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Thermodynamic modeling and molecular dynamics simulation of surfactant micelles

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Thermodynamic modeling and molecular dynamics simulation of surfactant micelles
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Farrell, Robert Anthony, 1956-
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English
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vii, 276 leaves : ill. ; 28 cm.

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Subjects / Keywords:
Free energy ( jstor )
Hydrocarbons ( jstor )
Micelles ( jstor )
Modeling ( jstor )
Molecular dynamics ( jstor )
Molecules ( jstor )
Monomers ( jstor )
Simulations ( jstor )
Surfactants ( jstor )
Thermodynamics ( jstor )
Chemical Engineering thesis Ph. D
Dissertations, Academic -- Chemical Engineering -- UF
Micelles ( lcsh )
Micelles -- Computer simulation ( lcsh )
Surface active agents ( lcsh )
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bibliography ( marcgt )
non-fiction ( marcgt )

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Thesis:
Thesis (Ph. D.)--University of Florida, 1988.
Bibliography:
Includes bibliographical references.
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Typescript.
General Note:
Vita.
Statement of Responsibility:
by Robert Anthony Farrell.

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University of Florida
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University of Florida
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Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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1
THERMODYNAMIC MODELING AND MOLECULAR DYNAMICS SIMULATION
OF SURFACTANT MICELLES
By
ROBERT ANTHONY FARRELL
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
1988


To Peggy, for her support,
encouragement, and patience


TABLE OF CONTENTS
Page
ABSTRACT V
CHAPTERS
1 INTRODUCTION 1
2 A MOLECULAR THERMODYNAMIC MODEL
OF MICELLE FORMATION 4
2.1 Background 4
2.2 Stoichiometry and Reaction Equilibria
for Multicomponent Micelles 8
2.3 Estimation of Free Energy Changes 11
2.4 The Calculational Technique 17
3 RESULTS OF THERMODYNAMIC MODELING 23
3.1 Parameter Estimation 23
3.2 Behavior of the Model for
Single-Component Nonionic Systems 30
3.3 Aggregate Size Distribution and
Concentration Behavior 40
3.4 Chain Length and Temperature Effects 43
4 MOLECULAR DYNAMICS SIMULATION
OF SURFACTANT MICELLES 52
4.1 Background 52
4.2 The Molecular Dynamics Method 55
4.3 The Model Surfactant Molecule 60
4.4 The Model Micelle 65
4.5 Summary of Computer Simulations 71
5 RESULTS OF MOLECULAR DYNAMICS SIMULATION 78
5.1 Mean Radial Positions of Groups 78
5.2 Probability Distributions of Group Positions. 90
5.3 Conformations of Chain Molecules 107
5.4 Shape Fluctuations 126
5.5 Pair Correlations of Groups 133
6 CONCLUSIONS 137
iii


APPENDICES
A MICELLE SIZE AND SHAPE 14 3
B HEAD GROUP INTERACTION IN A BINARY MICELLE .... 150
C PROGRAM LISTING FOR SINGLE-COMPONENT
NONIONIC MICELLE CALCULATION 160
D MOLECULAR DYNAMICS PROGRAM LISTING
FOR SIMULATIONS 1 AND 2 17 4
E DERIVATION OF HEAD GROUP
INTRAMOLECULAR INTERACTIONS 218
F MOLECULAR DYNAMICS PROGRAM LISTING
FOR SIMULATION 3 224
REFERENCES 2 68
BIOGRAPHICAL SKETCH 276
IV


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
THERMODYNAMIC MODELING AND MOLECULAR DYNAMICS SIMULATION
OF SURFACTANT MICELLES
By
ROBERT ANTHONY FARRELL
April, 1988
Chairman: John P. O'Connell
Major Department: Chemical Engineering
The association of surfactant molecules into aggregates
known as micelles gives them a broad range of applications.
In spite of the widespread use of this class of chemicals,
there is not yet sufficient scientific understanding to
predict their behavior in solution.
A molecular thermodynamic model has been developed to
describe the formation of micelles in multicomponent
surfactant solutions. Using a hypothetical, reversible
seven-step process, the total free energy of micellization
is calculated by summing the contributions due to
solvophobic interaction, mixing, surface formation, confor
mational change, head group interactions and electrostatics.
Distributions of micelle sizes and compositions can then be
generated through a set of reaction equilibria. Where
v


possible, the free energy contributions are related to
comparable processes on which experimental measurements have
been made. Aggregate size distributions have been generated
from the model for single-component solutions of nonionic
surfactants of different chain lengths and at different
temperatures and solution concentrations.
It has been found that a detailed description of micelle
structure and entropy effects on chain conformation is
necessary to fully describe the thermodynamics of micelle
formation without empirical parameterization. To this end,
computer simulations of model micelles have been conducted
by the molecular dynamics method. Micelles of three
different head group characteristics and a comparable
hydrocarbon droplet have been simulated. A spherical shell
is used to contain the aggregate, providing estimated
solvophobic interactions with the molecules.
The simulation results reveal that internal structure of
the aggregates is relatively insensitive to the head group
characteristics with the greatest effect resulting from head
group size. While the micelles all showed chain ordering
and the hydrocarbon droplet did not, the bond conformations
averaged approximately 71 percent trans in all cases.
Results of other simulations and experimental studies are
vi


generally similar to those of the present work for the
effects of chain length, aggregate size, and simulation
technique on static properties.
Vll


CHAPTER 1
INTRODUCTION
There are few classes of chemical species which have
received more attention in the scientific literature, or
have exhibited a more ubiquitous presence in everyday life,
than the amphiphilic molecules known as surfactants. Their
unique solution properties of association and adsorption at
interfaces give rise to a broad range of applications. Long
the essential component in detergency applications, in more
recent times surfactants have assumed roles of importance in
applications as diverse as enhanced oil recovery and
pharmacy.
The association of surfactant molecules in solution into
aggregates known as micelles is the primary attribute which
has garnered interest among the scientific community. Since
the discovery of micelles in solution (McBain and Salmon,
1920), many experimental and theoretical studies have been
conducted, yet full understanding of these systems of
molecules has not been achieved. True predictive capability
is not yet a reality.
This investigation takes a two-fold approach toward the
ability to make quantitative predictions of the behavior of
systems of surfactants in solution. Since the behavior of
1


2
the surfactant solution is a consequence of its thermodynam
ics, a model set in the framework of molecular
thermodynamics is developed to describe the formation of
micelles in a multicomponent surfactant solution.
A thermodynamic description of a micellar system is
limited by a precise knowledge of the structure of micelles.
To this end, computer simulations of model micelles by the
molecular dynamics method are conducted. By accurately
modeling the forces present, pertinent descriptions of
micelle structure are obtained.
In Chapter 2 of this work, further background on
micellar systems is given and the development of a model for
the free energy change upon formation of micelles in a
multicomponent surfactant solution is described. A multi-
step reversible process is employed to generate contribu
tions to the total free energy change due to hydrophobic
interaction, mixing effects, conformational change, head
group interaction, and electrostatic interaction.
Some brief results obtained with the thermodynamic model
are presented in Chapter 3. Distributions of free energy
change of micellization and aggregate size are calculated
for surfactants of different chain lengths and for solutions
of different temperature. Due to limitations in the scope


3
of this portion of the investigation, calculations were
carried out only for cases of single-component solutions of
nonionic surfactants.
The computer simulation of surfactant micelles is
described in Chapter 4. Following a concise description of
the molecular dynamics method, its application to the
simulation of micelles is discussed. A summary of the
computer simulations of the four modelsthree micelles and
one hydrocarbon dropletis given.
In Chapter 5, the results of analyses of the computer
simulations are presented. Elements of aggregate internal
structure, chain conformations, aggregate shapes, and
changes in the aggregate with time are investigated for the
four model aggregates.
Chapter 6 provides a summary of the significant
conclusions of this work. Recommendations are made toward
the future progress of both of the projects described in the
previous chapters.


CHAPTER 2
A MOLECULAR THERMODYNAMIC MODEL OF MICELLE FORMATION
2.1 Background
The forces present in liquid solutions dictate that
solution of a polar solute in a polar solvent is more
favored than is a polar solute in a nonpolar solvent.
Similarly, the nonpolar solvent is more accommodating toward
a nonpolar solute than a polar one. Therefore, molecules
which contain both polar and nonpolar groups exhibit a
unique behavior when present in a polar or nonpolar solvent.
Such molecules, known as surfactants, will tend to minimize
the unfavored contact (i.e., polar-nonpolar) while maximiz
ing the favored contact (i.e., polar-polar). At an
interface between polar and nonpolar liquids, the
surfactants will penetrate the interface to achieve "like"
interactions on both sides, reducing the "unlike"
interactions encountered in the bulk liquid. The polar
groups of a large number of surfactant molecules packed
closely at an interface will repel each other, producing a
spreading pressure which reduces the interfacial tension.
In the bulk liquid, surfactant molecules will aggregate
into structures known as micelles which can afford much the
same benefits as the interface. In the typical case, a
4
!


5
surfactant with a polar "head group" and a nonpolar "tail,"
when present in sufficient quantity in a polar solvent such
as water, will form micelles having an interior consisting
of tails and possibly some nonpolar solubilizate and a
surface comprised mostly of head groups. The head groups
remain in contact with the watera favored interac
tionwhile the tails reduce their contact with the
wateran unfavored interaction. When the solvent is
nonpolar, inverted micelles can form.
The ability of surfactants to adsorb at interfaces and
aggregate into micelles makes them a very useful class of
compounds. The reduction of interfacial tension has many
applications, ranging from oil recovery to biological
processes. In addition, micelles can solubilize other
solutes in their interior, as in drug delivery processes,
and reactions can even take place there, as in emulsion
polymerization. The earliest and best known use of
surfactants, detergency, uses both aspects of their
behavior. As useful as these phenomena are, they are not
understood to a degree that would allow their full potential
to be realized. The ability to predict behavior rather than
just explain it is the goal of this undertaking. This
requires a knowledge of the thermodynamics of surfactant
phenomena.


6
The formation of surfactant aggregates in solution
instills a certain ambiguity in the description of the
system by a traditional thermodynamic formalism. The
aggregation of surfactant monomers into micelle structures
has been treated as the formation of a "phase" (Blankschtein
et al., 1985; Kamrath and Franses, 1983; Matijevic and
Pethica, 1958) or as a stepwise association "reaction"
(Tanford, 1974; Mukerjee, 1972; Murray and Hartley, 1935).
Although the former description may aid in visualizing
certain aspects of micellar solutions, the thermodynamic
idea of a phase cannot be used in a rigorous manner. Its
requirements of continuity and homogeneity are not met by a
collection of micelles in solution and a single micelle
cannot be treated as a phase since its properties are
size-dependent.
The treatment of micelle formation as reaction
equilibrium is plagued by the lack of a single
stoichiometry. Since a distribution of products is formed
(Void, 1950), one must consider each micelle to be in
reaction equilibria with the dispersed monomers. The
determination of the many equilibrium constants by
experimental methods is impossible. Hall and Pethica (1967)
proposed using a small-systems thermodynamics approach to


7
avoid the difficulties of these two treatments. But their
approach cannot be used with ionic systems and is mainly
formal, not lending itself to practical use.
The thermodynamics of micelle formation remains an
interesting problem. The literature is abundant with
studies. In addition to the guantities of temperature,
pressure, and composition which typically define the
thermodynamic state of a typical solution, the thermodynamic
behavior of solutions containing surfactant species can
depend on the size, shape, and structure of the aggregates
which are formed. Thermodynamic properties have been
measured and correlated (Burchfield and Woolley, 1984;
Woolley and Burchfield, 1984, 1985). Aggregate formation
has been investigated from the points of view of classical
thermodynamics (Moroi et al., 1984; Muller, 1973) and
statistical thermodynamics (Hoeve and Benson, 1957; Owenson
and Pratt, 1984). Investigations have focused on size
distributions (Ruckenstein and Nagarajan, 1975; Ben-Naim and
Stillinger, 1980), the role of micelle shape (Tanford, 1974;
Israelachvili et al., 1976; Ljunggren and Eriksson, 1984,
1986; Eriksson and Ljunggren, 1985; Void, 1985), and shape
transitions (Van de Sande and Persoons, 1985; Ikeda, 1984;
Missel et al., 1983; Mukerjee, 1977).


8
While contributing to our understanding of the complex
nature of micelle formation, none of these works produced a
practical model with predictive capabilities. A semi-empir-
ical model for the thermodynamic properties of surfactant
aggregate formation based on molecular thermodynamic
processes was developed by Hourani (1984) and was successful
at predicting thermodynamic quantities and aggregate size
distributions for systems of a single surfactant species in
solution. Benedek (1985) developed a different model in the
framework of molecular thermodynamics. While it was
demonstrated successfully, the use of empirical parameters
was more extensive than in Hourani's work. The model of
Hourani showed promise of being extendable to multicomponent
systems and of being more closely related to other
observable molecular phenomena. The beginnings of such an
extension are given in this chapter.
2.2 Stoichiometry and Reaction Equilibria
for Multicomponent Micelles
Since the free energy of a process is independent of the
path chosen between the initial and final states, Hourani
proposed a process consisting of a series of steps for which
the free energy change can be modeled. In this process, the
monomers were removed from the solvent and placed in a
vacuum at their original densitya gaseous state. Cavities
of excluded volume remained in the solvent, to be coalesced


9
in a subsequent step. The monomer gas, considered ideal,
was compressed to micellar density, and placed into larger
cavities which had been formed in the solvent. Essential to
the modeling was the elimination and creation of the solvent
cavities. The counterions were handled in the same fashion,
with the addition of the necessary electrostatic calcula
tions. Extending Hourani's molecular thermodynamic model to
solutions containing two or more surfactant species required
the addition of new steps and modification of others. The
solvent cavity steps were eliminated and the monomers are
removed to a liquid state rather than the gaseous. These
changes facilitate the handling of multiple components. The
development of the multicomponent model is detailed below.
Micelle formation in solution yields a distribution of
micelle sizes. In addition, a multicomponent surfactant
solution has a distribution of compositions (Warr et al.,
1983; Scamehorn et al., 1982; Birdi, 1975; Moroi et al.,
1974, 1975a, 1975b; Rubingh, 1979; Clint, 1975). To
describe this, a set of equilibrium reactions can be written
for the formation of J micelles of distinct sizes and/or
compositions. For I different surfactant species, Zj_, and K
counterion species, B^, aggregating to form J micelles, Zj,


10
*1
NnZ
x + N2lZ7 + .,
+ N nZ ,+ M nB i
+ M 2l B 2 + ..
. + M K1B k 44 Z
2
Nl2Z
1 + N 22 Z 2 +
+ N ,2Z, + M l2B j
+ M 22B2 +..
, + M K2BK& Z2


(2
kj
NXJZ
i + N2JZ 2 + .
+ N UZ t + M Xj B
i + m2Jb2 + .
+ MkjbZJ
where N^j is the number of monomers of species i present in
the jth micelle and is the number of k counterions bound
to it. The equilibrium constant Kj for the formation of the
micelle is given by
[Z']
(2.2)
These are related to the standard state free energy of
micellization of the micelle by
AC = RT\n K j
The total number of monomers in this micelle is
(2.3)
i
so that
(2.4)
N jAGj = -RTlnK j (2.5)
where JGj is on a per monomer basis.
The mole fraction of monomers in the jth micelle is
found by combining equations (2.2) and (2.5). For the
dilute concentration range of micellar solutions, ideality


11
of the monomer solution can be assumed, and
where CD is the total solution concentration
[ ] indicates concentration of species
X is the monomer mole fraction in j^h micelle
Nj is the aggregation number of the jth micelle
Within the material balance constraint for each species,
(2.7)
equation (2.6) describes the distributions of micelle size
and composition in solution when provided with the free
energy change as a function of the size and composition of
the micelle formed.
2.3 Estimation of Free Energy Changes
The standard state free energy of micellization is
calculated via the seven-step process shown in Figure 2-1.
The standard state free energy of formation for a single
micelle in solution is the sum of the free energies of the
seven steps. Each step is modeled as closely as possible
from an observable phenomenon of a similar nature.
Certain free energy terms are dependent on the shape of
the micelle. The micelle grows as a roughly spherical
aggregate until the additional volume of one more monomer
would cause the radius of the sphere to exceed the length of
the longest all-trans chain. With the addition of the next


3
Figure 2-1. The seven-step reversible process used in the estimation of
contributions to the total Free Energy of micellization in a multicomponent
surfactant solution. From left to right: Head groups and counterions are
removed; surfactant chains (hydrocarbons) and solvent are separated into
discrete phases; the hydrocarbons are mixed into compositions of the micelles;
hydrocarbon droplets are placed in the solvent; chain ends are brought to the
surface; head groups are replaced; and counterions are replaced.


13
monomer the micelle must grow with a nonspherical geometry
to avoid the formation of a material-free core. Several
geometries have been proposed: spherical dumbells, oblate
ellipsoids, prolate ellipsoids, and spherocylindrical rods.
While Ljunggren and Eriksson (1984, 1986; Eriksson and
Ljunggren, 1985) have proposed that the shape fluctuates
between spherical, rod-shaped, and even disc-shaped, Void
(1985) has found little effect of the particular geometric
model on the thermodynamics of micelle formation. In this
work nonspherical micelles are modeled as prolate
spherocylindrical rods. The derivation of micelle dimen
sions and surface area based on this geometric model is
given in Appendix A.
In step 1, a standard state solution of surfactants is
transformed into a solution of hydrocarbon chains by
reversibly removing the head groups and counterions. Since
these are reversibly replaced in steps 6 and 7, the net free
energy change for the mere removal and replacement of the
head groups and counterions is zero. If no free energy
contributions due to the replacement of head groups and
counterions are contained in steps 6 and 7, then
AGX = 0 (2.8)
In step 2 the hydrocarbon solution is separated into I
pure hydrocarbon liquids and the pure solvent. This is the


14
reverse process of hydrocarbon solubility, so
(2.9)
In step 3 the I pure hydrocarbons are placed into J
ideal hydrocarbon mixtures of different compositions and
amounts. For this ideal mixing,
(2.10)
In step 4 the J hydrocarbon mixtures are formed into J
droplets and placed into the solvent. The free energy of
this step is the free energy of forming the hydrocarbon-sol-
vent interface. There are both surface area and curvature
contributions to this step. The expression for <34, based
on Buff (1955) and Stillinger (1973), is the surface area of
the droplet, S, times the planar interfacial tension, y, of
the hydrocarbon mixture, corrected for curvature:
(2.1 la)
The curvature effect is dependent on the parameter, g. The
surface area depends on the size and shape of the micelle.
For the spherocylindrical micelle, a second curvature
parameter, gc, is used for the cylindrical portion:
(2.1 lb)


15
This approximates the cylindrical curvature effect, whose
uncertainty has been discussed by Henderson and Rowlinson
(1984).
In step 5 conformational changes in the hydrocarbon
chains are made so that one end of each chain is at the
surface of the droplet. The contribution from this step is
entirely entropic and may only be estimated. The expression
used in this model is
(2.12)
where Sq is a parameter of the model. The squared ratio of
chain length, lc, to micelle radius, R, takes into account
the very severe conformational restrictions present when the
micelle radius is much smaller than the chain length.
As indicated in the discussion of step 1, step 6
contains no contribution due to the reattachment of head
groups. The quantity JGg is the free energy change due to
the interaction between the head groups in their positions
at the micelle surface. The head groups are modeled as
dipoles. For the ionic surfactant species, a charged head
group paired with a counter ion forms a strong dipole.
Nonionic head groups exhibit weak-to-moderate dipole
moments. The dipole-dipole interactions of adjacent pairs
are summed for the free energy contribution of this step.
The separation and orientation between two dipoles are


16
dependent on the size and shape of the micelle, with the
head groups evenly spaced over the surface of the micelle.
The derivation of the head group interactions is carried out
in Appendix B. The potential between a pair of adjacent
head groups is
(2.13)
where n is the dipole moment
r is the pair separation
R is the micelle radius
D is the dielectric constant of the solvent
As indicated in the derivation, this form of the potential
takes into account the angle between adjacent dipoles as a
function of micelle radius. The total energy contribution
from this step is found by summing the contributions from
the different pairs in the manner described by equations
(B.25) through (B.28) in Appendix B.
Step 7 contains no contribution for the replacement of
the counterions back into solution. The free energy of this
step is due to the difference between the original random
distribution of counterions in the micelle-free solution and
the final Poisson-Boltzmann distribution of counterions
around the surface of the micelle with a fraction bound in a
Guoy-Chapman electrical double layer. The derivation of
Hourani (1984) for the numerical solution of this charge


17
distribution model is applicable here. The entire step is
actually the process of discharging the counterions in their
original distribution, compressing them into the bound layer
and final distribution, and then recharging the counterions.
Therefore G70 contains an entropy contribution from the
compression and an enthalpy contribution from the
distribution. The bound layer will be populated with
dipoles formed by head group/counterion bound pairs and
unbound head groups. The charge interactions in the bound
layer are included in the dipole and point charge pairings
of the head group term, iGg0.
2.4 The Calculational Technique
The free energy of formation for a single mixed micelle
with nonionic head groups is obtained by summing the
contributions from steps 1 through 6:
AG
RT
= l^[xij\nCiq + xu\nx
2 n R ¡ y ¡
NkT
L\ i-2£
R,
2 R
g_
R
J / J
-5,
R
J
RT
(2.14)
To include the presence of ionic head groups and
counterions, the contributions of step 7 must be added to
equation (2.14). Such calculations were not carried out
here, so this section will pertain only to mixtures of
nonionic surfactants. Table 1 summarizes the model's
variables, parameters, and required data.


18
Table 1
Arguments of the Thermodynamic Model
Variables v
1/
Compositions of micelles
"i
Aggregate sizes of micelles
T
Temperature of system, Kelvin
Composition of system
nc
Number of carbons in surfactant chain
r
^ tot
Overall system concentration, moles/liter
Data
Solubility concentrations of
hydrocarbons, moles/liter
y i
Interfacial tensions of hydrocarbon
mixtures, dynes/cm
Vi
Dipole moments of head groups, Debyes
D
Dielectric constant of water
lc
Surfactant chain length, Angstroms
Surfactant chain volume, Angstroms3
Parameters g
Spherical curvature parameter, Angstroms
gc
Cylindrical curvature parameter, Angstroms
Sc
Entropy of conformation parameter
Parameters


19
The values of chain length (Ang.) and chain volume
(Ang.3) used in the determination of micelle size and shape
are calculated from Tanford's correlations (Tanford, 1972):
Zc = 1 .265nc+1 .5 (2.15)
vc = 26.9nc + 27.4 (2.16)
To facilitate the use of computer programs in carrying
out the calculations, correlations are used for the reguired
physical data. The aqueous solubility of hydrocarbons used
in the calculation of is obtained from a correlation
due to Leinonen et al. (1971):
The parameter K is fit to the solubility data of McAulliffe
(1966), Polak and Lu (1973), and Sutton and Calder (1974).
It is found to be a linear function of the hydrocarbon chain
length for the n-alkanes, but since hydrocarbon solubility
in water exhibits a break at decane, two linear
relationships for K are used, one for the longer chains and
one for the shorter chains. Equation (2.17) is solved
iteratively for the hydrocarbon solubility, xe<3.
For the hydrocarbon-water interfacial tension required
in the calculation of JG4", a correlation based on the works
of Aveyard et al. (1972), on the surface tension of
hydrocarbons, and Jasper (1972), on the surface tension of


20
water, is used:
57.868nc+ 1 17 .99 ( .059nc + 1768 T
y = 1.381 (2.18)
nc + 2.4
where the temperature is in Kelvin and the surface tension
is in dynes/cm.
The value of the dielectric constant of water used in
the evaluation of JGga is given by
D = 252.422- .806329 7+ .000746972 (2.19)
which is a polynomial fit of the data of Owen et al. (1961)
at atmospheric pressure with temperature in Kelvin. The
values of dipole moments used in this calculation are
estimated as the dipole moments of molecules of similar
structure to the head groups.
The three parameters, g, gc, and Sq, are fit to the
measurable data on the micellar system. These are the mean
aggregate size of the micelles and the set of mixture
critical micelle concentrations (CMCs) at the system
temperature. Equation (2.6) generates an I-dimensional
surface of the aggregate size distribution of the
multicomponent micelles. This is accomplished by first
choosing values of the temperature, T, the total
concentration, C^ot' and the system composition, x. Then
for each possible micelle composition, x^j, equation (2.6)
is evaluated at values of Nj from two to infinity (in
practice, until Xj/Nj becomes negligible). This generates


21
the distribution of aggregate sizes with micelle composi
tion. The parameters must be chosen such that the
distribution meets the material balance constraints of
equation (2.7).
To find the proper values of the parameters, the mean
size is calculated as
Zz
N = ~ (2.20)
I Z'/N,
and the CMC is taken to be the value for which
d{Ci+C,
lim
w,
dC,
= 0.5
(2.21)
as put forth by Hall and Pethica (1967). Here, C^ is the
free monomer concentration and C^ is the concentration of
micelles, calculated as
C,
C iof c
N
(2.22)
Once a set of parameters is determined for a system, the
response of the size distribution to changes in any of the
input variables can be investigated.
The program listing in Appendix C gives the FORTRAN
source for the interactive fitting of parameters and
calculation of aggregate size distribution on systems of a


22
single nonionic surfactant species in water. This is the
simplest application of the model and was used to generate
the results presented in the next chapter.


CHAPTER 3
RESULTS OF THERMODYNAMIC MODELING
3.1 Parameter Estimation
Three parameters of the model, g, gc, and Sq, must be
found by fitting the mean aggregate size generated by the
model to the experimental value at the critical micelle
concentration (CMC). The model output is considered to be
at the CMC when the total surfactant concentration is equal
to the experimental value of the CMC and the derivative of
the monomer concentration with respect to the total
concentration satisfies equation (2.21), indicating that one
out of two surfactant monomers added to the solution at this
concentration would join a micelle. The model convergence
is quite sensitive to the values of the parameters, so in
fitting them to the data, one must either use good initial
guesses or approach the values conservatively. Failure to
do either of these can result in an aggregate size
distribution which has infinite values of C^ot and ,
providing no useful information.
Each of the parameters has a physical significance which
can aid in the choice of the initial guesses. The
parameters g and gc are used in the curvature corrections to
the planar interfacial tension for the spherical and
23


24
cylindrical geometries, respectively. Physically, g is the
thickness of the spherical interface in Angstroms (Buff,
1955). Although experimental values are not available, it
is expected to be positive and small relative to the micelle
radius. The parameter gc does not have the same physical
connection to the interface (Henderson and Rowlinson, 1984),
but by comparison it is expected to be positive and less
than g. The parameter Sq is the (dimensionless) conforma
tional entropy change for a monomer joining a large micelle.
Conformational contributions of the aggregated monomers were
studied via a statistical thermodynamic theory by Ben-Shaul
and coworkers (Ben-Shaul, Szleifer and Gelbart, 1985;
Szleifer, Ben-Shaul and Gelbart, 1985) and values of the
conformational entropy were found to be in the range of -8
to -7 for a chain with seven bonds. While these values are
based strictly on theoretical considerations with no
experimental corroboration, the parameter Sc is expected to
be of the same sign and magnitude.
In approaching the parameter fitting conservatively, the
initial values of the parameters are chosen such that the
aggregate size distribution will definitely converge. That
is, the condition
d N
(3.1)


25
must be satisfied. Equation (2.6) defines the aggregate
size distribution. In this equation, ziGj is the only term
influenced by the parameters. In order for the distribution
to converge as N becomes large, this free energy change must
be greater than the logarithm of the monomer concentration.
Overestimating the parameters toward a less negative free
energy change will assure that the size distribution
converges. The parameters can then be adjusted toward a
more negative free energy change as they are fit to the
data.
The effect of adjusting the parameters can be foreseen
by analyzing the corresponding terms. The parameters g and
gc affect the behavior of the surface free energy, G40.
For g < R and gc < lc, the usual cases, increasing the
parameter decreases the free energy, consistent with
equation (2.11). In equation (2.12) it can be seen that
increasing the parameter Sq decreases the conformational
free energy, (25.
The aggregate size distribution can be divided into two
regions. By designating the value of N for which R = lc as
Nfrans' the transition of the micelle geometry from
spherical to spherocylindrical defines the two regions of
the size distribution. For N ^ N-^rans' the micelle is
spherical with radius R. For N > Ntrans, the micelle is
spherocylindrical with radius lc and length L. For sizes


26
above N-trans, the conformational free energy is constant
with a value of -Sq. At N-j-rans the surface free energy
makes the transition from being equal to the spherical
contribution to becoming dominated by the cylindrical
contribution for large N. Figures 3-1 and 3-2 show the
effect of aggregate size and parameter values on the surface
and conformational free energies.
A typical aggregate size distribution generated for a
nonionic surfactant is shown in Figure 3-3. A peak occurs
at Ntrans' beyond which the fraction of aggregate decreases
with increasing N. The total surfactant concentration is
proportional to the area under the distribution and is thus
influenced by both the height of the peak and the slope of
the distribution above N-f-rans. Because the distribution
below N-trans rises so rapidly, the mean aggregate size, ,
is dependent only on the slope of the distribution above
Ntrans*
The parameter Sq, since it contributes over the entire
range of N, affects both C^ot and . The parameter gc,
contributing only above N-j-rans, has a significant effect on
but only a slight effect on C-f-ot' while g influences only
ctot an<^ not . This causes such interplay between the
parameters that a range of parameter sets can produce
identical values of C-t-0t and . The derivative constraint,


Figure 3-1. The surface contribution to the free energy of micelle
formation for the C12 nonionic micelle at the CMC and a temperature
of 298.15 Kelvin. The lower curve was generated with Sc=-4.0,
g=6.9586, gc=3.6272. The upper curve was generated with Sc=-3.0,
g=4.8879, gc=0.51846. The lower curve demonstrates the effect of g
exceeding the micelle radius at low values of N.


14
13 -
12 -
11 -
10 -
9 -
8 -
AG4 7 -
RT
6 -
3 -
2 -
1 -
0 -
0 20 40 60 80 100 120 140 160 180 200
Aggregate Size
Figure 3-2. The conformational contribution to the free energy of
micelle formation for the C12 nonionic micelle at the CMC and a
temperature of 298.15 Kelvin. The parameters are as in Figure 3-1,
showing the effect of Sc.
SC = -4.0
SC= -3.0
n 1 i r r r i i i i i i i i i i


o
-10 -
o 200 400 600 800 1000
Aggregate Size
to
U3
Figure 3-3. Aggregate size distribution for the C12 nonionic
micelle at the CMC and a temperature of 298.15 Kelvin. Typical
behavior of the size distributions of nonionic surfactants is
exhibited.


30
(2.21), can also be satisfied by these parameter sets if
they are fit at a value of the free monomer concentration,
, just below the critical micelle concentration.
The method used to fit parameters consisted of first
choosing low values of the parameters, resulting in a C-j-0t
essentially egual to C;l and an insignificantly larger than
Nfrans* The parameter gc was increased until reached the
desired value, and then g was adjusted to result in the
proper C^ot-. This is repeated for different values of Sq to
generate the range of parameter sets fitting the data. The
experimental data used to generate the parameters is given
in Table 2.
3.2 Behavior of the Model for Single-Component
Nonionic Systems
Parameter fitting was carried out on systems of
different surfactant chain lengths and at different
temperatures. For each system, linear relationships were
found to exist between each pairing of the three parameters.
Corresponding to the manner in which the fitting was
accomplished, the curvature parameters g and gc can each be
expressed as a straight line plotted against the Sq
abscissa. Such a plot is given in Figure 3-4. These
expressions are of the form
g = mS c + b
gc=nSc+d
(3.2)
(3.3)


31
Table 2
Data Used in Fitting Model Parameters for Aqueous Solutions
of Nonionic
Surfactants
Surfactant
T (Kelvin)
CMC (M)
Mean Size
CioH2i(OC2H4)6OH
298.15
9.0E-04 a
73
a
308.15
6.6E-04
260
318.15
6.4E-04
640
c12h25(OC2H4)6OH
298.15
8.7E-05 b
400
c
C14H29(OC2H4)6OH
298.15
1.1E-05 a
3100
a
a:
Balmbra
et
al.
(1964)
b:
Corkill
et
al.
(1961)
c:
Balmbra
et
al.
(1962)


10
-1 H 1 1 t 1 1 1 1 1 1 1 i r
-5.2 -4.8 -4.4 -4 -3.6 -3.2 -2.8
Conformational Parameter
Figure 3-4. Interdependence of the model parameters for the C12
nonionic micelle at 298.15 Kelvin. The symbols represent the
values of the spherical (g) and cylindrical (gc) curvature
parameters. The lines give the regression results
g=-l.3235-2.0705SC and gc=-8.808-3.1087SC.


33
The values of the slopes and intercepts found for the
systems modeled are given in Table 3.
Many aspects of surfactant behavior have been found to
be linearly dependent on chain length and/or temperature
(Rosen, 1978). The information in Table 3 indicates that
such relationships are also possible for the interaction of
the parameters of this model. Though the three chain
lengths and three temperatures investigated cannot conclu
sively indicate linear behavior, they can indicate whether
this behavior is likely. Linear regression of the slopes and
intercepts of Table 3 versus chain length and temperature
resulted in the following equations and
correlation
coefficients:
m = -.00897+ .8817
R= .9998
(3.4)
6= .05497-20.20
R= .9778
(3.5)
n = -.01377+ 1 .421
R = .9999
(3.6)
d = 1 1257-45.09
R= .9810
(3.7)
for the 10 carbon nonionic, and
m = -.1464nc- .3170
R= .9999
(3.8)
6= 1 .418nc 18.10
R= .9988
(3.9)
/7 = -.2201nc .4676
R= 1 .000
(3.10)
d = 1 .553nc-27.29
R= .9991
(3.11)
at a temperature of 298.15 Kelvin.


34
Table 3
Slopes and Intercepts of Curvature Parameter Dependence
on Conformational Parameter
n£
T (Kelvin)
g slope
g int.
gc slope
gc int.
10
318.15
-1.96163
-2.80775
-2.94330
-9.43866
10
308.15
-1.86885
-3.15286
-2.80330
-10.1776
10
298.15
-1.78296
-3.90518
-2.66898
-11.6881
12
298.15
-2.07048
-1.32359
-3.10874
-8.80775
14
298.15
-2.36862
1.74162
-3.54946
-5.47762


35
Substituting (3.4) through (3.11) into (3.2) and (3.3),
the temperature relations for CIO nonionic are
g = (.0549- .0089SC)T- 20.20 + .881 7SC (3.12)
gc = (.1 125-.01 372SC)T-45.09 + 1 .421 45 c (3.13)
and the chain length relations at 298.15 Kelvin are
g = ( 1 .418- 14645C)nc- 1 8.1 0 .3 1705c (3.14)
gc = ( 1 .553- .2201 Sc)nc- 27.29 46765 c (3.15)
These relations are only valid for values of Sc which, for
each chain length, produce values of the curvature
parameters which are neither negative nor larger than the
micelle radius. In this sense, Sc is dependent on chain
length, but there remains a lack of uniqueness in the
parameter fitting for the single component case. This
results from the finding that the CMC and the concentration
derivative (2.21) are not independent of each other.
The free energy of micellization and its various
contributions, as given by the model, are shown in Figures
3-5 through 3-8. These plots were made for two chain
lengths at the same temperature, two temperatures with the
same chain length, and two parameter sets at the same chain
length and temperature. Figures 3-5 and 3-8 show the effect
of two different parameter sets for the same system. While
the surface and conformational contributions are different
due to the different parameters, the total free energy of
micellization is the same in the two figures. This follows


20
15 -
10 -
5 -
AT OH
RT
-5 -
-10
-15 -
-20
Hydrophobic
20
l 1 1 i r r
40 60 00 100
Aggregate Size
U>
G\
Figure 3-5. Free energy contributions for the CIO nonionic micelle
at the CMC and a temperature of 290.15 Kelvin. A value of -4.0 is
used for the conformational parameter, Sc, with the curvature
parameters fit to the data.


20
15
10 -
5 -
M o
RT
-5 -
-10 -
-15 -
-20
Surface
Conformational
Hydrophobic
20
40
\
60
1
BO
100
Aggregate Size
U)
Figure 3-6. Free energy contributions for the C14 non tonic micelle
at the CMC and a temperature of 29B.15 Kelvin. A value of -2.0 is
used for the conformational parameter, Sc.


20
15 -
-20
lydrophobic
20
40 60
Aggregate Size
BO
100
Figure 3-7. Free energy contributions for the CIO nonionic micelle
at the CMC and a temperature of 318.15 Kelvin. A value of -4.8 is
used for the conformational parameter, Sq.


Aggregate Size
Figure 3-8. Free energy contributions for the CIO nonionic micelle
at the CMC and a temperature of 298.15 Kelvin. A value of -4.0 is
used for the conformational parameter, Sc.


40
from the fact that the parameter sets were fit to the same
concentration and mean size data and generate identical size
distributions.
3.3 Aggregate Size Distribution and Concentration Behavior
With the parameter sets determined by the methods
described above, the aggregate size distributions and the
effects of surfactant concentration were investigated for
the nonionic alkyl surfactants. The size distributions for
this class of surfactants exhibit a rapid increase in the
spherical region of N, peaking at N-trans. In the
nonspherical region, the distribution decreases very slowly
through large values of N, producing the large mean
aggregate sizes which have been measured for the nonionic
surfactants.
The size derivative of the distribution in the
nonspherical region is negative. With increasing free
surfactant monomer concentration, it becomes less negative,
resulting in a higher mean aggregate size and total
surfactant concentration. The effect of surfactant concen
tration on the modeled size distribution and mean aggregate
size of the C12 nonionic surfactant is shown in Figures 3-9
and 3-10. Below the CMC, the model calculates a mean
aggregate size approximately equal to N-trans present as the
initial plateau in figure 3-10. This value is meaningless,
however, since the concentration of micelles in this range


-12
X
en
O
13 -
-14 -
15 -
-16 -
-17 -
18
-19
-20
5, N=694
1 ono
Aggregate Size
ii^
H
Figure 3-9. Effect of total surfactant concentration on the
aggregate size distribution of the C12 nonionic micelle at 298.15
Kelvin.


3200
3000
2800
2600
2400
2200
2000
1800
1600
1400
1200
1000
800
600
400
200
0
8.
F i a
E-5 8.80E-5 9.20E-5 9.60E-5 1.00E-4 1.04E-4
Total Surfactant Concentration, M.
1.0HE-4
3-10.
size
Effect
of the
of total surfactant concentration on the
Cl2 non ionic micelle at 298.15 Kelvin.
mean
N)


43
of surfactant concentration is insignificant. The result is
merely the ratio of two extremely small numbers, since the
model always produces a finite value for the concentration
of micelles (equation (2.6)).
Measurement of properties such as surface tension, which
depend on the free monomer concentration, show that beyond
the CMC the free monomer concentration is virtually
unaffected by an increase in the total surfactant
concentration. The relationship between the free and total
surfactant concentrations, as predicted by the model, is
exhibited in Figure 3-11. At total concentrations lower
than the CMC, the majority of any surfactant added to the
solution exists as free monomers, while at concentrations
higher than the CMC the bulk of any additional surfactant
micellizes. The definition of the CMC used in this work, as
expressed in equation (2.21), quantifies the break between
these two behaviors of the solution. Figure 3-12 shows the
concentration derivative of equation (2.21) for the data of
Figure 3-11 with the CMC identified.
3.4 Chain Length and Temperature Effects
It is known that the hydrophobic nature of the alkyl
chain increases with its chain length, as evidenced by the
decreasing solubility in water of the increasingly long


Free Monomer Concentration,
Total Surfactant Concentration, M.
Figure 3-11. Effect of total surfactant concentration on the free
monomer concentration of the C12 nonionic at 298.15 Kelvin. The
CMC of this system is 8.70E-5 M.


(wo + lo) e
Total Surfactant Concentration, M.
Figure 3-12. Effect of total surfactant concentration on the
concentration derivative of equation (21) for the C12 nonionic at
298.15 Kelvin. The dashed line shows that the value of this
derivative is 0.5 at the CMC of 0.7E-5 M.


46
chains (McAulliffe, 1966). With alkyl surfactants, this
increasing hydrophobicity promotes micellization. While no
significant aggregate formation is observed for chains of
six carbons or less, increasing chain length brings about
the formation of larger micelles at lower critical micelle
concentrations (Table 2).
The elements of the model which depend on chain length
are the hydrophobic free energy, taken directly from the
aqueous solubility, the curvature parameters, accounting for
the difference in micelle radius, and the conformational
free energy. Equations (3.14) and (3.15) show the
relationship of the parameters to the chain length. The
aqueous solubility of the alkane, which is chain-length
dependent, is used by the model in the fitting of the
parameters. The dependence of the aggregate size distribu
tion and free energy change on the surfactant chain length
is shown in Figures 3-13 and 3-14. The size distribution
broadens with increasing chain length, yielding a much
larger mean aggregate size with a less significant increase
in the CMC. The free energy of micellization becomes more
negative with increasing chain length, showing that micelle
formation by surfactants with more hydrophobic area is more
thermodynamically favored.


Aggregate Size
Figure 3-13. The effect of chain length on the aggregate size
distribution. Results are shown for the nonionic surfactants at
298.15 Kelvin and their respective CMCs (Table 2).
2000


I
O 200 A 00 600 800 1000 1200 1400 1600 1800 2000
Aggregate Size
Figure 3-14. The effect of chain length on the free energy of
micellization. Results are shown for the nonionic surfactants at
298.15 Kelvin and their respective CMCs.


49
Just as the aqueous solubility of an alkane decreases
with increasing temperature, the CMC of the corresponding
surfactant decreases and the mean aggregate size increases.
Again, by incorporating the solubility data into the model,
the proper temperature dependence is given to the
parameters. The temperature dependence of the parameters is
in the form of equations (3.12) and (3.13). The dependence
of the aggregate size distribution and free energy change on
temperature is shown in Figures 3-15 and 3-16. The effect
of increased temperature is similar to that of increasing
the surfactant chain length: a broadened size distribution
and a more negative free energy of micellization. But while
the first increase of ten degrees Kelvin has a much more
pronounced effect on the free energy change (and hence the
size distribution) than the next increase of ten degrees,
the mean aggregate size is more affected by the latter.


-8
-9 -
-10 -
-11 -
-12 -
CT>
O
_J -15 -
-16 -
-17 -
-18 -
-19 -
-20 -
0
Aggregate Size
298.15 K.
508.15 K.
518.15 K.
U1
o
Figure 3-15. The effect of temperature on the aggregate size
distribution. Results are shown for the CIO nonionic surfactant
at its CMC.


-6.4
-7.6 H 1 1 1 1 1i 1 1 1 1 1 1 1 1~i 1 1 i i I
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Aggregate Size
U1
H
Figure 3-16. The effect of temperature on the free energy of
micellization. Results are shown for the C]0 nonionic surfactant
at its CMC.


CHAPTER 4
MOLECULAR DYNAMICS SIMULATION OF SURFACTANT MICELLES
4.1 Background
The development of a predictive model of micelle
behavior, such as that of Chapter 2, requires a knowledge of
the structure of the aggregate in order to describe its
thermodynamic properties. The calculation of a surface free
energy contribution necessitates certain assumptions about
micelle size, shape, and surface roughness. The solvent
interface must be modeled and solvent penetration must be
estimated. Calculation of the conformational contribution
to the free energy calls for details of the molecular
conformations in the free and aggregated states. The head
group contribution depends on the average positions of the
head groups.
While experimental results are the most desirable basis
for thermodynamic models, measurements of the structure of
micelles are limited in the information they can provide.
Therefore, current models of micelle behavior, including
that derived in Chapter 2, are based primarily on
theoretical considerations.
The most informative techniques for studying micelles in
solution are spectroscopic. Light scattering, NMR, and
52


53
small-angle neutron scattering (SANS) have been used
extensively to study micelles. However, results of these
experiments often conflict, as in the question of whether
there is solvent penetration into the micelle core, and
their description of the micelle surface and internal
structure is incomplete (Tabony, 1984).
Quasi-elastic light scattering has been used effectively
to determine micelle size and shape (Corti and Degiorgio,
1981a, 1981b) and Raman light scattering has been used to
investigate chain conformations (Kalyanasundaram and Thomas,
1976). The absence of solvent from the micelle core was
determined with NMR techniques (Mitra et al., 1984), though
the opposite had been reported earlier (Menger, 1979). The
most promising experimental means available of studying
micelle structure is SANS. It has been used to study the
micelle surface (Hayter and Zemb, 1982), solvent penetration
(Tabony, 1984; Cabane et al., 1985), and chain conformations
in the micelle interior (Bendedouch et al., 1983a, 1983b).
The spectra generated by the above methods must be
carefully interpreted to yield correct information about the
micelle surface and interior. It is in this interpretation
of the results, particularly in accounting for intermicellar
effects, that inconsistencies arise (Cabane et al., 1985).
Computer simulation is an alternative means to answer
questions raised by the experimental measurements. For a


54
given model of surfactant and micelle behavior, simulation
can provide exact quantitative descriptions of micelle
structure.
Simulations of micelles have been conducted by both
Monte Carlo (Haan and Pratt, 1981a, 1981b; Owenson and
Pratt, 1984) and molecular dynamics (Haile and O'Connell,
1984; Woods et al., 1986; Jonsson et al., 1986) methods.
Thus far, results of the simulation studies have been
promising. Their contribution to a better understanding of
micelle structure has been limited only by the uncertainties
about the model micelle.
Monte Carlo calculations are carried out on a lattice of
sites. Movement of a molecular segment from one site to
another is allowed or disallowed based on the energy change
associated with the move. The restriction of segment
positions to the lattice sites limits the possible
conformations available to the molecules. To allow more
possible conformations, the number of lattice points, and
hence the computing effort, must be increased. This
restricts how far such a simulation can go in accurately
representing reality; in addition, the Monte Carlo method
precludes the opportunity to study the dynamics of a system
by simulation.


55
The molecular dynamics method provides a continuum of
sites for molecular positions and is limited only by the
accuracy of the forces built into the model. In the present
work, molecular dynamics models were developed based on the
work of Haile and coworkers (Haile and O'Connell, 1984;
Woods et al., 1986) to investigate the effects of surfactant
chain length, head group mass, and head group size.
4.2 The Molecular Dynamics Method
The molecular dynamics approach to computer simulation
consists of summing the forces on each particle in the
system and solving Newton's equations of motion for the
resultant positions and momenta (Haile, 1980). The problem
of constructing a realistic model system then becomes one of
supplying appropriate models for the intermolecular and
intramolecular potentials. It is assumed that a given
particle's interactions with its neighbors are pairwise
additive. Thus, the potential between two particles at time
t is of the form
£/(r£,(i)) = £/(r,.(0.r,(0) (4.1)
The force between the two particles is given by
?(r(0) = -V£/(r0.(0) (4.2)
and the total force on the particle of interest is the sum
of the forces resulting from interactions with all other
particles:
F.co-lFMo)
(4.3)


56
The resulting acceleration and velocity of the particle are
obtained from Newton's Second Law of Motion, relating force,
mass, and acceleration:
(4.4)
The set of second-order differential equations for all
of the particles in the system is solved at each time step
of the simulation. The means employed in this work for
determining the new positions of all of the particles from
the forces at the previous time step is a fifth-order
predictor-corrector algorithm (Gear, 1971). This algorithm
consists of three procedures which are repeated at each time
step. The position and its first five time derivatives at
time (t+zit) are predicted for all of the particles by
Taylor's series expansion of their values at time t:
d2 r ,()()2 d3 r ,{t){At)3
dt2 2~ + ~dt~3 3!~
d4 r ,(t)(At)4 | d5^,(0(df)5
dt4 44 + dt5 5*!
(4-5)
dr ,(f) d2 r ,(£) d3 r ,{t){At)2
dt dt2 ^ j dt3 2!
dV,(Q(zi03 | d5r,(0(zi04
dt4 3! + dt5 4!
(4.6)
d
r,(i + ZlO =
dt
2!
3!
(4.7)


57
(4.8)
(4.9)
d5r,(0
dt5
(4.10)
The forces Fj^(t+Jt) are then evaluated at the predicted
positions. Finally, the predicted values of the position
and its derivatives are corrected according to the error
between the acceleration predicted by the series expansion
and the acceleration calculated from F^(t+Jt). These
calculations are carried out by the subroutines PREDCT,
EVAL, and CORR in Appendix D.
The simulations described in this work were carried out
at constant temperature. The temperature of the system is
related to the velocities of the particles by
(4.11)
The velocities of all particles are scaled at each time step
so that the temperature is maintained at a constant value.
Subroutine EQBRAT in Appendix D accomplishes this task.
To start a simulation, values of the positions are
required to evaluate the initial forces. Initial values of
the five derivatives of position are required by the fifth
order predictor-corrector algorithm. In these simulations,
initial positions were assigned according to a lattice of


58
sites and random velocities were assigned the particles
(subroutines INTPOS and INTVEL). The momenta were scaled so
that there was no net linear momentum of the system. The
initial accelerations are calculated by equation (4.4) and
the third and higher derivatives of position are assigned
initial values of zero, since there is no way of evaluating
them. The algorithm recovers rapidly from this initial
condition, arriving at proper values of all derivatives
within 10-20 time steps.
Since intermolecular potentials are significant over a
pair separation which is usually small relative to the
dimensions of the molecular system, computing effort can be
reduced by employing truncated potentials (Haile, 1980) and
maintaining a neighbor list (Verlet, 1967). In a simulation
where the majority of possible pair separations are so large
that the corresponding pair potentials will be very small,
it is more efficient not to calculate the potential for pair
separations which are beyond a certain value and do not
contribute significantly to the system properties. Instead,
these contributions are neglected during the simulation and
a correction is made to the system properties. A cutoff
distance, rc, is set such that the potential can be
neglected for r^j > rc. The intermolecular forces are
truncated at rc and shifted vertically, going to zero
smoothly at rc (Nicolas et al., 1979). The truncated


59
potential is obtained from this shifted force by equation
(4.2). This eliminates a step change in the potential and
force at r-^j = rc, which could have a disruptive effect on
the energy conservation of the simulated system. The
general equations of a shifted force, Fs(r), and its
potential, Us(r), are given below:
ds£/(r)
(4.12)
r < r
d r
(4.13)
r < r
C
By checking the value of r-^j and bypassing the calculation
of F(rjLj) for r-[j > rc, a significant savings in computing
effort can be realized.
Further savings can be made by not checking pairs which
have a low probability of r^j < rc. This is accomplished
through the use of a neighbor list. A particle (i) in the
system is surrounded by a sphere of radius rc. Other
particles (j) inside this sphere will interact with it,
since r-¡i < rc. From one time step to the next, particles
will enter and exit this sphere. Over some short period of
time, all of the particles which have interacted with the
central particle will lie inside a larger sphere whose
radius is If, over this brief period of time, only
those particles inside the larger sphere were checked, all
of the particles interacting with the central particle would


60
be found without checking all of the other particles in the
system. Through a neighbor list a record is maintained of
all the "neighbors" within a distance r^ist of each particle
in the system. Only these neighbors are checked for r^j <
rc. The neighbor list is updated periodically based on the
value of rust and the dynamics of the system.
The molecular dynamics method represents a massive
computing effort for any system large enough to be of
interest. However, using the techniques described above to
improve efficiency, and with the development of supercomput
ers and parallel processors, it has become a practical tool
for quantitatively studying molecular behavior.
4.3 The Model Surfactant Molecule
The surfactant molecule studied in this work has a
linear alkyl chain of eight carbon groups. All of the
groups are considered to be methylenes. A polar head group
is attached to one chain end; head groups of different sizes
and masses were used in the simulations. The bond lengths
and angles along the chain are those of a normal alkane.
These are given in Figure 4-1.
Three types of intramolecular interaction take place in
the model surfactant molecule:
1. Bond vibration
2. Bond bending
3.
Bond rotation


61
Figure 4-1. The model surfactant molecule, consisting of an
eight-member alkane chain attached to a polar head group,
shown in the all-trans conformation. Bond angles and
lengths of the alkane chain are indicated. The bond to the
head group is head group dependent in the different
simulations conducted.


62
The bond vibration potential used is that of Weber (1978),
taken fro- a simulation of n-butane. It is a harmonic
potential about the equilibrium bond length. The potential
and the forces on the two groups are given by
(4.14)
(4.15)
(4.16)
where bj_ is the bond length between groups i and i+1, bQ is
the equilibrium bond length, and is the force constant.
The bond bending potential is also taken from the
simulation of Weber (1978). It is a harmonic potential
about the cosine of the equilibrium bond angle with forces
on the three groups forming the bond angle. For the angle e
formed by groups i, i+1, and i+2,
(4.17)
/ 9t. = y6 cos0o-cos0,
F'_ 6; =-y6 cos0O cos0,
(4.18)
(4.19)


63
^2(0J = yb(cos0o-cos0,.)
(4.20)
The model surfactant molecule is comprised of eight
bonds connecting its nine groups. Rotation about any of the
inner six bonds results in a change in the conformation of
the molecule and a change in its potential energy. A set of
four groups in sequence along the chain molecule contains a
dihedral angle formed by the three bonds connecting the
groups. This angle is a measure of rotation about the
middle bond, and the potential energy is a function of the
dihedral angle. The bond rotation potential chosen for
these simulations is that of Ryckaert and Bellemans (1975):
£/(0) = yr( 1 .1 16 1 .462 cos 0 1 .578 cos2 0 + 0.368 cos3 0
+ 3.156cos40 + 3.788cosj0)
(4.21)
where

potential is illustrated in Figure 4-2. The force on each
of the four groups resulting from this potential and
equation (4.2) has been derived by Woods (1985) and the
resulting calculations are carried out by subroutine RYFOR
in Appendix D.
The intramolecular potentials account for all of the
interactions among adjacent groups on a molecule as they are
taken two (vibration), three (bending), or four (rotation)
at a time. Two groups on the same molecule which are
separated by more than three intervening groups do not


Dihedral Angle//?,
degrees
X
o\
Figure 4-2. The bond rotation potential as a function of the
dihedral angle formed by four consecutive groups along the
surfactant chain. The diagrams below the plot depict the relative
orientations of the second and third groups and their attached
hydrogens (thin lines) and methylenes (heavy lines).


65
interact through any of these three potentials, but may
experience the intermolecular potential if they become
sufficiently close to one another.
4.4 The Model Micelle
The micelle used in the simulations contains 24
surfactant molecules. The individual groups of the
molecules interact through intermolecular potentials, and
the micelle is surrounded by a spherical shell which models
the solvophilic effect on the head groups and the
solvophobic effect on the chains. The intermolecular
interactions which take place in the micelle are
1. Head group-shell
2. Alkane group-shell
3. Head group-head group
4. Alkane group-alkane group
5. Head group-alkane group
The shell does not attempt to explicitly model the
solvent molecules. To do so greatly increases the computing
requirements of the simulation and a previous attempt at
such explicit modeling (Jonsson et al., 1986) did not show
it to be a clear improvement over the present poten
tial-field model. Rather, the solvophilic and solvophobic
natures of the head group and alkane group are modeled in a
simple manner intended to keep the head groups close to the
surface of the micelle and prevent the chains from extending


66
out from the micelle. This simulates the minimum
chain-solvent contact of accepted experimental results
(Tabony, 1984).
The head group interaction with the shell is through a
harmonic potential about an eguilibrium radial position:
^(r,)=yh(r,-r0)2 (4.22)
?(r,)-2yfc(r<-r0)^ (4.23)
' i
where r is the position of the head group and rQ is the
equilibrium position. The force constant is yft.
The alkane group interaction with the shell is through a
repulsive potential:
= (4-24)
V iw J
F{r iw)= 12e
(rm)12F,
(4.25)
(r lU/)13 r
where r^w is the quantity rw-r, rw being the radial
position of the spherical repulsive shell, and rm and e are
the radius and energy of minimum potential for the
alkane-alkane intermolecular potential.
The interaction between two head groups is described by
a potential comprised of both dipole (r-3) and soft-sphere
(r-12) repulsions:
(4.26)


67
This potential is designed to spread the head groups apart
on the exterior of the micelle. The force between a pair of
head groups is obtained by applying equation (4.2) to
equation (4.26) .
3 r3hh
12 r
12
hh
rjj
13 r
ij J U
(4.27)
Applying equations (4.12) and (4.13) to use the shift-
ed-force potential for r^j (4.28)
F,{r)~
(4.29)
The intermolecular potential between two alkane groups
is a (6,9) form of the Mie (m,n) potential (Reed and
Gubbins, 1973):
(4.30)
(4.31)
(4.32)


68
f,(r)-18e
(4.33)
This potential is also used for the interaction between a
head group and an alkane group, except that the radius of
minimum potential, rm, is adjusted to account for the
difference between the diameter of the head group and that
of the alkane group:
head-alkane
m
(4.34)
r
Figure 4-3 illustrates the intermolecular potentials used in
the model micelle, and Table 4 gives the values of the
parameters used in the intramolecular and intermolecular
potentials.
The model micelle is assembled by initially placing the
24 molecules with their head groups evenly spaced over the
surface of a sphere of twice the expected micelle radius.
Each chain is directed radially inward, in the all-trans
conformation. With the bond rotation force constant reduced
to one-tenth of its normal value to facilitate chain packing
during startup, the simulation is begun. The radius of the
confining sphere is reduced every ten time steps until the
appropriate radius is achieved. This is the value that
would give the density of the analogous liquid alkane,
adjusted to a pressure which fluctuates about zero with an
amplitude of less than ten atmospheres. After this, the


69
Figure 4-3. The intermolecular potentials for segment-seg
ment (6,9), segment-shell (12), and head-head (3,12). A
value of 419 Joule/mole was used for e; both rm and r^ were
assigned values of 4 Angstroms.


70
Table 4
Parameter Values for Potentials Used in Simulations
Potential
Parameter
Value
Units
Bond vibration +
y
9.25x10s
a
Joule/A2/mole
t>0
1.539
a
Angstrom
Bond bending +
yb
1.3x10s
a
Joule/mole
112.15
a
degree
Bond rotation +
yr
8313
b
Joule/mole
Head-shell
y
785
c
J oule/A2/mole
ra
4
c
Angstrom
Segment-shell
e
419
Joule/mole
rm
4
Angstrom
Head-head
e
419
Joule/mole
*
Angstrom
Segment-segment
6
419
a
Joule/mole
rm
4
a
Angstrom
Head-segment

419
Joule/mole
r h~a
1 ( \
Angstrom
2\ r m + rhh
a Weber (1978)
k Ryckaert and Bellemans (1975)
c Woods (1985)
* Parameter varies with head group size.
See also Table 6.


71
rotational force constant is restored to its desired value
and the system is allowed to equilibrate. When there is no
net drift in the energy of the system or in the fraction of
trans bonds, equilibrium is considered to have been reached.
The simulation is continued, periodically saving the
positions and velocities of all of the groups for subsequent
analysis.
The time step used in simulation must be small enough to
track the motions of the molecules and maintain stability,
yet large enough to avoid unnecessary computing effort. The
ideal time step can only be found by trial, which was
conducted on a simulated liquid alkane system by Weber
(1978). That result, Ait = .002 psec., is the basis for the
time steps used in the present work.
4.5 Summary of Computer Simulations
The following simulations were carried out on the model
micelle:
1. Head group mass and size same as chain segment.
2. Head group mass greater than segment, size the same.
3. Head group mass and size greater than segment.
In addition, a fourth simulation was carried out on a
hydrocarbon droplet of 24 nine-segment molecules by setting
the head group to the size and mass of a methylene and
replacing head-head and head-shell interactions with segment
interactions. From these simulations the independent


72
effects of head group size, mass, and solvophilic nature can
be observed, and by comparison with comparable simulations
of dodecyl surfactant micelles (Woods et al., 1986) the
effect of chain length can be evaluated. A brief summary of
the simulations is given in Table 5.
To carry out the first two simulations, 24 model
surfactant molecules were placed with their head groups on a
sphere of radius 24 Angstroms and their chains directed
radially inward in the all-trans configuration. The groups
were given initial velocities, the rotational force constant
was reduced to one-tenth of its normal value, and the
simulation was begun. Every ten time steps, the radius of
the confining sphere was decreased by .01 Angstrom. This
scaling down of the model micelle was continued until the
radius of the spherical shell reached a value of 12.00
Angstroms, slightly higher than the radius of an analogous
hydrocarbon droplet. Ten-thousand time steps were run at
this radius to allow the micelle to recover from the scaling
down procedure and observe the pressure at the shell. The
pressure fluctuated about zero with an amplitude of
approximately two atmospheres and analysis of the positions
of the groups revealed no effect of the initial conditions.
This model micelle was chosen as the starting point for
simulation 1.


Simulation
Head Group
Diameter
Summary of
Head Group
Mass
Table 5
Simulation
Micelle
Radius, A.
Computations
Equilibrium
Run,psec.
Time Step,
psec.
CPU Usage,
sec.
1
1
1
12.000
79
1.98x10-*
6719
2
1
7
.12.000
99
1.9 8x10-*
8602
3
2.45
7
12.800
28
1.40X103
8 699
4
1 a
1 a
11.928
119
1.9 8 X10 3
8342
a Hydrocarbon droplet, head group interactions same as chain segments.
Note: Head group diameter and mass are relative to chain segment (methylene.)


74
Simulation 2 was started from this same model micelle,
after increasing the mass of all head groups by a factor of
seven and allowing 10,000 time steps for equilibration.
This mass was chosen to represent a common ionic head group,
the sulfate ion. Only the mass of the head group was
changed; all of the intermolecular and intramolecular
interactions remained the same as in the previous
simulation. The expected result of this would be simply a
change in the dynamics of the head group. The program used
to compute simulations 1 and 2 is listed in Appendix D.
In simulation 3, the change in the head group was more
extensive. An attempt was made to represent the sulfate
head group in all of its intermolecular and intramolecular
interactions. The mass of the group remained at seven times
that of a methylene, while the diameter was increased to
2.45 times that of a methylene. The equilibrium bond length
and bond angle for the head group were changed, as were the
force constants for bond vibration, bond-angle bending, and
bond rotation involving the head group. These values are
given in Table 6 and compared with the methylene values.
The derivation of the head group intramolecular interaction
parameters based on the work of Muller and coworkers (Muller
and Nagarajan, 1967; Muller et al., 1968), Cahill et al.
(1968), and Blukis et al. (1963) is given in Appendix E.


75
Table 6
Bond Parameters of "Sulfate" and "Methylene" Groups
Parameter
"Methylene" Value
"Sulfate" Value
Units
y
9.25X105
2.7xl04
Joule/A2/mole
b0
1.539
2.6
Angstrom
yb
1.3x10s
9.1x10s
Joule/mole
0o
112.15
140
degree
yr
8313
20000
Joule/mole


76
Due to the increase in the size of the head group, the
range of head group interactions was significant in
comparison to the size of the micelle and truncation of the
potentials could no longer be justified. Therefore,
neighbor-listing and truncation of intermolecular potentials
were not employed in this simulation. It was also found to
be necessary to decrease the time step to maintain stability
in the energetics of the system. It is for these two
reasons that the computing efficiency (simulation time per
CPU time) of simulation 3 is dramatically lower than the
other simulations (Table 5). The program used to compute
simulation 3 is listed in Appendix F.
Simulation 4, the hydrocarbon droplet, was conducted in
the same manner as simulation 1, except that the head group
was replaced by a chain segment (all segments have the same
properties to simplify computation) in its interactions with
the shell and other groups. The radius was scaled down to
11.928 Angstroms to achieve the liquid hydrocarbon density
of .7176 g/cc. Since there was no sorting out of the head
groups for separate interactions, the efficiency of this
simulation was 18 percent greater than that of the first two
simulations.
The simulation computations were conducted on a Control
Data Corporation (CDC) Cyber 205 computer at the


77
Supercomputer Research Institute (SCRI) in Tallahassee,
Florida. Access to the SCRI facilities was provided through
a grant from the United States Department of Energy.
All of the simulation programs were modified for the CDC
FORTRAN 200 Vector-optimizing compiler. By employing
parallel processing technigues when possible, this compiler
provides very efficient operation from code which is highly
compatible with ANSI standard FORTRAN. One important
difference lies in the default word length of the Cyber 205.
Real numbers on this computer are eight bytes in length,
compared to four bytes on most other computers. To achieve
eight-byte precision in standard FORTRAN, the REAL*8
variable type is used. The programs in Appendices D and F
achieve this level of precision on the Cyber 205 with
default variable-type coding.
Further detail about the simulations, along with the
results of their analysis, is given in the next chapter.


CHAPTER 5
RESULTS OF MOLECULAR DYNAMICS SIMULATION
In this chapter the results of structural and dynamic
analyses performed on the output of the simulations are
reported. Certain of these results are reported as
time-averaged properties; others are given as the change in
a property with time. The former are averaged over the
entire length of the eguilibrium run (Table 5); the latter
appear in the figures over a common time period for ease of
comparison. The simulations are referenced by number, Run 1
being the case with head groups of the size and mass of a
chain segment (methylene); Run 2 the case with head groups
of the same size as a chain segment, but a mass seven times
greater; Run 3 the case with head groups of 2.45 times the
size and seven times the mass of a chain segment, and a
revised head group intramolecular potential; and in Run 4
the head groups replaced by chain segments to model a
hydrocarbon droplet.
5.1 Mean Radial Positions of Groups
The measurement of mean radial positions of the groups
on the surfactant chains is one of the simplest, yet most
78


79
telling, evaluations of micelle structure that can be made
on simulation output. While this measurement cannot be
obtained reliably for all groups by experiment, the position
of groups is a fundamental attribute of structural models of
micelles (Gruen, 1981). The mean radial position for group
i (i=l for head, i=9 for tail) is given by
(5.1)
The quantity N^(r) is the number of occurrences of the
center of a group i at a radial position (relative to the
center of mass of the micelle) r at time t and the angle
brackets denote the average over the time period of the
simulation. The unsubscripted N is the number of molecules
in the system, equal to the sum of N-^(r) over all values of
r and i. The standard deviation corresponding to this mean
is given by
2
(5.2)
The mean radial positions of the centers of the nine
groups bracketed by one standard deviation in each radial
direction are plotted for the four simulations in Figures
5-1 through 5-4. Assuming a normal distribution of a
group's position over time, the range shown for each group
in the figures includes 68 percent of its positions, while
twice this range includes 95 percent.


o<
in
x>
o
cr
20 -T
19 -
18 -
17 -
16 -
15 -
14 -
13 -
10 -
9 -
8 -
7 -
6 -
5 -
4 -
3 -
2 -
1 -
Head 2
12 -
11 -
3
1 1 1 i j
4 5 6 7 8 Tail
Group
00
o
Figure 5-1. Mean radial positions of the nine groups of the
surfactant molecules of Run 1. The vertical bars show one standard
deviation on either side of the mean.


19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
I 1 1 1 1 1 1 p-
Head 2 3 4 5 6 7 8 Tail
Group
re 5-2
actant
ation
. Mean radial positions of the nine groups of the
molecules of Run 2. The vertical bars show one standa
on either side of the mean.
rd


o<
en
D
X>
o
DI
co
to
Figure 5-3. Mean radial positions of the nine groups of the
surfactant molecules of Run 3. The vertical bars show one standard
deviation on either side of the mean.


20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
! 1 1 1 f 1 1 7 |-
Head 2 3 4 5 6 7 8 Toil
Group
ire 5-4. Mean radial positions of the nine group
ocarbon molecules of Run 4. The vertical bars s
idard deviation on either side of the mean.
s of the
how one


84
Although no group has its mean position in the center of
the micelle, this does not indicate a void within the
micelle. Rather, it is an artifact of the coordinate system
chosen. Each chain segment has a diameter of 3.56
Angstroms, and subsequently excludes a spherical volume of
this diameter to any other group. Since the spherical
coordinate system provides an available volume which
decreases with r, the excluded volume effect results in low
values of N(r) near the center of the micelle. Thus, no
group has its average position near the center, but the
center is always within the excluded volume of one of
several different groups. A range of three standard
deviations about the mean includes approximately all of a
group's positions, and, in Figures 5-1 through 5-4,
approaches within a segment diameter of the center for
several chain segments in each simulation.
The mean position results for Runs 1 and 2 are virtually
identical. This is to be expected, since the two models
differ only in the mass of the head group, a difference that
should manifest itself in the dynamic behavior of the
system, but not in a time-averaged result. In these two
simulations, the head groups are found at the surface of the
micelle with a relatively small deviation from the mean.
Progressing along the chain toward the tail, the mean radial
positions decrease and the deviations remain roughly


85
constant through group five. The mean positions of the last
four groups are at approximately the same radius, but toward
the tail the deviations increase, a consequence of the
excluded volume effect near the center of the micelle.
The results for Run 3 are markedly different. Figure
5-3 shows larger values of mean positions for the chain
segments and greater deviations for the head group and its
adjacent three groups. This simulation differs from Run 2
in the size of the head group and the nature of the
head-chain bond. With a diameter of 8.72 Angstroms, the
larger head group creates a steric effect which brings about
greater disorder in the micelle. The head groups are spread
over a greater range of radial positions, altering the
positions of the chain segments from those observed in Runs
1 and 2.
Figure 5-4 shows the results for Run 4, the hydrocarbon
droplet. The mean positions and deviations are symmetric
about group 5, the center of the molecule. This is the
proper result, since the two ends of a chain are
indistinguishable from one another. The chain ends have a
slightly larger mean position and deviation, a result of the
greater mobility of the chain end relative to the interior
groups of the chain. Overall, the means and deviations are
quite uniform, indicating a random arrangement of the


86
molecules within the droplet. Again, the excluded volume
effect places the mean positions of all groups over two
segment diameters away from the center of the droplet.
In a simulation of a micelle of forty dodecyl surfactant
chains, the same shell model was used to surround the
micelle and the head groups were identical to Run 1 in size,
mass, and potentials (Woods et al., 1986). Comparison with
this simulation reveals the effect of chain length on the
micelle model. In Figure 5-5, the average radial positions
of Woods' simulation are shown next to those of Run 1,
pairing the groups of Run 1 with the nine groups furthest
from the head group of the longer chain's thirteen. The
dodecyl micelle being larger than those of the present work,
its average radial positions are greater. Qualitatively,
however, the trend observed in the radial positions of the
groups and their standard deviations is quite similar in the
two simulations.
In Figure 5-6, the average radial positions of Run 1 are
compared to those resulting from a statistical model of 30
twelve-member chains with one end of each fixed at the
surface (Gruen, 1981). In this model, all possible
configurations of the surfactant chains were sampled.
Though Gruen's micelle is again larger than that of Run 1,
the results of the two models show notably similar radial
position behavior on a qualitative basis.


19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
n-12
n10
n-8
n6
n4
n2
Group
n
(Toil)
5-5. A comparison of the mean radial positions of groups in
and those obtained in a simulation of 40 dodecyl surfactant
les by Woods et al. (1906).


o<
(O
13
XJ
O
ct
20
19 -
18 -
17 -
16 -
15 -
14 -
13 -
12 -
11 -
10 -
9 -
8 -
7
6 -
5 -
4 -
3 -
2 -
1 -
0
n t
11 T
11 T
III
u
i 1 r
n-12 n10
i 1 1
n-8 n-6
i 1 1 1 r
n4 n-2 n
(Tail)
Group
Figure 5-6. A comparison of the mean radial positions of groups in
Run 1 and those obtained in a statistical model of 30 twelve-
segment chains proposed by Gruen (J901).
00
CO


20 1
19 -
18 -
17 -
16 -
15 -
14 -
13 -
il
II
II
III
II
II
II
I I
(I
cn 8
11
11
ii
7
ii
11
oo
1.0
6 -
5 -
4 -
3 -
2 -
1 -
0
n-8 n7 n6 n-5
n3 n2 n1
n
(rail)
Group
Figure 5-7. A comparison of the mean radial, positions of groups
in Run 1 and those obtained in a simulation of 1.5 sodium octanoate
molecules by Jonsson et al. (1986).


90
A model micelle of 15 sodium octanoate molecules, along
with surrounding molecular water, was the subject of another
molecular dynamics simulation (Jonsson et al., 1986). The
sodium carboxylate head group was explicitly modeled and
surrounding water molecules were included in the simulation.
Jonsson found it necessary to reduce the charge on the head
groups to produce acceptable results. Run 3, with the model
sulfate head group, is compared to Jonsson's reduced charge
simulation in Figure 5-7. Accounting for the difference in
the size of the two micelles, the radial positions of the
chain segments compare favorably.
5.2 Probability Distributions of Group Positions
The positions of groups within the micelle can be
studied in more detail by evaluating, at each radial
position, the probability of finding a given group at any
given time. The probability density function used in this
analysis is the time-averaged number of occurrences of the
center of a group i within a spherical volume element
centered at r:
P,(r) =

4/rr 2 Ar
(5.3)
Evaluating this function over all values of r generates a
probability density distribution for each molecular group.
The results of this analysis for each of the four
simulations are given in Figures 5-8 through 5-11. The


0.015
8 10 12
o
Radius, A.
20
Figure 5-8. Probability density distributions of the nine groups
of the surfactant molecules of Run 1. Group 1 is the head group
and group 9 is the tail group.


0.015
O
Radius, A.
Figure 5-9. Probability density distributions of the nine groups
of the surfactant molecules of Run 2. Group 1 is the head group
and group 9 is the tail group.


Full Text

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1
THERMODYNAMIC MODELING AND MOLECULAR DYNAMICS SIMULATION
OF SURFACTANT MICELLES
By
ROBERT ANTHONY FARRELL
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
1988

To Peggy, for her support,
encouragement, and patience

TABLE OF CONTENTS
Page
ABSTRACT V
CHAPTERS
1 INTRODUCTION 1
2 A MOLECULAR THERMODYNAMIC MODEL
OF MICELLE FORMATION 4
2.1 Background 4
2.2 Stoichiometry and Reaction Equilibria
for Multicomponent Micelles 8
2.3 Estimation of Free Energy Changes 11
2.4 The Calculational Technique 17
3 RESULTS OF THERMODYNAMIC MODELING 23
3.1 Parameter Estimation 23
3.2 Behavior of the Model for
Single-Component Nonionic Systems 30
3.3 Aggregate Size Distribution and
Concentration Behavior 40
3.4 Chain Length and Temperature Effects 43
4 MOLECULAR DYNAMICS SIMULATION
OF SURFACTANT MICELLES 52
4.1 Background 52
4.2 The Molecular Dynamics Method 55
4.3 The Model Surfactant Molecule 60
4.4 The Model Micelle 65
4.5 Summary of Computer Simulations 71
5 RESULTS OF MOLECULAR DYNAMICS SIMULATION 78
5.1 Mean Radial Positions of Groups 78
5.2 Probability Distributions of Group Positions. 90
5.3 Conformations of Chain Molecules 107
5.4 Shape Fluctuations 126
5.5 Pair Correlations of Groups 133
6 CONCLUSIONS 137
iii

APPENDICES
A MICELLE SIZE AND SHAPE 14 3
B HEAD GROUP INTERACTION IN A BINARY MICELLE .... 150
C PROGRAM LISTING FOR SINGLE-COMPONENT
NONIONIC MICELLE CALCULATION 160
D MOLECULAR DYNAMICS PROGRAM LISTING
FOR SIMULATIONS 1 AND 2 17 4
E DERIVATION OF HEAD GROUP
INTRAMOLECULAR INTERACTIONS 218
F MOLECULAR DYNAMICS PROGRAM LISTING
FOR SIMULATION 3 224
REFERENCES 2 68
BIOGRAPHICAL SKETCH 276
IV

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
THERMODYNAMIC MODELING AND MOLECULAR DYNAMICS SIMULATION
OF SURFACTANT MICELLES
By
ROBERT ANTHONY FARRELL
April, 1988
Chairman: John P. O'Connell
Major Department: Chemical Engineering
The association of surfactant molecules into aggregates
known as micelles gives them a broad range of applications.
In spite of the widespread use of this class of chemicals,
there is not yet sufficient scientific understanding to
predict their behavior in solution.
A molecular thermodynamic model has been developed to
describe the formation of micelles in multicomponent
surfactant solutions. Using a hypothetical, reversible
seven-step process, the total free energy of micellization
is calculated by summing the contributions due to
solvophobic interaction, mixing, surface formation, confor¬
mational change, head group interactions and electrostatics.
Distributions of micelle sizes and compositions can then be
generated through a set of reaction equilibria. Where
v

possible, the free energy contributions are related to
comparable processes on which experimental measurements have
been made. Aggregate size distributions have been generated
from the model for single-component solutions of nonionic
surfactants of different chain lengths and at different
temperatures and solution concentrations.
It has been found that a detailed description of micelle
structure and entropy effects on chain conformation is
necessary to fully describe the thermodynamics of micelle
formation without empirical parameterization. To this end,
computer simulations of model micelles have been conducted
by the molecular dynamics method. Micelles of three
different head group characteristics and a comparable
hydrocarbon droplet have been simulated. A spherical shell
is used to contain the aggregate, providing estimated
solvophobic interactions with the molecules.
The simulation results reveal that internal structure of
the aggregates is relatively insensitive to the head group
characteristics with the greatest effect resulting from head
group size. While the micelles all showed chain ordering
and the hydrocarbon droplet did not, the bond conformations
averaged approximately 71 percent trans in all cases.
Results of other simulations and experimental studies are
vi

generally similar to those of the present work for the
effects of chain length, aggregate size, and simulation
technique on static properties.
Vll

CHAPTER 1
INTRODUCTION
There are few classes of chemical species which have
received more attention in the scientific literature, or
have exhibited a more ubiquitous presence in everyday life,
than the amphiphilic molecules known as surfactants. Their
unique solution properties of association and adsorption at
interfaces give rise to a broad range of applications. Long
the essential component in detergency applications, in more
recent times surfactants have assumed roles of importance in
applications as diverse as enhanced oil recovery and
pharmacy.
The association of surfactant molecules in solution into
aggregates known as micelles is the primary attribute which
has garnered interest among the scientific community. Since
the discovery of micelles in solution (McBain and Salmon,
1920), many experimental and theoretical studies have been
conducted, yet full understanding of these systems of
molecules has not been achieved. True predictive capability
is not yet a reality.
This investigation takes a two-fold approach toward the
ability to make quantitative predictions of the behavior of
systems of surfactants in solution. Since the behavior of
1

2
the surfactant solution is a consequence of its thermodynam¬
ics, a model set in the framework of molecular
thermodynamics is developed to describe the formation of
micelles in a multicomponent surfactant solution.
A thermodynamic description of a micellar system is
limited by a precise knowledge of the structure of micelles.
To this end, computer simulations of model micelles by the
molecular dynamics method are conducted. By accurately
modeling the forces present, pertinent descriptions of
micelle structure are obtained.
In Chapter 2 of this work, further background on
micellar systems is given and the development of a model for
the free energy change upon formation of micelles in a
multicomponent surfactant solution is described. A multi-
step reversible process is employed to generate contribu¬
tions to the total free energy change due to hydrophobic
interaction, mixing effects, conformational change, head
group interaction, and electrostatic interaction.
Some brief results obtained with the thermodynamic model
are presented in Chapter 3. Distributions of free energy
change of micellization and aggregate size are calculated
for surfactants of different chain lengths and for solutions
of different temperature. Due to limitations in the scope

3
of this portion of the investigation, calculations were
carried out only for cases of single-component solutions of
nonionic surfactants.
The computer simulation of surfactant micelles is
described in Chapter 4. Following a concise description of
the molecular dynamics method, its application to the
simulation of micelles is discussed. A summary of the
computer simulations of the four models—three micelles and
one hydrocarbon droplet—is given.
In Chapter 5, the results of analyses of the computer
simulations are presented. Elements of aggregate internal
structure, chain conformations, aggregate shapes, and
changes in the aggregate with time are investigated for the
four model aggregates.
Chapter 6 provides a summary of the significant
conclusions of this work. Recommendations are made toward
the future progress of both of the projects described in the
previous chapters.

CHAPTER 2
A MOLECULAR THERMODYNAMIC MODEL OF MICELLE FORMATION
2.1 Background
The forces present in liquid solutions dictate that
solution of a polar solute in a polar solvent is more
favored than is a polar solute in a nonpolar solvent.
Similarly, the nonpolar solvent is more accommodating toward
a nonpolar solute than a polar one. Therefore, molecules
which contain both polar and nonpolar groups exhibit a
unique behavior when present in a polar or nonpolar solvent.
Such molecules, known as surfactants, will tend to minimize
the unfavored contact (i.e., polar-nonpolar) while maximiz¬
ing the favored contact (i.e., polar-polar). At an
interface between polar and nonpolar liquids, the
surfactants will penetrate the interface to achieve "like"
interactions on both sides, reducing the "unlike"
interactions encountered in the bulk liquid. The polar
groups of a large number of surfactant molecules packed
closely at an interface will repel each other, producing a
spreading pressure which reduces the interfacial tension.
In the bulk liquid, surfactant molecules will aggregate
into structures known as micelles which can afford much the
same benefits as the interface. In the typical case, a
4
!

5
surfactant with a polar "head group" and a nonpolar "tail,"
when present in sufficient quantity in a polar solvent such
as water, will form micelles having an interior consisting
of tails and possibly some nonpolar solubilizate and a
surface comprised mostly of head groups. The head groups
remain in contact with the water—a favored interac¬
tion—while the tails reduce their contact with the
water—an unfavored interaction. When the solvent is
nonpolar, inverted micelles can form.
The ability of surfactants to adsorb at interfaces and
aggregate into micelles makes them a very useful class of
compounds. The reduction of interfacial tension has many
applications, ranging from oil recovery to biological
processes. In addition, micelles can solubilize other
solutes in their interior, as in drug delivery processes,
and reactions can even take place there, as in emulsion
polymerization. The earliest and best known use of
surfactants, detergency, uses both aspects of their
behavior. As useful as these phenomena are, they are not
understood to a degree that would allow their full potential
to be realized. The ability to predict behavior rather than
just explain it is the goal of this undertaking. This
requires a knowledge of the thermodynamics of surfactant
phenomena.

6
The formation of surfactant aggregates in solution
instills a certain ambiguity in the description of the
system by a traditional thermodynamic formalism. The
aggregation of surfactant monomers into micelle structures
has been treated as the formation of a "phase" (Blankschtein
et al., 1985; Kamrath and Franses, 1983; Matijevic and
Pethica, 1958) or as a stepwise association "reaction"
(Tanford, 1974; Mukerjee, 1972; Murray and Hartley, 1935).
Although the former description may aid in visualizing
certain aspects of micellar solutions, the thermodynamic
idea of a phase cannot be used in a rigorous manner. Its
requirements of continuity and homogeneity are not met by a
collection of micelles in solution and a single micelle
cannot be treated as a phase since its properties are
size-dependent.
The treatment of micelle formation as reaction
equilibrium is plagued by the lack of a single
stoichiometry. Since a distribution of products is formed
(Void, 1950), one must consider each micelle to be in
reaction equilibria with the dispersed monomers. The
determination of the many equilibrium constants by
experimental methods is impossible. Hall and Pethica (1967)
proposed using a small-systems thermodynamics approach to

7
avoid the difficulties of these two treatments. But their
approach cannot be used with ionic systems and is mainly
formal, not lending itself to practical use.
The thermodynamics of micelle formation remains an
interesting problem. The literature is abundant with
studies. In addition to the guantities of temperature,
pressure, and composition which typically define the
thermodynamic state of a typical solution, the thermodynamic
behavior of solutions containing surfactant species can
depend on the size, shape, and structure of the aggregates
which are formed. Thermodynamic properties have been
measured and correlated (Burchfield and Woolley, 1984;
Woolley and Burchfield, 1984, 1985). Aggregate formation
has been investigated from the points of view of classical
thermodynamics (Moroi et al., 1984; Muller, 1973) and
statistical thermodynamics (Hoeve and Benson, 1957; Owenson
and Pratt, 1984). Investigations have focused on size
distributions (Ruckenstein and Nagarajan, 1975; Ben-Naim and
Stillinger, 1980), the role of micelle shape (Tanford, 1974;
Israelachvili et al., 1976; Ljunggren and Eriksson, 1984,
1986; Eriksson and Ljunggren, 1985; Void, 1985), and shape
transitions (Van de Sande and Persoons, 1985; Ikeda, 1984;
Missel et al., 1983; Mukerjee, 1977).

8
While contributing to our understanding of the complex
nature of micelle formation, none of these works produced a
practical model with predictive capabilities. A semi-empir-
ical model for the thermodynamic properties of surfactant
aggregate formation based on molecular thermodynamic
processes was developed by Hourani (1984) and was successful
at predicting thermodynamic quantities and aggregate size
distributions for systems of a single surfactant species in
solution. Benedek (1985) developed a different model in the
framework of molecular thermodynamics. While it was
demonstrated successfully, the use of empirical parameters
was more extensive than in Hourani's work. The model of
Hourani showed promise of being extendable to multicomponent
systems and of being more closely related to other
observable molecular phenomena. The beginnings of such an
extension are given in this chapter.
2.2 Stoichiometry and Reaction Equilibria
for Multicomponent Micelles
Since the free energy of a process is independent of the
path chosen between the initial and final states, Hourani
proposed a process consisting of a series of steps for which
the free energy change can be modeled. In this process, the
monomers were removed from the solvent and placed in a
vacuum at their original density—a gaseous state. Cavities
of excluded volume remained in the solvent, to be coalesced

9
in a subsequent step. The monomer gas, considered ideal,
was compressed to micellar density, and placed into larger
cavities which had been formed in the solvent. Essential to
the modeling was the elimination and creation of the solvent
cavities. The counterions were handled in the same fashion,
with the addition of the necessary electrostatic calcula¬
tions. Extending Hourani's molecular thermodynamic model to
solutions containing two or more surfactant species required
the addition of new steps and modification of others. The
solvent cavity steps were eliminated and the monomers are
removed to a liquid state rather than the gaseous. These
changes facilitate the handling of multiple components. The
development of the multicomponent model is detailed below.
Micelle formation in solution yields a distribution of
micelle sizes. In addition, a multicomponent surfactant
solution has a distribution of compositions (Warr et al.,
1983; Scamehorn et al., 1982; Birdi, 1975; Moroi et al.,
1974, 1975a, 1975b; Rubingh, 1979; Clint, 1975). To
describe this, a set of equilibrium reactions can be written
for the formation of J micelles of distinct sizes and/or
compositions. For I different surfactant species, Z^, and K
counterion species, B^, aggregating to form J micelles, Zj,

10
*>
NnZ
x + N2lZ7 + .,
• •+ NnZ,+ M UB j
+ M 2l B 2 + ..
. + M K1B k 44 Z
k2
Nl2Z
1 + N 22 Z 2 + • ■
••+ NI2Z, + M l2B j
+ M 22B2 +..
, + M K2BK& Z2
•
•
• (2
kj
NXJZ-
i + N2JZ 2 + .
••+ N UZ,+ M XJB
i + m2Jb2 + .
• • + M kjB Zj
where N^j is the number of monomers of species i present in
the jth micelle and is the number of k counterions bound
to it. The equilibrium constant Kj for the formation of the
micelle is given by
[Z'l
(2.2)
These are related to the standard state free energy of
micellization of the micelle by
AG° = - RT\n K j
The total number of monomers in this micelle is
(2.3)
i
so that
(2.4)
N jAGj° = -RTlnK j (2.5)
where JGj° is on a per monomer basis.
The mole fraction of monomers in the jth micelle is
found by combining equations (2.2) and (2.5). For the
dilute concentration range of micellar solutions, ideality

11
of the monomer solution can be assumed, and
where CD is the total solution concentration
[ ] indicates concentration of species
XÍ is the monomer mole fraction in j^h micelle
Nj is the aggregation number of the jth micelle
Within the material balance constraint for each species,
(2.7)
equation (2.6) describes the distributions of micelle size
and composition in solution when provided with the free
energy change as a function of the size and composition of
the micelle formed.
2.3 Estimation of Free Energy Changes
The standard state free energy of micellization is
calculated via the seven-step process shown in Figure 2-1.
The standard state free energy of formation for a single
micelle in solution is the sum of the free energies of the
seven steps. Each step is modeled as closely as possible
from an observable phenomenon of a similar nature.
Certain free energy terms are dependent on the shape of
the micelle. The micelle grows as a roughly spherical
aggregate until the additional volume of one more monomer
would cause the radius of the sphere to exceed the length of
the longest all-trans chain. With the addition of the next

3
H
to
Figure 2-1. The seven-step reversible process used in the estimation of
contributions to the total Free Energy of micellization in a multicomponent
surfactant solution. From left to right: Head groups and counterions are
removed; surfactant chains (hydrocarbons) and solvent are separated into
discrete phases; the hydrocarbons are mixed into compositions of the micelles;
hydrocarbon droplets are placed in the solvent; chain ends are brought to the
surface; head groups are replaced; and counterions are replaced.

13
monomer the micelle must grow with a nonspherical geometry
to avoid the formation of a material-free core. Several
geometries have been proposed: spherical dumbells, oblate
ellipsoids, prolate ellipsoids, and spherocylindrical rods.
While Ljunggren and Eriksson (1984, 1986; Eriksson and
Ljunggren, 1985) have proposed that the shape fluctuates
between spherical, rod-shaped, and even disc-shaped, Void
(1985) has found little effect of the particular geometric
model on the thermodynamics of micelle formation. In this
work nonspherical micelles are modeled as prolate
spherocylindrical rods. The derivation of micelle dimen¬
sions and surface area based on this geometric model is
given in Appendix A.
In step 1, a standard state solution of surfactants is
transformed into a solution of hydrocarbon chains by
reversibly removing the head groups and counterions. Since
these are reversibly replaced in steps 6 and 7, the net free
energy change for the mere removal and replacement of the
head groups and counterions is zero. If no free energy
contributions due to the replacement of head groups and
counterions are contained in steps 6 and 7, then
Z1G,° = 0 (2.8)
In step 2 the hydrocarbon solution is separated into I
pure hydrocarbon liquids and the pure solvent. This is the

14
reverse process of hydrocarbon solubility, so
(2.9)
In step 3 the I pure hydrocarbons are placed into J
ideal hydrocarbon mixtures of different compositions and
amounts. For this ideal mixing,
(2.10)
In step 4 the J hydrocarbon mixtures are formed into J
droplets and placed into the solvent. The free energy of
this step is the free energy of forming the hydrocarbon-sol-
vent interface. There are both surface area and curvature
contributions to this step. The expression for AG/£, based
on Buff (1955) and Stillinger (1973), is the surface area of
the droplet, S, times the planar interfacial tension, y, of
the hydrocarbon mixture, corrected for curvature:
(2.1 la)
The curvature effect is dependent on the parameter, g. The
surface area depends on the size and shape of the micelle.
For the spherocylindrical micelle, a second curvature
parameter, gc, is used for the cylindrical portion:
(2.1 lb)

15
This approximates the cylindrical curvature effect, whose
uncertainty has been discussed by Henderson and Rowlinson
(1984).
In step 5 conformational changes in the hydrocarbon
chains are made so that one end of each chain is at the
surface of the droplet. The contribution from this step is
entirely entropic and may only be estimated. The expression
used in this model is
(2.12)
where Sq is a parameter of the model. The squared ratio of
chain length, lc, to micelle radius, R, takes into account
the very severe conformational restrictions present when the
micelle radius is much smaller than the chain length.
As indicated in the discussion of step 1, step 6
contains no contribution due to the reattachment of head
groups. The quantity JGg0 is the free energy change due to
the interaction between the head groups in their positions
at the micelle surface. The head groups are modeled as
dipoles. For the ionic surfactant species, a charged head
group paired with a counter ion forms a strong dipole.
Nonionic head groups exhibit weak-to-moderate dipole
moments. The dipole-dipole interactions of adjacent pairs
are summed for the free energy contribution of this step.
The separation and orientation between two dipoles are

16
dependent on the size and shape of the micelle, with the
head groups evenly spaced over the surface of the micelle.
The derivation of the head group interactions is carried out
in Appendix B. The potential between a pair of adjacent
head groups is
(2.13)
where /¿ is the dipole moment
r is the pair separation
R is the micelle radius
D is the dielectric constant of the solvent
As indicated in the derivation, this form of the potential
takes into account the angle between adjacent dipoles as a
function of micelle radius. The total energy contribution
from this step is found by summing the contributions from
the different pairs in the manner described by equations
(B.25) through (B.28) in Appendix B.
Step 7 contains no contribution for the replacement of
the counterions back into solution. The free energy of this
step is due to the difference between the original random
distribution of counterions in the micelle-free solution and
the final Poisson-Boltzmann distribution of counterions
around the surface of the micelle with a fraction bound in a
Guoy-Chapman electrical double layer. The derivation of
Hourani (1984) for the numerical solution of this charge

17
distribution model is applicable here. The entire step is
actually the process of discharging the counterions in their
original distribution, compressing them into the bound layer
and final distribution, and then recharging the counterions.
Therefore AG70 contains an entropy contribution from the
compression and an enthalpy contribution from the
distribution. The bound layer will be populated with
dipoles formed by head group/counterion bound pairs and
unbound head groups. The charge interactions in the bound
layer are included in the dipole and point charge pairings
of the head group term, aGq° .
2.4 The Calculational Technique
The free energy of formation for a single mixed micelle
with nonionic head groups is obtained by summing the
contributions from steps 1 through 6:
AG
RT
= }^{xij\nCiq + xit\nx
2 n R ¡ y ¡
NhT
L\ i-2£
R,
2 R
g_
R
J J J
-5,
R
— J
RT
(2.14)
To include the presence of ionic head groups and
counterions, the contributions of step 7 must be added to
equation (2.14). Such calculations were not carried out
here, so this section will pertain only to mixtures of
nonionic surfactants. Table 1 summarizes the model's
variables, parameters, and required data.

18
Table 1
Arguments of the Thermodynamic Model
Variables v
1/
Compositions of micelles
"i
Aggregate sizes of micelles
T
Temperature of system, Kelvin
xt
Composition of system
nc
Number of carbons in surfactant chain
r
° tot
Overall system concentration, moles/liter
Data £eq
Solubility concentrations of
hydrocarbons, moles/liter
y¡
Interfacial tensions of hydrocarbon
mixtures, dynes/cm
Hi
Dipole moments of head groups, Debyes
D
Dielectric constant of water
lc
Surfactant chain length, Angstroms
Surfactant chain volume, Angstroms3
Parameters g
Spherical curvature parameter, Angstroms
gc
Cylindrical curvature parameter, Angstroms
Sc
Entropy of conformation parameter
Parameters

19
The values of chain length (Ang.) and chain volume
(Ang.3) used in the determination of micelle size and shape
are calculated from Tanford's correlations (Tanford, 1972):
le = 1 .265rcc+ 1 .5 (2.15)
vc = 26.9nc + 27.4 (2.16)
To facilitate the use of computer programs in carrying
out the calculations, correlations are used for the reguired
physical data. The aqueous solubility of hydrocarbons used
in the calculation of is obtained from a correlation
due to Leinonen et al. (1971):
The parameter K is fit to the solubility data of McAulliffe
(1966), Polak and Lu (1973), and Sutton and Calder (1974).
It is found to be a linear function of the hydrocarbon chain
length for the n-alkanes, but since hydrocarbon solubility
in water exhibits a break at decane, two linear
relationships for K are used, one for the longer chains and
one for the shorter chains. Equation (2.17) is solved
iteratively for the hydrocarbon solubility, xe<3.
For the hydrocarbon-water interfacial tension required
in the calculation of JG4°, a correlation based on the works
of Aveyard et al. (1972), on the surface tension of
hydrocarbons, and Jasper (1972), on the surface tension of

20
water, is used:
57.868nc+ 1 17 .99 - ( .059nc + . 1768 T
y = 1.381 — (2.18)
nc + 2.4
where the temperature is in Kelvin and the surface tension
is in dynes/cm.
The value of the dielectric constant of water used in
the evaluation of JGg° is given by
D = 252.422- .806329 7+ .000746972 (2.19)
which is a polynomial fit of the data of Owen et al. (1961)
at atmospheric pressure with temperature in Kelvin. The
values of dipole moments used in this calculation are
estimated as the dipole moments of molecules of similar
structure to the head groups.
The three parameters, g, gc, and Sq, are fit to the
measurable data on the micellar system. These are the mean
aggregate size of the micelles and the set of mixture
critical micelle concentrations (CMCs) at the system
temperature. Equation (2.6) generates an I-dimensional
surface of the aggregate size distribution of the
multicomponent micelles. This is accomplished by first
choosing values of the temperature, T, the total
concentration, C^ot' and the system composition, x¿. Then
for each possible micelle composition, x-j^j, equation (2.6)
is evaluated at values of Nj from two to infinity (in
practice, until Xj/Nj becomes negligible). This generates

21
the distribution of aggregate sizes with micelle composi¬
tion. The parameters must be chosen such that the
distribution meets the material balance constraints of
equation (2.7).
To find the proper values of the parameters, the mean
size is calculated as
N = ~— (2.20)
I Z'/N,
and the CMC is taken to be the value for which
lim
d(C, + C
M ,
dC,
= 0.5
(2.21)
as put forth by Hall and Pethica (1967). Here, C^ is the
free monomer concentration and C^ is the concentration of
micelles, calculated as
C,
C (oí c
N
(2.22)
Once a set of parameters is determined for a system, the
response of the size distribution to changes in any of the
input variables can be investigated.
The program listing in Appendix C gives the FORTRAN
source for the interactive fitting of parameters and
calculation of aggregate size distribution on systems of a

22
single nonionic surfactant species in water. This is the
simplest application of the model and was used to generate
the results presented in the next chapter.

CHAPTER 3
RESULTS OF THERMODYNAMIC MODELING
3.1 Parameter Estimation
Three parameters of the model, g, gc, and Sq, must be
found by fitting the mean aggregate size generated by the
model to the experimental value at the critical micelle
concentration (CMC). The model output is considered to be
at the CMC when the total surfactant concentration is equal
to the experimental value of the CMC and the derivative of
the monomer concentration with respect to the total
concentration satisfies equation (2.21), indicating that one
out of two surfactant monomers added to the solution at this
concentration would join a micelle. The model convergence
is quite sensitive to the values of the parameters, so in
fitting them to the data, one must either use good initial
guesses or approach the values conservatively. Failure to
do either of these can result in an aggregate size
distribution which has infinite values of C^ot and Ñ,
providing no useful information.
Each of the parameters has a physical significance which
can aid in the choice of the initial guesses. The
parameters g and gc are used in the curvature corrections to
the planar interfacial tension for the spherical and
23

24
cylindrical geometries, respectively. Physically, g is the
thickness of the spherical interface in Angstroms (Buff,
1955). Although experimental values are not available, it
is expected to be positive and small relative to the micelle
radius. The parameter gc does not have the same physical
connection to the interface (Henderson and Rowlinson, 1984),
but by comparison it is expected to be positive and less
than g. The parameter Sq is the (dimensionless) conforma¬
tional entropy change for a monomer joining a large micelle.
Conformational contributions of the aggregated monomers were
studied via a statistical thermodynamic theory by Ben-Shaul
and coworkers (Ben-Shaul, Szleifer and Gelbart, 1985;
Szleifer, Ben-Shaul and Gelbart, 1985) and values of the
conformational entropy were found to be in the range of -8
to -7 for a chain with seven bonds. While these values are
based strictly on theoretical considerations with no
experimental corroboration, the parameter Sc is expected to
be of the same sign and magnitude.
In approaching the parameter fitting conservatively, the
initial values of the parameters are chosen such that the
aggregate size distribution will definitely converge. That
is, the condition
d N
(3.1)

25
must be satisfied. Equation (2.6) defines the aggregate
size distribution. In this equation, ziGj° is the only term
influenced by the parameters. In order for the distribution
to converge as N becomes large, this free energy change must
be greater than the logarithm of the monomer concentration.
Overestimating the parameters toward a less negative free
energy change will assure that the size distribution
converges. The parameters can then be adjusted toward a
more negative free energy change as they are fit to the
data.
The effect of adjusting the parameters can be foreseen
by analyzing the corresponding terms. The parameters g and
gc affect the behavior of the surface free energy, ¿G40.
For g < R and gc < lc, the usual cases, increasing the
parameter decreases the free energy, consistent with
equation (2.11). In equation (2.12) it can be seen that
increasing the parameter Sq decreases the conformational
free energy, ¿(25°.
The aggregate size distribution can be divided into two
regions. By designating the value of N for which R = lc as
Nfrans' the transition of the micelle geometry from
spherical to spherocylindrical defines the two regions of
the size distribution. For N ^ N-^rans' the micelle is
spherical with radius R. For N > Ntrans, the micelle is
spherocylindrical with radius lc and length L. For sizes

26
above Ntrans, the conformational free energy is constant
with a value of -Sc. At Ntrans the surface free energy
makes the transition from being equal to the spherical
contribution to becoming dominated by the cylindrical
contribution for large N. Figures 3-1 and 3-2 show the
effect of aggregate size and parameter values on the surface
and conformational free energies.
A typical aggregate size distribution generated for a
nonionic surfactant is shown in Figure 3-3. A peak occurs
at N-trans' beyond which the fraction of aggregate decreases
with increasing N. The total surfactant concentration is
proportional to the area under the distribution and is thus
influenced by both the height of the peak and the slope of
the distribution above N-j-rans' Because the distribution
below N-trans rises so rapidly, the mean aggregate size, Ñ,
is dependent only on the slope of the distribution above
Ntrans*
The parameter Sq, since it contributes over the entire
range of N, affects both C^ot and Ñ. The parameter gc,
contributing only above N-j-rans, has a significant effect on
Ñ but only a slight effect on C-f-ot' while g influences only
ctot an<^ not Ñ. This causes such interplay between the
parameters that a range of parameter sets can produce
identical values of C-t-0t and Ñ. The derivative constraint,

Figure 3-1. The surface contribution to the free energy of micelle
formation for the C12 nonionic micelle at the CMC and a temperature
of 298.15 Kelvin. The lower curve was generated with Sc=-4.0,
g=6.9586, gc=3.6272. The upper curve was generated with Sc=-3.0,
g=4.8879, gc=0.51846. The lower curve demonstrates the effect of g
exceeding the micelle radius at low values of N.

Aggregate Size
Figure 3-2. The conformational contribution to the free energy of
micelle formation for the C12 nonionic micelle at the CMC and a
temperature of 298.15 Kelvin. The parameters are as in Figure 3-1,
showing the effect of Sc.

o
-10 -
o 200 400 600 800 1000
Aggregate Size
to
VO
Figure 3-3. Aggregate size distribution for the C12 nonionic
micelle at the CMC and a temperature of 298.15 Kelvin. Typical
behavior of the size distributions of nonionic surfactants is
exhibited.

30
(2.21), can also be satisfied by these parameter sets if
they are fit at a value of the free monomer concentration,
, just below the critical micelle concentration.
The method used to fit parameters consisted of first
choosing low values of the parameters, resulting in a C-j-0t
essentially egual to C;l and an ñ insignificantly larger than
Nfrans* The parameter gc was increased until Ñ reached the
desired value, and then g was adjusted to result in the
proper C^ot-. This is repeated for different values of Sq to
generate the range of parameter sets fitting the data. The
experimental data used to generate the parameters is given
in Table 2.
3.2 Behavior of the Model for Single-Component
Nonionic Systems
Parameter fitting was carried out on systems of
different surfactant chain lengths and at different
temperatures. For each system, linear relationships were
found to exist between each pairing of the three parameters.
Corresponding to the manner in which the fitting was
accomplished, the curvature parameters g and gc can each be
expressed as a straight line plotted against the Sq
abscissa. Such a plot is given in Figure 3-4. These
expressions are of the form
g = mS c + b
gc=nSc+d
(3.2)
(3.3)

31
Table 2
Data Used in Fitting Model Parameters for Aqueous Solutions
of Nonionic
Surfactants
Surfactant
T (Kelvin)
CMC (M)
Mean Size
CioH2i(OC2H4)6OH
298.15
9.0E-04 a
73
a
308.15
6.6E-04
260
318.15
6.4E-04
640
Ci2H25(OC2H4)6OH
298.15
8.7E-05 b
400
c
C14H29(OC2H4)6OH
298.15
1.1E-05 a
3100
a
a:
Balmbra
et
al.
(1964)
b:
Corkill
et
al.
(1961)
c:
Balmbra
et
al.
(1962)

10
-1 H 1 1 1 1 1 1 1 1 1 r~ i r
-5.2 -4.8 -4.4 -4 -3.6 -3.2 -2.8
Conformational Parameter
Figure 3-4. Interdependence of the model parameters for the C12
nonionic micelle at 298.15 Kelvin. The symbols represent the
values of the spherical (g) and cylindrical (go) curvature
parameters. The lines give the regression results
g=-l.3235-2.0705SC and gc=-8.808-3.1087SC.

33
The values of the slopes and intercepts found for the
systems modeled are given in Table 3.
Many aspects of surfactant behavior have been found to
be linearly dependent on chain length and/or temperature
(Rosen, 1978). The information in Table 3 indicates that
such relationships are also possible for the interaction of
the parameters of this model. Though the three chain
lengths and three temperatures investigated cannot conclu¬
sively indicate linear behavior, they can indicate whether
this behavior is likely. Linear regression of the slopes and
intercepts of Table 3 versus chain length and temperature
resulted in the following equations and
correlation
coefficients:
m = -.00897+ .8817
R= .9998
(3.4)
6= .05497-20.20
R= .9778
(3.5)
n = -.01377+ 1 .421
R = .9999
(3.6)
d = . 1 1257-45.09
R= .9810
(3.7)
for the 10 carbon nonionic, and
m=-.1464nc-.3170
R= .9999
(3.8)
6= 1 .418nc - 18.10
R= .9988
(3.9)
n = -.2201nc-.4676
R= 1 .000
(3.10)
d = 1 .553nc-27.29
R= .9991
(3.11)
at a temperature of 298.15 Kelvin.

34
Table 3
Slopes and Intercepts of Curvature Parameter Dependence
on Conformational Parameter
n£
T (Kelvin)
g slope
g int.
gc slope
gc int.
10
318.15
-1.96163
-2.80775
-2.94330
-9.43866
10
308.15
-1.86885
-3.15286
-2.80330
-10.1776
10
298.15
-1.78296
-3.90518
-2.66898
-11.6881
12
298.15
-2.07048
-1.32359
-3.10874
-8.80775
14
298.15
-2.36862
1.74162
-3.54946
-5.47762

35
Substituting (3.4) through (3.11) into (3.2) and (3.3),
the temperature relations for CIO nonionic are
g = (.0549- .0089SC)T- 20.20 + .881 7Sc (3.12)
gc = (. 1 125-.01 372SC)T-45.09 + 1 .421 45 c (3.13)
and the chain length relations at 298.15 Kelvin are
g = ( 1 .418- . 14645C)nc- 1 8.1 0 - .3 1705c (3.14)
gc = ( 1 .553- .2201 Sc)nc- 27.29 - .46765c (3.15)
These relations are only valid for values of Sc which, for
each chain length, produce values of the curvature
parameters which are neither negative nor larger than the
micelle radius. In this sense, Sc is dependent on chain
length, but there remains a lack of uniqueness in the
parameter fitting for the single component case. This
results from the finding that the CMC and the concentration
derivative (2.21) are not independent of each other.
The free energy of micellization and its various
contributions, as given by the model, are shown in Figures
3-5 through 3-8. These plots were made for two chain
lengths at the same temperature, two temperatures with the
same chain length, and two parameter sets at the same chain
length and temperature. Figures 3-5 and 3-8 show the effect
of two different parameter sets for the same system. While
the surface and conformational contributions are different
due to the different parameters, the total free energy of
micellization is the same in the two figures. This follows

20
15 -
10 -
5 -
AíT OH
RT
-5 -
-10
-15 -
-20
Hydrophobic
20
“i 1 1 i r r
40 60 80 100
Aggregate Size
U)
G\
Figure 3-5. Free energy contributions for the CIO nonionic micelle
at the CMC and a temperature of 290.15 Kelvin. A value of -4.0 is
used for the conformational parameter, Sc, with the curvature
parameters fit to the data.

20
15
10 -
A£° o
RT
-5 -
-10 -
-15 -
-20
Surface
Conformational
Hydrophobic
20
~T~
40
\
60
1
BO
100
Aggregate Size
U)
Figure 3-6. Free energy contributions for the C14 non tonic micelle
at the CMC and a temperature of 29B.15 Kelvin. A value of -2.0 is
used for the conformational parameter, Sc.

20
â– 15 -
-20
lydrophobic
20
40 60
Aggregate Size
BO
100
Figure 3-7. Free energy contributions for the CIO nonionic micelle
at the CMC and a temperature of 318.15 Kelvin. A value of -4.8 is
used for the conformational parameter, Sq.

Aggregate Size
Figure 3-8. Free energy contributions for the CIO nonionic micelle
at the CMC and a temperature of 298.15 Kelvin. A value of -4.0 is
used for the conformational parameter, Sc.

40
from the fact that the parameter sets were fit to the same
concentration and mean size data and generate identical size
distributions.
3.3 Aggregate Size Distribution and Concentration Behavior
With the parameter sets determined by the methods
described above, the aggregate size distributions and the
effects of surfactant concentration were investigated for
the nonionic alkyl surfactants. The size distributions for
this class of surfactants exhibit a rapid increase in the
spherical region of N, peaking at N-trans. In the
nonspherical region, the distribution decreases very slowly
through large values of N, producing the large mean
aggregate sizes which have been measured for the nonionic
surfactants.
The size derivative of the distribution in the
nonspherical region is negative. With increasing free
surfactant monomer concentration, it becomes less negative,
resulting in a higher mean aggregate size and total
surfactant concentration. The effect of surfactant concen¬
tration on the modeled size distribution and mean aggregate
size of the C12 nonionic surfactant is shown in Figures 3-9
and 3-10. Below the CMC, the model calculates a mean
aggregate size approximately equal to N-trans» present as the
initial plateau in figure 3-10. This value is meaningless,
however, since the concentration of micelles in this range

-12
X
en
O
— 13 -
-14 -
•15 -
-16 -
-17 -
•18
-19
-20
200
5, N=694
1 ono
Aggregate Size
ii^
H
Figure 3-9. Effect of total surfactant concentration on the
aggregate size distribution of the C12 nonionic micelle at 298.15
Kelvin.

3200
3000
2800
2600
2400
2200
2000
1800
1600
1400
1200
1000
800
600
400
200
0
8.‘
F i a
-5 8.80E-5 9.20E-5 9.60E-5 1.00E-4 1.04E-4 1.08E-4
Total Surfactant Concentration, M.
3-10.
size
Effect
of the
of total surfactant concentration on the
Cl2 non ionic micelle at 298.15 Kelvin.
mean
to

43
of surfactant concentration is insignificant. The result is
merely the ratio of two extremely small numbers, since the
model always produces a finite value for the concentration
of micelles (equation (2.6)).
Measurement of properties such as surface tension, which
depend on the free monomer concentration, show that beyond
the CMC the free monomer concentration is virtually
unaffected by an increase in the total surfactant
concentration. The relationship between the free and total
surfactant concentrations, as predicted by the model, is
exhibited in Figure 3-11. At total concentrations lower
than the CMC, the majority of any surfactant added to the
solution exists as free monomers, while at concentrations
higher than the CMC the bulk of any additional surfactant
micellizes. The definition of the CMC used in this work, as
expressed in equation (2.21), quantifies the break between
these two behaviors of the solution. Figure 3-12 shows the
concentration derivative of equation (2.21) for the data of
Figure 3-11 with the CMC identified.
3.4 Chain Length and Temperature Effects
It is known that the hydrophobic nature of the alkyl
chain increases with its chain length, as evidenced by the
decreasing solubility in water of the increasingly long

Free Monomer Concentration,
B.72E-5
Total Surfactant Concentration, M.
Figure 3-11. Effect of total surfactant concentration on the free
monomer concentration of the C12 nonionic at 298.15 Kelvin. The
CMC of this system is 8.70E-5 M.

(wo + lo) e
Total Surfactant Concentration, M.
Figure 3-12. Effect of total surfactant concentration on the
concentration derivative of equation (21) for the C12 nonionic at
298.15 Kelvin. The dashed line shows that the value of this
derivative is 0.5 at the CMC of 0.7E-5 M.

46
chains (McAulliffe, 1966). With alkyl surfactants, this
increasing hydrophobicity promotes micellization. While no
significant aggregate formation is observed for chains of
six carbons or less, increasing chain length brings about
the formation of larger micelles at lower critical micelle
concentrations (Table 2).
The elements of the model which depend on chain length
are the hydrophobic free energy, taken directly from the
aqueous solubility, the curvature parameters, accounting for
the difference in micelle radius, and the conformational
free energy. Equations (3.14) and (3.15) show the
relationship of the parameters to the chain length. The
aqueous solubility of the alkane, which is chain-length
dependent, is used by the model in the fitting of the
parameters. The dependence of the aggregate size distribu¬
tion and free energy change on the surfactant chain length
is shown in Figures 3-13 and 3-14. The size distribution
broadens with increasing chain length, yielding a much
larger mean aggregate size with a less significant increase
in the CMC. The free energy of micellization becomes more
negative with increasing chain length, showing that micelle
formation by surfactants with more hydrophobic area is more
thermodynamically favored.

Aggregate Size
Figure 3-13. The effect of chain length on the aggregate size
distribution. Results are shown for the nonionic surfactants at
298.15 Kelvin and their respective CMCs (Table 2).
2000

I
O 200 A 00 600 800 1000 1200 1400 1600 1800 2000
Aggregate Size
Figure 3-14. The effect of chain length on the free energy of
micellization. Results are shown for the nonionic surfactants at
298.15 Kelvin and their respective CMCs.

49
Just as the aqueous solubility of an alkane decreases
with increasing temperature, the CMC of the corresponding
surfactant decreases and the mean aggregate size increases.
Again, by incorporating the solubility data into the model,
the proper temperature dependence is given to the
parameters. The temperature dependence of the parameters is
in the form of equations (3.12) and (3.13). The dependence
of the aggregate size distribution and free energy change on
temperature is shown in Figures 3-15 and 3-16. The effect
of increased temperature is similar to that of increasing
the surfactant chain length: a broadened size distribution
and a more negative free energy of micellization. But while
the first increase of ten degrees Kelvin has a much more
pronounced effect on the free energy change (and hence the
size distribution) than the next increase of ten degrees,
the mean aggregate size is more affected by the latter.

Aggregate Size
Figure 3-15.
distribution.
at its CMC.
The effect of temperature on the aggregate size
Results are shown for the CIO nonionic surfactant

-6.4
-7.6 H 1 1 1 1 1——i 1 1 1 1 1 1 1 1—~i 1 1 i i I
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Aggregate Size
U1
H
Figure 3-16. The effect of temperature on the free energy of
micellization. Results are shown for the C]0 nonionic surfactant
at its CMC.

CHAPTER 4
MOLECULAR DYNAMICS SIMULATION OF SURFACTANT MICELLES
4.1 Background
The development of a predictive model of micelle
behavior, such as that of Chapter 2, requires a knowledge of
the structure of the aggregate in order to describe its
thermodynamic properties. The calculation of a surface free
energy contribution necessitates certain assumptions about
micelle size, shape, and surface roughness. The solvent
interface must be modeled and solvent penetration must be
estimated. Calculation of the conformational contribution
to the free energy calls for details of the molecular
conformations in the free and aggregated states. The head
group contribution depends on the average positions of the
head groups.
While experimental results are the most desirable basis
for thermodynamic models, measurements of the structure of
micelles are limited in the information they can provide.
Therefore, current models of micelle behavior, including
that derived in Chapter 2, are based primarily on
theoretical considerations.
The most informative techniques for studying micelles in
solution are spectroscopic. Light scattering, NMR, and
52

53
small-angle neutron scattering (SANS) have been used
extensively to study micelles. However, results of these
experiments often conflict, as in the question of whether
there is solvent penetration into the micelle core, and
their description of the micelle surface and internal
structure is incomplete (Tabony, 1984).
Quasi-elastic light scattering has been used effectively
to determine micelle size and shape (Corti and Degiorgio,
1981a, 1981b) and Raman light scattering has been used to
investigate chain conformations (Kalyanasundaram and Thomas,
1976). The absence of solvent from the micelle core was
determined with NMR techniques (Mitra et al., 1984), though
the opposite had been reported earlier (Menger, 1979). The
most promising experimental means available of studying
micelle structure is SANS. It has been used to study the
micelle surface (Hayter and Zemb, 1982), solvent penetration
(Tabony, 1984; Cabane et al., 1985), and chain conformations
in the micelle interior (Bendedouch et al., 1983a, 1983b).
The spectra generated by the above methods must be
carefully interpreted to yield correct information about the
micelle surface and interior. It is in this interpretation
of the results, particularly in accounting for intermicellar
effects, that inconsistencies arise (Cabane et al., 1985).
Computer simulation is an alternative means to answer
questions raised by the experimental measurements. For a

54
given model of surfactant and micelle behavior, simulation
can provide exact quantitative descriptions of micelle
structure.
Simulations of micelles have been conducted by both
Monte Carlo (Haan and Pratt, 1981a, 1981b; Owenson and
Pratt, 1984) and molecular dynamics (Haile and O'Connell,
1984; Woods et al., 1986; Jonsson et al., 1986) methods.
Thus far, results of the simulation studies have been
promising. Their contribution to a better understanding of
micelle structure has been limited only by the uncertainties
about the model micelle.
Monte Carlo calculations are carried out on a lattice of
sites. Movement of a molecular segment from one site to
another is allowed or disallowed based on the energy change
associated with the move. The restriction of segment
positions to the lattice sites limits the possible
conformations available to the molecules. To allow more
possible conformations, the number of lattice points, and
hence the computing effort, must be increased. This
restricts how far such a simulation can go in accurately
representing reality; in addition, the Monte Carlo method
precludes the opportunity to study the dynamics of a system
by simulation.

55
The molecular dynamics method provides a continuum of
sites for molecular positions and is limited only by the
accuracy of the forces built into the model. In the present
work, molecular dynamics models were developed based on the
work of Haile and coworkers (Haile and O'Connell, 1984;
Woods et al., 1986) to investigate the effects of surfactant
chain length, head group mass, and head group size.
4.2 The Molecular Dynamics Method
The molecular dynamics approach to computer simulation
consists of summing the forces on each particle in the
system and solving Newton's equations of motion for the
resultant positions and momenta (Haile, 1980). The problem
of constructing a realistic model system then becomes one of
supplying appropriate models for the intermolecular and
intramolecular potentials. It is assumed that a given
particle's interactions with its neighbors are pairwise
additive. Thus, the potential between two particles at time
t is of the form
í/(rí/(í)) = í/(r1.(0.r,(0) (4.1)
The force between the two particles is given by
?(ri;(0) = -V£/(r0.(0) (4.2)
and the total force on the particle of interest is the sum
of the forces resulting from interactions with all other
particles:
(4.3)

56
The resulting acceleration and velocity of the particle are
obtained from Newton's Second Law of Motion, relating force,
mass, and acceleration:
(4.4)
The set of second-order differential equations for all
of the particles in the system is solved at each time step
of the simulation. The means employed in this work for
determining the new positions of all of the particles from
the forces at the previous time step is a fifth-order
predictor-corrector algorithm (Gear, 1971). This algorithm
consists of three procedures which are repeated at each time
step. The position and its first five time derivatives at
time (t+zit) are predicted for all of the particles by
Taylor's series expansion of their values at time t:
d2 r ,(í)(üí)2 d3 r
dt2 2~ + ~dt~3 3!~~
dV,(0(^04 | d5r,(i)(df)5
dt4 44 + dt5 5*!
(4-5)
dr ,(f) d2 r , (i) d3 r ,{t){At)2
dt dt2 ^ j dt3 2!
dV,(Q(zi03 | d5r,(Q(zi04
dt4 3! + dt5 4!
(4.6)
d
73 r,(t + At) =
dt
2!
3!
(4.7)

57
(4.8)
(4.9)
d5r,(0
dt5
(4.10)
The forces Fj^(t+Jt) are then evaluated at the predicted
positions. Finally, the predicted values of the position
and its derivatives are corrected according to the error
between the acceleration predicted by the series expansion
and the acceleration calculated from F^(t+Jt). These
calculations are carried out by the subroutines PREDCT,
EVAL, and CORR in Appendix D.
The simulations described in this work were carried out
at constant temperature. The temperature of the system is
related to the velocities of the particles by
(4.11)
The velocities of all particles are scaled at each time step
so that the temperature is maintained at a constant value.
Subroutine EQBRAT in Appendix D accomplishes this task.
To start a simulation, values of the positions are
required to evaluate the initial forces. Initial values of
the five derivatives of position are required by the fifth
order predictor-corrector algorithm. In these simulations,
initial positions were assigned according to a lattice of

58
sites and random velocities were assigned the particles
(subroutines INTPOS and INTVEL). The momenta were scaled so
that there was no net linear momentum of the system. The
initial accelerations are calculated by equation (4.4) and
the third and higher derivatives of position are assigned
initial values of zero, since there is no way of evaluating
them. The algorithm recovers rapidly from this initial
condition, arriving at proper values of all derivatives
within 10-20 time steps.
Since intermolecular potentials are significant over a
pair separation which is usually small relative to the
dimensions of the molecular system, computing effort can be
reduced by employing truncated potentials (Haile, 1980) and
maintaining a neighbor list (Verlet, 1967). In a simulation
where the majority of possible pair separations are so large
that the corresponding pair potentials will be very small,
it is more efficient not to calculate the potential for pair
separations which are beyond a certain value and do not
contribute significantly to the system properties. Instead,
these contributions are neglected during the simulation and
a correction is made to the system properties. A cutoff
distance, rc, is set such that the potential can be
neglected for r^j > rc. The intermolecular forces are
truncated at rc and shifted vertically, going to zero
smoothly at rc (Nicolas et al., 1979). The truncated

59
potential is obtained from this shifted force by equation
(4.2). This eliminates a step change in the potential and
force at r-^j = rc, which could have a disruptive effect on
the energy conservation of the simulated system. The
general equations of a shifted force, Fs(r), and its
potential, Us(r), are given below:
d,i/(r)
(4.12)
r < r
dr
(4.13)
r < r
C
By checking the value of r-^j and bypassing the calculation
of F(rjLj) for r-[j > rc, a significant savings in computing
effort can be realized.
Further savings can be made by not checking pairs which
have a low probability of r^j < rc. This is accomplished
through the use of a neighbor list. A particle (i) in the
system is surrounded by a sphere of radius rc. Other
particles (j) inside this sphere will interact with it,
since r-¡i < rc. From one time step to the next, particles
will enter and exit this sphere. Over some short period of
time, all of the particles which have interacted with the
central particle will lie inside a larger sphere whose
radius is If, over this brief period of time, only
those particles inside the larger sphere were checked, all
of the particles interacting with the central particle would

60
be found without checking all of the other particles in the
system. Through a neighbor list a record is maintained of
all the "neighbors" within a distance r^g-t of each particle
in the system. Only these neighbors are checked for r^j <
rc. The neighbor list is updated periodically based on the
value of rnst- and the dynamics of the system.
The molecular dynamics method represents a massive
computing effort for any system large enough to be of
interest. However, using the techniques described above to
improve efficiency, and with the development of supercomput¬
ers and parallel processors, it has become a practical tool
for quantitatively studying molecular behavior.
4.3 The Model Surfactant Molecule
The surfactant molecule studied in this work has a
linear alkyl chain of eight carbon groups. All of the
groups are considered to be methylenes. A polar head group
is attached to one chain end; head groups of different sizes
and masses were used in the simulations. The bond lengths
and angles along the chain are those of a normal alkane.
These are given in Figure 4-1.
Three types of intramolecular interaction take place in
the model surfactant molecule:
1. Bond vibration
2. Bond bending
3.
Bond rotation

61
Figure 4-1. The model surfactant molecule, consisting of an
eight-member alkane chain attached to a polar head group,
shown in the all-trans conformation. Bond angles and
lengths of the alkane chain are indicated. The bond to the
head group is head group dependent in the different
simulations conducted.

62
The bond vibration potential used is that of Weber (1978),
taken from a simulation of n-butane. It is a harmonic
potential about the equilibrium bond length. The potential
and the forces on the two groups are given by
(4.14)
(4.15)
(4.16)
where b¿ is the bond length between groups i and i+1, bQ is
the equilibrium bond length, and is the force constant.
The bond bending potential is also taken from the
simulation of Weber (1978). It is a harmonic potential
about the cosine of the equilibrium bond angle with forces
on the three groups forming the bond angle. For the angle e
formed by groups i, i+1, and i+2,
(4.17)
0, =-yb cos0O — cos0,
9t = yb cos0o-cos0,
(4.18)
(4.19)

63
^2(0J= yb(cos0o-cos0,.)
(4.20)
The model surfactant molecule is comprised of eight
bonds connecting its nine groups. Rotation about any of the
inner six bonds results in a change in the conformation of
the molecule and a change in its potential energy. A set of
four groups in sequence along the chain molecule contains a
dihedral angle formed by the three bonds connecting the
groups. This angle is a measure of rotation about the
middle bond, and the potential energy is a function of the
dihedral angle. The bond rotation potential chosen for
these simulations is that of Ryckaert and Bellemans (1975):
£/(0) = yr( 1 .1 16 - 1 . 462 cos 0 - 1 .578 cos2 0 + 0.368 cos3 0
+ 3.156cos40 + 3.788cosj0)
(4.21)
where

potential is illustrated in Figure 4-2. The force on each
of the four groups resulting from this potential and
equation (4.2) has been derived by Woods (1985) and the
resulting calculations are carried out by subroutine RYFOR
in Appendix D.
The intramolecular potentials account for all of the
interactions among adjacent groups on a molecule as they are
taken two (vibration), three (bending), or four (rotation)
at a time. Two groups on the same molecule which are
separated by more than three intervening groups do not

Dihedral Angle//?,
/ \
degrees
X
o\
if*
Figure 4-2. The bond rotation potential as a function of the
dihedral angle formed by four consecutive groups along the
surfactant chain. The diagrams below the plot depict the relative
orientations of the second and third groups and their attached
hydrogens (thin lines) and methylenes (heavy lines).

65
interact through any of these three potentials, but may
experience the intermolecular potential if they become
sufficiently close to one another.
4.4 The Model Micelle
The micelle used in the simulations contains 24
surfactant molecules. The individual groups of the
molecules interact through intermolecular potentials, and
the micelle is surrounded by a spherical shell which models
the solvophilic effect on the head groups and the
solvophobic effect on the chains. The intermolecular
interactions which take place in the micelle are
1. Head group-shell
2. Alkane group-shell
3. Head group-head group
4. Alkane group-alkane group
5. Head group-alkane group
The shell does not attempt to explicitly model the
solvent molecules. To do so greatly increases the computing
requirements of the simulation and a previous attempt at
such explicit modeling (Jonsson et al., 1986) did not show
it to be a clear improvement over the present poten¬
tial-field model. Rather, the solvophilic and solvophobic
natures of the head group and alkane group are modeled in a
simple manner intended to keep the head groups close to the
surface of the micelle and prevent the chains from extending

66
out from the micelle. This simulates the minimum
chain-solvent contact of accepted experimental results
(Tabony, 1984).
The head group interaction with the shell is through a
harmonic potential about an eguilibrium radial position:
^(r,)=yh(r,-r0)2 (4.22)
?(r,)“-2yfc(ri-r0)^ (4.23)
' i
where r¿ is the position of the head group and rQ is the
equilibrium position. The force constant is yft.
The alkane group interaction with the shell is through a
repulsive potential:
= (4-24)
V ' iw J
?(/;„)= 12e
(rm)12F,
(4.25)
[l~ 13 r««"
where r^w is the quantity rw-r¿, rw being the radial
position of the spherical repulsive shell, and rm and e are
the radius and energy of minimum potential for the
alkane-alkane intermolecular potential.
The interaction between two head groups is described by
a potential comprised of both dipole (r-3) and soft-sphere
(r-12) repulsions:
(4.26)

67
This potential is designed to spread the head groups apart
on the exterior of the micelle. The force between a pair of
head groups is obtained by applying equation (4.2) to
equation (4.26) .
3 r3hh
12 r
12
hh
r_U
13 r
ij J ' U
(4.27)
Applying equations (4.12) and (4.13) to use the shift-
ed-force potential for r^j (4.28)
^s(ro) = e<
3 r3hh
12 r'h2h | r jj
r r c
3i ^y.12r^12
The intermolecular potential between two alkane groups
is a (6,9) form of the Mie (m,n) potential (Reed and
Gubbins, 1973):
(4.30)
(4.31)
(4.32)

68
f,(r„)-18e
(4.33)
This potential is also used for the interaction between a
head group and an alkane group, except that the radius of
minimum potential, rm, is adjusted to account for the
difference between the diameter of the head group and that
of the alkane group:
head-alkane
m
(4.34)
r
Figure 4-3 illustrates the intermolecular potentials used in
the model micelle, and Table 4 gives the values of the
parameters used in the intramolecular and intermolecular
potentials.
The model micelle is assembled by initially placing the
24 molecules with their head groups evenly spaced over the
surface of a sphere of twice the expected micelle radius.
Each chain is directed radially inward, in the all-trans
conformation. With the bond rotation force constant reduced
to one-tenth of its normal value to facilitate chain packing
during startup, the simulation is begun. The radius of the
confining sphere is reduced every ten time steps until the
appropriate radius is achieved. This is the value that
would give the density of the analogous liquid alkane,
adjusted to a pressure which fluctuates about zero with an
amplitude of less than ten atmospheres. After this, the

69
Figure 4-3. The intermolecular potentials for segment-seg¬
ment (6,9), segment-shell (12), and head-head (3,12). A
value of 419 Joule/mole was used for e; both rm and r^ were
assigned values of 4 Angstroms.

70
Table 4
Parameter Values for Potentials Used in Simulations
Potential
Parameter
Value
Units
Bond vibration +
y„
9.25x10s
a
Joule/A2/mole
t>o
1.539
a
Angstrom
Bond bending +
yb
1.3X105
a
Joule/mole
112.15
a
degree
Bond rotation +
yr
8313
b
Joule/mole
Head-shell
y„
785
c
J oule/A2/mole
4
c
Angstrom
Segment-shell
e
419
Joule/mole
rm
4
Angstrom
Head-head
£
419
Joule/mole
rhh
*
Angstrom
Segment-segment
£
419
a
Joule/mole
rm
4
a
Angstrom
Head-segment
£
419
Joule/mole
rh-a
1 ( \
Angstrom
2I r m + rhh
a Weber (1978)
k Ryckaert and Bellemans (1975)
c Woods (1985)
* Parameter varies with head group size.
See also Table 6.

71
rotational force constant is restored to its desired value
and the system is allowed to equilibrate. When there is no
net drift in the energy of the system or in the fraction of
trans bonds, equilibrium is considered to have been reached.
The simulation is continued, periodically saving the
positions and velocities of all of the groups for subsequent
analysis.
The time step used in simulation must be small enough to
track the motions of the molecules and maintain stability,
yet large enough to avoid unnecessary computing effort. The
ideal time step can only be found by trial, which was
conducted on a simulated liquid alkane system by Weber
(1978). That result, Ait = .002 psec., is the basis for the
time steps used in the present work.
4.5 Summary of Computer Simulations
The following simulations were carried out on the model
micelle:
1. Head group mass and size same as chain segment.
2. Head group mass greater than segment, size the same.
3. Head group mass and size greater than segment.
In addition, a fourth simulation was carried out on a
hydrocarbon droplet of 24 nine-segment molecules by setting
the head group to the size and mass of a methylene and
replacing head-head and head-shell interactions with segment
interactions. From these simulations the independent

72
effects of head group size, mass, and solvophilic nature can
be observed, and by comparison with comparable simulations
of dodecyl surfactant micelles (Woods et al., 1986) the
effect of chain length can be evaluated. A brief summary of
the simulations is given in Table 5.
To carry out the first two simulations, 24 model
surfactant molecules were placed with their head groups on a
sphere of radius 24 Angstroms and their chains directed
radially inward in the all-trans configuration. The groups
were given initial velocities, the rotational force constant
was reduced to one-tenth of its normal value, and the
simulation was begun. Every ten time steps, the radius of
the confining sphere was decreased by .01 Angstrom. This
scaling down of the model micelle was continued until the
radius of the spherical shell reached a value of 12.00
Angstroms, slightly higher than the radius of an analogous
hydrocarbon droplet. Ten-thousand time steps were run at
this radius to allow the micelle to recover from the scaling
down procedure and observe the pressure at the shell. The
pressure fluctuated about zero with an amplitude of
approximately two atmospheres and analysis of the positions
of the groups revealed no effect of the initial conditions.
This model micelle was chosen as the starting point for
simulation 1.

Simulation
Head Group
Diameter
Summary of
Head Group
Mass
Table 5
Simulation
Micelle
Radius, A.
Computations
Equilibrium
Run,psec.
Time Step,
psec.
CPU Usage,
sec.
1
1
1
12.000
79
1.98X10-3
6719
2
1
7
.12.000
99
1.9 8 X10 “ 3
8602
3
2.45
7
12.800
28
1.40xl0“3
8 699
4
1 a
1 a
11.928
119
1.98x10“3
8342
a Hydrocarbon droplet, head group interactions same as chain segments.
Note: Head group diameter and mass are relative to chain segment (methylene.)

74
Simulation 2 was started from this same model micelle,
after increasing the mass of all head groups by a factor of
seven and allowing 10,000 time steps for equilibration.
This mass was chosen to represent a common ionic head group,
the sulfate ion. Only the mass of the head group was
changed; all of the intermolecular and intramolecular
interactions remained the same as in the previous
simulation. The expected result of this would be simply a
change in the dynamics of the head group. The program used
to compute simulations 1 and 2 is listed in Appendix D.
In simulation 3, the change in the head group was more
extensive. An attempt was made to represent the sulfate
head group in all of its intermolecular and intramolecular
interactions. The mass of the group remained at seven times
that of a methylene, while the diameter was increased to
2.45 times that of a methylene. The equilibrium bond length
and bond angle for the head group were changed, as were the
force constants for bond vibration, bond-angle bending, and
bond rotation involving the head group. These values are
given in Table 6 and compared with the methylene values.
The derivation of the head group intramolecular interaction
parameters based on the work of Muller and coworkers (Muller
and Nagarajan, 1967; Muller et al., 1968), Cahill et al.
(1968), and Blukis et al. (1963) is given in Appendix E.

75
Table 6
Bond Parameters of "Sulfate" and "Methylene" Groups
Parameter
"Methylene" Value
"Sulfate" Value
Units
y„
9.25X105
2.7xl04
Joule/A2/mole
1.539
2.6
Angstrom
yb
1.3xl05
9.lxlO5
Joule/mole
112.15
140
degree
yr
8313
20000
Joule/mole

76
Due to the increase in the size of the head group, the
range of head group interactions was significant in
comparison to the size of the micelle and truncation of the
potentials could no longer be justified. Therefore,
neighbor-listing and truncation of intermolecular potentials
were not employed in this simulation. It was also found to
be necessary to decrease the time step to maintain stability
in the energetics of the system. It is for these two
reasons that the computing efficiency (simulation time per
CPU time) of simulation 3 is dramatically lower than the
other simulations (Table 5). The program used to compute
simulation 3 is listed in Appendix F.
Simulation 4, the hydrocarbon droplet, was conducted in
the same manner as simulation 1, except that the head group
was replaced by a chain segment (all segments have the same
properties to simplify computation) in its interactions with
the shell and other groups. The radius was scaled down to
11.928 Angstroms to achieve the liquid hydrocarbon density
of .7176 g/cc. Since there was no sorting out of the head
groups for separate interactions, the efficiency of this
simulation was 18 percent greater than that of the first two
simulations.
The simulation computations were conducted on a Control
Data Corporation (CDC) Cyber 205 computer at the

77
Supercomputer Research Institute (SCRI) in Tallahassee,
Florida. Access to the SCRI facilities was provided through
a grant from the United States Department of Energy.
All of the simulation programs were modified for the CDC
FORTRAN 200 Vector-optimizing compiler. By employing
parallel processing technigues when possible, this compiler
provides very efficient operation from code which is highly
compatible with ANSI standard FORTRAN. One important
difference lies in the default word length of the Cyber 205.
Real numbers on this computer are eight bytes in length,
compared to four bytes on most other computers. To achieve
eight-byte precision in standard FORTRAN, the REAL*8
variable type is used. The programs in Appendices D and F
achieve this level of precision on the Cyber 205 with
default variable-type coding.
Further detail about the simulations, along with the
results of their analysis, is given in the next chapter.

CHAPTER 5
RESULTS OF MOLECULAR DYNAMICS SIMULATION
In this chapter the results of structural and dynamic
analyses performed on the output of the simulations are
reported. Certain of these results are reported as
time-averaged properties; others are given as the change in
a property with time. The former are averaged over the
entire length of the eguilibrium run (Table 5); the latter
appear in the figures over a common time period for ease of
comparison. The simulations are referenced by number, Run 1
being the case with head groups of the size and mass of a
chain segment (methylene); Run 2 the case with head groups
of the same size as a chain segment, but a mass seven times
greater; Run 3 the case with head groups of 2.45 times the
size and seven times the mass of a chain segment, and a
revised head group intramolecular potential; and in Run 4
the head groups replaced by chain segments to model a
hydrocarbon droplet.
5.1 Mean Radial Positions of Groups
The measurement of mean radial positions of the groups
on the surfactant chains is one of the simplest, yet most
78

79
telling, evaluations of micelle structure that can be made
on simulation output. While this measurement cannot be
obtained reliably for all groups by experiment, the position
of groups is a fundamental attribute of structural models of
micelles (Gruen, 1981). The mean radial position for group
i (i=l for head, i=9 for tail) is given by
(5.1)
The quantity N^(r) is the number of occurrences of the
center of a group i at a radial position (relative to the
center of mass of the micelle) r at time t and the angle
brackets denote the average over the time period of the
simulation. The unsubscripted N is the number of molecules
in the system, equal to the sum of N-^(r) over all values of
r and i. The standard deviation corresponding to this mean
is given by
2
(5.2)
The mean radial positions of the centers of the nine
groups bracketed by one standard deviation in each radial
direction are plotted for the four simulations in Figures
5-1 through 5-4. Assuming a normal distribution of a
group's position over time, the range shown for each group
in the figures includes 68 percent of its positions, while
twice this range includes 95 percent.

o<
in
"xj
o
cr
20 -T
19 -
18 -
17 -
16 -
15 -
14 -
13 -
10 -
9 -
8 -
7 -
6 -
5 -
4 -
3 -
2 -
1 -
Head 2
12 -
11 -
3
1 1 1 i j—
4 5 6 7 8 Tail
Group
03
O
Figure 5-1. Mean radial positions of the nine groups of the
surfactant molecules of Run 1. The vertical bars show one standard
deviation on either side of the mean.

20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
oo
! ! [—
Head 2 3
4 5 6
Group
1 r~
7 8 Toil
re
5-2
Mean radial positions of the nine groups of the
molecules of Run 2. The vertical bars show one standard
on either side of the mean.

o<
cn
D
XJ
o
íY.
co
to
Figure 5-3. Mean radial positions of the nine groups of the
surfactant molecules of Run 3. The vertical bars show one standard
deviation on either side of the mean.

20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
! 1 1 1 f 1 1 7 J-
Head 2 3 4 5 6 7 8 Toil
Group
ire 5-4. Mean radial positions of the nine group
ocarbon molecules of Run 4. The vertical bars s
idard deviation on either side of the mean.
s of the
how one

84
Although no group has its mean position in the center of
the micelle, this does not indicate a void within the
micelle. Rather, it is an artifact of the coordinate system
chosen. Each chain segment has a diameter of 3.56
Angstroms, and subsequently excludes a spherical volume of
this diameter to any other group. Since the spherical
coordinate system provides an available volume which
decreases with r, the excluded volume effect results in low
values of N¿(r) near the center of the micelle. Thus, no
group has its average position near the center, but the
center is always within the excluded volume of one of
several different groups. A range of three standard
deviations about the mean includes approximately all of a
group's positions, and, in Figures 5-1 through 5-4,
approaches within a segment diameter of the center for
several chain segments in each simulation.
The mean position results for Runs 1 and 2 are virtually
identical. This is to be expected, since the two models
differ only in the mass of the head group, a difference that
should manifest itself in the dynamic behavior of the
system, but not in a time-averaged result. In these two
simulations, the head groups are found at the surface of the
micelle with a relatively small deviation from the mean.
Progressing along the chain toward the tail, the mean radial
positions decrease and the deviations remain roughly

85
constant through group five. The mean positions of the last
four groups are at approximately the same radius, but toward
the tail the deviations increase, a consequence of the
excluded volume effect near the center of the micelle.
The results for Run 3 are markedly different. Figure
5-3 shows larger values of mean positions for the chain
segments and greater deviations for the head group and its
adjacent three groups. This simulation differs from Run 2
in the size of the head group and the nature of the
head-chain bond. With a diameter of 8.72 Angstroms, the
larger head group creates a steric effect which brings about
greater disorder in the micelle. The head groups are spread
over a greater range of radial positions, altering the
positions of the chain segments from those observed in Runs
1 and 2.
Figure 5-4 shows the results for Run 4, the hydrocarbon
droplet. The mean positions and deviations are symmetric
about group 5, the center of the molecule. This is the
proper result, since the two ends of a chain are
indistinguishable from one another. The chain ends have a
slightly larger mean position and deviation, a result of the
greater mobility of the chain end relative to the interior
groups of the chain. Overall, the means and deviations are
quite uniform, indicating a random arrangement of the

86
molecules within the droplet. Again, the excluded volume
effect places the mean positions of all groups over two
segment diameters away from the center of the droplet.
In a simulation of a micelle of forty dodecyl surfactant
chains, the same shell model was used to surround the
micelle and the head groups were identical to Run 1 in size,
mass, and potentials (Woods et al., 1986). Comparison with
this simulation reveals the effect of chain length on the
micelle model. In Figure 5-5, the average radial positions
of Woods' simulation are shown next to those of Run 1,
pairing the groups of Run 1 with the nine groups furthest
from the head group of the longer chain's thirteen. The
dodecyl micelle being larger than those of the present work,
its average radial positions are greater. Qualitatively,
however, the trend observed in the radial positions of the
groups and their standard deviations is quite similar in the
two simulations.
In Figure 5-6, the average radial positions of Run 1 are
compared to those resulting from a statistical model of 30
twelve-member chains with one end of each fixed at the
surface (Gruen, 1981). In this model, all possible
configurations of the surfactant chains were sampled.
Though Gruen's micelle is again larger than that of Run 1,
the results of the two models show notably similar radial
position behavior on a qualitative basis.

19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
—! ! , 1 1 1 f
n-12 n-10 n-8 n-6
Group
I 1 1 1 T
n—4 n—2 ri
(Tail)
5-5. A comparison of the mean radial positions of groups in
and those obtained in a simulation of 40 dodecyl surfactant
les by Woods et al. (1906).

o<
CO
13
XJ
O
ct
20
19 -
18 -
17 -
16 -
15 -
14 -
13 -
12 -
11 -
10 -
9 -
8 -
7
6 -
5 -
4 -
3 -
2 -
1 -
0
n T
11 T
ID
U
—i 1 r
n-12 n—10
—i 1 1—
n-8 n-6
—i 1 1 1 r—
n—4 n-2 n
(Tail)
Group
Figure 5-6. A comparison of the mean radial positions of groups in
Run 1 and those obtained in a statistical model of 30 twelve-
segment chains proposed by Gruen (J901).
00
CO

20 1
19 -
18 -
17 -
16 -
15 -
14 -
13 -
il
n
li
II
ll
il
ll
11
o
ll
11
ll
7
n
11
oo
10
6 -
5 -
4 -
3 -
2 -
1 -
0
n-8 n—7 n—6 n-5
n—3 n-2 n-1
n
(rail)
Group
Figure 5-7. A comparison of the mean radial positions of groups
in Run 1 and those obtained in a simulation of 1.5 sodium octanoate
molecules by Jonsson et al. (19(16).

90
A model micelle of 15 sodium octanoate molecules, along
with surrounding molecular water, was the subject of another
molecular dynamics simulation (Jonsson et al., 1986). The
sodium carboxylate head group was explicitly modeled and
surrounding water molecules were included in the simulation.
Jonsson found it necessary to reduce the charge on the head
groups to produce acceptable results. Run 3, with the model
sulfate head group, is compared to Jonsson's reduced charge
simulation in Figure 5-7. Accounting for the difference in
the size of the two micelles, the radial positions of the
chain segments compare favorably.
5.2 Probability Distributions of Group Positions
The positions of groups within the micelle can be
studied in more detail by evaluating, at each radial
position, the probability of finding a given group at any
given time. The probability density function used in this
analysis is the time-averaged number of occurrences of the
center of a group i within a spherical volume element
centered at r:
P,(r) =

4/rr 2 Ar
(5.3)
Evaluating this function over all values of r generates a
probability density distribution for each molecular group.
The results of this analysis for each of the four
simulations are given in Figures 5-8 through 5-11. The

0.015
0.014
0.013
0.012
0.011
0.010
0.009
m
I .
o< 0.008
0.007
0.006
0.005
0.004
0.003
0.002
0.001
0.000
0 2 4 6 8 10 12 14 16 18 20
o
Radius, A.
' D
H
Figure 5-8. Probability density distributions of the nine groups
of the surfactant molecules of Run 1. Group 1 is the head group
and group 9 is the tail group.

0.015
Radius, A.
Figure 5-9. Probability density distributions of the nine groups
of the surfactant molecules of Run 2. Group 1 is the head group
and group 9 is the tail group.

0.015
0.014 -
0.013 -
0.012 -
0.011 -
0.010 -
0.009 -
ro
o< 0.008 -
c¿ 0.007 -
0.006 -
Radius, A.
Figure 5-10. Probability density distributions of the nine groups
of the surfactant molecules of Run 3. Group 1 is the head group
and group 9 is the tail group.

I
Figure 5-11. Probability density distributions of the five
indistinguishable groups of the hydrocarbon molecules of Run 4.
The label 1/9 denotes the terminal groups of the linear molecule
and group 9 is the tail group.

95
probability density function is indefinite at the center of
the micelle, but by choosing a finite value of Ar the
indefinite result is avoided. Some scatter is introduced in
the distributions at small r, however. Division by the
volume element eliminates the excluded volume effect that
produced no mean radial positions near the center of the
micelle, as described in Section 5.1. Rather, the
probability density distributions have peaks throughout the
micelle for the various groups.
As in the case of the mean positions, the results for
Runs 1 and 2 are quite similar and show the same degree of
order. The head groups in these two simulations remain near
the surface of the micelle. Moving along the chain toward
the tail, the peak for each group occurs closer to the
center and the distribution broadens. There is some
probability of finding each of the groups at the surface,
but there is no probability of finding the head group and
the adjacent three groups within a segment diameter of the
center. Two molecular configurations can lead to this
result. First, with the head group at the surface, the
chain is directed inward from the surface and then is bent
around such that its tail is at the surface. Second, a
molecule can lie along the surface. The output shows that
both of these configurations do occur in Runs 1 and 2.

96
The results of two analogous simulations of micelles of
40 and 52 dodecyl surfactants (Woods et al., 1986) give
distributions which are, except for micelle radius and
surfactant chain length, remarkably similar to the results
of Runs 1 and 2. These simulations also exhibited
comparable average radial position behavior in the previous
section. For the micelle model used in these four
simulations, the difference in surfactant chain length,
other than increasing the size of the micelle, has a
negligible effect on the arrangement of chain segments in
the micelle interior.
Figure 5-10 better displays the disorder present in Run
3 which was evidenced by the mean positions in Figure 5-3.
The distribution of head group probabilities has two major
peaks: one at the surface, and one approximately one head
group diameter from the surface. Groups two through four
have roughly the same peak arrangement, but the distribu¬
tions become more random toward the end of the chain. With
the larger head group size, the excluded volume effect at
the surface forces one or more head groups into the micelle
interior and disrupts the orderly arrangement of chain
segments found in Runs 1 and 2.
Figure 5-11 shows the distributions for Run 4, the
hydrocarbon droplet. Since the symmetry of the chains was
demonstrated in the previous section, the combined

97
probabilities for the indistinguishable pairs of groups are
plotted. All groups have a finite probability of being
found anywhere between the center and the surface of the
droplet, and, within statistical uncertainties, the
distributions indicate a random arrangement of the
molecules.
While the probability density function serves well to
remove the effect of excluded volume at the center of the
micelle in evaluating group positions, the r2 dependence
causes some difficulties in interpreting the results. The
distribution curves do not enclose equal areas, and they do
not correspond to the mean radial positions of Section 5.1.
In that section it was assumed that the distribution of
Nj^(r) was normal in assigning a percentage of the
distribution to the standard deviation. By multiplying the
probability density function by r2, a distribution is
generated which will enclose equal areas for each group, and
will serve to evaluate the assumption of symmetry made in
the previous section. It is apparent from the probability
densities that the three groups adjacent to the head group
follow the head group while the three or four groups
adjacent to the tail follow the tail. By plotting the
quantity p, r2 for only the heads and tails in Figures 5-12
through 5-15, clarity is improved without excessive loss of
information.

0.7
Figure 5-12. The distributions of probability density multiplied
by the square of the radial position for the heads and tails of the
surfactant molecules of Run 1. The plotted curves enclose equal
areas and correspond to the mean radial positions of Figure 5-1.

0.7
Figure 5-13. The distributions of probability density multiplied
by the square of the radial position for the heads and tails of the
surfactant molecules of Run 2. The plotted curves enclose equal
areas and correspond to the mean radial positions of Figure 5-2.
V)

0.7
Figure 5-14. The distributions of probability density multiplied
by the square of the radial position for the heads and tails of the
surfactant molecules of Run 3. The plotted curves enclose equal
areas and correspond to the mean radial positions of Figure 5-3.
100

0.7
0.6 -
0.5 -
i
Figure 5-15. The distributions of probability density multiplied
by the square of the radial position for the terminal groups of the
hydrocarbon molecules of Run 4. The curve corresponds to the mean
radial positions groups 1 and 9 in Figure 5-4.
101

102
These plots for Runs 1 (Figure 5-12) and 2 (Figure 5-13)
show the head groups to have a very nearly normal
distribution, centered at the mean radial position, with a
range of less than three standard deviations in each
direction along the abscissa. The tails have broader
distributions, not quite as symmetric, but with a range
within three standard deviations in each direction. The
tails have a finite probability of radial positions well
within a segment diameter of the center, confirming the
statistical assumptions of the previous section. •
The p,r2 plot for Run 3 (Figure 5-14) reveals a
symmetric peak for the head group, centered just above the
mean position. It is broader than those of Runs 1 and 2,
with a smaller peak in the interior of the micelle, as
discussed above. A comparison of the areas of the two peaks
indicates that, on the average, two head groups can be found
in the interior of the micelle. The tail distribution is
skewed toward the outer portion of the micelle since fewer
groups can be accommodated in the center than in the outer
portion. The tail distribution does, however, show finite
probability of radial positions within a segment diameter of
the center. The result for the chain ends of Run 4 is
similar to that of the tail in Run 3. There is probability

103
of end groups occurring throughout the hydrocarbon droplet,
with the curve skewed toward the outer portion of the
droplet.
Simulations of a micelle of 15 sodium octanoate
molecules in water have been carried out by groups led by
Jonsson (Jonsson et al., 1986) and Watanabe (Watanabe et
al., 1988). Jonsson reduced the charge on the head groups,
while Watanabe used the Ewald method to handle the long
range electrostatic effects. The chain length of the
surfactant monomer of these simulations differs from that of
the present work by only one chain segment. By scaling the
probability density function with respect to the aggregate
size, the result from these previous simulations can be
compared to that of this work. Such a comparison for the
tail groups is made in Figure 5-16. The larger micelle of
Run 3 exhibits a distribution with a higher mean value of
the radial position. It is not symmetric, but skewed toward
the surface of the micelle. The tails of Run 3 are
distributed over a range egual to that of the other two
simulations.
The experimental technique of small angle neutron
scattering (SANS) can be used to obtain information which
may be indicative of micelle structure. Scattering

0.045
pr
Ñ^3
0.040 -
0.035 -
0.030 -
2 0.025 -
0.020 -
0.015 -
0.010 -
0.005 -
0.000
Run 3
Jonsson
Watnnabe
Figure 5-16. Probability distributions of the tail group position
in three micelle simulations. Results are scaled to remove the
impact on the distribution of the aggregate number, N. The solid
line is the result obtained for Run 3, the dashed line for Jonsson
et al. (1986) and the dash-dot line for Watanabe et; al. (1987).
104

105
measurements were made on aqueous solutions of lithium
dodecyl sulfate above the critical micelle concentration
(Bendedouch et al., 1983a). The results of this study
showed the alkyl core to have more order than the liquid
state, yet less than the all-trans model, and to be free of
water. One result of the SANS study is the Fourier
transform of the scatterinq amplitude due to the
distribution of methyl tails. This can be written as
(5.4)
where A(Q) is the Fourier transform of the scattering
amplitude
pMe(r) is the probability density of the tail group
position
Q is the scattering vector
Plots of A(Q)/A(0) for the SANS result and Runs 1
through 4 are given in Figure 5-17. Again the results for
Runs 1 and 2 confirm nearly identical structure. A plot of
A(Q)/A(0) for a uniform distribution of tails through the
micelle nearly coincides with the result for Run 4, the
hydrocarbon droplet. As the size of the aggregate
increases, the A(Q)/A(0) curve is shifted to lower values of
Q. The order present in the aggregate affects the shape of
the curve, as seen in the curves for the highly-ordered Run
1 and the uniform result without order. The SANS result,

Figure 5-17. Fourier Transform of tall group density lor Ihinn 1-4,
the SANS data of Itondodouch ot at. (l!)M3n), and the reoult lor >i
uniform distribution oi tails throughout t tie aggrognto.

107
for an aggregate of approximately 78 dodecyl chains, falls
off much more rapidly than the result for the much smaller
micelles of Runs 1 and 2. Its shape, however, is quite
similar to the curves for Runs 1 and 2, suggesting that the
structure of Runs 1 and 2 may be more like the experimental
result than that of Run 3.
The A(Q)/A(0) curve for Run 3 falls off more rapidly
than those of the other three simulations, due to the larger
size of the Run 3 micelle. Its shape is very much like the
Run 4 result. The disorder in structure of the Run 3
micelle may be greater than is indicated for this
experimental result, though it is difficult to make any real
conclusion regarding this since the two micelles are of such
drastically different size. Such a size difference could
affect internal structure, and, in fact, the experimentally
determined aggregate number of 78 for dodecyl chains yields
a nonspherical micelle.
5.3 Conformations of Chain Molecules
From the analysis of the positions of the groups along
the surfactant chain within the micelle, it is apparent that
the majority of the molecules are not in the all-trans
conformation of the simulation's initial condition. In
order to achieve the distributions of positions observed,
chain bending involving the transition of bonds from the
trans to the gauche conformation must occur. The

108
conformations of the chains, in terms of numbers of gauche
and trans bonds, gives further information about the
internal structure of the micelle.
In this analysis, a bond is considered to be in the
trans conformation if the cosine of its dihedral angle is
less than -0.5 (Weber, 1978; Woods et al., 1986):
cos < -0.5 (5.5)
Thus, a trans bond in this analysis has a rotational
potential between zero and the surrounding relative maxima
(see Figure 4-2).
The instantaneous percentage of trans bonds versus time
for each of the four simulations is given in Figures 5-18
through 5-21. With six dihedral angles per molecule, the
transition of one bond is represented by a change of 0.69
percent in these figures. Changes of this magnitude occur
almost continuously (within two femtoseconds), while changes
on the order of five percent take place over a range of five
to ten picoseconds. The fluctuations in trans percentage
have a mean value which is dependent on the simulation
model. The mean value for Run 1 is 68±2, for Run 2, 70±3,
for Run 3, 74±2, and for Run 4, 73±3. In their simulation
of 15 sodium octanoate molecules, Watanabe et al. (1988)
reported a value of 78±2 percent. Jonsson et al. (1986), in
a simulation of the same system, reported a value of 50±3
percent for their reduced charge model. In the

% Trans Bonds
100
90 -
80 -
T
4
_i | , | | | | p- -j | r~
8 12 16 20 24 26
Time, psec.
Figure 5-18. The percentage of surfactant molecule bonds of Run 1
which are in the trans conformation. The percentage is reported at
intervals of .0198 picosecond. The mean value is 67.7 percent with
a standard deviation of 2.2 percent.
109

% Trans Bonds
100
Figure 5-19. The percentage of surfactant molecule bonds of Run 2
which are in the trans conformation. The percentage is reported at
intervals of .0190 picosecond. The mean value is 70.1 percent with
a standard deviation of 2.6 percent.
110

% Trans Bonds
100
80
50
%
p— 1 -r -- -T—
,/a/" Wv,
- ' r r —r —r - r ' i r i
12 16 20 24 26
Time, psec.
Figure 5-20. The percentage of surfactant molecule bonds of Run 3
which are in the trans conformation. The percentage is reported at
intervals of .0140 picosecond. The mean value is 74.3 percent with
a standard deviation of 1.6 percent.
Ill

% Trans Bonds
100
90 -
80 -
70
60 -
50 -| 1 1 1 1 1 1 r i i i i i i i
0 4 8 12 16 20 24 28
Time, psec.
Figure 5-21. The percentage of hydrocarbon molecule bonds of Run 4
which are in the trans conformation. The percentage is reported at
intervals of .0190 picosecond. The mean value is 73.0 percent with
a standard deviation of 2.8 percent.
112

113
simulations of dodecyl surfactants (Woods et al., 1986), a
mean value of 72±3 percent for aggregates of both 40 and 52
surfactant molecules.
With the exception of the group led by Jonsson, all of
the above-mentioned simulations found a percentage of bonds
in the trans conformation roughly within one standard
deviation of one another. It is interesting to note that
the hydrocarbon droplet, even at liguid density, produced a
mean trans percentage in the range corresponding to the
micelles, indicating that no significant straightening
occurs when the head group is constrained near the surface.
Whether chain straightening occurs due to the spherical
geometry of the aggregate is not known, since no simulations
of hydrocarbon molecules of this length in the bulk liguid
or aqueous solution phases have been reported.
Figures 5-22 through 5-25 break the trans fraction down
by bond number, each plot showing the fraction of the
twenty-four bonds of that number which is in the trans
conformation. The bond numbering scheme sets Bond 1 as the
bond between the head group (or segment 1) and chain segment
2. Bond 2 is the bond between segments 2 and 3, and its
dihedral angle is formed by groups 1 through 4. The data
for these figures was sampled at one-tenth the frequency
used in the previous four figures, causing the plots to

Fraction of Trans Bonds
114
Figure 5-22. Fraction of trans conformations for each of
the rotational bonds of Run 1. Result is shown for every
hundredth time step.

Fraction of Trans Bonds
115
1.0
C.3
CL8
0.7
0.S
0^
0.4
1.0
C.9
O P
C.7
0.5
0.5
0.4
Ecnc 2
A aJ
\-r
/V
eona j
'S/'
Figure 5-23. Fraction of trans conformations for each of
the rotational bonds of Run 2. Result is shown for every
hundredth time step.

Fraction of Trans Bonds
116
1.0
C.9
C.S -i
0.7
“ene ¿i
Figure 5-24. Fraction of trans conformations for each of
the rotational bonds of Run 3. Result is shown for every
hundredth time step.

Fraction of Trans Bonds
117
Figure 5-25. Fraction of trans conformations for each of
the rotational bonds of Run 4. Result is shown for every
hundredth time step.

118
appear smoother. One significant result apparent in these
figures is the very high fraction of trans bonds near the
head group in Run 3. In this simulation the head group size
and bond rotational force constant were greater than in Runs
1 and 2.
Another indicator of micelle structure is the bond order
parameter, S(r), defined by
(5.6)
where e is the angle formed by the bond vector between
adjacent groups on a chain and the radial vector from the
aggregate center of mass to the center of the bond. The
parameter is averaged over all bonds whose centers are found
at a distance r and over time. An average over completely
random bond orientations would result in an order parameter
of zero.
Plots of S(r) are given for the four simulations in
Figures 5-26 through 5-29. In all cases little ordering of
bond orientations exists in the intermediate portion of the
aggregate between the center and the surface. In Runs 1, 2,
and 3, the micelle simulations, S(r) approaches unity at the
surface, indicating that the bonds near the surface tend to
parallel the radial vector. This also occurs at the center
of the micelle, where a group whose center is coincident

1.0
0.9
0.8
0.7
0.6
0.5
0.4
g °-3
U) 0.2
0.1
0.0
-0.1
-0.2
-0.3
-0.4
-0.5
0 2 4 6 0 10 12 14 16 10 20
O
Radius, A.
Figure 5-26. The distribution of the bond order parameter, S(R),
through the interior of the micelle of Run 1. A value of 1.0
indicates bonds aligned in the radial direction, while a value of
0.0 results from no preferential order in the bonds.
D n
Rj
n
n ct,
-a—Etr
â–¡
r-Flq rrfPctanjj fxFlrr|, jn
’ll
cP
iiu ruin.
119

Figure 5-27. The distribution of the bond order parameter, S(R),
through the interior of the micelle of Run 2. A value of 1.0
indicates bonds aligned in the radial direction, while a value of
0.0 results from no preferential order in the bonds.
120

Figure 5-28. The distribution of the bond order parameter, S(R),
through the interior of the micelle of Run 3. A value of 1.0
indicates bonds aligned in the radial direction, while a value of
0.0 results from no preferential order in the bonds.
121

1.0
Radius, Á.
Figure 5-29. The distribution of the bond order parameter, S(R),
through the interior of the droplet of Run 4. A value of 1.0
indicates bonds aligned in the radial direction, a value of
0.0 results from no preferential order in the bonds, and a value of
-0.5 indicates bonds perpendicular to the radial direction.
122

123
with the center of the micelle would require that S(r)=l for
a bond attaching it to another segment. The same behavior
of S(r) was found by Woods et al. (1986).
In Run 4, the hydrocarbon droplet simulation, S(r) goes
to -0.5 at the surface, indicating that most bonds at the
surface are perpendicular to the radial vector. Similar
behavior to the other simulations is observed in the
interior of the aggregate. The difference in the bond
orientations at the surface of the micelle and the
hydrocarbon droplet lies in the head group interaction with
the shell. In the micelle, forces act to maintain the head
groups at the surface while the chain segments are repelled
from the shell. This tends to orient the bond between the
head group and its adjacent segment perpendicularly to the
shell. In the droplet all of the chain segments experience
the same repulsive force at the shell, resulting in bonds
which lie parallel to the shell.
Gruen (1981) has obtained the bond order parameter as a
function of the carbon number from his statistical model.
Chevalier and Chachaty (1985) also express a result of their
NMR study of sodium mono-n-octyl hydrogen phosphate micelles
in this way. The bond order parameter as a function of
group number, S¿, can be approximated from S(r) by

124
S
N,(r)S(r)dr
/V, ( r ) d r
(5.7)
The approximation is due to the values being taken at
group centers while the S values are taken at bond centers.
Since the difference between the two can be no longer than
one-half the bond length, accuracy does not suffer greatly.
Comparisons between these two studies and Run 3 are made in
Figure 5-30. The NMR result was given as values relative
to that of the head group bond without a value for the head
group bond; in the figure the same head group value as Run 3
was used. The simulation result shows considerable radial
orientation of the head group bond, with groups removed from
the head group showing no preferential orientation. This
corresponds to the high trans fraction observed near the
head group in Figure 5-24. The NMR result shows a decrease
in radial orientation going from head to tail, but more
gradually than the simulation. The statistical model does
not show this trend in bond orientation along the chain.

125
Carbon Number
Figure 5-30. Bond order parameter by chain
segment. Results are shown for the statistical
model of Gruen (1981), the NMR study of
Chevalier and Chachaty (1985), and Run 3 of the
present work.

126
5.4 Shape Fluctuations
The shape of the model micelles and droplet can be
investigated through the moment of inertia tensor, I:
/ =
I XX
^ yx
xy
lXZ
Iy,
I XX
(5.8)
whose diagonal and off-diagonal elements are calculated from
the mass and position of each of the 216 particles in the
system as follows:
/„= Y.mv[y2r + z2v) * V'=1’2 216 (5.9)
V
I xy = ~Y.mvxvyv , v = 1,2,...,216 (5.10)
V
The eigenvalues of this tensor are the principal moments of
inertia, Ilf I2, and I3. In the case of all particles being
of equal mass, such as in Runs 1 and 4, a spherical micelle
or droplet would result in Ii=l2=I3- In Runs 2 and 3, where
the mass of the head group is different than that of a chain
segment, a spherical micelle with the head groups equally
spaced over the surface would yield equality of the
principal moments.
Figures 5-31 through 5-34 show the change over time of
the ratio of the maximum principal moment to the minimum
principal moment. Any deviation from the conditions given
above for equality of the principal moments will result in
this ratio being greater than unity. In all four of the
simulations, the ratio Imax/^min never attained a value of

2.0
1.9 -
1.8 -
1.7 -
Time, psec.
Figure 5-31. The ratio of the largest principal moment of inertia
to the smallest for Run 1. A value of unity would be necessary for
a spherical aggregate. The mean value of Imax/^min •2 8 with a
standard deviation of 0.07.
127

2.0
1.8 -
1.1 -
1.0 -| 1 1 1 1 1 1 1 1 r 1 r 1 1 1
0 4 8 12 16 20 2A 28
Time, psec.
Figure 5-32. The ratio of the largest principal moment of inertia
to the smallest for Run 2. A value of unity would be necessary for
a spherical aggregate. The mean value of llnax/^min 1*49 with a
standard deviation of 0.10.
128

2.0
1.9 -
1.8 -
Time, psec.
Figure 5-33. The ratio of the largest principal moment of inertia
to the smallest for Run 3. A value of unity would be necessary for
a spherical aggregate. The mean value of Iinax/I1T1in is 1.35 with a
standard deviation of 0.05.
129

2.0
1.9 -
1.8 -
1.7 -
1.6 -
'max
1.5 -
'min
1.4 -
1.0
i r
4
i r
—i 1 i~
12 16
20
24
28
Time, psec.
H
u>
o
Figure 5-34. The ratio of the largest; principal moment of inertia
to the smallest for Run 4, the hydrocarbon droplet. A value of
unity would be necessary for a spherical aggregate. The mean value
of Imax/Imin i-25 with a standard deviation of 0.06.

131
unity, despite the aggregate being confined in a spherical
shell. Rather, it fluctuates about a mean value slightly
greater than 1.0.
The result for Run 1 shows a mean value of Imax/Imin °f
1.28, with an approximate range of values from 1.1 to 1.4.
The large fluctuations are fairly regular, with a period of
about five picoseconds. Smaller fluctuations are minimal,
on the order of 0.05 on the Imax^min scale- The result for
Run 4 is quite similar to that of Run 1, with a mean of 1.25
and range from approximately 1.1 to 1.4. There is not a
regular frequency with which fluctuations occur, however.
Run 2 experienced fluctuations in the approximate range
of 1.2 to 1.7 with a mean value of Imax/Imin of 1-5. Large
fluctuations occurred with a period of about four
picoseconds. Smaller fluctuations were distinct and
regular, occurring with a period of less than one picosecond
and an amplitude of approximately 0.01 Imax/Imin* The
result for Run 3 produced a mean value of Imax/^min °f 1.35
and a range within 1.1 and 1.5. Fluctuations of varying
amplitude occurred with a period ranging from one to four
picoseconds.
Woods et al. (1986) obtained mean values of Imax/^min °f
1.25 and 1.18, respectively, for their 40 and 52 member

132
dodecyl micelles. Jonsson et al. (1986) obtained a value of
1.49 in their octanoate simulation using the reduced charge
model.
The micelle simulations, Runs 1 through 3, exhibited
greater regularity in the fluctuations of Imax/Imin than did
Run 4, the hydrocarbon droplet. The head group interaction
with the confining shell must be considered largely
responsible for this difference. Runs 2 and 3, with the
head groups of higher mass, had higher values of Ijnax/^^min
than Runs 1 and 4, in which all groups were of equal mass.
The higher mass head groups contribute to greater inequality
among the principal moments of inertia if they are
asymmetrically arranged about the center of mass of the
micelle. This was much more prevalent in Run 2 than in Run
3. The smaller diameter of the head groups in Run 2 allows
a higher local concentration of the mass on the micelle
surface than does the larger diameter of the head groups of
Run 3.
Neutron scattering experiments conducted by Cabane et
al. (1985) on sodium decyl sulfate micelles concluded that
the average shape of the micelles must be spherical, with
nonspherical shapes possible as fluctuations. This result
is consistent with the results of the simulations. A
theoretical study of shape fluctuations in spherical
micelles (Ljunggren and Eriksson, 1984) estimated shape

133
fluctuations due to capillary wave effects on a time scale
of 100 picoseconds. The simulations of this work were not
of sufficient length to comment on fluctuations of this long
a period, but, as demonstrated, shape fluctuations of
significant proportion did occur on a time scale of one
order of magnitude less.
5.5 Pair Correlations of Groups
An experimental measurement that can be used to test the
validity of structural models of micelles is the pair
correlation distribution function. This is a measure of the
probability of a pair of groups being separated by a given
distance. Cabane et al. (1985) have measured the pair
correlations for certain groups in sodium dodecyl sulfate
micelles of mean size 74. Using deuterium labeling and
small angle neutron scattering (SANS), they obtained results
for tail-tail and yCH2-yCH2 correlations.
Comparing their SANS result with the theoretical
predictions of several structural models, they found that
the models consistently predict the tails to be closer to
each other and the yCH2 groups to be further from each other
than the experimental result. The models predicted a
tendency for tails to be concentrated near the center of the
micelle and yCH2 groups to remain in a spherical shell near
the micelle surface.

I
r/N
1/3
Figure 5-35. Tail-tail pair correlation function distributions
resulting from Run 2, Run 3, and the SANS study of Cabane et al .
(1985). Scaling of axes places all results on an equivalent basis
in aggregate size.

r/N '/3
Figure 5-36. Correlation function distributions for CII2 —CII2
(gamma position) pairs resulting from Run 2, Run 3, and the SAMS
study of Cabane et al. (1985). Scaling of axes places all results
on an equivalent basis in aggregate size.

136
In Figures 5-35 and 5-36, the tail-tail and yCH2~yCH2
pair correlation results from Runs 2 and 3 are plotted along
with the SANS result of Cabane et al. The axes are scaled
to put the results on an equivalent basis in aggregate size.
Run 3, with its larger head groups, is of a greater radius
(Table 5) and a lower number density than Run 2. The
experimental results lie between those of Run 2 and Run 3,
with Run 2 giving slightly lower pair separations and Run 3
slightly higher. There is good agreement between Run 3 and
the experimental result for the tail-tail correlation. The
agreement between Run 3 and the experimental result for the
yCH2-yCH2 correlation is better than that of Run 2.
The SANS results show that tails are not concentrated in
the center of the micelle, but rather are distributed
throughout. It is also indicated by experiment that the
yCH2 groups are not strictly in the surface region, but are
distributed throughout the micelle. The simulation results,
particularly Run 3, exhibit this same behavior.

CHAPTER 6
SUMMARY AND CONCLUSIONS
A molecular thermodynamic model has been developed to
describe the formation of micelles in a solution of multiple
surfactant components. Each micelle of distinct size and
composition is treated as the product of a reaction whose
equilibrium constant is the result of the total free energy
of micellization for the micelle. A seven-step, reversible
process is employed to calculate the change in free energy
for the surfactant molecules changing their state from free
monomers in solution to aggregated molecules in micelles of
distributed sizes and compositions. Contributions to the
total free energy are obtained due to solvophobic
interaction, mixing, surface formation, conformational
change, head group interactions and electrostatics. With
the model for the free energy as a function of temperature
and composition, distributions of micelle sizes and
compositions can be generated through a set of reaction
equilibria relationships.
Where possible, the free energy contributions are
related to comparable processes on which experimental
measurements have been made. The solvophobic term is
obtained from hydrocarbon solubility; the surface term
137

138
is related to the interfacial tension between hydrocarbon
and solvent phases. In addition, reasonable assumptions of
ideal mixing of hydrocarbon chains and ideal solution
behavior of the dilute surfactant solution are made.
Aggregate size distributions have been generated from the
model for single-component solutions of nonionic surfactants
of different chain lengths and at different temperatures and
solution concentrations.
Three parameters are required by the model: one to
estimate the entropy change due to conformational changes in
the surfactant molecules upon micellization and two to
estimate the curvature effects on the surface free energy of
spherical and cylindrical interfaces. The parameters can be
fitted to the mean aggregate size at the critical micelle
concentration—only two independent results. The consequent
interrelation among the parameters gives curvature parame¬
ters which are linearly dependent on the conformational
parameter. Within the physically plausible range of values,
the conformational parameter has a range which gives
identical size distributions when corresponding values of
the curvature parameters are used.
This variation of the parameters reveals a need for more
fundamental information to establish a truly predictive
model for the thermodynamics of micelle formation. While
structure and geometry of the micelle are fundamental to the

139
calculation of free energy contributions, experimental
evidence is insufficient to completely define the
thermodynamics of micelle formation.
In order to acquire some of the detailed molecular-level
information necessary for the modeling, the structure of the
micelle has been investigated through the molecular dynamics
computer simulation of model micelles. Simulations of three
micelles with different head group attributes and one
hydrocarbon droplet were conducted. In each case, the
aggregate consisted of 24 nine-member, linear chain
molecules. The solvent was not explicitly modeled.
Instead, the aggregate was surrounded by a spherical shell.
Interactions of the molecules with the shell were designed
to mimic the appropriate interaction of a chain segment or
head group with solvent.
The results of the simulations have been analyzed for
the distributions of molecular groups within the aggregate,
bond conformations and orientations, and shape fluctuations.
Comparisons have been made among the runs to study the
effects of head group size and mass, with other simulations
to study the effects of chain length, aggregate size, and
simulation technique, and with experimental results to
reveal the differences and similarities between the
simulations and experiment.

140
An increase in head group mass by a factor of seven had
no effect on the structure of the micelle. The results for
Runs 1 and 2 are virtually identical for the mean radial
positions of groups, probability distributions of group
positions, percentage of bonds in the trans conformation,
bond order parameter, and Fourier transform of tail
distribution. There was an effect of head group mass on the
dynamic quantities. More rapid gauche/trans transitions
were observed in the first dihedral angle of Run 2 than in
Run 1. Larger and more rapid fluctuations were observed in
the principal moments of inertia of Run 2 than in Run 1.
The model sulfate head group of Run 3, with its higher
mass, larger diameter, and different intramolecular
potentials, had a great effect on the structure. With the
larger head group size, the excluded volume effect at the
surface forced an average of two head groups into the
interior of the micelle, disrupting the order displayed by
the chains in Runs 1 and 2. While this result is entirely
possible, the shell around the micelle did not allow other
possible outcomes of the excluded volume effect at the
surface, such as head groups moving into the solvent.
Though the distributions are similar to other simulations,
the internal structure of Run 3 may not be accurate.

141
The hydrocarbon droplet simulation resulted in a nearly
uniform distribution of groups within the droplet, and a
percentage of trans bonds eguivalent to that of the three
micelle simulations. This is in contrast to statements
based on other micelle simulations which suggest there is a
higher percentage of trans bonds than that found
experimentally in bulk liquid alkanes as proof of order in
the micelle. The present results may mean that chain
straightening is more a product of the spherical geometry
than head group interactions, or it may just be an artifact
of simulation. Simulations of aggregates cannot be compared
with experimental bulk liquid results until the same
surfactant molecules are simulated under "bulk" conditions.
Comparisons of the simulation results of this work with
other simulations were generally consistent. In spite of
different surfactant molecules, aggregate sizes and
simulation techniques, the results are qualitatively quite
similar.
Comparisons with experimental results also show
similarities, but there is not complete agreement. Since
the experimental quantities are normally interpretations of
the measurements which often conflict from one study to
another, the disagreements may not reflect adversely on the
simulation results.

142
The key to the success of the thermodynamic modeling is
better information about the entropy change in transforming
the hydrocarbon droplet into the head-group-free micelle.
The results of the present simulations are inadequate to
determine chain conformations of monomers in micelles,
droplets, and bulk solvents. This can serve as the basis
for future efforts on both of these projects. Chain
conformations and their energy and entropy effects must be
better understood. Simulations of these molecules as bulk
liquids, in vacuum, and at infinite dilution in water would
be quite beneficial in understanding this phenomena.
Future micelle simulations should incorporate an
improved solvent model, with less restrictions on head group
movement and shape fluctuations. Since changes in head
group size had such dramatic effects, the size of the
terminal methyl tail group should also be reconsidered.

APPENDIX A
MICELLE SIZE AND SHAPE
As the aggregate size of a micelle increases, a shape
transition must occur so that the volume of the additional
monomers can be accomodated while maintaining a uniform
density througout. In the case of a single surfactant
species, a micelle can remain spherical only until the
radius of the sphere reaches the length of an extended,
all-trans, surfactant chain. In the multicomponent case,
the composition of the micelle also plays a role in
determining its shape. The geometric model for a binary
micelle will be derived to demonstrate this effect.
The two surfactant components will be designated 1 and
2, each with an aggregate number, N, an extended chain
length, 1, and a chain volume, v. Component 2 will have the
longer chain length and the larger chain volume. As in the
single component case, the maximum possible radius of a
spherical micelle is the length of the longest chain present
in the micelle, so
N N 2v7 = — R3 , R<12 (A. 1 )
3
143

144
When the radius of the sphere is greater than the
shorter chain length, there is a spherical "core" in the
center of the micelle which must be devoid of component 1.
In order to form a sphere of such a radius, a sufficient
amount of component 2 must be present to completely fill the
core. This is illustrated in Figure A-l. The radius of the
core, r, is given by
r = R-l, , /?>/, (A.2)
A fraction, f, of the total volume of component 2 in the
micelle will make up the core.
/W2u2 = ^(/?-Z,) (A .3)
Since a chain of component 2 extending into the core must
pass through the outer portion of the micelle, there is an
upper limit on the fraction of component 2's volume that can
make up the core.
f<\-~ (A .4)
Í 2
If both (A.1) and (A.4) are satisfied, a spherical
micelle will form. Combining (A.l) and (A.3) yields the
expression for f:
4 n
3
/ =
yv 2^2
(A .5)

145
Figure A-l. Schematic of a binary micelle with monomers of
different chain lengths. Inner dashed circle depicts the
micelle core devoid of the shorter chains.

146
A "core fraction constraint" on spherical micelle
formation is obtained by substituting (A.4) for f and
solving for N]_:
4 n
3
yv,<
3
4 n
3
- N 2u 2
(A .6)
From A-l a "radius constraint" is written:
y132~N2v2
A,<- (A .7)
V\
The possible combinations of and N2 which will form a
spherical micelle are bounded by these two constraints, as
shown in Figure A-2. There are four regions defined by the
two constraints:
I)Both constraints met — spherical micelle.
II)Only radius constraint met — nonspherical.
Ill)Only core fraction constraint met — nonspherical.
IV)Neither constraint met — nonspherical.
Nonspherical micelles are modeled as prolate spherocylin-
ders. The spherical caps and their cores have radii of R
and r, as in the spherical case. The cylindrical portion
has these same radii and length L. These dimensions are
shown in Figure A-3.

147
N2
Figure A-2. Possible combinations of shorter chains (1)
longer chains (2) in a binary micelle. Four regions are
defined by the two composition constraints on micelle
geometry.
and

H- tJ' (I)
143
A-3. Side view cf the prolate spherocylinde
inary micelle. The micelle is of radius R an
le the core is of radius r and length L.
M 'U

As in the spherical case, equations can be written for
the total volume
2
(A.8)
and the core volume
fN2v2 = ^(R-ll)3 + nL(R-llY
(A .9)
The two volume equations can be combined to eliminate L,
resulting in a polynomial in R:
4 8/r
~3
l,R3 + ^Z?-(1-/)N2ü2-^V1i;i R2
+ 2 Z, (AZ, y, + N 2v 2) R - l lvl + N 2v2) = 0 (A.10)
For the spherical micelle of region I, L is zero and
(A.1) gives the value of R. In region III, the R is set
equal to I2 and L is given by (A.9). In regions II and IV,
f is set equal to 1 - I2/I1 and (A.10) is solved for its
root between 1^ and I2.

APPENDIX B
HEAD GROUP INTERACTION IN A BINARY MICELLE
The potential energy due to the interaction of polar
head groups at the surface of the micelle can be calculated
as the sum of the potentials of all possible interacting
pairs. Considering ionic head groups to be point charges
and nonionic head groups to be dipoles, the types of pairs
possible are dipole/dipole, charge/dipole, and
charge/charge. The potentials for all three are functions
of the inter-group distance, and for the first two are also
functions of the orientations of the dipole(s).
Assuming that the mean distance between adjacent head
groups is the inter-group distance for every adjacent pair,
head goups can be arranged hexagonally on the surface
(Figure B-l.) In this arrangement, each head group has six
adjacent groups, each at a distance "r". If each group is
taken as the center of a hexagon and the adjacent pairs
formed with it are summed, each pair will be counted twice
in the total of all adjacent pairs formed by all groups.
150

151
Figure B-l. Hexagonal arrangement of head
micelle surface, with equal separation, r,
groups.
groups on the
between adjacent

152
For a micelle of N head groups,
m n
(B.l)
n = 1 ,2 6
The distance r will first be determined for head groups
on a spherical surface, as in a spherical micelle or the
spherical portions of a spherocylindrical micelle. Taking a
great circle of the sphere such that adjacent head groups
lie on the circle (Figure B-2), R is the radius of the
sphere and <5(N) is the angle formed by two radii meeting the
surface at the adjacent groups, as a function of N. From
planar trigonometry,
(B.2)
Each triad of mutually adjacent groups defines an
equilateral spherical triangle with sides of arclength R6.
Each group is a vertex of six triangles. If six triangles
are assigned to each group and all groups are summed each
triangle is counted three times, so the number of spherical
triangles is 2N.
From spherical trigonometry, the area of each triangle
A = (3a - n)R2
(B.3)
where a is the angle at each vertex of the equilateral

153
Figure B-2. Vectors and angles used in the calculation of
dipole - dipole interaction potential. The magnitude of the
separation vector, r^, is the separation given in Figure
B-l.

154
spherical triangle. From the Law of Cosines for sides of
spherical triangles,
cos 6 - cos2 6
cosa =
sin <5
(B.4)
cos
2sin"‘(^)
cos(2sin-‘(¿))_
(B.5)
and from the double-angle relation,
2sin "' | —— ) = 1 - - f —
2 R 2\R
(B.6)
Combining (B.3), (B.5), and (B.6) yields an expression for
the area of a triangle in terms of r and R:
A = ( 3cos
-1
¡(ir
- n ) R'
(B.7)
The area of a triangle is also equal to the total area
divided by the number of triangles:
4 nR2
A
2N
(B.8)
Eliminating the area in these two equations and solving for
r gives the distance between groups for the spherical
surface:
r = R
/ 4cos
' n 2 n
, 3 + 3Ñ
I"2
cos
l 3 3 N )
- 1
The surface of the spherocyUnder can be separated
(B.9)
into
those of a sphere of radius R and cylinder of radius R and
length L. The number of groups on the surface of the

155
micelle is the sum of the numbers of groups on the two
surfaces:
A/ = A¡sph + Ncyl (B.10)
From (B.7) and (B.8) the number of groups on the spherical
portion is given by
N
sph
2 n
3cos 1
r^r
- n
(B.ll)
On the cylindrical portion, the hexagonal pattern will
have the groups lying on circles evenly spaced along the
length of the cylinder. In this arrangement,
# of circles = 2L
7ir
#
of groups per circle = n
sin
-1
so that
N
cyl
2 n L
^rsin'l{h)
(B. 1 2)
Equation (B.10) can then be written as the following
nonlinear equation in r,
2 n
3cos 1
r-un
- n
2 n L
^rsin''(21)
-N
(B. 1 3)

156
which can be solved for r by iterative techniques. The
values of R and L are calculated by the method given in
Appendix A.
For the calculation of pair potentials, Weston and
Schwarz (1972) give:
E
charge/charge
ZiZ2e2
Dr 12
(B.14)
E
charge/dipole
-z^eii 2cos02
Dr2X2
(B.15)
where e2 is the angle between the directions r^2 and a<2 •
Murrell et al. (1978) give:
E
dipole/dipole
E\'E2 3{e\• ri2)[n2 • r,2)
D r ?2 E) r\2
(B. 16)
Assuming all dipoles to be radially oriented, all of the
dot products are two-dimensional, so
dipole/dipole
Dr3i2
COS0-3cOS0!COS02j
(B. 1 7)
where the angles are those shown in Figure B-2. Only
adjacent pairs are being considered in this model, and all
pair separations are equal to r, so the subscripts on r will
not be used. Since 6 is known (B.2), the angle terms in
(B.15) and (B.17) can be eliminated:
_ ~ z\eE2
A*
^ charge/dipole
2 DRr
(B.18)

157
T 1 +
Dr3
2
f
dipole/dipole
(B. 1 9)
The total number of adjacent pairs formed among N head
groups is 3N. For a binary micelle,
n =yv, + yv2
N,
Assuming only adjacent pairs contribute to the total
interaction potential, the single component limit is
E tot = 3 n E„
(B.20)
At X2=0, replacing a "1" with a "2" replaces six "1-1" pairs
by six "1-2" pairs. With replacements distributed evenly
over the surface, this continues until X2= 1/4. The next
replacement of "1" by "2" replaces four "1-1" pairs by two
"1-2" pairs and two "2-2" pairs. This continues until X2=
1/2. Beyond this, two "1-1" and two "1-2" pairs are
replaced by four "2-2" pairs until X2= 3/4. Finally, six
"1-2" pairs are replaced by six "2-2" pairs until X2= 1.

158
dE,
d x
-6N(E12-Eu)
0 < x2 < -
2 4
(B.21)
= N [2 E 72 + 2 E 12_4£'11)
1 < < 1
4 2 2
(B.22)
= n{4E22~2E ,2-2£m)
1 < <3
2 2 4
(B.23)
= 6N[E 22-E 12)
- 4 2
(B.24)
Integrating the above and writing on a per monomer basis:
E_ = F11(3-6x1) + F12(óX|)
^ 1 1 ( 2 ^ X \ \ + ^ \2{^ + 2-X x) + E 22\2.X \ 2
E 1 1 | 2 2 X 1 J + ^ 12 ( ^ 2x ! ) + £*22^ 4x ] 2
¿ri2(6 6X|)+F22(6-V] 3
0 < x2 ^ -
2 4
(B.25)
1 < < 1
(B.26)
4 2
2 2 4
(B.27)
VI
CN
X
VI
CO 1 m
(B.28)
By substituting in the appropriate pair potentials from
(B.14) through (B.16) the total energy of interaction of
adjacent pairs of a micelle of binary composition is be
calculated with one of equations (B.25) through (B.28.)

APPENDIX C
PROGRAM LISTING FOR SINGLE-COMPONENT
NONIONIC MICELLE CALCULATION

REAL*8 XSUM,XN,ARG,X
CHARACTER CONT,P,DISP,WHAT,SHOW,SAV,NAME*8
REAL N,NAVG,SAMP(3,5),NSAM(5)
COMMON N,DGM/I/G,GC,SCONF/J/C1,CO
COMMON /K/U,T,ZZ/L/GVAL(6)
WRITE(*,*)'NUMBER OF CARBONS'
READ(*,*)CN
OPEN(9,FILE='INVALS',FORM='FORMATTED',ACCESS
&'SEQUENTIAL',STATUS='OLD')
READ(9,*) T,U,C0,C1,G,GC,SCONF
READ(9,*)IMIN,IMAX,(NSAM(J),J=1,5)
5 WRITE(*,100) Cl,CO,U,T,G,GC,SCONF,IMIN,IMAX
READ(*,'(A)')WHAT
IF(WHAT.EQ•'D') THEN
WRITE(*,*)'Enter Cl, CO, U, T:'
READ(*,*) C1,C0,U,T
ELSEIF(WHAT.EQ.'P') THEN
WRITE(*,*)'Enter G, GC, SCONF:'
READ(*,*)G,GC,SCONF
ELSEIF(WHAT.EQ.'R') THEN
WRITE(*,*)'Enter min and max values of N:'
READ(*,*) IMIN,IMAX
ELSEIF(WHAT.EQ.'G') THEN
GO TO 6
ENDIF
160

161
GO TO 5
6 WRITE(*,*)'PRINT GVALUES TO FILE? (Y/N)'
READ(*,'(A)') P
IF(P.EQ.'Y') THEN
WRITE(*,*) 'ENTER NAME OF FILE (1-8 CHAR.):'
READ(*,'(A)') NAME
OPEN(UNIT=8,FILE=NAME,FORM='FORMATTED',ACCESS
&'SEQUENTIAL',STATUS='NEW')
WRITE(8,81)' ','N','G2','SC','G4','G6','GM',
&'LOG(X/N)','X/N'
ENDIF
WRITE(*,*)'CALCULATING DISTRIBUTION. . .'
COLG=ALOG(CO)
C1LG=AL0G(Cl)
CALL HCSOL(XEQ,CN)
DG2=ALOG(XEQ*C0)
X1=C1/C0
XSUM=0.
L0=1
SUM1=0.0
SUM2=0.0
IF(P.EQ.'Y') THEN
KOUNT=0
KOUNT2=l
KUP=0

162
DO 8 I=IMIN,IMAX
N=I
IF(KOUNT2.EQ.80) THEN
WRITE(8,81)' 1,'N','G2','SC','G4','G6',1GM',
&'LOG(X/N)','X/N'
KOUNT2=l
ENDIF
CALL DELTAG(DG2,CN)
ARG=N*(-DGM+C1LG)-COLG
IF(DABS(ARG).GT.174.673) THEN
IF(ARG.GT.0.0) THEN
WRITE(*,85) N,DGM
GO TO 11
ELSE
XN=0.0
ENDIF
ELSE
XN=DEXP(ARG)
ENDIF
IF(LO.GT.5) GO TO 7
IF(N.EQ.NSAM(LO)) THEN
SAMP(1,LO)=N
SAMP(2,LO)=GVAL(6)
SAMP(3,LO)=XN
L0=L0+1

163
ENDIF
7 X=XN*N
XSUM=XSUM+X
KOUNT=KOUNT+1
IF(N.EQ.100.) THEN
KOUNT=ll
KUP=1
ENDIF
IF(N.EQ.1000.) THEN
KOUNT=101
KUP=2
ENDIF
IF(KUP.EQ.0)THEN
WRITE(8,82)' ',(GVAL(M),M=1,6),ARG/2.303,XN
KOUNT2=KOUNT2+l
ELSEIF(KUP.EQ.1.AND.FOUNT.EQ.11) THEN
WRITE(8,82)' ',(GVAL(M),M=1,6),ARG/2.303,XN
KOUNT2=KOUNT2+l
KOUNT=l
ELSEIF(KUP.EQ.2.AND.FOUNT.EQ.101) THEN
WRITE(8,82)' ',(GVAL(M),M=1,6),ARG/2.303,XN
KOUNT2=KOUNT2+l
KOUNT=l
ENDIF
SUM1=SUM1+XN

164
SUM2=SUM2+X
IF(XN.LT.1.OD-24.AND.I•GT•10000) GOTO 10009
8 CONTINUE
10009 NAVG=SUM2/SUM1
CMON=XSUM*CO + Cl
WRITE(6,84) Cl,CO,U,T
WRITE(6,101) G,GC,SCONF,NAVG,CMON
WRITE(*,*)'DISPLAY SAMPLE GVALUES? (Y/N)'
READ(*,'(A)')DISP
IF(DISP.EQ.'Y*)WRITE(6,83)((SAMP(J,K),J=l,3),K=1,5)
11 WRITE(*,*)'CONTINUE? (Y/N)'
READ(*,'(A)') CONT
WRITE(8,84) Cl,CO,U,T
WRITE(8,101) G,GC,SCONF,NAVG,CMON
C ENDFILE(UNIT=8)
CLOSE(UNIT=8)
ELSE
DO 10 I=IMIN,IMAX
N=I
CALL DELTAG(DG2,CN)
ARG=N*(-DGM+C1LG)-COLG
IF(DABS(ARG).GT.174.673) THEN
IF(ARG.GT.0.0) THEN
WRITE(*,85) N,DGM
GO TO 12

165
ELSE
XN=0.0
ENDIF
ELSE
XN=DEXP(ARG)
ENDIF
IF(LO.GT.5) GO TO 9
IF(N.EQ.NSAM(LO)) THEN
SAMP(1,LO)=N
SAMP(2,LO)=GVAL(6)
SAMP(3,LO)=XN
LO=LO+l
ENDIF
9 X=XN*N
XSUM=XSUM+X
SUM1=SUM1+XN
SUM2=SUM2+X
IF(XN.LT.1.OD-24.AND.I.GT.1000) GOTO 10000
10 CONTINUE
10000 NAVG=SUM2/SUM1
CMON=XSUM*CO + Cl
WRITE(6,84) Cl,CO,U,T
WRITE(6,101) G,GC,SCONF,NAVG,CMON
WRITE(*,*)'DISPLAY SAMPLE GVALUES? (Y/N)'
READ(*,'(A)') DISP

166
IF(DISP.EQ.'Y')WRITE(6,83)((SAMP(J,K),J=l,3),K=1,5)
12 WRITE(*,*)'CONTINUE? (Y/N)'
READ(*,'(A)') CONT
ENDIF
IF(CONT.EQ.'Y') GO TO 5
WRITE(*,*)'SAVE DATA AND PARAMETER VALUES?'
READ(*,'(A)')SAV
IF(SAV.EQ.'Y') THEN
REWIND(9)
WRITE(9,*)T,U,CO,Cl,G,GC,SCONF
WRITE(9,*)IMIN,IMAX,(NSAM(J),J=1,5)
ENDIF
CLOSE(9)
STOP
81 FORMAT(A,2X,A,6X,A,6X,A,6X,A,6X,A,8X,A,5X,A,4X,A,
&5X,A,E10.4,A,F4.1,A,F6.2)
82 FORMAT(A,F6.0,1X,F6.2,2X,F6.3,2X,F6.3,2X,F6.5,3X,
&F8.4,2X,F6.1,2X,E9.2)
83 FORMAT(5X,'N=',F6.0,5X,'DGM=',F7.3,5X,'XN/N=',E9.2)
84 FORMAT(IX,'Cl=',E12.5,4X,'C0=',F5.1,4X,'U=',F4.1,4X,
&'T=',F7.2)
85 FORMAT(IX,'FOR N=',F6.0,'DG)M=',E10.3/IX,
&'OVERFLOW ON ','XN/XN CALCULATION')
86 FORMAT(IX,'FOR N=',F6.0,'DG)M=',E10.3/1X,
&'X/N SET TO 0 ','TO AVOID UNDERFLOW')

167
100 FORMAT(IXCurrent values of Data and Parameters are:'
&/6X,'DATA: Cl=',E12.5,2X,'C0=',F5.1,2X,'U=',F4.1,2X,
&'T=',F7.2/6X,'PARAMETERS: G=',F7.4,3X,'GC=',F7.5,3X,
&'SC0NF=',F8.4/6X,'RANGE of N is from ',13,' to ',16//
& IX,'Change values of D)ata, P)arameters, R)ange,
& or G)o?'/)
101 FORMAT(IX,'G=',F7.4,3X,'GC=',F7.5/1X,'SC0NF=',F8.4
& /IX,'MEAN AGGREGATE SIZE IS',F9.2,5X,
& 'SURFACTANT CONCENTRATION IS',E13.6)
END
Q •k’k'krk-k-k-k'k’k’k-k'k’k-k-k'k’k-k'k'k-k-k-k-k’k'k-k’k'k’k’k-k'kjc-k-k-k'k'k-k'k-k’k’k'k-k-k'k'k-k-k-k’k-k’k-k
SUBROUTINE DELTAG(DG2,CN)
REAL NTRANS
COMMON EN,DGM/H/R,XLM/I/G,GC,SCONF/J/Cl,CO/K/U,T,DG6
COMMON /L/GVAL(6)
CALL SHAPE1(XLC(CN),VC(CN))
NTRANS=4.1888*XLC(CN)**3/VC(CN)
IF(EN.LE.NTRANS) THEN
SURF=12.5664*R*R*(NTRANS/EN)**(1./3.)
DG4=SURF*GAM(CN)*(1.-G/R)/(1.381*EN*T)
ELSE
DG4=6.2832*R*GAM(CN)*(XLM*(1-GC/R)+2*R*(1-G/R))/
& (1.381*EN*T)
ENDIF
CALL DHHEDS(D)

168
SC=-SCONF*(XLC(CN)/R)**2
DGM=DG2+DG4+SC+DG6
GVAL(1)=EN
GVAL(2)=DG2
GVAL(3)=SC
GVAL(4)=DG4
GVAL(5)=DG6
GVAL(6)=DGM
RETURN
END
Q •k’k'k'k-k-k'k'k'k'k’k-k-k’k'k'k-k’k-k'k'k-k-k-k'k'k'k-k'k'k'k'kJc’k-k'k-k-k'k'k'k-k'k'k’k'k-k’k-k'k-k-k-k-k'k
FUNCTION XLC(CN)
XLC=1.265*CN + 1.5
RETURN
END
C *******************************************************
FUNCTION VC(CN)
VC=2 6.9*CN + 27.4
RETURN
END
Q *******************************************************
FUNCTION GAM(CN)
COMMON /K/Z,T,ZZ
GAM=1.381*(57.868*CN + 117.99 -
& (CN + 2.4)
(.059*CN + .1768)*T)/

169
RETURN
END
0 •k'k’k'kic'k'k’kic'k’k'k'k'k'k’k’k'k'k'k'k'k'k'k'k'kic'k'k'k’k'k'k'kic'k-kJc’k'k’k’k'k-k'k'k'k’k’k'kic'k-k’k'k
SUBROUTINE HCSOL(XEQ,CN)
REAL K
REAL*8 XI(10),EXPK
DIMENSION A(2),B(2)
COMMON /K/ZO,T,Z1
DATA A/997.098,258.015/B/1603.72,9686.96/R/1.987/
IF(CN.LT.10.) THEN
1 = 1
ELSE
1=2
ENDIF
K=A(I)*CN + B(I)
EXPK=EXP(-K/(R*T))
XI(1)=EXPK/(1.+EXPK)
DO 60 J=2,10
XI(J)=DEXP(-(1.-2.*X1(J-l))*K/(R*T)+DLOG(1.-XI(J-l)))
TEST=DABS(XI(J)-XI(J-l))/Xl(J)
IF(TEST.LT.1.OE—8) GO TO 61
60 CONTINUE
61 XEQ=X1(J)
RETURN
END

170
C
*******************************************************
SUBROUTINE SHAPE1(XLC,VC)
COMMON AG,Z/H/R,XL
DATA B/4.18879/
XL=0.0
R=(AG*VC/B)**(1./3•)
IF(R.GT.XLC) THEN
R=XLC
XL=(AG*VC-B*XLC**3)/(3.14159*XLC**2)
ENDIF
RETURN
END
Q •k'k'k’k-k’k'k-k'k’k'k-k-k’k'k'k'k’k-k-k-k-k’k-k’k’k-k-k’k-k-k-k-k'k-k'k’k’k'k-k-k’k’k-k’k-k'k'k-k'k'kic’k'k-k
SUBROUTINE HEDSEP(D)
CHARACTER*1 FLAG
COMMON AG,Z/H/R,XL
EXTERNAL F,DF
IF(XL.GT.0.) THEN
CALL NEWTON(F,DF,D,.01,DO,50,FLAG)
IF(FLAG.EQ•'N') GO TO 49
D=D0
RETURN
49 WRITE(6,400) AG
RETURN
ENDIF

171
T=COS(1.04720*(1.+2./AG))
D=SQRT((4.*T-2.)/(T-l.))*R
RETURN
400 FORMAT(IX,'NO HEAD SEP. FOUND, N=',F7.0)
END
C *******************************************************
FUNCTION F(D)
COMMON AG,XZ/H/R,XL
DATA E/6.28319/
Z=(D/R)**2
F=E/(3*ACOS((2.—Z)/(4.-Z))-3.14159)+E*XL/(1.732*D
& *ASIN(D/(2.*R))) - AG
RETURN
END
C *******************************************************
FUNCTION DF(D)
COMMON/H/R,XL
Z=D/(2.*R)
Z2=(D/R)**2
DF=-3.6276*XL*(1./(D*D*ASIN(Z)) +1./(2.*R*D*(ASIN(Z))
&**2*SQRT(1.-Z*Z)))-75.398*D/(R*R*(4.-Z2)**2*(3.*ACOS(
& (2.-Z2)/(4.-Z2))-3.14159)**2*SQRT(l.-((2.-Z2)/(4.-Z2
& ))**2))
RETURN
END

172
C *******************************************************
SUBROUTINE DHHEDS(D)
COMMON /H/R,XL/K/U,T,DG6
DI(T)=(.0007469*T-.8063294)*T+252.422
CALL HEDSEP(D)
DG6=3*U*U*(1.+(D/(2.*R))**2)/(DI(T)*D**3*T*1.38IE-23)
& *9.9907E-20
RETURN
END
Q ******************************************************
SUBROUTINE NEWTON(F,DFDX,Al,EPS,XO,J,FLAG)
CHARACTER*1 FLAG
FLAG='Y'
X=A1
DO 11 I=1,J
FX=F(X)
IF(ABS(FX).LT.EPS) THEN
X0=X
RETURN
ENDIF
X=X-FX/DFDX(X)
11 CONTINUE
FLAG='N'
RETURN
END

Page
Missing
or
Unavailable

c
c
c
c
c
c
c
c
c
c
c
c
c
c
MOLECULAR DYNAMICS SOURCE PROGRAM FOR A SIMPLE
MODEL OF A MICELLE ENCLOSED IN A SHELL. A
FIFTH-ORDER PREDICTOR-CORRECTOR ALGORITHM IS
USED TO SOLVE THE EQUATIONS OF MOTION. A
SKELETAL MODEL DUE TO WEBER IS USED FOR EACH
ALKANE MOLECULE.
INTENDED FOR LONGER-RANGE POTENTIALS. NO
TRUNCATION OF POTENTIALS OR NEIGHBOR-LISTING
IS EMPLOYED.
ALL HEAD GROUP ATTRIBUTES CAN BE SET.
PARAMETER(MHEAD=7,HDSGMA=2.45,BLHEAD=.65,GLHEAD=
& 1.03103E3,BAHEAD=-.82904,GAHEAD=2.17184E3,GTHEAD=
& 47.733)
COMMON/POS/XO(216),Y0(216),Z0(216)
COMMON/VEL/X1(216),Y1(216),Z1(216)
COMMON/DER/X2(216),Y2(216),Z2(216),X3(216),Y3(216),
& Z3(216),X4(216),Y4(216),Z4(216),X5(216),Y5(216),
& Z5(216)
COMMON/FOR/FX(216),FY(216),FZ(216)
COMMON/DISP/DAX(216),DAY(216),DAZ(216),X0L(216),
& YOL(216),ZOL(216)
174

175
& FORCON
COMMON/ENERGY/TOTE,TOTLJ,ETOR,EBON,EBEN,EINTR,TRFRAC
& ,EXTPRS,FTOTWL
COMMON/NUM/NM,NAM,NP,NP1,NP2,NP22,KSORT,KB,NABTOT
COMMON/SCAL/RSPHER,RWALL,DELRS,ISCALE
COMMON/MASS/MASS(216)
C
C FLAG, IF IFLAG.NE.l USES INTPOS AND RANDOM VELOCITIES
C IF IFLAG.EQ.l XO,YO,ZO,AND ALL DERIVATIVES ARE READ
C FROM PREVIOUS RUN.
C
IFLAG=1
C
C === SET NUMBER OF PARTICLES IN PRIMARY CELL
NAM=9
NM=24
NP=NM*NAM
PART=NP
PART=NP
NP1=NP-1
NP2=NP-2
NP22=.5*PART+.01
C
C === SET RELATIVE MASS OF PARTICLES
DO 20 1=1,NP,NAM

176
MASS(I)=MHEAD
20 CONTINUE
C
C === SET RADIUS OF SPHERE ON WHICH HEAD GROUPS ARE ATTACHED
RSPHER= 6.000
DELRS=0.00025
ISCALE=0
RWALL =RSPHER+1.0
REQUIL=1.
FORCON=30.
C
C === SET VALUES OF PHYSICAL CONSTANTS
XNAM=FLOAT(NAM)
WTMOL=XNAM*14.+2.
WTPART=WTMOL/XNAM
RSTAR=0.4
EPS=419.
GABB=9.25E7
GAB=GABB/EPS*RSTAR*RSTAR
THA=112.15*3.14159/180.
CT0=COS(THA)
GTHAA=1.3E5
GTHA=GTHAA/EPS
C ROTATIONAL PARAMETER DIVIDED BY 10 FOR SCALE-DOWN
GTHEE= 8.314E3

177
IF(ISCALE.EQ.1)GTHEE=GTHEE/10.
GTHE=GTHEE/EPS
BB=0.1539
B0=BB/RSTAR
B2I=1./B0/B0
AV0=6.0225E+23
B0LZ=1.38054D-23
EPSI=EPS/BOLZ/AVO
THIRD=l./3•
PI=3.1415926535
C
C === SET DESIRED FLUID STATE CONDITION
TR=5.913
T=TR*EPSI
VOL=1.333333*PI*RSPHER**3
DR=FLOAT(NM)/VOL
C
C === SET RUN FLAGS AND PARAMETERS
IFLG=-1
KB=0
KSAVE=10
KSORT=10
KWRITE=10
MAXKB=90000
XDIST=0.1

178
C
C === SET TIME-STEP AND ITS MULTIPLES
DELTA=0.000850
DELSQ=DELTA*DELTA
DELTSQ=.5*DELSQ
TSTEP=SQRT(WTPART*RSTAR**2/EPS/1.E21)*1.D+12
TT1=TSTEP*DELTA*l.D-12
C
c === SET PARAMETERS IN PREDICTOR-CORRECTOR METHOD
F02=3./16.
F12=251./360.
F3 2 = 11•/18.
F42=l./6.
F52=l./60.
C
C === SET DISTANCES FOR POTENTIAL CUT-OFF, VERLET LIST,
CUBE=2.*RSPHER
CUBE2=.5*CUBE
RC=2.5
RCHH=2.5
RLIST=(RC+.25)**2
RDMAX=CUBE2 *CUBE2
IF(RLIST.GT.RDMAX) RLIST=RDMAX
IF(RC.GT.CUBE2) RC=CUBE2
C
ETC.

179
C === SCALE FACTOR FOR VELOCITIES DURING EQUILIBRATION
AHEAT=DE LS Q * PART * 3.*TR
C
c ===== WRITE TAPE HEADING
OPEN(20,FILE='C811RUN1',ACCESS='SEQUENTIAL',
& FORM='UNFORMATTED' )
WRITE(20) NP,NAM,NM,DR,TR,EPS,RSTAR
WRITE(20) CUBE,VOL,RC,RLIST,RDMAX,DELTA,TSTEP
C
C === SHIFTED-FORCE CONSTANTS
RRMAX=1./RC
RRMAX6=RRMAX**6
RRMAX9=RRMAX**9
ESHFT=21.*RRMAX6-20.*RRMAX9
ESHFTA=18.*RRMAX*(RRMAX9-RRMAX6)
FSHFT=ESHFTA
C SHIFTED FORCE POTENTIALS FOR HEAD-HEAD REPULSIVE
C INTERACTIONS
C
RCI=1./RCHH
RCI2=RCI*RCI
RCI3=RCI2 *RCI
RCI4=RCI3 *RCI
RCI9=RCI3*RCI3*RCI3
RCI13=RCI4*RCI*RCI4*RCI4

180
HSHFT=RCI3*(13.*RCI9+4.)
HSHFTA=3.*RCI4 *(4.*RCI9 + 1.)
HFSHFT=HSHFTA
C
C === CORRECTIONS FOR LONG-RANGE INTERACTIONS
RC3=RC**3
RC6=RC3 *RC3
CORE=4.*PI*DR*(1./6./RC6-0.5/RC3)
DE=CORE
C
c === INITIALIZE SUM ACCUMMULATORS
XSUM=0.
SUME=0.
c
C === PRINT PARAMETERS
WRITE(*,900)
900 FORMAT(1H1///)
WRITE(*,902)
902 FORMAT(7X,49('*') )
WRITE(*,904)
904 FORMAT(7X, '*',T56, '* ' )
WRITE(*,906) NM
906 FORMAT(7X,'*',2X,'MOLECULAR DYNAMICS FOR',13,
& ' N-ALKANE MOLECULES',T56,'*')
WRITE(*,904)

181
WRITE(*,917) NAM
917 F0RMAT(7X,'*',5X,'WITH',13,' PARTICLES PER
&MOLECULE',T56,'*' )
WRITE(*,904)
WRITE(*,902)
WRITE(*,904)
WRITE(*,908) EPSI,RSTAR
908 FORMAT(7X,'*',2X,'EPSI/K = ',F7.3,T36,'RSTAR
&F7.3,T56,'*')
WRITE(*,910) TR,T
910 FORMAT(7X, '*',2 X, 1TR = ',F7.3,T36,' T
&F7.3,T56,'*')
WRITE(*,912) DR,VOL
912 FORMAT(7X,'*',2X,'DR = ',F7.3,T36,'VOL
&F8.3,T56,'*')
WRITE(*,914) CUBE
914 FORMAT(7X,'*',2X,'CUBE = ',F7.3,T56,'*')
WRITE(*,916) RC
916 FORMAT(7X,'*',2X,'RC = ',F7.3,T56,'*')
WRITE(*,918) RLIST,RDMAX
918 FORMAT(7X,'*',2X,'RLIST = ',F7.3,T36,'RDMAX
&F7.3,T56,'*')
WRITE(*,920) DELTA
920 FORMAT(7X,'*',2X,'DELTA = ',F9.5,T56,'*')
WRITE(*,904)

182
WRITE(*,922) TSTEP
922 FORMAT(7X,,2X,'TIME UNIT = ',F6.3,'E-12 SEC',
&T56,'*')
WRITE(*,924) TT1
924 FORMAT(7X,,2X,'TIME STEP = ',1PE10.3,' SEC',
&T56,'*')
WRITE(*,904)
WRITE(*,926) DE
926 FORMAT(7X,'*',2X,'ENERGY CORRECTION = ',F7.3,T56,
&'*')
WRITE(*,904)
WRITE(*,902)
WRITE(*,932) RSPHER
932 FORMAT(////10X,'RADIUS OF SPHERE = ',F6.2)
WRITE(*,934) GTHE
934 FORMAT(//lOX,'ROTATIONAL POTENTIAL PARAMETER = ',
*F7.3/)
C
C
C
C
OPEN(30,FILE='LASTDAT',ACCESS='SEQUENTIAL',FORM=
&'UNFORMATTED')
IF(IFLAG.EQ.1) GOTO 199
C

183
C === LOAD INITIAL POSITIONS OF ATOMS
CALL INTPOS(RSPHER)
CALL POSPRI(KB,NM,NAM)
C
C — PRINT RUN-TABLE HEADING
WRITE(*,930)
930 FORMAT(1H1////4X,'KB',5X,'RSPH',4X,'ENRG',5X,'El',
& 4X,'DIST',6X,'TEMP',3X,'TRFRC',3X,'NAB',3X,
&'TOT ENR', 3X,'FTOTWL',3X,'PRESSURE'/)
C
C === LOAD INITIAL VELOCITIES OF ATOMS
CALL INTVEL(AHEAT,PART)
C
C = ASSIGN INITIAL ACCELERATIONS BASED ON INITIAL
C POSITIONS
CALL EVAL(RWALL)
C
C === SCALE ACCELERATIONS AND STORE STARTING POSITIONS
DO 530 1=1,NP
X2(I)=FX(I)*DELTSQ/MASS(I)
Y2(I)=FY(I)*DELTSQ/MASS(I)
Z 2(I)=FZ(I)*DELTSQ/MASS(I)
530 CONTINUE
GOTO 188
C

184
C READ POSITIONS AND DERIVATIVES INSTEAD OF USING
C FCC, INVEL, ACCELERATION
c
199 WRITE(*,930)
READ(30) KB,IFLG,NM,NAM,RSPHER,XSUM,SUME
RWALL=RS PHER+1.0
DO 200 K=1,NP
READ(30) X0(K),Y0(K),Z0(K)
READ(30) X1(K),Y1(K),Z1(K)
READ(30) X2(K),Y2(K),Z2(K)
READ(30) X3(K),Y3(K),Z3(K)
READ(30) X4(K),Y4(K),Z4(K)
READ(30) X5(K),Y5(K),Z5(K)
READ(30) DAX(K),DAY(K),DAZ(K)
200 CONTINUE
188 DO 377 1=1,NP
X0L(I)=X0(I)
Y0L(I)=Y0(I)
ZOL(I)=Z0(I)
377 CONTINUE
C
C === ENTER MAIN LOOP OF SIMULATION
IF(IFLAG.NE.1) GOTO 777

185
777
C
540
C
c ===
CALL PREDCT(NP)
CALL EVAL(RWALL)
CALL CORR(DELTSQ)
IF(ISCALE.EQ.1) RSPHER=RSPHER+DELRS
RWALL=RSPHER+1.0
NS=KB+1
DO 599 NTIMES=NS,MAXKB
KB=KB+1
CALL PREDCT(NP)
CALL EVAL(RWALL)
CALL CORR(DELTSQ)
CALCULATE MEAN SQUARE DISPLACEMENT & KINETIC ENERGY
SUMVEL=0.
TDIST=0•
DO 540 1=1,NP
TDIST=TDIST+DAX(I)**2+DAY(I)**2+DAZ(I)**2
SUMVEL=SUMVEL+(XI(I)**2+Yl(I)**2+Zl(I)**2)
&*MASS(I)
CONTINUE
TDIST=TDIST/PART
EK=SUMVEL/(2.*PART*DELSQ)
ACCUMMULATE SUMS FOR PROPERTY AVERAGES
XSUM=XSUM+SUMVEL

186
C
C ===
C
940
C
C ===
550
C
C
SUME=SUME+TOTE
PROPERTY CALCULATION & PRINT-OUT AT INTERVALS
IF(MOD(KB,KWRITE).NE.O) GOTO 550
FKB=FLOAT(KB)*PART
TMP=XSUM/(3.*DELSQ*FKB)
ENR=(SUME/FKB+CORE)
El=TOTE/PART+CORE
ET0T=E1+EK
RLTIM=DELTA*FLOAT(KB)*TSTEP
WRITE(*,940) KB,RSPHER,ENR,El,TDIST,TMP,TRFRAC,
NABTOT,ETOT,FTOTWL,EXTPRS
IF(MOD(KB,1000).NE.O) GOTO 550
CALL POSPRI(KB,NM,NAM)
WRITE(*,930)
FORMAT(1H ,I6,4F8.3,F10.3,1X,F6.3,I6,2X,F8.4,
: 2(3X,F9.3))
DURING FIRST OF RUN, SCALE VELOCITIES FOR TEMPERATURE
IF(IFLG.LT.l) CALL EQBRAT(SUMVEL,AHEAT,TDIST,XDIST
:,NP,IFLG,KB,NAM)
â–  WRITE DATA ONTO TAPE FOR LATER USE
IF(KB.EQ.0) GOTO 777

187
IF(IFLG.EQ.O) GOTO 588
IF(MOD(KB,10).NE.0) GOTO 588
WRITE(20) KB,TMP,ETOT,El,EK,TOTE,EINTR,TOTLJ,ETOR,
&EBON,EBEN
WRITE(20) XO,YO,ZO
WRITE(20) XI,Y1,Z1
WRITE(20) X2,Y2,Z2
C WRITE(20) X3,Y3,Z3
C WRITE(20) X4,Y4,Z4
C WRITE(20) X5,Y5,Z5
588 IF(MOD(NTIMES,100).NE.0) GOTO 599
REWIND 30
WRITE(30) KB,IFLG,NM,NAM,RS PHER,XSUM,SUME
DO 598 K=1,NP
WRITE(30) X0(K),YO(K),Z0(K)
WRITE(30) X1(K),Y1(K),Z1(K)
WRITE(30) X2(K),Y2(K),Z2(K)
WRITE(30) X3(K),Y3(K),Z3(K)
WRITE(30) X4(K),Y4(K),Z4(K)
WRITE(30) X5(K),Y5(K),Z5(K)
WRITE(30) DAX(K),DAY(K),DAZ(K)
598 CONTINUE
599 CONTINUE
C CLOSE(20)
CLOSE(30)

188
C
942
935
STOP
END
SUBROUTINE POSPRI(KB,NM,NAM)
COMMON/POS/XO(216),Y0(216),Z0(216)
11 = 0
WRITE(*,942) KB
FORMAT(1H1///7X,'POSITIONS OF GROUPS AT TIME-STEP
&',16//)
DO 404 JJ=1,NM
WRITE(*,935)
FORMAT(//2 X, 'MOLECULE',7X, 'X',11X, 'Y',12X, 'Z',
&12X,'R',
& 9X,'BOND LEN'/)
DO 404 KK=1,NAM
11=11+1
RR=SQRT(X0(II)**2+Y0(II)**2+Z0(II)**2)
XXX=X0(II)-X0(II-l)
YYY=Y0(II)-YO(II-l)
ZZZ=Z0(II)-ZO(II-l)
BB=SQRT(XXX*XXX+YYY*YYY+ZZZ*ZZZ)
IF(KK.EQ.l) WRITE(*,909) JJ,XO(II),YO(II),
&Z0(II),RR
IF(KK.GT.l) WRITE(*,911) XO(II),YO(II),ZO(II)
&RR,BB

189
404 CONTINUE
909 FORMAT(5X,I2,4(5X,F8.4))
911 FORMAT(7X,5(5X,F8.4))
RETURN
END
SUBROUTINE INTPOS(RSPHER)
C
C SET-UP INITIAL POSITIONS OF PARTICLES. HEAD GROUPS ARE
C ASSIGNED TO FIXED POSITIONS ON THE SURFACE OF A SPHERE
C OF RADIUS "RSPHER".
C
COMMON/POS/XO(216),Y0(216),Z0(216)
COMMON/NUM/NM,NAM,NP,NP1,NP2,NP22,KSORT,KB,NABTOT
COMMON/CONT/GAB,CTO,GTHA,GTHE,BO,B2I,A1,AO,EPS
DIMENSION THETA(50),PHI(50)
C
PI=3.1415926535
PI180=PI/180.
C
C TWENTY-FOUR MOLECULES
THETA(1)=0.
PHI(1)=0.
THETA(24)=PI

190
70
71
72
73
PHI(24)=0.
PH24=0•
DO 70 1=2,5
PHI(I)=PH24
PH24=PH24+PI/2.
THETA(I)=3 6.*PI180
DELPHI=2.*PI/7
PH24=DELPHI/2.
DO 71 1=6,12
PHI(I)=PH24
PH24=PH24+DELPHI
THETA(I)=72.*PI180
PH24=0.
DO 72 1=13,19
PHI(I)=PH24
PH24=PH24+DELPHI
THETA(I)=108.*PI180
PH24=PI/4.
DO 73 1=20,23
PHI(I)=PH24
PH24=PH24+PI/2.
THETA(I)=144.*PI180
C
60
909
WRITE(*,909)
FORMAT(1H1////7X,'INITIAL POSITIONS OF HEAD GROUPS'

191
& //7X,'MOLECULE', 5X,'THETA',8X,'PHI'/)
DO 101 1=1,NM
TTT=THETA(I)/PI180
PPP=PHI(IJ/PI180
WRITE(*,900) I,TTT,PPP
900 FORMAT(10X,I3,2(6X,F7.3))
101 CONTINUE
C
c
C STANDARD DISTANCES ALONG TRANS CHAIN
XLEN=0.5*B0*SQRT(2.*(1.-CTO))
ZLEN=SQRT(B0*B0-XLEN*XLEN)
ZLENSQ=ZLEN*ZLEN
C
c === ADD ATOMS TO FORM MOL IN TRANS POS =====
DO 120 1=1,NM
IADD=(1-1)*NAM
THR=THETA(I)
PHIR=PHI(I)
SR=SIN(THR)
CR=COS(THR)
SPR=SIN(PHIR)
CPR=COS(PHIR)
C
C ODD NUMBERED GROUPS ON A MOLECULE

192
92
C
C EVEN
95
120
DO 92 J=1,NAM,2
K=IADD+J
RK=RSPHER-FLOAT(J-l)*XLEN
X0(K)=RK*SR*CPR
YO(K)=RK*SR*SPR
ZO(K)=RK*CR
CONTINUE
NUMBERED GROUPS ON A MOLECULE
DO 95 J=2,NAM,2
K=IADD+J
RDIS=RSPHER-FLOAT(J—1)*XLEN
RK=SQRT(RDIS*RDIS+ZLENSQ)
BETA=ASIN(ZLEN/RK)
THE=THR-BETA
ST=SIN(THE)
XO(K)=RK*ST*CPR
YO(K)=RK*ST*SPR
ZO(K)=RK*COS(THE)
CONTINUE
CONTINUE
RETURN
END
SUBROUTINE INTVEL(AHEAT,PART)
C

193
C === ASSIGN INITIAL VELOCITIES TO ATOMS
C
COMMON/NUM/NM,NAM,NP,NP1,NP2,NP22,KSORT,KB,NABTOT
COMMON/VEL/X1(216),Y1(216),Z1(216)
COMMON/MASS/MASS(216)
SUMX=0.
SUMY=0.
SUMZ=0.
DO 200 1=1,NM
XX=RANF(DUM)
YY=RANF(DUM)
ZZ=RANF(DUM)
NX=(1-1)*NAM
DO 200 J=1,NAM
XI(J+NX)=XX
Y1(J+NX)=YY
Z1(J+NX)=ZZ
SUMX=SUMX+X1(J+NX)*MASS(J+NX)
SUMY=SUMY+Y1(J+NX)*MASS(J+NX)
SUMZ=SUMZ+Z1(J+NX)*MASS(J+NX)
200 CONTINUE
C
C === SCALE VELOCITIES SO THAT TOTAL MOMENTUM=ZERO
X=0.
DO 210 1=1,NP

194
XI(I)=X1(I)-SUMX/PART/MASS(I)
Y1(I)=Y1(I)-SUMY/PART/MASS(I)
Z1(I)=Z1(I)-SUMZ/PART/MASS(I)
X=X+(X1(I)**2+Yl(I)**2+Zl(I)**2)*MASS(I)
210 CONTINUE
C
C === SCALE VELOCITIES TO DESIRED TEMPERATURE
HEAT=SQRT(AHEAT/X)
DO 220 1=1,NP
XI(I)=X1(I)*HEAT
Y1(I)=Y1(I)*HEAT
Z1(I)=Z1(1)*HEAT
220 CONTINUE
RETURN
END
SUBROUTINE PREDCT(NP)
C
C === USE TAYLOR SERIES TO PREDICT POSITIONS & THEIR
C DERIVATIVES AT NEXT TIME-STEP
C
COMMON/POS/XO(216),Y0(216),Z0(216)
COMMON/VEL/X1(216),Y1(216),Z1(216)
COMMON/DER/X2(216),Y2(216),Z2(216),X3(216),Y3(216),

195
& Z3(216),X4(216),Y4(216),Z4(216),X5(216),Y5(216),
&Z5(216)
COMMON/FOR/FX(216),FY(216),FZ(216)
DO 300 1=1,NP
XO(I)=X0(I)+X1(I)+X2(I)+X3(I)+X4(I)+X5(I)
YO(I)=Y0(I)+Y1(I)+Y2(I)+Y3(I)+Y4(I)+Y5(I)
ZO(I)=Z0(I)+Z1(I)+Z2(I)+Z3(I)+Z4(I)+Z5(I)
XI(I)=X1(I)+2.*X2(I)+3.*X3(I)+4.*X4(I)+5.*X5(I)
Y1(I)=Y1(I)+2.*Y2(I)+3.*Y3(I)+4.*Y4(I)+5.*Y5(I)
Z1(I)=Z1(I)+2.*Z2(I)+3.*Z3(I)+4.*Z4(I)+5.*Z5(I)
X2(I)=X2(I)+3.*X3(I)+6.*X4(I)+10.*X5(I)
Y2(I)=Y2(I)+3.*Y3(I)+6.*Y4(I)+10.*Y5(I)
Z2(I)=Z2(I)+3.*Z3(I)+6.*Z4(I)+10.*Z5(I)
X3(I)=X3(I)+4.*X4(I)+10.*X5(I)
Y3 (I) =Y3 (I) +4 . *Y4 (I) +10 . *Y5 (I)
Z3(I)=Z3(I)+4.*Z4(I)+10.*Z5(I)
X4(I)=X4(I)+5.*X5(I)
Y4(I)=Y4(I)+5.*Y5(I)
Z4(I)=Z4(I)+5.*Z5(I)
FX(I)=0•
FY(I)=0.
FZ(I)=0.
300 CONTINUE
RETURN
END

196
SUBROUTINE CORR(DELTSQ)
C
c === CORRECT PREDICTED POSITIONS AND THEIR DERIVATIVES
C
COMMON/POS/XO(216),Y0(216),Z0(216)
COMMON/VEL/X1(216),Y1(216),Z1(216)
COMMON/DER/X2(216),Y2(216),Z2(216),X3(216),Y3(216),
& Z3(216),X4(216),Y4(216),Z4(216),X5(216),Y5(216),
& Z5(216)
COMMON/FOR/FX(216),FY(216),FZ(216)
COMMON/DISP/DAX(216),DAY(216),DAZ(216),X0L(216),
& Y0L(216),ZOL(216)
COMMON/PARM/FO2,F12,F32,F42 , F52
COMMON/NUM/NM,NAM,NP,NP1,NP2,NP22,KSORT,KB,NABTOT
COMMON/PROP/SUME,XSUM
COMMON/SCAL/RSPHER,RWALL,DELRS,ISCALE
COMMON/MASS/MASS(216)
C
IF(MOD(KB,10).NE.0.OR.ISCALE.NE.1) GOTO 610
C IF(RSPHER .LE. 2.800) GO TO 610
RS PHER=RSPHER-DELRS
RWALL =RWALL -DELRS
C

197
610 DO 690 1=1,NP
XERR=X2(I)-DELTSQ*FX(I)/MASS(I)
YERR=Y2(I)-DELTSQ*FY(I)/MASS(I)
ZERR=Z2(I)-DELTSQ*FZ(I)/MASS(I)
XO(I)=X0(I)-XERR*F02
XI(I)=X1(I)-XERR*F12
X2(I)=X2(I)-XERR
X3(I)=X3(I)—XERR*F32
X4(I)=X4(I)-XERR*F42
X5(I)=X5(I)-XERR*F52
YO(I)=Y0(I)-YERR*F02
Y1(I)=Y1(I)-YERR*F12
Y2(I)=Y2(I)-YERR
Y3(I)=Y3(I)-YERR*F32
Y4(I)=Y4(I)-YERR*F42
Y5(I)=Y5(I)-YERR*F52
ZO(I)=Z0(I)-ZERR*F02
Z1(I)=Z1(I)-ZERR*F12
Z2(I)=Z2(I)-ZERR
Z3(I)=Z3(I)-ZERR*F32
Z4(I)=Z4(I)-ZERR*F42
Z5(I)=Z5(I)-ZERR*F52
690 CONTINUE
IF(MOD(KB,KSORT).NE.0.OR.ISCALE.NE.1) GOTO 680
DO 691 1=1,NP

198
RSQ=XO(I)*X0(I)+Y0(I)*YO(I)+ZO(I)*ZO(I)
R=SQRT(RSQ)
REACT = 1.O-DELRS/R
XO(I) = XO(I)*RFACT
YO(I) = YO(I)*RFACT
ZO(I) = ZO(I)*RFACT
691 CONTINUE
C
C === DISPLACEMENTS
680 DO 692 1=1,NP
DAX(I)=DAX(I)-XO(I)+XOL(I)
DAY(I)=DAY(I)-YO(I)+YOL(I)
DAZ(I)=DAZ(I)-ZO(I)+ZOL(I)
C
C === STORE NEW POSITIONS
XOL(I)=X0(I)
YOL(I)=Y0(I)
ZOL(I)=Z0(I)
692 CONTINUE
RETURN
END
&
SUBROUTINE EQBRAT(SUMVEL,AHEAT,TDIST,XDIST,NP,
IFLG,KB,NAM)

199
C
c === SCALE VELOCITIES DURING INITIAL TIME-STEPS
C
COMMON/VEL/XI(216),Y1(216)fZ1(216)
COMMON/PROP/SUME,XSUM
COMMON/SCAL/RSPHER,RWALL,DELRS,ISCALE
C
IF(IFLG.EQ.-l) GOTO 720
IF(TDIST.GT•XDIST.OR.IFLG.EQ.-2) GOTO 750
720 HEAT=SQRT(AHEAT/SUMVEL)
DO 730 1=1,NP
XI(I)=X1(I)*HEAT
Y1(I)=Y1(1)*HEAT
Z1(I)=Z1(I)*HEAT
730 CONTINUE
RETURN
C
C === AT END OF EQUILIBRATION STAGE, SET PROPERTY SUMS
C TO ZERO
750 IFLG= 1
KB=0
SUME=0.
XSUM=0.
RETURN
END

200
SUBROUTINE EVAL(RWALL)
C
COMMON/POS/XO(216),Y0(216),Z0(216)
COMMON/FOR/FX(216),FY(216),FZ(216)
COMMON/NABLST/LIST(12000),NABORS(216)
COMMON/CONT/GAB,CTO,GTHA,GTHE,BO,B2I,A1,AO,EPS
COMMON/PROP/SUME,XSUM
COMMON/PROPA/RC,RLIST,FSHFT,ESHFT,ESHFTA,CUBE,CUBE2,
& RDMAX,RCHH,HSHFT,HSHFTA,HFSHFT,REQUIL,FORCON
COMMON/ENERGY/TOTE,TOTLJ,ETOR,EBON,EBEN,EINTR,TRFRAC,
& EXTPRS,FTOTWL
COMMON/NUM/NM,NAM,NP,NP1,NP2,NP22,KSORT,KB,NABTOT
DIMENSION B(216),BX(216),BY(216),BZ(216)
C
C INITIALIZE ACCUMULATORS
NABTOT=0
ICOUNT=0
K=0
ITRAN =0
TOTE=0.0
TOTLJ=0.0
FTOTWL=0.0
ETOR=0.0

201
EBON=0.0
EBEN=0.0
EINTR=0.0
DO 150 1=1,NP
FX(I)=0.0
FY(I)=0.0
FZ(I)=0.0
150 CONTINUE
C
C CALCULATE LJ FORCES
LCHK=2
IF(MOD(KB,KSORT).EQ.O) LCHK=1
C
C OUTER LOOP OVER PARTICLES
DO 300 1=1,NP1
ICHK=I-1+NAM
XI=X0(I)
YI=Y0(I)
ZI=Z0(I)
IF(I•GE.NP2) GOTO 788
IF(LCHK.EQ.1) GOTO 410
JBEGIN=NABORS(I)
JEND=NABORS(1+1)-1
GOTO 415
410
NABORS(I) =K+1

202
JBEGIN=I+1
JEND=NP
415 CONTINUE
C
C DECIDE WHICH MOLECULE ATOM I IS ON
IM=(I—1)/NAM+1.0001
C
C INNER LOOP
DO 290 JX=JBEGIN,JEND
J=JX
IF(LCHK.EQ.2) J=LIST(JX)
C
C IF I & J ARE HEAD GROUPS, CALCULATE HEAD-HEAD REPULSIONS
J CHK=J+NAM-1
IF(MOD(ICHK,NAM).NE.0.OR.MOD(JCHK,NAM).NE.O)GOTO 614
X=XI-X0(J)
Y=YI-Y0(J)
Z=ZI-Z0(J)
RSQ=X*X+Y*Y+Z*Z
RIJ=SQRT(RSQ)
RI=1.0/RIJ
R3=RI*RI*RI
R4=R3 *RI
R8=R4 *R4
C

203
ELJ=R3*(R8*RI + 1.0) + (RIJ *HSHFTA)-HSHFT
FLJS=3.0*R4*(4.0*R8*RI+1.0)-HFSHFT
FXD=FLJS*X*RI
FYD=FLJS*Y*RI
FZD=FLJS*Z*RI
FX(I)=FX(I)+ FXD
FX(J)=FX(J)-FXD
FY(I)=FY(I)+FYD
FY(J)=FY(J)-FYD
FZ(I)=FZ(I)+FZD
FZ(J)=FZ(J)-FZD
GO TO 290
C
C DECIDE WHICH MOLECULE ATOM J IS ON
614 JM=(J-l)/NAM+1.0001
C
C CHECK WHETHER I AND J ARE ON SAME MOL
IF(IM.NE.JM) GOTO 289
C
C CHECK IF I AND J ARE FOURTH NEIGHBORS OR GREATER
KIJ=J—I
IF(KIJ.LE.3) GOTO 290
C
289 X=XI-X0(J)
Y=YI-Y0(J)

204
Z=ZI-Z0(J)
c
RSQ=X*X+Y*Y+Z*Z
RIJ=SQRT(RSQ)
IF(LCHK.NE.l) GOTO 419
IF(IM.EQ.JM) GOTO 701
IF(RSQ•GT.RDMAX) GOTO 290
IF(RSQ.GT.RLIST) GOTO 290
701 K=K+1
LIST(K)=J
419 IF(RIJ.GT.RC) GOTO 290
RI=1./RIJ
R3=RI**3
R6=R3 *R3
ELJ=2.0*R6*(R3-1.50)+ESHFT+RIJ*ESHFTA
TOTLJ=TOTLJ+ELJ
FLJS=18.*RI*R6*(R3-1.0)-FSHFT
FXD=FLJS*X*RI
FYD=FLJS*Y*RI
FZD=FLJS*Z*RI
FX(I)=FX(I)+FXD
FX(J)=FX(J)-FXD
FY(I)=FY(I)+FYD
FY(J)=FY(J)-FYD
FZ(I)=FZ(I)+FZD

205
FZ(J)=FZ(J)-FZD
290 CONTINUE
C
C CALCULATE BOND LENGTHS
C CHECK IF LAST ATOM ON MOLECULE
788 IF(MOD(I,NAM).EQ.O) GOTO 300
BX(I)=XI-X0(1+1)
BY(I)=YI—YO(1+1)
BZ(I)=ZI-Z0(1+1)
BSQ=BX(I)**2+BY(I)**2+BZ(I)**2
B(I)=SQRT(BSQ)
300 CONTINUE
C
IF(LCHK.NE.l) GOTO 499
NABORS(NP2)=K+1
NABTOT=K
C
C LOOP CALCULATES INTRAMOLECULAR FORCES
499 DO 600 1=1,NP1
C
C CHECK IF LAST ATOM ON MOLECULE
IF(MOD(I,NAM).EQ.O) GOTO 600
C
C CALCULATE BOND FORCES
BDIFF=B(I)-BO

206
EBON=EBON+0.5*GAB*BDIFF*BDIFF
FBON=GAB*BDIFF
BI=1./B(I)
BXI=BX(I)*BI
BYI=BY(I)*BI
BZI=BZ(I)*BI
FXB=FBON*BXI
FYB=FBON*BYI
FZB=FBON*BZI
FX(I) =FX(I)-FXB
FX(1+1)=FX(1+1)+FXB
FY(I) =FY(I)-FYB
FY(1+1)=FY(1+1)+FYB
FZ(I) =FZ(I)-FZB
FZ(1+1)=FZ(1+1)+FZB
C
C CALCULATION BENDING FORCE
C CHECK IF NEXT-TO-LAST ATOM ON MOLECULE
IF(MOD(1+1,NAM).EQ.O) GOTO 600
BII=1./B(I+1)
BXJ=BX(1+1)*BII
BYJ=BY(1+1)*BII
BZJ=BZ(1+1)*BII
CTI=-(BXJ*BXI+BYJ*BYI+BZJ*BZI)
CDIFF=CTO-CTI

207
EBEN=EBEN+0.5*GTHA*CDIFF*CDIFF
FT=GTHA*CDIFF
FT1=FT*BI
FT2=FT*BII
FX1=FT1*(BXJ+CTI*BXI)
FY1=FT1*(BYJ+CTI*BYI)
FZ1=FT1*(BZJ+CTI*BZI)
FX2=FT2*(BXI+CTI*BXJ)
FY2=FT2*(BYI+CTI*BYJ)
FZ2=FT2 *(BZI+CTI*BZJ)
FX(I) =FX(I) -FX1
FX(1+1)=FX(1+1)+FX1-FX2
FX(1+2)=FX(1+2)+FX2
FY(I) =FY(I) -FY1
FY(1+1)=FY(1+1)+FY1-FY2
FY(1+2)=FY(1+2)+FY2
FZ(I) =FZ(I) -FZ1
FZ(1+1)=FZ(I+1)+FZ1-FZ2
FZ(1+2)=FZ(1+2)+FZ2
C
C CALCULATION OF TORSION FORCES
C CHECK WHETHER FIRST ATOM OF MOLECULE
IF(MOD(I+NAM-1,NAM).EQ.O) GOTO 600
CALL RYFOR(BX(1-1),BY(I-1),BZ(I-1),BX(I),BY(I),BZ(I),
& BX(1+1),BY(1+1),BZ(1+1),FX1,FY1,FZ1,FX2,FY2,FZ2,

208
& FX3,FY3,FZ3,FX4,FY4,FZ4,ETQ,CODI,GTHE)
C
ICOUNT=ICOUNT+l
IF(CODI.LT.-0.50) ITRAN=ITRAN+1
FX(I-l)=FX(I-1)+FX1
FY(1-1)=FY(1-1)+FY1
FZ(1-1)=FZ(1-1)+FZ1
FX(I) =FX(I) +FX2
FY(I) =FY(I) +FY2
FZ(I) =FZ(I) +FZ2
FX(1+1)=FX(1+1)+FX3
FY(1+1)=FY(1+1)+FY3
FZ(1+1)=FZ(I+1)+FZ3
FX(1+2)=FX(1+2)+FX4
FY(1+2)=FY(1+2)+FY4
FZ(1+2)=FZ(1+2)+FZ4
ETOR=ETOR+ETQ
600 CONTINUE
C
C CALCULATE GROUP-SHELL INTERACTION
EWALL=0.0
RCHKSQ=(RWALL-2.0)**2
DO 650 1=1,NP
ICHK=I-1+NAM
C

209
XI=X0(I)
YI=Y0(I)
ZI=Z0(I)
c
C DISTANCE FROM CENTER TO GROUP
RSQ=XI*XI+YI*YI+ZI*ZI
C IF HEAD GROUP, CALCULATE INTERACTIONS WITH WALL USING A
C POTENTIAL OF THE FORM — GAMMA*(R-RO)**2
IF(MOD(ICHK,NAM).NE.O) GO TO 384
RR=SQRT(RSQ)
RI=1.0/RR
RD=RWALL-RR
C
EWALL=EWALL+FORCON*(RD-REQUIL)*(RD-REQUIL)
FWALL=-2.0*FORCON*(RD-REQUIL)
FTOTWL=FTOTWL+FWALL
C
FX(I)=FX(I)-FWALL*XI*RI
FY(I)=FY(I)-FWALL*YI*RI
FZ(I)=FZ(I)-FWALL*ZI*RI
GO TO 650
C IF NOT A HEAD GROUP, CALCULATE REPULSIVE INTERACTION WITH
C THE WALL
384 IF(RSQ.LT.RCHKSQ) GOTO 650
C

210
C DISTANCE FROM GROUP TO SHELL
RR=SQRT(RSQ)
RI=1.0/RR
RD=RWALL-RR
RDR=1.0/RD
R3 =RDR*RDR*RDR
R6 =R3 *R3
R12=R6*R6
C
C FORCE AND ENERGY FOR R-12 WALL
EWALL=EWALL+R12
FWALL=12.0*R12 *RDR
FTOTWL=FTOTWL+FWALL
FX(I)=FX(I)-FWALL*XI*RI
FY(I)=FY(I)-FWALL*YI*RI
FZ(I)=FZ(I)-FWALL*ZI*RI
650 CONTINUE
C
C CALCULATE WALL PRESSURE IN ATMOSPHERES
EXTPRS=0.0085382* FTOTWL/RWALL** 2
C
C CALCULATE INTRAMOLECULAR AND TOTAL ENERGY
EINTR=ETOR + EBON + EBEN
TOTE = TOTLJ + EINTR + EWALL
TRFRAC=FLOAT(ITRAN)/FLOAT(ICOUNT)

211
RETURN
END
SUBROUTINE RYFOR(BONX1,BONY1,BONZ1,BONX2,BONY2,BONZ2,
& BONX3,BONY3,BONZ3,FORX1,FORY1,FORZ1,FORX2,FORY2,
& FORZ2,FORX3,FORY3,FORZ3,FORX4,FORY4,FORZ4,ETOR,
& CODI,GR)
C
DATA Al,A2,A3,A4,A5/-1.462,-1.578,0.368,
& 3.156,3.788/
C
C CALCULATE NORMAL TO PLANE OF BONDS 1 AND 2
E=BONYl*BONZ2-BONZl*BONY2
D=BONZ1* BONX2-BONX1* BONZ 2
C=BONX1* BONY 2-BONY1* BONX2
ANORM2=C*C+D*D+E*E
ANORM =SQRT(ANORM2)
ANORMR=l./ANORM
AX=E*ANORMR
AY=D*ANORMR
AZ=C*ANORMR
C
C CALCULATE NORMAL TO PLANE OF BONDS 2 AND 3
H=BONY2* BONZ 3-BONZ 2 * BONY 3

212
G=BONZ2*BONX3-BONX2*BONZ3
F=B0NX2 *B0NY3-B0NY2 *B0NX3
BN0RM2=F*F+G*G+H*H
BNORM =SQRT(BNORM2)
BNORMR=l./BNORM
BX=H*BNORMR
BY=G*BNORMR
BZ=F*BNORMR
C
C DETERMINE COSINE OF THE DIHEDRAL ANGLE, PHI
CODI= AX * BX+AY * BY+AZ * BZ
C
C DETERMINE TORSIONAL ENERGY
CODISQ=CODI*CODI
CODI3 =CODI*CODISQ
ETOR = GR*(1.116+Al*CODI+A2*CODISQ+A3*CODI3
& +A4 *CODISQ*CODISQ+A5*CODISQ*CODI3)
C
C CALCULATE DU/COS(PHI)
DUDCO=(Al+2.*A2 *CODI + 3.*A3*CODISQ+4.*A4*CODISQ*CODI
& +5.*A5*CODISQ*CODISQ)*GR
C
C START EVALUATION OF FORCE ON GROUP 1
C CALCULATE DANORM/DR1
FACT1=(C*BONY2-D*BONZ2)/ANORM2

213
DADX1X=
= -FACT1*AX
DADX1Y=
= -FACTl*AY-BONZ2*ANORMR
DADX1Z=
= -FACTl*AZ+BONY2*ANORMR
FACT2 =
=(E*BONZ2-C*BONX2)/ANORM2
DADY1X=
= -FACT2 *AX+BONZ2 *ANORMR
DADY1Y=
= -FACT2 *AY
DADY1Z=
= -FACT2 *AZ-BONX2 *ANORMR
FACT3 =
=(D*BONX2-E*BONY2)/ANORM2
DADZ 1X=
= -FACT3 *AX-BONY2 *ANORMR
DADZ1Y=
= -FACT3 *AY+BONX2 *ANORMR
DADZ1Z=
= -FACT3*AZ
C
C CALCULATE D(AB)/DR1
DABDX1=
=BX*DADX1X+BY*DADX1Y+BZ*DADX1Z
DABDY1=
=BX*DADY1X+BY*DADY1Y+BZ*DADY1Z
DABDZ1=
=BX*DADZ1X+BY*DADZ1Y+BZ*DADZ1Z
C CALCULATE FORCE ON GROUP 1
FORXl=
-DUDCO*DABDXl
FORYl=
-DUDCO*DABDYl
FORZl=
-DUDCO*DABDZ1
C
C START EVALUATION OF FORCE ON GROUP 2

214
C CALCULATE DAN0RM/DR2
FACT4 =(D*BONZ1-C*B0NY1)/AN0RM2
DADX2X= -FACT4 *AX-DADX1X
DADX2Y= -FACT4 *AY+BONZ1*AN0RMR-DADX1Y
DADX2Z= -FACT4 *AZ-B0NY1*AN0RMR-DADX1Z
C
FACT5 =(C*B0NX1-E*B0NZ1)/AN0RM2
DADY2X= -FACT5*AX-B0NZ1*AN0RMR-DADY1X
DADY2Y= -FACT5*AY-DADY1Y
DADY2Z= -FACT5*AZ+B0NX1*AN0RMR-DADY1Z
C
FACT6 =(E*B0NY1-D*B0NX1)/AN0RM2
DADZ 2 X= -FACT6 *AX+BONY1*ANORMR-DADZIX
DADZ2Y= -FACT6*AY-B0NX1*AN0RMR-DADZ1Y
DADZ2Z= —FACT6*AZ-DADZ1Z
C
C EVALUATE D(BNORM)/DR2
FACT7 =(F*BONY3-G*BONZ3)/BNORM2
DBDX2X= -FACT7 *BX
DBDX2Y= -FACT7 *BY-BONZ3*BNORMR
DBDX2Z= -FACT7 *BZ+BONY3 *BNORMR
C
FACT8 =(H*BONZ3-F*BONX3)/BNORM2
DBDY2X= -FACT8*BX+BONZ3*BNORMR
DBDY2Y= -FACT8 *BY

215
DBDY2Z= -FACT8 *BZ-B0NX3 *BNORMR
C
FACT9 =(G*BONX3-H*BONY3)/BN0RM2
DBDZ2X= -FACT9*BX-BONY3 *BNORMR
DBDZ2Y= -FACT9*BY+BONX3*BNORMR
DBDZ2Z= -FACT9*BZ
C
C CALCULATE D C0S(PHI)/DR2
DABDX2=BX* DADX2 X+BY * DADX2 Y+BZ* DADX2Z
& +AX* DBDX2 X+AY * DBDX2 Y+AZ * DBDX2 Z
DABDY2=BX*DADY2X+BY*DADY2Y+BZ*DADY2Z
& +AX*DBDY2X+AY*DBDY2Y+AZ*DBDY2Z
DABDZ2=BX*DADZ2X+BY*DADZ2Y+BZ*DADZ2Z
& +AX*DBDZ2X+AY*DBDZ2Y+AZ*DBDZ2Z
C
C FORCE ON GROUP 2
FORX2 = -DUDCO*DABDX2
FORY2 = -DUDCO*DABDY2
FORZ2 = -DUDCO*DABDZ2
C START EVALUATION OF FORCE ON GROUP 4
C CALCULATE D(BNORM)/DR4
FACT1 =(F*BONY2-G*BONZ2)/BNORM2
DBDX4X= —FACT1*BX
DBDX4Y= -FACTl*BY-BONZ2 *BNORMR

216
DBDX4 Z= -FACT1*BZ+B0NY2*BN0RMR
C
FACT2 =(H*BONZ2-F*BONX2)/BN0RM2
DBDY4X= -FACT2 *BX+B0NZ2 *BNORMR
DBDY4Y= -FACT2*BY
DBDY4Z= -FACT2*BZ-BONX2*BNORMR
C
FACT3 =(G*BONX2-H*BONY2)/BN0RM2
DBDZ4X= -FACT3 *BX-B0NY2 *BNORMR
DBDZ4Y= -FACT3 *BY+B0NX2 *BNORMR
DBDZ4Z= -FACT3 *BZ
C
C EVALUATE D C0S(PHI)/DR4
DABDX4 =AX* DBDX4 X+AY * DBDX4 Y+AZ * DBDX4 Z
DABDY4=AX*DBDY4X+AY*DBDY4Y+AZ*DBDY4Z
DABDZ4=AX*DBDZ4X+AY*DBDZ4Y+AZ*DBDZ4Z
C
C EVALUATE FORCE ON GROUP 4
FORX4= -DUDCO*DABDX4
FORY4= -DUDCO*DABDY4
FORZ4= -DUDCO*DABDZ4
C
C EVALUATE FORCE ON GROUP 3 BY NEWTON'S THIRD LAW
FORX3 = —FORX1-FORX2-FORX4
FORY3
-FORY1-FORY2-FORY4

217
F0RZ3 = -FORZ1-FORZ2-FORZ4
RETURN
END
BLOCK DATA
COMMON/DER/X2(216),Y2(216),Z2(216),X3(216),Y3(216),
& Z3(216),X4(216),Y4(216),Z4(216),X5(216),Y5(216),
& Z5(216)
COMMON/FOR/FX(216),FY(216),FZ(216)
COMMON/DIS P/DAX(216) ,DAY(216) ,DAZ(216) fX0L(216) ,
& YOL(216),ZOL(216)
COMMON/PROP/SUME,XSUM
COMMON/MASS/MASS(216)
DATA X3,Y3,Z3,X4,Y4,Z4,X5,Y5,Z5/1944*0./
DATA DAX,DAY,DAZ,FX,FY,FZ/1296*0./
DATA MASS/216*!/
END

APPENDIX E
ESTIMATION OF RUN 3 HEAD GROUP PARAMETERS
In Run 3 of the molecular dynamics simulations of
micelles, the head group was given the mass and diameter of
a sulfate ion. While it was desirable to model this ion as
the polar head due to its common usage in surfactants,
information on its intramolecular potentials is incomplete.
Data pertaining to other molecular groups was used to
estimate intramolecular potential parameters necessary to
complete the model of the sulfate head group (O'Connell,
1987) .
Muller and coworkers have measured the mean bond
lengths, bond angles, and the vibrational and bending
amplitudes for the HSO4- ion (Muller and Nagarajan, 1967;
Muller et al., 1968). The values are assigned to the bonds
and angles in Figure E-la. Blukis et al. (1963) have
measured the C-0 bond length and C-C-0 angle of dimethyl
ether. These values are given in Figure E-lb. These are
used in the estimation of bond lengths, angles and
amplitudes for the C-O-S-O3- arrangement of the sulfate head
group and the alpha carbon of the surfactant chain.
218

219
Figure E-l. a.) Schematic representation of the the HSO4-
ion, showing mean bond lengths, bond angles, and vibrational
and bending amplitudes (Muller and Nagarajan, 1967; Muller
et al., 1968). b.) The C-C-0 bonds of dimethyl ether,
showing the mean angle and the mean length of the C-0 bond
(Blukis et al., 1963).

220
In the simulation model, the head group is represented
as a homogeneous sphere connected to the surfactant chain
through a single bond. The center of mass of the atoms of
the sulfate group determines the length and angle of this
bond. In Figure E-2, the sulfate group and alpha carbon are
schematically represented. The appropriate bond lengths and
angles have been used from Figure E-l. The center of mass
of the sulfate group is shown in Figure E-2, and the dashed
line connecting it to the alpha carbon is the bond for the
model head group. Its length, 2.6 Angstrom, and angle, 146
degrees, are indicated in the figure.
The force constant for the harmonic bond vibration
potential used in the simulation is determined from the mean
amplitude of the vibration:
y = 1 .68 x 10"4-^t (E.l)
a
where y is the force constant in mdyne/Angstrom.
/i is the reduced mass of the two molecular groups,
a is the mean amplitude.
For the head group bond (dashed line in Figure E-2) a mean
amplitude of vibration is calculated from its components to
be .078 Angstrom. In equation E.l this yields vibrational
force constant for the head group bond to the alpha carbon
of .37 mdyne/Angstrom.

221
Figure E-2. The sulfate head group, shown with two segments
of the surfactant chain. Its center of mass is designated
by the "*" and the dashed line connecting it to the alpha
carbon designates the bond length and angle of the head
group of Run 3. In this simulation, the sulfate group is
modeled by a single sphere, centered at the sulfate center
of mass, with intramolecular potentials derived from the
sulfate group.

222
The force constant for the bond angle bending potential
is proportional to the mass of the head group. It is
therefore seven times greater in Run 3 than in the other
runs. Note that although Run 2 used the same head group
mass, only the mass effect was investigated in that
simulation; all intramolecular potentials were the same as
in Run 1.
Rotational potential barriers were obtained for
molecular groups of geometry similar to that of the sulfate
group (Cahill et al., 1968). They are, in kcal/mole:
methyl formate 1.2
propionaldehyde 2.3
methyl vinyl ether 3.8
cis-l-butene 4.0
By comparison to these values the rotational barrier for the
larger, more polar sulfate group is estimated to be 5.0
kcal/mole.

APPENDIX F
MOLECULAR DYNAMICS PROGRAM LISTING
FOR SIMULATION 3

c
MOLECULAR DYNAMICS SOURCE PROGRAM FOR A SIMPLE
C
MODEL OF A MICELLE ENCLOSED IN A SHELL. A
C
FIFTH-ORDER PREDICTOR-CORRECTOR ALGORITHM IS
C
USED TO SOLVE THE EQUATIONS OF MOTION. A
C
SKELETAL MODEL DUE TO WEBER IS USED FOR EACH
C
ALKANE MOLECULE
C
C
HEAD GROUP MASS CAN BE SET. SHIFTED-FORCE
C
POTENTIALS AND NEIGHBOR LIST ARE USED
C
PARAMETER(MHEAD=1)
COMMON/POS/XO(216),Y0(216),Z0(216)
COMMON/VEL/Xl(216),Y1(216),Z1(216)
COMMON/DER/X2(216),Y2(216),Z2(216),X3(216),Y3(216),
& Z3(216),X4(216),Y4(216),Z4(216),X5(216),Y5(216),
& Z5(216)
COMMON/FOR/FX(216),FY(216),FZ(216)
COMMON/DISP/DAX(216),DAY(216),DAZ(216),X0L(216),
& Y0L(216),ZOL(216)
COMMON/NABLST/LIST(12000),NABORS(216)
COMMON/CONT/GAB,CTO,GTHA,GTHE,BO,B2I,A1,AO,EPS
COMMON/PARM/FO2,F12,F32,F42,F52
COMMON/PROP/SUME,XSUM
COMMON/PROPA/RC,RLIST,FSHFT,ESHFT,ESHFTA,
& CUBE,CUBE2,RDMAX,RCHH,HSHFT,HSHFTA,HFSHFT,REQUIL,
224

225
COMMON/CONT/Al,AO,EPS
COMMON/PARM/FO2,F12,F32,F42,F52
COMMON/PROP/SUME,XSUM
COMMON/PROPA/REQUIL,FORCON,HDSIG,SIG12
COMMON/ENERGY/TOTE,TOTLJ,ETOR,EBON,EBEN,EINTR,TRFRAC,
& EXTPRS,FTOTWL
COMMON/NUM/NM,NAM,NP,NP1,NP2,NP22,KSORT,KB
COMMON/SCAL/RSPHER,RWALL,DELRS,ISCALE
COMMON/MASS/MASS(216)
COMMON/INTRA/BO(216),GAB(216),CT0(216),GTHA(216),
& GTHE(216)
C
C FLAG, IF IFLAG.NE.1 USES INTPOS AND RANDOM VELOCITIES
C IF IFLAG.EQ.l XO,YO,ZO,AND ALL DERIVATIVES ARE READ
C FROM PREVIOUS RUN.
C
IFLAG=1
C
C === SET NUMBER OF PARTICLES IN PRIMARY CELL
NAM=9
NM=2 4
NP=NM*NAM
PART=NP
NP1=NP-1
NP2=NP-2

226
NP22=.5*PART+.01
C
C === SET HEAD GROUP MASS, BOND, AND FORCE PARAMETERS
DO 20 1=1,NP,NAM
MASS(I)=MHEAD
BO(I)=BLHEAD
GAB(I)=GLHEAD
CTO(I)=BAHEAD
GTHA(I)=GAHEAD
GTHE(I)=GTHEAD
20 CONTINUE
C
C === SET RELATIVE SIGMA OF HEAD-ALKANE PAIR
HDSIG=HDSGMA
SIG12=(1.+HDSIG)/2
C
C === SET RADIUS OF SPHERE ON WHICH HEAD GROUPS ARE ATTACHED
RSPHER= 6.000
DELRS=0.0025
ISCALE=0
RWALL =RSPHER+1.0
REQUIL=1.
FORCON=3 0.
C
C === SET VALUES OF PHYSICAL CONSTANTS

227
XNAM=FLOAT(NAM)
WTM0L=XNAM*14.+2.
WTPART=WTMOL/XNAM
RSTAR=0.4
EPS=419.
AV0=6.0225E+23
B0LZ=1.38054D-23
EPSI=EPS/BOLZ/AVO
THIRD=1./3.
PI=3.1415926535
C
c === SET DESIRED FLUID STATE CONDITION
TR=5.913
T=TR*EPSI
VOL=l.333333*PI*RSPHER**3
DR=FLOAT(NM)/VOL
C
C === SET RUN FLAGS AND PARAMETERS
IFLG=-1
KB=0
KSAVE=10
KS0RT=10
KWRITE=10
MAXKB=4 0000
XDIST=0.1

228
C
C === SET TIME-STEP AND ITS MULTIPLES
DELTA=0.00060
DELSQ=DELTA*DELTA
DELTSQ=.5*DELSQ
TSTEP=SQRT(WTPART*RSTAR**2/EPS/1.E21)*1.D+12
TT1=TSTEP*DELTA*1.D-12
C
C === SET PARAMETERS IN PREDICTOR-CORRECTOR METHOD
F02=3./16.
F12=251./360.
F32=ll./18.
F42=l./6.
F52=l./60.
C
C === SCALE FACTOR FOR VELOCITIES DURING EQUILIBRATION
AHEAT=DELSQ*PART*3.*TR
C
c === WRITE TAPE HEADING
OPEN(20,FILE='C872DAT2',ACCESS='SEQUENTIAL',
& FORM='UNFORMATTED')
WRITE(20) NP,NAM,NM,DR,TR,EPS ,RSTAR
WRITE(20) VOL,DELTA,TSTEP
C
C === INITIALIZE SUM ACCUMMULATORS

229
XSUM=0.
SUME=0.
c
C === PRINT PARAMETERS
WRITE(*,900)
900 FORMAT(1H1///)
WRITE(*,902)
902 FORMAT(7X,49('*'))
WRITE(*,904)
904 FORMAT(7X,•*',T56,'*')
WRITE(*,906) NM
906 FORMAT(7X,'*',2X,'MOLECULAR DYNAMICS FOR',13,
A ' N-ALKANE MOLECULES',T56,'*')
WRITE(*,904)
WRITE(*,917) NAM
917 FORMAT(7X,'*',5X,'WITH',13,' PARTICLES PER MOLECULE'
& ,T56,'*')
WRITE(*,904)
WRITE(*,902)
WRITE(*,904)
WRITE(*,908) EPSI,RSTAR
908 FORMAT(7X,'*',2X,'EPSI/K = ',F7.3,T36,'RSTAR = ',
& F7.3,T56,'*')
WRITE(*,910) TR, T
910 FORMAT(7X,'*',2X,'TR = ',F7.3,T36,' T= ',

230
& F7.3,T56,'*')
WRITE(*,912) DR,VOL
912 FORMAT(7X,,2X,'DR = ',F7.3,T36,'VOL =
& F8.3,T56,'*')
WRITE(*,920) DELTA
920 FORMAT(7X,'*',2X,'DELTA = ',F9.5,T56,'*')
WRITE(*,904)
WRITE(*,922) TSTEP
922 FORMAT(7X,'*',2X,'TIME UNIT = ',F6.3,'E-12 SEC',
& T56,'*')
WRITE(*,924) TT1
924 FORMAT(7X,'*',2X,'TIME STEP= ',1PE10.3,' SEC',
& T56,'*')
WRITE(*,904)
WRITE(*,926) DE
926 FORMAT(7X,'*',2X,'ENERGY CORRECTION = ',F7.3,
& T56,'*')
WRITE(*,904)
WRITE(*,902)
WRITE(*,932) RSPHER
932 FORMAT(////lOX,'RADIUS OF SPHERE = ',F6.2)
WRITE(*,934) GTHE(2)
934 FORMAT(//10X,'ROTATIONAL POTENTIAL PARAMETER = ',
& F7.3/)
C

231
C
c
c
OPEN(30,FILE='LASTDAT1,ACCESS='SEQUENTIAL',
& FORM='UNFORMATTED')
IF(IFLAG.EQ.1) GOTO 199
C
C === LOAD INITIAL POSITIONS OF ATOMS
CALL INTPOS(RSPHER)
CALL POSPRI(KB,NM,NAM)
C
C === PRINT RUN-TABLE HEADING
WRITE(*,930)
930 FORMAT(1H1////4X,'KB',5X,'RSPH',4X,'ENRG',5X,'El',
A 4X,'DIST',6X,'TEMP',3X,'TRFRC',3X,'TOT ENR',
A 3X,'FTOTWL',3X,'PRESSURE'/)
C
C === LOAD INITIAL VELOCITIES OF ATOMS
CALL INTVE L(AHEAT,PART)
C
C === ASSIGN INITIAL ACCELERATIONS BASED ON INITIAL
C POSITIONS
CALL EVAL(RWALL)
C
C === SCALE ACCELERATIONS AND STORE STARTING POSITIONS

232
DO 530 1=1,NP
X2(I)=FX(I)*DELTSQ/MASS(I)
Y2(I)=FY(I)*DELTSQ/MASS(I)
Z2(I)=FZ(I)*DELTSQ/MASS(I)
530 CONTINUE
GOTO 188
C
c READ POSITIONS AND DERIVATIVES INSTEAD OF USING
c FCC, INVEL, ACCELERATION
c
199 WRITE(*,930)
READ(30) KB,IFLG,NM,NAM,RS PHER,XSUM,SUME
RWALL=RS PHER+1.0
DO 200 K=1,
NP
READ(30)
XO(K),YO(K),ZO(K)
READ(30)
XI(K),Y1(K),Z1(K)
READ(30)
X2(K),Y2(K),Z2(K)
READ(30)
X3(K),Y3(K),Z3(K)
READ(30)
X4(K),Y4(K),Z4(K)
READ(30)
X5(K),Y5(K),Z5(K)
READ(30)
DAX(K),DAY(K),DAZ(K)
CONTINUE
REWIND 30
188 DO 377 1=1,NP
XOL(I)=X0(I)

YOL(I)=YO(I)
ZOL(I)=ZO(I)
233
377 CONTINUE
C
c
c
C === ENTER MAIN LOOP OF SIMULATION
IF(IFLAG.NE.1) GOTO 777
CALL PREDCT(NP)
CALL EVAL(RWALL)
CALL CORR(DELTSQ)
IF(ISCALE.EQ.1) RSPHER=RSPHER+DELRS
RWALL=RSPHER+1.0
777 NS=KB+1
DO 599 NTIMES=NS,MAXKB
KB=KB+1
CALL PREDCT(NP)
CALL EVAL(RWALL)
CALL CORR(DELTSQ)
C
C === CALCULATE MEAN SQUARE DISPLACEMENT & KINETIC ENERGY
SUMVEL=0.
TDIST=0.
DO 540 1=1,NP
TDIST=TDIST+DAX(I)**2+DAY(I)**2+DAZ(I) **2

234
SUMVEL=SUMVEL+(X1(I)**2+Yl(I)**2+Zl(I)**2)*
& MASS(I)
540 CONTINUE
TDIST=TDIST/PART
EK=SUMVEL/(2.*PART*DELSQ)
C
C === ACCUMMULATE SUMS FOR PROPERTY AVERAGES
XSUM=XSUM+SUMVEL
SUME=SUME+TOTE
C
c === PROPERTY CALCULATION & PRINT-OUT AT INTERVALS
IF(MOD(KB,KWRITE).NE.O) GOTO 550
FKB=FLOAT(KB)*PART
TMP=XSUM/(3.*DELSQ*FKB)
ENR=(SUME/FKB)
El=TOTE/PART
ET0T=E1+EK
RLTIM=DELTA*FLOAT(KB)*TSTEP
WRITE(*,940) KB,RSPHER,ENR,El,TDIST,TMP,TRFRAC,
& ETOT,FTOTWL,EXTPRS
C
IF(MOD(KB,1000).NE.0) GOTO 550
CALL POS PRI(KB,NM,NAM)
WRITE(*,930)
940
FORMAT(1H ,16,F6.3,3F9.2,F10.3,IX,F6.3,IX,F9.3,

235
& 2(2X,F9.3))
C
C === DURING FIRST OF RUN, SCALE VELOCITIES FOR TEMPERATURE
550 IF(IFLG.LT.l) CALL EQBRAT(SUMVEL,AHEAT,TDIST,
& XDIST,NP,IFLG,KB,NAM)
C
C WRITE DATA ONTO TAPE FOR LATER USE
IF(KB.EQ.0) GOTO 777
C IF(IFLG.EQ.0) GOTO 588
IF(MOD(KB,10).NE.O) GOTO 588
WRITE(20) KB,TMP,ETOT,El,EK,TOTE,EINTR,TOTLJ,ETOR,
& EBON,EBEN
WRITE(20) XO,YO,ZO
WRITE(20) XI,Y1,Z1
WRITE(20) X2,Y2,Z2
C WRITE(20) X3,Y3,Z3
C WRITE(20) X4,Y4,Z4
C WRITE(20) X5,Y5,Z5
588 IF(MOD(NTIMES,100).NE.0) GOTO 599
WRITE(30) KB,IFLG,NM,NAM,RSPHER,XSUM,SUME
DO 598 K=1,NP
WRITE(30) X0(K),Y0(K),Z0(K)
WRITE(30) XI(K),Y1(K),Z1(K)
WRITE(30) X2(K),Y2(K),Z2(K)
WRITE(30) X3(K),Y3(K),Z3(K)

236
WRITE(30) X4(K),Y4(K),Z4(K)
WRITE(30) X5(K),Y5(K),Z5(K)
WRITE(30) DAX(K),DAY(K),DAZ(K)
598 CONTINUE
REWIND 30
599 CONTINUE
CLOSE(20)
CLOSE(30)
STOP
END
SUBROUTINE POSPRI(KB,NM,NAM)
COMMON/POS/XO(216),Y0(216),Z0(216)
C
11 = 0
WRITE(*,942) KB
942 FORMAT(1H1///7X,'POSITIONS OF GROUPS AT TIME-STEP
& ,16//)
DO 404 JJ=1,NM
WRITE(*,935)
935 FORMAT(//2 X, 'MOLECULE',7X, 'X' ,11X, 'Y',12X, 'Z',
& 12X,'R',9X,'BOND LEN'/)
DO 404 KK=1,NAM
11=11+1
RR=SQRT(XO(II)**2+Y0(II)**2+Z0(II)**2)
XXX=X0(II)-X0(II-l)

237
YYY=YO(II)-YO(II-l)
ZZZ=ZO(II)-ZO(II-l)
BB=SQRT(XXX*XXX+YYY*YYY+ZZZ*ZZZ)
IF(KK.EQ.l) WRITE(*,909) JJ,XO(II),YO(II),
& ZO(II),RR
IF(KK.GT.l) WRITE(*,911) XO(II),YO(II),ZO(II),
& RR,BB
404 CONTINUE
909 FORMAT(5X,12,4(5X,F8.4))
911 FORMAT(7X,5(5X,F8.4))
RETURN
END
SUBROUTINE INTPOS(RSPHER)
C
C SET-UP INITIAL POSITIONS OF PARTICLES. HEAD GROUPS ARE
C ASSIGNED TO FIXED POSITIONS ON THE SURFACE OF A SPHERE
C OF RADIUS "RSPHER".
C
COMMON/POS/XO(216),Y0(216),Z0(216)
COMMON/NUM/NM,NAM,NP,NP1,NP2,NP22,KSORT,KB
COMMON/CONT/A1,AO,EPS
COMMON/INTRA/BO(216),GAB(216),CT0(216),GTHA(216),
&
GTHE(216)

238
C
DIMENSION THETA(50),PHI(50)
PI=3.1415926535
PI180=PI/180.
C
C TWENTY-FOUR MOLECULES
THETA(1)=0.
PHI(1)=0.
THETA(24)=PI
PHI(24)=0.
PH24=0.
DO 70 1=2,5
PHI(I)=PH24
PH24=PH24+PI/2.
70 THETA(I)=36.*PI180
DELPHI=2.*PI/7
PH24=DELPHI/2.
DO 71 1=6,12
PHI(I)=PH24
PH24=PH24+DELPHI
71 THETA(I)=72.*PI180
PH24=0.
DO 72 1=13,19
PHI(I)=PH24
PH24=PH24+DELPHI

72
239
THETA(I)=108.*PI180
PH24=PI/4.
DO 73 1=20,23
PHI(I)=PH24
PH24=PH24+PI/2.
73 THETA(I)=144.*PI180
C
60 WRITE(*,909)
909 FORMAT(1H1////7X,'INITIAL POSITIONS OF HEAD GROUPS'
& //7X,'MOLECULE',5X,'THETA',8X,'PHI'/)
DO 101 1=1,NM
TTT=THETA(I)/PI18 0
PPP=PHI(I)/PI180
WRITE(*,900) I,TTT,PPP
900 FORMAT(10X,13,2(6X,F7.3))
101 CONTINUE
C
c
C STANDARD DISTANCES ALONG TRANS CHAIN
XLEN=0.5*B0(2)*SQRT(2.*(1.-CTO(2)))
ZLEN=SQRT(BO(2)*B0(2)-XLEN*XLEN)
Z LENSQ=Z LEN * Z LEN
C
C HEAD-TO-ALPHA DISTANCE
XLEN1=0.5*B0(1)*SQRT(2.*(1.-CTO(1)))

240
ZLEN1=SQRT(BO(1)*B0(1)-XLEN1*XLEN1)
ZLN1SQ=ZLEN1*ZLEN1
C
C === ADD ATOMS TO FORM MOL IN TRANS POS =
DO 120 1=1,NM
IADD=(I—1)*NAM
THR=THETA(I)
PHIR=PHI(I)
SR=SIN(THR)
CR=COS(THR)
SPR=SIN(PHIR)
CPR=COS(PHIR)
C
C HEAD GROUP ON A MOLECULE
J=1
K=IADD+J
RK=RSPHER
XO(K)=RK*SR*CPR
YO(K)=RK*SR*SPR
ZO(K)=RK*CR
C
C ALPHA GROUP ON A MOLECULE
J=2
K=IADD+J
RDISA=RSPHER-XLEN1

241
RK=SQRT(RDISA*RDISA+ZLN1SQ)
BETA=ASIN(ZLEN1/RK)
THE=THR-BETA
ST=SIN(THE)
XO(K)=RK*ST*CPR
YO(K)=RK*ST*SPR
ZO(K)=RK*COS(THE)
C
C ODD NUMBERED GROUPS ON A MOLECULE
DO 92 J=3,NAM,2
K=IADD+J
RK=RDISA-FLOAT(J-2)*XLEN
XO(K)=RK*SR*CPR
YO(K)=RK*SR*SPR
ZO(K)=RK*CR
92 CONTINUE
C
C EVEN NUMBERED GROUPS ON A MOLECULE
DO 95 J=4,NAM,2
K=IADD+J
RDIS=RDISA-FLOAT(J-2)*XLEN
RK=SQRT(RDIS*RDIS+ZLENSQ)
BETA=ASIN(Z LEN/RK)
THE=THR-BETA
ST=SIN(THE)

242
95
120
C
c ==
c
X0(K)=RK*ST*CPR
YO(K)=RK*ST*SPR
ZO(K)=RK*COS(THE)
CONTINUE
CONTINUE
RETURN
END
SUBROUTINE INTVEL(AHEAT,PART)
ASSIGN INITIAL VELOCITIES TO ATOMS
COMMON/NUM/NM,NAM,NP,NP1,NP2,NP22,KSORT,KB
COMMON/VEL/X1(216),Y1(216),Z1(216)
COMMON/MASS/MASS(216)
SUMX=0.
SUMY=0.
SUMZ=0.
DO 200 1=1,NM
XX=RANF(DUM)
YY=RANF(DUM)
ZZ=RANF(DUM)
NX=(1-1)*NAM
DO 200 J=1,NAM

243
XI(J+NX)=XX
Y1(J+NX)=YY
Z1(J+NX)=ZZ
SUMX=SUMX+X1(J+NX)*MASS(J+NX)
SUMY=SUMY+Y1(J+NX)*MASS(J+NX)
SUMZ=SUMZ+Z1(J+NX)*MASS(J+NX)
200 CONTINUE
C
C === SCALE VELOCITIES SO THAT TOTAL MOMENTUM = ZERO
X=0.
DO 210 1=1,NP
XI(I)=X1(I)-SUMX/PART/MASS(I)
Y1(I)=Y1(I)-SUMY/PART/MAS S(I)
Z1(I)=Z1(I)-SUMZ/PART/MASS(I)
X=X+(XI(I)**2+Yl(I)**2+Zl(I)**2)*MASS(I)
210 CONTINUE
C
C === SCALE VELOCITIES TO DESIRED TEMPERATURE
HEAT=SQRT(AHEAT/X)
DO 220 1=1,NP
XI(I)=X1(I)*HEAT
Y1(I)=Y1(1)*HEAT
Z1(I)=Z1(I)*HEAT
220 CONTINUE
RETURN

244
END
SUBROUTINE PREDCT(NP)
C
C === USE TAYLOR SERIES TO PREDICT POSITIONS & THEIR
C DERIVATIVES AT NEXT TIME-STEP
C
COMMON/POS/XO(216),Y0(216),Z0(216)
COMMON/VEL/Xl(216),Y1(216),Z1(216)
COMMON/DER/X2(216),Y2(216),Z2(216),X3(216),Y3(216),
& Z3(216),X4(216),Y4(216),Z4(216),X5(216),Y5(216),
& Z5(216)
COMMON/FOR/FX(216),FY(216),FZ(216)
DO 300 1=1,NP
XO(I)=X0(I)+X1(I)+X2(I)+X3(I)+X4(I)+X5(I)
YO(I)=Y0(I)+Y1(I)+Y2(I)+Y3(I)+Y4(I)+Y5(I)
Z0(I)=Z0(I)+Z1(I)+Z2(I)+Z3(I)+Z4(I)+Z5(I)
XI(I)=X1(I)+2.*X2(I)+3.*X3(I)+4.*X4(I)+5.*X5(I)
Y1(I)=Y1(I)+2.*Y2(I)+3.*Y3(I)+4.*Y4(I)+5.*Y5(I)
Z1(I)=Z1(I)+2.*Z2(I)+3.*Z3(I)+4.*Z4(I)+5.*Z5(I)
X2(I)=X2(I)+3.*X3(I)+6.*X4(I)+10.*X5(I)
Y2(I)=Y2(I)+3.*Y3(I)+6.*Y4(I)+10.*Y5(I)
Z2(I)=Z2(I)+3.*Z3(I)+6.*Z4(I)+10.*Z5(I)
X3(I)=X3(I)+4.*X4(I)+10.*X5(I)

245
Y3
(I)
=Y3
(I)+4.
. *Y4
(I)+10.
,*Y5(I)
Z3
(I)
=Z3
(I)+4.
. *Z4
(I)+10.
,*Z5(I)
X4
(I)
=X4
(I)+5.
, *X5
(I)
Y4
(I)
=Y4
(I)+5.
. *Y5
(I)
Z4
(I)
= Z4
(I)+5.
, *Z5
(I)
FX
(I)
= 0.
FY
(I)
=0.
FZ
(I)
= 0.
300 CONTINUE
RETURN
END
SUBROUTINE CORR(DELTSQ)
C
C === CORRECT PREDICTED POSITIONS AND THEIR DERIVATIVES
C
COMMON/POS/XO(216),Y0(216),Z0(216)
COMMON/VEL/X1(216),Y1(216),Z1(216)
COMMON/DER/X2(216),Y2(216),Z2(216),X3(216),Y3(216),
& Z3(216),X4(216),Y4(216),Z4(216),X5(216),Y5(216),
& Z5(216)
COMMON/FOR/FX(216),FY(216),FZ(216)
COMMON/DISP/DAX(216),DAY(216),DAZ(216),X0L(216),
&
YOL(216),ZOL(216)

246
C
C
C
610
COMMON/PARM/FO2,F12,F32,F42,F52
COMMON/NUM/NM,NAM,NP,NP1,NP2,NP22,KSORT,KB
COMMON/PROP/SUME,XSUM
COMMON/SCAL/RSPHER,RWALL,DELRS,ISCALE
COMMON/MASS/MASS(216)
IF(MOD(KB,10).NE.0.OR.ISCALE.NE.1) GOTO 610
IF(RSPHER .LE. 3.6) GO TO 610
RS PHER=RSPHER-DELRS
RWALL =RWALL -DELRS
DO 690 1=1,NP
XERR=X2(I)-DELTSQ*FX(I)/MASS(I)
YERR=Y2(I)-DELTSQ*FY(I)/MASS(I)
ZERR=Z2(I)-DELTSQ*FZ(I)/MASS(I)
XO(I)=X0(I)-XERR*F02
XI(I)=X1(I)-XERR*F12
X2(I)=X2(I)-XERR
X3(I)=X3(I)-XERR*F32
X4(I)=X4(I)-XERR*F42
X5(I)=X5(I)-XERR*F52
YO(I)=Y0(I)-YERR*F02
Y1(I)=Y1(I)-YERR*F12
Y2 (I)=Y2(I)-YERR
Y3(I)=Y3(I)-YERR*F32

247
Y4(I)=Y4(I)-YERR*F42
Y5(I)=Y5(I)-YERR*F52
ZO(I)=Z0(I)-ZERR*F02
Z1(I)=Z1(I)-ZERR*F12
Z2(I)=Z2(I)-ZERR
Z3(I)=Z3(I)-ZERR*F32
Z4(I)=Z4(I)-ZERR*F42
Z5(I)=Z5(I)-ZERR*F52
690 CONTINUE
IF(MOD(KB,KSORT).NE.0.OR.ISCALE.NE.1) GOTO 680
DO 691 1=1,NP
RSQ=X0(I)*X0(I)+Y0(I)*Y0(I)+Z0(I)*Z0 (I)
R=SQRT(RSQ)
RFACT = 1.0-DELRS/R
XO(I) = XO(I)*RFACT
YO(I) = YO(I)*RFACT
Z0(I) = ZO(I)*RFACT
CONTINUE
691
C === DISPLACEMENTS
680 DO 692 1=1,NP
DAX(I)=DAX(I)-XO(I)+X0L(I)
DAY(I)=DAY(I)-YO(I)+YOL(I)
DAZ(I)=DAZ(I)—ZO(I)+ZOL(I)
C

248
C === STORE NEW POSITIONS
XOL(I)=X0(I)
YOL(I)=Y0(I)
ZOL(I)=Z0(I)
692 CONTINUE
RETURN
END
SUBROUTINE EQBRAT(SUMVEL,AHEAT,TDIST,XDIST,NP,IFLG,
& KB,NAM)
C
C === SCALE VELOCITIES DURING INITIAL TIME-STEPS
C
COMMON/VEL/X1(216),Y1(216),Z1(216)
COMMON/PROP/SUME,XSUM
COMMON/SCAL/RSPHER,RWALL,DELRS,ISCALE
C
IF(IFLG.EQ.-l) GOTO 720
IF(TDIST.GT.XDIST.OR.IFLG.EQ•—2) GOTO 750
720 HEAT=SQRT(AHEAT/SUMVEL)
DO 730 1=1,NP
XI(I)=X1(I)*HEAT
Y1(I)=Y1(I)*HEAT
Z1(I)=Z1(1)*HEAT

249
730 CONTINUE
RETURN
C
C === AT END OF EQUILIBRATION STAGE, SET PROPERTY SUMS
C TO ZERO
750 IFLG= 1
KB=0
SUME=0.
XSUM=0.
RETURN
END
SUBROUTINE EVAL(RWALL)
C
COMMON/POS/XO(216),Y0(216),Z0(216)
COMMON/FOR/FX(216),FY(216),FZ(216)
COMMON/CONT/A1,AO,EPS
COMMON/PROP/SUME,XSUM
COMMON/PROPA/REQUIL,FORCON,HDSIG,SIG12
COMMON/ENERGY/TOTE,TOTLJ,ETOR,EBON,EBEN,EINTR,
& TRFRAC,EXTPRS,FTOTWL
COMMON/NUM/NM,NAM,NP,NP1,NP2,NP22,KSORT,KB
COMMON/INTRA/BO(216),GAB(216),CTO(216),GTHA(216),
& GTHE(216)

250
DIMENSION B(216),BX(216),BY(216),BZ(216)
C
C INITIALIZE ACCUMULATORS
ICOUNT=0
K=0
ITRAN =0
TOTE=0.0
TOTLJ=0.0
FTOTWL=0.0
ETOR=0.0
EBON=0.0
EBEN=0.0
EINTR=0.0
DO 150 1=1,NP
FX(I)=0.0
FY(I)=0.0
FZ(I)=0.0
150 CONTINUE
C
C CALCULATE LJ FORCES
C
C OUTER LOOP OVER PARTICLES
DO 300 1=1,NP1
ICHK=I-1+NAM
XI=X0(I)

251
YI=YO(I)
ZI=ZO(I)
IF(I.GE.NP2) GOTO 788
C
C DECIDE WHICH MOLECULE ATOM I IS ON
IM=(I-1)/NAM+1.0001
C
C INNER LOOP
DO 290 J=I+1,NP
C
C
IF I & J ARE HEAD GROUPS, CALCULATE HEAD-HEAD REPULSIONS
J CHK=J+NAM-1
IF(MOD(ICHK,NAM).NE.0.OR.MOD(JCHK,NAM).NE.O)GOTO 614
X=XI-X0(J)
Y=YI-Y0(J)
Z=ZI-Z0(J)
RSQ=X*X+Y*Y+Z*Z
RIJ=SQRT(RSQ)
RI=HDSIG/RIJ
R3=RI*RI*RI
R4=R3 *RI
R8=R4 *R4
C
HS3=1.0/HDSIG**3
HS4=HS3/HDSIG

252
ELJ=R3*(R8*RI+HS3)
FLJS=3.0*R4*(4.0*R8*RI+HS4)
FXD=FLJS*X*RI
FYD=FLJS*Y*RI
FZD=FLJS*Z*RI
FX(I)=FX(I)+FXD
FX(J)=FX(J)-FXD
FY(I)=FY(I)+FYD
FY(J)=FY(J)-FYD
FZ(I)=FZ(I)+FZD
FZ(J)=FZ(J)-FZD
GO TO 290
C
C DECIDE WHICH MOLECULE ATOM J IS ON
614 JM=(J—1)/NAM+1.0001
C
C CHECK WHETHER I AND J ARE ON SAME MOL
IF(IM.NE.JM) GOTO 289
C
C CHECK IF I AND J ARE FOURTH NEIGHBORS OR GREATER
KIJ=J—I
IF(KIJ•LE.3) GOTO 290
C
289 X=XI-X0(J)
Y=YI—YO(J)

253
Z=ZI-ZO(J)
C
RSQ=X*X+Y*Y+Z*Z
RIJ=SQRT(RSQ)
C
C IF ONE IS A HEAD GROUP, CALCULATE HEAD-ALKANE INTERACTION
IF(MOD(ICHK,NAM).EQ.0.OR.MOD(JCHK,NAM).EQ.O) THEN
RI=SIG12/RIJ
R3=RI**3
R6=R3*R3
ELJ=2.0*R6*(R3-1.50)
FLJ=18.*RI*R6*(R3-1.0)
C
C IF NOT, CALCULATE ALKANE-ALKANE INTERACTION
ELSE
ri=i./RU
R3=RI**3
R6=R3 *R3
ELJ=2.0*R6*(R3-1.50)
FLJS=18.*RI*R6*(R3-1.0)
ENDIF
TOTLJ=TOTLJ+ELJ
FXD=FLJS*X*RI
FYD=FLJS*Y*RI
FZD=FLJS*Z*RI

254
FX(I)=FX(I)+FXD
FX(J)=FX(J)-FXD
FY(I)=FY(I)+FYD
FY(J)=FY(J)-FYD
FZ(I)=FZ(I)+FZD
FZ(J)=FZ(J)-FZD
290 CONTINUE
C
c
c
3
C
C
C
CALCULATE BOND LENGTHS
CHECK IF LAST ATOM ON MOLECULE
788 IF(MOD(I,NAM).EQ.O) GOTO 300
BX(I)=XI-X0(1+1)
BY(I)=YI-Y0(1+1)
BZ(I)=ZI-Z0(I+1)
BSQ=BX(I)**2+BY(I)**2+BZ(I)**2
B(I)=SQRT(BSQ)
00 CONTINUE
LOOP CALCULATES INTRAMOLECULAR FORCES
499 DO 600 1=1,NP1
C
C CHECK IF LAST ATOM ON MOLECULE
IF(MOD(I,NAM).EQ.O) GOTO 600
C

255
C CALCULATE BOND FORCES
BDIFF=B(I)-BO(I)
EBON=EBON+0.5*GAB(I)*BDIFF*BDIFF
FBON=GAB(I)*BDIFF
BI=1./B(I)
BXI=BX(I)*BI
BYI=BY(I)*BI
BZI=BZ(I)*BI
FXB=FBON*BXI
FYB=FBON*BYI
FZB=FBON*BZI
FX(I) =FX(I)-FXB
FX(1+1)=FX(1+1)+FXB
FY(I) =FY(I)-FYB
FY(1+1)=FY(1+1)+FYB
FZ(I) =FZ(I)-FZB
FZ(1+1)=FZ(1+1)+FZB
C
C CALCULATION BENDING FORCE
C CHECK IF NEXT-TO-LAST ATOM ON MOLECULE
IF(MOD(1+1,NAM).EQ.O) GOTO 600
BII=1./B(I+1)
BXJ=BX(1+1)*BII
BYJ=BY(1+1)*BII
BZJ=BZ(1+1)*BII

256
CTI=-(BXJ*BXI+BYJ*BYI+BZJ*BZI)
CDIFF=CTO(I)-CTI
EBEN=EBEN+0.5*GTHA(I)*CDIFF*CDIFF
FT=GTHA(I)*CDIFF
FT1=FT*BI
FT2=FT*BII
FX1=FT1*(BXJ+CTI*BXI)
FY1=FT1*(BYJ+CTI*BYI)
FZ1=FT1*(BZJ+CTI*BZI)
FX2=FT2 *(BXI + CTI*BXJ)
FY2=FT2 *(BYI+CTI*BYJ)
FZ2=FT2*(BZI+CTI*BZJ)
FX(I) =FX(I) -FX1
FX(1+1)=FX(1+1)+FX1-FX2
FX(1+2)=FX(1+2)+FX2
FY(I) =FY(I) -FY1
FY(1+1)=FY(1+1)+FY1-FY2
FY(1+2)=FY(1+2)+FY2
FZ(I) =FZ(I) —FZ1
FZ(1+1)=FZ(1+1)+FZ1-FZ2
FZ(1+2)=FZ(I+2)+FZ2
C
C CALCULATION OF TORSION FORCES
C CHECK WHETHER FIRST ATOM OF MOLECULE
IF(MOD(I+NAM-1,NAM).EQ.O) GOTO 600

257
CALL RYFOR(BX(1-1),BY(I-1),BZ(I-1),BX(I),BY(I),BZ(I),
& BX(1+1),BY(1+1),BZ(1+1),FX1,FY1,FZ1,FX2,FY2,FZ2,
& FX3,FY3, FZ3 ,FX4,FY4,FZ4,ETQ,CODI,GTHE(1-1))
C
ICOUNT=ICOUNT+1
IF(CODI.LT.-0.50) ITRAN=ITRAN+1
FX(1-1)=FX(I—1)+FX1
FY(1-1)=FY(1-1)+FY1
FZ(1-1)=FZ(I-1)+FZ1
FX(I) =FX(I) +FX2
FY(I) =FY(I) +FY2
FZ(I) =FZ(I) +FZ2
FX(1+1)=FX(1+1)+ FX3
FY(1+1)=FY(1+1)+FY3
FZ(1+1)=FZ(1+1)+FZ3
FX(1+2)=FX(1+2)+FX4
FY(1+2)=FY(1+2)+FY4
FZ(I+2)=FZ(I+2)+FZ4
ETOR=ETOR+ETQ
600 CONTINUE
C
C CALCULATE GROUP-SHELL INTERACTION
EWALL=0.0
RCHKSQ=(RWALL-2.0)**2
DO 650 1=1,NP

258
ICHK=I-1+NAM
C
XI=XO(I)
YI=YO(I)
ZI=ZO(I)
c
C DISTANCE FROM CENTER TO GROUP
RSQ=XI*XI+YI*YI+ZI*ZI
C IF HEAD GROUP, CALCULATE INTERACTIONS WITH WALL USING
C A POTENTIAL OF THE FORM — GAMMA*(R-RO)**2
IF(MOD(ICHK,NAM).NE.O) GO TO 384
C DURING SCALE-DOWN, DO NOT CALCULATE HEAD-WALL INTERACTION
C GOTO 650
C
RR=SQRT(RSQ)
RI=1.0/RR
RD=RWALL-RR
C
EWALL=EWALL+FORCON*(RD-REQUIL)*(RD-REQUIL)
FWALL=-2.0*FORCON*(RD-REQUIL)
FTOTWL=FTOTWL+FWALL
C
FX(I)=FX(I)-FWALL*XI*RI
FY(I)=FY(I)-FWALL*YI*RI
FZ(I)=FZ(I)-FWALL*ZI*RI

259
GO TO 650
C IF NOT A HEAD GROUP, CALCULATE REPULSIVE INTERACTION
C WITH THE WALL
384 IF(RSQ.LT.RCHKSQ) GOTO 650
C
C DISTANCE FROM GROUP TO SHELL
RR=SQRT(RSQ)
RI=1.0/RR
RD=RWALL-RR
RDR=1.0/RD
R3 =RDR*RDR*RDR
R6 =R3 *R3
R12=R6*R6
C
C FORCE AND ENERGY FOR R-12 WALL
EWALL=EWALL+R12
FWALL=12.0*R12*RDR
FTOTWL=FTOTWL+FWALL
FX(I)=FX(I)-FWALL*XI*RI
FY(I)=FY(I)-FWALL*YI*RI
FZ(I)=FZ(I)-FWALL*ZI*RI
650 CONTINUE
C
C CALCULATE WALL PRESSURE IN ATMOSPHERES
EXTPRS=0.0085382*FTOTWL/RWALL**2

260
C
C CALCULATE INTRAMOLECULAR AND TOTAL ENERGY
EINTR=ETOR + EBON + EBEN
TOTE = TOTLJ + EINTR + EWALL
TRFRAC=FLOAT(ITRAN)/FLOAT(ICOUNT)
RETURN
END
SUBROUTINE RYFOR(BONX1,BONY1,BONZ1,BONX2,BONY2,
& BONZ2,BONX3,BONY3,BONZ3,FORX1,FORY1,FORZ1,FORX2,
& FORY2,FORZ2,FORX3,FORY3,FORZ3,FORX4,FORY4,FORZ4,
& ETOR,CODI,GR)
C
DATA Al,A2,A3,A4,A5/-1.462,-1.578,0.368,
& 3.156,3.788/
C
C CALCULATE NORMAL TO PLANE OF BONDS 1 AND 2
E=BONY1* BONZ 2-BON Z1* BONY 2
D= BON Z1* BONX2-BONXl*BONZ2
C=BONX1* BONY 2-BONY1* BONX2
ANORM2=C*C+D*D+E*E
ANORM =SQRT(ANORM2)
ANORMR=l./ANORM
AX=E*ANORMR

261
AY=D*ANORMR
AZ=C*ANORMR
C
C CALCULATE NORMAL TO PLANE OF BONDS 2 AND 3
H=BONY2 *BONZ3-BONZ2 *BONY3
G=BONZ 2 * BONX3-BONX2 *BONZ 3
F=BONX2 *BONY3-BONY2 *BONX3
BNORM2=F*F+G*G+H*H
BNORM =SQRT(BNORM2)
BNORMR=l./BNORM
BX=H*BNORMR
BY=G*BNORMR
BZ=F*BNORMR
C
C DETERMINE COSINE OF THE DIHEDRAL ANGLE, PHI
CODI= AX*BX+AY*BY+AZ*BZ
C
C DETERMINE TORSIONAL ENERGY
CODISQ=CODI*CODI
CODI3 =CODI*CODISQ
ETOR = GR*(1.116+Al*CODI+A2*CODISQ+A3*CODI3
& +A4*CODISQ*CODISQ+A5*CODISQ*CODI3)
C
C CALCULATE DU/COS(PHI)
DUDCO=(Al+2.*A2*CODI+3.*A3*CODISQ+4.*A4*CODISQ*

262
& CODI+5.*A5*C0DISQ*C0DISQ)*GR
C
C START EVALUATION OF FORCE ON GROUP 1
C CALCULATE DANORM/DR1
FACT1=(C*BONY2-D*BONZ2)/ANORM2
DADX1X= -FACT1*AX
DADX1Y= -FACTl*AY-BONZ2*ANORMR
DADX1Z= -FACTl*AZ+BONY2*ANORMR
C
FACT2 =(E*BONZ2-C*BONX2)/ANORM2
DADY1X= -FACT2 *AX+BONZ2*ANORMR
DADY1Y= -FACT2 *AY
DADY1Z= -FACT2 *AZ-BONX2 *ANORMR
C
FACT3 =(D*BONX2—E*BONY2)/ANORM2
DADZ1X= -FACT3 *AX-BONY2 *ANORMR
DADZ1Y= -FACT3 *AY+BONX2 *ANORMR
DADZ1Z= -FACT3 *AZ
C
C CALCULATE D(AB)/DR1
DABDX1=BX*DADX1X+BY*DADX1Y+BZ*DADX1Z
DABDY1=BX*DADY1X+BY*DADY1Y+BZ*DADY1Z
DABDZ1=BX*DADZ1X+BY*DADZ1Y+BZ*DADZ1Z
C CALCULATE FORCE ON GROUP 1

263
F0RX1= -DUDC0*DABDX1
F0RY1= -DUDC0*DABDY1
F0RZ1= -DUDCO*DABDZ1
C
C START EVALUATION OF FORCE ON GROUP 2
C CALCULATE DANORM/DR2
FACT4 =(D*BONZl-C*BONYl)/ANORM2
DADX2X= -FACT4 *AX-DADX1X
DADX2Y= -FACT4*AY+BONZ1*AN0RMR-DADX1Y
DADX2 Z= -FACT4*AZ-B0NY1*AN0RMR-DADX1Z
C
FACT5 =(C*BONXl-E*BONZ1)/ANORM2
DADY2X= -FACT5*AX-B0NZ1*AN0RMR-DADY1X
DADY2Y= -FACT5*AY-DADY1Y
DADY2Z= -FACT5*AZ+B0NX1*AN0RMR-DADY1Z
C
FACT6 =(E*BONYl-D*BONXl)/ANORM2
DADZ 2 X= -FACT6 *AX+BONY1*ANORMR-DADZIX
DADZ2Y= -FACT6*AY-BONX1*ANORMR-DADZ1Y
DADZ2 Z= -FACT6*AZ-DADZ1Z
C
C EVALUATE D(BNORM)/DR2
FACT7 =(F*BONY3-G*BONZ3)/BNORM2
DBDX2X= -FACT7 *BX
DBDX2Y= -FACT7*BY-BONZ3*BNORMR

264
DBDX2 Z= -FACT7 *BZ + B0NY3 *BNORMR
C
FACT8 =(H*BONZ3-F*BONX3)/BN0RM2
DBDY2X= -FACT8*BX+BONZ3 *BNORMR
DBDY2Y= -FACT8*BY
DBDY2Z= -FACT8 *BZ-B0NX3 *BNORMR
C
FACT9 =(G*BONX3-H*BONY3)/BN0RM2
DBDZ2X= -FACT9 *BX-B0NY3 *BNORMR
DBDZ2Y= -FACT9 *BY+B0NX3 *BNORMR
DBDZ2Z= —FACT9*BZ
C
C CALCULATE D C0S(PHI)/DR2
DABDX2=BX*DADX2X+BY*DADX2Y+BZ*DADX2Z
& +AX*DBDX2X+AY*DBDX2Y+AZ*DBDX2Z
DABDY2=BX*DADY2X+BY*DADY2Y+BZ*DADY2 Z
& +AX * DBDY 2 X+AY * DBDY 2 Y+AZ * DBDY 2 Z
DABDZ2=BX*DADZ2X+BY*DADZ2Y+BZ*DADZ2Z
& +AX*DBDZ2X+AY*DBDZ2Y+AZ*DBDZ2Z
C
C FORCE ON GROUP 2
FORX2 = -DUDCO*DABDX2
FORY2 = -DUDCO*DABDY2
FORZ2 = -DUDCO*DABDZ2
C

265
C START EVALUATION OF FORCE ON GROUP 4
C CALCULATE D(BNORM)/DR4
FACT1 =(F*BONY2-G*BONZ2)/BNORM2
DBDX4X= -FACT1*BX
DBDX4Y= -FACTl*BY-BONZ2*BNORMR
DBDX4Z= -FACTl*BZ+BONY2*BNORMR
C
FACT2 =(H*BONZ2-F*BONX2)/BNORM2
DBDY4X= -FACT2*BX+BONZ2*BNORMR
DBDY4Y= -FACT2 *BY
DBDY4Z= -FACT2 *BZ-BONX2 *BNORMR
C
FACT3 =(G*BONX2-H*BONY2)/BNORM2
DBDZ4X= -FACT3 *BX-BONY2 *BNORMR
DBDZ4Y= -FACT3 *BY+BONX2 *BNORMR
DBDZ4Z= —FACT3 *BZ
C
C EVALUATE D COS(PHI)/DR4
DABDX4=AX*DBDX4X+AY*DBDX4Y+AZ*DBDX4Z
DABDY 4 =AX* DBDY 4 X+AY * DBDY 4 Y+AZ * DBDY 4 Z
DABDZ4=AX*DBDZ4X+AY*DBDZ4Y+AZ*DBDZ4Z
C
C EVALUATE FORCE ON GROUP 4
FORX4= -DUDCO*DABDX4
FORY4= -DUDCO*DABDY4

266
F0RZ4= -DUDC0*DABDZ4
C
C EVALUATE FORCE ON GROUP 3 BY NEWTON'S THIRD LAW
FORX3 = -FORX1-FORX2-FORX4
FORY3 = -FORY1-FORY2-FORY4
FORZ3 = -FORZ1-FORZ2-FORZ4
C
RETURN
END
BLOCK DATA
COMMON/DER/X2(216) ,Y2(216) ,Z2 (216) ,X3(216) ,Y3(216) ,
& Z3(216),X4(216),Y4(216),Z4(216),X5(216),Y5(216),
& Z5(216)
COMMON/FOR/FX(216),FY(216),FZ(216)
COMMON/DISP/DAX(216),DAY(216),DAZ(216),X0L(216),
& Y0L(216),Z0L(216)
COMMON/PROP/SUME,XSUM
COMMON/MASS/MASS(216)
COMMON/INTRA/BO(216),GAB(216),CTO(216),GTHA(216),
& GTHE(216)
DATA X3,Y3,Z3,X4,Y4,Z4/X5,Y5,Z5/1944*0./
DATA DAX,DAY,DAZ,FX,FY,FZ/12 9 6 * 0./
DATA MASS/216*!/

267
C FOR SCALE-DOWN, GTHE SET TO GTHE/10.
DATA BO,GAB,CTO,GTHA,GTHE/216*.38475,216*35322.1957,
& 216*-.377032668,216*310.26253,216*19.84248/
C NORMAL GTHE IS 19.84248
END

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BIOGRAPHICAL SKETCH
The author was born in 1956 in Orlando, Florida, the son
of Richard T. Farrell and the former Teresa Corcoran. He
was graduated from Winter Park High School in 1974 and
entered the Georgia Institute of Technology in the fall of
that year. During his undergraduate years, he joined the
staff of the campus newspaper as a music writer and
progressed to the position of managing editor. This period
also included intern positions in engineering with the Union
Carbide Nuclear Division in Oak Ridge, Tennesee, and the
Procter and Gamble Paper Products Division in Albany,
Georgia. Taking the degree of Bachelor of Chemical
Engineering in 1979, he was subsequently employed by Procter
and Gamble in various production engineering capacities.
He entered graduate study in the Department of
Chemical Engineering at the University of Florida in the
fall of 1982, receiving a Master of Science the following
year. Upon completion of the doctoral program, he will hold
a position with Westvaco Research in Laurel, Maryland.
276

I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
/ O 'C~.^ut4C
John P. O'Connell, Chairman
Professor of Chemical
Enqineering
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
— 0 .
Dinesh 0. Shah
Professor of Chemical
Engineering
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
Brij ijm. Moudgil
Professor of Materials
Science and Engineering
This dissertation was submitted to the Graduate Faculty
of the College of Engineering and to the Graduate School and
was accepted as partial fulfillment of the requirements for
the degree of Doctor of Philosophy.
April, 1988
iLxj
Dean, College of Engineering
Dean, Graduate School

UNIVERSITY OF FLORIDA
3 1262 08556 7708


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