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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[26] A.M. Somoza and P. Tarazona, J. Chem. Phys. 91, 517 (1989).

[27] R. Dickman and C.K. Hall, J. Chem. Phys. 85, 4108 (1986).

[28] O.G. Mouritsen, Cornputer Studzes of Phase Transitions and Cn'tical

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).

[42] W.G. Chapman, G. Jackson and K. Gubbins, Mol. Phys. 65, 1037 (1988).

[43] P. Bolhuis and D. Frenkel: J. Chern. Phys. 106, 666 (1997).

[44] J. Wickert, H.S. Gulati and C.K. Hall, J. Chem. Phys. 105, 7669 (1996).

[15] J.D. McCoy, K G . Honnell, K.S. Schweizer and J.G. Curro, J. Chem. Phys. 95,

9348 (1991).

[46] D. Henderson, J. Quintana and S. Sokoloivski, J. Chem. Phys. 102, 1991 (1993)-

[47] T. Boublik. C. Vega and 11. Diaz-Pefia. J. Chern. Phys. 93, 730 (1990).

[48] .LM. Somoza and P. Tarazona. Phys. Rev. A. 41 965 (1990).

[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