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Structural and dynamics studies of surfactants and micelles

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Structural and dynamics studies of surfactants and micelles
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Karaborni, Sami
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xv, 164 leaves : ill. ; 29 cm.

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Chemicals ( jstor )
Hydrocarbons ( jstor )
Micelles ( jstor )
Modeling ( jstor )
Molecular dynamics ( jstor )
Molecular interactions ( jstor )
Molecules ( jstor )
Simulations ( jstor )
Sulfates ( jstor )
Surfactants ( jstor )
Chemical Engineering thesis Ph. D ( lcsh )
Chemical structure ( lcsh )
Dissertations, Academic -- Chemical Engineering -- UF ( lcsh )
Micelles ( lcsh )
Molecular dynamics ( 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.
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Vita.
Statement of Responsibility:
by Sami Karaborni.

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STRUCTURAL AND DYNAMICS STUDIES
OF SURFACTANTS AND MICELLES









By

SAMI KARABORNI


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY








UNIVERSITY OF FLORIDA


1990

































To

my Father









ACKNOWLEDGMENTS


To my Father Tijani, God rest his soul, my mother Jamila, my brother Mustapha

and his wife, my sisters Najet, Badiaa and Sihem and their husbands, and all my

nephews and nieces, thank you for all the love and care. To June Rarick who has

given me her endless support and sympathy during the past four years, thanks for

everything.

I would like to thank Professor O'Connell for his guidance and support and ex-

pertise and encouragement. Through his hard work and dedication John O'Connell

taught me to be my best at whatever I do.

I wish to thank Professors Shah, Moudgil, 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




Full Text
127
Table 6.4 presents the probabilities for finding a particular bond on the chain
in the trans conformation. Results from simulations 5, 8 and that of Woods et
al. (1986) show an essentially flat distribution among all bonds, while run 5 has
bond 2 (dihedral angle involving the head and segments 2-4) with a distinctly higher
probability to be in the trans conformation. The nonionic sulfate mass, size and
bond forces apparently affect the first dihedral bond conformation. Chevalier and
Chachaty (1985) obtained similar results from NMR measurements. They suggest
that the increase is due to micellization, but we believe that it is due to the nature of
the head group. A single surfactant molecule with a large head group in a hydrocarbon
fluid of methane groups showed the same increase in trans fraction around the head
group.
Bond Orientation
An indicator of the chain conformation is the bond order parameter, S(r), defined
by:
S(r) =< ^(3 cos2 0 1) > (6.1)
where 6 is the angle formed by the bond vector connecting two adjacent groups and
the radius vector from the aggregate center of mass to the center of the bond. A value
of 1 for S(r) indicates that bonds are parallel to the micelle radius, and a value of 0.5
indicates that bonds are normal to the micelle radius, while a value of 0 indicates no
preferential ordering. Figure 6.13 shows that for run 5 the bond parameter is positive
for r > 10 A indicating an ordering of bonds parallel to the micelle radius, while for


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; Bnon 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.
3


106
Figure 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.


APPENDIX
WATER STRUCTURE IN THE PRESENCE OF AN ANIONIC METHYL
SURFACTANT
A.l Background
Since the advent of molecular dynamics there have been numerous simulations
to study the structure of water in the presence of a solute given a reliable effective
water pair potential. Polar (Rao and Berne, 1981) and nonpolar solutes (Geiger et
ah, 1979; Okazaki et ah, 1981; Rappaport and Scherega, 1982; Remerie et ah, 1984;
Jorgensen et ah, 1985; Zichi and Rossky, 1986b; Watanabe and Andersen, 1986;
Tanaka, 1987) have received equal emphasis in simulations due to the importance of
water as a solvent in many applications. While section 3.5 thoroughly analyzed the
conformation of an ionic surfactant in water, this appendix is an in-depth study of
the structure of the water in the presence of the ionic surfactant since surfactants are
a different type of molecules.
Many postulates have been made about the configuration of water molecules
around nonpolar solutes. Most suggest that water molecules around nonpolar so
lutes form icebergs (Frank and Evans, 1945; Nemethy and Scheraga, 1962) or
clathrate-like environments (Glew, 1962) around the inert solute, but there seem
to be no experiments that have strongly supported this picture. Only NMR studies
136


41
Figure 3.3: End-to-end distribution for the octyl ionic methyl surfactant in a
Lennard-Jones fluid of segments.


52
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


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
1


21
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 gg* 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 p, 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.


114
20
15-
r (A) 10-
5-
[]
[],
[]i
[}
[J
[J


run 5
Jonsson
Watanabe
1 CT
M
[]i
Segment Number
Figure 6.6: Scaled average radial positions for run 5, and from the micelle simulations
of Jonsson et al. (RC model) (1986) and of Watanabe et al. (1988).


3.10Radius 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^afi r*)-12 potential b) finite
energy barrier, U* = U/e, e = 419 J/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¡t,t = 419 J/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 S(r) throughout the micelle for runs 4,
5 and 6 99
5.11 Individual bond order parameter 5, 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


2
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.


145
ference in magnitude between the shell and the bulk. In this analysis the peak of
the shell distribution is consistently higher than that of the bulk distribution. This
indicates a slight structure enhancement in the shell over the bulk. To measure the
degree of structure, a ratio of the heights of the first maximum and the following
minimum is calculated. Table A.2 shows that this ratio is always higher by a factor
of 1.03 to 1.1 for the shell over the bulk. This is not a very large effect. Geiger et
al. (1979) have found that the shell-bulk enhancement factor varies from 1.3 to 1.57,
but with a shell that is smaller than the one used in this analysis. (Geiger et al. used
a shell of 4.5 radius).
Water-Surfactant radial pair correlation functions. The water-surfactant pair cor
relations can be calculated on a segment-to-segment basis. In particular, the oxygen-
chain segment, oxygen-head group, hydrogen-chain segment and the hydrogen-head
group pair correlation functions have been found. In Figure A.5 the hydrogen-head
group distribution is plotted. The first peak at r=1.82 has a magnitude of 5.21
and the second peak at r=3.3 has a magnitude of 1.89. The first minimum is at
r=2.64 and has a magnitude of 0.29. The large difference in heights of the first
peak and the following minimum indicates a high degree of structure of hydrogen
atoms around the head group, as expected from their opposite charges.
Figure A.6 shows the oxygen-head group pair correlation function. This distri
bution has a very high first peak (magnitude of 4.81) at r=2.82 , but a very small
second peak (magnitude of 1.20) at r=5.00 . The first minimum is at r=3.54 and


CHAPTER 7
CONCLUSIONS AND RECOMMENDATIONS
Molecular dynamics methods have been used to study micellar structure, and
alkane and surfactant conformation in dilute solutions. A summary of the most
important conclusions that have been reached during the course of this study are
listed in this chapter.
In general, n-alkanes in a monatomic fluid of methylene segments were found to
have the same trans fraction independent of chain length, while other properties such
as radius of gyration and end-to-end distance are linear with chain length. From
the simulations of surfactant molecules, it is apparent that ionic surfactants behave
differently in a nonpolar fluid than in water.
The conformation of ionic surfactants in nonpolar fluids was found to be similar to
that of n-alkanes in the same fluids, but the average trans fraction of these surfactants
was found to be significantly smaller in water than in a monatomic fluid of segments.
The general results from the micelle simulations indicate that structural results
are insensitive to different micelle-solvent interaction models, although there are no
ticeable effects on head group distributions. Simulation results show movements of
heads into the core that could be best explained by an excluded volume effect at
the micellar surface resulting from the large size of these head groups. Surfactant
chains in micelles are not in the all-trans conformation but are curled toward the
133


144
OH Pair
Correlation
Figure A.4: Intermolecular oxygen-hydrogen pair correlation functions.


29
U () = 7r( 1.116 1.462 cos (j> 1.578 cos^ (f> + 0.368 cos3 (f>
+3.156 cos4 4> + 3.788 cos5 )
(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 :
6'
(3.4)
The parameters involving all segments on the octyl ionic methyl and the octyl
nonionic sulfate surfactants (Muller et ah, 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-CH2-0. 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.


84
Table 5.1: Average Radial Position R, 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


88
P(r)
Figure 5.5: Distribution of distances between head groups


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. OConnell
Cochairman: Dinesh O. 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.
xiv


92
Figure 5.8: Ratio of moments of inertia from runs 4, 5 and 6.


107
clue to head groups excluding tails from the palisade region in the first case and the
tendency of tails to collect at the interface in the second.
To test whether the results of our solvent field models are consistent with full
simulations having solvent molecules, figure 6.3 compares scaled tail distributions
from our run 5 with those of Watanabe et al. (1988) and of Jnsson et al. (RC model)
(1986). The micelle radius is scaled by (1/iV)1/3 to account for the difference in
aggregate numbers (N = 24 for run 5 and N = 15 for Watanabe et al. and Jnsson
et al.) The agreement is excellent.
The head group distributions of the micelles are shown in figure 6.4 where the
scaling by (N/N\)*/3 is used on the results from Woods et al. (1986). The head
group distribution is sharp for the smaller head groups, regardless of chain length
and interaction whereas larger head groups are broadly distributed with a second
peak at the micelle center. As explained in the previous chapter this is an excluded
volume effect.
The peak heights of runs 7 and Farrell (1988) (octyl chains) are slightly displaced
from those of Woods et al. (1986) (dodecyl chains) even though the head group
distribution for the dodecyl polar methyl micelle is scaled by the one-third power
of the aggregate number ratio, and is normalized to achieve an equal area under all
curves. Whether this effect is significant or not is unknown.


Figure 3.2: Model poly (oxyethylene) molecule.


86
are segments 6, 7 and 8. The results from run 2 differ slightly in that all segments are
closer to the center. This result is probably due to the strong head group harmonic
potential keeping the head on the surface of the micelle, and forcing segments into
the core at smaller average radial positions. Also, the half harmonic potential has the
opposite effect because head groups can easily move away from the micelle. Therefore
the average positions of all groups in run 3 are slightly higher than in the other runs.
5.3 Distributions of tail groups
Another measure of local structure is the distribution of tail groups as determined
from scattering amplitudes of tails (a Fourier transform of the singlet density p,(r)).
A(Q)
-4(0)
= 4* rPi(r)^lr
J la/
Q
(5.8)
Figure 5.4 presents the scattering amplitude for tails in runs 1-6. The simulation
results show very good agreement for all runs except run 2.
5.4 Distribution of Distances Between Groups
A pair correlation function can be used to determine the distribution of distances
between groups by calculating the average number of groups on molecule i that are
at a distance r from groups on molecule j.
P(r) =
EiEjiM r)
(5.9)
N(N 1)
These distributions for pairs of heads and tails are shown in figures 5.5 and 5.6
respectively. The results are the same for all runs. The distributions in figure 5.5 have


43
P(r 19)
Figure 3.4: End-to-end distribution for the octyl nonionic sulfate surfactant in a
Lennard-Jones fluid of segments.


CHAPTER 6
EFFECTS OF CHAIN LENGTH AND HEAD GROUP CHARACTERISTICS
In the previous chapters, molecular dynamics simulations were used to determine
the effect of micelle-solvent models on the micelle behavior. In this chapter we
turn to the effect of chain length and head group properties on the internal micelle
structure, micellar shape and chain conformation inside the micelle. The results from
one molecular dynamics simulation of a octyl polar methyl micelle and one of a
nonane hydrocarbon droplet are compared with an octyl nonionic sulfate micelle
from the previous chapter (Run 5) and the dodecyl polar methyl micelle described
by Woods et al. (1986)
Local structure
One primary measure of micellar molecular structure is the set of spatial proba
bility distributions of the chain segments. A convenient form to use is r2p(r) versus
r, which is within a multiplicative constant of the true probability, P,(r) (see section
5.1). This particular form yields an equal area under the curves for all segments in
all runs, and so is a good basis for comparison. When comparing runs with different
aggregation numbers a suitable scaling for the micelle radius or any other distance is
the ratio of the aggregate numbers to the one-third power. This scaling procedure
103


94
spherical (ratio ~ 2.0). The shape fluctuations are not very large (er ~ 0.1) except for
run 3 (<7 ~ 0.2). The different results from run 3 are caused by the half-harmonic head
group potential which puts a high energetic wall on head groups from the micellar
side, while allowing the head groups to move freely away from the micelle surface. The
kinetic effect of the head group movements seems to create stronger shape fluctuations
of the micelle.
Chain Conformation. Trans Bond Distributions
A 9-member chain has 6 dihedral angles. A bond is considered to be in a trans
position whenever cos<^ < 0.5 where is the dihedral angle. Table 5.3 gives the
average value of the trans fraction. This value is similar for all runs (67% or about
2 out of 3 bonds are in trans conformation). A similar value was found before for
micelles of longer chains (Woods et ah, 1986), as well as for octyl micelles with smaller
head groups (see Chapter 5) and single chains of various lengths in nonpolar media
(see Chapter 2).
Table 5.4 presents the average probability of finding a given number of trans bonds
on a chain. The uncertainties may be as large as 0.1. Four out of the six simulations
consistently show a most probable value of 4 trans out of 6 bonds. In runs 3 and
6, the most probable value is 5. The probability of finding a chain in total gauche
conformation is always zero, while the probability of finding a chain in total trans
conformation is small. The distribution among numbers of bonds varies significantly
among the runs. In particular, the distribution of run 3 is quite sharp while that


19
2.4.3 Trans Bond Distribution
Trans bond distributions have been calculated but are of limited quantitative
value. Simulations must be significantly longer for any statistically meaningful con
clusions to be made. For example, symmetry in the bonds was sometimes not fully
reached. However, the analyses can provide some insights.
Several trends can be seen in the distributions of tables 2.9 and 2.11. Table 2.9
shows that for most chain lengths, the most probable number of trans bonds agrees
with the average trans fraction. The distribution widens as the chain length decreases,
and a few states are rarely reached, especially those with more gauche than trans
bonds. The results from these probability distributions can be used to calculate a
conformational entropy associated with the runs as measured by
n3
S/k = J^p. lnp, (2.5)
t=0
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


157
Cabane, B., 1987. Small Angle Neutron Scattering. In Surfactant Solutions,
New Methods of Investigation; Raoul Zana ed.; Dekker, New York
Cabane, B., R. Duplessix, and T. Zemb, 1985. High Resolution Neutron Scat
tering on Ionic Surfactant Micelles: SDS in Water. Journal de Physique, 46,
133
Candau, J., 1987. Light Scattering. In Surfactant Solutions, New Methods
of Investigation; Raoul Zana ed.; Dekker, New York
Cantor, R. S., and K. A. Dill, 1984. Statistical Thermodynamics of Short Chain
Molecule Interphases 2. Configurational Properties of Amphiphilic Aggregates.
Macromolecules, F7, 380
Chang, N. J., and E. W. Kaler, 1985. The Structure of Sodium Dodecyl Sulfate
Micelles in Solutions of H20 and D20. Journal of Physical Chemistry, 89, 2996
Chen, S.-H, 1986. Small Angle Neutron Scattering Studies of the Structure and
Interaction in Micellar and Microemulsion Systems. Annual Review of Physical
Chemistry, 37, 351
Chevalier, Y., and C. Chachaty, 1985. Hydrocarbon Chain Conformation in
Micelles. A Nuclear Magnetic Relaxation Study. Journal of Physical Chemistry,
89, 875
Clarke, J. H. R., and D. Brown, 1986. Molecular Dynamics Computer Sim
ulation of Chain Molecule Liquids. I. The Coupling of Torsional Motions to
Translational Diffusion. Molecular Physics, 58, 815
Degiorgio, V., 1983. Nonionic Micelles. In Physics of Amphiphiles: Micelles
Vesicles, and Microemulsions; V. Degiorgio and M. Corti, eds; North-Holland,
Amsterdam
Dill, K. A., 1982. Configurations of the Amphiphilic Molecules in Micelles.
Journal of Physical Chemistry, 86, 1498
Dill, K. A., 1984a. Molecular Organization in Amphiphilic Aggregates. In
Surfactants in Solution, Vol. 1; K. L. Mittal and B. Lindmann, eds.; Plenum
Press, New York
Dill, K. A., 1985. Reply to F. M. Menger. Nature, 313. 603
Dill, K. A., and P. J. Flory, 1980. Interphases of Chain Molecules: Monolayers
and Lipid Bilayer Membranes. Proceedings of the National Academy of Sciences
U.S.A., 77, 3115


38
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.


11
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


56
Table 3.17: Lennard-Jones Parameters for Interacting Atoms and Segments, a is
Given in and e is Given in J/mol. Net Charges are Given in Units of the Elementary
Charge e=1.602xl0~19esu.
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
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


66
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 OConnell, 1984; Woods et al.,
1986; Farrell, 1988) while Jonsson 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,


I certify that I have read this study and that in my opinion it conforms to accept
able standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
ytfi Z\y b^
Gerald B. Westermann-Clark
Associate Professor of Chemical
Engineering
This dissertation was submitted to the Graduate Faculty of the College of En
gineering and to the Graduate School and was accepted as partial fulfillment of the
requirements for the degree of Doctor of Philosophy.
May, 1990
LO S^TPwr
,3finfrecPM. Phillipfe
Dean, ^College of Engineering
Madelyn M. Lockhart
Dean, Graduate School


99
S(r)
Figure 5.10: Overall bond order parameter S(r) throughout the micelle for runs 4, 5
and 6.


91
I max/
Figure 5.7: Ratio of moments of inertia from runs 1, 2 and 3.


159
Gruen, D. W. R., 1985a. A Model for the Chains in Amphiphilic Aggregates.
1. Comparison with a Molecular Dynamics Simulation of a Bilayer. Journal of
Physical Chemistry, 89, 146
Gruen, D. W. R., 1985b. A Model for the Chains in Amphiphilic Aggregates.
1. Thermodynamics and Experimental Comparisons for Aggregates of Different
Shape and Size. Journal of Physical Chemistry, 89, 153
Haan, S. W., and L. R. Pratt, 1981a. Monte Carlo Study of a Simple Model for
Micelle Structure. Chemical Physics Letters, 79, 436
Haan, S. W., and L. R. Pratt, 1981b. Errata of 79, 436. Chemical Physics
Letters, 81, 386
Haile, J. M., 1980. A Primer on the Computer Simulation of Atomic Fluids
by Molecular Dynamics, Unpublished.
Haile, J. M., and J. P. OConnell, 1984. Internal Structure of a Model Micelle
via Computer Simulation. Journal of Physical Chemistry, 88, 6363.
Hartley, G. S., 1935. Transactions of The Faraday Society, 31, 31 The Applica
tion of the Debye-Huckel Theory to Colloidal Electrolytes
Hayter, J. B., M. Hayoun, and T. Zemb, 1984. Neutron Scattering Study of
Pentanol Solubilization in Sodium Octanoate Micelles. Colloid and Polymer
Science, 262. 798
Hayter, J. B., and J. Penfold, 1981. Self-Consistent Structural and Dynamic
Study of Concentrated Micelle Solutions. Journal of the Chemical Society, Fara
day Transactions 1, 77, 1851
Hayter, J. B., and T. Zemb, 1982. Concentration-Dependent Structure of
Sodium Octanoate Micelles. Chemical Physics Letters, 93, 91
Helfland, E., 1979. Flexible vs Rigid Constraints in Statistical Mechanics. Jour
nal of chemical Physics, 71, 5000
Jnsson, B., O. Edholm, and O. Teleman, 1986. Molecular Dynamics Simula
tion of a Sodium Octanoate Micelle in Aqueous Solution. Journal of Chemical
Physics, 85, 2259
Jorgensen, W. L., 1981a. Internal Rotation in Liquid 1,2-Dichloroethane and
n-Butane. Journal of the American Chemical Society, 103. 677
Jorgensen, W. L., 1981b. Pressure Dependence of the Structure and Properties
of Liquid n-Butane. Journal of the American Chemical Society, 103. 4721


117
0.08-
0.06-
P(r)
0.04-
0.02-
T
20
Distance r (A)
40
Figure 6.8: Distribution of distances between tail groups of a model hydrocarbon
droplet.


17
Figure 2.1: The end-to-end distance for hydrocarbon chains as a function of chain
length


119
core is a hydrocarbon-like interior. The level of agreement between these micelle
simulations and the tail-tail distributions is different from other models. Cabane
et al. (1985) noted that most other models (Fromherz, 1981; Dill and Flory; 1981;
Gruen; 1985a) predict too sharp a distribution of distances that is shifted toward
smaller distances.
In figure 6.10 the distributions of distances between all groups within the core
from runs 5, 7 and 8 are compared with the same SANS data of Cabane et al. (1985).
This quantity differs from P in equation 5.9 only in that i ranges over all groups
on a molecule and j covers all groups on all other molecules. The denominator is
(/ l)2iV(./V 1), where l is the number of groups on a chain. The agreement
between the micelle simulations from runs 5 and 7 and the hydrocarbon simulation of
run 8 is excellent, indicating no head group size effect on this particular distribution
and confirming the hydrocarbon-like core for micelles. The peak positions from our
runs are slightly shifted toward smaller distances compared to those of the data, even
after using the usual scaling by the aggregation number to the one-third power. This
may either suggest that this particular distribution is affected by chain length, or
that a different scaling should be used. If the distribution of Cabane et al. (1985) is
scaled by the total number of segments instead of the aggregation number, excellent
agreement occurs (Figure 6.10).
Figure 6.11 shows that the head group pair distributions in runs 5, 7 and 8 are
significantly different. In particular, the head group size affects the distribution since


32
For j i > 3 :
U(r,)=^ + 4e
J
12
Jij,
Jij.
(3.5)
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((f)) = D0 + Di cos + T>2 cos 2 + D3 cos 3 (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.
(3.7)
J
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.


108
1 1 1
0 2 4 6 8
/
Figure 6.3: Group probability distributions of tails from run 5 and from Watanabe
et al. (1988) and Jnsson et al. (1986)


138
Table A.l: Computed Coordination Numbers for an Octyl Anionic Methyl Surfac
tant in Water.
Segment
1
2
3
4
5
6
7
8
9
Total
Period 1
12.21
4.01
3.30
2.99
3.31
2.75
3.20
5.41
9.04
46.22
Period 2
12.29
3.46
3.72
2.29
3.58
3.47
2.64
5.31
9.49
46.27
Period 3
12.56
3.92
3.79
3.12
3.11
3.44
3.40
4.16
9.50
47.00
Period 4
12.04
4.15
3.82
3.24
3.21
2.63
3.28
4.92
10.62
47.90
Period 5
13.17
4.13
2.76
3.51
3.26
2.89
3.69
3.90
10.72
48.03
Average
12.46
3.94
3.48
3.03
3.29
3.04
3.24
4.74
9.88
47.09
Std. Dev.
0.39
0.25
0.41
0.41
0.16
0.35
0.34
0.61
0.67
0.77


85
1/2
Table 5.2: Mean Radial Position (i?) 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


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


100
where 0, is the angle formed by a bond vector between segment centers and the radius
vector from the aggregate center of mass to the bond center. Figure 5.11 shows that
the individual bond order parameters from all six runs are very similar.
The bond between the head group and the first segment shows the most prefer
ential ordering, generally normal to the micelle surface, but the other bonds on the
chain show little preferred order.
Conclusions
Six molecular dynamics simulations have been made to study the effects of dif
ferent interaction models between micelles and their environment. In general, core
structure results are insensitive to different forms. There are some effects on head
group distributions, but few differences appear for segments along the chain.
While the head distributions differ among the runs, all show movement of heads
into the core. This is probably caused by excluded volume effects at the surface when
the head groups are larger than the segments. The excluded volume effect is also seen
in the pair correlations between head groups.
The average radial positions of the segments are generally similar. In all runs the
average tail group position is further from the center than the adjacent segments,
implying some curled chain conformation.
Micellar shape is generally nonspherical to some degree even though the force field
representing the micelle-environment interactions is spherical.


67


39
Table 3.8: Average Properties for the Octyl Ionic Methyl Surfactant in a Lennard-
Jones Fluid of Segments.
Property
Value
Units
% Trans
74 6
< R >
8.20


68.7
2

2.70


8.76
2
Table 3.9: Average Properties for the Octyl Nonionic Sulfate Surfactant in a
Lennard-Jones Fluid of Segments.
Property
Value
Units
% Trans
73 5
< R >
9.96


100.
2

3.53


16.1
2
Table 3.10: Average Properties for Poly (Oxyethylene) in a Lennard-Jones Fluid of
Segments.
Property
Value
Units
% Trans
46 1
< R>
11.9


148.
2

4.13


20.2
2


129
Figure 6.13: Bond order parameter S(r) throughout the micelle, for runs 5, 7 and 8.


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-Diifusion coefficients for Bulk and Shell Water Molecules in units
of 10 ~5cm2/sec 153
Xlll


132
Conclusions
The results from several molecular dynamics simulations have shown the effects
of head group size and chain length on micellar behavior. From simulations of other
workers and the present study including comparison with experimental data, several
significant conclusions can be drawn concerning micellar interior structure and shape
as well as chain conformation.
The results from the total segment distribution indicate a hydrocarbon-like in
terior in the micelle core. Consistent with SANS data (Cabane et al., 1985), the
tail-tail distribution of distances is not like those of all trans conformations nor sim
plified statistical models which are shifted to smaller distances and are sharper. The
average position of segments in the micelle is independent of chain length and head
group. It is consistent with other simulation (Jnsson et ah, 1986), while tail singlet
distributions compare well with two full simulations (Jonsson et ah, 1986; Watanabe
et ah, 1988).
The micelle shape is not affected by chain length or head group size, and on the
average is slightly nonspherical with small, but not negligible, shape fluctuations.
The conformation of chains is independent of head group size or chain length. The
trans fraction in the chains was found in all runs to be around 72%. This value was
found to be independent of rotational potential or intermolecular potential.


Table 3.4: Angle Parameters for Poly (oxyethylene).
Angle
7b
J/mol
$0
degree
CH2-CH2-0
1.651 xlO5
109.5
CH2-0-CH2
2.067 xlO5
111.8
CH2-0-H
1.135 XlO5
108.5
Table 3.5: Torsion Parameters for Poly (oxyethylene).
Bond
7r
J/mol
Do
Dr
d2
Ds
CH2-CH2-0-CH2
8314
1.053
1.250
0.368
0.675
0-CH2-CH2-0
8314
1.078
0.355
0.068
0.791
CH2-CH2-0-H
8314
1.053
1.250
0.368
0.675


13
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 >x/2 A
5.56
6.16
% Trans
68
69


60
Figure 3.10: Radius of gyration distribution for the octyl ionic methyl surfactant
in water.


115
Figure 6.7 presents the scattering amplitudes for tails, from runs 5, 7 and 8, from
Woods et al. (1986) and from Bendedouch et al. (1983a). The results from Bende-
douch et al. are truncated at the likely limits suggested by Chevalier and Chachaty
(1985). The scattering amplitude plot shows excellent agreement with the results of
Woods et al. (1986) and experimental data, and small differences with the results of
runs 5 and 7 and experimental data. The fact that figure 6.2 shows such dramatic dif
ferences among runs, while figure 6.7 does not, confirms the insensitivity of scattering
amplitudes (Cabane et al., 1985; Chevalier and Chachaty, 1985).
Distribution of Distances Between Groups
The distribution of distances between tail groups is shown in Figures 6.8 and 6.9.
Figure 6.8 is a test of the symmetry between the chain tails in the hydrocarbon
droplet. The difference between the three distributions gives a quantitative idea of
the error that would be expected in micelle distributions. In this case it was found
to be about 5%.
In figure 6.9 the distributions of distances between tail groups from runs 5, 7
and 8 are compared with the SANS data of Cabane et al. (1985) that was scaled by
(N/Ni)1/3 to account for the difference in aggregation numbers ( fV = 24 for runs
5, 7 and 8, N\ = 74 for Cabane et al.). The difference between the distributions of
runs 5 and 7 and the SANS data is very small. Apparently the head group size does
not make much difference. The distributions from the hydrocarbon droplet are also
slightly shifted from those of the SANS data, enhancing the idea that the micelle


27
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.
Uv (bi) = (bi b0)2 (3.1)
Ub (Oi) = ^76 (cos 0O cos 0,)2 (3.2)
where bi is the bond length between segments i and i+1, b0 is the equilibrium length,
7 is the bond vibration force constant, 0O is the equilibrium bond angle, 0, is the
angle between segments i, i+1 and i+2, and 7is the bending vibration force constant.
The bond rotational potential chosen for these simulations is that of Ryckaert and
Bellemans (1975):


125
Table 6.2: Average Trans Fraction and Average Ratio of Moments of Inertia
Run
run 5
run 7
run 8
Woods et al.
% Trans
I max/1 min
69 4
1.4 0.1
74 2
1.4 0.1
70 2
1.3 0.1
72 3
1.2 0.1


146
Table A.2: Ratios of the Heights of the First Maximum and the Following Minimum
for Various Water-Water Pair Correlation Functions in Bulk and Shell.
g max
groin
9max
enhancement
goo
Shell
2.65
0.91
2.91
1.10
bulk
2.40
0.91
2.64
g OH
Shell
1.28
0.26
4.92
1.05
Bulk
1.17
0.25
4.68
g HH
Shell
1.19
0.79
1.51
1.03
Bulk
1.11
0.76
1.46


82
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 OConnell, 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, R,,
and the mean radial position, (Rf)1/2, for each group relative to the aggregate center
of mass are calculated by
Ri = -jy J p, (r)r,-47rr2dr (5.6)
R? = J Pi(r)r24irr2dr (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


7
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.
Newtons second differential equations of motion were solved for each of the Na
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 1.395xl0-15 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 -3.


118
Figure 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.


5
soft spheres representing methyl tails or methylene segments. The bond vibration
and angle bending potentials are
Uv (M = ^7v (bi b0)2 (2.1)
Ub {Oi) = ^76 (cos 0o cos Oif (2.2)
where 2>, is the bond length between segments i and i+1, b0 is the equilibrium length,
7v is the bond vibration force constant, 6o is the equilibrium bond angle, 0, is the
angle between segments i, i+1 and i+2, and 7¡, is the bending vibration force constant.
The bond rotational potential chosen for these simulations is that of Ryckaert and
Bellemans (1975):
U ((f>) = 7r( 1.116 1.462 cos 1.578 cos2 (j> + 0.368 cos3
+3.156 cos4 + 3.788 cos5 )
(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.
r / \ 9 / \ 61
U(r0)=e
(2.4)
The parameters are listed in Table 2.1.


Ill
Figure 6.5: Hydrocarbon distributions for runs 5, 7 and 8, and from the micelle
simulation of Jnsson et al. (RC model) (1986). The Jnsson distribution is scaled
by (24/15)1/3.


104
seems appropriate because the micelle volume is proportional to the aggregation num
ber.
Results for the tail group distributions for the hydrocarbon droplet (run 8) are
shown in figure 6.1. This plot must show symmetry for both ends of the chain; any
discrepancy gives a measure of uncertainty in the distributions. The figure indicates
that the agreement is excellent at large and small values of r, and small differences
appear at the peak.
Results for the tail group distributions for runs 5, 7 and 8, are shown in figure 6.2
along with the results from a micelle with 24 octyl polar methyl heads (Farrell,
1988) and a micelle with 52 dodecyl polar methyl heads (Woods et al., 1986). Both
simulations have an infinite wall chain-solvent interaction model (Equation 4.4) and a
harmonic potential head-solvent interaction model (Equation 4.6). The micelle radius
for Woods et al. (1986) has been scaled by the (N/Ni)1/3 to account for the difference
in aggregate numbers (N = 24 for runs 5, 7 and 8 and for Farrell, and N\ = 52 for
Woods et al.), and the distribution is normalized. The curves for all systems appear
statistically different though the most important feature is that tail groups are found
everywhere in the micelle in all cases. On the other hand it is difficult to separate
the effects of chain length, head size and chain-solvent interaction model. It appears
that the only major difference is that the broad distributions of the small heads (runs
7, Woods et al. (1986) and Farrell (1988)) can be distinguished from the sharper
nonionic sulfate (run 5) and hydrocarbon droplet (run 8) results. The latter are


22
Table 2.10: Randomness of Conformation: Ratio of Equations 2.5 and 2.6.
Chain Length
7
9
11
13
15
17
21
- £ Pi In Pi
ln(n 2)
- Vp. lnp,
ln(n2)
1.06
1.61
0.66
1.44
1.95
0.74
1.50
2.20
0.68
1.51
2.40
0.63
1.35
2.56
0.53
1.67
2.71
0.62
2.00
2.94
0.68


154
Figure A.9: Mean square displacements of water molecules in the shell and the bulk.


55
The parameters for all the potentials are shown in Table 3.17. These parameters
were used earlier by Jnsson 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


162
Stillinger, F. H., and A. Rahman, 1978. Revised Central Force Potentials for
Water. Journal of Chemical Physics, 68, 666
Strauch, H. J., and P. T. Cummings, 1989. Computer Simulation of the Dielec
tric Properties of Liquid Water. Molecular Simulation, 2, 89
Szczepanski, R., and G. C. Maitland, 1983. Influence of Flexibility on the
Properties of Chain Molecules. In Molecular-Based Study of Fluids, Advances
in Chemistry Series, 204; J. M. Haile, and G. A. Mansoori eds.; Washington
Tabony, J., 1984. Structure of the Polar Head Layer, and water Penetration in
a Cationic Micelle. Contrast Variation Neutron Small Angle Scattering Experi
ments. Molecular Physics, 51, 975
Tanaka, H., 1987. Integral Equation and Monte Carlo Study on Hydrophobic
Effects: Size Dependence of Apolar Solutes on Solute-Solute Interactions and
Structures of Water. Journal of Chemical Physics, 86, 1512
Taupin, C., and M. Dvolaitzky, 1987. Spin Labels. In Surfactant Solutions,
New Methods of Investigation; Raoul Zana ed.; Dekker, New York
Toxvaerd, S., 1987. Comment on Constrained Molecular Dynamics of Macro
molecules. Journal Of Chemical Physics, 87, 6140
Toxvaerd, S., 1988. Molecular Dynamics of Liquid Butane. Journal of Chemical
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Van Gunsteren, W. F., J. C. Berendsen, and J. A. C. Rullmann, 1981. Stochastic
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Vilallonga, F. A., R. J. Koftan, and J. P. OConnell, 1982. Interfacial Ten
sions and Partition Coefficients in Water and n-Heptane Systems Containing
n-Alkanols, Alkylketones, Alkylamides, and Alkylmonocarboxylic Acids. Jour
nal of Colloid and Interface Science, 90, 539
Ulmius, J., and B. Lindmann, 1981. 19F NMR Relaxation and Water Penetra
tion in Surfactant Micelles. Journal of Physical Chemistry, 85, 4131
Watanabe, K., and H. C. Andersen, 1986. Molecular Dynamics Study of the
Hydrophobic Interaction in an Aqueous Solution of Krypton. Journal of Physical
Chemistry, 90, 795
Watanabe, K., M. Ferrario, and M. L. Klein, 1988. Molecular Dynamics Study
of a Sodium Octanoate Micelle in Aqueous Solution. Journal of Physical Chem
istry, 92, 819


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 Jnsson 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 Jnsson et al. (RC model) (1986). The Jnsson distribu
tion is scaled by (24/15)1/3 Ill
6.6 Scaled average radial positions for run 5, and from the micelle simu
lations of Jnsson 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 5,- for individual bonds on the N-member chains
for runs 5, 7 and 8 and from Woods et al. (1986) 131
Vlll


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


141
water oxygen unit vector such that:
uos
rp rs
IrQ rs|
(A.3)
The cutoff distance |rb rs| = 5.35 .
The distribution results show a pronounced preference for positive values of cos(a).
In other words, orientations where both hydrogens of the water are pointing towards
the segment (cos(a) = 1) are avoided, and while those where both hydrogens are
partially pointing away (cos(a) = 0.6) are favored. Similar results were previously
found by Geiger et al. (1979) in their study of the hydration of a pair of Lennard-Jones
solutes.
Water-Water Radial Pair Correlation Functions. Water-water correlation func
tions are calculated on a atom-atom basis near the surfactant molecule and in the
bulk. As before, water molecules are considered to be in the shell if they are below
a cutoff distance of 5.35 from the surfactant molecule. All other molecules are
considered to be in the bulk. In cases where one water molecule is in the shell and
another in the bulk, the atomic distances arising from such a pair contribute to both
the shell and the bulk pair correlations.
Figures A.2, A.3 and A.4 show the radial distributions of oxygen-oxygen, hydrogen-
hydrogen and oxygen-hydrogen distances in the shell and in the bulk. All distribu
tions show great similarity between the bulk and the shell with the peaks not shifted
but only having different magnitudes. This is consistent with the results found by
Geiger et al. (1979) that show no discernible shift in peak position but only a dif-


25
The end-to-end distance and radius of gyration are linear functions of chain length
for chains of 7 to 21 segments.


152
shell and bulk water can provide a good basis for comparing the motions of both
kinds of water molecules.
Since water molecules can move in and out of the shell around the surfactant
molecule, separation of the total diffusion of certain molecules into shell or bulk
diffusion is difficult. Only those water molecules that are within the shell at time
t = 0 count towards the shell diffusion, and those which enter the shell after the
beginning of the run are not counted. To limit the number of water molecules moving
in and out of the shell, the simulation was divided into several small samples of the
order of 3 picoseconds, and self diffusion coefficients are calculated only during these
small periods. The bulk and shell diffusion coefficients for seven periods are shown
in Table A.3. Over the simulation length the results indicate that bulk diffusion is
faster than shell diffusion by 14%. Geiger et al. (1979) found a 16% difference.
The results from period 4 are plotted in Figure A.9. The shell and bulk displace
ment curves have different shapes. The bulk variation is almost linear, but that of
the shell is irregular, mainly due to surfactant motions transporting water molecules
in the shell.
A.3 Conclusions
Several conclusions can be drawn about the structure and dynamics of water
molecules in the proximity of an amphiphilic molecule. It was found that hydration
numbers of end segments are much higher than those of the interior segments, and
that hydrogen atoms tend to point outward from the nonpolar part of the surfac-


109
rfp{r¡)
(A-')
Figure 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.


128
Table 6.4:
Probability of Finding a Particular Bond in the Trans Conformation
bond no.
run 5
run 7
run 8
Woods et al.
2
0.21
0.16
0.15
0.09
3
0.16
0.17
0.15
0.08
4
0.16
0.16
0.17
0.10
5
0.15
0.18
0.15
0.09
6
0.15
0.18
0.18
0.09
7
0.17
0.15
0.18
0.13
8
0.10
9
0.11
10
0.09
11
0.12


35
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.
Parameter
Methylene Value
Sulfate Value
Units
rm
4.0
10

t
419
419
J/mol
-1
0
electrons


139
end and interior segments is due to the shielding of these segments. Jorgensen et al.
(1985) found coordination numbers of 10.7 for the pentane end carbons, 4.7 for the
a-carbons and 4.1 for the middle carbon.
The difference in the coordination numbers of the head group and the tail group
can be explained by the hydrophilicity of the ionic head group. In fact, this difference
would have been much larger without the presence of a counterion which is coupled
with with the anionic head group most of the time.
The total coordination number (47.09) is slightly larger than the coordination
number of n-nonane (46.00) obtained by extrapolation of the coordination numbers
for ethane, propane, butane and pentane as calculated by Jorgensen et al. (1985).
Orientational Structure. Figure A.l shows the distribution of water dipole vector
orientations with respect to segment-water center-to-center vectors for all water
molecules present within one cutoff distance of the surfactant molecule. The relative
orientation of these vectors are calculated by computing the cosine between the above
mentioned vectors.
cos(a) Udipole UOS (A.l)
where Udipoie is the unit vector for the water dipole moment such that:
^dipole
E grt
l£;
(A.2)
and the summation over i is over the atoms in the water molecule, uqs is the segment-


6
Table 2.1: Intermolecular and Intramolecular Potential Parameters.
rm
c
0
#0
7v
76
7r
A
J/mol
A
degree
J/(mol 2)
J/mol
J/mol
4.00
419
1.539
112.15
9.25x10s
1.3x10s
8313


149
has a magnitude of 0.25. The first peak in the oxygen-head group distribution occurs
about 1 further away from the head group than the first peak in the hydrogen-
head group distribution as expected from the intratomic distance between the oxygen
and hydrogen. Thus, oxygen atoms are at the maximum distance away from the
head group given that the positively charged hydrogen is strongly attracted to the
head group while the negatively charged oxygen atom is strongly repelled. The ar
rangement of water molecules around the head group is such that only one hydrogen
from each water molecule is next to the head group. The distance of the the other
hydrogen to the head group is associated with the second peak in the hydrogen-head
group distribution.
The hydrogen-chain segment and the oxygen-chain segment distributions are
shown in Figures A.7 and A.8. Both distributions show little structure of water
molecules around the chain segments. The peak for the hydrogen-chain segment
distribution occurs at a much larger distance than the peak for the oxygen-chain
segment distribution, confirming that hydrogen atoms tend to point outward from
the chain segments.
Self Diffusion. The most significant measure of water dynamics around the sur
factant molecule and in the bulk is water self diffusion. The self-diffusion coefficient
can be determined by the long time slope of the mean square displacement. Since
the present simulation is of the order of 20 picoseconds, calculating an accurate self
diffusion is uncertain. However comparisons of the the mean square displacement of


58
Table 3.18: Average Properties for the Octyl Ionic Methyl Surfactant in Water.
Property
Value
Units
% Trans
60 3
< R >
8.18
A

67.4
A2

2.59
A

8.07
A2


126
Table 6.3:
Probability of Finding a Given Number of Trans Bonds on One Chain
no. of trans
bonds
run 5
run 7
run 8
Woods et al.
0
0.00
0.00
0.00
0.00
1
0.01
0.01
0.01
0.00
2
0.08
0.04
0.06
0.00
3
0.18
0.13
0.15
0.00
4
0.32
0.33
0.38
0.02
5
0.29
0.32
0.30
0.09
6
0.12
0.18
0.10
0.17
7
0.30
8
0.27
9
0.12
10
0.03


49
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


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
xi


24
~lP¡
Figure 2.3: Entropy S/k = Inp, 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.


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 OConnell for his guidance and support and ex
pertise and encouragement. Through his hard work and dedication John OConnell
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.
m


12
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


83
Figure 5.3: Group probability distributions for head groups


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


76
System preparation for all runs. Newtons 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*3 and rh> 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.


STRUCTURAL AND DYNAMICS STUDIES
OF SURFACTANTS AND MICELLES
By
SAMIKARABORNI
xzr
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


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
ss 0
0
1
0
0
No. of Molecules
24
24
24
24
24
24
24
24
Packing Fraction0
0.70
0.70
0.70
0.70
0.70
0.70
0.70
0.70
Packing fraction is


59
Figure 3.9: End-to-end distribution for the octyl ionic methyl surfactant in water.


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
i


Table 5.5: Probability of a Particular Bond Being Trans.
bond no.
run 1
run 2
run 3
run 4
run 5
run 6
2
0.14
0.24
0.20
0.23
0.21
0.20
3
0.20
0.16
0.15
0.15
0.16
0.14
4
0.18
0.17
0.13
0.15
0.16
0.18
5
0.18
0.13
0.17
0.15
0.15
0.16
6
0.17
0.17
0.16
0.20
0.15
0.18
7
0.14
0.14
0.20
0.13
0.17
0.13


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
64


131
Si
run 5
o run 7
x run 8
Woods et al.

0
1=1 fi g
5 8 8 8 8 8 8 1
J 1 1 1 I I I I I I I
3 6 9 12
Bond Number
Figure 6.14: Bond order parameter 5, for individual bonds on the N-member chains
for runs 5, 7 and 8 and from Woods et al. (1986).


98
S(r)
-0.5
Figure 5.9: Overall bond order parameter S(r) throughout the micelle for runs 1, 2
and 3.


123
beyond the micelle surface. In table 6.2, the average values of the ratio of moments
of inertia maxima are listed with a corresponding standard deviation. The results
show that on the average, the micellar shape is slightly non-spherical, and the shape
fluctuations are generally small, as from experiment (Hayter, 1981).
Chain Conformation. Trans Bond Distributions
A N-member chain has N-3 dihedral angles. A bond is considered to be in a trans
position whenever coscf> < 0.5 where is the dihedral angle. Table 6.2 gives average
values of the trans fraction for all systems. The value is essentially the same for all
runs (72%). We found a similar value with different rotational potentials (Jorgensen,
1984) in a 24 octyl polar methyl micelle. Other micelle simulations have yielded
different average trans fractions. Watanabe et al. (1988) reported a value of 78% while
Jnsson et al. (1986) reported a value of 50%. As discussed above, the discrepancy
between our results and those of Watanabe et al. (1988) could be a result of their
chain angle constraints. The origin of the large difference in results from our runs
and those of Jnsson et al. (1986) is unknown.
Table 6.3 presents the average probability of finding a given number of trans
bonds on one chain, as found from all four simulations. The most probable value in
all simulations is 4 out of 6 bonds (7 out of 10 for Woods et al. (1986)), consistent
with the average trans fraction. Table 5.4 also shows that the probability of finding
a chain in the all gauche conformation is zero.


40
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.


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 = £p,lnp, 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 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
vi


137
have showed a slowing of molecular motions around the nonpolar solute (Goldammer
and Hertz, 1970).
This analysis is intended to show what simulations indicate about this issue. The
model and simulation details for the present study are those described in section 3.5.
A.2 Results and Discussion
Hydration Numbers. The computed primary coordination numbers for the octyl
anionic methyl surfactant are listed in Table A.l. Coordination numbers for non-
spherical solutes can be computed according to the definition of Mehrotra and Bev
eridge (1980) and Jorgensen et al. (1985) which requires a cutoff distance of 5.35 be
tween CHn groups and water molecules. This value corresponds to the location of the
first minimum in the C-0 radial distribution functions of methane and other alkanes
in water at 25 C. Water molecules that are within the cutoff distance of two or more
surfactant segments are counted towards the coordination number of the nearest seg
ment. The total coordination number of the surfactant molecule is the sum of all
segment coordination numbers.
The average coordination numbers are shown in Table A.l for successive periods
in the simulations and for the whole duration. The coordination numbers of the head
group and the chain end are much higher than those of the interior segments. In
fact the coordination number of the ionic head group (12.46) is about three times
higher than for any interior segment, while the tail group coordination number (9.88)
is more than a factor of two greater. The difference in the coordination numbers of


69
Figure 4.2: Chain-solvent interaction models a) (r^a/i r*) 12 potential b) finite
energy barrier, U* = U/e,t = 419J/mol


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


142
00 Pair
Correlation
Figure A.2: Intermolecular oxygen-oxygen pair correlations function.


158
Dill, K. A., and P. J. Flory, 1981. Molecular Organization in Micelles and
Vesicles. Proceedings of the National Academy of Sciences U.S.A, 78, 676
Dill, K. A., D. E. Koppel, R. S. Cantor, J. A. Dill, D. Bendedouch, and S.
H.-Chen, 1984b. The Molecular Conformations in Surfactant Micelles. Nature
(London), 309. 42
Edberg, R., D. J. Evans, and G. P. Morriss, 1986. Constrained Molecular
Dynamics: Simulations of Liquid Alkanes with a New Algorithm. Journal of
Chemical Physics, 84, 6933
Edberg, R., G. P. Morriss, and D. J. Evans, 1987. Rheology of n-Alkanes by
Nonequilibrium Molecular Dynamics. Journal of Chemical Physics, 86, 4555
Enciso, E., J. Alonso, N. G. Almarza, and F. J. Bermejo, 1989. Statistical
Mechanics of Small Chain Molecular Liquids. I. Conformational Properties of
Modeled n-Butane. Journal of Chemical Physics, 90, 413
Evans, D. J., 1977. On the Representation of Orientation Space. Molecular
Physics, 34) 317
Farrell, R. A., 1988. Thermodynamic Modeling and Molecular Dynamics Sim
ulation of Surfactant Micelles. Ph. D. Dissertation, University of Florida
Frank, H. S., and M. W. Evans, 1945. Free Volume and Entropy in Condensed
Systems. Journal of Chemical Physics, 13, 507
Fromherz, P., 1981. Micelle Structure: A Surfactant Block Model. Chemical.
Phys. Lett, 77, 460
Gear, C. W., 1971. Computational Methods in Ordinary Differential Equations;
Prentice-Hall, Englewood Cliffs, NJ.
Geiger, A., A. Rahman, and F. H. Stillinger, 1979. Molecular Dynamics Study
of the Hydration of Lennard-Jones Solutes. Journal of Physical Chemistry, 70,
263
Glew, D. N., 1962. Aqueous Solubility and the Gas-Hydrates. The Methane-
Water System. Journal of Physical Chemistry, 66) 605
Goldammer, E. V., and H. G. Hertz, 1970. Molecular Motion and Structure of
Aqueous Mixtures with Nonelectrolytes as Studied by Nuclear Magnetic Relax
ation Methods. Journal of Physical Chemistry, 74, 3734
Gruen, D. W. R., 1981. The Packing of Amphiphile Chains in a Small Spherical
Micelle. Journal of Colloid and Interface Science, 84, 281


140
P(a)
Figure A.l: Distribution function for the angle cosines describing the orientation of
the water molecule dipole moment with respect to the segment-oxygen vector.


112
Table 6.1: Average Radial Position for Each Group After Scaling (See Text), i?,, ()
Relative to the Aggregate Center of Mass.
group no
run 5
run 7
run 8
Woods et al.
scaled
Jnsson et al.
scaled
Watanabe et al.
scaled
1
12.5
11.4
9.4
11.9
11.1
14.2
2
10.6
10.8
9.0
11.1
10.4
13.2
3
9.7
10.1
8.6
10.3
9.6
12.2
4
9.2
9.4
8.5
9.5
9.0
11.3
5
8.4
8.9
8.4
8.7
8.5
10.5
6
8.2
8.5
8.5
8.3
8.3
9.8
7
7.9
8.4
8.8
8.0
8.2
9.1
8
8.0
8.2
9.1
8.0
8.2
8.7
9
8.4
8.3
9.6
8.0
8.4
8.3


30
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
bo
1.539
2.6
X
0o
112.15
140
degree
7
9.25x10s
2.7xl04
J/(mol 2)
76
1.3x10s
9.1x10s
J/mol
It
8313
20000
J/mol


9
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 ah, 1986; Wielopolski and Smith, 1986; Toxvaerd, 1988; Rebertus et ah, 1979) and
statistical mechanics calculations (Enciso et al., 1989; Zichi and Rossky, 1986a), but
comparable to BD (Van Gunsteren et ah, 1981) and MC results (Jorgensen 1981a,
1981b; Jorgensen et ah, 1981e; Bigot and Jorgensen, 1981; Bnon et ah, 1985). Several
factors may have affected the MD work, especially the limited duration of some


96
of run 6 is quite broad. Entropy values calculated from S/R = J2PilnR; are also
given in table 5.4 and appear to be similar, though run 3 is smaller while run 6 is
much larger.
Table 5.5 presents the probabilities of finding a particular bond on the chain in
the trans conformation. Results from five out of six simulations show that bond
number 2 (dihedral angle involving groups 1-4) has the highest probability among all
bonds to be in trans conformation with the others being about equal. This probably
arises from the different bond and head group potential models. Run 1 appears to be
different, perhaps due to poorer statistics.
Bond Orientation
An indicator of the chain conformation is the overall bond order parameter, S(r),
defined by:
S(r) =< ^(3cos2 0 ~ 1) > (5.10)
where 9 is the angle formed by the bond vector connecting two adjacent groups and the
radius vector from the aggregate center of mass to the center of the bond. Figures 5.9
and 5.10 show that for all runs the overall bond parameter is positive, reaching a
high preferential ordering at the micelle surface and a somewhat preferential ordering
at the micelle center.
Individual bond order parameters 5,- can be calculated for each bond on the chain
Si =< (3cos29i l) >
(5.11)


15
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
8.87
8.87
8.82
8.72
9.06
< S > A
3.11
-
-
3.07
2.90
< S2 > V3 A
3.16
3.11
3.12
3.08
3.18
% Trans
60.4
62.4
62.4
60
69


101
Figure 5.11: Individual bond order parameter 5,- for bonds on the 9-member chains
from runs 1-6.


160
Jorgensen, W. L., 1981c. Transferable Intermolecular Potential Functions for
Water, Alcohols and Ethers. Application to Liquid Water. Journal of the Amer
ican Chemical Society, 103. 335
Jorgensen, W. L., 1982. Revised TIPS for Simulations of Liquid water and
Aqueous Solutions. Journal of Chemical Physics, 77, 4156
Jorgensen, W. L., R. C. Binning, and B. Bigot, 1981e. Structure and Properties
of Organic Liquids: n-Butane and 1,2-Dichloroethane and Their Conformational
Equilibria. Journal of the American Chemical Society, 103. 4393
Jorgensen, W. L., J Chandrasekhar, J. Madura, W. Impey, and M. Klein, 1983.
Comparison of Simple Potential Functions for Simulating Liquid Water. Jour
nal of Chemical Physics, 79, 926
Jorgensen, W. L., J. D. Madura, and C. J. Swenson, 1984. Optimized Inter
molecular Functions for Liquid Hydrocarbons. Journal of the American Chem
ical Society, 106. 6638
Jorgensen, W. L., J. Gao, and C. Ravimohan, 1985. Monte Carlo Simulations
of Alkanes in Water: Hydration Numbers and the Hydrophobic Effect. Journal
of Physical Chemistry, 89, 3470
Jorgensen, W. L., and M. Ibrahim, 1981d. Structure and Properties of Organic
Liquids: n-Alkyls Ethers and Their Conformational Equilibria. Journal of the
American Chemical Society, 103. 3976
Lindman, B., O. Soderman, and H. Wennerstrom, 1987. NMR Studies of Sur
factant Systems In Surfactant Solutions, New Methods of Investigation; Raoul
Zana ed.; Dekker, New York
Matsouka, O., E. Clementi, and Y. Yoshimine, 1976. Cl Study of the Water
Dimer Potential Surface. Journal of Chemical Physics, 64, 1351
Mehrotra, P. K., and D. L. Beveridge, 1980. Structural Analysis of Molecular
Solutions Based on Quasi-Component Distribution Functions. Journal of The
American Chemical Society, 102. 4287
Menger, F. M., 1979. On The Structure of Micelles. Accounts of Chemical
Research. 12. Ill
Menger, F. M., 1985. Molecular Conformations in Surfactant Micelles. Nature,
313. 603


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. Given in and t is Given in J/mol. Net Charges are Given in Units
of the Elementary Charge e=1.602x 10_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, e^, 7 and /? Are in Units of e.
rhhi r*hs and r*s are in Units of rm 75
4.2 Temperatures and Pressures for Molecular Dynamics Simulations. . 77
5.1 Average Radial Position Ri for Each Group, Measured Relative to the
Aggregate Center of Mass 84
5.2 Mean Radial Position (i?) 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), R¡,
() 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.l Computed Coordination Numbers for an Octyl Anionic Methyl Sur
factant in Water 138
xii


121
P(r)
Figure 6.11: Distribution of distances between head groups


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 Model 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 WATER 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
formation 49
IV


53
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 ah, 1981),
TIPS (Jorgensen, 1981c), TIPS2 (Jorgensen, 1982), and TIP4P (Jorgensen et ah,
1983). Overall the SPC, ST2, TIPS2 and TIP4P models give reasonable structural
and thermodynamic descriptions of liquid water (Jorgensen et ah, 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


42
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 4 to 11. Basically the end-to-end distance
samples all available conformational space from 4 (rmtn in the Lennard-Jones po
tential) to 11 (the all-trans end-to-end distance).
The end-to-end distance distribution for the nonionic sulfate surfactant extends
from about 7 to about 12.3. 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 rmtn 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
<7 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


113
of aggregate numbers. The result of this comparison is good agreement between the
average positions of run 7 and the scaled positions of Woods et al. (1986). (Statistical
error of 0.1 ). Thus the relative effect of chain length on average position is
negligible. If similar scaling is used on the average positions reported by Jnsson et
al. (1986) and by Watanabe et al. (1988), it can be seen from figure 6.6 that there is
excellent agreement between those of Jnsson et al. and the present ones while those
from the Watanabe et al. are different, being monotonic and further out. This may be
due to a higher trans fraction in the chain that could be an artifact of their simulation
that had constrained chain angles (Helfland, 1979; Rallison, 1979). Toxvaerd (1987)
has reported that the number of trans-gauche transitions per picosecond decreased
from 0.333 to 0.231 when constraints are applied to a decane molecule, resulting in
greater trans fractions in shorter runs. The same effect may be present here.
Distributions of tail groups
The tail distribution of figure 6.2 can be compared with measurements of Bende-
douch et al. (1983a) using small angle neutron scattering (SANS) on lithium dodecyl
sulfate. This system has a head group with similar size and mass to that in run 5
and a chain length similar to that of Woods et al. (1986). The scattering vector Q is
scaled with (N\/N)1/3 to account for the difference in aggregation numbers (Ai=78
for Bendedouch et al. (1983a), Nj=52 for Woods et al. and N=24 for runs 5, 7 and
8).


79
The singlet density pi(r) is related to P(r) by:
Pi(r)4nr2Ar
Pi{r) =
(5.4)
When Ar 0
J pi(r)4-Kr2dr = N (5.5)
Thus, a convenient form to use is r2pi(r), which is within a multiplicative constant
of the true probability P,(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.


4
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 OConnell, 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


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


BIOGRAPHICAL SKETCH
Sami Karaborni was born in 1964 in Sousse, Tunisia, the son of Tijani Karaborni
and the former Jamila Kedadi. He graduated from the Lygee de Garmons de Sousse in
1982. After pursuing a brief intensive English program in State College, Pennsylva
nia, he joined the Pennsylvania State University College of Engineering in December
of 1982. In 1986 he received with high distinction the degree of Bachelor of Sci
ence in chemical engineering. Subsequently, he joined the Graduate Program at the
University of Florida, receiving a Master of Science degree in 1987.
164


Table 2.11: Probability of Finding a Particular Dihedral Angle in the Trans Confor
mation.
Dihedral Angle
7
9
11
13
15
17
21
fa
0.25
0.15
0.11
0.11
0.06
0.03
0.05
2
0.26
0.19
0.16
0.12
0.12
0.07
0.05
4> 3
0.25
0.15
0.12
0.11
0.07
0.08
0.07
4
0.24
0.18
0.13
0.08
0.11
0.06
0.03
fa
0.15
0.13
0.13
0.09
0.07
0.05
6
0.18
0.06
0.08
0.07
0.08
0.07
fa
0.16
0.11
0.05
0.08
0.07
fa
0.13
0.09
0.08
0.05
0.08
fa
0.05
0.07
0.07
0.04
fao
0.13
0.09
0.08
0.08
fal
0.08
0.10
0.03
12
0.11
0.08
0.05
fas
0.05
0.08
14
0.10
0.05
0.03
fas
0.07
fal
0.05
fas
0.06
Uniform
0.25
0.17
0.13
0.10
0.08
0.07
0.06


37
Table 3.7: Simulation Details for Runs in Lennard-Jones Fluid of Segments.
Simulation
Time Step
secs
Equilibration
Steps
Equilibrium
Run
ionic methyl surfactant
1.395x 10-15
50,000
150,000
nonionic sulfate surfactant
1.395X10-15
40,000
130,000
poly (oxyethylene)
1.331xl0~15
10,000
75,000


3.4.5Probability of Finding a Particular Angle in the Trans Confor
mation 49
3.5 Ionic Methyl Octyl Surfactant in Water 52
3.5.1 Model 52
3.5.2 Results 57
3.6 Conclusions 63
4 MODEL MICELLE 64
4.1 Background 64
4.2 Micelle Models 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 Micelle 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 Micelle 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
v


68
Uuirij)
(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:
headsegment
'm
= j (r + rkh)
(4.2)
Head-head interactions are modeled by a purely repulsive potential which includes
both dipole-like repulsion and excluded volume effects:
Uhh ( f'ij )
(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
ttf" 00 = (r;. r-)
-12
(4.4)


51
Table 3.13: Probability of Finding a Number of Bonds in the Trans Conformation on
a Poly (Oxyethylene).
Number of Bonds
0
1
2
3
4
5
6
7
8
Probability
0.0
0.0
0.0
0.01
0.04
0.09
0.15
0.21
0.21
Number of Bonds
9
10
11
12
13
14
15
16
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



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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 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


155
tant molecule. Around the surfactant anionic head group only one hydrogen of the
water molecule points inward. The water-water pair correlation functions show a
slight structure enhancement in the shell over the bulk, while the water self diffusion
coefficients are somewhat smaller in the shell than in the bulk.


87
Figure 5.4: Scattering amplitude for methyl tail groups.


116
Q
Figure 6.7: Scattering amplitude from methyl tails for runs 5, 7 and 8, from Woods
et al. (1986) and from Bendedouch et al. (1983a)


72
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.
u2) ir) = 7(rL -r*)2
r* < n.
(4.7)
uh2) (r) = - r*> rhs
As with equation 4.6, the value of the repulsive energy at the center of the shell is 7
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.
2,3|(0 = //(l- + />(2-r*/rUT) r-<2 r\, (4.8)
The values of p and r were chosen to provide a sharp (4.5 ) transition while (3
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
ah, 1982) were used. The potential changes rapidly from zero in the bulk solvent to a


134
surface. On average, the tail group position is further from the micelle center than
the adjacent segments. Analysis of moment of inertia shows that micellar shape is
on the average somewhat nonspherical, and is strongly affected by the nature of the
head group-solvent interaction model.
Analysis from micelle and hydrocarbon droplet simulations with different chain
lengths and head sizes shows that properties such as average radial positions of seg
ments, micellar shapes and chain conformations are not affected by the size of the
head group or the length of the chain, while other properties such as singlet radial
distributions and distributions of distances between segments do depend on the sur
factant characteristics.
In general, results from all micelle simulations indicate that the micellar core is
hydrocarbon like, and that the conformation of micellar surfactants is much like that
of ionic surfactants in nonpolar fluids.
The results found here have been generally consistent with experiment and other
simulations yielding a consistant understanding about micellar structure. Nonethe
less, to fully understand the process of micelle formation future research should ex
amine in more details the conformation of ionic and nonionic surfactants prior to
micellization, and study the structure of water in the presence of these surfactants.
The micelle model that has been originally developed by Woods et al. (1985) and
improved in this study might be usable to study the solubilization of nonpolar species
in micellar solutions and to investigate the catalytic capabilities of micelles.


33
Table 3.2: Lennard-Jones and Coulombic Interaction Parameters for Poly (oxyethy-
lene).
Site
q
electrons
t
J/mol
a
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
Table 3.3: Bond Parameters for Poly (oxyethylene).
Bond
J/(mol 2)
bo
A
CH2-CH2
9.25 xlO5
1.526
CH2-0
1.14 xlO6
1.425
O-H
1.97 xlO6
0.960


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:
< JV¡(r) >
N
(51)
Where < Ni(r) > is the average number over time of groups i that are found in shell
of thickness Sr with radius r, and N is the total number of molecules, with the sum
over all shells being:
E < Ni(n) > = N (5.2)
3
and
EP(ri) = 1 (5.3)
78


122
two peaks are present in run 5 while only one peak is present in run 7. There is a
crowding of the larger head groups around the micelle surface in run 5. The first peak
is associated with the pair potential while the second peak (located where the smaller
head peak is) is from a reasonably uniform distribution of heads around the micelle
interface. The distribution of distances between tails in the hydrocarbon droplet is
shown for comparison. As expected, the peak for non surfactant chains occurs at a
shorter distance due to their freedom to travel throughout the hydrocarbon droplet.
Micelle Shape
The shape of the model micelles can be indicated by values of the principal mo
ments of inertia. One measure of nonsphericity is the ratio of the largest moment of
inertia to the smallest moment of inertia which is unity for a sphere and greater than
one for nonspherical shapes. A good measure of shape fluctuations is the standard
deviation from the mean of this ratio. In Figure 6.12, a plot is given of this ratio
as a function of time. It can be noted that both the period and the amplitude of
fluctuations for run 5 are somewhat larger than for runs 7 and 8. Peaks for run 5 are
separated by about 10 picoseconds while peaks for runs 7 and 8 are separated by only
5 picoseconds. The harmonic potential for the heads groups in run 7 is responsible for
keeping them a short distance from the micelle surface, i.e. the frequency imposed
by the harmonic potential controls the period of shape fluctuations. In run 8 the
lack of head-solvent interactions results in very small fluctuations about the average
aggregate shape because chains can extend only a small fraction of their total lengths


124
7 max/
Figure 6.12: Ratio of moments of inertia from runs 5, 7 and 8.


71
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/t^t = 419J/mol


62
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
0.17


no
Hydrocarbon distribution
The hydrocarbon distribution is the sum of all chain segment distributions. A
plot of this distribution (figure 6.5) shows no significant difference between the octyl
micelles and the hydrocarbon droplet. Thus the micelle core structure is essentially
that of a hydrocarbon. The scaled and normalized results from Jnsson et al. (1986)
show excellent agreement with our results.
Average Chain Segment Positions
Further information on the local structure of groups is given by the average radial
position for each group. The results for all simulations are shown in Table 6.1. For run
7, the average position decreases faster closer to the head group and is nearly constant
for the last four groups. However, in run 5 the tail group is significantly further from
the micelle center than are groups 6, 7 and 8. This appears to be due to the large
head group having some tendency to be in the center. The hydrocarbon droplet (run
8) shows significant backward flux with tails being at the solvent interface. The
required symmetry is shown.
To study the effect of aggregate number and chain length on the average position
of groups, a proper basis for comparisons is required. In particular, the tail-to-
tail or head-to-head bases are not appropriate. However the chain positions can
be treated as continuous variables and average positions can be calculated for the
same fractional location on a longer or shorter chain that corresponds to segments in
the octyl chain. These positions can be scaled by the one-third power of the ratio


81
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)


46
P(r)
Figure 3.6: Radius of gyration distribution for the octyl ionic methyl surfactant in
Lennard-Jones fluid of segments.


102
The chain conformation results show great similarity among the runs with an av
erage trans fraction of 67% and chains are never in a completely gauche conformation.
The first bond on the chain tends to have the highest probability among all bonds to
be in trans conformation, and is usually the most preferentially ordered among the
bonds.


151
Figure A.8: Intermolecular oxygen-chain segment pair correlation functions.


36
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.
Newtons 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


105
Figure 6.1: Group probability distributions of chain ends of a model hydrocarbon
droplet.


163
Weber, T. A., 1978. Simulation of n-Butane Using a Skeletal Alkane Model.
Journal of Chemical Physics, 69, 2347
Weiner, S. J., P. A. Kollman, D. A. Case, U. C. Singh, C. Ghio, G. Alagona,
S. Profeta, and P. Weiner, 1984. A New Force Field for Molecular Mechanical
Simulation of Nucleic Acids and Proteins. Journal of The American Chemical
Society, 106. 765
Wielopolski, P. A., and E. R. Smith, 1986. Dihedral Angle Distribution in Liquid
n-Butane: Molecular Dynamics Simulations. Journal of Chemical Physics, 84.
6940
Woods, M. C., J. M. Haile, and J. P. OConnell, 1986. Internal Structure of
a Model Micelle via Computer Simulation. 2. Spherically Confined Aggregates
with Mobile Head Groups. Journal of Physical Chemistry 90, 1875
Zana, R., 1987. Luminescence Probing Methods. In Surfactant Solutions, New
Methods of Investigation; Raoul Zana ed.; Dekker, New York
Zemb, T., and C. Chachaty, 1982. Alkyl Chain Conformations in a Micellar
System From the Nuclear Spin Relaxation Enhanced by Paramagnetic Ions.
Chemical Physics Letters, 88, 68
Zemb, T., and P. Charpin, 1985. Micellar Structure From Comparison of X-ray
and Neutron Small-Angle Scattering. Journal de Physique, 46, 249
Zichi, D. A., and P. J. Rossky, 1986a. Molecular Conformational Equilibria in
Liquids. Journal of Chemical Physics, 84, 1712
Zichi, D. A., and P. J. Rossky, 1986b. Solvent Molecular Dynamics in Regions
of Hydrophobic Hydration. Journal of Chemical Physics, 84, 2814


47
P(r)
Figure 3.7: Radius of gyration distribution for the octyl nonionic sulfate surfactant
in a Lennard-Jones fluid of segments.


50
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


63
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.


20
Table 2.9: Probability of Finding a Given Number of Trans Bonds on the Chain.
Number of
Trans Bonds
7
9
11
13
15
17
21
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
-£p,lnp,
1.06
1.44
1.50
1.51
1.35
1.67
2.00


80
rfpin
i-i
Figure 5.1: Group probability distributions for tail groups


90
two peaks. The first peak is associated with the pair potential and is due to excluded
volume effects, while the second peak is from a reasonably uniform distribution of
heads around the micelle surface.
The distributions of distances between tail groups are similar for all runs except
run 2. The distribution of distances between tails in run 2 exhibits a peak shifted
towards smaller distances. It is probably due to the strong head group harmonic
potential keeping the head on the surface of the micelle and forcing all segments into
the core. This forces the tail groups into a smaller volume than in other runs.
Micelle Shape
The shape of micelle models can be determined from the principal moments of
inertia. We used the ratio of the largest moment of inertia to the smallest moment
of inertia where a sphere has a value of unity. A good measure of shape fluctuation
is the standard deviation of this ratio. In Figures 5.7 and 5.8, a plot is given of this
ratio as a function of time. It can be noted that, to the degree it exists, the period
of fluctuations for runs 1 and 2 are somewhat smaller (~ 3 ps) than for runs 4, 5 and
6 (~ 10 ps). The presence of the harmonic potential greatly affects the micelle shape
by keeping head groups a short distance from the micelle surface. The head-solvent
potential has a significant effect on the instantaneous micellar shape.
In table 5.3, average values are listed with their corresponding standard deviations.
The results show that, on the average, the micellar shape is slightly non-spherical
(ratio ~ 1.3) except in run 3 (half-harmonic) in which the shape is highly non-


147
Q(r)
Figure A.5: Intermolecular hydrogen-head group pair correlation function.


44
0.008
r^,^9(A)
Figure 3.5: End-to-end distribution for poly (oxyethylene) in a Lennard-Jones fluid
of segments.


153
Table A.3: Self-Diffusion coefficients for Bulk and Shell Water Molecules in units of
10 ~5cm2/sec.
Period
1
2
3
4
5
6
7
Total
Dbuik
D shell
Dbulk
D.ihell
6.9
5.4
1.28
7.3
6.1
1.20
6.5
5.6
1.16
6.9
5.7
1.21
6.5
6.4
1.02
6.6
5.9
1.12
5.3
5.4
0.98
6.6
5.8
1.14


A.l 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
IX


75
Table 4.1: Intermolecular Potential Parameters, e/,/,, 7 and /? Are in Units of e.
r£s and r*a are in Units of rm.
Run
run 1
run 2
run 3
run 4
run 5
run 6
run 7
run 8
e(J/mol)
419
419
419
419
419
419
419
419
()
4.0
4.0
4.0
4.0
4.0
4.0
4.0
4.0
<7 ()
3.5
3.5
3.5
3.5
3.5
3.5
3.5
3.5
hh
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

rL
3.2
3.2
3.2
4.72
4.72
2.85
3.00

rc,
4.2
4.2
3.35
4.2
4.2
3.35
3.50
3.48
P
-
-
-
8.27
82.7
82.7
-
-


73
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 rTO,
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 (7 = 300) applied to the head groups. The solvophobic potential of
equation 4.5 was applied to the segments ( =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


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 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 7.
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.


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
formation 23
x


I certify that I have read this study and that in my opinion it conforms to accept
able standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
John P. OConnell, Chairman
Professor of Chemical Engineering
I certify that I have read this study and that in my opinion it conforms to accept
able standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
0.
Dinesh O. Shah, Co-Chairman
Professor of Chemical Engineering
I certify that I have read this study and that in my opinion it conforms to accept
able standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
Brij M^loudgil
Professor of Materials Science
and Engineering


28
Figure 3.1: Model octyl surfactants.


120
Figure 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.


95
Table 5.4: Probability of a Given Number of Trans Bonds on One Chain
no. of trans
bonds
run 1
run 2
run 3
run 4
run 5
run 6
0
0.00
0.00
0.00
0.00
0.00
0.00
1
0.04
0.01
0.01
0.02
0.01
0.05
2
0.13
0.09
0.06
0.11
0.08
0.30
3
0.28
0.28
0.22
0.26
0.18
0.28
4
0.37
0.32
0.25
0.35
0.32
0.23
5
0.16
0.26
0.41
0.20
0.29
0.33
6
0.02
0.05
0.05
0.07
0.12
0.07
- E PilnPi
1.49
1.48
1.41
1.55
1.53
1.76


45
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 , while the main peak oc
curs at 4.2 . 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((f>,) in equations 3.3 and 3.6 is less than -0.5. For all other values of cos (fa) the


To
my Father


74
equation 4.8 (/? = 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 /? 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 (7 = 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.


18
Figure 2.2: The radius of gyration for hydrocarbon chains as a function of chain
length


143
HH Pair
Correlation
Figure A.3: Intermolecular hydrogen-hydrogen pair correlation function.


89
Figure 5.6: Distribution of distances between tail groups


54
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.
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.


161
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1656
Nemethy, G., and H. A. Scheraga, 1962. Structure of Water and Hydrophobic
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Okazaki, S., K. Nakanishi, H. Touhara, N. Watanabe, and Y. Adachi, 1981. A
Monte Carlo Study on the Size Dependence in Hydrophobic Hydration. Journal
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Owenson, B., and L. R. Pratt, 1984. Molecular Statistical Thermodynamics of
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Pitzer, K.S., 1940. The Vibration Frequencies and Thermodynamic Functions
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Rao, M., and B. J. Berne, 1981. Molecular Dynamic Simulation of the Structure
of Water in the Vicinity of a Solvated Ion. Journal of Physical Chemistry, 85.
1498
Rapaport, D. C., and H. A. Scheraga, 1982. Hydration of Inert Solutes. A
Molecular Dynamics Study. Journal of Physical Chemistry, 86, 873
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J. B. F. N. Engberts, 1984. Molecular Dynamics Computer Simulation of the
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Butane Near its Boiling Point. Chem. Phys. Lett., 30, 123
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nes. Discuss. Faraday Soc., 66, 95
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by Molecular Dynamics. Journal of Chemical Physics, 60, 1545


16
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
(Bnon 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.


70
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
ah, 1982).
U*2) (r) = k/ (1. + p (r*/r*ca)T) (4.5)
This potential changes rapidly from zero in the core to a higher value outside the core.
The value of k was chosen to match the free energy of solubilization of methylenes
in water (Vilallonga et ah, 1982), while the steepness of the potential was controlled
by p and r to make 90% of the change in 4.5 () 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.
Uhs1)(r) = l (r*hs-r*)2 (4.6)
The harmonic potential constant 7 controls the amplitude of normal movement of


148
g(r)
Figure A.6: Intermolecular oxygen-head group pair correlation functions.


130
runs 7 and 8 the bond parameter is zero except at the surface and the center. The
scatter in the values at very small and very large r seen in the figure is a consequence
of the relatively small number of bonds to be sampled there. The preferential ordering
of run 5 is due to excluded volume effects since head groups totally occupy the micelle
surface forcing the first few bonds to be parallel to the micelle radius.
Another indicator of chain conformation is the bond order parameter Si which
can be calculated for each bond, i, on the chain from
Si =< (1/2) (3cos26i l) > (6.2)
where 0,- is the angle formed by the bond vector connecting the adjacent segment
centers and the radius vector from the aggregate center of mass to the center of
the bond. Figure 6.14 shows that for runs 7 and 8, the bonds in the chain show no
preferred order, while for run 5 the bond between the head group and the first segment
shows the most preferential ordering, and bonds along the chain show less, but finite,
preferred order. The latter is consistent with the first dihedral bond having a higher
fraction of trans conformation and the segments being excluded from the surface. In
Woods et al. (1986) all bonds have a positive order parameter, suggesting a slight
preference of bonds to be parallel to the micelle radius. This arises from the infinite
wall chain-solvent interaction model used in their model tending to keep segments
from reaching the micelle surface. The finite barrier potential of runs 7 and 8 does
not exhibit any order.


135
The effective solvent environment intermolecular potentials may be pertinent to all
surfactant solutions for application to reverse micelles, microemulsions and polymer-
surfactant mixtures.


93
Table 5.3: Average Trans Fraction and Average Ratio of Moments of Inertia
Run
run 1
run 2
run 3
run 4
run 5
run 6
% Trans
1max/1 min
59 2
1.3 0.1
64 2
1.2 0.1
69 2
2.0 0.2
63 1
1.3 0.1
69 4
1.4 0.1
66 2
1.3 0.1


Bibliography
Allen, M. P., and D. J. Tildesley, 1987. Computer Simulation of Liquids, Claren
don Press, Oxford.
Alper, H. E., and R. M. Levy, 1989. Computer Simulations of the Dielectric
Properties of Water: Studies of the Simple Point Charge and Transferable Inter-
molecular Potential Models. Journal of Chemical Physics, 91, 1242
Baon, A., F. Serrano Adan, and J. Santamara, 1985. The Effect of Inter-
molecular Potential Model on the Structure and Conformational Equilibrium of
Liquid n-Butane. Journal of Chemical Physics, 83, 297
Bendedouch, D., S.-H. Chen, and W. C. Koehler, 1983a. Structure of Ionic
Micelles from Small Angle Neutron Scattering. Journal of Physical Chemistry,
87, 153
Bendedouch, D., S.-H. Chen, and W. C. Koehler, 1983b. Determination of In
terparticle Structure Factors in Ionic Micellar Solutions by Small Angle Neutron
Scattering. Journal of Physical Chemistry, 87, 2621
Ben-Shaul, A., and W. M. Gelbart, 1985. Theory of Chain Packing in Am
phiphilic Aggregates. Annual Review of Physical Chemistry, 36, 179
Berendsen, H. J. C., J. P. M. Postma, W. F. Van Gunsteren, and J. Her
mans, 1981. Interaction Models for Water in Relation to Protein Hydration.
In Intermolecular Forces. B. Pullman ed.; Reidel, Dordrecht, Holland
Bernal, J. D., and R. H. Fowler, 1933. A Theory of Water, and Ionic Solution,
with Particular Reference to Hydrogen and Hydroxyl Ions. Journal of Chemical
Physics, 1, 515
Bigot, B., and W. L. Jorgensen, 1981. Sampling Methods for Monte Carlo
Simulations of n-Butane in Dilute Solution. Journal of Chemical Physics, 75.
1944
Cabane, B., 1981. Structure of the Water/Surfactant Interface in Micelles:
An NMR Study of SDS Micelles Labeled with Paramagnetic Ions. Journal de
Physique, 42, 847
156


65
Small-Angle Scattering is the only method that allows distances to be measured in
the range 5 to 500 (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 ah, 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 ah, 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


14
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 >J/2 A
7.28
7.57
< S >
2.59
2.67
< S2 >1'2 A
2.60
2.67
% Trans
64
69