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THE NEMATIC-ISOTROPIC PHASE TRANSITION IN RIGID
LINEAR FUSED HARD-SPHERE CHAIN FLUIDS
A Thesis
Presented to
The Faculty of Graduate Studies
of
The University of Guelph
In partial fulfilment of requirements
for the degree of
Master of Science
March: 1999
OKar im M. Jaffer, 1999
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ABSTRACT
THE NEMATIC-ISOTROPIC PHASE TRANSITION IN RIGID
LINEAR FUSED HARD-SPHERE CHAIN FLUIDS
Karim M. Jaffer University of Guelph, 1999
Advisor: Professor D.E. Sullivan
In this study, a modification of the generalized F loq dimer theory is employed to
investigate the nematic (Y) to isotropic (1) phase transit ion in chain fluids. focusing
on rigid linear fused hard-sphere (LFHS) chah molecules. A generalized density
functional theory is developed, which involves an angular weighting of the dimer
reference fluid as suggested by decoupling theory, to accommodate nematic ordering
in the systern. A key ingredient of this theory is the calculation of the exact exciuded
volume for a pair of molecules in an arbitrary relative orientation. mhich extends the
work of Williamson and Jackson[l] for linear tangent hard-sphere chain molecules to
the case of linear fused hard-sphere chains wit h arbitrary intrarnolecular bondlength.
The present results for the N-I transition are compared with previous theories and
with computer simulations. In cornparison with previous studies? the results show
much better agreement with simulations for both the coexistence densities and the
nematic order parameter a t the transition.
Acknowledgement s
". . .faciliS descensus At-emo"
-Virgil, TheÆneid (Skth Book)
The above passage is translated loosely as ". . .easy is the descent into hell," which
is quite indicative of the mood a t times on the Qh floor of the MacWaughton Building.
For those not inundated with the nuances of the University of Guelph, the famed 4lth
floor is the home of the theoretical physics graduate students, amongst whom 1 have
lived, learned and earned over the duration of my h1.S~. Degree. To keep rnyself from
this downhill course (sic), 1 have been fortunate enough t o have many individuals to
rely upon. It has been their task to keep m y sanity in check. and perhaps those of
o u fortunate enough to meet me will be able to find thern and berate them for a job
poorly done.
The first group of people 1 want to mention in this unenviable position is, of course,
my fellow inrnates. The physics undergraduate and graduate students rvhom 1 have
had occasion to meet over the past ferv years have al1 found thernselves subject to my
loud and abrasive behaviour and have put up with it using a surprising amount of
grace and good humour. To al1 of them 1 extend my rhanks for making mlr tenure here
fun and enjoyable. Specifically, 1 want to thank Stephen Leonard for his support and
understanding in aiding me through a difficult time. 17d like to extend a very serious
and heartfelt thank you to Sheldon B. Opps for his guidance and encouragement
throughout the course of this study. Thank you for sharing your knowledge and
for assistance in learning how to zsk the correct questions. which provided the key
turning point in this investigation. As well? I wish to single out Christian Schroeder
for standing by me, which of course diverts al1 the attention away from me.
17d like to also take this opportunity to thank my many non-physics friends. From
Friday night outings to midnight nature hikes (procrastination is always the key) , it's
been delightful. 1 must make mention of the tremendous amount of experience 17ve
achieved through competing in the World Debating Circuit as a representative for the
University of Guelph. These pursuits into philosophy and politics provided a unique
counterbalance to my scientific endeavours.
A special thanks to Professors C.G. Gray and B.G. Nickel for serving on m,v
advisory and examining cornmittees. Through their ability to ask insightful questions,
1 was able to understand both the qualitative and quantitative features of my work. I
am also grateful to Professor Kickel for aiding us in the calculation of the second virial
coefficient, and for his continued guidance in our extensions of this investigation.
A tremendous amount of gratitude goes out to Don Sullivan: my supervisor for this
MSc. Thesis, for his support. guidance, encouragement and patience over the past
few years. It's hard to put into words the knowledge that I have gained through the
course of this investigation, but as a result 1 can finally consider rnyself a researcher
as well as a student. Thank you Don.
A final thank o u to rny f a m i l . Thanks for being there through the highs and the
lows, though 1 didn't give any of you much choice. Your support and encouragement
vas priceless. Tbank you.
Contents
1 Introduction
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Previous S tudies
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Nematic Ordering
. . . . . . . . . . . . . . . . . . . . . . . 1.3 Density-Functional Uethods
. . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Mean-Field Methods
. . . . . . . . . . . . . . . . 1.4.1 Generalized Flow Dimer T h e o l
. . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Monte Car10 Simulations
2 Proposed Theory 10
. . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Generai Formulation 10
. . . . . . . . . . . . . . 2.1.1 Monomer Decoupling Approximation 13
. . . . . . . . . . . . . . . . 2.1.2 Dimer Decoupling .Approximation 15
. . . . . . . . . . . . . . . . . . . . 2.1.3 Derivation of GFD Theory 16
. . . . . . . . . . . . . . . . . . . . . . 2.2 Specialization to LFHS Chains 17
. . . . . . . . . . . . . . 2.2.1 Generalized Second Virial Coefficient 17
. . . . . . . . . . . . . . . . . . . . 2.2.2 Free Energy Minimization 24
3 Excluded Volume CalcuIation 30
3.1 Barrett 's Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2 Exact Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2.1 Alternate Formulation . . . . . . . . . . . . . . . . . . . . . . 39
4 Results and Cornparisons 41
4.1 LTHS Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.2 LFHS Chainc; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5 Conclusions and F'uture Work 61
Bibliography 65
A l -Alternative Excluded Volume Calculation . . . . . . . . . . . . . . . . 69
A2 Cornparison of Virial Coefficients . . . . . . . . . . . . . . . . . . . . 71
CI . . . . . . . . . . . . . . . . . . . . . . . A3 Density Functional Program (2
List of Tables
4.1 Coexistence results from simulation and theory. . . . . . . . . . . . . 47
AP.1 Cornparison of second and third reduced virial coefficients between
the present t h e o s exact calculation[49] and Monte Car10 simulation
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . data[lô]. 71
List of Figures
3.1 Diagrammatic representation of the exchded volume for n = 2- The
slice is taken through the z = O plane. where the radius of each circle
is the monomer diameter, d. . . . . . . . . . . . . . . . . . . . . . . . 33
3.2 Two types of partial spheres contributing to the excluded volume ex-
terior to the central region. . . . . . . . . . . . . . . . . . . . . . . . . 34
3.3 Second case for two-body overlap, when O < OI2 < 7~13. . . . . . . . . 36
4.1 Variation of order parameter S2 with volume fraction q , comparing
present theory with the Parsons theory and Monte Carlo data[9, 221,
. . . . . . . . for (a)8-mer LTHS chains, and (b)20-mer LTHS chains. 44
4.2 Variation of the reduced pressure with volume fraction 7, comparing
present theory with the Parsons theory and Monte Car10 data[9, 221,
. . . . . . . . for (a)8-mer LTHS chains, and (b)20-mer LTHS chains. 46
4.3 Order parameter S2 VS. volume fraction 17 for LTHS 7-mers, comparing
. . . . . . . . . . . . . . . present theory with Monte Car10 data[21]. 49
4.4 Reduced pressure for LTHS 7-mers as a function of volume fraction
71, comparing present theory with Monte Carlo data and the modified
Vega-Lago theory from Ref. [2 11. . . . . . . . . . . . . . . . . . . . . . 50
4.5 Comparison of volume fractions of the isotropic and nematic phases at
coexistence for LTHS n-mers. as a function of the number of rnonomers
n, between the present theory and Monte Carlo simulation data. . . . 32
4.6 Comparison of the reduced pressure between the present theory, the
Mehta and Honnell GFD theory and TPT in Ref.[16], and Monte Carlo
simulations[4l] for (a) LFHS 6-mers with bondlengt h to diameter ratio
. . 1' = 0.5. (b)LFHS 8-mers. 1' = 0.5, and (c)LFHS 8-mers. 1' = 0.6. 56
4.7 fractions of the isotropic and nematic phases a t coexistence as
a function of the reduced bondlength I * : for (a)LFHS 8-mers. (b)LFHS
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20-mers. 58
4.8 Volume fractions of the isotropic and nematic phases at coexistence
as a function of number of monomers for LFHS chains of constant
length rvith Lld = 19. The spherocylinder limits are given by the Lee
theory[21] and by the Monte Carlo simuLation results of Bolhuis and
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FrenkeI[43]. 60
-41.1 Depiction of escluded regions between monociers in different molecules
for FHS dimers at arbitrary orientation el?. The monomers in contact
over each region (bounded by solid lines) are denoted. The dashed
. . . . . . . . . . . . . . . . . . . . lines indicate quadrant symmetry. 70
vii
Chapter 1
Introduction
1.1 Motivation
Liquid crystalline materials possess orient ational ordering wit h or wit hout long-range
positional ordering. Nematic materials are the simplest class of liquid crystals,
exhibiting no long-range p ositional ordering. The result ing pro pert ies lie between
those of Iiquids and solids, and are characterized b - liquid-like flow properties and
anisotropic optical propenies. Polyneric fluids which exhibit nematic ordering are
of particular importance in industry offering the advantages of being stable and rel-
atively inexpensive, and these materials can be fabricated in thin films with ease. In
addition! liquid crystalline polymers (LCPs) are of importance in biological systems
due to their ability to form semi-rigid structures with some fluid properties. These
characteristics become important in the formation of membranes and are evidenced in
phospholipid bilayers. Liquid crystalline behaviour is also believed to play an impor-
tant role in blood clotting, atherosclerosis, and sickle ce11 anemia. Although there are
a plethura of practical applications of liquid crystals which are well known: the phase
behaviour of these systems is still not completely understood. An understanding of
the phase behaviour is of practicai interest for industry in determining mhich LCPs
are best suited for applications.
This study focuses on the nematic to isotropic ( X I ) phase transition in athermal
liquid crystalline polymers' which elchibit t his behai-ior independent of temperature.
This is the simplest phase transition for these polymers and can be driven exclusively
by entropic effects, as is elaborated in Section 1.2. Despite the simplicity of this
model, no quantitatively accurate theory describing the Pl-I behaviour of LCPs has
been developed. As a further simplification. the polymeric fluid will be modeled by a
system of rigid linear fused hard-sphere (LFHS) chains. The theory that is developed
in this study can be extended to semiflexible chains. as well as to non-uniform systems.
The hard-sphere chain model has an intuitive appeal as a starting point for the
study of real chain fluids. The hard-body monomers interact through site-site po-
tentials which results in an independence of the thermodynamic properties for the
system on temperature. The use of this model simplifies the intermolecular interac-
tions, without sacrificing the geometric features of the chain. The hard-body model
allows for attractive forces to be added into the system as perturbations, thus allow-
ing the hard-sphere c h a h model to play a direct role in the study of more realistic
chain fluids.
1.1.1 Previous S t udies
The initial rnicroscopic study of the N-1 transition was performed by Onsager[2] using
a model of hard rod-like molecules. This investigation opened interest in using hard-
body systems as models for realiçtic systems. In recent years? the hard-sphere chain
fluid has been widely studied as a model for polymeric fluids. This model has been
used as a framework in thermodynamic perturbation theories (TPT)[3, 4, 5, 6. 71,
scaled particle t heories (SPT) [8, 91, integral-equation theories[l0, 11 , 2 131 and
applications of the generalized Flory dimer (GFD) theory[ll, 15? 161. The theories
which have been developed suffer from many limitations, including an inability to
account for liquid-crystalline ordering[.l, 5: 16: 131 and a difficulty in extending these
theories to non-uniform systems. Several densit-functional approaches have been
examined for the latter situation[lï, 18, 191, but have been unable to account for
ordering in the system.
In this study, the theory that is applied to determine the S I phase behaviour
of LFHS chains is a density-functional modification of GFD t h e o r . Several recent
studies[9; 13: 16' 20: 21: 221 have examined this topic. Mehta and Honnell(l6j applied
GFD theory to the LFHS fluid, but were restricted to examining the isotropic phase.
This is also true of the SPT and integral-equation approaches in reference [13], al-
though these investigations yielded some insight into the phase behaviour through
stability analysis of the isotropic phase at high densities. Many recent investigations
have focused on one limit of the LFHS chains, that of linear tangent hard-sphere
(LTHS) chains where the molecular bondlength is equivalent to the monomer diame-
ter. References [9, 20, 21; 221 studied the N-1 phase transition in LTHS chain fluids:
incorporating a density-functional approach based on the decoupling approximation
introduced by Parsons['23] and Lee(241 for hard ellipsoids and spherocylinders. This
approximation consists of a rescaling of the virial series, originally introduced by
Onsager[2], for determining the thermodynamic properties of the rigid-rod fluid.
The formulation employed in this study is sirnilar to tha t of references[g, 20' 21,
221, but incorporates a picture of molecular interactions closely related to that of
the TPT and GFD theories. The thermodynamic properties of the full molecule
are described through a judicious combination of the properties characterizing "ref-
erence" fluids composed of smaller sub-units, narnely monomers and dimers. In a
forma1 and intuitive sense, this picture of molecular interaction is well suited to sys-
tems of flexible chain molecules. The escluded volume for the molecules in previous
studies[l4, 15. 16' 18. 19. 201 has been determined t hrough approximate calculations
or through numerical integration methods. Recently, 'IVilliamson and Jackson[l] de-
rived an analytic expression for the escluded volume of LTHS chains and used this
expression to improve the precision of previous theones for LTHS chain fluids. In
t his study the calculation[l] is estended to LFHS chains of arbitra- intramolecular
bondlength. The resultant esact escluded volume for LFHS chains is a cornerstone
of the approach taken to mode1 the N-1 transition. The concept of exchded volume
is discussed to a gea te r extent in the next section.
1.2 Nematic Ordering
The nematic phase in liquid crystals is characterized by the absence of long-range
positional ordering, mhile the system maintains a degree of orientational ordering.
In effect, the centres of rnass of the molecules have translational syrnmetry as in an
isotropic fluid while the unique axes of the molecules remain: on average, parallel
to each other. This direction of the alignment is referred to as the director, and is
distinguished by a unit vector n. The States described by n or -n are observed to
be indistinguishable in a homonuclear chain.
In order to quanti& the degree of ordering in a systern? it is necessary to define a
nematic order parameter. The Legendre polynomials have the characteristic of van-
ishing when integrated over al1 orientations in an isotropic system. Nematic systems
involve a normalized angular distribution which is not isotropic, with an orientation
parallel or antiparallel to n being preferred over al1 others. It then foilows that the
order parameter can be defined as the integral over the product of the angular dis-
tribution function and a Legendre polynomial. The first Legendre polynomial does
not preserve the aforementioned inversion syrnmet- and will always yield an order
parameter of O. The second Legendre polynornial is thus chosen for the nematic order
parameter,
where f (w ) is the angular distribution function and P2 (cos O) = 112 (3 cos2 19 - 1) is
the second Legendre polynomial. In the isotropic phase, S2 = O: while in a completely
ordered nematic phase, the parameter has a value of S2 = 1. Thus, S2 d l yield the
relative degree of ordering in the nematic phase of the system.
As mentioned previously, the transition between nematic and isotropic phases can
be understood in terms of entropy alone, as a cornpetition between translational and
orientational entropy. At this juncture, it is important to introduce the concept of
molecular excluded volume. The excluded volume is defined as that volume which is
inaccessible to the centre of mass of one molecule due to the presence of another. This
quantity is minimized when a pair of rod-like molecules are in parallel orientations.
For hard-body models, the generalized second virial coefficient is expressed in terms
of this excluded volume between the molecules: weighted by p2. This contribution
to the free energy is minimized when the molecules exhibit orientational ordering.
However, the free energy also involves an entropy term dependent on the angular
distribution function, which is minimized by an isotropic distribution. The latter
term has a linear dependence on density and therefore dominates over the virial
terms at low density, leading to an isotropic phase. At higher densities, molecules
are sufficiently close together that the excluded volume must be taken into account
and the total free energy becomes minimized only by sacrificing some orientational
entropy in favour of translational entropy Thus the excluded volume will have a
direct influence on the degree of nematic ordering in systems above a certain density
The calculation of excluded volume is therefore intrinsic to any theory modeling the
N-1 phase behaviour. As is covered in Chapter 3- the excluded vohme calculation
in this study is performed exactly fm a given angle, 812 between two molecules. The
exact weighted angle-average of the excluded volume is then incorporated into the
theory, as in the theories of references (9, 20. 21, 221. to determine the orientational
ordering in the nematic phase.
1.3 Density- Functional Met hods
There are many different approaches which can be taken to determine thermodynarnic
potentials in fiuid systems. Of these, the most intuitively appealing choices are the
density-functional methods, which are based on the idea that the free energy of a
Buid can be espressed as a functional of the density[2S]. Through modeling the free
energy as a functional of density, for example through a virial expansion, al1 relevant
information pertaining t o the fluid can be obtained[26]. For example, t hermody-
namic functions can be calcu!ated to obtain interfacial tensions. solvat ion forces can
be cornputed, and phase transitions can be investigated through the behaviour of the
pressure and chemical potential. Determination of the exact free energy funct ional
would be equivalent to calculating the partition function for the fluid under inves-
tigation, which for most systems is practically impossible. The density-functional
approach allows for explicit approximations for the free energy that will accornodate
straightfonvard calculations for a multitude of systems. This is made possible due
to the uniqueness of the Helmholtz free e n e r g functional for a given intermolecular
potential[25]. The portion of the free energy of a system which is not direct- associ-
ated with an esternal potential is then entirely independent of that potential for al1
values of density. This enables a calculation of the free energy which is valid whether
the system is isolated, or is subject to external constraints.
1.4 Mean-Field Methods
In previous studies of monatomic systems. a successful approach has been the corn-
bination of a theory for the local structure of the fluid with a statistical mechanical
relation between the local structure and the reduced pressure. This general procedure
requires two key steps[ll]. First, a theory or simulation is performed which deter-
mines the radial distribution function, summarizing the pair correlations between
neighbouring molecules. Second, the radial distribution function is applied in either
the determination of the reduced pressure or compressibility in terms of the virial
theorem to obtain a prediction for these functions. This method yields tremendous
amounts of information about these simple monatomic systems.
In more complicated systems, such as the molecular fluids under study, both the
correlation functions and the viriai expansions become considerably more difficult to
evaluate. These difficulties are; of course, due to the additional degrees of freedom
(i.e., orientational) which must be considered. The technical difficulties which mise
in the "traditional" paths to deterrnining the pressure lead to the conclusion that a
simpler theory may yield more progress to determining the behaviour of these fluids.
It is in this Iight that mean-field methods are applied to determine the thermodynamic
functions in molecular fluids.
The mean-field approach t reats the interaction between the molecule and al1 neigh-
bouring molecules as a mean effect on the molecule. Thus any fluctuations in the
radial distributions for the neighbouring molecules are absorbed into the mean-field
and become averaged out. This approximation limits investigation of the critical
erponents, which depend on local fluctuations, but allows for the determination of
thermodynamic functions such as free energy, pressure, and chemicai potential.
1.4.1 Generalized Flory Dimer Theory
Two theories originally formed the b a i s for mean-field approaches to chain fluids, the
Flory and Flory-Huggins theories. These theories were originally developed for chains
on a lattice, but proved useful in modeling many polymer systems[lil]. In cornparison
with Monte Carlo simulations (to be discussed in Section 1.3), both theories proved
to severely underestimate the pressure. Dickman and Ka11[27] extended the Flory
theory to a continuous-space analogue, taking esplicit account of the real nature
of chains in a fluid state. This generalized Flory (GF) theory was based on an
approximate relationship between the properties of a full chain fluid and those of
a reference fluid cornposed of monomers, and gave fair agreement with simulation
results. Honnell and Ha11[14] built upon the GF theory and derived a new equation
of state for athermal chains employing information from both monomer and dimer
reference fluids. The resulting e4xpressions relate the compressibility factor for the
fluid to the respective monomer and dimer compressibility factors. The weighting
parameters for the reference fiuids were determined uniquely and were consistent
with boundary conditions, as will be discussed in Section 2.1. This study will employ
a modification to the generalized Flory dimer (GFD) theory to provide weighting
parameters which result from the geometric properties of the LFHS chah molecules:
and a separate modification to account for nematic ordering.
1.5 Monte Carlo Simulations
Numerical simulation of statisticai rnechanics problems are performed by taking ran-
dom samples from systems undergoing fluctuations. The dependence on cornputer
generated pseudo-random numbers led to calling these rnethods Monte CarZo meth-
ods. These methods are used to solve problems too complex to allow for ana-
lytic treatment. Simulations of molecular fluids involve numerical evahation of
multi-dimensional integrals which arise in the st a t istical mechanical treatments of
many-body systerns. The Monte Carlo importance sarnpling method introduced bÿ
Metropolis et al. in 1953[28] has proven particularly successful in statistical mechanics
and is the root of most procedures used to this day.
S ince the present study examines the idealized limit of athermal chains interacting
by purely hard-body forces, which does not strictly apply to real fluids. the results of
Our work will be compared with Monte Carlo simulations of such fluids rather than
with direct experimental results. These cornparisons wïll be used to determine the
accuracy of our theory.
Chapter 2
Proposed Theory
-4s mentioned in the previous chapter, the proposed theory is a modification ta GFD
t heory[l4], accounting for nematic ordering in LFHS chain fluids. .Ut hough the actual
study was limited to rigid LFHS chains in simple systems, the general formulation
derived below allows for application to systems of semiflexible chains as ive11 as non-
uniform systems.
2.1 General Formulation
Consider a one-component fluid containing an average of N semi-flexible molecules.
Each molecule is pictured to consist of a chain with n atomic sites. The configuration
of chain i is then specified by the set of positions of these n sites, eYl, Cs, . . - <,., which
is denoted as in. The intramolecular potential energy of an individual chain, which
accounts for al1 interna1 bonding constraints and non-bonding interactions between
atomic sites (monomers) on the same chain, as well as for any interactions with
external fields, is denoted as U(ll (in) . The total intermolecular potential energÿ for N
molecules is assumed to be pairwise additive, and the pair potential between molecules
i and j is denoted as U(*) (in. j,).
The probability density for finding any chain in the configuration in is denoted as
p(in), which is normalized such that
Here, din ZE d e y l , dev2, -. - dc,,. The integration over positions of each atomic site
is unconstrained, apart fiom being restricted to lie within the s-tem volume V . In
density-functional theories, the grand canonical potential of the fluid, R, is expressed
as a functional of p(in). .-\ formallp exact, generahed virial expansion of 0 in powers
of p(i,) can be obtained by straightfonvard adaptation of the expansion for rnonatomic
fluids[29], and is given by[30, 311
where T is the temperature. k is Boltzmann's constant, = ( k T ) - ' , p is the chernical
potential, and u is the thermal de Broglie "volume" of a rnolecule[32]. i l F represents
the excess (over a n ideal gas) Helmholtz free energy of the sustem; and is given by
the generalized virial series expansion
The generalized virial coefficients, B ~ , have a standard diagrammatic representation
in terms of irreducible cluster integrals, where the vertices of the diagrams represent
the products of p(l,), p(2,), . . . p(mn) and the bonds denote the Mayer function
f ( i d n ) = e-9u(2) (in ~ n ) - 1. (2.4)
and the higher order tirial coefficients can be expressed in an analogous fashion.
In the standard application of the decoupling approximation to u n i f o n fluids[23:
24,381, the 2nd virial coefficient is treated exactly while the m t h-order virial coeflicient
is approlamated as
L
where Bzf is the virial coefficient of sorne suitably chosen reference fluid. In previous
investigations, the choice of reference fluid has been limited to either a hard-sphere
fluid[23, 241 or the isotropie phase of the actual system being considered[20. 21, 381.
In both cases, resumrnation of the virial series (2.3) yields
L
In the customary approach of density-functional methods, the distribution func-
tion p(in) in the expressions (2.5) and (2.6) is considered to be a n arbitrary function
of the molecular configuration in. The equiiibriurn distribution function p,,(i,) (and
corresponding equilibriurn grand potential Re,) is O btained by functional minimiza-
tion of 0 with respect to p(in). This leads to the following "self-consistency" equation
for p,, (in) (the subscript "eq" is dropped) :
where C G e b ~ / v is the fugacity and
where A is the functional derivative. In principle, a virial expansion can be ob- 6Ain
tained for C(i,) from (2.3) and (2.10): although it is not necessary to examine this
here.
2.1.1 Monomer Decoupling Approximation
The intermolecular pair potential between chains i and j , LQ2) (in, jn), can be approx-
imated as a sum of atom-atom (or site-site) potentials. i.e.,
By inserting (2.11) into the relation for Mayer functions (2.1) and expanding f (in, j,)
into sums of products of site-site Mayer functions
the generalized virial expansion in (2.2)-(2.6) can be shown to generate the Chandler-
Pratt[30, 311 theory of polyatomic fluids, from nhich in turn can be derived[4, 51
Wertheim's thermodynamic perturbation theory (TPT) [3].
For the purposes of this study, an alternative formulation is considered. Following
the substitution of (2.11) into (2.4), linearization of f (in, j,) in terms of site-site
Mayer functions yields
n.w nnr
f (in: j n ) x 1 C fkl(I6,k - T;,lOr
where it has been açsumed that it might become appropriate to replace the true num-
ber of monomers per chaiq n, by an effective number of monomers, n.bf to prevent the
volume fraction of the reference fluid from exceeding that of the system. Substitution
of (2.13) into (2.5) yields the following approximation for B?,
where
The summations involving the indices cr and 6 in (2.14) are over al1 distinct types of
atoms within a molecule, with atoms of the same type experiencing identical forces.
Summations involving the index k in (2.15) are over all atoms within the molecule
of type a. The quantity p, (q is the site-density of atoms of type a. In the approxi-
mation (2.14). B2 is equal to the generalized second virial coefficient of a mixture of
monomers M t h the set of densities {p,(7)). If an analogous approximation is applied
to al1 higher-order virial coefficients, such that the latter are approximated by their
analogous coefficients for a monomeric mixture, resummation of the series (2.3) yields
Here A F ( ~ ) is the excess Helmholtz free energy of a monomer mixture interacting
through pair potentials {Li,p (r) ) and characterized by densities { p , (3).
The approximation in (2.16) is the same as one described by Woodward[33] for
the case of a homonuclear c h a h fluid containing a single tvpe of site. Woodward also
pointed out that most "self-consistent field" theories of polymer solutions are based on
an essentially equivalent approximation, i.e., one in which the non-ideal effects of the
fluid surrounding any rnolecule are approximated by those of a monomeric fluid. In
this approximation, regardless of how b . ~ ( ~ ) ( { p , ( r ' ) } ) is evaluated: it can be shown
that the function C(i,) defined in (2.10) takes the form of a sum of single-site func-
tiens: Ck(eyk ) . In a uniform fluid this becomes independent of molecular con-
formation. In dense polymer liquids this is not necessarily a bad approximation[34],
however in some cases it is a serious drawback. In relation to this study, the monomer
decoupling approximation alone cannot predict nematic orientational ordering in a
bulk uniform fluid, which should occur for fairly stiff chain molecules[4, 51.
2.1.2 Dimer Decoupling Approximation
Suppose successive pairs of monomers in a chain molecule are grouped into dimers
(assurning for the moment that n is an even number). Then the pair potential between
chains i and j (2.11) can be rewritten exactly as
where the indices k and 1 are summed over al1 dimers in chains i and j , respectively,
and n~ is the number of dimers per molecule. Strictly, no = 4 2 , but as with the
replacement of n by nibf earlier. n~ shall be considered to be an "effective" parameter.
The argument denotes the configuration of dimer k in molecule i, which can be
represented by the positions of the two constituent monomers within the dirner. Al1
of the previous steps and approximations leading to (2.16) c m now be repeated in
terms of the dimer representation, with the site-site Mayer functions replaced by those
between dimers,
and atornic site-densit ies replaced bu dimer densit ies.
where c r ~ labels a distinct type of dimer and CkEQD denotes the summation over al1
dimers k of type &D within a molecule. The dimer analogue of (2.16) is then
A F = A F ( ~ ) ( { ~ , , (R))), (2.20)
where A F ( ~ ) is the excess Helmholtz free energy for a miuture of diatomic molecules
with pair potentials {u;:LD (R. RI)} and the set of densities (R) }.
2.1.3 Derivation of GFD Theory
Due to the non-dependence on molecular conformations in the monomer decoupling
approximation, i t can be conjectured that a judicious combination of the monomer
and dirner approximations could lead to a n improved approximation which mould ac-
count for nematic orientational ordering in chain fluid. At the level of the generalized
second virial coefficient. t his combinat ion can be expressed as
mhere BiD) is the dimer approximation to È2, giwn by the analogue of (2.11) with
{ p a ( 3 ) and {fa&)) replaced by their dimer counterparts. It is then assurned that
al1 higher-order generalized virial coefficients B, can be written in a similar rnanner.
Resummation of the series in (2.3) yields
h possible condition for determining the mixing parameters aiLr) and ai*) is sug-
gested by decoupling theory[2Oo 21: 241, namely that (2.21) be satisfied by the exact
B2. The standard GFD theory in the case of an isotropie, homonuclear hard-sphere
c h a h fluid followç from a further approximation whereby & and BkD) in (2.21) are
replaced by the generalized cross second virial coefficient between a monomer and the
full chain molecule and between a monomer and a dimer, respectively. An additional
condition is then required to fix a;''[) and an) uniquely, which turns out to be
This relation is an extrapolation which satisfies GFD theory, consistent with the
"initial conditions" that a n ) = 1, a,) = O when n = n~ = 1 and = O, an) = 1
when n = n~ = 2nD = 2. With these approximations, one obtains
where Yn is giwn by[16] - -
Here, v,(n) represents the average excluded volume of an n-mer; defined as the anount
of space excluded by the n-mer to a monomer averaged over al1 conformations of the
n-mer.
The remainder of this chapter will be devoted to specializing the theory to the
case of a uniform fluid of rigzd homonuclear hard-sphere chains. The present study
incorporates an exact calculation of the excluded volume between tmo n-mers, detailed
in Section 3.2, which yields a unique choice for the parameters aih') and a n ) depending
only on geometric properties of the molecules.
2.2 Specialization to LFHS Chains
2.2.1 Generalized Second Virial Coefficient
The system is now considered to consist of Iinear chain molecules, where each chain is
considered as a rigid rod composed of n identical hard-sphere atoms, each of diameter
d, with adjacent atoms separated by bondlength 1. It can be assumed that the
molecular rigidity constraints are provided by an appropriate "bonding Boltzmann
factor" [3O, 311 e-Pu(l)(ln), as in (2.9). When the rigidity is taken into account from
the outset, the configuration of any molecule i is specified uniquely by the position
6 of some chosen %enter" of the molecule, and by the Euler angles (Oi, +i) ui of
the molecular a u s relative to some space-hed frame. -411 the formulas of Section 2.1
can be camed over by replacing in with (6, mi) and omitting the intramolecular part
of U(1) (in) -
The one-molecule probability density is now denoted ~ ( F w ) . In a uniform but
possibly orient ationally ordered fluid, this takes the form
where p = $ is the molecular number density and f (w) is the normalized angular
distribution function for the fluid. The generalized second virial coefficient B, defined
in (2.5) becomes
For hard-body interactions, t his rediices to
where @)(Ol2) is the escluded volume between two rigid n-mers. which depends on
the relative angle 012 between their aues. Analogous expressions hold for the monomer
(M) and dirner virial coefficients, B, and B$? In particular?
where p~ = n ~ p is the number density of dimers and uL2) (Ol2) is the corresponding
excluded volume. (Note that for rigid linear rods, the same angular distribution
function f ( w ) characterizes the full rod and any diatornic subunit of the rod.) For
the monomer fluid,
where ph* = n ~ r p is the monomer number d e n s i - The monomer excluded volume
uL1), of course. has no angular dependence. The requirement that (2.21) hold exactly
t hen implies the following relation between the excluded volumeso independent of the
angular distribution function and number densities,
In this study it will be shown that the relation (2.32) is satisfied exactly using
appropriate values of a;'') and a n ) . These values are obtained from the analytic
results for the angle-dependent excludecl volume between rigid n-mers first deriwd
by Williamson and Jackson (WJ)[l], and are geometric results derived for this specific
case. The work of reference[l] was restricted to linear tangent hard-sphere chains,
but it is straightforward to generalize their analysis to LFHS chains of a rb i t rav
intramolecular bondlength, 1. Details are contained in Section 3.2. The basic result
is (see (3.6))
mhere u?) (O) is the excluded volume for two parallel n-mers (OL2 = O ) , given by (3.2)
and vL2)(O12) is the contribution from the so-called ';central region" (in the t e r m i n o l o ~
of WJ) of the overlap volume between two diatomic molecules. This is given by (3.18)
in Section 3.2.
Before comparing (3.32) and (2.33) to obtain the appropriate mLuing parame-
ters, one rewriting of the former relation is required. -4s already noted, the " d e r -
ence" monomer and dimer fluids rnay be characterized by "effective" values of the
numbers nhr and no. Similarly, it may be allowed that the sizes, i.e., diameters
and bondlengths, of the referenee particles differ from those of the original chain
molecule. This feature arises in GFD theory[15, 353 due to the conjecture tha t the
volume fractions of the reference fluids should be equal to that of the original n-mer
fluid. q = pu,, mhere on is the n-mer molecular volume. This treatment is adhered
to in this stud- although no physically rigorous arguments have been made for its
support. The equality between the volume fractions of the reference fluids and n-
mer fluid arises automatically in the case of tangent hard-sphere chains with 1 = d
and no adjustment. of the number and sizes of reference particles is needed in this
case. This is no longer true when a chain of fused hard spheres (1 < d) is decom-
posed into reference monomers and dimers[l3, 351. Thus. the characteristic values for
the reference fluids are taken to be effective values. with the diameter of a reference
monomer denoted as dl1,[, the diameter of a monomer in a reference dimer is denoted
as d D , tvhile the bondlength of the reference dimer is denoted I D . It is these lengths
which characterize the excluded volumes vL1) and u r ) ( Q l 2 ) in (2.32), whereas al1 terrns
in (2.33) for u ~ ) ( B , , ) are characterized by the diameter d and bondlength 1 of the
original n-mer. Therefore. it is appropriate to rewrite (2.32) in terms of the original
molecular dimensions. On dimensional grounds, as confirmed by the equations in the
Section 3.2. the dimer escluded volume scales as
v:2)(@L27 dD7 1 D ) = d i @ ( 012, 2) (2 -34)
where the dependence of uL2)(&) on molecular lengths has been indicated. Then
(2.32) can be remritten as
where v(') and v(*) (Ol2) now refer to the original monomer and dimer inside
Comparing (2.33) and (2.35) yields the relations for mixing parameters.
(2.35)
the n-mer.
where the second line in (2.37) follows from (3.2) and (3.3) in Section 3.2. From these
relations. nDaiD) and n,crak"':c'i are seen to depend only on the geometric properties of
the molecules. -4s d l be seen in Section 2-22? only the combined parameters nD@)
and nhfa:'f) enter into the thermodpamic functions of uniform phases.
Of the prescriptions introduced in GFD theoqfl5, 351 to equate the volume Frac-
tions bp- adjusting the effective parameter values. reliance on the scaling relation (2.31)
restricts the present theory to those methods mhich conserve the reduced bondlength?
= $ E 1.. These are the approaches analogous to the A, C? and -4C versions of
GFD theory[l5]. The GFD-A approach corresponds to taking d = clhf = d ~ , while the
effective numbers of monomers nhl and dirners n~ are adjusted t o conserve volume
fractions. In the GFD-C approach the effective diameters are adjusted to conserve
voiume fractions (with the reduced bondlength conserved), while the relative nurnbers
are taken as n = n.b,r = 2nD. In GFD-AC! the effective diameters and numbers are al1
adjusted to equate the volume fractions and molecular surface areas: while conserving
1'. In the present study, i t turns out that al1 three presciptions yield identical ther-
modynamic results. This follows from the fact that n,brakW) and noan) depend on
the diameters and numbers (for h e d 1') only through the combinations niW(dhf/d3)
and r ~ ~ ( d ~ / d ) ~ , respectively. The values of these combinations are uniquely deter-
mined by the condition of equal volume fractions in each respective prescription,
V, = nbfvl = m2;2 which gives
It is useful to briefly compare the present expressions for aiD) and ae) in (2.36)
and (2.37) with those given by the standard GFD theory for LFHS chain. The
latter were derived recently by Mehta and Honnell[lG] using GFD-.A theory, which
corresponds to taking dhr = d D = d while the effective numbers of monomers ni1f
and dimer no are adjusted to conserve volume fraction. For this case. Mehta and
Honnell[l6] showed that the parameter Y, in (2.26) has the value (n - 2 ) and hence
relation (2 -25) becomes
independent of bondlength. (As will be seen shortly, only the combined parame-
ters nMaiM) and nDaiD) enter expressions for thermodynamic functions of uniform
phases.) For simplicity, compare (2.39) with (2.36) and (2.37) in the case of tangent
spheres, for which n~ = n and n~ = ;. Equations (2.36) and (2.37) then give
where the leading asymptotic dependence for large n is also indicated. In this limit,
the parameters nMakVf) and noan) from the present theory have nearly double the
magnitude of those given by the standard GFD theory. This leads, via nDahD) to
a significantly stronger angular dependence of the excluded volume than would be
predicted by GFD theory, according to (2.32). One notes that nhfaY1 is negative for
n > 2 according to both (2.39) and (2.40), and thus the non-ideal thermodynamic
behaviour of the fluid can be said to involve a subtraction of the behaviour of the
monomer fluid from tha t of the dimer fluid.
By construction, the present theo- yields exact values of the generalized second
virial coefficient. 6r2: in both isotropic and nematic phases of LFHS Chain fluids
with arbitrary n. It is straightformard to show that the approsimate factor of two
difference beh-een GFD and the present theory for the separate parameters ai") and
a n ) largely cancels out in the B? for an isotropic phase, consistent with the findings
of Mehta and Honnell[lG]. The proof niil be investigated here for the case of LTHS
chains.
The relation for B2 in terms of the generalized second virial coefficients of the
reference fluids is given by (2.21). b i n g the results from equations (2.30) and (2.31).
this relation can be rewritten as
where B2, B2D and BaM are the conventional second virial coefficients for an isotropic
hard-sphere chah fiuid, eg.,
For tangent-sphere chains the effective parameters have the values n~ = n / 2 and
n~ = n. The reduced parameter B; is defined as
where vl is the volume of a monomer. In an analogous rnanner, reduced parameters
can be defined for the B coefficients for the reference fluids, B& B 2 ~ / 2 v i and
BZM B2M/u1 = 4. Equation (2.43) now reduces to
The miung parameters are replaced by the constant values from (2.39) for the GFD-
A theory[lô] and by the values from (2.40) for the present theory. Hence. for the
GFD theory in the isotropic limit with n + co:
B,' -+ n[B&-, - 41 - 1.43.n' (2.45)
where the isotropic limit of B;, is taken from Tildesley-Street(361, GD - 5-45 The
present t heory yields
- - The last term in (2.46) evaluates to B;, - a.s r= -0.05
difference from the GFD-..A value in (2 .45) .
I - (2.46)
: which is a fairly small
2.2.2 Free Energy Minimization
The total Helmholtz free energ-, F for a uniform one-component LFHS chain fluid
follows from (2.2): (2.23): and (2.27) (see also the first paragraph of Section 2 . 2 4 :
F = k S p V dw f (w) [ 1 n ( 4 a v ~ ~ f ( w ) ) - 11 + ~P%F(" ) ( p l ~ I ) + a L D ) ~ ~ ( D ) [ j ( w ) ] ) - J (2.47)
The conversion from grand canonical to canonical po tential amounts to omit ting the
chernical potential term in (2 .2 ) . The additional factor of 4x in the argument of the
logarithm in (2.47) is consistent with changing from a site representation to that of
a rigid molecular auis. The notation in (2.47) for nFCM) and AF(O) indicates that
A F ( ~ ) is a function of the monomer number density pbf, white A F ( ~ ) is both a
function of p~ and a functional of the angular distribution f (w) .
For isotropic phases, the previous practice of GFD theory shall be adhered to and
the "exact" Carnahan-Starlingr37 and TiIdesleyStreett[36] equations of state will be
used to prescribe values for A F ( ~ ) and A F ( ~ ) :
A F ( ~ ) ( P ~ ) VkT
H f ln(1- q) + "2(~'+ H f ) - ( F I - G'+3Ht)g] 2(1 - TI)* 1
where
(the quantities F, G, and H denoting the same quantities defined by Tildesley and
Streett [36] ) . As stated earlier, the volume fraction 7 is held to be the same for the n-
mer, dimer. and monomer fluids. In a uniform nematic fluid, the excess monomer free
energy is still given by (2.48). The simplest conceiwble ansatz for A F ( ~ ) (pD; [f ( w ) ] )
is suggested by decoupling theory (2.8): narnely
where J is given by
J l f (41 = I h l d W 2 f ( d f (w)~:~)(Q12) (2.52) * I d W d ~ 2 1 - L ~ ) (612)
The functional J[f ( w ) ] is equivalent t o the ratio between the second virial coefficients
of the nematic and isotropic dirner fluids (see (2.30)). At the dimer level, the form
of decoupling approximation in (2.32) is equivalent to those described (but applied
to the full n-mer) in refs. [20: 21; 381. This difTers from the original formulation of
the decoupling approximation[23, 2.11 in using the isotropic phase of the actual (i.e..
dimer) fluid as a "reference" sustem rather than a hard-sphere reference fluid.
It is of interest to investigate the degree of effect that the functional J[f (w) ] has
by evaluating the integrals in (2.52) in the "perfectly aligned" (PAL) limit. This
limit is induced at high enough density where the orientational order is expected to
saturate, Le., f (a) -+ 6(w - un) where wn is the orientation of the director. In this
limit, the integral in the numerator of (2.52) becomes
For two parallel dimers. the excluded volume is given by equations (3.2) and (3.3) in -
Section 3.2,
The denominator in (2.52) represents the angle-averaged excluded volume of a dimer
in the isotropic phase. This is equivalent to 2BZD: 11-here the notation is the same as
in the previous section. Vsing the notation for the reduced coefficient, B;, = B2D/u2
where 272 is the volume of a dimer, the denominator can be easily evaluated:
Evaluating the ratio of (2.54) and (2.55) requires an expression for B;D for arbitrary
bondlength. This value is derived in Section 2.2.1 for the tangent limit, 1' = 1.0. One
possible method for extending t his result to arbitrary bondlengths is analogous to
the approach applied by Williamson and Jackson[Z] for numerical fits to the excluded
volume between dimers a t arbitra- orientations. Alternatively, the Tildesley and
Street[36] fit can be used as an approximation:
where F(1') is given by ( U O ) . For the tangent case, the ratio is e a d y evduated to
yield
The "exact" value, B;D = 5.4439184[39], fields a value of ( J [ f ( u ) ] ) ~ A L = 0.872533.
It should be noted that this is a fairly small deviation from the isotropie lirnit of
J [ f (41 = 1-
For the case of partial alignment, the angular distribution function is not known à
priori and must be evaluated. -4 self-consistent equation determining the equilibrium
form of the angular distribution function follows by functionally minimizing F with
respect to f (w) , subject to the normalization condition / dwf (w) = 1- The result is
where
and the fact that p~ = n ~ p has been used. Strictly7 the excluded volume v(?) (Ol2) in
(2.52) and (2.59) is that for reference dirners of diameter dD and bondlength l D , but
explicit factors of d; cancel in the ratio of integrals due to the scaling law (2.34) and
only the reduced bondlength Z* = $ = is required in specifying those integrals.
For a given volume fraction 7, the integral equation (2.58) was solved numerically
by iteration as in reference 1301. To evaluate the angular integral involving vL2)(012)
in (2.59), the molecular orientation is espressed as wi (Bi, #*) in the director frame
of reference, Le., with polar angle Bi measured from the nernatic director axis. Then
f (di) = /(Bi) is independent of the azirnuthal angle &. The angle 012 between the
two rod axes can be expressed as
0L2 = sin el sin & cos 02 + COS BI COS 02,
where \ve arbitrarily set giL = O. Then
/ d u 2 f (4~:~) (O,,) = /Ir sin &d& f (O,)@ (0,. O ? ) !
where
Using the analytic formulae for u ! ~ ) ( & ~ ) from Section 3.2. the integrations over O2 and
42 in (2.62) are performed nurnerically by the trapezoid rule: for each pair of O1 and
82, the integration over b2 has to be done only once. The variables cos 19 and 4 were
discretized on grids of typical stepsizes A(cos 6) = 0.005 and A# = &. A change in
frame to Bl2 could be applied to reduce the number of integration variables involved
in the denominator of (2.59): but for numerical consistency and with a vient towards
cancellation of errors. that integral mas evaluated in a manner analogous to (2.61)
(noting that the integration j 2 becomes redundant ) . Sirnilar numerical integrarion
techniques were used to evaluate the denominator of (2.58) and the order parameter
Sz (defined in (1.1) in Section 1.2).
Once the numerical solution of (2.38) is found, the corresponding equilibrium
therrnodynamic potentials can be evaluated from the Helmholtz free energy given by
eqs(2.47)-(2.52). Substituting (2.58) into the logarithrn of (2.47), one obtains
The chemical potential p follows from
Hence
(2.65)
where A p M and Apo are the excess chernical potentials of the monomer and isotropic
dimer fluids respectively,
F where a' (7) G y. Finally. the equilibrium pressure is obtained from P = pp - 7.
which gives
where AZM and AZD are excess rnonomer and isotropic dirner compressibility factors,
respectivelx given by
The isotropic limits of the preceeding equations are obtained by setting f (w) = &. In this lirnit, J[f (w ) ] = 1. Coexistence between isotropic(1) and nematic(N) phases
is then evaluated by solving P(qr) = P(qN) and p(qr) = p(qN), which is done by a
Newton-Raphson procedure.
Chapter 3
Excluded Volume Calculat ion
For the purposes of this study, the specialization of the present theory to LFHS
chains utilizes the exact calculation of the excluded volume (see Section 2.2.1). This
denvation is presented in Section 3.2 of this chapter as an extension of the Williamson
and Jackson[l] calculation for LTHS n-mer chains. Initial investigations in this study
were performed using a hlonte Carlo technique based on Barrett's hlgorithm[40].
This technique is described in the first section of this chapter and can be used for the
general formulation of the theory for semi-flexible chains. for which exact results for
the excluded volume are unavailable.
3.1 Barrett 's Algorithm
Barrett's Algorithm[40] is best viewed as a deterministic Monte Carlo method em-
ploying a random walk through the sample space. The procedure used to apply
this algorithm to the calculation of an excluded volume for hard-sphere dimers is
a straightforward adaptation of the method described in reference [40] and will be
described in this section. The procedure yields accurate results in cornparison to the
exact calculation described in Section 3.2.
For LFHS dimers, the excluded volume will wry as the intramolecular bondlength.
1, is changed. For 1 = 0, the case of superimposed monomers' the excluded volume
between a pair of chains is equivalent to that between two monorners
where d is the monomer diameter in the dimer. This value is independent of orien-
tation and represents the smallest value for the excluded volume between ttvo LFHS
dimers in a given orientation. To determine the maximum possible value of the ex-
cluded volume, consider the case of I >> 1 and & = ~ / 2 . The excluded volume
for this orientation is due to four spheres of equal volume, each given by (3.1). Each
sphere corresponds to the trace of the center of m a s for dimer j when a single
monomer in j is excluded by a monorner in dimer i.
For al1 other cases of 1 and 012, the value of the excluded volume will lie in the range
@ ( l = 0) + 4u(n)(l = O). The excluded volume can be described as a region of space
traced out by dimer j for a h e d position of dimer i, whenever a t least one monomer
in j overlaps with a t least one monomer in i. Randomly generated configurations
of the two dimers: with the constraint that there is at least one monomer-monomer
overlap between i and j , will generally involve several monomer-monomer overlaps.
Barrett's -4lgorithm[40] involves applying a statistical weighting to each configuration
which is sampled between dimers i and j. Each sample is given a weight of lin, where
n is the number of overlapping pairs resulting from this sampling. If only one overlap
occurs for each configuration, the final result of the excluded volume corresponds to
the maximum value. The constraint of pre-determined overlaps ensures that n lies
within the range 1 + 4.
The general procedure of applying Barrett's i21gorithm[40] to the calculation of
an excluded volume between LFHS dimers in a fived relative orientation, BL2, is
straightfonvard. -4 dimer labelled i is chosen to lie in some orientation Bi with respect
to the space-fixed axes' and the positions of the corresponding monomers in i are
determined. An orientation is chosen for dimer j? and the relative positions of the
rnonomers in j are determined. The position of j relative to the origin is left as a
parameter for sampling. Monomer 1 in dimer j is then placed in random positions
within the excluded-voiume sphere of monomer 1 in dirner il and the total number
of overlaps between monomers in i and j is then calculated. The reciprocal of this
number? l / n , is determined and is accumulated over the sample space. The same
procedure is performed for the 3 remaining pairs of monorners in dimers i and j .
The average value of the accumulated weights < l / n > is then multiplied by the
maximum excluded volume to yield the excluded volume between dimers in the given
orientation.
3.2 Exact Calculat ion
Williamson and Jackson (WJ) [1] recently derived an expression for the escluded vol-
ume between a pair of LTHS n-mers at an arbitrary relative orientation Bt2 . In this
study, the derivation is extended to LFHS n-mers of arbit rary bondlength.
The excluded volume for LFHS n-mers in a parallel orientation, OI2 = O? is a
straightforward e-xtension of the WJ result. and is described by a chain of (2n - 1)
overiapping spheres of radius d separated . the n-mer bondlength Z. The total
volume for this chain is given by the relation
where us = h d 3 / 3 is the volume of a single sphere. The overlap volume between a
32
Figure 3.1: Diagrammatic representation of the excluded volume for n = 2. The slice is taken through the z = O plane. where the radius of each circle is the monomer diameter, d.
pair of adjacent spheres, v,, is dependent on the bondlength-to-diameter ratio l l d ,
and is given by
For a pair of LFHS n-mer chains in an arbitrary orientation BL2, the excluded
volume is represented diagrammatically as n2 overlapping spheres of radius d whose
centers Iie on a rhombus (Fig. 3.1). The centers of adjacent spheres are separated
by 1 and the rhornbus has angle The "central region" of the rhombus (in the
terminology of WJ) is defined as the parallelpiped based on the rhombus, taken to lie
in the xy plane and extending along the z avis to distances I d . As in WJ[1], it can
be shown that the excluded volume outside the central region is equal to the excluded
volume in the parallel orientation. For the dimer case. n = 2, the volume outside the
central region has contributions from 2 types of partial spheres, shown in Fig. 3.2.
Each shape is given by overlapping spheres, with the sector traced by the corners of
the parallelpiped removed. There are two contributions from each shape, as can be
Figure 3.2: Two types of partial spheres contributing to the excluded volume exterior to the central region.
seen from Figs. 3.1 and 3.2. When the contributions are summed. with the overlap
portions (each of volume v&) subtracted off, the exterior volume is given as
For arbitrary n: the volume of this exterior region miil have contributions from the
four "corner spheres" as well as from 4(n - 2) partial spheres along the sides of the
rhombus. The contribution from the corners is given by (3.4) . Each partial sphere
along the sides yields a contribution equal to uside = (us - u,)/2. Thus the exterior
volume for arbitrary n is a v e n as
Auezt (n) = 3vS - 221, + 2 (n - 2) (us - v,)
= (2n - l ) v s - 2(n - l ) v , uLn)(0). (3.5)
From the symmetry of the parallelpiped, the excluded volume from the central
region can be related to that for a dimer as v?) (&~) = (n - L ) ~ U ? ) ( O l 2 ) . The total
excluded volume for LFHS n-mers at arbitrary 012 is thus given by
where @)(O) is given by (3.2) and is the contribution from the central region
to the excluded volume between two diatomic molecules.
Following WJ [l], it is convenient to evaluate v?)(Ol2) by considering infinitely
thin slices parallel to xy a t v a ~ i n g heights z. Due to mirror symmetry in the xy
plane, it is only necessary to evaluate distances O 5 z 5 d and then multiply the
results by 2. There are three distinct ranges of z which must be considered. The
outermost range of z is characterized by the absence of two-body overlaps between
the circular cross-sections through the spheres. The total excluded area within a slice
parallel to xy within this range is that of a circle.
where z ranges from d to the height at which two-body overlaps begin ta occur.
The excluded volume for this outer range is thus given by
where the factor of 2 in (3.8) is due to the mirror s y m m e t . The distance where
two-body overlaps begin to occur? hrYII , is dependent on the magnitude of the angle
between the molecular aues, OI2. For large enough OI2 ( ~ / 2 2 OI2 2 7r/3), overlap
between the circular sections first occurs for adjacent pairs of spheres. At this height ,
the radius of a circular cross-section is r = 112, and the corresponding value of
For smaller OI2 (O 5 ûI2 5 ~ / 3 ) , the first two-body overlaps occur between opposing
spheres (Figure 3.3). This height is given by the relation
Figure 3.3: Second case for two-body overlap, when O < O,, c a/3.
The second range of the dimer central region is characterized bÿ two-body over-
laps between the circular cross-sections through the spheres. For both angular cases
evaluated above, this second range extends until z = hrrtrrr, where the circular cross-
sections through the spheres begin overlapping in a three-body configuration. The
value for this height is given by
Throughout the second range, the total excluded area of a parallel slice can be es-
pressed as
A:'(=) = dé(z) - &(z ) (3.12)
where Aé(z) is giwn by (3.7). A0 (2) is the total two-body overlap area between the
circles at height z. This overlap area has distinct contributions depending on whether
the overlapping spheres are adjacent or opposite from each other(Figure 3.3). The
total overlap area is then
iio(z) = 2ao(l, r ) + ao(î, z ) ,
where 1 = 21 sin (OI2/2). Here, a,([', z ) is the overlap area for a pair of circles of radius
r , separated by 1': given by
with r = dd2=tT. The relation (3.14) differs from equation (7) of WJ by a factor
of 2 because WJ calculate only half of the overlap area. Remembering that the
contributions of the two types of overlap begin at different values of 2' the excluded
volume of this region can be erpressed as
by cornbining (3.12) and (3.13). The difference between the two ranges of OI2 enters
in the first term of (3.15) and eventually cancels out with the contribution (3.8)
from the outermost region. The integrah involving aks are quoted by WJ as being
"intractable", but the results of this study have yielded a compact and tractable form
for the indefinit e integral:
One can verifi that the z-derivative of the right-hand side of (3.16) equals a,(lr, z).
The innermost part of the central region is defined by z 5 hrr,rrr7 and is charac-
terized by three-body overlaps. The entire volume of this region is excluded. Thus,
the contribution -from this region to $1 is equal to
The excluded volume in the central region is thus given by a sum of the three
parts,
where the various contributions are given by (3.8), (3.15) and (3.17), respectively, for
012 5 7~12 . Due to chain inversion symmetry, the excluded volume for 012 > 7r/2
is given by uL2) (0'2) = ui2) (K - 19,~). Substituting the expressions for h(lfiI, h&
and hrI,rrr from (3.9), (3.10) and (3.11) into (3.15) and (3.16), and applying sev-
eral trigonometric identities (most importantly the relation tan-'(A) - tane'(B) =
tan-' [(i4-B)/(1 +AB)]), we are able to reduce the result for vL2) (012) to the following
compact form:
v ) ( 0 ) = (;l2d) sin (Bi&) S(B12)
1 sin (Ol2/2) + 4 1 [ c ~ - ; ( t ) ~ ] t a n - ~ [ - d ~ 1 2 ) ] +41 sin (O&) [ 3 d2 - - sin2 (OL2 1 2 )
- ( 3 3 ) tan-1 [ l 2 sin (e12 1 2 ) S(BI2)
4dZ - l2 (1 + 2 sin2 (oL2/2))
where
It can be verified that the preceding result for the angle-dependent excluded vol-
ume of LFHS n-mers reduces t o that for hard spherocylinders in the limit that n + cm,
1 + 0, and L r (n - 1)l remains finite, where is identified with the cylinder length.
In this limit, (3.2) and (3.3) give
where us= is the spherocylinder molecular volume. For s m d 1 , (3.1'7) reduces to
~ : ' * ( 8 ~ 2 ) = 212d sin eL2 + 0 (14) Y (3.21)
whae it cari be s h o m that uL(012) + ui1(012) is of leading order 0(14). Hence; in this
limit, (3.6) becomes
L J ! ~ ) ( e l2 ) -+ 8vSC + 2 ~ ~ d sin 012 , (3.22)
agreeing wit h the result derived for hard spherocylinders[2, 2.13.
3.2.1 Alternate Formulation
Dr. B.G. Nickel has recently derived an alternate formulation for the excluded volume
between LFHS n-mer chains, which is detailed in Appendix hl. The excluded volume
between two LFHS n-mer chains at an arbitrary orientation 612, is (Al-4)
which agrees with the result in (3.1gaob). This result can be integrated
exact generalized second virial coefficient for an isotropie phase. defined
to yield the
in (2.42),
This latter result agrees exactly with the previous calculations (3.6) and (3.18): as
well as with the Isihara calculations[39].
Chapter 4
Results and Cornparisons
The primary limitation to the analysis of the present theory is the lack of extensive
Monte Carlo simulation data available (see Section 1.5) for comparison. Of the various
Monte Carlo studies which have examined systems of hard-sphere chains, the majority
have dealt with semi-flexible, tangent hard-sphere chains. Several of these studies have
included investigations of the limit of infinite rigidit. The results yielded by these
latter simulations are used for comparison of the predictions by the present theory
in the first section of this chapter. In particular, recent studies by Yethiraj and
F-ynewever[9, 221 have provided substantial analyses of the %mer and 20-mer LTHS
chains. Williamson and Jackson[21] recently exarnined systems of 7-mer LTHS chains
and performed cornparisons wit h t heoretical treatments involving the exact excluded
volume for the chains[l]. These studies are the primary sources for cornparisons with
the present theory. The second section focuses on cases of LFHS chain systems. The
results from this study are compared with the only available Monte Carlo data for
LFHS chains[4i], as well as with a modified form of GFD theorv[16] and a modified
TPT[16, 31.
4.1 LTHS Chains
The conventional orientational order parameter S2 is defined by (1.1) from Section
1.2. Figures 4.l(a) and (b) depict S2 as a function of the volume fraction q for 8-
mers and 20-mers: respectively The Monte Carlo simulation data[9] is obtained using
both constant-pressure (NPT) and constant-volume ('NT) methods, which are seen
to be in very good agreement. This data is compared with the present theow and
with the Parsons theory[23], as employed by Yethiraj and Fynewever[g, 221 (using a
simple hard-sphere reference fluid and incorporating the excluded volume derived by
Williamson and Jackson [l]) .
In general, the present theory is in excellent agreement with the simulation data
for the degree of ordering in each system. The order parameter at which the nematic
phase first appears is of the order S2 > 0.5. The degree of ordering continues to
increase dramatically over an extremely short density range, until S2 = 0.8, after
which the value of the order parameter begins to level off. This general behavior
is also evide~it in the Parsons theory, although the values obtained for S2 and the
coexistence densities are less accurate. For the &mer LTHS chain systems' the values
of the order parameter predicted by the present theory are in excellent agreement with
the values obtained by simulation. The results for the 20-mers exhibit slightly lower
values of S2 in cornparison with the simulations, although the amount of data available
is much smaller in this case. The Parsons theory systematically underestimates the
degree of ordering in the nematic phase, as is evidenced in Figs. 4.l(a) and (b) .
In Figs. 4.2(a) and (b), the reduced pressure P' = PvJkT is plotted as a function
of volume fraction for the %mer and 20-mer LTHS chains, respectively. The simu-
lation data clearly show the coexistence densities between the isotropic and nematic
branches. The values for these densities as yielded by both theory and simulation
- Present theory _ _ _ _ Parsons theory
MC-NPT data o MC-NVT data
- Present theory - - - - Parsons theory
MC-NPT data O MC-NVT data
Figure 4.1: Variation of order parameter S2 with volume fraction 7, comparing present theory with the Parsons theory and Monte Carlo data[9, 221, for (a)&mer LTHS chains, and (b) 20-mer LTHS chains.
- lsotropic branch, present theory - - - Nernatic branch, present theory
Coexistence region, present theory ------------ Isotropic branch, Parsons theory --- Nematic branch, Parsons theory
o lsotropic branch, MC-NPT data O Nematic branch, MC-NPT data
- Isotropie branch, present theory - - - Nematic branch, present theory
Coexistence region, present theory ------------ Isotropie branch, Parsons theory --- Nematic branch, Parsons theory
O lsotropic branch, MC-NPT data O Nematic branch, MC-NPT data
I
Figure 4.2: Variation of the reduced pressure with volume fraction 11, comparing present theory with the Parsons theory and Monte Carlo data[9, 221, for (a)&mer LTHS chains, and (b)20-mer LTHS chains.
LFHS Chain: Sources 1 r)(iso) ~ ( n e m ) &(nem) P* 7-mer. 1' = 1.0 1 Present theory M C N P T data[21] Vega-Lago t heory[2 11 Parsons t heo- [2 11 8-mer, 1* = 1.0 Present theory MC-NPT data[9] Parsons theory[9, 221 20-mer. Z' = 1 .O Present theory WC-NPT data[9]
0.2903 3.2989 0.6491 4.94 0.266-0.303 0.285-0.312 0.64-0.66 3.15-3.78
0-355 0.273 25 0.7 2-78 0.303 0.319 N/A 3.12
0.2601 0.2689 0.6538 3.95 0.251 O. 271 4 . 7 2.63 0.280 0.305 = 0.7 2.6
O. 1158 O. 1243 0.694'7 0.97 0.105 O. 120 30.7 0-62
Parsons t heor-y[$ 221 8-mer. Z* = 0.5
0.115 O. 140 h: 0.75 0.69
1
Present theory MC-NPT data[4l] 8-mer, 1' = 0.6
MC-NPT data[41] 1 0.419 0.624 5.7
0.4687 0.4768 0.6168 13.1 No transition found
1
Table 4.1: Coexistence results from simulation and theory
Present theory
are given in Table 4.1, along mith the order parameter S2 in the coesisting nematic
0.4167 0.4251 0.6224 9.8
phase. The coexistence densities predicted by the present theor- agree very ive-11
with simulation. although the reduced pressure at any density shows poor agreement.
The present theory overestimates the value of P* throughout the densit- range, a
discrepancy which becomes more pronounced at higher densities. In cornparison, the
Parsons theory predicts coexistence densities substantially higher than those observed
in simulation as well as a larger coexistence range, as is depicted on both graphs. The
Parsons theory underestimates the value of the reduced pressure throughout the den-
sity range for both systems. Yethiraj and Fynewever[9] note that SPT treatments also
underestimate the pressure for the %mer systeml while overestimating it for larger
molecules.
The ,-mer LTHS chah system has been recently examined by Williamson and
Jackson [W.J][21] in ïight of their explicit calculation of the excluded volume for
LTHS chains[l]. WJ performed Monte Carlo simulations for a system of N = 576
molecules. The results of the simulation were compared with several theories, after
modifications to incorporate the calculation of the exact excluded volume. In partic-
ular, W J compare the simulation results with those obt ained from a modified version
of the Vega and Lago theo~[20] . The original Vega and Lago theory is based on the
form of the decoupling approximation in (2.9) and (2.10), where the reference fluid
is the isotropic phase of the n-mer fluid. JVJ modifS. the original Vega and Lago
theory to incorporate their esact analytic calculation of the excluded volume into
$, and obtain the isotropic phase free energy from TPT[3, 421. Figure 4.3 depicts
the comparison of the results for S2 from the present theory with the Monte Carlo
simulation data[2l]. As in Fig. 4.1, the present theory predicts the order parameter
with excellent accuracy The general behavior of S2 is similar to the previous graphs.
The cornparison of reduced pressure versus volume fraction for the 7-mer system
is shown in Fig. 4.4. The modified Vega and Lago theory yields accurate pressures in
comparison with simulation, but the predicted transition densities (evidenced by the
plateau in the trace) are too low. This is also indicated in Sable 4.17 mhich s h o w
that the present theory yields coexistence densities lying within the ranges obtained
through the simulations. The Vega-Lago theory also overestimates the value of S2
in the nematic phase. However, as in Fig. 4.2 the values of reduced pressure given
by the present theory clearly exceed the reported simulation data. This discrepancy
appears to be systernatic in the theory and will be discussed in Chapter 5. It should
be noted here that the WJ simulation data predict a smectic phase to occur a t volume
Figure 4.3: Order parameter S2 vs. volume fraction for LTHS 7-mers, comparing present theory with Monte Carlo data[21).
1
0.8
- Present theory oO O MC-NPT data (expansion} 0 MC-NPT data (compression)
-
-
- Isotropic branch, present theory --- Nematic branch, present theory
Coexistence region, present theory - --- Vega & Lago theory
MC-NPT data (expansion) 0 MC-NPT Data {compression}
Figure 4.4: Reduced pressure for LTHS ?-mers as a function of volume fraction q; comparing present theory with Monte Carlo data and the modified Vega-Lago theory from Ref. [ai].
fractions greater than 0.37.
The relationship between the length of molecules and the range of the coexistence
region is depicted in Fig. 4.3. The general trend is an increaçing difference between
the densities of the isotropic and nematic branches a t coexistence as the number of
rnonomers constituting the molecule increases. This trend is consistent with simda-
tion, as is shown in this figure for T-mers. &mers and 20-mers. The Parsons theoq
and various SPT treatments have yielded a similar trend[9], although these theories
tend to overestimate the width of the coexistence region- In addition, the values of
the coexistence densities obrained from these theories are in poor agreement with
simulation, while the present theory provides a quantitatively accurate description of
t hese densit ies.
4.2 LFHS Chains
Simulations of LFHS chains have been performed by Whittle and Masters[.ll] for the
cases of 6-mers wit h a reduced bondlength of 1' = 0.5 and 8-mers with 1' = 0.5 and 0.6.
These three systems were recently investigated by Mehta and Honnell[lGI using GFD
theoq, comparing their results mith the Whittle and Masters[.ll] simulation data and
with a modification of TPT[3]. The GFD theory and the modified TPT are unable
to treat ordering, and therefore are inappropriate at densities above the isotropic-
nematic transition which was evidenced in the simulation of the 8-mer systern with
l* = 0.6. Figure 4.6 depicts an analogous study of the three fluids. comparing the
previous results[l6] with those of the present t h e o - Figure 4.6(a) shows the reduced
pressure as a function of volume fraction for the 6-mer LFHS c h a h with 1' = 0.5.
There is no evidence of a nematic transition in the simulations, nor is there any
- Isotropie branch - - -t Nematic branch
Isotropic branch, MC data O Nematic branch, MC data
\
6 9 12 15 18 21 Number of Monomers
Figure 4.5: Cornparison of volume fractions of the isotropic and nematic phases at coexistence for LTHS n-mers, as a function of the number of monorners n, between the present theory and Monte Carlo simulation data.
indication through the t heoretical treatments. The present t heory and the modified
TPT both overestimate the values of the reduced pressure, while the GFD theory
appears to predict the reduced pressure to a fair degree of accuracy. This trend is
also apparent in Figs. 4.6(b) and (c) for the 8-mer cases. In the latter plots, the
present theory is seen to predict a stable nematic branch a t sufficiently high volume
fractions. The simulation da ta in Fig. 4.6(b) elchibit no such transition, and me are
unaware of any other simulations which have been done for this system. It should
be noted that the monomeric reference fiuid does not exist as a stable fluid above
7 x 0.494[16] and as such the theory is probably invalid at these high densities.
Figure 46(c) investigates the &mer fluid with 1' = 0.6; a system which clearly
exhibits nematic ordering. At first glance it appears that the present theory, whilc
succeeding in predicting a stable nernatic brancha predicts that it occurs at volume
fractions much greater t han the simulations indicate. However. it should be noted that
although Whittle and Masters[4l] report a nematic branching at q z 0.33, t he order
parameter was not found to be stable until much higher volume fractions, 7 2 0.419.
The predictions of the present theory for transition properties are compared with the
simulation results in Table 4.1. Once again, the present theory is seen to be in good
agreement for the value of the nematic order parameter S2.
Figures 4.7(a) and (b) investigate the relationship between the length of the
molecule and the width of the coexistence region for a constant number of monomers.
Figure 4.7 (a) studies 8-mers with varying reduced bondlength values, from 1' = 0.5
to 1* = 1.0. The resulting trend in the coexistence region is analogous to tha t shown
in Figure 4.5. As the length of the molecule increases, the coexistence densities of
the branches decrease, while the differences between the isotropic and nematic values
increase. Figure 4.7 (b) shows a n identical study for the 20-mer LFHS systems with
- Present theory --- Mehta & Honnell GFD theory --*--*.----- Modified TPT
A MC data
Isotropic branch, present theory --- Nematic branch, present theory
Coexistence reg ion, present theory Mehta & Honnell GFD theory Modified TPT MC data
Isotropic branch, present theory --- Nematic branch, present theory
Coexistence region, present theory --- Mehta & Honnell GFD theory ------------ Modified TPT
A Isotropic branch, MC data O Nematic branch, MC data
Figure 4.6: Cornparison of the reduced pressure between the present theory, the Mehta and Honnell GFD theory and TPT in Ref.[lG], and Monte Carlo simulations[41] for (a)LFHS 6-mers with bondlength to diameter ratio 1' = 0.5, (b)LFHS 8-mers, 2' = 0.5, and (c)LFHS &mers, 1' = 0.6.
u Isotropic branch, present theory t - + Nernatic branch, present theory branch, present theory
M Isotropic branch, present theory t - + Nematic branch, present theory
Figure 4.7: Volume fractions of the isotropie and nematic phases at coexistence as a function of the reduced bondlength l * , for (a)LFHS 8-mers, (b)LFHS 20-mers.
reduced bondlengths 1' = 0.2 to 1' = 1.0.
It is of interest to examine the present theory in the '%pherocy1inderz limit, which
is obtained in the limits n -+ xo 1 -t 0. such that L z (n - 1)l remains h i t e . -4s
discussed in Section 3.2. the analytical result for che LFHS excluded volume reduces
(as should be expected) to that for hard spherocylinders of cylinder length L in
this limit. Figure 4.8 shows the nematic and isotropic coexistence volume fractions
for LFHS n-mers as a function of n. for a fked value of Lld = 19, in cornparison
with the values for spherocylinders obtained from the Lee theory[24] and from Monte
Carlo calculations (using the Gibbs-D uhem integration procedure) by Bolhuis and
Frenkel[43]. Along the coexistence region. bot h the individual volume fractions and
their differences increase slomly mith n, approaching the asymptotic spherocylinder
values although underestimating the coexistence width in the limit. In more detail.
it can be shown that the present theory in the spherocylinder limit yields exactly the
same dependence of the angular distribution function f ( w ) on volume fraction 7) as
in the Lee theori[Z4]. However. the free energy and pressure are actually predicted
to diverge in this limit. mhich accounts for the narrower coexistence width found in
Fig. 4.8.
I Isotropic branch, present theory
--- Nematic branch, present theory
4 Isotropie spherocylinder value, Lee theory 4
4 Nematic spherocylinder value, Lee theory O lsotropic spherocylinder value, Bolhuis and Frenkel
Nematic spherocylinder value, Bolhuis and Frenkel 1 -
30 40 Number of Monomers
Figure 4.8: Volume fractions of the isotropic and nematic phases a t coexistence as a function of number of monorners for LFHS chains of constant length with Lld = 19. The spherocylinder limits are given by the Lee theory(241 and by the Monte Car10 simulation results of Bolhuis and Frenkel[43].
Chapter 5
Conclusions and Future Work
In this study, two key modifications have been introduced into the generalized Flory-
dirner (GFD) theory to describe nematic behavior in hard-sphere chain fluids of arbi-
tra- intramolecular bondlength. The first modification is the inclusion of the exact
excluded volume and second virial coefficient for LFHS chain molecules, based on
generalizing the earlier calculation of Williamson and Jackson[l]. This procedure
results in rnixing parameters for the reference monorner and dimer fluids which are
dependent only on the geometric properties of the molecules (see (2.36) and (2.37)).
A related feature is that the ansatz of equal volume fractions uniquely determines the
values of the relevant combinations of the effective reference parameters 7 2 ~ , dg,
and dhf, in contrast to previous studies based on GFD and related theories[7, 15: 351.
The second modification of GFD theory, in order to account for the possibility of ne-
matic ordering in the system, is the weighting of the excess dimer free energy by the
ratio between the second vinal coefficients of the nematic and isotropie dimer fluids
(see (2.51) and (2.52)). This ansatz, suggested by decoupling theory, is dependent on
the angular distribution function, which is determined self-consistently-
The present theory is found to be in excellent agreement with simulations of
LTHS fluids in determining coexistence densities and the nematic order parameter as a
function of densi t . As is seen clearly in Sable 1 and Fig. 5 , the transition densities for
the 7-mer, 8-mer and 20-mer fluids predicted by the theory fa11 within the coexistence
range found in simulations. In particular: the 7-mer system studied by Williamson
and Jackson[Pl] shows much better agreement with the present theory than with the
Vega and Lago[PO] and the Parsons[P3] theories for the coexistence densities as well as
the nematic order parameter a t the transition. The present theory is able to account
for stable nematic branches in LFHS chains of sufficient length? for which theoretical
work has been lacking up t o now. The 8-mer with 1' = 0.6 simulated by Whittle
and Masters[4l] yielded transitions a t lower densities t han predicted by the present
theory, although the simulated nematic branch did not become stable until densities
similar to those predicted in our work. In addition, transitions between the isotropic
and nematic phases are predicted by the present theory for several systems which
have yet to be simulated (see Figs. 4.5, 4.7 and 4.8). The agreement between the
present theory and simulations is promising and it is hoped that further work d l be
encouraged.
For al1 systems studied, the present theory yields values for the pressure which
exceed those given by available simulation data as well as previous theories. This
discrepancy is not accounted for within this study, and further work needs to be done
to resolve the problem. It was noted in Section 4.2 that the free energy and pressure
given by the present theory actually diverge in the spherocylinder limit n + cm,
1 -+ O, with (n - 1)1 -= L finite. This divergence can be traced to the incorrect
limiting behavior of the Tildesley-S treet t (TS) [36] dimer equation of state for small
reduced bondlength 1'. The TS equation of state predicts that the reduced isotropic
second virial coefficient (as well as higher virial coefficients) varies linearly with 1' (see
(2.50) J , in contrast with the correct leading-order variation proportional to (1*)2 (see
(3.21))- The generalized second virid coefficient determined by the present theory is
accurate to within a few percent for 1' > 0.5 (see Appendix -42). However, for small
l*, as in the results s h o m in Figs. 4.7(b) and 4.8, this limitation prevents accurate
investigation of the c h a h fluids.
Despite the mived agreement of the resuits from this study with those of available
simulations, the approach described here has distinct advantages over ot her current
density functional theories of chain Buids[4_ 5 : 9? 17- 18: 19. 20, 21, 221, particularly
for considering extensions of the theory to non-uniform fluids and ones containing
semi-flexible molecules. These generalizations of the t h e o q are briefly indicated in
Section 2.1, although the details of such extensions clearly require further work. In
particular, while the inclusion of differing species of monomers and dimers enters into
the theory in (2.15) and (2.19), the evahation of the densities and the corresponding
effects on the free energy functional require additional study. The flexibility of chains
can be introduced directly into the intrarnolecular potential energy, U(&). The
effect of flexibility on the escluded volume between two hard-sphere chain molecules
must be determined nurnerically and thus the exact calculation detailed in Section
3.2 is invalid in tQe case of semi-flexible chains. The derivation of mixing parameters
for the reference fluids: which for LFHS chains arose from the exact second virial
coefficient, also requires additional investigation.
The present formuiation retains the geometrically motivated spirit of the GFD and
TPT theories in utilizing reference fluids composed of monomer and dirner subunits,
whose properties can be more readily determined. This contrasts with the form of the
decoupling approximation used in refs. (20, 2 11, which requires à pn'ori information
about the thermodynamics of the isotropie phase of the full system being considered,
which may be either unavailable or computationally difficult to obtain in the more
general cases. At the same tirne? it is crucial that the reference systems involve
orientational degrees of fieedom, as does the dirner fhid. This feature corrects a
limitation of previous density-functional approaches for nûn-uniforrn c h a h fluids[4, 5 ,
17, 18, 191 based on a purely monomeric reference Buid, which are unable to account
for orientational ordering effects.
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D. Henderson. ed. (Dekker. England. 1992).
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Phenornena. (Springer-Verlag, Berlin, 1984).
[29] G. Stell, in The Equilibrium Theory of Classical Fluids.
H.L. Frisch and J.L. Lebowitz, eds. (Benjamin, New York, 1964).
[30] D. Chandler and L.R. Pratt, J. Chem. Phys. 65, 2925 (1976).
66
(311 L.R. Prat t and D. Chandler: J. Chem. Phys. 66, 147 (1977).
[32] M.P. Allen, G.T. Evans, D. Frenkel and B.M. Mulder, in Advances in Chernical
Physics. 1. Prigogine and S.A. Rice. ed. (John Wiley and Sons; Torontoo 1993).
[33] C.E. Woodward, J. Chem. Phys. 94, 3183 (1991).
[34] S. Neyertz, D. Brown and J.H.R. Clarke. J. Chem. Phys. 105, 2076 (1996).
[35] A- Yethiraj, J.G. Curro, K.S. Schweizer and J.D. McCoy, J. Chem. Phys. 98,
1635 (1993).
(361 D.J. Tildesley and W.B. Streett. Mol. Phys. 41, 85 (1980).
(371 N.F. Carnahan and K.E. Stariing, J. Chem. Phys. 51, 635 (1969).
1381 P. Padilla and E. Velasco, J. Chem. Phys. 106, 10299 (1997).
[39] A. Isihara, J. Chem. Phys. 19, 397 (1951).
[40] A.J. Barrett, Macromolecules. 18, 196 (1985).
[41] M. Whittle and -4.J. Slasters. Nol. Phys. 72, 247 (1991).
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9348 (1991).
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[49] K.M. Jaffer, S.B. Opps and D.E. Sullivan: J. Chem. Phys. t o be published
(June 15, 1999).
A l Alternative Excluded Volume Calculat ion
Dr. B.G. Nickel recently derived a formulation for determining the excluded volume
between FHS dimers at arbitrary orientations using st rictly geometric arguments.
This derivation can be extended to LFHS n-mer chains using the scaliog arguments
from Section 3.2. The procedure is detailed in this appendix.
Consider two FHS dimers. taken to lie in the xy plane. The center of one dimer
is Exed at the origin. The excluded volume boundary wiIl be given by the set of
surfaces f,,(xyz) = O' Ja6(xyz) = 0, etc. where each surface is defined by monomers
from different dimers touching. In total, there n-il1 be four surfaces. Each pair of
surfaces involving the same monomer meet on planes perpendicular to the xy plane
(see Fig. Al) , where that monomer touches both monomers on the other dimer. These
planes can be used to subdivide the xyz space for convenience.
A fifth plane can be constructed? defining the region where al1 four monomers
are in contact painvise. This plane is the bisector of the intermolecular angle. This
results in four separate regions, which can be split by syrnrnetry to give the final
result
(Al. 1)
where V i and dB are the volumes of Region A and B in Fig. A l respectively.
The evaluation of the volumes for each region is involved, and will not be detailed
here. The result for the dimer case is given by
Figure Al.1: Depiction of excluded regions between monomers in different molecules for FHS dimers at arbitra- orientation 812. The monomers in contact over each region (bounded by solid lines) are denoted. The dashed Iines indicate quadrant s y m m e t -
for 2' 5 fi, where the diameter is taken as d = 1. The escluded volume for two
dimers in a parallel orientation can be readily derived from this relation
~ ( ~ ' ( 0 ) 4~ -- d3
- -+ 3
and can be generalized to the n-mer case using equation (3.5) from Section 3.2.
The general expression for LFHS n-mers a t arbitrary orientation OI2 c m deter-
mined in an analogous manner to (3.6) from Section 3.2,
using the terms derived in this appendiu.
70
A2 Cornparison of Virial Coefficients
Table A2.1: Cornparison of second and third reduced virial coefficients between the present theory, exact calculation[49] and Monte Carlo simulation data[l6].
n-mer 3
The present theory uses the exact calculation of the generalized second virial co-
efficient (see Section 3.2) t O determine the weighting parameters in the decoupling
I f 1.0
approximation (2.47). These parameters are then applied to the Carnahan-Starling[G]
h ~2~ emy ) B;(eract) B;(themy) B;(MC)
6.885 6.85 6.85 27.43 24.47
isotropic fluids. respectivel. It is of interest to determine the accuracy of this pro-
cedure in calculating the virial coefficients for LFHS n-mer chain Buids. The theo.
does not precisely reproduce the exact second virial coefficient, because F' in the
Tildesley-Street equation of state (2.49) in Section 2.2.2 is not consistent with the
exact B; (discussed at the end of Section 2.2.1 on page 24), and is a particularly poor
approximation for small 1'. The discrepancy in the third virial coefficient, noted in
Table A2.1, could arise from the approximation to Bg in (2.49). For srnaIl 1': nrher-
Monte Carlo data is not available, results from the present theory diverge and are
unable to reproduce the exact virial coefficients, as mted in Chapter 3.
A3 Density Functional Program
PROGMM DENSITY
IMPLICIT NONE
! Karim Jaffer MSc. 1999. Supervisor: Don Sullivan
! This program incorporates a modified GFD t h e o ~ to calculate the
! isotropie and nematic branches of free energy, reduced pressure, and
! reduced chemical potential as a function of volume fraction for a
! system of linear fused hard-sphere chains. -4 Newton-Raphson
! algorithm is then used to determine the coexistence densities.
!
! DECLARATION OF V.4RIA4BLES
INTEGER Nmol,Nm,a~b,k,17x,y1z,zl,zla7zlb,z2,z2a,z2b
REAL pi,tol,deIJ~jint,Vsys~VmollVpn~p2,ratiom,ratiod,fuse
REAL fusecube,fuselengt h,alphmd~vfrac~Fprime ,Gprime,Hprime
REAL betanm,~1uum~~~uud,delzm,delzd,Sfactor,Sfactor3,maxit2
REAL angc,angp1angf,isoc,isop,isof~iso~,angf2,isop2,an~2
REAL isoc2,angc2,freedif,chemdif,pressdif,iprime,aprime7isofd
REAL ini, ani,diffc,diffp,dchemildchema,dpressi,dpressa7denom
REAL ifree(500) ,afree(5OO) ,ipress(500) ,apress(300) ,S (500)
REAL ichem(500) ,achem(500) ,angle(40I) ,ext2(401)
REAL cext2(401) ,sext2 (401) ,int (401,401) ,F (500,401)
DOUBLE PRECISIOW fini(4Ol) ,fend(401) ,S2(4Ol)
!
! MAIN CODE
! Open output files.
0PEN(G7FILE='865001.dat')
0PEN(7,FILE,='865001f.dat7)
OPEN(8,FILE='S6ûOOlp.dat~)
OPEN(9,FILE=%65001~.dat')
OPEN(lO,FILE=736500l~.dat')
! Define al1 constants and parameters.
pi=3.141592654
tol=0.000001
k = l
1=401
maxit2=10000
vsys=1000.0
Vmol= 1. O
! Define bondlength to diameter ratio.
fuse= (0.6)
fusecube=fuse*fuse*fuse
! The following functions will be used to calculate wrious isotropic
! thermodynamic variables which depend also upon density.
Fprime=4.0+0.37836*fuse+ 1.07860*fusecube
Gprime=-2.0+l.30376*fuse+1.80010*fusecube
Hprirne=2.39803*fuse+0.35700*fusecube
! Equate volume fractions using constant ratios.
Nm=8
ratiod=(l.O+(Nm-l.0)*(1.5*fus~fusecube/2.0))/(l.O+l.û*~se
-fusecube/2.O)
ratiom=l .O+(i\im-1 .O) *(1 .Phise-fusecube/2.0)
! Determine mising coefficients for the monomer and dimer fluids.
Vpn=(2.0*Nm-1.0)-2.O*(Nm-1 .O) *(1.0-3.0+fuse/4.O+fusecube/l6.0)
Vp2=3.O-2.O*(l.O-3.O*fuse/.I.O+fusecube/l6.O)
alphand=(Nm-1 .O) *(Nm-1 .O) /ratiod
betanm=(Vpn-(Xm-1.0) *(Nm-1.0) *Vp2)/ratiom
! Define the theta angles for integration.
do a=0,400
b=a+l
angle(b)=acos(l.O-(a+O.O)/200.0)
end do
! Determine the excluded volume as a function of the orientation of
! each rnolecule.
CALL exvol(k,l,pi,fuse,angle,int,elt2)
! Calculate the Stability Factor.
do a=1,401
S2(a)=dble(5.0/(8.0*pi) *(3.0*cos(angle(a))*cos(angle(a))-1.0))
end do
GALL calcj (k,l,pi,S2,ext:!,int ,sext2,delj)
WRITE(G,*) 'Stability Factor=',delj
! Define guess function, with order parameter of approximately 0.5.
do a=1,401
fini(a)=3*(l.0-15.0*cos(angle(a))*cos(angle(a)))/(4*pi*(-12.0))
end do
!
! Begin density loop.
do x=0,499
Nrnol=500-x
vfrac=Nmol/Vsys
! Calculate the various isotropic t hermodynamic functions needed.
CALL t funcs (vfrac,Fprime. Gprime,Hprime,isofd.LIuum, kluud,
delzm,delzd)
! Calculate the angular distribution function.
CALL angint (k,L.pi?tol.isofd.alphandfini,e.ut2,int&nd)
! Calculate t hermodynamic functions for angular distribution function.
CALL calcj (k,l,pi7fend,ext2,int,ceut2jint)
CALL calce(k,l,pi,jint7vfrac~alphand,betanm~~1uum~~Iuud7
isofd,delzd,delzm,isof,angf,isop~angp,isoc~angc,Sfactor,
cext 2 ,ert 2, angle ,fend)
! Store values in arrays for Newton-Raphson interpolation-
ifree(Nmo1) =isof
afree(Nmo1) =an@
ipress(Nmo1) =isop
apress(Nmo1) =angp
ichern(Nmo1) =isoc
achem(Nmo1) =angc
S(Nmol)=Sfactor
WRITE (7, *) vfrac,isof,angf
WFUTE (8, *) vfiac,isop,angp
WRITE(S,*) vfrac,isoc:angc
WRITE(lO,*) vfrac,Sfactor
do a=1,401
F (Nmo1.a) =fend(a)
end do
! Use Newt on-Rap hson interpo!ation to determine coexistence densities.
z=o
! Select first value by comparing free energies for each phase.
do a=0,499
b=500-a
freedif=ifree(b)-afree(b)
if (freedif .gt. 0.01) then
z=b
end if
end do
C'VRITE(G,*) 'Nmol:',~,' Order Parameter:'?S(z)
! Define the guess for the isotropie density as slightly less than z.
! Recall the points higher and lower for calculation of the derivative.
z I = z - ~
zla=zl-1
zlb=zl+l
! Define the guess for the anisotropic density as the guess value, z.
! Same derivative considerations.
22=2
z2a=z2-1
z2b=z2+1
ini=zl+0.0
ani=z2+0.0
!
! Convergence hop for the Newton Raphson interpolation.
y=O
30 y=y+l
! For isotropie density, calculate al1 values as in main code.
vfrac=ini/Vsys
CALL tfuncs(vfrac,Fprime,Gprime,Hprime,isofd~um,~Iuud,
delzm,delzd)
do a=1,401
fend(a) =1.0/ (4.O*pi)
end do
CALL calcj (k,l,pi,fend7ext2,int ,cext2:jint)
CALL calce(k,l,pi7jint ,vfrac,alphand, betanm,&Iuurn.lf uud:
isofd,delzd,delzm,isof,angf.isop,angp,isoc~angc,Sfactor.
cext2 ,ext2 ,angle,fend)
! For anisotropic density, calculate al1 values as in main code.
vfrac=ani/Vs ys
CALL tfuncs(vfrac,Fprime,Gprime,Hprime,isofd,~Iuum,NIuud,
delzm,delzd)
do a=1,401
fin@) =F(z2b,a)
end do
C ALL ang& (kJYpi, tol,isofd,alphandfini.eut2,int ,fend)
GALL calcj(k,l,pi,fend,ext2,int,ceiut2,jint)
CALL calce(k,l,pi~jint,vfrac,alphand~betanm,Muum~Muud,
isofd,delzd,delzm,isoD~an@,isop2,angp2,isoc~,angc2,
Sfactor2,cext2,ext2,augle,fend)
! Calculate the difference in chemical potential and pressure, and the
! square of the diKerence for use in convergence determination.
chemdif=isoc-angc2
pressdif=isop-angp2
diffc=chemdiFchemdif
difTp=pressdif*pressdif
! Find partial derivatives of chemdif and pressdif with respect to the
! isotropie and anisotropic densities.
dchemi=(ichem(zl b)-ichem(z1a)) *Vsys/2.0
dchema=(achern(z2 b)-achem(z2a) ) *Vsys/2.0
dpressi=(ipress (zl b)-ipress(z1a)) *Vsys/2 -0
dpressa= (apress(z2 b)-apress(z2a)) *Vsys/2.O
! Jacobian of the matrix of partial derivatives.
denorn=dpressi*dchema-dchemi*dpressa
! Calculate new values of densities.
iprime=-dpressa*chemdif/denom+dchemaf pressdifldenom
aprime=-dpressi*chemdif/denom+dcherni*pressdif/denom
ini=ini-iprime
ani=ani-aprime
! Check for convergence of both chemical potential and pressure.
if (diffc .gt. tol .or. diffp .gt. tol) then
if(y .It. maxit2) then
goto 30
end if
end if
WRITE(G,*) 'The densities for transition are:',ini.ani
WRITE(6,*) 'with order parameters:',Sfactor,Sfactor2
WRITE(6, *) 'The nurnber of iterations ~ a s : ' ~ y
WRITE (6, *) 'The difference in chemical potent ial: ',cherndif
WRITE (6, *) 'The difference in pressure: ' ,pressdif
CLOSE(6)
CLOSE(7)
cLOSE(8)
CLOSE(9)
CLOSE(10)
END
! SUBROUTINES
!
SUBROUTINE tfuncs(vfrac,Fprime.GprimeoHprimejsofd.Muum~Muud~
delzm,delzd)
! Calculates the thermodynamic functions using the coefficients of
! Carnahan-Starling for the monomer fluid and Tildesley-Streett for
! the fused hard-sphere dimer fluid.
IMPLICIT NONE
REAL vfrac,isofd,Fprirne,GprimeoHprirne,h1uum1~1u~d
REAL vfracsq,vfracube,delzm,delzd
vfracsq=(l .O-vfrac) * (1 .O-vfrac)
vfracube=(l.O-vfiac)*(l.O-vfrac) *(l.O-vfrac)
isofd=(Hprimelalog(l .O-vfrac) +(vfrac/ (2 .~*vfracsq)) *
(2.0* (Fprime+Hprime)-(Fprime-Gprime+3.0*Hprime) *vfrac) )
Muum=vfrac*(8.0-9.O*vfrac+3.0*vfrac+~~ac) lvfracube
Muud=Hprime*alog(i.O-vfrac) +(vfrac/(2*vfracube)) *(4O*Fprime
+2.01Hprime-vfrac* (3.0fFprime-3.0% prime+S.Of Hprime) + vfraclvfrac* (Fprime-G prime+ Hprime) )
delzm=2.0*vfrac* (2.0-vfrac) /vfracube
delzd=vfrac* (Fprime+vfrac*Gprime-vfrac*vfYac*Hprime) /vfracube
PLETURN
END SUBROUTINE tfuncs
1
SUBROUTINE exvol(k,l,pi,fuse,angle~int,ext2)
! Calculates the excluded volume as a function of molecular orientations
! using the extension to the Williamson and Jackson calculation. Each
! of the four regions is evaluated separatell
IMPLICIT NOM3
INTEGER a,b,c,k,l,p,q
REAL fuse,fcube,tlim,numl.estO,dis~Ydi~,Zdis,azi~h3,inttemp,temp
REAL pi, tempo, tempi,temp2,temp3,temp2bi: temp%f,temp2ci,temp2cf
REAL angle(401) ,int (401,401) ,ext2(401)
DOUBLE PRECISION t12,lcap,lsrnall,h23
tlim=pi/3.0
fcube=fuse*fuse*fuse
do a=1,401
do b=1,401
inttemp-0.0
do c=1,400
numl=O.O+c
azi=(num1/200.0)*pi
Ydis=sin(angle(a)) *sin(angle (b)) *cos jazi)
Zdis=cos(angle( a) ) * cos(angle(b) )
dis=abs (YdistZdis)
t 12=dble(acos(dis))
h23=dsqrt(l.O-(fuse/(2.O*dcos(t12/2.0)))
*(fuse/(2.O*dcos(tl2/2.0))))
temp0=4.0*~i-(8.0*~i/3.0) *(1 .O-(3.0/4.0) *fuse+fcube/l6.0)
tempi=2.0*pi*((l.O-h23)-(1.O-h23*h23*h23) 13.0)
lsmall=dble (fuse)
lcap=dble(2.0*fuse*dble(sin(dble(t 12/2 -0)))
CALL aoint(t 12,1srnaillh23, ternp2bi)
temp2bf=-lsrnall* (1 .O-lsmali*~small/l2.0) *~i/2.0+2 .O*pi/3.Q
C ALL aoint (t 12Jcap, h23, temp2ci)
q=l
if (temp2ci .le. 0.0 .or. temp2ci g t . 0.0) then
q=8
end if
if (q .eq. 1) then
temp2ci=-lcap*(l.O-lcap*lcap/l2.0)*pi/2.0+2.0*pi/3.0
end if
temp2cf=-Icap* (l .O-lcap*!cap/l2.0) *pi/2.0+2.O*pi/3.0
temp2=-4.0*(temp2bf-temp2bi)-2.0*(temp2cf-temp2ci)
temp3=2 .O*fuse*fuse*sqrt (sin( t 12) *sin (t 12)-fuse*fuse*
sin(t 12/2.0)*sin(t12/2.0))
temp=temp0+templ+temp2+temp3
p=l
if (temp2bi .ge. 0.0 .or. temp2bi .It. 0.0) then
p=8
end if
if (p .eq. 1) then
temp=tempO
end if
inttemp=inttemp+temp
end do
int(a,b)=inttemp*pi/200.0
end do
end do
do a=1,401
int(a;k)=O.Z*int(a,k)
int(a,l) =0.SSint (aJ)
end do
do a=1,401
ext0=0.0
do b=1,401
extO=extO+int(a,b)/(4.0*pi)
end do
ext2(a)=ext0/200.0
end do
RETURN
END SUBROUTINE exvol
!
SUBROUTINE aoint ( t 12 ocapl.z.intrecall)
! Calculates the volume of the region where the large spheres overlap in
! the central region of the parallelpiped.
IMPLICIT NONE
REAL intrecall,A1,.42,.43,.44,pi
DOUBLE PRECISION t 12,capl,z7denl.den2.den3
pi=3.141592654
denl=dbIe(2.0*dsqrt (1.0-z*z))
den2=dsqrt (1.0-capltcapl/4.0)
den3=dble(capt*capl* (1 .O-z*z-capl*capl/4.0))
A1=2.0*(z-z*z*z/3.0)*dble(acos(capl/denl))
A2=-(2.0/3.0)*capl*z*sqrt (Z .O-z*z-capl*capl/4.0)
A3=-capl*(l .O-caplrcapl/ 12.0) *dble(asin(z/den2))
.44=2.0/3.0*atan(dble(2.0*(1.0+z-capl*capl/4.0) /sqrt (den3)))
-2.0/3.O*dble(atan(2.0* (1 .O-z-caplfcapl/4.0) /sqrt (den3)))
intrecall=A 1 +A2+A3tt44
RETURV
END SUBROUTINE aoint
!
SUBROUTINE angint (k,l,pi,tol7isofd~alphand,fini7ext2~int7fend)
! Apply self-consistent equations to solve for the angular distribution
! function.
IMPLICIT NONE
INTEGER a, k,l,y,maxit
REAL pi1dummy,nurn6,num7.to1,isofd,a1phand,dum
REAL Cd(4Oi) ,dumCd(401) ,ext2(401) ,int (401,iiOi)
DOUBLE PRECISION nrg,etemp,Eint
DOUBLE PRECISION fini(4Ol) ,fend(401) ,energy($Ol)
maxit=1OOO
! Convergence loop for calculating the angular distribution function.
y=o
20 y=y+l
CALL calcj(k,l,pi,fini,e~t2~int,dumCd,dumrny)
! Calculate the " reduced energy" for each molecular orientation.
Cd(a) =2.O*isofd*dumCd(a)/e~2 (a)
dum=dum+Cd(a)/401 .O
end do
energy(a) =dble(e-y(-alphand*Cd(a) +alphand*dm) )
end do
! Integrate the "energy" using the trapezoid method.
nrg=O. 0
do a=1,401
if (a .eq. k .or. a .eq. 1) then
etemp=dble(O .5*etemp)
end if
nrg=d ble (nrg+et ernp)
end do
Eint=dble(nrg*pi/100.0)
! Calculate the new angular distribution function frorn the " energies" .
do a=1,401
fend (a) =dble (energ-y(a) /Eint )
end do
num6=O.O
! Determine the convergence.
do a=1,401
num7= ( fend(a)- fini (a) ) * (fend (a) -fini (a) ) / (fini (a) *fini(a) )
if (mm7 .gt. numô) then
num6=num7
end if
end do
do a=1,401
h ( a ) =fend(a)
end do
if (y .lt. maxit) then
if (numô .gt. toi) then
goto 20
end if
end if
RETURN
END SUBROUTINE angint
I
SUBROUTINE calcj (k,l,pi,weight ,ext2,int ,sext2,value)
! Determines the ratio of integrals which defines the excess Helmholtz
! free energy ansatz, denoted as J.
IMPLICIT NONE
INTEGER a, b,k,l
REAL pijextojext 1 ,seaxtO,sext 1 ,value
REAL ext2(401) ,sext2(401) ,int (401,401)
DOUBLE PRECISION nreight(l01)
jext0=0.0
do a=1,401
sext0=0.0
do b=1,401
sext l=weight (b) *int (a7 b)
sextO=sext Otsext 1
end do
sext2(a)=sext0/200.0
jext i=weight (a) *sext 2 (a) /ext 2 (a)
if (a .eq. k .or. a .eq. 1) then
jextl=0.5*jextl
end if
jextO=jextO+jextl
end do
value=jextO*pi/lOO .O
RETURN
END SUBROUTINE calcj
!
SUBROUTINE calce(k,l,pi,jint~vfrac~alphand~betanm,~fuum~1\~Iuud~
i~ofd,delzd,delzm~isof~angf~isop,angp,isoc~angc,Sfa~t~r,
cext2 ,ext2,angle,fend)
! Calculation of the "energy" for each orientation. to determine the
! isotropic and anisotropic chernical potential, pressure, and free
! energy. The order parameter for the given density ir calculated.
IMPLICIT NONE
INTEGER a,k,l
RE4L pijint ,vfrac,alphand'betanm?Muum, bluud:delzd,deIzmlisofd
REAL isof~angEisop,angp,isoc~angc~Sfactor~sorder~empodum
REAL Cd(40 1) ,cext2 (40 1) ,ex12 (40 1) ,order(40 1) ,angle(40 1)
DOUBLE PRECISION isoenergy,isoetemp,etemp ,nrg,isonrg
DOUBLE PRECISION isoEint,Einttisozeta~zeta
DOUBLE PRECISION energy(401) ,fend(401)
dum=O. 0
do a=1,401
Cd(a) =jint*Muud+2.O*isofd*(cext2(a)/ext2(a)-jint)
dum=dum+Cd(a)/401 .O
end do
do a=1,401
energy(a) =dble(exp(-alphandrCd(a) +alphand*dum) )
order(a) =(3 .O*cos(angie(a)) *cos(angle(a))-1 .O) *fend@)
end do
isoenergy=l.O
sorder=O .O
isonrg=O .O
nrg=O. 0
do a=1,401
stemp=order (a)
et emp=energy (a)
isoetemp=isoenergy
if (a .eq. k .or. a .eq. 1) then
etemp=dble(0.5*etemp)
isoetemp=dble(0.5*isoetemp)
stemp=0.5*stemp
end if
sorder=sorder+stemp
nrg=nrg+etemp
isonrg=isonrg+isoetemp
end do
Sfactor=sorderf pi/200.0
isoEint=dble(isonrg/400.0)
Eint =cible (nrg/400.0)
isozeta=dble(vfrac/isoEint)
zeta=dble(vfrac/Eint )
isoc=log(isozeta) +alphand*Muud+b
isop=vfrac* (1 .O+alphand*delzd+betanddeizm)
angp=vfrac* (1 .~+j in t *alphand*delzd+betanmCdelzm)
isof=vfrac*isoc-isop
angf=vfrac*angc-angp
RETURN
END SUBROUTINE calce