibrium chain structures are essentially the same63,64,69,70,72-75 though the solvent force field models cannot yield any information about solvent or salt effects, except in an indirect way. However, such a model is considerably less computationally intensive and thus more useful to explore the properties it can provide. We use the “solvent force field” model in this work. In contrast to the extensive investigations of PHA micelles, knowledge of PFA micelles is limited.4,93 We have done a systematic simulation study of PHA and PFA chain systems to determine what simulation might show about the similarities and differences of these chains in different environments. The work described above has usually focused on individual cases, but we have attempted to elucidate the general context of these systems. This paper describes chain conformations and internal structure of pure component chains and aggregates using a variety of intra- and intermolecular force models. The influence of composition in mixed aggregates will be investigated in succeeding papers. First, single PHA and PFA molecules with chain length ranging from 7 to 29 were simulated in segment solvents to study the effects of chain length on conformational characteristics. Second, simulations of PHA and PFA droplets of 24 monomers of 9 segments were performed with the static and dynamic structure analyzed. Third, micelles consisting of 24 PHA octyl and of 40 PFA heptyl chains with “nonionic sulfate” headgroups were simulated.

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Figure 1. Torsional potential models for PHA chains.

2. Interaction Potential and Aggregate Model 2.1. Segment Interaction Potentials. The chain interaction potential plays a key role in molecular simulation studies. We chose relatively simple unitedatom chain potentials to describe and reflect the properties of interest with sufficient accuracy to provide reliable images and statistical results of micelle cores. For the single chain investigations, we used the potentials of Table 1. The CH2 and CH3 segments, or CF2 and CF3 segments, are given the same interaction parameters since test simulations showed no difference in results when the segments were given different interaction parameters. For the aggregate studies, we used three PHA chain potential models and five PFA chain potential models (see Table 2).18,69,70,72,75,94-96 They all have the same general form as the equations of Table 1, though the parameters may differ. Most of these chain potentials have been used in the simulations of vapor-liquid phase equilibria and interfacial structure where such difference can significantly affect simulation results, but their influences on molecular behavior have not been investigated. We were particularly interested in the torsional potential, since it may affect conformational properties (average and dynamic flexibility) and might lead to different steric (93) Hoffmann, H.; Wurtz, J. J. Mol. Liq. 1997, 72, 191. (94) Hariharan, A.; Harris, J. G. J. Chem. Phys. 1994, 101, 4156. (95) Shin, S.; Collazo, N.; Rice, S. A. J. Chem. Phys. 1992, 96, 1352. (96) Bates, S. W.; Stockmayer, W. H. Macromolecules 1968, 1, 12, 17.

Figure 2. Torsional potential models for PFA chains.

arrangements of surfactants in micelles (internal structure). Figures 1 and 2 illustrate our torsional potential models for PHA and PFA chain molecules. The main differences are the positions of minimum gauche energy, the difference between trans and gauche energies and the energy barrier for trans-gauche transformations. Figure 1 shows that among the PHA models the positions of gauche and trans conformation states for the three PHA torsional potentials are very similar to somewhat different transition energy barriers. We expect that this would yield different dynamics, but similar equilibrium structure. However, as Figure 2 illustrates, there are significant differences among the published PFA models in all respects, influencing both static and dynamic properties, if they are sensitive to the potential. For example, the torsional potential of the Cui-T PFA model,18 which was recently used for vapor-liquid phase equilibria simulation calculations has two gauche minima. The claim is that the helical superstructure of PFA chain is an important difference between PHA and PFA chain molecules. Although the helical chain conformation in the PFA simulations does not appear with our united-atom po-

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Table 1. Potential Parameters for PHA and PFA Chains

Table 2. Summary of Chain Potential Models model RB SKS HH94 Cui-T Cui-M HH94 Shin BS

sources PHA chains Karaborni and O’Connell69,70,72,75 Siepmann, Karaborni and Smit8 Hariharan and Harris94 PFA chains Cui et al.18 Cui et al.18 Hariharan and Harris94 Shin et al.95 all intra- and intermolecular interactions are the same as Cui-T model but the torsional potential is from Bates and Stockmayer96

tentials, we still can examine if this variation changes the aggregate shapes and chain structures. The headgroup on both PHA or PFA surfactant molecules was a single soft sphere of larger size and mass, having an additional repulsive term varying as r-3 representing a fully neutralized sulfate group, which is dipolar rather than ionic. 2.2. Droplet and Micelle Aggregate Potential Models. The interaction between surfactant molecules and the surrounding solvent molecules, such as counte-

rions and water, was represented by a solvent force field model which has been proven to be efficient and reasonable in our previous micelle simulations.69,70,72,75 There are two parts to the interactions: segment-solvent, Uss, and headgroup-solvent interactions, Uhs as shown in for spheres in Figures 3 and 4. There is a shell with rapidly changing potentials to inhibit movement of segments to outside the droplet or micelle core and of headgroups into the micelle core. This represents the solvophobic and solvophilic interactions between surfactant and solvent molecules. For segments the energy rises sharply at the shell and then at infinite separation is the Gibbs energy of transfer of a CH2 or CF2 segment from liquid oil droplet to water based on experimental solubility and surface tension data,97 which showed that such transfer energies are independent of attached entities. An estimate was made of the Gibbs energy of transfer of headgroups from water to oil as about 1 order of magnitude greater than that for segments to transfer to solvent, as also found in the experiments. The parameter values and formula for solvent force field models are given in Table 3. We have (97) Vilallonga, F. A.; Koftan, R. J.; O’Connell, J. P. J. Colloid Interface Sci. 1982, 90, 539.


Figure 3. Segment-solvent interaction models (solvent force field model).

Figure 4. Headgroup-solvent interaction models (solvent force field model).

found that our results are quite insensitive to variations of the values and positions. 3. Simulation Details 3.1. Single Chains. Our simulation procedure for single chain in its solvent segments was the same as in our previous simulation study on hydrocarbon chains.39 Briefly, we created a face-centered cubic simulation cell composed of Ns individual CH2 or CF2 solvent segments and one PHA or PFA chain containing Nc CH2 or CF2 segments with Ns chosen to achieve the desired segment reduced density of 0.0109 as in previous work39 (Ns varied from 100 with the 7-segment chains to 835 for the 29segment chains). The initialization for all simulation runs arranged the positions of the chain segments in the alltrans conformation in the center of the cell, and then assigned the other solvent segments randomly in the cell. The minimum image convention with a cutoff distance of half of the side length of the simulation cell was used for all solvent segments. The cell boundary was updated every

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10 time steps in order to keep the center-of-mass of the chain molecule in the center of the simulation cell. The equations of motion for all segments on the chains were integrated using Gear’s fifth-order predictor-corrector algorithm98 for PHA chain simulation and the RATTLE algorithm98 for PFA chain simulation since the PFA bond length was fixed to eliminate vibrational contributions. The time steps were the same for all chain simulations (1.4 fs). All the conformational properties analyzed were average values of 10 computational blocks of 5000 time steps each, except for the shortest chains where longer simulation times were used in the equilibrium and production phases to obtain better statistics in averaging the conformational properties. The run was 700 ps for the 7-segment chains and 140 ps for chains longer than 11 segments. 3.2. Droplets and Micelles. The simulation procedure used in this work for droplet and micelle systems was the essentially the same as in our previous simulation studies.69,70,72,75 All the simulations are prepared with initial positions of all segments on the chains being in the all-trans configuration and radially directed to the centerof mass of the aggregate which is the center of the spherical shell. The centers of one end of each of the chains or of the headgroup of each of the surfactants were distributed uniformly on a sphere shell of a radius that was 1.5 times the ultimate outside barrier shell radius. The initialization potential allowed a small amount of vibration normal to the shell through out the shrinking process of 30 00050 000 time steps, depending on the final shell radius (see Figures 3 and 4). The barriers remained infinite while the volume of the shell is decreased during initialization, but then they were made finite in fixed position for the equilibration and production runs. Nonspherical aggregate initialization began the same way as for spherical aggregates. But, while the potential remained infinite and the shell volume was decreased over about 50 000-80 000 time steps, the location of the confining potential was simultaneously changed to the desired ellipsoid shape. When the final equilibrium size and shape was reached, the “finite nonspherical shell” potentials of solvent force field model were used for equilibration over another 250 000 time steps. After that, production runs of about 100 000 time steps were performed. The final PHA and PFA aggregate shell sizes were chosen to achieve the overall segment number density suggested by experimental cmc data. We also tested to see the influence of different radii of the aggregates, and found little effect on the internal structure and aggregate chain conformation. The only exception was when the radius became so small that monomers drifted away from the aggregate because too little attractive energy existed inside the shell. In this work, the final radius of aggregates of 24 nonane chains in the PHA droplet and 24 octyl sulfate surfactants in the PHA micelle was 3.5 σCH2 which corresponds to a packing fraction of 0.6. For PFA aggregates, the final radius was 3.5 σCF2 for 24 fluorononanes in the PFA droplet and 4.0 σCF2 for 40 perfluoroheptyl sulfate surfactants in the PFA micelle. All the calculated conformational and structural properties are average values over instantaneous sample values extracted every 100 time steps. The temperature of all simulation systems is equivalent to 300 K maintained by isokinetic scaling at each time step.98 All the simulation systems and conditions in this work are summarized in (98) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Clarendon: Oxford, England, 1987.

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Table 3. Parameters for Solvent Force Field Modelsa

a

Note: The quantities with asterisks are in reduced units: distance units by r/σss; energies by u/ss. Table 4. Summary of Simulation Systems and Conditions run no.

imposed shape

ellipticity, ()Rb/Ra)

no. of surfactant monomers

no. of segments on monomers

radius of shell potential (Å)

1 2 3 4 5

sphere oblate oblate prolate prolate

1.0 0.5 0.67 1.5 2.0

PHA Droplet 24 24 24 24 24

9 9 9 9 9

R ) 13.67 Ra ) 17.22, Rb ) 8.61 Ra ) 15.67, Rb ) 10.42 Ra ) 11.95, Rb ) 17.92 Ra ) 10.86, Rb ) 21.72

6

sphere

1.0

PFA Droplet 24

9

R ) 16.10

9 9 9 9 9


8 8 8 8 8


7 8 9 10 11


1.0 0.5 0.67 1.5 2.0

PHA Micelle 24 24 24 24 24

12 13 14 15 16


1.0 0.5 0.67 1.5 2.0

PFA Micelle 40 40 40 40 40

Table 4. It should be mentioned that no difficulties were encountered in obtaining rapid transitions among configurations regardless of the energy barrier between gauche and trans minima. In these systems, there is

apparently enough energy passed among the chain modes of motion to overcome even the largest barriers. The equations of motion for all segments on the chains and surfactants in aggregates were integrated using Gear’s


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Table 5. Simulation Results for a Single PHA Chain in Solvent of Segmentsa length of chain

7

9

11

13

15

17

21

29

trans fraction 〈R〉/σss 〈R2〉/σss2 〈S〉/σss 〈S2〉/σss2 - ∑ Pi ln Pi ln(n - 2) - ∑ Pi ln Pi/[ln(n - 2)]

0.7019 1.7117 2.9558 0.561 0.380 1.153 1.609 0.717

0.6515 2.1124 4.511.02 0.691 0.540 1.308 1.959 0.668

0.6811 2.6427 7.031.38 0.831 0.841 1.307 2.197 0.595

0.739 3.0240 9.282.28 0.971 1.131 1.380 2.398 0.580

0.7110 3.3498 11.403.22 1.071 1.361 1.534 2.565 0.598

0.6619 3.4919 13.136.11 1.153 1.583 1.451 2.708 0.536

0.677 4.4093 20.217.85 1.303 2.013 1.558 2.944 0.539

0.695 5.9781 34.969.24 1.513 2.824 1.711 3.296 0.519

a

Note: The subscripted numbers give the standard deviation of the last decimal(s); e.g., 0.743 is 0.74 ( 0.03. Table 6. Simulation Results for a Single PFA Chain in Solvent of Segmentsa length of chain

7

9

11

13

15

17

21

29

trans fraction 〈R〉/σss 〈R2〉/σss2 〈S〉/σss 〈S2〉/σss2 - ∑ Piln Pi ln(n - 2) - ∑ Pi ln Pi/[ln(n - 2)]

0.873 1.589 2.5230 0.490 0.300 0.863 1.609 0.536

0.873 2.0716 4.2665 0.620 0.490 1.108 1.959 0.566

0.872 2.5518 6.5391 0.750 0.710 1.225 2.197 0.558

0.912 3.0626 9.411.45 0.891 0.990 1.218 2.398 0.508

0.872 3.4135 11.732.29 1.001 1.240 1.504 2.565 0.586

0.882 3.8445 14.923.20 1.121 1.571 1.442 2.708 0.532

0.872 4.6143 21.393.82 1.261 1.911 1.507 2.944 0.512

0.891 5.891.14 35.9412.34 1.483 2.644 1.611 3.296 0.489

a

Note: The standard deviations are as in Table 5.

fifth-order predictor-corrector algorithm with a time step interval of 0.62 fs for PHA aggregates, and the RATTLE algorithm with time steps of 0.66 fs for PFA aggregates. 4. Results and Discussion 4.1. Single Chains. Molecular dynamics simulations of single PHA and PFA chain molecules in their solvent segments were run with 8 different lengths ranging from 7 to 29 using the RB (PHA) and Cui-T (PFA) models (Table 1). To verify reproducibility, we choose the same simulation conditions as Karaborni and O’Connell.39 The fluctuations of chain conformational properties for single chains systems are about 10-15% for most quantities obtained in this work, larger than those from aggregate simulations. The principal results are summarized in Tables 5 and 6. The trans bond fraction was 0.69 ( 0.02 for PHA and 0.88 ( 0.01 for PFA, regardless of chain length. The higher trans bond fraction for single PFA chains compared to single PHA chains arises mainly from its larger trans/ gauche transition energy difference. The most probable number of trans bonds agrees with the average trans fraction for all the chain lengths, which generally confirms a random distribution of trans bonds even though individual chains showed somewhat variable distributions. As the chain gets longer, the distribution becomes more normal. The PHA values are quite consistent with those of previous work.39 The conformational entropy for different chain lengths is found from

S k

n-3

)-

pi ln pi ∑ i)0

(1)

where k is the Boltzmann constant, pi is the probability of finding a particular number of bonds of type i in the trans conformation on the chain molecule and n - 3 is the total number of dihedral angles of the chain. The maximum conformational entropy from a uniform distribution is ln(n - 2). Our simulations give ratios of the two quantities that are generally independent of chain length even though a larger entropy ratio might be expected for longer chains (Tables 5 and 6).

The probability of finding a dihedral angle in the trans conformation for all angles on a chain was found to be equal for all angles in a chain and the conformational entropy calculated from eq 1 is less than to that of a fully random distribution, ln(n - 2), for all chains of both types. This was found previously39 and consistent with the “pentane interference”. Besides trans bond fraction and distribution, two other characteristic properties of the size and shape of a flexible chain are the end-to-end distance, 〈R〉, and the radius of gyration about the center-of-mass of the chain 〈S〉. Tables 5 and 6 show average and mean squared values of these properties. End-to-end distance or radius of gyration will depend on chain length. According to scaling laws,38 the end-to-end distance and radius of gyration of very long chains will follow exponential relations:

〈R〉 ∝ Ncν or 〈S〉 ∝ Ncν

(2)

in which ν is a universal scaling constant with ν ) 0.6 for a single chain in good solvent, ν ) 0.5 for single chain in an unperturbed or “theta” solvent, and ν ) 0.38 for a single chain in a poor (collapsed chains) solvent. Figures SI-1 and SI-2 (Supporting Information) show the simulation results of 〈R〉 and 〈S〉 for PHA and PFA chains as a function of length. For both types of segments, values for chains with n ) 11 or longer are consistent with scaling in a good solvent for 〈R〉, though the relation is not followed as well for 〈S〉. As expected, the variation with shorter chains involves a larger exponent. 4.2. Droplets and Micelles: Chain Conformations. As in Tables 5 and 6 for single chains in segments, Tables SI-1 and SI-2 (Supporting Information) give PHA and PFA droplet averages while Tables SI-3 and SI-4 (Supporting Information) give micelle averages. These include trans bond fractions, end-to-end distances, 〈R〉 and 〈R2〉, radii of gyration, 〈S〉 and 〈S2〉, and conformational entropies for the RB potential and the Cui-T potential. The PHA chains in a droplet are slightly more straightened than single chains while those of spherical micelles ( ) 1.0) are even more straightened. The PFA chains in a droplet show less trans conformations than those for nine-segment chains while those in micelles show more trans. The effect of chain potential model on the droplet chain conformations is noticeable. The average trans fractions

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Figure 5. Bond orientation parameters as a function of distance from center-of-mass of micelles of different shapes: (a) PHA; (b) PFA.

of the chains correlate precisely with the trans-gauche energy difference in the torsional potential and all our calculated averages are the same, within uncertainty, as those computed from Boltzmann partitioning. The results of end-to-end distances and radii of gyration of chain molecule are fully consistent with the trans fraction trends. There is much more gauche and bent character to the RB potential compared to the other PHA models. The PFA models are of two types: three with similar results to the PHA and two with nearly completely trans character (HH94 and Shin models). Interestingly, the probability of finding a particular dihedral angle in the trans conformation is uniform on all bonds for all potential models. Trans bond distribution provides more information about chain conformation in the aggregates. There is nearly uniform distribution among all bonds though, there is a slightly higher probability for a trans bond next to the headgroup in the micelles. This is consistent with NMR experiments of PHA micelles.48 The most probable number of trans bonds on chains and surfactants agrees with the average trans fraction value. There is zero probability of finding chains or surfactants in an all-gauche conformation or with only one trans bond.

Mei and O’Connell

Figure 6. Segment-bond order parameters as a function of bond number in spherical droplets for different potential models: (a) PHA; (b) PFA.

Tables SI-1-SI-4 (Supporting Information) show the conformational entropy calculated from bond probabilities for chains in all the aggregates. Consistent with the changes in trans fractions upon aggregation, the PHA chains have lowest entropy in the micelle and slightly higher entropy in the segment solvent than in the droplet. The conformational entropies of the PFA systems are also consistent with their trans fractions with the HH94 and Shin models showing much lower entropy with high trans fraction. 4.3. Droplets and Micelles: Bond Orientation Parameters. It is useful to evaluate some statistical measures of ordering in aggregates. Bond order parameters can be obtained from dynamic 13C NMR methods.46,48,61 and related to simulations,72,75 by taking a long time average of

S)

〈23cos θ - 21〉 2

i

(3)

where θi is the angle formed by the bisector of two bonds among three consecutive segments of the chain as time evolves. We have not yet been able to complete the evaluation of these quantities for our systems. However,


we have obtained bond orientation parameters,63,69 So, which facilitate comparisons with visualizations of micelle structure, and a “segment-bond order parameter”, S′, which we have found to be similar to NMR results. The value of So is found by a spatial averages, in thin shells beginning at the center of mass, of the same function in (3), of the angle formed between a vector directed from the center-of-mass of the aggregate to the midpoint of the vector of the bond between two segments. So is unity if the bond vectors average to parallel to the radius vector while a value of -0.5 means the bonds average to perpendicular to the radius vector. An average value close to zero reflects statistically random bond orientations except for some perfectly aligned configurations which were not observed here. We discuss here the results for spherical aggregates where the ellipticity, , is unity; nonspherical systems are described in the next section. Figure SI-3 (Supporting Information) show bond orientation parameters, So, as a function of distance from the aggregate center-of-mass in PHA and PFA droplets. Except near the surface and the center, where the statistics are poor, there is basically random bond orientation for all the models. Figure 5 shows So as a function of radius for micelles. There are poor statistics in regions close to the centerof-mass and the surface, but the spherical core is basically random with outward preference appearing only at about 80-90% of the distance from the center of mass to the surface. Values of S′ are found from an average over all chains of the same angle of the bond and center of mass vectors; the result is a variation over bond number with the interpretation of the signs being the same as for So. Figure 6 shows values of S′ for droplets. Similar to previous simulations, the variations of S′ show mostly random structure, independent of the potential model chosen. Figure 7 shows S′ as a function of bond number for micelles. In contrast to spherical droplets, where no preferred orientation exists, the bonds in spherical PHA micelles connecting headgroups and first segments (bond 1) show some preferential radial ordering. This is consistent with the somewhat higher probability of first dihedral angle being in trans conformation for surfactant chains in the micelles. These trends and values generally agree with the bond order parameter values, S, from NMR experimental data48 and previous MD simulation results.69,70,72,75 The variation of S′ for PFA micelles has a clear alternating pattern of nearly random orientation for the even bonds and preferred outward bonds for the odd numbered bonds, which decreases toward the chain end. This alternation does not seem to be observed experimentally61 and is absent from our preliminary time correlation functions that give values of S. Perhaps the analogy of S′ and S breaks down in PFA systems with their combination of more trans bonds and the helical torsional potential. 4.4. Aggregate Shape. The shapes of the aggregates can be described by three principal moments of inertia. The ratio of largest moment of inertia Imax to the smallest moment of inertia Imin is commonly used for determining shape. For a spherical aggregate, Imax/Imin is unity. In the present simulations, even when the confining shell is spherical, aggregate shape fluctuations occur and their average is slightly nonspherical. For the PHA droplet and micelle the ratios are 1.24 ( 0.07 and 1.20 ( 0.08. For the PFA droplet and micelle, they are 1.26 ( 0.11 and 1.13

Langmuir, Vol. 18, No. 23, 2002 9075

Figure 7. Segment-bond order parameters as a function of bond number in aggregates of different shapes: (a) PHA droplets; (b) PHA micelles; (c) PFA micelles.

( 0.05. This conclusion is consistent with SANS experimental evidence56 and MD simulation results of hydro-

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Table 7. Simulations of PHA Droplets of Various Shapes with RB Chain Potential ellipticity trans fraction Imax/Imin 〈R〉/σss 〈R2〉/σss2 〈S〉/σss 〈S2〉/σss2 - ∑ Pi ln Pi ln(n - 2) - ∑ Pi ln Pi/[ln(n - 2)] a

) 0.5 ) 0.67 ) 1.0 ) 1.5 ) 2.0 0.732 1.4510 2.205 4.9319 0.6972 0.5930 1.289 1.946 0.665

0.782 1.6111 2.293 5.3114 0.7073 0.6180 1.1810 1.946 0.615

0.743 1.247 2.235 5.0221 0.7862 0.8101 1.2410 1.946 0.645

0.754 1.3311 2.236 5.0124 0.6992 0.5971 1.2116 1.946 0.625

0.742 1.5412 2.245 5.0621 0.6992 0.5972 1.2310 1.946 0.635

Note: The standard deviations are as in Table 5.

Figure 9. Radial distribution functions for pairs of chain ends in PHA droplets for different potential models.

Figure 10. Radial distribution functions for pairs of chain ends in PFA droplets for different potential models.

Figure 8. Singlet density distribution functions for chain ends in spherical droplets for different potential models: (a) PHA; (b) PFA.

carbon micelles using full micelle models.65-68,71,73,74,76-81 Our simulation technique also allows us to force the aggregate shape to be nonspherical so we can study shape effects on internal structure and chain conformationproperties. We chose three kinds of aggregates: a PHA droplet with 24 nonyl monomers, a PHA micelle containing 24 octyl sulfate surfactants, and a PFA micelle containing 40 heptyl sulfate surfactants. The RB chain model was used for the PHA systems and the Cui-T model for the

PFA systems. For each kind of aggregate, four aggregate shapes with different ellipticity have been considered. The ellipticity used here is the ratio of major to minor semiaxes of the imposed ellipsoid. A value of unity is for a sphere, values larger than unity are prolate (egg-like), and values smaller than unity are oblate (donutlike). The same number densities are used for all aggregates. Table 4 presents the set of aggregates studied and Tables SI-1-SI-4 (Supporting Information) and Table 7 give their conformational properties. The average trans fractions, bond distributions on each chain, end-to-end distances, and radii of gyration are essentially independent of shape. Bond orientation parameters, So, for all droplets show random behavior over position and shape. Figure 5 for micelle So values show that the PHA spherical micelle has a more extensive random core than either ellipsoid (Figure 5a). Apparently forcing some chains to be shortened in the direction of the minor axis forces the chains to be oriented more outward deeper within the core. On the other hand, So values of Figure 5b show that the random PFA micelle core increases in size from oblate to


Figure 11. Singlet density distributions of head and end groups micelles of different shapes: (a) PHA; (b) PFA.

spherical to prolate shape suggesting that the headgroups seek a higher curvature environment in nonspherical PFA systems. Figure 6 shows S′ values for droplets. there is little effect of potential model. Figure 7 shows S′ vs bond number for aggregates of different shapes. Figure 7a shows that droplet segment-bond order parameters, S′, do not depend on shape. Nonspherical PHA micelles are all the same and are less ordered near the headgroup than in nonspherical shapes (Figure 7b). This may be due to the selection of the number of monomers to match that for an equilibrium sphere. Nonspherical PFA micelles have the same unexpected behavior as spherical PFA micelles (Figure 7c). 4.5. Droplets and Micelles: Segment Spatial Distributions. The primary measures of the internal structure of droplet and micelle aggregates are the probability distributions of segments of chain molecules and surfactants. A convenient quantity is ri2F(ri), which is within a multiplication constant of the true singlet probability, Pi(r), and yields an area under the curve that is proportional to the total number of segments in the aggregate. This particular form can provide a good basis for comparing simulation results of different total numbers of segments

Langmuir, Vol. 18, No. 23, 2002 9077

Figure 12. Pair distribution functions of micelle head-head pairs in micelles of different shapes: (a) PHA; (b) PFA.

and simulations can be tested for consistency by the requirement that both segments of a chain equidistant from the ends must have the same peak. Figure 8a shows the results for end-segment distributions for the PHA droplet with the three potentials, RB, SKS, and HH94, while Figure 8b shows the distributions for the PFA droplet with the five potentials, Cui-T, Cui-M, HH94, Shin, and BS. For PFA, all but the HH94 and Shin models (with symbols) have statistically similar peaks. The Shin model is consistent, but the HH94 model is not. There is no obvious reason. As with single chains bond distributions, these models give different singlet density results; this is seen at distances shorter than the maximum in Figure 8b. Further information about local structure of PHA and PFA droplets is given by pair radial distribution functions (rdf) for headgroups and end groups which we believe are most representative of chain structure in droplets. Figure 9 shows end-end pair rdf simulation results of PHA droplets. To test symmetry, one end was selected from each chain to form the pair 1-1 and the other ends were selected to form the pair 9-9. All the rdfs are the same within uncertainties and symmetry, indicating that the

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Figure 13. Pair distribution functions of micelle segment 4-segment 4 for PHA micelles of different shapes.

small differences in the chain potential model make no significant effect on the internal structure of PHA droplets. The rdf results for PFA droplets (except for the nonsymmetric HH94 model) are shown in Figure 10 where the 1-1 and 9-9 pair results are separated for clarity. The symmetry is satisfactory and curves are quite similar except for the small shift in the Shin model95 showing the pairs being closer together. This model does not have the helical portion of the torsional energy. Thus, the simulation results indicate that all forms of available chain potential models give essentially the same internal structure of chain aggregates. In contrast to the droplets, the probability distributions for the head and end segments on the spherical micelles ( ) 1.0) of PHA in Figure 11a and PFA in Figure 11b show more sharp peaks for the headgroups near the surface and broader distribution of the chain ends. Chain bending apparently allows groups near the chain end to occupy many different radial positions while the headgroups are confined near the surface of the micelles. In fact, visualizations confirm that end groups can reach the outer shell of the aggregate. For both PHA and PFA micelles, the average segment radial position decreases from headgroup to end group. These results are in good agreement with previous simulation results67-75 and also is consistent with the traditional picture of a micelle that has all headgroups distributed close the surface of micelle surrounding a liquidlike core of tail segments. As expected, nonspherical micelles show shoulders. While the end distributions are similar for the PHA and PFA, the nonspherical PFA heads are distributed more broadly. Again this may be an indication that the nonspherical shape is preferred by the PFA micelle for this number of monomers. The radial distribution functions of head-head pairs for micelles are shown in Figure 12. For all shapes, the PHA heads are somewhat more broadly paired than are the PFA heads, probably due to the greater amount of trans chains of the PFA. Figure 13 presents the results of segment-4/segment-4 pair distributions in the PHA micelle, while Figure 14 shows the end-end rdfs. In all cases, the shape is similar to that of the heads with a broader distribution in the PHA micelle, but the peaks are at smaller distances from the center of mass, the further they are from the headgroup.

Figure 14. Pair distribution functions of end-end groups in micelles of different shapes: (a) PHA; (b) PFA.

For nonspherical shapes, the rdfs are narrower than those of spherical micelles, and their peaks are at smaller separations in all cases. The oblate is between prolate and sphere in breadth and separation at the maximum. There seems to be less difference between the nonspherical shapes than between them and the spherical shape. Compared to the PFA systems, the PHA peaks become much sharper from head to end indicating that higher curvature confines the ends more. This is not likely to be preferred in reality, especially given the amount of conformational freedom that the PHA chains seek as evidenced by their low trans fractions. The rdf peaks for PFA are much more symmetric, and there is little effect of shape. It seems that the PFA micelle can adjust to any curvature, perhaps because its chains are more trans. For the PHA micelle, plots of scaled experimental SANS data from Cabane et al.47 (see Karaborni and O’Connell69 for the technique used) show good agreement with the simulations. Figure 15 shows the singlet density distributions of head and end groups, and Figure 16 shows the head-head rdf for oblate, spherical, and prolate PHA droplets. The symmetry shown by the headgroups and end groups is not strongly affected by shape though the positions of the


Figure 15. Singlet density distributions of head and end groups in PHA droplets of different shapes.

nonspherical peaks are shifted to slightly shorter range, as expected. 5. Conclusions A series of molecular dynamics simulations have been performed to study the chain conformation and internal structural characteristics for three types of systems: single chains in a solvent of segments, and small aggregates of chains (droplets) and of model surfactants (micelles). The chains model n-alkanes (PHA) and n-perfluoroalkanes (PFA) and the surfactants have a “nonionic sulfate” head with dipolar repulsion. Several different chain potential models were used. The aggregates are confined by a shell potential whose shape included spheres and oblate and prolate ellipsoids. For single chain systems trans-bond fractions and distributions are independent of chain length and the scaling exponent for PHA and PFA chains longer than 11 segments is consistent with a “good solvent” scaling law exponent of 0.6. The simulation results show that the conformational and structural properties for the PFA chains, droplets and micelles are similar to those of PHA systems except for having higher trans-bond fractions from the larger PFA torsional trans-gauche energy difference.Variations of the chain potential affect only the average trans conformation and average chain extension consistent with the trans-gauche energy difference. All other aggregate and averaged chain properties do not vary

Langmuir, Vol. 18, No. 23, 2002 9079

Figure 16. Pair distribution functions of head-head groups in PHA droplets of different shapes.

with the torsional potential. Segment positions and segment-segment radial distribution functions suggest that the headgroups of nonspherical PHA micelles reside in regions of higher curvature, though no such preference is found in PFA systems. However, the saw tooth pattern of bond orientation parameters with bond number in PFA micelles is not found in the PHA systems probably related to both its higher trans fraction and the helical portion of its torsional potential. These subtle differences on different chain systems from imposed micelle shape may be of significance in mixed systems including those with solutes and other chain types. Acknowledgment. This work was supported by the Division of Chemical, Biochemical, and Thermal Engineering of the National Science Foundation. We are grateful to J. M. Haile and S. Karaborni for making available their original and modified MD programs and to S. Wonczak for assistance in revising code and analyzing some of the results. Supporting Information Available: Text giving details of the analyses, figures showing simulation results and bond orientation parameters, and tables giving simulation results. This material is available free of charge via the Internet at http://pubs.acs.org. LA0201826

Molecular Dynamics Simulations of Model Perhydrogenated and

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