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Page i Dedication Page ii Acknowledgement Page iii Table of Contents Page iv Page v List of Figures Page vi Page vii Page viii Page ix List of Tables Page x Page xi Page xii Page xiii Abstract Page xiv Page xv Chapter 1. Introduction Page 1 Page 2 Chapter 2. Dilute Nalkane simulations Page 3 Page 4 Page 5 Page 6 Page 7 Page 8 Page 9 Page 10 Page 11 Page 12 Page 13 Page 14 Page 15 Page 16 Page 17 Page 18 Page 19 Page 20 Page 21 Page 22 Page 23 Page 24 Page 25 Chapter 3. Simulations of surfactants in a monatomic fluid and in water Page 26 Page 27 Page 28 Page 29 Page 30 Page 31 Page 32 Page 33 Page 34 Page 35 Page 36 Page 37 Page 38 Page 39 Page 40 Page 41 Page 42 Page 43 Page 44 Page 45 Page 46 Page 47 Page 48 Page 49 Page 50 Page 51 Page 52 Page 53 Page 54 Page 55 Page 56 Page 57 Page 58 Page 59 Page 60 Page 61 Page 62 Page 63 Chapter 4. Model Michelle Page 64 Page 65 Page 66 Page 67 Page 68 Page 69 Page 70 Page 71 Page 72 Page 73 Page 74 Page 75 Page 76 Page 77 Chapter 5. Effects of Micellesolvent interaction Page 78 Page 79 Page 80 Page 81 Page 82 Page 83 Page 84 Page 85 Page 86 Page 87 Page 88 Page 89 Page 90 Page 91 Page 92 Page 93 Page 94 Page 95 Page 96 Page 97 Page 98 Page 99 Page 100 Page 101 Page 102 Chapter 6. Effects of chain length and head group characteristics Page 103 Page 104 Page 105 Page 106 Page 107 Page 108 Page 109 Page 110 Page 111 Page 112 Page 113 Page 114 Page 115 Page 116 Page 117 Page 118 Page 119 Page 120 Page 121 Page 122 Page 123 Page 124 Page 125 Page 126 Page 127 Page 128 Page 129 Page 130 Page 131 Page 132 Chapter 7. Conclusions and recommendations Page 133 Page 134 Page 135 Appendix. Water structure in the presence of an "anionic methyl" surfactant Page 136 Page 137 Page 138 Page 139 Page 140 Page 141 Page 142 Page 143 Page 144 Page 145 Page 146 Page 147 Page 148 Page 149 Page 150 Page 151 Page 152 Page 153 Page 154 Page 155 Bibliography Page 156 Page 157 Page 158 Page 159 Page 160 Page 161 Page 162 Page 163 Biographical sketch Page 164 Page 165 Page 166 
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STRUCTURAL AND DYNAMICS STUDIES OF SURFACTANTS AND MICELLES By SAMI KARABORNI A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1990 To my Father ACKNOWLEDGMENTS To my Father Tijani, God rest his soul, my mother Jamila, my brother Mustapha and his wife, my sisters Najet, Badiaa and Sihem and their husbands, and all my nephews and nieces, thank you for all the love and care. To June Rarick who has given me her endless support and sympathy during the past four years, thanks for everything. I would like to thank Professor O'Connell for his guidance and support and ex pertise and encouragement. Through his hard work and dedication John O'Connell taught me to be my best at whatever I do. I wish to thank Professors Shah, Moudgil, WestermannClark and Bitsanis for serving on the thesis supervisory committee. I would like to express my gratitude to the University Mission of Tunisia for their support of my education in the U.S.A. I thank Professor Cummings for his help, and for providing the best atmosphere around the lab for work and for laughter. I also thank B. Rodin for all the help he has given me, H. Das, with whom I had many philosophical discussions, B. Wang, for teaching me some of his most amazing moves, J. Rudisill, who has given me a good appreciation of the South, and R. Osborne, for providing a feminine touch around the lab. Finally, I would like to express my special thanks to D. Ayres, T. Daley, H. Strauch, D. Stubbs and M. Tandon, and I ask forgiveness from those I have not mentioned. TABLE OF CONTENTS ACKNOWLEDGMENTS ............................. iii LIST OF FIGURES ................................ vi LIST OF TABLES ................................. x ABSTRACT .................................... xiv CHAPTERS 1 INTRODUCTION ............................... 1 2 DILUTE NALKANE SIMULATIONS ....................... 3 2.1 Background ................................ 3 2.2 Chain M odel ............................... 4 2.3 Simulation Details ............................ 7 2.4 Results . . . . . . . . . . . . . . . . . .. 9 2.4.1 Chain Conformation. Average Trans Bond Fraction ...... 9 2.4.2 EndtoEnd Distance and Radius of Gyration .......... 16 2.4.3 The Trans Bond Distribution ..................... 19 2.5 Conclusions .. .... ..... ...... ... .... .. ..... 21 3 SIMULATIONS OF SURFACTANTS IN A MONATOMIC FLUID AND IN W ATER ................................... 26 3.1 Background ................................ 26 3.2 Potential Model for Segmented Molecules .................. 27 3.2.1 Octyl Surfactants ......................... 27 3.2.2 Poly (Oxyethylene) Molecule ...................... 29 3.3 Interaction Models for Molecules in a LennardJones Fluid of Segments and Simulation Details ........................... 32 3.4 Results for Molecules in a LennardJones Fluid of Segments . . 38 3.4.1 Average and Mean Values ....................... 38 3.4.2 EndtoEnd Distance . . . . . . . . . ... .. 41 3.4.3 Radius of Gyration . . . . . . . . . . ... .. 45 3.4.4 Probability of Finding a Number of Angles in the Trans Con form ation . . . . . . . . . . . . . . .49 3.4.5 Probability of Finding a Particular Angle in the Trans Confor m ation . . . . . . . . . . . . . . .. .. 49 3.5 "Ionic Methyl" Octyl Surfactant in Water . . . . . . .... .. 52 3.5.1 M odel . . . . . . . . . . . . . . ... .. 52 3.5.2 Results . . . . . . . . . . . . . . ... .. 57 3.6 Conclusions . . . . . . . . . . . . . . ... .. 63 4 MODEL MICELLE . . . . . . . . . . . . . . ... .. 64 4.1 Background . . . . . . . . . . . . . . ... .. 64 4.2 M icelle M odels . . . . . . . . . . . . . ... .. 66 4.2.1 ChainSolvent Interaction . . . . . . . . ... .. 68 4.2.2 HeadSolvent Interaction . . . . . . . . . ... .. 70 4.3 Simulations . . . . . . . . . . . . . . ... .. 73 5 EFFECTS OF MICELLESOLVENT INTERACTION .. ........ 78 5.1 Local Structure. Probability Distributions . . . . . . .... .. 78 5.2 Average Positions of Groups . . . . . . . . . . ... .. 82 5.3 Distribution of Tail Groups . . . . . . . . . . ... .. 84 5.4 Distribution of Distances Between Groups . . . . . . .... .. 84 5.5 M icelle Shape . . . . . . . . . . . . . . ... .. 88 5.6 Chain Conformation. Trans Bond Distributions . . . . ... .. 93 5.7 Bond Orientation . . . . . . . . . . . . . ... .. 96 5.8 Conclusions . . . . . . . . . . . . . . ... .. 100 6 EFFECTS OF CHAIN LENGH AND HEAD GROUP CHARACTERISTICS 103 6.1 Local Structure . . . . . . . . . . . . . ... .. 103 6.2 Hydrocarbon Distribution . . . . . . . . . . .. .. 110 6.3 Average Chain Segment Positions . . . . . . . . .. .. 110 6.4 Distributions of Tail Groups . . . . . . . . . . ... .. 113 6.5 Distributions of Distances Between Groups . . . . . . ... .. 115 6.6 M icelle Shape . . . . . . . . . . . . . . ... .. 122 6.7 Chain Conformation. Trans Bond Distributions . . . . .... .. 123 6.8 Bond Orientation . . . . . . . . . . . . . ... .. 127 6.9 Conclusions . . . . . . . . . . . . . . ... .. 132 7 CONCLUSIONS AND RECOMMENDATIONS . . . . . . ... .. 133 APPENDIX . . . . . . . . . . . . . . . . . . 136 BIBLIOGRAPHY . . . . . . . . . . . . . . . ... .. 156 BIOGRAPHICAL SKETCH . . . . . . . . . . . . ... .. 164 LIST OF FIGURES 2.1 The endtoend distance for hydrocarbon chains as a function of chain length . . . . . . . . . . . . . . . . . . 17 2.2 The radius of gyration for hydrocarbon chains as a function of chain length . . . . . . . . . . . . . . . . . . 18 2.3 Entropy S/k = Eplnpi is plotted as a function of ln(n 3). For an nalkane there are n3 dihedral bonds and pi is the probability of finding a bond in trans conformation . . . . . . . . .... .. 24 3.1 Model octyl surfactants . . . . . . . . . . . .... .. 28 3.2 Model poly (oxyethylene) molecule . . . . . . . . ... .. 31 3.3 Endtoend distribution for the octyl "ionic methyl" surfactant in a LennardJones fluid of segments . . . . . . . . . ... .. 41 3.4 Endtoend distribution for the octyl "nonionic sulfate" surfactant in a LennardJones fluid of segments . . . . . . . . ... .. 43 3.5 Endtoend distribution for poly (oxyethylene) in a LennardJones fluid of segments . . . . . . . . . . . . . .... .. 44 3.6 Radius of gyration distribution for the octyl "ionic methyl" surfactant in LennardJones fluid of segments . . . . . . . . ... .. 46 3.7 Radius of gyration distribution for the octyl "nonionic sulfate" surfac tant in a LennardJones fluid of segments . . . . . . ... .. 47 3.8 Radius of gyration distribution for poly (oxyethylene) in a Lennard Jones fluid of segments . . . . . . . . . . . ... .. 48 3.9 Endtoend distribution for the octyl "ionic methyl" surfactant in water. 59 3.10 Radius of gyration distribution for the octyl "ionic methyl" surfactant in water . . . . . . . . . . . . . . . . . .. 60 4.1 Model for intermolecular interactions in micelles . . . . ... ..67 4.2 Chainsolvent interaction models a) (r'an r*)12 potential b) finite energy barrier, U* = U/e, c = 419J/mol . . . . . . . ... .. 69 4.3 Headsolvent interaction models a) harmonic potential b) finite energy barrier. Half harmonic potential has same form as harmonic potential for a radius less than the equilibrium radius, and is equal to zero for a radius greater than the equilibrium radius. U* = U/e, c = 419J/mol . 71 5.1 Group probability distributions for tail groups . . . . . .... .. 80 5.2 Group probability distributions for middle segments (segment 5 from the top of the chain with the head group numbered 1 and the tail group 9) . . . . . . . . . . . . . . . . . . . 81 5.3 Group probability distributions for head groups . . . . ... .. 83 5.4 Scattering amplitude for methyl tail groups . . . . . ... .. 87 5.5 Distribution of distances between head groups . . . . . .... .. 88 5.6 Distribution of distances between tail groups . . . . . .... .. 89 5.7 Ratio of moments of inertia from runs 1, 2 and 3 . . . . ... .. 91 5.8 Ratio of moments of inertia from runs 4, 5 and 6 . . . . ... .. 92 5.9 Overall bond order parameter S(r) throughout the micelle for runs 1, 2 and 3 . . . . . . . . . . . . . . . . . .. 98 5.10 Overall bond order parameter 5S(r) throughout the micelle for runs 4, 5 and 6 . . . . . . . . . . . . . . . . . .. 99 5.11 Individual bond order parameter Si for bonds on the 9member chains from runs 16 . . . . . . . . . . . . . . ... .. 101 6.1 Group probability distributions of chain ends of a model hydrocarbon droplet . . . . . . . . . . . . . . . . . .. 105 6.2 Group probability distributions of tails for systems 5, 7 and 8, and from Woods et al. (1986). The distribution by Woods et al. is scaled by (24/52)1/3 . . . . . . . . . . . . . . . ... .. 106 6.3 Group probability distributions of tails from run 5 and from Watanabe et al. (1988) and Jonsson et al. (1986) . . . . . . . ... ..108 6.4 Probability distributions of head groups for systems 5 and 7, Farrell (1988) and from Woods et al. (1986). The probability distribution of Woods et al. is scaled by (24/52)1/3 . . . . . . . . ... .. 109 6.5 Hydrocarbon distributions for runs 5, 7 and 8, and from the micelle simulation of J5nsson et al. (RC model) (1986). The Jonsson distribu tion is scaled by (24/15)1/3 . . . . . . . . . . . . 111 6.6 Scaled average radial positions for run 5, and from the micelle simu lations of J5nsson et al. (RC model) (1986) and of Watanabe et al. (1988) . . . . . . . . . . . . . . . . . .. .. 114 6.7 Scattering amplitude from methyl tails for runs 5, 7 and 8, from Woods et al. (1986) and from Bendedouch et al. (1983a) . . . . ... ..116 6.8 Distribution of distances between tail groups of a model hydrocarbon droplet . . . . . . . . . . . . . . . . . .. 117 6.9 Distribution of distances between tail groups as determined from runs 5, 7 and 8 and the SANS data of Cabane et al. (1985). The Cabane distribution is scaled by (24/74)1/3 . . . . . . . . ... .. 118 6.10 Distribution of distances within the whole core as determined from runs 5, 7 and 8 and the scaled SANS data of Cabane et al. (1985). The Cabane (Scaled 1) distribution is scaled by (24/74)1/3 and the Cabane (scaled 2) by (216/962)1/3 . . . . . . . . . . . ... .. 120 6.11 Distribution of distances between head groups . . . . . .... ..121 6.12 Ratio of moments of inertia from runs 5, 7 and 8 . . . . ... .. 124 6.13 Bond order parameter S(r) throughout the micelle, for runs 5, 7 and 8. 129 6.14 Bond order parameter Si for individual bonds on the Nmember chains for runs 5, 7 and 8 and from Woods et al. (1986) . . . . ... .. 131 A.1 Distribution function for the angle cosines describing the orientation of the water molecule dipole moment with respect to the segmentoxygen vector . . . . . . . . . . . . . . . . . . .. 140 A.2 Intermolecular oxygenoxygen pair correlations function . . ... .. 142 A.3 Intermolecular hydrogenhydrogen pair correlation function . . 143 A.4 Intermolecular oxygenhydrogen pair correlation functions . ... .. 144 A.5 Intermolecular hydrogenhead group pair correlation function . 147 A.6 Intermolecular oxygenhead group pair correlation functions . . . 148 A.7 Intermolecular hydrogenchain segment pair correlation function. . 150 A.8 Intermolecular oxygenchain segment pair correlation functions .. 151 A.9 Mean square displacements of water molecules in the shell and the bulk.154 LIST OF TABLES 2.1 Intermolecular and Intramolecular Potential Parameters . . . .. 6 2.2 Summary of Simulations . . . . . . . . . . . . .. 8 2.3 Average Structural Values for All Hydrocarbon Chains . . ... 10 2.4 Average Values for Neat NButane as extrapolated from simulations, and as calculated by Ryckaert and Bellemans (1978), Edberg et al. (1986), Toxvaerd (1988), Jorgensen (1981a), Banon et al. (1985) and Wielopolski and Smith (1986) . . . . . . . . . . ... .. 11 2.5 Average Values for Single and Dilute NButane as extrapolated from simulations, and as calculated by Rebertus et al. (1979), Bigot and Jorgensen (1981), Zichi and Rossky (1986a), Enciso et al. (1989) and Van Gunsteren et al. (1981) . . . . . . . . . . ... .. 12 2.6 Average Values for NHexane as extrapolated from Our Linear Fits and as Determined from Other Workers: Clarke and Brown (1986) . 13 2.7 Average Values for NOctane as Interpolated from Linear Fits and as Determined from Other Workers: Szczepanski and Maitland (1983). . 14 2.8 Average Values for NDecane as Interpolated from Linear Fits and as determined by Ryckaert and Bellemans (1978), Edberg et al. (1987), Toxvaerd (1987) and Van Gunsteren et al. (1981) . . . . ... .. 15 2.9 Probability of Finding a Given Number of Trans Bonds on the Chain. 20 2.10 Randomness of Conformation: Ratio of Equations 2.5 and 2.6 . 22 2.11 Probability of Finding a Particular Dihedral Angle in the Trans Con form action . . . . . . . . . . . . . . . . .. .. 23 3.1 Bond Parameters of "Methylene" and "Sulfate" Groups. "Sulfate" Parameters are Used when an Intramolecular Interaction Involves a "Nonionic Sulfate" Head Group. "Methylene" Parameters are Used with All Other Intramolecular Interactions . . . . . . ... .. 30 3.2 LennardJones and Coulombic Interaction Parameters for Poly (oxyethy lene) . . . . . . . . . . . . . . . . . . .. 33 3.3 Bond Parameters for Poly (oxyethylene) . . . . . . .... .. 33 3.4 Angle Parameters for Poly (oxyethylene) . . . . . . .... .. 34 3.5 Torsion Parameters for Poly (oxyethylene) . . . . . . .... .. 34 3.6 Intermolecular Potential Parameters for "Methylene" and "Sulfate" Groups. "Sulfate" Parameters are Used when an Intermolecular In teraction Involves a "Nonionic sulfate" Head Group. "Methylene" Pa rameters are Used with All Other Intermolecular Interactions . . . 35 3.7 Simulation Details for Runs in LennardJones Fluid of Segments. . 37 3.8 Average Properties for the Octyl "Ionic Methyl" Surfactant in a Lennard Jones Fluid of Segments . . . . . . . . . . . ... .. 39 3.9 Average Properties for the Octyl "Nonionic Sulfate" Surfactant in a LennardJones Fluid of Segments . . . . . . . . ... .. 39 3.10 Average Properties for Poly (Oxyethylene) in a LennardJones Fluid of Segments . . . . . . . . . . . . . . . ... .. 39 3.11 Probability of Finding a Number of Bonds in the Trans Conformation on the Octyl "Ionic Methyl" Surfactant . . . . . . . .... .. 49 3.12 Probability of Finding a Number of Bonds in the Trans Conformation on the Octyl "Nonionic Sulfate" Surfactant . . . . . . ... .. 49 3.13 Probability of Finding a Number of Bonds in the Trans Conformation on a Poly (Oxyethylene) . . . . . . . . . . . ... .. 51 3.14 Probability of Finding a Particular Bond in the Trans Conformation on the Octyl "Ionic Methyl" Surfactant . . . . . . . .... .. 51 3.15 Probability of Finding a Particular Bond in the Trans Conformation on the Octyl "Nonionic Sulfate" Surfactant . . . . . . ... .. 52 3.16 Probability of Finding a Particular Bond in the Trans Conformation on the Poly (Oxyethylene) Molecule in a LennardJones Fluid of Segments. 52 3.17 LennardJones Parameters for Interacting Atoms and Segments. a is Given in A and f is Given in J/mol. Net Charges are Given in Units of the Elementary Charge e=1.602x1019esu . . . . . .... .. 56 3.18 Average Properties for the Octyl "Ionic Methyl" Surfactant in Water. 58 3.19 Probability of Finding a Number of Bonds in the Trans Conformation on the Octyl "Ionic Methyl" Surfactant in Water . . . . ... .. 62 3.20 Probability of Finding a Particular Bond in the Trans Conformation on the Octyl "Ionic Methyl" Surfactant in Water . . . . ... .. 62 4.1 Intermolecular Potential Parameters. Chh, 7 and 3 Are in Units of C. rhh, rh5 and r* are in Units of rm ... . . . . . . . ..... . 75 4.2 Temperatures and Pressures for Molecular Dynamics Simulations. . 77 5.1 Average Radial Position R/ for Each Group, Measured Relative to the Aggregate Center of Mass . . . . . . . . . . . ... ..84 5.2 Mean Radial Position () 1/2 for Each Group, Measured Relative to the Aggregate Center of Mass . . . . . . . . . . ... ..85 5.3 Average Trans Fraction and Average Ratio of Moments of Inertia . 93 5.4 Probability of a Given Number of Trans Bonds on One Chain . .. 95 5.5 Probability of a Particular Bond Being Trans . . . . . .... .. 97 6.1 Average Radial Position for Each Group After Scaling (See Text), A., (A) Relative to the Aggregate Center of Mass . . . . . ... .. 112 6.2 Average Trans Fraction and Average Ratio of Moments of Inertia . 125 6.3 Probability of Finding a Given Number of Trans Bonds on One Chain 126 6.4 Probability of Finding a Particular Bond in the Trans Conformation 128 A.1 Computed Coordination Numbers for an Octyl "Anionic Methyl" Sur factant in W ater . . . . . . . . . . . . . ... .. 138 A.2 Ratios of the Heights of the First Maximum and the Following Mini mum for Various WaterWater Pair Correlation Functions in Bulk and Shell . . . . . . . . . . . . . . . . . . .. 146 A.3 SelfDiffusion coefficients for Bulk and Shell Water Molecules in units of 105 cm 2/sec . . . . . . . . . . . . . . ... .. 153 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy STRUCTURAL AND DYNAMICS STUDIES OF SURFACTANTS AND MICELLES By Sami Karaborni May, 1990 Chairman: John P. O'Connell Cochairman: Dinesh 0. Shah Major Department: Chemical Engineering Micelles are an important class of molecular aggregates that have growing uses in industry. Yet there is still an absence of good structural or thermodynamic models due to the lack of a thorough understanding of micellar behavior and micelle formation. Micelle structure has long been known to be very complex due to the amphiphilic nature of surfactants. The presence of ions, hydrocarbon chains and water makes the micellar aggregate difficult to study theoretically since the contributions from each factor are not known. Presently there are no single experimental or theoretical methods that can comprehensively study micelles. In this work several molecular dynamics simulations have been used to study both the statics and dynamics of micelles and hydrocarbon droplets as well as the conformation of alkanes and surfactants in water and nonpolar environments. Micelle and oil droplet simulations have been performed using a segment force model for intra and intersurfactant interactions while micellesolvent interactions have been modeled using several field potentials that realistically describe surfactant interactions with polar solvents. Dilute solutions of surfactants and nalkanes in a monatomic nonpolar fluid and in water were performed using conventional intermolec ular interactions. In general, the results show the insensitivity of micellar structure and chain con formation to micellesolvent interaction models regardless of chain length or head group characteristics, while aggregate shape was found on the average to be some what nonspherical with significant fluctuations. In all instances the micelle core was found to be like the oil droplet and the chain conformation to be similar to that of surfactants in nonpolar media. In general, local structure results were similar to experimental and other simulation data. The conformation of alkanes in a fluid of nonpolar segments closely resembles the conformation of surfactants. In addition, alkanes exhibit some characteristics that are independent of chain length, such as the average trans fraction, and other properties that are proportional to chain length, such as radius of gyration and endtoend distance. The conformation of ionic surfactants in water was found to be significantly differ ent from that in nonpolar fluids. For example, the trans fraction of ionic surfactants was smaller in water than in the nonpolar segment fluid and in micelles. xv CHAPTER 1 INTRODUCTION Surfactants are an important species of amphiphilic molecules that over the years have received great attention from many industries and researchers. When present at high enough concentrations in certain solvents, some surfactants form complex structures known as micelles. Micelles are an important class of aggregates with wide theoretical and practical use, yet the behavior of micelles in polar fluids is still not well understood. In this study molecular dynamics methods have been used to investigate micellar structure and behavior. The molecular dynamics method has been shown to be a very useful tool in the study of complex molecular systems and is presently the only method to study both the statics and dynamics of micellar solutions. Nonetheless, no explanation of molec ular dynamics methods is given in this thesis, but exact details are found elsewhere (Allen and Tildesley, 1987; Haile, 1980). The purpose of the present work has been to determine the conformation of model surfactant molecules in nonpolar and polar fluids as well as in micellar solutions, and to study the effect of head group size, surfactant chain length and micellesolvent interaction models on micellar structure and shape via molecular dynamics. In chapter 2 a molecular dynamics investigation of the conformation of nalkanes in a monatomic fluid of methylene segments is described. In particular, properties such as trans fraction, radius of gyration and endtoend distance have been calcu lated for seven different chain lengths. In chapter 3 the conformation of two octyl surfactants and a poly (oxyethylene) head group in a monatomic fluid of methylenes are examined. The conformation of an "ionic methyl" octyl surfactant in water is also considered and the effect of the surfactant molecule on water structure is discussed in Appendix A. In chapter 4 a complete description of all intramolecular and intermolecular in teractions present in micelles and hydrocarbon droplets are given with a summary of all micellesolvent interaction models used. In chapter 5 the effects of micellesolvent interaction models on the internal struc ture and shape of the model micelles, as well as the conformation of surfactants inside the micelles, are analyzed. In chapter 6 the effect of surfactant chain length and head group characteristics on the micellar behavior are given, and results are compared with those of a hydrocarbon droplet as well as with experimental and other simulation results. Finally, In chapter 7 some general conclusions are given along with a few recom mendations on future work. CHAPTER 2 DILUTE NALKANE SIMULATIONS 2.1 Background Over the past few years there have been several molecular simulations and statisti cal mechanics calculations of model nalkanes. Molecular dynamics (MD) (Ryckaert and Bellemans, 1975, 1978; Weber, 1978; Edberg et al., 1986, 1987; Wielopolski and Smith, 1986; Toxvaerd, 1987, 1988; Clarke and Brown, 1986; Szczepanski and Maitland, 1983; Rebertus et al., 1979), Monte Carlo (MC) (Jorgensen, 1981a, 1981b; Jorgensen et al., 1981e; Bigot and Jorgensen, 1981; Banon et al., 1985), Brownian dynamics (BD) (Van Gunsteren et al., 1981), and statistical mechanics (SM) (Enciso et al., 1989; Zichi and Rossky, 1986a) have been used to determine the conformation of liquid, isolated and dilute nalkanes. However, none have examined the confor mation of long chain molecules mixed with segment molecules, as might be related to dilute polymer/monomer solutions, supercritical extraction and to micelle forming surfactant monomers. Also, little analysis of chain length effect on the conformation of nalkanes has been made. Molecular simulation is a powerful tool to investigate the chain conformation, yet results are usually subject to the effect of force field models, computational methods and simulation duration. Previously, molecular simulations have concentrated on short nalkanes, especially nbutane. Simulations of butane appear to be very simple, since they involve only one di hedral angle, but they are extremely difficult to run because they require a large amount of computation time for any statistically meaningful conformational results. Despite the abundance of nbutane simulations, there is no clear conclusion about its conformation in liquid or in dilute solutions. In general, all chain simulations should be carefully undertaken if an analysis on conformation is intended. In particular, special care should be given to the applica tion of constraints (Toxvaerd, 1987; Rallison, 1979; Helfland, 1979) and preferential sampling methods (Bigot and Jorgensen, 1981). We report here the results of a series of molecular dynamics of seven different model nalkanes having from 7 to 21 carbons in LennardJones monatomic fluids, without the application of chain constraints or preferential sampling. The objectives were to study chain length effects on structure and to determine the dominant ef fects on chain conformation. Results from these simulations may give some insight on the conformations of chains with fewer carbons without actually performing the simulations. 2.2 Chain Model The interaction potential model used here has been previously applied to micellar aggregates of model chain surfactants (Haile and O'Connell, 1984; Woods et al., 1986). Except for the rotational potential, it is similar to the one described by Weber (1978). The chain molecule is represented by a skeletal chain composed of n equaldiameter soft spheres representing methyl tails or methylene segments. The bond vibration and angle bending potentials are U (b) = (b, bo)2 (2.1) Ub () = 1 (cos o cos 0,O)2 (2.2) where bi6, is the bond length between segments i and i+1, bo is the equilibrium length, yj is the bond vibration force constant, 00 is the equilibrium bond angle, 0i is the angle between segments i, i+1 and i+2, and 7b is the bending vibration force constant. The bond rotational potential chosen for these simulations is that of Ryckaert and Bellemans (1975): U(O) = yr( 1.116 1.462cos 1.578cos2 +0.368cos3 0 +3.156 cos4 + 3.788 cos5 0) (2.3) Following Weber (1978) the intramolecular potential also includes a (69) Lennard Jones interaction between segments on the chain that are separated by at least three carbons, and for all intermolecular interactions. U = [2 (E) 3 (2.4) L ku ri \ r^ ) The parameters are listed in Table 2.1. Table 2.1: Intermolecular and Intramolecular Potential Parameters. rm f bo 00 1 N ir A J/mol A degree J/(mol A2) J/mol J/mol 4.00 419 1.539 112.15 9.25x105 1.3x105 8313 2.3 Simulation Details In all simulations, a box was created with N, particles having the size and mass of a methylene group along with the nalkane in the middle, and periodic boundary conditions were applied to solvent segments. The box boundaries moved with the chain to keep its center of mass always in the middle. Newton's second differential equations of motion were solved for each of the N, plus n soft spheres by using a fifthorder predictorcorrector algorithm due to Gear (1971). The number of solvent groups was chosen so that the chain was entirely in the box when fully extended. The time step used in solving the equations of motion was 1.395x1015 secs. The preparation procedure for all runs was to assign initial positions to all seg ments including those of the chain, which was not in the all trans conformation. The simulation was then run until equilibrium was reached, and the analysis performed on samples of 105 to 698 picoseconds. The state conditions for all runs are listed in table 2.2. The temperature is the same for all runs at 298 K, and the reduced density is 0.7 which corresponds to a number density of 0.0109 A3. Table 2.2: Summary of Simulations. Chain Length 7 9 11 13 15 17 21 Equilibrium Run psec 698 530 209 140 112 140 105 Number of solvent segments 101 99 245 243 485 483 479 2.4 Results 2.4.1 Chain Conformation. Average Trans Bond Fraction. The trans bond fraction was calculated for the seven different chain lengths and is shown in Table 2.3. In general the trans fraction does not show any trends, and the mean values differ. The uncertainty in all simulations is less than 10%. Although the trans fraction is not constant, the variation among chains is smaller than the statistical fluctuations. An average value of about 69% can be used for all chains from nbutane to nuneicosane. The results from our simulations can be compared to those from MD simulations of hydrocarbon fluids, dilute solutions and single molecules, and with BD simulations of single chains as well as statistical mechanics calculations. Comparisons include n butane (Tables 2.4 and 2.5), nhexane (Table 2.6), noctane (Table 2.7) and ndecane (Table 2.8). The following discussion assumes that the fraction of trans bonds is equal to the average value of 69% for all hydrocarbon chains up to 21 carbons. This value for n butane is higher than other MD simulations (Ryckaert and Bellemans, 1978; Edberg et al., 1986; Wielopolski and Smith, 1986; Toxvaerd, 1988; Rebertus et al., 1979) and statistical mechanics calculations (Enciso et al., 1989; Zichi and Rossky, 1986a), but comparable to BD (Van Gunsteren et al., 1981) and MC results (Jorgensen 1981a, 1981b; Jorgensen et al., 1981e; Bigot and Jorgensen, 1981; BAnon et al., 1985). Several factors may have affected the MD work, especially the limited duration of some Table 2.3: Average Structural Values for All Hydrocarbon Chains Chain Length 7 9 11 13 15 17 21 % Trans 71 67 68 68 71 68 70 6 8 9 8 4 7 7 < R > A 6.78 8.19 9.54 11.5 12.4 14.7 16.4 < R2 > A2 52.8 68.4 93.9 135.8 157.4 220.6 286.1 < S > A 2.18 2.66 3.13 3.63 4.06 4.63 5.36 < S2 > A2 5.82 8.54 11.7 15.9 19.5 26.0 34.7 Table 2.4: Average Values for Neat NButane as extrapolated from simulations, and as calculated by Ryckaert and Bellemans (1978), Edberg et al. (1986), Toxvaerd (1988), Jorgensen (1981a), Banon et al. (1985) and Wielopolski and Smith (1986). Author Method Temperature (K) % Trans Ryckaert MD 291 54 Edberg MD 291 60.6 Toxvaerd MD 291 62.6 Jorgensen MC 273 67.1 Banon MC 298 67.5 Wielopolski MD 285 58.5 Extrapolated MD 298 69 Table 2.5: Average Values for Single and Dilute NButane as extrapolated from simulations, and as calculated by Rebertus et al. (1979), Bigot and Jorgensen (1981), Zichi and Rossky (1986a), Enciso et al. (1989) and Van Gunsteren et al. (1981). Rebertus Bigot Zichi Enciso Van Gunsteren Extrapolated Method MD MC SM SM BD MD T (K) 298 298 298 300 291.5 298 % Trans 57 65 38 58.5 66 69 Table 2.6: Average Values for NHexane as extrapolated from Our Linear Fits and as Determined from Other Workers: Clarke and Brown (1986) ______ Clarke Extrapolated Method MD MD T (K) 300 298 < R2 >1/2 A 5.56 6.16 % Trans 68 69 Table 2.7: Average Values for NOctane as Interpolated from Linear Fits and as Determined from Other Workers: Szczepanski and Maitland (1983). Szczepanski Interpolated Method MD MD T (K) 396 298 < R > A 7.24 7.59 < R2 >1/2 A 7.28 7.57 < S > A 2.59 2.67 < S2 >1/2 A 2.60 2.67 % Trans 64 69 Table 2.8: Average Values for NDecane as Interpolated from Linear Fits and as determined by Ryckaert and Bellemans (1978), Edberg et al. (1987), Toxvaerd (1987) and Van Gunsteren et al. (1981). Ryckaert Edberg Toxvaerd Van Gunsteren Interpolated Method MD MD MD BD MD T (K) 481 481 481 481 298 < R > A 8.81 8.64 9.00 < R2 >1/2 A 8.87 8.87 8.82 8.72 9.06 < S > A 3.11 3.07 2.90 < S2 >1/2 A 3.16 3.11 3.12 3.08 3.18 % Trans 60.4 62.4 62.4 60 69 simulations and the use of constraints on the angles that reduces the rate of trans gauche transitions (Toxvaerd, 1987). It is also possible that some of the differences between simulations arises from variations in the intermolecular potential models (Banon et al., 1985). For pure nhexane the MD results of Clarke and Brown (1986) at 300 K gave a similar value to ours. For pure noctane the only available results are those of Szczepanski and Maitland (1983) at 394 K. They found 64% trans, a value consistent with ours at 298 K. For ndecane there are several available MD and BD results, but most were per formed at high temperatures. Again, however, the somewhat reduced trans fractions are consistent with our lower temperature result. 2.4.2 EndtoEnd Distance and Radius of Gyration The end to end distance < R > and the radius of gyration < S > were calculated for the different chains. Figure 2.1 shows how the average endtoend distance is a linear function of chain length. The standard deviation increases with the carbon number. Figure 2.2 shows that the average radius of gyration for all seven chains is also proportional to the chain length. In tables 2.42.8 the radius of gyration and the endtoend distance from the linear fits are compared to other simulation data. In general the agreement is good, particularly considering the differences in temperature for noctane and ndecane. 25 20 3 15 End to End Distance 10 5 0 I I I I I I I 7 9 11 13 15 17 21 Carbon Number Figure 2.1: The endtoend distance for hydrocarbon chains as a function of chain length Radius of Gyration g0 I I I I I I 5 7 9 11 13 15 Carbon Number Figure 2.2: The radius of gyration for hydrocarbon chains length as a function of chain 2.4.3 Trans Bond Distribution Trans bond distributions have been calculated but are of limited quantitative value. Simulations must be significantly longer for any statistically meaningful con clusions to be made. For example, symmetry in the bonds was sometimes not fully reached. However, the analyses can provide some insights. Several trends can be seen in the distributions of tables 2.9 and 2.11. Table 2.9 shows that for most chain lengths, the most probable number of trans bonds agrees with the average trans fraction. The distribution widens as the chain length decreases, and a few states are rarely reached, especially those with more gauche than trans bonds. The results from these probability distributions can be used to calculate a conformational entropy associated with the runs as measured by n3 S/k = pi lnpi (2.5) i=O where p, is the probability of finding a number of bonds, i, in the trans conforma tion, and n 3 is the number of dihedral angles on a chain of n segments. The uniform distribution entropy resulting in the highest conformational entropy can also be calculated: S/k = ln(n 2) (2.6) The ratio of equations 2.5 and 2.6 is a measure of randomness with respect to the uniform distribution with a value of unity showing maximum randomness. Values for Table 2.9: Probability of Finding a Given Number of Trans Bonds on the Chain. Number of 7 9 11 13 15 17 21 Trans Bonds 0 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1 0.01 0.01 0.00 0.00 0.00 0.00 0.00 2 0.30 0.09 0.00 0.00 0.00 0.00 0.00 3 0.51 0.20 0.03 0.00 0.00 0.00 0.00 4 0.18 0.34 0.22 0.02 0.00 0.00 0.00 5 0.32 0.28 0.11 0.00 0.05 0.00 6 0.05 0.22 0.23 0.01 0.00 0.00 7 0.24 0.37 0.11 0.04 0.00 8 0.01 0.22 0.38 0.26 0.01 9 0.05 0.38 0.22 0.03 10 0.00 0.11 0.30 0.07 11 0.02 0.09 0.16 12 0.00 0.04 0.24 13 0.00 0.17 14 0.00 0.16 15 0.08 16 0.05 17 0.02 18 ________ ___ ____ 0.00 ,pilnpi 1.06 1.44 1.50 1.51 1.35 1.67 2.00 the entropy ratio for all chain lengths is shown in Table 2.10. The entropy ratio of the chains in dense LJ fluid is not affected by chain length (about 0.65 0.06), even though longer chains have larger numbers of available states and might be expected to have a much higher entropy ratio. Apparently, all chains have the same constraint from reaching some of the states such as the g~g: conformations. These were shown by Pitzer (1940) to be unfavored by an overlapping called the "pentane interference." Table 2.11 shows that within the statistics of 0.03 the probability of finding a di hedral angle in trans conformation is essentially equal for all angles on the chain. The uncertainties can be estimated from comparing results for bonds in the same position relative to the chain end. In particular there seem to be no trends of probabilities from the ends to the middle of the chains. The probability distributions can be used to calculate another conformational entropy for the different alkane chains, using pi in equation 2.5 as the probability of a particular dihedral angle, i, to be in trans conformation. Figure 2.3 shows that the entropy is equal to the logarithm of the number of states, n 3, confirming the equiprobability of all angles to be in trans conformation. 2.5 Conclusions The conformation of isolated chains of segments in fluids of segments at liquid densities have been examined by molecular dynamics. The trans fraction is about 2/3 with uniform distribution among the dihedral angles. Table 2.10: Randomness of Conformation: Ratio of Equations 2.5 and 2.6. Chain Length 7 9 11 13 15 17 21 Epilnpi 1.06 1.44 1.50 1.51 1.35 1.67 2.00 ln(n 2) 1.61 1.95 2.20 2.40 2.56 2.71 2.94 ,ni npi 0.66 0.74 0.68 0.63 0.53 0.62 0.68 Table 2.11: Probability of Finding a Particular Dihedral Angle in the Trans Confor mation. Dihedral Angle 7 9 11 13 15 17 21 01 0.25 0.15 0.11 0.11 0.06 0.03 0.05 02 0.26 0.19 0.16 0.12 0.12 0.07 0.05 03 0.25 0.15 0.12 0.11 0.07 0.08 0.07 04 0.24 0.18 0.13 0.08 0.11 0.06 0.03 05 0.15 0.13 0.13 0.09 0.07 0.05 06 0.18 0.06 0.08 0.07 0.08 0.07 07 0.16 0.11 0.05 0.08 0.07 0s 0.13 0.09 0.08 0.05 0.08 09 0.05 0.07 0.07 0.04 1io 0.13 0.09 0.08 0.08 Oil 0.08 0.10 0.03 012 0.11 0.08 0.05 013 0.05 0.08 414 0.10 0.05 415 0.03 416 0.07 17 0.05 18 ________________0.06 Uniform 0.25 0.17 0.13 0.10 0.08 0.07 0.06 24 3 2 YPilnpi 1 0 i 0 1 2 3 In(n3) Figure 2.3: Entropy S/k = Ipilnpi is plotted as a function of ln(n 3). For an nalkane there are n3 dihedral bonds and p, is the probability of finding a bond in trans conformation. 25 The endtoend distance and radius of gyration are linear functions of chain length for chains of 7 to 21 segments. CHAPTER 3 SIMULATIONS OF SURFACTANTS IN A LENNARDJONES FLUID OF SEGMENTS AND IN WATER 3.1 Background In the last chapter we have discussed the conformation of nalkanes, and have shown some important properties of nalkanes. In this chapter we direct our attention to the study of ionic and nonionic surfactants. Surfactants are an important class of molecules due to their amphiphilic behavior. They are used in the formation of many colloidal solutions, and have applications in enhanced oil recovery, detergency, catalysis and many other industries, however there has been no detailed simulation studies of dilute surfactant solutions. A conformational study of free surfactants is particularly important for comparison with micellar surfactants, as well as with free alkanes and those in hydrocarbon droplets. Many experimental studies of micellar solutions and thermodynamic studies of micelles have claimed that surfactant chains change conformation upon micellization by making the trans fraction higher in micelles than in hydrocarbon fluids or water. To date, no well documented molecular simulations have been performed to verify this assertion. In this chapter a series of molecular dynamics simulations have been performed to study the conformation of an octyl "ionic methyl" surfactant in a LennardJones 26 fluid of methylene segments and in water, and an octyl "nonionic sulfate" surfactant in a LennardJones fluid of segments. We also describe a simulation of a nonionic surfactant head group (poly (oxyethylene)) in a LennardJones fluid of segments. 3.2 Potential Model for Segmented Molecules 3.2.1 Octyl Surfactants The interaction potential model for the surfactant used in these simulations, ex cept for rotational effects, is similar to the one described by Weber (1978). The surfactant molecule is represented by a skeletal chain composed of 8 equaldiameter soft spheres each representing a methyl tail or methylene segment and a soft sphere representing the head group (Figure 3.1). The bond vibration and angle bending potentials for groups other than the head group are those of Weber (1978) taken from a simulation of nbutane. U,, (bi) = (bi bo)2 (3.1) Ub (Oi) = I7b (cos 00 cos Oi)2 (3.2) where bi6, is the bond length between segments i and i+1, bo is the equilibrium length, y,, is the bond vibration force constant, 00 is the equilibrium bond angle, 0i is the angle between segments i, i+1 and i+2, and 7b, is the bending vibration force constant. The bond rotational potential chosen for these simulations is that of Ryckaert and Bellemans (1975): Octyl "Ionic Methyl" Head 112.15o 1.539 A Octyl "Nonionic Sulfate" 2.6 A 1 Head 1400 Tail Tail Figure 3.1: Model octyl surfactants. U(O) =yr( 1.116 1.462coso1.578cos2 + 0.368cos3 +3.156 cos4 4 + 3.788 cos5 q) (3.3) The intramolecular potential also includes a (69) LennardJones interaction between segments on the chain that are separated by at least three carbons. For j i >3 : ULJ (rij) = [2 3) r 3 6] (3.4) The parameters involving all segments on the octyl "ionic methyl" and the octyl "nonionic sulfate" surfactants (Muller et al., 1968) are listed in Table 3.1. 3.2.2 Poly (Oxyethylene) Molecule The model used for the poly (oxyethylene) chain is composed of six oxyethylene segments. As shown in Figure 3.2 each oxyethylene segment is a CH2CH2O. The first carbon on the poly (oxyethylene) has three hydrogens and the last oxygen on the chain has one hydrogen. Nineteen soft spheres with different sizes and masses were made to represent methyl, methylene and oxygen groups. The hydrogen atom attached to the last oxygen on the chain is also represented by a soft sphere. All groups interact via bond vibration, angle bending and rotation, as well as (612) LennardJones and electrostatic interactions between groups that are separated by at least three groups. Table 3.1: Bond Parameters of "Methylene" and "Sulfate" Groups. "Sulfate" Pa rameters are Used when an Intramolecular Interaction Involves a "Nonionic Sulfate" Head Group. "Methylene" Parameters are Used with All Other Intramolecular In teractions. Parameter "Methylene" Value "Sulfate" Value Units b0 1.539 2.6 00 112.15 140 degree I 9.25x105 2.7x104 J/(mol A2) 1b 1.3x105 9.1 x105 J/mol 7r ___ 8313 20000 J/mol Poly (Oxyethylene) ^ CH3 ) CH2 1.526A 0 CH2 ) CH2 1.425A 0 CH2 CH2 0 CH2 ) CH2 0 CH2 ) CH2 ^111.80 0 CH2 ) CH2 0 H Figure 3.2: Model poly (oxyethylene) molecule. 109.5" For j i > 3 U ) \12 \ 61 +(q4) + 4E (3.5) 3 jrij ri The values for coulombic interaction parameters (Table 3.2) are similar to those by Jorgensen (1981c) from a study of alcohols and ethers. The bond vibration and angle bending parameters were those extracted from molecular mechanical studies by Weiner et al. (1987) and used in equations 3.1 and 3.2. The rotational potential is from a Monte Carlo study of nalkyl ethers by Jor gensen and Ibrahim (1981d): V(O) = D0 + D1 cos 0 + D2 cos 20 + D3 cos 30 (3.6) A complete list of intramolecular parameters is given in Tables 3.3, 3.4 and 3.5. 3.3 Interaction Models for Molecules in a LennardJones Fluid of Segments and Simulation Details In simulations involving the "ionic methyl" and "nonionic sulfate" surfactants in a LennardJones fluid of segments a LennardJones (69) potential (equation 3.4) is used for all surfactant segmentfluid segment interactions. In addition a coulombic interaction is used to model the head groupcounterion attraction in the case of the "ionic methyl" simulation. U(r =qje2 (3.7) t)r? As shown in Table 3.6 all segments on the surfactant or in the fluid have the same LennardJones parameters except the "nonionic sulfate" surfactant head group which has different parameters. Table 3.2: LennardJones and Coulombic Interaction Parameters for Poly (oxyethy lene). Table 3.3: Bond Parameters for Poly (oxyethylene). Site q f o electrons J/mol A CH2,CH3 0.29 480 4 0 0.58 811 3.05 0 (of OH) 0.69 811 3.05 H 0.40 0 0 Bond 7^ bo __ __ J/(mol A2) A CH2CH2 9.25 x 105 1.526 CH20 1.14 xl06 1.425 0H 1.97 x 106 0.960 Table 3.4: Angle Parameters for Poly (oxyethylene). Table 3.5: Torsion Parameters for Poly (oxyethylene). Bond l Do D1 D2 D3 __ J/mol CH2CH2OCH2 8314 1.053 1.250 0.368 0.675 OCH2CH2O 8314 1.078 0.355 0.068 0.791 CH2CH2OH 8314 1.053 1.250 0.368 0.675 Angle 7o 0o J/mol degree CH2CH2O 1.651 x105 109.5 CH2OCH2 2.067 x105 111.8 CH2OH 1.135 x105 108.5 Table 3.6: Intermolecular Potential Parameters for "Methylene" and "Sulfate" Groups. "Sulfate" Parameters are Used when an Intermolecular Interaction Involves a "Nonionic sulfate" Head Group. "Methylene" Parameters are Used with All Other Intermolecular Interactions. In the poly (oxyethylene) simulation a LennardJones (612) potential plus an electrostatic interaction is used to model all pair potentials. In each of the simulations involving an octyl surfactant a box with 108 particles each with a size and mass of a methylene group was created, then 9 particles in the middle of the box are replaced by the surfactant chain. In the octyl "ionic methyl" surfactant simulation, one methylene group is also replaced by a counterion that has the same intermolecular potential as other solvent groups, but with a positive charge of le. When the simulation is started the surfactant chain is not in the alltrans conformation, and periodic boundary conditions are applied to the solvent segments and to the counterion, but not to the surfactant molecule. The box is moved according to the movements of the surfactant molecule so that its center of mass is always in the middle. In the simulation involving the poly (oxyethylene) chain, the simulation box in cluded 500 particles of which 19 were replaced by the poly (oxyethylene) molecule. Newton's second differential equations of motion were solved for all segments in the solvent and on the chain by using a fifthorder predictorcorrector algorithm due to Gear (1971). All simulation runs consisted of a large number of steps until equilibrium was reached as determined by constant average energy and temperature. A sample of subsequent time steps is then used to calculate the average properties. Simulation details for all runs are shown in Table 3.7 Table 3.7: Simulation Details for Runs in LennardJones Fluid of Segments. Simulation Time Step Equilibration Equilibrium secs Steps Run "ionic methyl" surfactant 1.395x 1015 50,000 150,000 "nonionic sulfate" surfactant 1.395x 1015 40,000 130,000 poly (oxyethylene) 1.331 x 1015 10,000 75,000 3.4 Results for Molecules in a LennardJones Fluid of Segments In this section we report results on endtoend distance and radius of gyration distributions, and the probability distribution of the number of bonds in trans frac tion, and the probability of a bond to be in trans fraction. Average values for the trans fraction, endtoend distance and radius of gyration are also reported. 3.4.1 Average and Mean Values In Tables 3.8, 3.9 and 3.10 we show the average and mean values for trans frac tions, endtoend distances and radii of gyration. The trans percentage for both the "ionic methyl" and "nonionic sulfate" surfactants is about 736%. A value that is similar to the trans fraction of nonane in dilute solution (see chapter 2) and that in micelles of "polar methyl" and "nonionic sulfate" surfactants (see chapters 5 and 6). The trans fraction in poly (oxyethylene) is 461% indicating a mostly gauche conformation. Apparently neither the size of the head group on the octyl "nonionic sulfate" surfactant nor the added negative charge on the head group of the octyl "ionic methyl" surfactant have an effect on the average trans fraction as compared to a 9carbon nalkane. On the other hand the trans fraction for poly (oxyethylene) is considerably different from the corresponding 19carbon nalkane. There may be several factors affecting the conformation of this molecule, but the dominant one is probably the presence of charges on different segments of the molecule. The distri bution of charges on the chain yield several extra interactions such as dipoledipole, quadrupolequadrupole, hydrogen bonding or any combination of these interactions. Table 3.8: Average Properties for the Octyl "Ionic MNlethyl" Surfactant in a Lennard Jones Fluid of Segments. Table 3.9: Average Properties for the Octyl LennardJones Fluid of Segments. Table 3.10: Average Properties for Poly (Oxyethylene) Segments. Property % Trans <$2> Value 46 1 11.9 148. 4.13 20.2 Units A A2 A A2 "Nonionic Sulfate" Surfactant in a in a LennardJones Fluid of Property Value Units %Trans 74 6 < R > 8.20 A < R2 > 68.7 A2 < S > 2.70 A < S2 > 8.76 A2 Property Value Units %Trans 73 5 < R > 9.96 A < R2 > 100. A2 < S > 3.53 A < S2 > 16.1 A2 The larger LennardJones energy parameters for chain oxygen and different rota tional potentials for oxygens and methylenes also would lead to differences between the 19carbon nalkane conformation and that of poly (oxyethylene). The endtoend distance and radius of gyration for the octyl "ionic methyl" sur factant are similar to those of the 9carbon nalkane (see chapter 2), while the "non ionic sulfate" surfactant shows a larger endtoend distance and radius of gyration. Apparently the presence of a negative charge on the "ionic methyl" surfactant head group has little effect on these quantities in contrast to the influence of head group size and mass. The radius of gyration shows a 26% increase for the "nonionic sulfate" surfactant over that of the "ionic methyl" surfactant. Part of this disparity in radius of gyration could be accounted for by the mass of the head group which is seven times that of the "ionic methyl" surfactant head group, and by the longer chain from end to end when in the alltrans conformation. Additionally the mean endtoend distance for the "nonionic sulfate" molecule is 21% larger than that of the "ionic methyl" surfactant mainly because small separations are not accessible (see below). The end toend distance and the radius of gyration for the poly (oxyethylene) molecule are much smaller than for the corresponding 19carbon nalkane, suggesting a bunched up conformation consistent with a small average trans fraction. 3.4.2 Endtoend Distance The plot for the endtoend distance of the "ionic methyl" surfactant is shown in Figure 3.3. The distribution is skewed, though it has a single most probable peak. 0.02 P(ri9) 0.01 0.005 0 7 4 6 8 10 rg(A) Figure 3.3: Endtoend distribution for the octyl "ionic methyl" surfactant in a LennardJones fluid of segments. Several small peaks arising from allowed and forbidden conformations are present. The most probable value for the endtoend distance is higher than the average value. The endtoend distance distribution for the "nonionic sulfate" surfactant is shown in figure 3.4. This distribution is also skewed. The occurrence of small peaks is not as frequent as in the distribution for the "ionic methyl" surfactant. In both cases, extra peaks in the distributions at distances below the peak are sharper than those above the peak. The endtoend distance distribution for the "ionic methyl" surfactant chain ex tends from quite small distances of 4A to 11A. Basically the endtoend distance samples all available conformational space from 4A (rmin in the LennardJones po tential) to 11A (the alltrans endtoend distance). The endtoend distance distribution for the "nonionic sulfate" surfactant extends from about 7A to about 12.3A. Here again the long range part of the distribution is indicative of the alltrans endtoend distance, while the short range part is indicative of the head and tail approaching each other to rmin in the LennardJones potential. Figure 3.5 shows that the endtoend distance for the poly (oxyethylene) molecule is a fairly symmetric distribution which reaches from values around the LennardJones cr to values less than the alltrans endtoend distance. The difference between the average and the most probable values of the endtoend distance is less than 5 %. The short range limit indicates that hydrogen bonding may occur between the terminal 0.025 P(r19) 'U 0.01 0.005 0 T  5 6 8 10 12 13 r19(A) Figure 3.4: Endtoend distribution for the octyl "nonionic sulfate" surfactant in a LennardJones fluid of segments. 0.008 P(r1, 1) 0.004 Figure 3.5: Endtoend distribution for poly (oxyethylene) in a LennardJones fluid of segments. hydrogen atom and the first occurring oxygen atom on the other side of the molecule. Several sharp peaks are present, a result of the observed motions of the chain among its 16 bonds (which should be compared to only 6 for the octyl surfactants). The distribution has large amplitude spikes, particularly around the peak. 3.4.3 Radius of Gyration In figures 3.6, 3.7 and 3.8 are shown the radii of gyration for all three simulations. The radius of gyration distributions for the "ionic methyl" and the "nonionic sul fate" surfactants are fairly symmetric and smooth with some small peaks at discrete positions on the chains, indicating different conformations. The difference in both simulations between the average and most probable values of the radius of gyration is less than 2%. The radius of gyration distribution for the poly (oxyethylene) molecule is not symmetric, and has a distinctive shoulder at 3.6 A, while the main peak oc curs at 4.2 A. There are also many more extra peaks than for the octyl surfactants. The particular conformation of the shoulder in this distribution is uncertain; it could be due to dipoledipole, quadrupolequadrupole, chargecharge or hydrogen bonding interactions. 3.4.4 Probability of Finding a Number of Bonds in the Trans Conformation. The probabilities of finding a number of bonds in the trans conformation for simu lations in the LennardJones fluid of segments are shown in Tables 3.11, 3.12 and 3.13. (In the analysis that follows, a bond is considered to be in trans conformation if cos(q#) in equations 3.3 and 3.6 is less than 0.5. For all other values of cos(0,) the 0.06 0.04 / P (r) 0.02 0 2 2.5 3 3.5 r(A) Figure 3.6: Radius of gyration distribution for the octyl "ionic methyl" surfactant in LennardJones fluid of segments. 0.06 0.04 P(r) 0.02 0 2.7 3 3.5 4 4.3 r(A) Figure 3.7: Radius of gyration distribution for the octyl "nonionic sulfate" surfactant in a LennardJones fluid of segments. 0.025 0.02 0.015 P(r) 0.01 0.005 0  2.5 3 4 5 5.5 r(A) Figure 3.8: Radius of gyration distribution for poly (oxyethylene) in a LennardJones fluid of segments. Table 3.11: Probability of Finding a Number of Bonds in the Trans Conformation on the Octyl "Ionic Methyl" Surfactant. Number of bonds 0 1 2 3 4 5 6 Probability 0.0 0.0 0.0 0.12 0.39 0.44 0.05 Table 3.12: Probability of Finding a Number of Bonds in the Trans Conformation on the Octyl "Nonionic Sulfate" Surfactant. Number of bonds 0 1 2 3 4 5 6 Probability 0.00 0.00 0.03 0.12 0.33 0.51 0.01 bond is considered gauche.) This probability is similar for both octyl surfactants indicating that states with high gauche conformations are not accessible, while states with one or two gauche bonds are the most probable. The trans bond probability distribution for poly (oxyethylene) is fairly symmetric, and states with 7 or 8 trans bonds are most probable. This probability is consistent with the average trans fraction found earlier. 3.4.5 Probability of Finding a Particular Bond in the Trans Conformation. This particular probability looks at each bond separately. The results for this particular property have significant statistical uncertainty due to large fluctuations in the average values. The standard deviation on these values can be as high as the average values. Nonetheless it can be seen that all bonds on the "ionic methyl" surfactant have a similar probability to be in the trans conformation (Table 3.14). The probability of being in trans conformation for each single bond on the "nonionic sulfate" surfactant is highest for bond 1 (bond involving head groups and segments 2, 3 and 4) and then follows a somewhat decreasing probability toward the tail (Ta ble 3.15). This result is probably due to different head group mass, size and rotational potential. Table 3.16 shows that torsional bonds on poly (oxyethylene) of the groups XC OX (1,2,4,5,7,8,10,11,13,14,16) generally have a higher probability to be in the trans conformation than bonds of the goups XCCX (3,6,9,12,15). The exception is at Table 3.13: Probability of Finding a Number of Bonds in the Trans Conformation on a Poly (Oxyethylene). Number of Bonds 0 11 2 3 4 5 6 1 8 Probability 0.0 0.0 0.0 0.01 0.04 0.09 0.15 10.21 0.21 Number of Bonds 9 10 1 1 12 13 14 15 1 1 Probability 0.16 0.08 0.03 0.01 0.0 0.0 0.0 0.0 Table 3.14: Probability of Finding a Particular Bond in the Trans Conformation on the Octyl "Ionic Methyl" Surfactant. Bond Number 1 2 3 4 5 6 Probability 0.14 0.19 0.18 0.15 0.17 0.18 Table 3.15: Probability of Finding a Particular Bond in the Trans Conformation on the Octyl "Nonionic Sulfate" Surfactant. Bond Number 1 2 3 4 5 6 Probability 0.23 0.19 0.18 0.15 0.12 0.13 Table 3.16: Probability of Finding a Particular Bond in the Trans Conformation on the Poly (Oxyethylene) Molecule in a LennardJones Fluid of Segments. Bond Number 1 2 3 4 5 6 7 8 Probability 0.07 0.07 0.05 0.07 0.07 0.06 0.07 0.07 Bond Number 9 10 11 12 13 14 15 16 Probability 0.06 0.07 0.07 0.06 0.06 0.08 0.04 0.05 the terminal hydrogen end (bond 16) where the hydrogen bonding probably affects the conformation. 3.5 Octyl "Ionic Methyl" Surfactant in Water 3.5.1 Model In the previous sections of this chapter and in chapter 2 we have discussed the simulation of solutes in a LennardJones fluid of segments. In this section we turn our attention to simulations of aqueous solutions. Simulations involving water are usually uncertain since there is no generally valid potential for water. Rather, there are several effective pair potentials such as the BF (Bernal and Fowler, 1933), ST2 (Stillinger and Rahman, 1974, 1978), MCY (Matsuoka et al., 1976), SPC (Berendsen et al., 1981), TIPS (Jorgensen, 1981c), TIPS2 (Jorgensen, 1982), and TIP4P (Jorgensen et al., 1983). Overall the SPC, ST2, TIPS2 and TIP4P models give reasonable structural and thermodynamic descriptions of liquid water (Jorgensen et al., 1983), but the simplicity of SPC from a computational point of view makes it attractive. It is not clear yet which model predicts the best dynamics, though it seems that SPC has a slight edge over TIPS2 and TIP4P (Strauch and Cummings, 1989; Alper and Levy, 1989) in predicting the dielectric constant. Consequently the SPC potential is used here to model water. The octyl "ionic methyl" surfactant molecule is similar to the one described in 3.2.1 except that the interactions between chain segments are modeled by a (612) LennardJones potential instead of the (69) potential. This should not affect the conformation of the surfactant since the excluded volume effects for nbutane have been modeled equally well by an r12 or an r9 contribution to the LennardJones potential (Weber, 1978; Ryckaert and Bellemans, 1978), and no difference was found here for surfactants in micelles (see below). The water potential used was originally given by Berendsen et al. (1981), and consists of two parts: 1) a soft sphere interac tion between oxygen atoms on the water molecule and 2) a coulombic potential that involves oxygenoxygen, oxygenhydrogen and hydrogenhydrogen interactions. q i~e [ ( \7 12 or \ 61 (rj) qje2 = + 4c (3.8) rij i i Effectively the SPC model consists of 10 interactions, of which one is Lennard Jonesian while the remaining nine contributions are coulombic. In our present model for the dilute solution of the octyl "ionic methyl" surfactant, there are basically six kinds of interactions: 1) the waterwater interaction which is modeled by the SPC potential; 2) the waterchain segment interaction which is modeled by a (612) LennardJones term; 3) the waterhead group interaction that is modeled by a LennardJones interaction plus a coulombic term to account for charges on the surfactant head and the water molecules; 4) the watercounterion interaction that is modeled similar to the waterhead group interaction; 5) the head groupcounterion interaction that is similar to the watercounterion interaction; 6) the chain segmentcounterion interaction which is modeled by a (612) Lennard Jones potential. The parameters for all the potentials are shown in Table 3.17. These parameters were used earlier by Jonsson et al. (1986) in their study of an octyl surfactant micelle. The counterion has the size of a sodium ion, while the head group is a methylsized segment that is negatively charged. The simulation techniques chosen here are different from those described in prior simulations. In particular the bonds and angles in the water molecule are held rigid using a quaternion method (Evans, 1977). A fourth order predictorcorrector method is used to solve the translational and rotational equations of motion, and a gaussian thermostat is used to keep a constant temperature. In this simulation the box contained 206 water molecules, one counterion and the octyl "ionic methyl" chain. The surfactant molecule is free to wonder around the box. Periodic boundary conditions are applied to all molecules including the surfactant. There is no clear way on how to apply periodic boundary conditions to the surfactant molecule once one of its segments leaves the main box. Therefore, when this occurred the simulation was stopped and restarted from the previous configuration with the surfactant molecule moved to the middle of the box. The minimum image criteria is used to evaluate all interactions, except for interactions on the surfactant chain, and a spherical cutoff distance is used for all short and long range interactions. This apparently crude assumption was used in the original development of the SPC model. The time step used in this simulation (0.5 fs) is relatively short compared to prior simulations. The run proceeded with 20000 time steps until the usual criteria of Table 3.17: LennardJones Parameters for Interacting Atoms and Segments. cr is Given in A and c is Given in J/mol. Net Charges are Given in Units of the Elementary Charge e=1.602x 1019esu. a chain head oxygen sodium hydrogen segment group on water ion on water Chain segment 3.92 3.92 3.279 2.667 0.00 head group 3.92 3.279 2.667 0.00 oxygen on water 3.165 2.667 0.00 sodium ion 2.667 0.00 hydrogen on water 0.00 e chain head oxygen sodium hydrogen segment group on water ion on water Chain segment 697 697 702 313 0.00 head group 697 702 313 0.00 oxygen on water 650 313 0.00 sodium ion 313 0.00 hydrogen on water 0.00 qjqj chain head oxygen sodium hydrogen segment group on water ion on water Chain segment 0.00 0.00 0.00 0.00 0.00 head group 1.00 0.82 1.00 0.41 oxygen on water 0.672 0.82 0.336 sodium ion 1.00 0.41 hydrogen on water 0.168 57 equilibrium was reached followed by an additional 39000 time steps to calculate the equilibrium and dynamic properties of the solution. 3.5.2 Results The average values for the endtoend distance, the radius of gyration and the trans fraction are shown in Table 3.18. The trans percentage for the "ionic methyl" surfactant in water is 60%. This value is significantly different from the trans fraction of the octyl "ionic methyl" surfactant in a LennardJones fluid of segments (see section 3.4.1) and the "polar methyl" octyl surfactant in micelles (see chapter 6). Apparently the presence of an aqueous solvent significantly affects the surfactant conformation. The average endtoend distance and the radius of gyration for the "ionic methyl" surfactant in water are similar to those of the same surfactant in the LennardJones fluid of segments, but the distribution and most probable values are significantly different in the two cases. In general, the average results are consistent with the concept of chain straightening upon micellization. Plots for the endtoend distance and the radius of gyration are shown in Fig ures 3.9 and 3.10. The endtoend distribution is irregular though fairly symmetric. There are two major peaks near 8.4 A with several other peaks at shorter distances. When compared to the endtoend distribution for the octyl "ionic methyl" surfac tant in a LennardJones fluid of segments, the distribution is narrower, and does not reach either the alltrans endtoend distance or the LennardJones parameter a. The large gauche fraction explains the first effect and a possible explanation for the Table 3.18: Average Properties for the Octyl "Ionic Methyl" Surfactant in Water. Property Value Units %Trans 603 < R2 > 67.4 A2 < S > 2.59 A < S2 > 8.07 A2 0.03 0.02 P (r 19) 0.01 ri19(A) Figure 3.9: Endtoend distribution for the octyl "ionic methyl" surfactant in water. 0.08 0.06 P(r) 0.04  0.02 0 2 2.2 2.4 2.6 2.8 3 r(A) Figure 3.10: Radius of gyration distribution for the octyl "ionic methyl" surfactant in water. 61 short range difference could be the presence of water molecules that are consistently around the negatively charged head group preventing close contact with the tail seg ment. Thus, there is a peak at 6.5 A corresponding to the distance between two methylene segments separated by a water molecule. The presence of other waters around the head may force the distance to be larger than 7A. The radius of gyration distribution is fairly symmetric, but has several more peaks and is narrower than the corresponding distribution in the LennardJones fluid of segments. Its slight skewness is towards shorter distances rather than longer ones as in the LennardJones fluid. Such details of the difference between segments and water environments are uncertain. The probability of a number of bonds to be in the trans conformation and the probability of a bond to be trans are shown in Tables 3.19 and 3.20. From Table 3.19 it can be seen that all states are accessible except the allgauche conformation, and states with two or three gauche bonds have the highest probabilities. The distribution is similar to that of Table 3.11 for the LennardJones fluid but the number of bonds is decreased by one with the alltrans configuration being quite improbable. It can be seen in Table 3.20 that all bonds have about the same probability to be in trans conformation. Thus, the water solvent shows no bond preference as does the LennardJones fluid. Table 3.19: Probability of Finding a Number of Bonds in the Trans Conformation on the Octyl "Ionic Methyl" Surfactant in Water. Number of Bonds 0 1 2 3 4 5 6 Probability 0.00 0.04 0.09 0.31 0.42 0.14 0.02 Table 3.20: Probability of Finding a Particular Bond in the Trans Conformation on the Octyl "Ionic Methyl" Surfactant in Water. Bond Number 1 2 3 4 5 6 Probability 0.19 0.15 0.16 0.14 0.20 0.q17 3.6 Conclusions From these simulations of surfactant molecules, it is apparent that ionic surfac tants in nonionic fluids behave differently from those in water. In particular the surfactant conformation is more trans in nonpolar fluids (73%) than in water (60%). The conformation of ionic surfactants in a LennardJones fluid of segments is very similar to that of a hydrocarbon chain of corresponding length regardless of head group size, mass and charge. The conformation of surfactants in micelles (chapters 5 and 6) is more like that in nonionic fluids than in water. On the other hand simu lations involving poly (oxyethylene) in a LennardJones fluid of segments show that the gauche conformation is preferred (46% trans), and that the distribution of bond orientations is considerably different from that of a model methylene chain. CHAPTER 4 MODEL MICELLE 4.1 Background Micelles are an important class of aggregates with wide theoretical and practical use, yet the behavior of micelles in polar fluids is still not well understood. Over the past few years, considerable experimental work has examined micellar structure, mi celle shape and fluctuations, the micellar chain conformations, and water penetration in the micelle core. Experimental methods that are used to study the micellar behavior involve spec troscopic techniques such as Small Angle Neutron Scattering (SANS) (Bendedouch et al., 1983a, 1983b; Tabony, 1984; Cabane et al., 1985; Chen, 1986; Hayter and Pen fold, 1981; Hayter and Zemb, 1982; Hayter et al., 1984), Nuclear Magnetic Resonance (NMR) (Cabane, 1981; Chevalier and Chachaty, 1985; Ulmius and Lindmann, 1981; Zemb and Chachaty, 1982), Light Scattering (Candau, 1987; Chang and kaler, 1985), Luminescence Probing (Zana, 1987), Spin Labeling (Taupin and Dvolaitzky, 1987) and Xray Scattering (Zemb and Charpin, 1985). Due to the limitations of some spectroscopic techniques on resolution of time and space, and the wide distribution of micellar size and shape in solution (BenShaul and Gelbart, 1985; Degiorgio, 1983), experimental results often disagree. At present SmallAngle Scattering is the only method that allows distances to be measured in the range 5 to 500 A (Cabane, 1987). It has been suggested that SANS, particularly with careful isotropic substitution, is the most promising technique for the study of local structure, degree of water penetration in the micelle core, and micellar shape. Although NMR may be the most powerful and versatile spectroscopic technique for studying systems in the liquid state, the interpretation of primary spectroscopic data is difficult (Lindmann et al., 1987). In fact, NMR is only unambiguous in describing the chain conformation (Cabane et al., 1985). Micelles have been also studied by structural models. Many of the modeling efforts suggest a structure that differs from the original "pincushion" image of Hartley (1935), but most models make simplifying assumptions ranging from a simple "matchstick" construction (Fromherz, 1981) and a "brush heap" configuration (Menger, 1979, 1985) to a more complex statistical lattice theory (Dill, 1982, 1984a, 1985; Dill and Flory 1980, 1981; Dill et al., 1984b; Cantor and Dill, 1984) and an equal density micelle model (Gruen, 1981, 1985a, 1985b). Unfortunately, the quantitative, and even qual itative model descriptions of micelle behavior may not be accurate because of the assumptions used in their development and the apparent complexity of the micelle structure. There also have been a few attempts to study micelle structure by computer simu lations, mainly Molecular Dynamics and Monte Carlo simulations. The advantage of computer simulations over structural models and experimental methods is that their only assumption involves the intramolecular and intermolecular potentials while de tailed molecular information can be obtained. Haan and Pratt (1981a, 1981b) used Monte Carlo Methods to simulate a micelle with a mean interaction between surfac tants. Molecular Dynamics simulations have attempted to model the micellesolvent interaction without including a solvent (Haile and O'Connell, 1984; Woods et al., 1986; Farrell, 1988) while J6nsson et al. (1986) and Watanabe et al. (1988) have sim ulated sodium octanoate micelles of 15 members in model water. The results from all simulations appeared to be different, apparently because of their use of different force field models and computational methods. 4.2 Micelle Models The model used in these simulations is similar to the one described in Woods et al. (1986): a skeletal chain composed of 8 equaldiameter soft spheres for the methyl tail or methylene segments and a soft sphere for the head group. Methyl, methylene segments and head group on the same chain interact via bond vibrational, bending (Weber, 1978) and torsional forces (Ryckaert and Bellemans, 1975) as well as a (6 9) LennardJones potential between segments that are separated by at least three carbons (See Chapter 2). The intermolecular interactions can be modeled by using five potentials. The different interactions are shown in figure 4.1. Segmentsegment and headsegment interactions are modeled by a pairwise additive LennardJones (69) form, HeadHead Interaction ChainChain Interaction Hydrophilic Interaction Figure 4.1: Model for intermolecular interactions in micelles [( .)9 /r \ 6 ULJ (rj) = 2 () 3 (r)1 (4.1) For headsegment interactions the radius of the minimum potential, rm,, is adjusted to account for the difference between the diameter of the head group and that of the chain segment: ,headsegment 1 rhm 2 (rm + rhh) (4.2) Headhead interactions are modeled by a purely repulsive potential which includes both dipolelike repulsion and excluded volume effects: Uhh (rij)= [2 3+3r (4.3) The micellesolvent interactions are not modeled on a particle basis. Rather, the surfactant molecules are surrounded with a varying thickness spherical shell used to mimic a polar solvent. The micellesolvent interactions can be divided into two contributions, the chainsolvent and headsolvent interactions. 4.2.1 ChainSolvent Interaction Two models are proposed to account for the chainsolvent interaction (Figure 4.2). First, an r12 potential on a spherical shell whose center is the aggregate center of mass was applied on the methyl and methylene segments to prevent chains from leaving the micelle u.:) (r) = (r:, r)12 (4.4) 69 20 a U 10 b I 0 5 10 15 20 25 Micelle radius A Figure 4.2: Chainsolvent interaction models a) (r, r*)12 potential b) finite energy barrier, U* = U/e, c = 419J/mol This potential was previously used by Woods et al. (1986) and Farrell (1988). It is considered to be unrealistic because it assumes that methylene segments are com pletely insoluble in water. A more realistic approach has been to impose a finite barrier on the hydrocarbon chain that mimics the barrier for solubilization of methylene in water (Vilallonga et al., 1982). U*(2) (r) = (1.+ p(r+/r:,)) (4.5) This potential changes rapidly from zero in the core to a higher value outside the core. The value of r was chosen to match the free energy of solubilization of methylenes in water (Vilallonga et al., 1982), while the steepness of the potential was controlled by p and r to make 90% of the change in 4.5 (A) as suggested by neutron scattering (Hayter and Penfold, 1981) (p = 0.76 and r = 46). This potential is more realistic than the infinite wall potential and allows methylene segments to leave the aggregate. 4.2.2 HeadSolvent Interaction Three models have been used to account for the headsolvent interactions (Fig ure 4.3). First, there was the harmonic potential (Woods et al., 1986; Farrell, 1988) about an equilibrium radial position to limit head group movement to a short dis tance normal to the micelle surface, with free movement along the micellesolvent boundary. U')1 (r) y (r* r*)2 (4.6) The harmonic potential constant 7 controls the amplitude of normal movement of 150 100 a U b 50 0 0 5 10 15 20 25 Micelle Radius A Figure 4.3: Headsolvent interaction models a) harmonic potential b) finite energy barrier. Half harmonic potential has same form as harmonic potential for a radius less than the equilibrium radius, and is equal to zero for a radius greater than the equilibrium radius. U* = U/e, c = 419J/mol the head groups, effectively controlling the interfacial area. The application of this potential implies that head groups prefer the interfacial area over both the micelle core and the bulk solvent. Although it can be true that head groups prefer polar over nonpolar environment, it is not known if head groups prefer the interface over the bulk solvent. Assuming that the bulk solvent is as equally favorable to heads as is the interface, two other potentials have been used. A halfharmonic potential was used to put a high energy barrier on head groups from the micelle side, and no energy barrier imposed from the solvent side. ( (r) = (rh*, r 2 <(r.7 (4.7) .2 (r) =0. r* > rh J As with equation 4.6, the value of the repulsive energy at the center of the shell is r. Although this model has the required characteristics, the potential is not twice differentiable at rL,. Finally, a continuously differentiable potential that imposed a finite energy barrier on head groups from the micelle side and no energy barrier from the bulk solvent side was used. (3 ) " hs (r) = O/ (1. + p(2 r*/,)) r* < 2r, (4.8) The values of p and r were chosen to provide a sharp (4.5 A) transition while / controlled the barrier height. In particular, values of the order of the segment hy drophobic barrier and of the dehydration free energy of head groups (Vilallonga et al., 1982) were used. The potential changes rapidly from zero in the bulk solvent to a higher value in the micelle core, and is more realistic than the harmonic potential in that the heads do not feel a continuously changing repulsion either inside the micelle or in the solvent. All parameter values for the micellesolvent interaction models are in units of r,, the radius that corresponds to the minimum of the segmentsegment (69) Lennard Jones potential, and of e, the energy value for the potential minimum. 4.3 Simulations Molecular Dynamics simulations of seven model micelles and one hydrocarbon droplet were performed using the above models. In each of the runs, one chainsolvent interaction potential was combined with one headsolvent interaction potential to complete the micelle force field. Run 1. A micelle of 24 octyl "nonionic sulfate" monomers with the weak harmonic potential of equation 4.6 (7 = 30) applied to head groups, and the infinite wall potential of equation 4.4 for the chainshell interactions. Run 2. A micelle of 24 octyl "nonionic sulfate" monomers with a stronger har monic potential (7y = 300) applied to the head groups. The solvophobic potential of equation 4.5 was applied to the segments (ic =8.27). Run 3. A micelle of 24 octyl "nonionic sulfate" monomers with the halfharmonic potential of equation 4.7 (7 = 300) applied to the head groups. The potential energy for chain segments was the solvophobic potential of run 2. Run 4. A micelle of 24 octyl "nonionic sulfate" monomers with the potential of equation 4.8 (03 = 8.27) applied to the head groups. This value of # gives the same barrier for head groups into the core as for segments into the solvent. The head group energy at the center is about 1/360 that of run 3. The segment potential was the solvophobic potential of run 2. Run 5. A micelle of 24 octyl "nonionic sulfate" monomers with the potential of equation 4.7 applied to the head groups. The value of 6 was an order of magnitude greater (/3 = 82.7) an estimate of the free energy of transfer of sulfate groups from an aqueous to a hydrocarbon environment. The segment potential was the was the solvophobic potential of run 2. Run 6. A micelle of 24 octyl "nonionic sulfate" monomers with the head group and segment potentials the same as run 5, but at a high pressure of about 1 bar. Run 7. A micelle of 24 octyl "polar methyl" monomers with the head group po tential of equation 4.6 (^1 = 30), and the chain segment potential was the solvophobic potential of run 2. Run 8. A hydrocarbon droplet with 24 nonyl chains with the finite barrier segment potential of equation 4.5. The headhead and headsolvent interactions used in the model micelles are segment interactions. The parameters of equations 4.14.8 for all simulations are listed in Table 4.1. These eight simulations provide a basis to check the effect of micellesolvent models, head group characteristics and chain length on the micellar behavior, and provide a base for comparison between micelles and hydrocarbon droplets. Table 4.1: Intermolecular Potential Parameters. rh and r* are in Units of r,. hh, 7 and P Are in Units of c. rhh, Run run 1 run 2 run 3 run 4 run 5 run 6 run 7 run8 f(J/mol) 419 419 419 419 419 419 419 419 rm (A) 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 o (A) 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 Chh 1 1 1 1 1 1 1  rhh 2.45 2.45 2.45 2.45 2.45 2.45 1  7 30 300 300 30  K 8.27 8.27 8.27 8.27 8.27 8.27  r, 3.2 3.2 3.2 4.72 4.72 2.85 3.00  r, 4.2 4.2 3.35 4.2 4.2 3.35 3.50 3.48 / 8.27 82.7 82.7  System preparation for all runs. Newton's second differential equations of motion were solved for each of the 216 soft spheres by using a fifthorder predictorcorrector algorithm due to Gear (1971). The time step used in solving the equations of motion was equivalent to 1.5 fs for runs 16 and 2.0 fs for runs 78. The procedure used in preparing all micelle runs started with initial positions of all segments on each chain in the alltrans conformation and the head group centers distributed about a sphere about twice the final micelle size. The initial steps of the simulation consisted of decreasing the radius of the micelle from the initial to the intended radius while applying an infinite wall potential (Equation 4.4) on the chains and the appropriate head group potential. During the initial steps, the rotational barrier was decreased to one tenth the desired value, and then raised to the final value. The next steps in the micelle simulation changed the chainsolvent interaction to the intended models and finetuned the micellesolvent interaction models radii, r* and r*, to reach the intended pressure (Woods et al., 1986). Finally 12000 time steps constituted the equilibrium run. The same procedure was used in preparing the hydrocarbon droplet simulation. The state conditions for all runs are listed in table 4.2. The temperature was 298 K in all cases. Table 4.2: Temperatures and Pressures for Molecular Dynamics Simulations. Run 1 2 3 4 5 6 7 8 Temp (K) 298 298 298 298 298 298 298 298 Pressure(atm) 0 0 0 0 0 1 0 ,0 No. of Molecules 24 24 24 24 24 24 24 24 Packing Fractiona 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70 . Packing fraction is v 6V CHAPTER 5 EFFECTS OF MICELLESOLVENT INTERACTION In this chapter we describe the effect of the micellesolvent interaction models on the micelle behavior by comparing six molecular dynamics simulations of "nonionic sulfate" micelles (Runs 16) that have the same inter and intramolecular potentials but different micellesolvent interaction models. 5.1 Local Structure. Probability Distributions The primary measures of local structure are the probability distributions of seg ments. The singlet probability Pi(r) can be determined directly from the simulation by: Pi(r) = N(r) > (5.1) N Where < Ni(r) > is the average number over time of groups i that are found in shell of thickness 6r with radius r, and N is the total number of molecules, with the sum over all shells being: < Ni(rj) > = N (5.2) and EPi(r) = 1 (5.3) 3 The singlet density pt(r) is related to Pi(r) by: P,(r) = Pi(r)4?rr2Ar (54) N g,= ) (5.4) When Ar * 0 / p,(r)47rr2dr = N (5.5) Thus, a convenient form to use is r2p,(r), which is within a multiplicative constant of the true probability Pi(r), and yields an area under the curve that is proportional to N. This particular form is a good basis for comparison between runs. Results for the tail group distributions for runs 16 are shown in figure 5.1. The curves for all six runs are very similar, suggesting that neither the chainsolvent nor the headsolvent models affect the tail distributions. In all cases, tail groups have a finite probability of being found at any distance from the aggregate center, including the micelle surface. Results for the middle group (segment 5) distributions are shown in figure 5.2. The effect of micellesolvent interaction models can be seen. Although the peak for all curves occurs at about the same distance from the micelle radius, the heights vary among the runs. In particular, the curves with the harmonic headsolvent models (run 1 and run 2) exhibit a higher peak than the rest of the curves. The height of these curves increases with increasing harmonic constant 7, i.e., the segments peak height increases with greater limitations on the head motion about its equilibrium position. Middle segments, like tail groups, have a finite probability of being found everywhere in the micelle, but the distribution is not as wide as the tail group distribution. r2p(ri) 0.2 AI' .2 Figure 5.1: Group probability distributions for tail groups r~p(ri) (A1) 0.2 04 0 5 10 15 20 25 Micelle Radius (A) Figure 5.2: Group probability distributions for middle segments (segment 5 from the top of the chain with the head group numbered 1 and the tail group 9) The effect of micellesolvent models is greatest for the head group distributions (Figure 5.3). In this plot, neither the positions nor the height of the peaks are similar, and some of the curves exhibit multiple peaks. Again runs 1 and 2 have the highest peaks. In all runs, the head groups are predominately in the palisade region, but some are found in the micelle core. Runs with a finite energy barrier tend to have lower primary peaks, and have second peaks at the micelle center. Even the run with the half harmonic potential shows a small peak at the micelle center. This particular result may be caused by head group repulsion forcing heads to be in the micelle core and the chains need not be fully stretched to fill the core space. This is unlike previous work (Haile and O'Connell, 1984; Woods et al., 1986) where the heads were small, and movement into the center would not yield such energy and entropy advantages for the system. 5.2 Average Positions of Groups Further information is given on the local structure of groups by calculating the average and the mean radial position for each group. The average radial position, Ri, and the mean radial position, (R?)1/2, for each group relative to the aggregate center of mass are calculated by B, = p,(r)r47rr2dr (5.6) R? = 1 J Pi(r)r4rr2dr (5.7) The results shown in Tables 5.1 and 5.2 do not show a sizable difference among the runs. On the average, the tail groups are further from the center of mass than (A 1) Figure 5.3: Group probability distributions for head groups Table 5.1: Average Radial Position A. for Each Group, Measured Relative to the Aggregate Center of Mass group no run 1 run 2 run 3 run 4 run 5 run 6 1 12.6 12.2 13.2 12.6 12.5 12.4 2 10.6 10.1 11.2 10.6 10.6 10.6 3 9.7 9.2 10.4 9.8 9.7 9.8 4 9.1 8.5 9.7 9.1 9.1 9.2 5 8.6 7.9 9.1 8.6 8.4 8.7 6 8.3 7.6 8.7 8.2 8.2 8.3 7 8.2 7.4 8.4 8.2 7.9 8.3 8 8.2 7.4 8.4 8.2 8.0 8.3 9 8.5 7.7 8.5 8.4 8.4 8.5 Table 5.2: Mean Radial Position (f)1/2 for Each Group, Measured Relative to the Aggregate Center of Mass group no run 1 run 2 run 3 run 4 run 5 run 6 1 13.0 12.4 13.7 13.0 12.9 12.9 2 10.9 10.2 11.6 10.9 10.9 10.9 3 10.0 9.3 10.7 10.1 10.0 10.1 4 9.4 8.6 10.0 9.3 9.3 9.4 5 8.9 8.1 9.4 8.8 8.5 8.9 6 8.6 7.8 9.0 8.5 8.3 8.6 7 8.5 7.7 8.7 8.4 8.1 8.5 8 8.6 7.8 8.7 8.5 8.2 8.5 9 8.9 8.1 8.9 8.7 8.8 8.8 