Structural and dynamics studies of surfactants and micelles

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Structural and dynamics studies of surfactants and micelles
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xv, 164 leaves : ill. ; 29 cm.
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Karaborni, Sami
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Surface active agents   ( lcsh )
Micelles   ( lcsh )
Molecular dynamics   ( lcsh )
Chemical structure   ( lcsh )
Chemical Engineering thesis Ph. D   ( lcsh )
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Thesis:
Thesis (Ph. D.)--University of Florida, 1990.
Bibliography:
Includes bibliographical references (leaves 156-163).
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Typescript.
General Note:
Vita.
Statement of Responsibility:
by Sami Karaborni.

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Table of Contents
    Title Page
        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 N-alkane simulations
        Page 3
        Page 4
        Page 5
        Page 6
        Page 7
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        Page 9
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        Page 25
    Chapter 3. Simulations of surfactants in a monatomic fluid and in water
        Page 26
        Page 27
        Page 28
        Page 29
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    Chapter 4. Model Michelle
        Page 64
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        Page 70
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    Chapter 5. Effects of Micelle-solvent interaction
        Page 78
        Page 79
        Page 80
        Page 81
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    Chapter 6. Effects of chain length and head group characteristics
        Page 103
        Page 104
        Page 105
        Page 106
        Page 107
        Page 108
        Page 109
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    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
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    Bibliography
        Page 156
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    Biographical sketch
        Page 164
        Page 165
        Page 166
Full Text










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, Westermann-Clark 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 N-ALKANE 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 End-to-End 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 Lennard-Jones Fluid of Segments
and Simulation Details ........................... 32
3.4 Results for Molecules in a Lennard-Jones Fluid of Segments . . 38
3.4.1 Average and Mean Values ....................... 38
3.4.2 End-to-End 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 Chain-Solvent Interaction . . . . . . . . ... .. 68
4.2.2 Head-Solvent Interaction . . . . . . . . . ... .. 70
4.3 Simulations . . . . . . . . . . . . . . ... .. 73

5 EFFECTS OF MICELLE-SOLVENT 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 end-to-end 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 n-alkane there are n-3 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 End-to-end distribution for the octyl "ionic methyl" surfactant in a
Lennard-Jones fluid of segments . . . . . . . . . ... .. 41

3.4 End-to-end distribution for the octyl "nonionic sulfate" surfactant in
a Lennard-Jones fluid of segments . . . . . . . . ... .. 43

3.5 End-to-end distribution for poly (oxyethylene) in a Lennard-Jones
fluid of segments . . . . . . . . . . . . . .... .. 44

3.6 Radius of gyration distribution for the octyl "ionic methyl" surfactant
in Lennard-Jones fluid of segments . . . . . . . . ... .. 46

3.7 Radius of gyration distribution for the octyl "nonionic sulfate" surfac-
tant in a Lennard-Jones fluid of segments . . . . . . ... .. 47

3.8 Radius of gyration distribution for poly (oxyethylene) in a Lennard-
Jones fluid of segments . . . . . . . . . . . ... .. 48

3.9 End-to-end 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 Chain-solvent interaction models a) (r'an r*)-12 potential b) finite
energy barrier, U* = U/e, c = 419J/mol . . . . . . . ... .. 69

4.3 Head-solvent 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 9-member chains
from runs 1-6 . . . . . . . . . . . . . . ... .. 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 N-member 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 segment-oxygen
vector . . . . . . . . . . . . . . . . . . .. 140

A.2 Intermolecular oxygen-oxygen pair correlations function . . ... .. 142

A.3 Intermolecular hydrogen-hydrogen pair correlation function . . 143

A.4 Intermolecular oxygen-hydrogen pair correlation functions . ... .. 144

A.5 Intermolecular hydrogen-head group pair correlation function . 147

A.6 Intermolecular oxygen-head group pair correlation functions . . . 148

A.7 Intermolecular hydrogen-chain segment pair correlation function. . 150

A.8 Intermolecular oxygen-chain 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 N-Butane 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 N-Butane 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 N-Hexane as extrapolated from Our Linear Fits
and as Determined from Other Workers: Clarke and Brown (1986) . 13

2.7 Average Values for N-Octane as Interpolated from Linear Fits and as
Determined from Other Workers: Szczepanski and Maitland (1983). . 14

2.8 Average Values for N-Decane 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 Lennard-Jones 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 Lennard-Jones 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
Lennard-Jones Fluid of Segments . . . . . . . . ... .. 39

3.10 Average Properties for Poly (Oxyethylene) in a Lennard-Jones 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 Lennard-Jones Fluid of Segments. 52

3.17 Lennard-Jones 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=1.602x10-19esu . . . . . .... .. 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 Water-Water Pair Correlation Functions in Bulk and
Shell . . . . . . . . . . . . . . . . . . .. 146

A.3 Self-Diffusion 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 micelle-solvent interactions

have been modeled using several field potentials that realistically describe surfactant

interactions with polar solvents. Dilute solutions of surfactants and n-alkanes 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 micelle-solvent 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 end-to-end

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 micelle-solvent

interaction models on micellar structure and shape via molecular dynamics.

In chapter 2 a molecular dynamics investigation of the conformation of n-alkanes

in a monatomic fluid of methylene segments is described. In particular, properties











such as trans fraction, radius of gyration and end-to-end 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 micelle-solvent interaction models used.

In chapter 5 the effects of micelle-solvent 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 N-ALKANE SIMULATIONS

2.1 Background

Over the past few years there have been several molecular simulations and statisti-

cal mechanics calculations of model n-alkanes. 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 n-alkanes. 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 n-alkanes 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 n-alkanes, especially n-butane.











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 n-butane 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 n-alkanes having from 7 to 21 carbons in Lennard-Jones 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 equal-diameter










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 (6-9) 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 n-alkane 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 fifth-order predictor-corrector 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 1.395x10-15 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 A-3.































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 n-butane to n-uneicosane.

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), n-hexane (Table 2.6), n-octane (Table 2.7) and n-decane

(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 N-Butane 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 N-Butane 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 N-Hexane 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 N-Octane 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 N-Decane 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 n-hexane the MD results of Clarke and Brown (1986) at 300 K gave a

similar value to ours.

For pure n-octane 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 n-decane 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 End-to-End 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 end-to-end 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 2.4-2.8 the radius of gyration

and the end-to-end distance from the linear fits are compared to other simulation

data. In general the agreement is good, particularly considering the differences in

temperature for n-octane and n-decane.
















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 end-to-end 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


n-3
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(n-3)


Figure 2.3: Entropy S/k = Ipilnpi is plotted as a function of ln(n 3). For an
n-alkane there are n-3 dihedral bonds and p, is the probability of finding a bond in
trans conformation.








25

The end-to-end 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 LENNARD-JONES FLUID
OF SEGMENTS AND IN WATER


3.1 Background


In the last chapter we have discussed the conformation of n-alkanes, and have

shown some important properties of n-alkanes. 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 Lennard-Jones

26










fluid of methylene segments and in water, and an octyl "nonionic sulfate" surfactant

in a Lennard-Jones fluid of segments. We also describe a simulation of a nonionic

surfactant head group (poly (oxyethylene)) in a Lennard-Jones 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 equal-diameter

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 n-butane.



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- 1.462coso-1.578cos2 + 0.368cos3

+3.156 cos4 4 + 3.788 cos5 q) (3.3)


The intramolecular potential also includes a (6-9) Lennard-Jones 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 -CH2-CH2-O. 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 (6-12)

Lennard-Jones 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 n-alkyl 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 Lennard-Jones
Fluid of Segments and Simulation Details

In simulations involving the "ionic methyl" and "nonionic sulfate" surfactants in

a Lennard-Jones fluid of segments a Lennard-Jones (6-9) potential (equation 3.4) is

used for all surfactant segment-fluid segment interactions. In addition a coulombic

interaction is used to model the head group-counterion 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

Lennard-Jones parameters except the "nonionic sulfate" surfactant head group which

has different parameters.




















Table 3.2: Lennard-Jones 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
CH2-CH2 9.25 x 105 1.526
CH2-0 1.14 xl06 1.425
0-H 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
CH2-CH2-O-CH2 8314 1.053 1.250 0.368 0.675
O-CH2-CH2-O 8314 1.078 0.355 0.068 0.791
CH2-CH2-O-H 8314 1.053 1.250 0.368 0.675


Angle 7o 0o
J/mol degree
CH2-CH2-O 1.651 x105 109.5
CH2-O-CH2 2.067 x105 111.8
CH2-O-H 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 Lennard-Jones (6-12) 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 all-trans

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 fifth-order predictor-corrector 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 Lennard-Jones Fluid of Segments.

Simulation Time Step Equilibration Equilibrium
secs Steps Run
"ionic methyl" surfactant 1.395x 10-15 50,000 150,000
"nonionic sulfate" surfactant 1.395x 10-15 40,000 130,000
poly (oxyethylene) 1.331 x 10-15 10,000 75,000











3.4 Results for Molecules in a Lennard-Jones Fluid of Segments


In this section we report results on end-to-end 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, end-to-end 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, end-to-end 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 9-carbon n-alkane. On the other hand the trans fraction for poly (oxyethylene)

is considerably different from the corresponding 19-carbon n-alkane. 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 dipole-dipole,

quadrupole-quadrupole, 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
Lennard-Jones 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 Lennard-Jones 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 Lennard-Jones energy parameters for chain oxygen and different rota-

tional potentials for oxygens and methylenes also would lead to differences between

the 19-carbon n-alkane conformation and that of poly (oxyethylene).

The end-to-end distance and radius of gyration for the octyl "ionic methyl" sur-

factant are similar to those of the 9-carbon n-alkane (see chapter 2), while the "non-

ionic sulfate" surfactant shows a larger end-to-end 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 all-trans conformation. Additionally the mean end-to-end 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-

to-end distance and the radius of gyration for the poly (oxyethylene) molecule are

much smaller than for the corresponding 19-carbon n-alkane, suggesting a bunched

up conformation consistent with a small average trans fraction.

3.4.2 End-to--end Distance

The plot for the end-to-end 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: End-to-end distribution for the octyl "ionic methyl" surfactant in a
Lennard-Jones fluid of segments.











Several small peaks arising from allowed and forbidden conformations are present.

The most probable value for the end-to-end distance is higher than the average

value.

The end-to-end 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 end-to-end distance distribution for the "ionic methyl" surfactant chain ex-

tends from quite small distances of 4A to 11A. Basically the end-to-end distance

samples all available conformational space from 4A (rmin in the Lennard-Jones po-

tential) to 11A (the all-trans end-to-end distance).

The end-to-end 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 all-trans end-to-end distance, while the short range part is indicative

of the head and tail approaching each other to rmin in the Lennard-Jones potential.

Figure 3.5 shows that the end-to-end distance for the poly (oxyethylene) molecule

is a fairly symmetric distribution which reaches from values around the Lennard-Jones

cr to values less than the all-trans end-to-end distance. The difference between the

average and the most probable values of the end-to-end 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: End-to-end distribution for the octyl "nonionic sulfate" surfactant in a
Lennard-Jones fluid of segments.


















0.008


P(r1, 1) 0.004


Figure 3.5: End-to-end distribution for poly (oxyethylene) in a Lennard-Jones 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 dipole-dipole, quadrupole-quadrupole, charge-charge 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 Lennard-Jones 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
Lennard-Jones 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 Lennard-Jones 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 Lennard-Jones
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 X-C-

O-X (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 X-C-C-X (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 Lennard-Jones 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 Lennard-Jones 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 (6-12)

Lennard-Jones potential instead of the (6-9) potential. This should not affect the










conformation of the surfactant since the excluded volume effects for n-butane have

been modeled equally well by an r-12 or an r-9 contribution to the Lennard-Jones

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 oxygen-oxygen, oxygen-hydrogen and hydrogen-hydrogen 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 water-water interaction which

is modeled by the SPC potential; 2) the water-chain segment interaction which is

modeled by a (6-12) Lennard-Jones term; 3) the water-head group interaction that

is modeled by a Lennard-Jones interaction plus a coulombic term to account for

charges on the surfactant head and the water molecules; 4) the water-counterion

interaction that is modeled similar to the water-head group interaction; 5) the head

group-counterion interaction that is similar to the water-counterion interaction; 6)

the chain segment-counterion interaction which is modeled by a (6-12) 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 methyl-sized

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 predictor-corrector 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: Lennard-Jones 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 10-19esu.

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 end-to-end 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 Lennard-Jones 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 end-to-end distance and the radius of gyration for the

"ionic methyl" surfactant in water are similar to those of the same surfactant in the

Lennard-Jones 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 end-to-end distance and the radius of gyration are shown in Fig-

ures 3.9 and 3.10. The end-to-end 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 end-to-end distribution for the octyl "ionic methyl" surfac-

tant in a Lennard-Jones fluid of segments, the distribution is narrower, and does not

reach either the all-trans end-to-end distance or the Lennard-Jones 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
8.18 A
< 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: End-to-end 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 Lennard-Jones fluid of

segments. Its slight skewness is towards shorter distances rather than longer ones

as in the Lennard-Jones 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 all-gauche 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 Lennard-Jones fluid but the number of bonds

is decreased by one with the all-trans 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

Lennard-Jones 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 Lennard-Jones 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 Lennard-Jones 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 X-ray 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 (Ben-Shaul

and Gelbart, 1985; Degiorgio, 1983), experimental results often disagree. At present











Small-Angle 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 micelle-solvent

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 equal-diameter 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) Lennard-Jones 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. Segment-segment and head-segment

interactions are modeled by a pairwise additive Lennard-Jones (6-9) form,
























Head-Head Interaction


Chain-Chain Interaction Hydrophilic Interaction
Figure 4.1: Model for intermolecular interactions in micelles












[( .)9 /r \ 6
ULJ (rj) = 2 (-) -3 (r)1 (4.1)

For head-segment 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:


,head-segment 1
rhm 2 (rm + rhh) (4.2)

Head-head interactions are modeled by a purely repulsive potential which includes

both dipole-like repulsion and excluded volume effects:



Uhh (rij)= [2 3+3r (4.3)

The micelle-solvent 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 micelle-solvent interactions can be divided into two

contributions, the chain-solvent and head-solvent interactions.

4.2.1 Chain-Solvent Interaction

Two models are proposed to account for the chain-solvent interaction (Figure 4.2).

First, an r-12 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: Chain-solvent 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 Head-Solvent Interaction

Three models have been used to account for the head-solvent 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 micelle-solvent

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: Head-solvent 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 half-harmonic 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 micelle-solvent interaction models are in units of r,,

the radius that corresponds to the minimum of the segment-segment (6-9) 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 chain-solvent

interaction potential was combined with one head-solvent 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 chain-shell 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 half-harmonic

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 head-head and head-solvent interactions used in the

model micelles are segment interactions.

The parameters of equations 4.1-4.8 for all simulations are listed in Table 4.1.

These eight simulations provide a basis to check the effect of micelle-solvent 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 fifth-order predictor-corrector

algorithm due to Gear (1971). The time step used in solving the equations of motion

was equivalent to 1.5 fs for runs 1-6 and 2.0 fs for runs 7-8.

The procedure used in preparing all micelle runs started with initial positions of

all segments on each chain in the all-trans 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 chain-solvent interaction

to the intended models and fine-tuned the micelle-solvent 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 MICELLE-SOLVENT INTERACTION




In this chapter we describe the effect of the micelle-solvent interaction models on

the micelle behavior by comparing six molecular dynamics simulations of "nonionic

sulfate" micelles (Runs 1-6) that have the same inter and intramolecular potentials

but different micelle-solvent 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 1-6 are shown in figure 5.1. The

curves for all six runs are very similar, suggesting that neither the chain-solvent nor

the head-solvent 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 micelle-solvent 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 head-solvent 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
A-I' .2


Figure 5.1: Group probability distributions for tail groups






























r~p(ri)
(A-1)




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 micelle-solvent 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