UNIVERSAL CRITICAL MOLECULE · scaling amplitudes bA/bR were also calculated with a very good precisiun. The results are u = 0.58756(5), A = 0.5310(33), ** = 0.23221(11) and bA/bR

RENORMALIZATION GROUP CALCULATION OF THE

UNIVERSAL CRITICAL EXPONENTS OF A POLYMER MOLECULE

A Thesis

Presented to

The Faculty of Graduate Studies

of

The University of Guelph

by

PETER BELOHOREC

In partial fulfilment of requirements

for the degree of

Doctor of Philosophy

May, 1997

@Peter Belohorec, 1997

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ABSTRACT

RENORMALIZATION GROUP CALCULATION OF THE

UNIVERSAL CRITICAL EXPONENTS OF A POLTM33R MOLECULE

Peter Belohorec University of Guelph, 1997

Advisor: Professor B. G. Nickel

In this work the excluded volume problem of a linear flexible polyrner molecule in a

solution was investigated using a new method. The Domb-Joyce (DJ) lattice mode1

[Domb C. and Joyce G. S. (1972). J. Phys. Cr Sol2d State Phys. 5, 9561 was used

to describe the polymer chah with a varying excluded volume parameter w and

bond nuniber N. Monte Carlo (MC) generated data for the mean square end-to-end

distance R: and the second virial coefficient A2,N were analyzed by a renormaliza-

tion g o u p technique that is a generalization of the one-parameter recursion mode1

[Nickel B. G. (1991). Macromolecules 24, 13581. By defining the effective expo-

nents v R ( N i I)) and vA(N, $) using 22"R = G N / R & and 23"A = A2,2N/A2,N where

t,h = ' 4 n (c)~'* A2.N is the interpenetration function, the corrections varying as N - A

were eliminated from vR(N, $) and v A ( N , $) and both universal critical exponents

v and A of the expected long chain behaviors R$ OC a R p Y ( l + b R W A + - - -) and

AZ,N OE aAN3"( l + bA + - .) were determined very accurately. The problems

encountered by standard methods when extracting the values of the leading exponent

Y and the correction to scaling exponent A from the finite chah data were eliminated

by the simultaneous use of mauy modeis (i.e., w in the range of O < w < 1) and by

the use of the effective exponent transformation. Other universal quantities such as

the asymptotic value .Sr* of the interpenetration function proportional to the dimen-

sionless ratio of leading scaling amplitudes ar/a3R/2 as well as the ratio of correction to

scaling amplitudes b A / b R were also calculated with a very good precisiun. The results

are u = 0.58756(5), A = 0.5310(33), ** = 0.23221(11) and b A / b R = -0.9028(132).

The numerical solution of the DJ model, obtained as a fit to both the MC and exact

data, allowed us to generate recursively d l values of R$(w) and (ID) for chahs of

length N = 2 x 2n or N = 3 x 2n and for any value of DJ parameter w using the in-

verse of the effective exponent transformation. This was used to evaluate the Ieading

non-universal scaling amplitudes aR jw) , a A (w) and the non-universal correction to

scaling amplitudes bR(w), bA(w) as well as to compare our results to those of others.

In the self-avoiding walk limit (w = 1) our generated data for Rg(1) and A2,N(1) very

well agree with the MC data of Li et ai. [Li B., Madras N., and Sokal A. D. (1995).

J. Stat. Phys. 80, 6611. Also in the two-parameter mode1 limit (w + O and N + m

with r cc W N I / ~ = const.) our result for the expansion factor &(z) agrees very well

with the previous high precision estimate of des Cloizeaux e t al. [des Cloizeaux J.,

Conte R. and Jannik G. (19%). Journal de Physique Lettres 46, L-5951 in the range

z 5 1. The two-parameter result for the linear expansion factor a;(z ) is new.

To my wife Katarina

1 would like to express my most sincere gratitude to Prof. B. G. Nickel for his

patient guidance and his generous help with any problern that 1 have encountered

throughout this work. In addition, 1 would also like to thank him for the financial

assistance in the form of research assistantship.

I would also like to thank the members of my supervisory cornmittee, Prof. D. E.

Sullivan and Prof. C. G. Gray for their help.

Financial support from the Department of Physics in the form of a teaching as-

sistantship is greatly acknowledged, as the University Graduate Scholarships and the

Fee WaiverIVisa Scholarship.

Last but not least: 1 would like to thank my wife Katarina for her help with the

proof-reading of the manuscript but most importantly for her patience and constant

support during the years of my studies and for her always being there.

Contents

1 Introduction 1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Background 1

. . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Present thesis research 10

2 Polymer models 15

. . . . . . . . . . . . . . . 2.1 Polymer models of an ideal polyrner chain 16

. . . . . . . . . . . . . . . . . . . . . . 2.2 The equivalent Gaussian chain 20

. . . . . . . . . . . . . 2.3 Two-parameter modei of a real polymer c h a h 22

. . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Self-avoiding walk mode1 26

Methods of calculating polymer properties 30

. . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Mean field calculations 31

. . . . . . . . . . . . . . . . . . . . . . . . 3.2 Lattice mode1 calculations 35

. . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Exact counts 35

. . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Monte Carlo method 37

. . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Field theory calculations 38

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 E-expansions 39

. . . . . . . . . . . . . . . . . . . . . 3.3.2 PoIymer-magnet analogy 39

. . . . . . . . . . . . . . . . . . . . . 3.4 Perturbation theory calcuIations 40

9 Conclusions

A History of polymer science

B Generating functions 164

C Monte Car10 Method 167

C.1 Monte Carla method . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

C.3 Random number generator . . . . . . . . . . . . . . . . . . . . . . - . 171

D Blocking method of calculating standard errors 174

E Fitting functions 179

E.1 The choice of fitting functions . . . . . . . . . - . . . . . - - . . - - . 179

E.2 The optimal fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

F Monte Carlo data

List of Tables

Universality classes for different physical systems and their mode1 coun-

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . terparts 14

. . . . . . . . . . . . . . . . . . Exact counts for chain length of N = 4 75

Values of perturbation series coefficients of a;., (w) and a i Y N ( w ) . . . 81

. . . . . . . . . The derivatives of U R ( $ ) and Y A ( $ J ) for small N values 83

Fits of UR data (total # of MC data 60 . N=12.16.24.32) . . . . . . . .

) . . . . Fits of V A data (total # of MC data 90 N = 6.8.12.16.24.32)

Y . . . . . . Fits of VA data (total # of MC data 78 Ar=6.8.12J6724.32)

Improving x2 by adding one parameter . " P" represents the polynomial

. . . . . . . . . . . . . fit and "PP" the "polynomial Nith the pole" fit

Nonlinear fit of UR data (m! = 5. m, = 5) and UA data (ml = 8.

mg = 5) . The Iast line represents the fit with the corrected constraints

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . at 11 = O (see text)

Nonlinear fit of UR data (mf = 5. mg = 6 ) and UA data (mi = 8. mg = 6)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Our results

. . . . . . . Nonuniversal scaling amplitudes and their statistical errors

. . . . . . . . . . . . . . . . . . . . . . . Cornparison to previous work 130

Cornparison between results obtained from the "old" data set and from

the "SAW enhanced" data set . . . . . . . . . . . . . . . . . . . . . . .

Cornparison of v* and A values obtained by our method to those ob-

tained by different methods . . . . . . . . . . . . . . . . . . . . . . . .

Relationship between the TPM values of linear expansion factors o i ( z ) ,

a;(z) and the effective exponents VR. V A for various vaIues of log, (2) .

Fitting parameters of linear and nonlinear fit . . . . . . . . . . . . . .

Fitting parameters of linear and nonlinear fit (continued) . . . . . . . .

Monte Carlo data for w = 0.01 and w = 0.03.







Monte Car10 data for w = 1.00. . . . . . . .

List of Figures

.4n example of a short SAW in two dimensions. . . . . . . . . . . . . 27

The data for polystyrene (PS) in various solvents: PS in toluene (empty

diamonds) and in methyl ethyI ketone, Canne1 et al. (1987) (fuIl di-

amonds); in benzene at 25OC Miyaki et al. (1978) (full squares); in

benzene, in toluene and in dichlorethane at 30°C Yamamoto e t al.

(1971) (squares) and PS in O-solvents: cyclohexane at 34.5"C and

trans-decalin at 20.4'C Miyaki et ai. (1978) (circles). . . . . . . . . . 49

The data for poly methyl met hacrylate taken from F'ujita and Norisuye

(1985) and the references therein. The solid line is the estimate of the

slope of measured values at + m. . . . . . . . . . . . . . . . . . . 51

Berry's data (1966) for polystyrene in various solvents. . . . . . . . . 53

Huber's SANS data for short chain polystyrene in good solvents. . . 54

Flow lines of $N for different initial conditions $[ as obtained by tbe

one-parameter recursion model. The flow lines are: (a) is the "near"

two-parameter solution, (b) the two-parmeter solution and (c) SAW-

like solution of the OPFUI. . . . . . . . . . . . . . . . . . . . . . . . 68

Graphs in the first and second order of perturbation expansion. . . . 77

The residues of the fits compared to the exact data plotted for N.

Individual lines correspond to 107 x A(*'$(")) (zero a t N = 15),

' d 2 u n ~ ( ' ) ) (zero a t N = 14) and 106 x A ( ~ ~ ~ ~ $ ' ) zero at 1 0 5 x A ( 2 432 1 ( N = 20). Symbol A represents the residue. . . . . . . . . . . . . . . 84

Residues of MC generated data compared to exact data for (R:,(w)). 87

Residues of MC generated data compared to exact data for second

virial coefficient. . . . . . . - . . . . . . . . . . . . . . . . . . . . . . . 88

Typical renomalization group flow pattern. . . . . . . . . . . . . . . 94

Monte Carlo data of effective exponents UR and nu^ versus I/.J (without

transformation). Set of curves with the common value of 0.5 in the

limit ?,b + O is the nu^ set. . . . . . . . . . . . . . . . . . . . . . . . 96

Monte Carlo data of effective exponents v versus S, after the transfor-

mation was applied. . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

The flow Iines constmcted as connecting lines between successive Monte

Carlo data for u at the same value of W. . . . . . . . . . . . . . . . . 99

Zoom into the region of criticality. Both N-curves as well as the flow

lines are plotted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

The plot of dx(N) = VX,N ($4 1*,0.24 where X = R or A. . . . . . . 102

7.7 The plot of s ( N ) = . . . . . . . . . . . . . . . . . . . . . 103 dr(t @=0.24'

7.8 Unusual behavior of the effective exponent variables V A near the SAW

limit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

7.9 Residues of ln RL(w) for different values of W. . . . . . . . . . . . . . 116

7.10 Residues of ln A 2 , N ( ~ ) for different values of W. . . . . . . . . . . . . 117

7.11 Fitted functional form of UR and UA for different values of N used in

MCsimulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

vii

7.12 Zoom into region of criticality of u-fits. . . . . . . . . . . . . . . . . 119

7.13 Global view of calculated 0ow. The recursion was carried to very high

values of LV. . - . . - . . . . - . - . . . . - . . . . . . . . . . . . . . . 121

7.14 Zoom of the calculated flow showing the region close to the critical

point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

7.15 The plot of nonuniversal scaling amplitudes aR(w) and a.&). . . . . 127

7.16 The plot of nonuniversal scaling amplitudes bR(w) and ba ( w ) . . . . . 128

The comparison of Li et al. (1995) bln(R$) data to our data. . . . . 132

The comparison of Li et al. (1995) bln(Az,N) data to our data. . . . 133

The comparison of Li et al. (1995) 6ln(APVN) data to our own data

after including the SAW dataof N = 192, 256, 384, 512, 768, 1024. . 136

The results O btained from various two-parameter theories (see text for

details). . . . . . . . . . . . . . . . . . . . . . . . . . . . . - - . . . - 139

The relative error of TPM results of des Cloizeaux, Conte and Jannink

(1985) and hluthukumar and Nickel (1987) compared to our exact re-

cursion results plotted versus log,(z). .4t log, z = O the functions fiom

top are dCCJ, our approximate formula (8.7) and MN, respectively.

The function with less than 0.1% relative error in the region of inter-

mediate values of z corresponds to our approximate formula given by

eq. (8.7). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

Plot of a,(w) versus W. . . . . . . . . . . . . . . . . . . . . . . . . . . 143

Plot of b,(w) versus W. . . . . . . . . . . . . . . . . . . . . . . . . . . 144

The results of various two-parameter theories (see text for details).

Plot of cri(z) versus z. . . . . . . . . . . . . . . . . . . . . . . . . . 146

viii

D.l The blocking method of estimating the statistical errors of (R&(w))

. . . . . . . . . . . . . . . . . . . . . . . . . . . for various values of w 177

D.2 The blocking method of estimating the statistical mors of (A&(w))

. . . . . . . . . . . . . . . . . . . . . . . . . . . for various values of w 178

Chapter 1

Introduction

LVe are surrounded by plastic materials with unique properties. In teflon, polyester,

nylon, neoprene, styrofoam, PVC or other synthetic materials polymer is the main

constituent. Even such items of everyday life as clothes, wood and paper are polymer-

based materials +th cellulose (polymerized sugar) being the main component. Poly-

mers are also parts of our bodies. They are the the basis of the living ce11 and par-

ticipate in m a q important biochemical processes. Without the genetic information

carrier, the deoxyribonucleic acid, the reproduction and the evolution of the species,

as well as the synthesis of proteins would be impossible. Proteins, copolymers of

L-amino acids, on the other hand, participate in many biochemical and transport

processes and are important constituents of biological membranes.

1.1 Background

A polymer is a large molecule consisting of many repeating units cdled monomers.

The simplest polymer structure is that of a linear homopolymer where chemically

identical monomers are bound together in a linear fashion. More complicated forms

such as star, comb, randomly branched and ladder structures or even interconnected

1

three-dimensional polymer networks can also be formed. The present knowledge

about polymers is based on the macromolecular concept that is fundamental for

explaining polymer properties. However, this concept gained attention only very

slowly (see Appendix A) and for over hundred years struggled against other concepts

such as the association hypothesis.

One of the principal physical characteristics of a polymer is the molecular weight

M. It is a product of the degree of polymerization (n) and the molecular weight

of a monomer, ,W1 = nM,. The moIecular weight of a synthesized linear polyrner

is typically between 104g x mol-' and 10" x but the laboratory samples of

ultrahigh molecular weight of up to 1 0 ~ ~ x mol-' are not uncommon. The stability

of the structure of a polymer macromolecule is maintained by covalent bonds be-

tween successive repeat units. For example, polwinyl chloride with chernical formula

[-CH2 - CHCI-], is formed by the so called addition polymerization reaction where

monomers (molecules of vinyl chloride CH2 = CHCZ) are simply added together in

a process that is initiated by an active species (molecule with an unpaired electron).

Such covalent structure is very stable and its behavior can be studied under various

conditions.

One of the main objectives of studying polymers is to find out how different

polymer properties Vary with the degree of polymerization. Even a single polymer

molecule can have thousands of degrees of freedom, so it is extremely difficult to

describe physical properties of dense assemblies of polymer molecules. In this thesis,

the attention is focused on dzlute polyrner solutions. In such systems different macro-

molecules are separated apart sufficiently far from each other so that the polymer-

polymer interactions are very rare. Dilute polymer solutions are the experimental

realization of the single chah limit of polymer interactions. The two most important

physical quantities one is interested in are the mean square end-to-end distance (R2)

and the second osmotic virial coefficient (A") representing the mean dimensions of

a polymer molecule in the solution and the effective polymer-polymer interaction1 ,

respectively. These quantities are thermal averages t hat can be theoretically calcu-

lated as weil as be expenmentally measured. One of the methods frequently used

for the measurements is the light scattering. For practical purposes experimentalists

prefer the radius of gyration (S2) as a convenient measure of the polymer dimensions.

Theoretical investigations concentrate on both (S2) and (R2) . Since (R2) is easier to

calculate than (S2) , more theoretical information about (R2) h a accumulated over

the years. The theoretical calculations of (R2) and (A") follow the fomuIae

and

where fik and gk represent the coordinates of monomers of individual polymer chains

and Pl and P2 are the distribution functions for a single polymer molecule and for a

pair of polymer molecules, respectively. The sample volume, molecular weight and the

Avogadro's number are denoted as V, !VI and NA. The integration is over the whole

coordinate space of individual macromolecules. In order to simplify the evaluation of

these averages, various theoretical models for polymers in solution are used.

The theoretical study of linear flexible polymer chains is almost exclusively con-

cerned with the N-dependence of the properties (R:) and (A2,N) . Here N is the

number of links between monomers (N = n - 1). In any polymer model such as the

stick-bead model where spheres of a finite volume are connected by zero volume bonds

the distinction between N and n is clear. For a real polymer chah the introduction of

'Superscript "expn is used because the quantity (AFP) represents the second viriai measured in

the experiment.

the parameter such as N is somewhat artificial, because the effective thickness along

the c h i n is more or less uniform and the distinction between "sticks" and "beads"

becomes a problem. This is because part of the molecdar weight is distributed along

the bonds rather than concentrated in a well defined region around monomers. In

the Iattice mode1 we use, the distinction is clear and in our caIcdations we used the

bond number N as a parameter. Also, the second virial coefficient (.42,N)1 that we are

interested in, is transformed from the "experimental" one (A") using the formula

i t follows from renormalization group arguments that the scaling of global p r o p

erties (of polymers in "good" solvents) in the long chain limit (iV + w) is

where v is the critical exponent called the correlation length exponent. The eqs. (1.4)

and (1.5) can be understood by noting that properties (RN) and (A2,N) at large N

are dominated by a common correlation length < oc N" and that R2 x c2 and A2 cx c3 where R2 represents the area and A:! represents the volume. It is generaily believed

that this so called h-vperscaling argument is true even though it has not yet been

proven rigorously. The renormalization group goes beyond the simple proportionality

of eqs. (1.4) and (1.5) predicts that the ratio ( A ~ , ~ ) / ( R & ) ~ / ~ approaches a universal

limit as N + m. This is why the quantity

called the "interpenetration function" is so popular with theoreticians. Value of v

has been the primary focus of many theoretical studies and different models have

been proposed to calculate v. in an ideal case aii the important effects that can

be seen in the experiment will be explained within the frarnework of a particular

model. One of the first polymer scientists to begin with the modem analytical and

numerical calculations of polymer properties was Kuhn [II in 1934. He used the

random flight mode1 that takes into account only interactions between neighbouring

monomers dong the c h a h These interactions are called short-ranged as opposed

to the long-range interactions that act between monomers far apart from each other

along the chain. The random flight model predicts u = 1/2. This result is a direct

analogue of the diffusion problem where the average spread of a diffusing particle is

proportionai to ( t measures the time since the diffusion process has started). The

important distinction between the process of diffusion and the spread of the polymer

chain is that a difhsing particle can return to the position already visited before, but

the monomer of a polymer chain can not occupy the same space another monomer

already occupies. Therefore this analogue is valid only under special circumstances.

From the rnicroscopic point of view the forces experienced by a single macrornolecule

in a solution are averaged over al1 positions of solvent molecules. Also, for a given

polymer, these forces depend on chemicai composition of a solvent, and most impor-

tantiy, the temperature. Some solvent molecules are highly attracted by monomers

and the resulting effect is that different parts of the polymer chain try not to be in

a near contact with each other. Solvent of this type is called a good solvent. On the

other side, in solvents with smaller attraction to monomers, called poor soivents, the

solvent effects on polymer chains can be regulated by a change of the temperature.

For poor solvents there usually exists a temperature, called O-temperature, a t which

a l the effects of attraction and repulsion in the system average out so that the overall

effect is zero. At these conditions the diffusion analogue is valid and the chain is

said to have a Markov character. The polymer chain is called ideal (unperturbed).

At T > 8 the chain experiences "restriction" or "self-avoidance" effects. The self-

avoidance effect is a manifestation of the long-chin interactions commody referred

to as the ezcluded volume effect. In this case the chah is called real (perturbed) . The

presence of these excluded volume interactions significantly increases the difficulty of

the theoretical evaluation of the averages in eqs. (1.1) and (1.2).

Let us imagine a polymer coil consisting of one thousand monomers with the

relative monomer volume density of the coil being one percent. The probability, that

any monomer randomly placed into the region of the coil does not interfere with any

other monomer is 0.99. For any random configuration of the chain to be accessible, al1

monomers simultaneously have to satisfy this requirement. However, the probability

of this to happen is very small, namely 0 . 9 9 ~ ~ ~ ~ = 4 x 10-~. Therefore a large fraction

of the " random fiight" configurations are excluded from the configuration space of the

real polymer chain. In order to accomodate the interacting monomers, the polymer

coil expands and the overall nurnber density of the coil decreases. The expansion of

the polymer coil causes the increase of the value of v so that eqs. (1.4) and (1.5) are

no longer of a random flight type with v = 112. This qualitative argument was first

presented in 1934 by Kuhn [l]. Even though this is an estirnate for hard-sphere liquid,

i t illustrates the effects of the excluded volume very well. A decade later Flory [2]

estimated effects of the excluded volume on the average dimension of a polymer coil by

mean field theory. The critical exponent he obtained was vp = 0.6. Later Fisher [52]

applied Flory's theory to d dimensions and obtained the result v~ = 3 / ( d + 2). For

d = 1 and d = 4 one gets the "straight rod" and the "random walk" results (R2) - N2

and (R2) N , respectively. Flory's predicted the value of v = 0.6 was a very good

approximation at that time and is still used today for qualitative estimates of excluded

volume effects. Since then many other more sophisticated methods of estimating v

were presented [48, 581 and we will discuss them in greater detail later.

Over the years an agreement has been reached that the value of the critical ex-

ponent u is a universal quantity. That means the value of v is independent of the

details of the interactions between monomers and depends only on the dimensionality

of space d. There are physicd systems such as ferromagnets, simple fluids, 'binary

alloys that despite t heir different rnicroscopic structure exhibit striking similarities

in their behavior near the critical point. This has lead scientists to the formulation

of the hypothesis of the critical universality. According to this hypothesis the only

two quantities that the critical behavior of the system depends on are the dimension-

ality of the space d and the dimensionality of the order parameter2 n. The details

of particle-particle interactions as well as the type of lattice used in the mode1 are

not important. Based on the universality hypothesis systems can be grouped into so

called universality classes (see Table 1.1)-

The experimental manifestation of universality in the polymer-solvent systems is

that in the limit of long chains the slope of ln(R$) versus ln N is the same for many

different polymer-solvent systems. There is a discussion in polymer community (see

e.g. [15]) on which aspects of the long-chain behavior are universal and which are

detail-dependent. For example, within the two-parameter theory (TPT), the most

trusted theory of dilute polymer solutions, the interpenetration function $ ( z ) is a

universal function of the excluded volume variable z that approaches its asymptotic

value @* from below. The numerical results of lattice self-avoiding walk rnodels, on

the other hand, are in contradiction with TPT since they predict that q!~ approaches

11' from above. More thorough discussion will be presented in Chapter 4.

According to the renormalization group theory [71] any global observable X N in

*The symbol usually used is n which, in this chapter, is unfortunately in confiict with the symbol

for the polymerization index.

the long chah limit behaves as

where the scaling is governed by the leading exponent a and the rest are the cor-

rections to scaling. The corrections of the form N l k where k is the positive integer

are the analytic corrections whereas the terms N-A-k with A being a non-integer

are the non-analytic corrections to scaling. For different quantities X the leading

exponent a, is different but corrections to scaling exponents are the same. In practice

it is virtually impossible to calculate more than just the first non-analytic correction

because the effects of others are diminishing very fast. In context of eq. (1.7) both

eqs. (1.4) and (1.5) can be rewritten into the form

where a ~ , a~ are the leading scaling amplitudes and bR, bA are the correction to

scaling amplitudes. Over the years many theoretical models have been applied to the

problem of excluded volume such as the self-avoiding lattice walk (S-4W) [58], bead-

rod model [72], continuous Edwards' model [73] or two-parameter model (TPM) [6].

Just as the detailed behavior of a real polymer depends on the details of its chemical

structure the behavior of a model polymer depends on the particular mode1 chosen.

The confusion as to which aspects of the long-chah behavior (see eqs. (1.8) and

(1.9)) are universal and which are detail (model) dependent was recently adressed

by Nickel [4]. By using a simple one-parameter recursion model of flexible-chain

polymer Nickel illustrated some of the universal/non-universal details of polymer-

solvent behavior and showed that the two-parameter model is just a special case of

the whole family of models. Without going into d e t d s of his one-parameter recursion

mode1 one gets a rough idea of which aspects are universal and which are not by

plotting the experimentai interpenetration function

versus N. By using the scaling for (SL) similar to that of eq. (1.8) one gets

where $* = ~ ~ / 4 ( s a ~ ) ~ / * and b* = UA - i b s . If the accurate data of self-avoiding

walks (SAW) are used [41, 491 one gets

whereas by using the results of the two-parameter mode1 (TPM) and applying the

"direct renormalization" scheme [75] to extract the large-N behavior, one gets

Here $* value was fixed to that of the SAW. Despite a slightly different values of ex-

ponents A one can cIearly see that the approach to the limiting value $* = 0.2465 for

these two models is from the opposite directions. In fact this behavior is detectabIe

also experimentally. As measured by Berry [SI, for polysty-rene in trans-decalin (poor

solvent) where the effects of the excluded volume are s m d , the experimental data

are well fitted by TPM. On the other side Huber et al. [19] carried out an exper-

iment on polystyrene of relatively short chains in toluene (good solvent) and found

that approached its Iimiting value from above rather than frorn below as was

the case of poor solvent. For discussion on comparison of theory versus experiment,

see Section 4.3. From these experiments one can infer that the correction to scal-

ing amplitude b , ~ is not a universal quantity. Also, the leading ampiitudes are not

universal quantities, however their dimensionless ratio zs a universal quantity. The

leading amplitudes as well as the correction to scaling amplitudes are therefore model-

dependent and for real polyrners in solvents their magnitude depends on the strength

of the excluded volume interactions.

1.2 Present t hesis research

In this thesis a new approach to the excluded-volume problem is presented. For this

purpose the generahation

of the one-parameter recursion method (41 is used where the 1/N analytic corrections

are included explicitly. To model flexible polymer chain with excluded volume effects,

the Dornb-Joyce (DJ) lattice model [5] of N steps on a simple cubic lattice was used

and its behavior was investigated by a suggested Monte Car10 renormalization c o u p

technique. There are many advantages of this approach:

0 The effective exponent transformation

eliminates the AT-* corrections to scaling. Here XN represents either (R2) or

(A2). As a result the simultaneous use of DJ model for many different values

of excluded volume parameter w allows us to determine universal quantities v7

$J;, and A with high precision from relatively short chahs. It can be shown

exactly, in the limit of St + O, that ueff does not contain the 1/m correction

terms if the scaling variable $J is introduced.

Al1 analytic I/N corrections in the limit w + O are included exactly. This is

possible because the retum-to-origin generating functions (those appearing in

the perturbation expansion of DJ model near UJ = 0) are known exactly [68].

The 1/N corrections in the case of large-w are included approximately through

the choice of fitting functions.

The solution of the excluded volume problem that we get is the full approximate

numerical solution of the DJ model for al1 values of parameter w and al1 chah

lengths N . This is obtained as a combined fit to both Monte Carlo data for long

chains and exact data for short chains. The iteration of this solution (similar to

Nickel's one-parameter iteration) allows us to estimate the nonuniversal scaling

amplitudes as functions of w with great precision. It also allows us to find the

d u e of w* at which the "two-parameter" regime changes to the "self-avoiding

waik" regime as the excluded volume w is increased.

0 Any attempt to solve nurnerically the two-parameter model by direct Monte

Carlo simulation of chains a t small w is doomed to fail because the flow of

the renormalization group along the two-parameter curve (w + O) requires

extremely long chains in order to get close to the h e d point. Our approximate

DJ mode1 solution in the limit w + O, on the other hand, is (to our knowledge)

the only numerical solution of TPM available in the literature. It allows us to

reproduce the two-parameter curves for both (RN) and The result for

(R:) can be compared with that of des Cloizeaux et al. [75] and Muthukumar

and Nickel [74] while the result for (A2,N) is essentially new.

In the next chapter we introduce some of the excluded volume models and con-

centrate on two most important models, the two-parameter model [6], currently the

most widely used model for interpreting the light scattering experimental data, and

11

the self-avoiding lattice walk model. In Chapter 3 vanous techniques that c m be used

to extract the universal Features of the polymer-in-solvent behavior (such as the value

of Y) are described. Chapter 4 is the introduction into the light scattering technique

used to measure both polymer coi1 dimensions as well as the second virial coefficient.

Recent experimental data on both poor solvents (Berry [8]) and good solvents (Hu-

ber [19]) are aIso summarized. In Chapter 5 the problem of the excluded volume is

explained in the context of the Renormalization group theory of Wilson. Nickel's one-

parameter recursion mode1 of flexible polymer is formulated in the effective exponent

variables UR and UA and generalized to include the analytic 1V corrections. Chapter 6

focuses on the Domb-Joyce lattice mode1 [5]. First, the model is defined and then

the perturbation theory in small w parameter is formulated in terms of generating

functions. The exact counts allow us to find functions vR(N, @) and UA (N, @) exactly

for short chains. The quantities avA/a$, avR/8$ and a 2 Y R / a 2 @ are evaluated for any

finite N and their asymptotic expansions are also evaluated. The exact elimination of

terms l / f i from asymptotic expansions is shown. This 1/m elimination is crucial

for the success of the subsequent analysis since by analogy we expect al1 1/IVA terms

to be eliminated near the h e d point .Sr = .Sr'.

Chapter 7 explains how the critical exponents and the universal scaling amplitude

ratios as well as the non-universal scaling amplitudes were calculated. The details of

the analysis are presented in this chapter. For better clarity other technical details

such as the random number generator or the blocking method of estimating the

variance are referred to the appendices. Chapter 8 compares our results to other

results in the literature. The last chapter is the summary of our contribution to the

field of long flexible polymer chains.

Presence of the corrections to scaling poses many problems when values of the

critical exponents are to be extracted from the finite chain data. In our method

the analysis of multiple model data3 by using the effective exponent transformation

allowed us to eliminate the corrections to scaling effects and calculate the leading

exponent v, the correction to scaling exponent 4 and various other universal and

non-universal quantities with a very good precision. For v the precision was up to 10

times better compared to the next most precise numerical estimate of v [58] with much

less CPU time used. Our numerical solution of the DJ model allowed us to generate

all values of R$(w) and A2,N(w) for chains of length N = 2 x 2" or 1V = 3 x 2" and

any value of the parameter W. These generated data were used for cornparison to

other results available in the Iiterature both in the SAW limit (w = 1) and the TPM

limit (w + 0, N + CG and z cc wN112 = const.). In both limits the agreement of

our data with the most precise estirnates from the literature is excellent. The new

method presented in this thesis proved to be a powerful approach to the numericd

solution of the excluded volume problem.

3difTering by the strength of the excluded volume

PHYSICAL SYSTEM MODEL SYSTEM

(ORDER PARAMETER)

adsorbed films

(surface density)

He4 films

(condensate wavefunction)

flexible long chah polymers

(density of chah ends)

a uniaxial ferromagnet

(magnet izat ion)

0 0uid near critical point

(density difference btw. phases)

planar ferromagnet (magnet izat ion)

isotropie ferromagnet

(magne t izat ion)

Ising model in 2-d

X Y model in 2-d

SAW on a regular 3-d lattice

--

Ising model in 3-d

-

XY-mode1 in 3-d

Heisenberg model in 3-d

Table 1.1: Universality classes for different physical systems and t heir model coun-

terparts.

Chapter 2

Polymer models

When the macromolecular hypothesis got wider acceptance and researchers realized

theoretical models could be used to describe long chain molecules, analytical and

numerical aspects of polymer science started to develop. In 1934 Kuhn [l] was the

first one to explore this area. He used a random walk to model a polyrner chain. Over

the years other more redistic models were introduced.

There are lattice and off-lattice models of polymer molecules. In the case of lattice

models the angles between individual links of a modeled polymer are such that the

chah can be embedded into a lattice of a particular type. The lattice random walk,

correlated random waik, self-avoiding walk, or its generalization, called the Domb-

Joyce model, are examples of lattice models. The random flight model (i. e. freely

jointed chain), freely rotating chain, spring-bead or stick-bead model as well as the

continuous Edwards' model are the most widely used off-lattice models. An ideal

polymer (i. e. a chain with negligible excluded volume interactions) can be modeled

by a random walk with short range correlations, whereas in case of a real polymer

(i.e. chain with excluded volume effects) the effective volume of the segments has to

be taken into account and long range correlations have to be included into the model.

This chapter is a short review of the models of linear polymers (i. e. polymers

consisting of sequentially connected links). First we describe some off-lattice polymer

models with an emphasis on the two-parameter model and then we focus on the self-

avoiding walk (SAW). The Domb-Joyce model 151, the lattice model used in our work,

is described in greater detail in Chapter 6.

2.1 Polymer models of an ideal polymer chain

A polymer chah is a statistical object with an enormous number of internal degreees

of freedom. It is usually modeled as a chain consisting of N bonds connecting N + 1

monomers that interact with each other. Let us consider a polymer molecule in a solu-

tion. The probability of a polyrner configuration {a} = ((Q, go, z0), ...( zN, y ~ , r N ) }

where {fi = {(qzl qyl , qz l ) , ...( q z ~ , : Q ~ N , , IN, ) } is a set of coordinates of solvent

molecules and d{q3 = nZ1 dqZjdq,dqzj. T and ks are the absolute temperature

and the Boltzmann constant, respectively. The configuration partition function of

the polymer-solvent system is

where the integration runs over the whole coordinate space of the system. The po-

tential of the mean force u({&}), describing the interaction energy between the

monomers of the chah mediated by the solvent molecules, can be defined by

where the normalization constant Z is called the partition function of the "polymer

in a soIventn system and is given by the multiple integral

In general, the effective interaction energy u({Rk}) consists of short-range interac-

tions ~ ( f 7 , - ~ , 8,) of two consecutive units within the backbone and long-range inter-

actions w({Rk}) between units that are more than one bond length apart from each

other along the chain, i.e.

The standard approximation for ~ ( ( 8 ~ ) ) used for low density polymers is

where w ( ~ ) is a pair potential of the Lennard-Jones type with a short-range repulsive

hard-core part and a long-range attractive soft-tail part. To calculate w ( ~ ) from the

first principles is a difficult many-body problem. Fortunately, there are universal

aspects of the long-scale polymer properties such as the radius of gyration (S2), the

mean square end-to-end distance (R2) or the second osmotic virial coefficient (Az)

that are independent of the details of the W(2) potential. This is the basis of the

universality hypothesis allowing the use of simple models such as the Domb-Joyce

mode1 [5] to calculate critical exponents and other universal quantities.

Let us assume, for now, that there are no long range interactions in the polymer

chain so that w({Rk}) contribution to u({&}) is negligible. Under this assumption

the expression for P({&}) (see eq. (2.3)) can be written in an ideal chain form

4 4

where the bond vector Tk = & - Rk-l and

is the nomalized bond probability distribution. Any global polymer property t hat

depends on chah conformations can be calcdated using P(')({&}). If one is inter-

ested in the quantities depending o d y on the end-to-end distance fi = RN - &? it is

sufficient to know ~ r ) ( d ) , the probability distribution of the N-th step reaching the 4

point RN = 2. This can be formally mitten as

Then the mean square end-to-end distance (RC), representing the characteristic size

of a polymer, is given by the integral

If individual bond vectors Fk of the polymer, each of the length 1, are freely jointed

(i. e. have arbitrary relative orientations), the bond probability is

and we get the random flight (FU?) mode1 of a polymer proposed by Kuhn in 1934 [l].

The probability density P F ) ( ~ ) for this mode1 was first evaluated by Kuhn and

Gmn [29] in 1942. It is of the form

where C is the normalization constant and L-'(2) is the inverse Langevin hinction.

It can be shown that the asymptotic form of P ~ ) ( R ) becornes

The evaluation of (R2) for the case of the asymptotically Gaussian limit of distribu-

tion (2.13) is trivial and one h d s that

However, the random flight model is clearly not an adequate representation of an

ideal polyrner chain since a polymer is not a sequence of randomly oriented bonds.

On the contrary, the neighboring bonds form certain valence angles. Moreover, due

to the overlaps of side groups, monomers are allowed to occupy only certain positions.

In the more realistic freely rotating chain mode1 the valence angle' is restricted to

(T -a) and monomers are free to rotate around the bonds. The Iong chain result for

(RN) of this model

first derived by Eyring in 1932 [30], can be rewritten into the form of the random

flight model (see eq. (2.14)) by transforming I as 1 + = 1J(l + cos a ) / ( l - cos a).

If, in addition, the bond rotation is restricted by some potential U(4) where 4 is the

angle of relative rotation of neighboring side groups, the result for (RL) in the long

chain limit [31] becornes

-2 1 + (cos 4) (RS) = N1 1 - (cos 4)

where

(COS 6) = 2" cos (4) e-u(@)'k~Td4

JoZn e - u ( # ) / k ~ T d #

By redefining ï again, the above result can be written in the random flight form of

eq. (2.14). Al1 the effects mentioned above cause the short range correlations between

the segments to change, but they do not affect the asymptotic form of the global chain

properties

(RN) = aN

'The angle created by three neighboring carbon atorns of the polymer backbone.

19

where only the constant a, proportional to the square of the effective Kuhn lenght,

is affected by the correlations.

Chandrasekhar [27] was the fkst one to show that limN,,(R$)/N was a finite

number for any bond distribution function r(fl as long as the individual bonds were

not correlated. In the context of lattice walks Montroll [32] found that (R$)/N

converges asyrnptotically as long as correlations are of a finite range. Later it was

shown that this result is valid for any type of walk as long as the correlations are

of a finite range. On the other side, if correlations are of an infinite range the ratio

(R$)/N diverges and the problern of exact evaluation of ~ ~ ( 8 ) and (RN) becornes

hopelessly difficult .

2.2 The equivdent Gaussian chah

In the case of independent probabilities (such as those in the random flight model)

the evaluation of P ~ ) ( E ) is relatively simple. To evduate ~ ! ) ( d ) one fixes $ = 5

and RN = 8 and integrates over al1 intermediate coordinates

4 4

where dr{&} = d { & } / d & ~ ~ . By introducing the Fourier transforrn of dRF)

one gets

After integrating most of the " intermediate" degrees of fieedom (by doing the double

integrals J 1 .. .dRid&+, ) and leaving only every 6,-t h pair of ($, kj+,) in the integral

one gets

where the integration variables are now relabeled. The Fourier transform of the bond

probability dW)(k) can be approximated by the Fourier transform of the Gaussian

bond probability in the following way

Even if one integrates out only very few degrees of freedom (Le. 6, is a very smdl

nurnber) the above approximation is very good (e-g. for & = 4 the relative error is

less then 3%) and most importantly the correction to eq. (2.14) is of higher order in

N-' . Finally we get

This derivation in fact shows that an ideal chah of 1V steps and of finite-variance

step probability distribution c m be replaced by an equivdent Gaussian chah of N / C ~ ~

steps with Gaussian probability distribution of segments

where Z, = 1J6; is the mean square dispIacement of one step. This so caIled Gaussian

mode1 is very convenient for theoretical calculations.

Cornmon feature of al1 the theoretical models described above is that they can

be replaced by Gaussian mode1 and in the long chain Iimit (R$ ) /N converges to a

fixed number. This characteristic of a flexible chah is referred to as the Markov

nature of the ideal chain. Under the O-temperature conditions, the Markov nature

of polymer chains can be seen experimentally and therefore i t is assumed that the

Gaussian mode1 is adequate for the description of large scale properties of polymers in

solution when, on average, the excluded volume effects cancel out with the attraction

effects.

2.3 Two-parameter model of a real polymer chah

The polymer models presented in the previous section can not be used to describe

real polymer chains in good solvents since in these chains the long-range interactions

w ( ~ ) can not be neglected. Unfortunately, the complications created by including the

w ( ~ ) are enormous and so far no analytic solution of the excluded volume problern

has been presented. There is, however, no shortage of approximate solutions based

on different models. In the following we focus our attention on one of the models, the

two-parameter model of a polymer chain.

Let us rewrite ~ ( d ) (i. e. the perturbed analogue of p(O)(l?) of eq. (2.9)) within

the fomalism used in eqs. (2.24) and (2.25) into the following form

where L = NI and As = Ids and their dimension is that of the length. This form is

ideal to make the transition to the continuous chain limit (As + 0) and replacing

As with ds. The sums of eq. (2.26) becorne integrals and one gets

where the continuous chain is pararnetrized by a function d(s). The normalization

constant is included into the "differential" fi]. The standard assurnption used in the

theory of polymer solutions approximates the potential W(2)( f l , ) by a short-ranged

potential of the delta function type as follows

where the vector aj represents the separation between the monomers i and j of the

chah and

is the binary cluster integral representing the effective monomer-monomer interaction.

Parameter ,û is temperature dependent and its dimension is that of the volume. At

a certain temperature (T = 0) the effects of the interaction potential average out so

that p = O. At this, the so called Flory O-temperature, the pol_wier in solution be-

haves as an ideal one and bas al1 characteristics of a Markov chain (as discussed in the

previous section). Let us rewrite ~ ( 8 ) uusing a newly defined interaction parameter

w = B/12 in the following way

where

is the Edwards' Hamiltonian of the continuous chain representation [73]. The trans-

formation s = Lt mas used to rewrite the Hamiltonian into the above forrn with E(t)

being the new parametrization of the continuous c h a h To calculate an average of a

physical quantity one needs the partition function Z (see eq. (2.4)). In the continuous

chain limit, Z is a function of three parameters, 1 (the Kuhn length), L (the total

chah length) and w as follows

Z is of such a form that only combinations IL and wL2 of these parameters appear

explicitly. The theories with t h s characteristic feature are referred to as the two-

parameter t heories. Because Z is a dirnensionless quantity, only a dimensionless

combination of these two parameters appears in the functional form of 2.

This dimensionless combination is called the excluded volume variable and for prac-

tical purposes it is defined as

3/2 p p ' = (k) (PN)V2

for the discrete chah case.

In the theory of poIymer solutions one focuses on the effects of escluded volume

interactions on various quantities. In this work Our interest is to find the dependence

of the mean square end-to-end distance ( R i ) and the second virial coefficient (A2,L)

on the excluded volume (represented by parameter ,O) as well as the length L of the

c h a h These quantities can be t heoretically evaluated using formulae (1.1) and (1.2)

which can be rewritten in the context of the two-parameter mode1 as

and

where

is the nomalized probabiiity distribution for a single chain, w12(R, 8') is the in-

teraction potential between the two chains and @t) and R'(tr) are the continuous

representations of chains one and two, respectively. Since the interchain interactions

are of the same type as the intrachain ones, one c m write W12 in the f o m

which is the continuous chain analogue of the discrete potential

The evaluation of (Ri) and (AzvL) in the two-parameter limit2 leads to

where (Ri )e = IL and (Az,L)e = $wL2. The use of z cc WL''~ (see eq. (2.34)) allows

us to rewrite eqs. (1.4) and (1.5) as scaling laws for the linear expansion factors cuR(z)

and <yi(z). These are

depending only on a single dimensionless excluded volume variable z (in the Limit

of L -+ oa and w fxed), which allow us to make a connection between approximate

functional forms for c&z) and cri(z) proposed in the early days (see e.g. Flory formula

eq. (3.5)).

2The limit of L + oo and w + O such that t = m s t .

2.4 Self-avoiding walk model

The self-avoiding walk (SAW) mode1 is the most widely used lattice model of real

polyrner chahs. It models the excluded volume effect using the self-avoidance con-

dition [54] and it is highly suitable for computer analysis. The analogy between the

Ising model and SAW, first noticed by Temperley in 1956 [33], caused an increase of

interest of scientists in SAW. This analogy has been clarified in 1972 by de Gennes [34]

who noticed the equivalence between S-4FV and the n = O limit of the n-vector model.

Since then SAW has been also an important testing model in the theory of critical

phenornena. Detailed review of S.4W can be found in 1371 and the references therein.

A self-avoiding walk is a correlated lattice walk Mth infinite memory, Le. the

walker is not allowed to visit the same lattice site again during the walk. More

precisely, the walker being at the lattice site A can not step into sites D and E but

only to sites B and C (see Figure 2.1). Quite generally, a SAW can be defined for

any lattice type in any dimension as a sequence of sites (labels) {Ai) such that

D(Ai - l ,A i ) = a for al1 i = 1 ,..., N

A, # A, for al1 i # j

where D(A, B), a and N are the distance between sites A and B, lattice spacing,

and the number of steps of the walk, respectively. The problem of SAW can be

summarized in the following two questions

a How many different self-avoiding walks are there on an infinite lattice of a given

type and dimension?

What is the probability distribution function of the end points of such walks'?

Despite the simplicity of the formulation the S.4W problem is enormously complex

and it has not yet been solved. There are three types of approaches to the S.4W

Figure 2.1: An example of a short SAW in two dimensions.

problem: the rigorous analysis, direct enumeration techniques and the Monte Carlo

method. The most important characteristics of SAW are the total number of walks

CNy the number of walks cN(R) with the end point a t 8 (these walks c m be used to

find the probability distribution of end points P$*~( I? ) = c N ( f i ) / ~ N the number

of closed polygons UN, and the averages such as the mean square end-to-end distance

(RC) and the radius of gyration (Sc). The total number CNi,N2 of the pairs of self-

avoiding walks (of respective lengths Nl and 1V2) sharing at least one lattice site

is important for the calculation of the second virial coefficient. In this chapter we

briefly mention the rigorous analysis and leave the exact enumeratioas and Monte

Carlo method for the next chapter.

Rigorous analysis relies rnostly on inequalities derived by counting SAW-s of spe-

cial properties. The trivial result for CN, obtained by counting al1 non-reversa1 ran-

dom walks3 on a lattice of the coordination number4 q, is

A less trivial result for k(n), d e h e d by equation CN = exp(Nk(N)), was obtained

by Hammersley [35] and later improved [36] as

O < k ( N ) - k < y ~ - ' / 2 + N-' lnd (2.46)

where k and y are constants. Restricted walks of the order r are obtained by exduding

the polygons of length r from the configurations of the walk (the non-reversd wdk

.being the restricted walk of the order one). As long as r is finite the walk can, a t

least in principle, be studied by the Markovian transition matriu the size of which

increases very rapidly as qr. From the theory of Markov chains we know that for any -

3~ the non-reversal random waiks the walker is not allowed to retum immediately to its previous

lattice position.

4The number of nearest neighbors; for hypercubic lattice of dimension d, it is q = 2d.

finite r the total number of walks of the order r is given by the formula

(r) IV (4 N 4 N cg) = Ai A,,, + A2 Az,r + ... + AN

that is asymptotically dominated by the largest eigenvalue, Say XI,,. Based on the

numerical evidence it is believed that

where p is some constant. This can be used to conjecture the asymptotic form

The same asymptotic form of CN can be also obtained from exact counts and Monte

Car10 simulations.

In the next chapter we briefly mention some of the techniques used to extract the

universal mode1 properties of polymers in solution.

Chapter 3

Methods of calculating polymer

properties

Over the years many models of polymers in solution have been suggested. Two of

them, the two-parameter and the self-avoiding walk (SAW) models were described in

the previous chapter in greater detail. The ultimate goal in theoretical modeling of

polymers in solution is to calculate physical characteristics that can also be experi-

mentaIly measured. These properties are the exponents v and A and the crossover

behavior of a i or ai. In the next chapter it wiil be shown how u and a;(z) can be

extracted from experimental data, but prior to that various methods of obtaining this

information from mode1 calculations will be discussed. Chronologically, these meth-

ods are based on the mean field theory (19507s), computer generated lattice models

(late 19507s), field theory (19707s), and the perturbation theory and related series

analysis techniques (1980's).

3.1 Mean field calculations

Mean field theory calculations are based on various smoothed density rnodels of poly-

mers. In ali these models a polymer chah is represented by a continuous distribution

of segments. The density fluctuations are ignored. The first mean fieId calculation in

1949 was done by Flory [2] who derived a relationship between the Iinear expansion

factor as, defined as a: = (S2)/(S2)a as in eq. (2.41), the molecular weight M and

other important molecular parameters of a polymer in solvent systems. In his work,

Flory used the Gaussian segment distribution to evaluate the elastic contribution to

the free energy (proportional to the entropy decrease due to the sweIling of a chain)

and the contribution to the free energy from the excluded volume effects (arising

from solvent-mediated monomer-monomer interactions). Balance between these tnro

competing effects allowed him to derive the equation

where CM is a constant independent of 121 for rnost of the polymers with high molec-

ular weight, Q is the so-cdled entropy parameter and O is the FIory temperature. -4

number of interesting conclusions c m be d r a m from eq. (3.1) that are al1 in qualita-

tive agreement with the current knowledge of polymer solutions:

rn Assuming that Q(1- O/T) is positive, <ri - ai$ increases proportionaily to a. In the long chah limit we can approximate a: M ' / ~ which gives us the

M-dependence of the radius of gyration as

where ~ ~ ( 3 ) = 0.6 is the critical exponent in the 3-dimensional' space. The

most precise current estimates of Y are around 0.59 so that the Flory mean field

estimate of value of v is excellent.

At Q-temperature as does not depend on Al, Le. as = 1 for any M value. This

overall cancellation of the excluded volume effect is a consequence of two corn-

peting effects. One is the monomer-monomer repulsion mediated by monomer-

solvent interactions and the other is monomer-monorner van der Wads attrac-

tion. At a certain temperature (that is different for various solvents) these

effects are equal and of the opposite sign and so the chain is unperturbed.

.4t T > 8 the temperature dependence of as is mainly due to the factor Q(1-

OIT). By increasing the temperature, this factor increases and consequently as

increases. The chain of the same M becomes more or less swollen depending on

the value of Q(l - O I T ) . In solvents whose @-temperature is below the room

temperature (called good solvents) the chain is significantly more swollen than

in solvents with O above the room temperature (called poor solvents) as can be

seen in Figure 4.1 in Chapter 4.

IOne can do a very simple qualitative calculation of Flory's exponent uF(d) for any dimension d

by minimixing the free energy of a polymer coi1

where S represents the linear size of the polymer coil. The constants C i and C l s i depend

on rnicroscopic details of the system but not on hl and S. F ( S ) reaches a minimum at

and therefore the Flory exponent in the d-dimensional space is uF(d) = 3 / ( d + 2).

a The most important conclusion of the Flory calculation is that the Markov

character of the unperturbed c h i n (S2 - M) in the presence of excluded volume

effects changes into non-Markov behavior S2 - Ml.?

Many attempts have been made to find linear expansion factors and &z)

(defined by eqs. (2.41) and (2.42)) for all values of the excluded volume parameter

z (see eq. (2.34)). For the smoothed density mode12 the Flory equation (3.1) can be

written into the z-form as

Flory's approach as well as other original approaches, worked within the framework

of the mean-field theory. Later, the combination of mean-field and the two-parameter

theory (see Section 3.4) became popular. Based on an equivalent ellipsoid model of

a polyrner c h a h Kurata, Stockmayer and Roig [60] derived an expression

Flory and Fisk [59] derived a serniempirical formula for cui(z) of the follonring form

where f (x) is a function quickly decreasing to its asyrnptotic value with increasing x.

The experirnental justification of their formula came shortly afterwards when Berry [B]

has shown that the data on dilute polystyrene solutions near &temperature in various

solvents could be well fitted by the Flory-Fisk formula.

The second vina1 coefficient evaluation was approached by Flory and Krigbaum [24]

within the same model and later rewritten by Orofino and Flory into a semiempirical

form. Stockmayer corrected it so that it yields the correct first-order perturbation

2Here the factors as and c r ~ are equal to each other.

(see Section 3.4)

The subscript O denotes that the individual chains were assumed to be unperturbed.

The differential equation approach developed by Kurata et al. [16] and subsequently

generalized by Yamakawa [17] yielded the result

It has been believed for a long time that the asymptotic form of a R ( z ) is of the

where A is a multiplicative constant and w is the exponent related to the critical

exponent (defined by (Ri ) - L2u) Y by formula 2v = 1 + l/w (see eqs. (2.43) and

(2.44)). The asymptotic theories of aR(z) can be grouped according to their prediction

of w vaIue. The original Flory formula (eq. (3.5)) belongs to the fifth-power type

(w = 5, v = 3/51 whereas equation of Kurata et al. (eq. (3.6)) is of the third-power

type (w = 3, v = 2/3). Bueche [61] obtained fourth-power result (w = 4, u = 5/8).

An extensive account of many important resuIts of theory of poIyrner solutions is

discussed by Yamakawa [6]. Only recently an agreement has been reached on the

approximate value of the exponent W. This is due to the modern concepts such

as the connection between the self-avoiding walk mode1 and the n = O component

44 field theory (Le Guillou and Zinn-Justin [48], w = 5.68) or the renormalization

group theory and the related eexpansions (Douglas and Freed [65], w = 5.45 and

Le Guillou and Zinn-Justin [84], w = 5.65). These results will be discussed in more

detail in Section 3.3. The result for a i ( z ) equivalent to Flory result of eq. (3.5) was

much less understood. Neither eq. (3.8) nor eq. (3.9) are consistent with the Flory

result for exponent v = 3/5 that predicts a i ( z ) oc z -~ /= .

3.2 Lat tice mode1 cdculations

There are two major approaches based on lattice mode1 caiculations, the exact count

of short chahs and Monte Carlo simulations of Iong chahs.

3.2.1 Exact counts

The number of self-avoiding lattice walks CN is a fast increasing function of the length

N, e.g. there are over four million distinct SAW-s of the length M = 10 on a trianguIar

lattice- If one wants to count al1 the walks exactly, clever counting methods (such

as those of Sykes [38]) have to be used to speed up the computations. Even with

this improvement exact counting done by Martin and Watts 1391 reached only up to

N = 15 for a simple cubic lattice (d = 3). The recent work by MacDonald et al.

[40] extends the length of chain to N = 23. If the second virial coefficient data are

desired, the length of the walk gets even shorter, Le. N = 7 on a cubic Iattice [38].

In this section we do not intend to give the literature review on the exact counts, we

just want to mention couple of methods usually used in connection with the exact

counts and typical results obtained by these methods.

It is generally believed that the long scale properties of SAW such as CN, (R$)

and A 2 , ~ obey the following asymptotic Iaws

where y, v and a are the critical exponents3. Various methods of estimating these

exponents and the connective constant p from exact counts have been reviewed by

3According to hyperscaling assumption 2 - a = 3v as in eq. (2.5).

Gaunt and Guttmann [42]. By plotting h(CN/CN-I) against 1/lv [43, 441, one cm

estimate l n p and y - 1 from the intercept and the slope of the curve, respectively.

The result for face centered cubic lattice is y = 1.1663(3) (d = 3). Similady, by

plotting the quantity UN = N((R$+~)/(RN) - 1)/2 versus 1/N an estimate of u c m

be obtained from the intercept of such graph [45, 46, 471. To srnooth out even-odd

oscillations due to the lattice structure for loose-packed lattices the average exponent

can be conveniently defined as UN = 1/2(vN + u + ~ ) Result v = 0.60(2) (d = 3)

obtained by the method of exact counts is identical to the Flory result UF = 3/(2 + d).

This also presents a strong numerical evidence that (RL) /N diverges. There are other

rnethods of extrapolating series results such as the method of Pade approximants, but

we will not go into any details.

To estimate the second virial coefficient of chains of length N one draws two self-

avoiding walks on a lattice and counts the number of forbidden configurations (i.e.

the configurations in which the walks overlap each other a t least once). The number

of forbidden configurations CN,N for two tvalks of an equal length nT is related to the

second virial coefficient in the following way A2,N - C N , N / ~ C i and it was found by

McKenzie and Domb [41] that the value of exponent a in the asymptotic formula for

(see eq.(3.13)) is estimated to be 0.28(2) for three dimensionai lattice4.

Early 1960's exact and Monte Carlo calculations suggested that the probability

distribution function of the end points p p W ( R ) rnight not be Gaussian as it was

thought before. Later more precise histogram plots of exact enurnerations on square

and simple cubic lattices by Domb, Gillis and Wilmers [46] showed the scaling of the

4Note that (2 - a)/3 = 0.573 # v. This was one of the numerical results that started the debate

over whether or not the hyperscaling assumption is correct. This assumption was not generally

believed before the advent of renormalization group methods. In this work we assume it to be true.

distribution function to be

where f (x) xKexp(-Dxd) for x » 1 and f (x) .v xe for x « 1. D is the lattice de-

pendent constant. Their results and the results of more recent work by McKenzie [51]

are ~1 = 113, 6 = 512 (d = 3). These results also indicated that the distribution func-

tions become spherically syrnmetric which aIlowed Fisher [52] to obtain the relation

6 = 1/(1 - v) . This relation is satisfied for both the Flory result v~ with the values

of 6 given above and Gaussian values v = 112, d = 2. The results of the most recent

work by Dayantis and Palierne [53] are K = 0.27, 6 = 2.45.

3.2.2 Monte Carlo method

The Monte Carlo (MC) method is suitable for estimating the ensemble averages with-

out generating al1 walks of a given length N. The MC method was developed by Wall

in late 1950's in the context of polymers. The aim of the method is to generate an

ensemble of configurations of the mode1 physical system that can be considered as

a representative sample ensemble. Its advantage is that it aIlows to generate longer

chains than the exact counts method, but its accuracy is restricted by statistical fluc-

tuations. A detailed account of the MC method applied to linear polymers is given

by Wall, Windwer and Gans [55].

First indications that (RN)/N diverges were obtained from MC calcuIations of

Wall et al. [54, 56, 551 and later confirmed by Gans [57] using a new technique to

overcome the sample attrition. The most impressive lengths of SAW-s ever used

in MC calculations can be found in the work by Li, Madras and Sokal [58]. They

generated SAW-s of up to N = 80000 on the simple cubic lattice and evaluated

the averages (R:), (SN) and also the second virial coefficient (4,~). The use of

a newly discovered pivot algorithm for generating SAW-s (for more details on this

algorithm, ako used in this work, see [66]) dlowed them to effectively simulate such

long walks, elirninate the corrections to scaling and estimate the exponent u to very

good statistical accuracy. Their result of v = 0.5877(6) agrees well wïth the Le

Guillou and Zinn-Justin result [48] and shows that difFerent models can be used to

obtain the universal quantities with a very good precision.

3.3 Field t heory calculations

In 1971 Wilson introduced the renormalization group (RG) into critical phenom-

ena [77] and showed how scaling can be explained in terms of RG differential equa-

tions and their solution near the fixed point. By doing a phase space analysis on the

generalized Ising mode1 he showed that the effective interactions between block spins

are of the Landau-Ginzburg form

The effective block Hamiltonian 3CL is not of the Kadanoff form with two parameters

KL and hl, but of more complicated form with an infinite number of parametersS.

Wilson succeeded in evahating the critical exponents such as y = 1.22 and v = 0.61

for the Esing mode1 in three dimensions. He also showed that their values in five

dimensions are equd to the values obtained from the meaa field theory i.e. y = 1 and

Y = 0.5 [78]. Wilson's work initiated the field theory approach to critical phenomena

that was later developed by others including Brezin, Le Guillou and Zinn-Justin [81].

5For brief explanation of RG theory and the Kadanoff block spin picture, see Chapter 5.

Important developments followed after the discovery of Wilson and Fisher [82] that

E = 4 - d can be used as an expansion parameter (d is the dimension of the space).

The E-expansion of a variety of quantities with respect to E = O (Landau's mean field

theory in d = 4) followed. Currently the e-expansion is known up to the fifth order

in E [83] and recently it was used by Le Guillou and Zinn-Justin 1841 to evahate

the critical exponents for difTerent values of n (in n-vector model) in two and three

dimensions. The three dimensional results for n = O are y = l.l60(4), v = 0.5885(25)

and A = 0.482(25).

-4 completely new way of calculating the critical exponent v, based on the polyrner-

magnet analogy, was Erst presented by de Gennes in 1972 [34]. In fact, it is the exact

forma16 mathematical equivalence between the SAW problem and the n-vector rnodel

of interacting fields. The derivation of this analogy can be found in (791. Here we will

only mention that the susceptibility of the n-vector model defined by the Hamiltonian

in the limit of n = O is given by the formula

where CN is the number of self-avoiding walks on a lattice. The equations (3.17) and

(3.18) form the essence of the polymer-magnet analogy. From eq. (3.11) we know that

%I the evaluation of the n-vector mode1 one has to take the limit n = O to arrive at the analogy.

39

since the asymptotic behavior of CN is - pNN7-L, by substituthg it into eq. (3.18),

near the critical temperature Tc, one gets

The above analogy shows that in the n = O limit the susceptibility critical exponent

y is in fact equal to the SAW critical exponent y kom eq. (3.11) and that the long

chain limit N -t oo corresponds to the limit IT - TcI + O in the magnetic systems.

Other analogies can be drawn, e.g. the critical exponent v, also called the correlation

length exponent, is analogous [79] to the exponent governing the asymptotic behavior

of the correlation lenght of a magnetic system near Tc

Another method of estimating u and A is based on the expansions of these ex-

ponents in the renormalized coupling constant g. The series in g, derived Baker et

al. [85, 861 for the n-vector model up to the sixth order are divergent and the Borel

summation techniques have to be applied to obtain the values of the exponents for

critical (fbced) point value g' at b e d dimension d = 3. LeGuillou and Zinn-Justin

used such summation [48, 841 to calculate both the critical exponent u and the cor-

rection to scaling exponent A. The values they obtained for n = O are u = 0.5880(15)

and A = OMO(25).

3.4 Perturbation t heory calculat ions

In the limit of small z (z < 1) both expansion factors (eqs. (2.41) and (2.42)) can be

written as series in powers of z

O&) = 1 + c [ ~ ) ~ + ciRb2 + ciRb3 + . . .

c u l ( z ) = 1 + cIA)z + c~)z* + ciA)z3 + . . .

40

For the linear term ciR) = 413 was found in 1953 (e. g. [61]). In 1955 Fixman [62], us-

ing the Urseli-Mayer type cluster theory, derived the quadratic term ciR) = -(16/3 -

281r/27). Later, cluster analysis of ( r i ( z ) was used by Kurata and Yamakawa [16] and

they determined c[") = -2.8654. The cubic terrn ciR) was determined by Yamakawa

and Tanaka [18], but later it was found to be incorrect [89]. The cluster theory evalu-

ation of perturbation series of eqs. (2.36) and (2.37) is straightforward but extremely

complicated. A much simpler method of the &(z) evaluation based on the diagram-

matic field-theoretic techniques was presented by Muthukumar and Nickel [64]. They

calculated cui(z) up to the sixth order in z based on the perturbation series for the

inverse Laplace transforms. Later, they applied the same method to aA(z) [74] and

obtained the series up to CF). Their results

are currently the most accurate perturbation formulas available in the literature. The

absolute values of Ci in both series increase so rapidly with the order of z that it is

assumed that the series converge only for very small values of z or do not converge

a t all, i.e. they are only asymptotic. This is why specialized series analysis techniques

are needed to extract the behavior of c&(z) and CE$(Z) for al1 values of Z.

The approximate closed expressions of Section 3.1 are based on various mean-

field, self-consistent, or variational calculations where the approximations can not be

controlled and therefore uncertainties are hard to estimate. Recently, Muthukumar

and Nickel [74] performed extensive analysis of the intrinsic errors of various crossover

formulae for o&(z). They derived &(z) for al1 values of z based on their previous

power series results (641 using the Borel summation technique. Their result can be

well approximated by formula

and is in excellent agreement with the previous result of des Cloizeaux et al. [75]

arrived at by a direct renormalization method. The asymptotic (z + ca) result of

eq. (3.25) is

a;(z) = 1 . 5 3 1 0 r ~ . ~ ~ ~ ~ ( 1 + + ...) (3.26)

The most recent results of the theory of excluded voIume problern are reviewed by

des Cloizeaux and Jannink [80].

In this thesis we present a method for the numerical solution of the Domb-Joyce

polymer lattice mode1 [5] presented in the form of recurrence equations. It has the

advantage of providing the solution for any value of z so that the expansion factors

ag(r) and o i (z) for al1 values of parameter z can be evaluated with great precision.

The result a;(z) is later compared to the closed form interpoIation formula (3.25)

and the direct renormalization resuIt of des Cloizeaux et al. [75]. Our method also

provides very precise estimate of critical exponent v = 0.58756(5) (w = 5.710(4)).

Chapter 4

Experimental met hods

The chemical structure of a polymer chain is, without any doubt, major contributing

factor to the specific chemical behavior of a polyrner. I t c m affect such geometric

characteristics of a polymer chah as the angle 0 between successive C-C bonds in

hydrocarbon chains or the rotation angles 4 about neighboring C-C bonds. These

characteristics have a direct influence on polymer macroscopic properties such as the

molecular dimension or the second vinal coefficient, and subsequently on mechanical

and thermodynamic properties Like elasticity of bulk polymers, intrinsic viscosity or

the osmotic pressure of polymer solutions.

One of the systems attracting a lot of attention of polymer scientists is a polymer in

solution. When an isolated chain molecule is placed into a solution, it passes through

an enormous number of conformations due to interactions with solvent molecules.

Despite the complexity of such a system, certain aspects of its behavior can be suc-

cessfully studied using various experimentd techniques. The two most important

techniques that c m be used to probe equilibrium properties of long flexible polymers

in dilute solutions are the osmotic pressure and the light scattering measurements.

The osmotic pressure of a polymer solution is analogous to the pressure of an imper-

fect gas. The theory of the osmotic pressure was derived by McMillan and Mayer [ï].

By measuring the pressure difference across the semipermeable membrane1 for vari-

ous solute concentrations one can determine both the average rnolecular weight and

the second vinal coefficient of a polymer chain. Comprehensive treatment of osmotic

pressure from both theoretical and experimental point of view can be found in [80].

In this chapter we concentrate on light scattering experiments in ivhich the mea-

surement of absolute intensity and the angular dependence of the scattered light

provides information about a system. We describe basic physical characteristics of

polymers and briefly explain the physical principles used to measure these character-

istics. The universality is discussed from the experimental point of view. The results

of two important experiments (a polymer in good and in poor solvent) are outlined

and the predicting ability of the two-parameter theory is discussed.

4.1 Polymer characteristics

The average molecular weight, the molecular dimension, and the second osmotic virial

coefficient are the three basic molecular characteristics of a polymer chain. The re-

lations between them are the main focus of experirnental work. Molecular weight

M is the most important molecular characteristic of a polymer chain. For a single

polymer molecule it is a product of the degree of polymerization n and the molecular

weight of a monomer Mm, M = nM,. Due to the finite sensitivity of instruments,

measurements on a single polymer molecule are can not be performed and instead

samples consisting of rnany polymer molecules are investigated. Polymerization meth-

ods, however, cannot precisely control the termination process and thus all polymer

samples are polydisperse to some degree. Polydispersity refers to the finite width of --

'with a solvent on one side and polymer solution on the other side

44

the distribution of molecular weights and is often a source of controversies where the

interpretation of experimental results is concerned. By performing rneasurements on

polymer samples the effects of al1 pol_vrner molecules in a sample are averaged and

thus average quantities are obtained. In light scattering experiments the so calIed

weight average molecular weight

is obtained where Ni is the number of chains of the molecular weight Mi. The ratio

MW /Mn where

indicates the degree of polydispersity of a sample (Mn is the number average molecuIar

weight that is experimentally accessible from the osmotic pressure measurements).

The degree of polydispersity for a typical sample of a good quality is on the order of

1.1 or less.

An experimentally useful measure of the molecular dimension of a polymer chain

is the radius of gyration S2 which represents the distribution of the monomers around

the centre of mass R C M of a polymer coi1 and is defined as

where is the position of i-th monorner and pi({&}) is the distribution function for

a single polymer molecule. The second osmotic virial coefficient Aiq is the measure

of the interaction between two polymer molecules in a solution and is defined by the

formulae (1.2) and (1.3) except that in this section we will drop ( ) as averaging is

to be understood. The second virial coefficient can be combined with S2 to form the

interpenetration function $ J ~ given by the formula (1.10) which in terms of AYP is

Note that l(ls is a dimensionless measure of the strength of interaction that is useful

for the cornparison between theory and experiment.

4.2 Light scat t ering measurements

In a light scattering experiment, equilibrium properties of dilute polymer solutions

such as the characteristics &, S2 and can be determined from the magni-

tude and the angle dependency of the scattered light intensity. The theory of light

scatterin$ was developed by Debye [22] and Zimm [21] in late 1940's. They evahated

the influence of polyrner molecules on fluctuations of the refractive index of solvent

and related the osmotic pressure .rr to the optical constant H and Rayleigh ratio R(6)

in the following way

The optical constant H depends on the refractive index of the solution n, its concen-

tration derivative anlac and the wavelength of the scattered light A. The Rayleigh

factor R(B) is proportional to the ratio of scattered and incident light intensities Io/IO.

Equation (4.5) can be written into the form

where P(8) is the scattering form factor and K is a constant given by

For a random coi1 P(0) is given by the formula

2For textbook reference, see [23].

where Si is the unperturbed radius of gyration and q is the momentum transfer wave

vector

47r q = - sin(O/2)

X (4-9)

at the scattering angle 0. Smith and Carpenter [-O] have shown that the eq. (4.8) can

be also used for a non-random coi1 provided Si is replaced by S2. Eq. (4.6) in the

limit of the infinite dilution and zero scattering angle gives the following two formulas

Usually, the measured values of Kc/R(B) are plotted versus concentration c for fixed

values of the scattering angle and versus sin2(0/2) for ûxed values of concentration

c into the same graph. This, so called Zimm plot [21] provides reliable estimates of

a, A2 and S2 in the limit of an infinite dilution (c + 0) and zero scattering angle

(e + O).

Light scattering experiments on high molecular weight polymers are crucial in

elucidating the excluded volume effects. In the mid 1960's and early 1970's light

scattering experiments were performed on polymers of molecular weights on the order

of 1 x 106gmol-' [8, 9, 101. The first group to perform ultra high Ad measurements

was Slagowski et al. [Il, 121. Their polystyrene samples reached up to M = 50 x

l ~ ~ ~ r n o l - ~ . However, due to the bad scatter of data [12] it was nearly impossible

to draw any conclusions. Since then there were other reports on the static scding

behavior of polymers of very high M in solutions [13, 141. One of the polymers

frequently used in light scat tering experiments is polystyrene. Its advantage is t hat

it creates perfectly linear backbones with molecular weights up to 1 0 ~ ~ x mol-'. It

is easily dissolved in various solvents such as benzene, toluene and dichlorethane at

room temperatures, and it is a good light scatterer due to the presence of benzene

rings in styrene. The molar mass of styrene is about Mm = 104.159 x mol-1 so that

polystyrene with the molar m a s of M = 1 0 ~ ~ x mol-' consists of about n = 0.96 x 10'

monomer units, each of length of approximately 1 = 0.3nm. Thus the total length of

the chah is about L = 2.9pm. The thickness of the chain is only on the order of a

few Angstroms and therefore the chain cm be considered to be a linear object.

It was found that the dependence of both the molecular size and the second virial

coefficient on the molecular weight is a power law and the exportent is universal in

the long chain limit. The universdity can be graphically represented by the follow-

ing two plots that show the typical results of the light scattering experiments. In

Figure 4.1 the measurements of S are logarithmicdly plotted versus 1M, for different

solvents. An interesting feature of the graph can be noticed, namely that the different

solvents have various effects on the size of a polymer molecule. In poor solvent (e.g.

methyl-ethyl ketone, full diamonds) chains are less swollen than in a good solvent

(e.g. toluene, empty diamonds) since the excluded volume has a larger value in good

solvents compared to poor solvents. The most important feature, however, is a very

similar slope of the lines in the upper part of the graph. This means that in the long

chah limit (M + oo) the exponent Y in the power law behavior

is of the same value (- 0.59) for any solvent as long as the effective monomer-

monomer interactions are repulsive (Le. nonzero excluded volume effect). Therefore

al1 perturbed chains belong to the same universali@ class no matter what the details

of the interactions are. The bottom line on Figtire 4.1 has a slightly lower dope

because for a polymer in the B-solvent (polystyrene in cyclohexane) the excluded

volume effects cancel on average and the chah becomes an unperturbed one. Then

it can be described by the random flight mode1 (S2 - N, Y = 0.5). The universal-

Figure 4.1: The data for polystyrene (PS) in various solvents: PS in toluene (empty

diamonds) and in methyl ethyl ketone, Canne1 et al. (1987) (full diamonds); in benzene

at 25°C Miyaki et al. (1978) (full squares); in benzene, in toluene and in dichlorethane

at 30°C Yamamoto et al. (1971) (squares) and PS in O-solvents: cyclohexane at

34.5"C and trans-decalin a t 20.4"C Miyaki et al. (1978) (circles).

ity is a widely accepted experimental fact. However, there are some aspects of the

behavior of polymers in solutions that are non-universal. Various theories provide

different predictions as to which aspects of the behavior of a polymer chah in so-

lution are universal and which are mode1 dependent. As i t was rnentioned before,

the two-parameter theory predicts the interpenetration function $(z) to be a univer-

sa1 function of the excluded volume variable z. Recent experirnents on short chain

polymers in solvents show, however, that this prediction is wrong. More detailed

discussion of the universal/non-universal behavior follows in Section 4.3.

Molecular weight dependence of A2 in good solvents is also of interest both the-

oretically and experimentally. The first theoretica1 prediction of il2 dependence on

hf was suggested by Flory and Krigbaum in 1950 [24]. Using the mean-field theory

they predicted that A2 should decrease with increasing M. The ac tud decrease of A2

measured in the experiment, however, is faster than the predicted one. Al1 the exper-

imental data available for A2 (such as those shown in Figure 4.2) can be empiricdly

described by the formula

A2 = M -CA ( M ) (4.13)

where the M dependence of the exponent E A is just a statement of the experimental

fact that A2 plotted double-logarithmically versus M (see Fig. 4.2) does not follow

a straight line, but rather slightly convex-downward curve with the negative slope

é*(M) slowly decreasing. The hint of the curvature was visible in some of the previous

data such as [14], but due to the lack of sensitivity of experiments the small M sarnples

could not be measured before and the curvature ivas con£irmed only recently by Fujita

and Norisuye [25]. In their work, poly methyl methacrylate in acetone (good solvent)

was used. The asymptotic value of eA(M) is estimated to be around zz 0.2 that turns

out to be about 2 - 3v as expected theoretically.

4.3 Two-parameter t heory versus experiment

The dependence of both A" and 9 on M was formulated within the two-parameter

theory of polyrner solutions3. The theory predicts t hat the interpenetration function

T/.J~ (see eq. (4.4) is a universal function of the excluded volume parameter z defined

by the formula

where /3 is the binary cluster integral between polymer segments and !V is the number

of statistically independent segments of an effective length 1. The parameter z cannot

be measured directly and thus indirect methods have to be used to confirm the theo-

retical prediction for +s(z). There are other quantities for which the two-parameter

theory gives prediction in terrns of z, e.g. the expansion factor as(%) of the radius of

gyration defined by the formula

The unperturbed (ideal) dimensions S: can be obtained from the light scattering mea-

surements at the O-ternperature. Moreover, the two-parameter theory predicts that

as(z) is also a uni~wsal function of thc variable z , therefore by plotting @(z) against

crs(z) one can eliminate the z-dependence and fi becomes a universal function of as.

To confirm the universality, the measured data of are usually plotted versus ai for

many different polymer-solvent conditions. Such a plot is shown in Figure 4.3. One of

very few theories that are consistent in treating the intramoIecular and intermolecular

interactions when calculating .iCls and as are the Kurata-Yamakawa theory [16, 171 of

$.Js and Yamakawa-Tanaka theory [18] of as. The theoretical predictions

3For a thorough review of theoreticd methods see Yamakawa [6] or references therein.

52

Figure 4.3: Berry's data (1966) for poiystyrene in various solvents.

obtained fiom these theories can be used for cornparison with the experiment. The

experimental data obtained from the high molecular weight measurements in poor

solvents [BI fa11 within the experimental error of the curve given by eqs. (4.16) and

(4.17).

There are different ways of varying the excluded volume parameter z. Let us

concentrate on varying z by changing the b i n a l cluster integral P. This can be done

either by changing the temperature (the value of is sensitive to the temperature

change in poor soIvents) or changing the nature of the solvent (i.e. one can use a

better solvent to increase the value of p). In the former approach (used by B e r l [8])

Figure 4.4: Huber's SANS data for short chah polystyrene in good solvents.

the zero value of z is obtained when the temperature is exactly equal to the Flory

temperature 0 at which crg(0) = 1 by definition. For temperatures T > O higher

values of z are obtained. The latter approach was used by Miyaki et al. [14] as well

as Huber et al. [19]. The first group concentrated their attention on the ultrahigh

molecular weight range. On the other extreme, Huber et al. used very short chah

polymers of molecular weight as low as M = 1 x 103gmol-'. Measurements on

polymers of such a small M would not be possible using the standard technique

of light scattering. Instead, they used the small angle neutron scattering (SANS)

and obtained a surprising result. The plot of @s versus ai did not exhibit the shape

predicted by the two-parameter theory. Instead, the decrease of $JS with increasing ai

was measured. The decrease of $s with increasing cri was measured d s o by Myiaki et

al. [14] but their data were not conclusive as to whether Ils will eventually decrease in

the region of small ai as predicted or it will continue increasing. SANS data of Huber

et al. on polymers in good solvents (191 reaching the region of a, = 1.1 seem to be

conclusive enough (see Fig. 4.4) to say that for polystyrene in a good solvent there is

a sharp increase of $s as as + 1. The short chah polystyrene data in a good solvent

therefore show behavior contradicting to that predicted by the two-parameter theory:

there is a decrease rather than an increase of $ J ~ as as + CU. Huber and Stockmayer

attributed the increase of +s as as + 1 to the fact that the two-parameter theories

are an approximation of some "three-parameter" theory with the third parameter

related to the stiffness of a polymer chah [19]. Nickel [4], however, showed that

this is not the case, since within a one-parameter recursion mode1 the decrease of $s

follows naturally even for completely flexible polymer chains.

The fact that the data in Figures 4.3 and 4.4 do not overlap indicates that qs is no t

a universal function of as. This breakdown of the two-parameter theory's predicting

power is due to the failure to recognize which quantities are universal and which

are detail-dependent. The renormalization group arguments predict the asymptotic

behavior

where A2 = M2Ayp/NA and n = M/M, is the polymerization index. It was pointed

out by Nickel [4] that the exponents v and A as well as the dimensionless amplitude

ratios aA/ay2 and ba/bs are universal, but that the amplitudes as, bs, aa and bA

thernselves are not. The amplitudes bA,bs were shown to have an arbitrary sign.

That explains why predictions of $(m) based on both poor solvent and good solvent

data are the same (ideally) while the approach of @&) to Srs(oo) (governed by the

combination b* = bA - 3/2bs ) can be from below (two-parameter, poor solvent) or

from above (good solvent). The two-iiarameter theory, the most trusted theory of

dilute p o l p e r solutions, can not explain al1 experimental data and therefore more

generd approaches such as the one-parameter recursion mode1 [4] or the renormal-

ization group Monte Car10 method for Domb-Joyce model, presented in this thesis,

have to be used to explain both good and poor solvent measurements.

Chapter 5

Renormalization group met hod

The physical world that we live in allows us to separate the phenomena occurring on

different length scales and thus to easier formulate physical theories. In the case of

critical phenomena, however, the separation of length scales is impossible and theories

describing the state of matter near the continuum phase transition are difficult to

formulate. There are many physical systems that undergo a phase transition in which

some quantities such as the heat capacity are divergent. One example is the liquid-gas

system near the critical point (Pc,Tc,pc)- In 1869 Andrews discovered an interesting

behavior of carbon dioxide near the temperature of 31°C and the pressure of 73

atm. The properties of vapour and liquid phase of CO2 at this temperature and

pressure became indistinguishable. In 1895 Pierre Curie noticed similar behavior

in ferromagnetic iron. Iron displayed spontaneous magnetization below a specific

temperature, later caIled the Curie temperature. The similarities between the Curie

point of a magnetic system and the critical point of a liquid-gas system were studied

ever since.

The reason why the formulation of a successful theory of critical phenomena is

so difficult can be understood if one looks a t the case of COn. If the container of

CO2 that is under the pressure P, = 73 atm is continuously heated, a t a certain

temperature the transparent liquid becomes milky in color. This phenornenon, c d e d

critical opalescence, can be explained as follows. By increasing the temperature,

significant density fluctuations start to occur and they spread over larger and larger

regions as T + Tc- The fluctuating regions, however, contain fluctuating regions

of progressively smaller sizes within thernselves. At T = Tc the fluctuations spread

over the whole macroscopic size of the system that now contains fluctuations of al1

sizes. This is why the scattered Iight passing through the sample is white rather

than of any specific color. The critical phase transitions constitute a special class of

physical phenomena where length scaies differing by orders of magnitude have equally

important contributions 1761.

5.1 Wilson's renormdization group

The renorrnalization group method 1771 introduced in 1971 by Wilson proved ta be

an ideal computational tool to tacide problems of multiple length scales. Wilson used

the conceptual picture of continuous phase transitions based on Kadanoff block spins

and showed that the renormalization group differential equations lead to the singu-

larities of the Widom-Kadanoff scaling laws quite naturally. The singular behavior

of the partition function Z (or of the free energy F) is very difficult to obtain using

the older methods in which the evaluation of Z is approached directly. Once the

problem is recast in the differential form it is easy to see how the singular behavior

arises. For that purpose Wilson used the Ising model of uniaxial ferromapet. This

model, introduced in 1920's by Lenz and Ising, describes a real magnet by an array

of spins pointing up or d o m , located on the sites of a regular lattice. Interactions

in the system are modelled by coupling of neighboring spins with an overall aligning

tendency. The Hamiltonian of the Ising model is

where J is the positive coupling constant, H is the magnetic field strength and si are

the dynamic variables of the model with only two possible values {si = +1, si = -1).

The first sum goes over al1 the nearest neighbors and the second sum goes over al1

individual spins. In the problem of ferromagnet one evaluates the free energy densityl

hnction f defined by the following formula

where K = J / k B T , h = H / k B T and 1V is the total number of spins in the system

(N + w, in the thermodynarnic limit). The partition function Z(K , h) of the systern

is on the right hand side of eq. (5.2). The summation C{,} goes over al1 possible

configurations {s) of the spins. If some intermediate degrees of freedom in eq. (5.2)

are integrated out in such a way that only block spins s', (obtained by averaging the

degrees of freedom within a cube of side L) will be left, the Ising model problem will

be recast in different dynamic variables, but it will still be a representation of the

same ferromagnetic system. In these new variables the definition of the free energy

is equivalent2 to eq. (5.2). Now, the number of biock spins is smaller (Le. N/ L3) and

the coupling constants Kr, and hL are different from K and h (of eq. (5.2)) since the

block spins interact in some "effective" way depending on L. One can rephrase this

by saying that for any K, h and for any length L there exists a pair of (effective)

'free energy per spin

"This is only an approximation that negIects higher order coupling terms of type s l , s l , s~s~ etc.

coupling constants KG and hL such that for the free energy density one obtains

Another important characteristic of the system is the correlation length J (measured

in the units of lattice spacing) for which the following relationship holds

If one performs blocking of size 2L after blocking of size L was done, the coupling

constants K2', h2L that result depend on Kt and hL but not on L because a t the

E t h level the block Hamiltonian does not explicitly depend on L. Based on this the

differential equations for KL and hL can be written as

where both functions ul and u2 depend only on KL and hL and are analytic3. These

equations are the differential equations of the renormalization group method.

Let us consider the problem of a polymer in solution. In the Gaussian equiva-

lent chain the long chain limit result for the randorn flight mode1 (R2) = Z2N can

be rewritten into the form (R2) = (1J63)~(N/b,) = Z ~ N G where lc and lVG are the

3These functions can be formaiiy derived under the assumption that the exact solution of the

problem is known and both functions f (K, h) and ( ( K , h ) are available. Then by taking the deriva-

tives with respect to L of both eqs. (5.4) and (5.5) and rearranging the results afterwards one

can express ui and u2 in terms of ~ ( K L , h ~ ) , ~ ( K L , h ~ ) and their derivatives. This suggests that

h c t i o n s u1 and u2 are just as singular a t the critical point as the functions f (K, h) and <(K, h).

The purpose of the renormalization group method was to show the construction of ut and u2 by

operations that do not introduce any singularities. En fact, approximations were made in the renor-

rnalization group derivation and a remaining challenge is to determine how singular (non-singular)

the functions ui and u2 really are.

Gaussian equivalent link length and number of Gaussian ünks, respectively. By inte-

grating out some intermediate degrees of freedom (called the "decimation" procedure)

certain nonuniversal aspects of the chah behavior are eliminated. One can apply this

approach to perturbed chains [79] as well. In this case the three most important

parameters in the excluded volume problem are 1, P and N so we expect that

in which both expansion factors a~ and a~ depend on a dimensionless interaction

constant defined as u = P/13. Rather than trying to End the functions a&, N) and

aa(u, N ) directly, an alternative approach can be used. It is based on finding how the

basic parameters of the polymer chain are changed under the grouping transformation.

This transformation that consists of grouping g segments into one subunit is simple

to perform in the ideal chah case where the results are 1, = lgLI2 and u, = ugl/*. In

the real chain these relations are of the form

1, = lgL'*(l + f 1 (u))

ug = ugl/2(l -f2(u))

where fi(u) and f2(u) are corrections to the ideal chah case. The evaluation of

the functions fi(u) and f2(u) requires a direct (i.e. numerical) calculation of 1, and

u, in which al1 interactions between segments inside a subunit must be taken into

consideration. However, this calculation is much simpler task than the evaluation of

both c u ~ and a ~ . Origindly, the decimation procedure was performed only once. In

the renormalization group method it is performed repeatedly until N, is a relatively

small number (i.e. the polymer is represented by a short chain). Since short chain

polymers in solutions behave as hard spheres, the excluded volume P scales as P and

61

therefore B/13 = u* has a 6xed value that can be determined from the equation (5.11)

in the following way

u* = ~ ' ~ ' ~ ~ ( 1 - f2(u8)) (5.12)

where u* is so called fixd point. As a result, a~ and a~ in the long chain limit are

where

The interpretation of the symbols in going from eqs. (5.10) and (5.1 1) to eqs. (5.12)-

(5.16) has changed completely. The original u is a "bare" interaction parameter that

is defined by the mode1 and does not have a fixed point value. The final u is a

"renormalized" (rescaled u,) that changes as repeated scalings are performed and

can tend to a fixed point value. This is essentially the difference between z and

variables defined in the following section.

To summarize, the solution of a complex problem of h d i n g a large-iV behavior of

aR(u, N) and a~ (21, N) was replaced by a much simpler problem of finding the fixed

point u* of the transformation u + ~ ~ ' ' ~ ( 1 - f2(u)). One possible implementation

of these renormalization group ideas was described by Kremer e t al. [63]. In the

following section another approach is given.

5.2 The one-parameter recursion model

The two-parameter model (TPM) series expansions of R2 and A2 (see Chapter 2) are4

where

In this section we are going to do two things. First, we will show, that eqs. (5.17),

(5.18) and (5.19) can be cast into the form of recursion relations which do not depend

on the bare parameters 1 and W. Second, we will use the series equations (5.17) and

(5.18), which are exact but limited to weak coupling, and rewrite them in terms of

simple analytical expressions which are approximately valid for al1 coupiing. A com-

plete (approximate) solution to the TPM is then obtained by iterating the recursion

relations from initial conditions that depend on I and W.

The elimination of explicit 1 and w dependence is achieved by defining reIative

functions

which are the analogs of the scaling equations (5.10) and (5.11). Equations (5.20) and

(5.21) are not yet useful, because of the dependence on z that still explicitly depends

on 1 and W. For that reason the interpenetration function .St = $R is defined as

41n expansion (5.17) the first six terms are known and in expansion (5.18) only the 6 r s t two are

known.

so that

dr = z (1 - 4.8653776011 z + 26.1049923666 zZ) (5.23)

follows by substituting for A2 = A2(z) and R2 = R2(z) from eqs. (5.17): (5.18) and

(5.19). Equation (5.23) can be inverted so that we get

and by sustituting eq. (5.24) into & and f A we get

Equations (5.25) and (5.26) are analogous to equations (5.10) and (5.11) in a sense

that both these pairs of equations relate characteristics of a polymer chah before and

after the g ~ o u ~ i n g ' of segments. Also in equations (5.25) and (5.26) the global quan-

tities R2 and A2 were chosen as parameters as opposed to the effective segment length

1 and dimensionless interaction parameter IL being the parameters of equations (5.10)

and (5.11).

It $vas shown by Nickel [4] that the renormalization group method applies equally

welI in this formulation. He assumed that the behavior of a flexible polymer in a good

solvent can be described by the recursion mode16

51n eqs. (5.25) and (5.26) grouping corresponds to dimerization and therefore g = 2.

6Nickel used the radius of gyration S2 instead of R2

with only one parameter $. He showed that the one-parameter recursion model

(OPRM) reproduces both the two-parameter model and the self-avoiding mode1 re-

sultç very well. Also the experimental data of Huber at al. [19] on short polystyrene

chains in a good solvent can be fitted surprisingly well by this model.

Let us now present the results of the OPRI1 calculations for R2 and A2. First,

let us rewrite Nickel's OPRM equations into an effective exponent form with a new

variable ax defined as

where XL stands either for Ri or for A ~ J . The effective exponent ax represents the

dope of a log-log plot of (XL) versus L. To be consistent with equations (5.25) and

(5.26) and equations (1.8) and (1.9), the effective exponents for RN and A 2 , ~ in this

thesis7 will be defined in the following way

providing

In the context of effective exponent variables the recursion equations of Nickel (see

eqs. (5.27) and (5.28)) can be written into the form

7Zn the lattice model with the lattice spacing 1 = 1 the length of the chain L is identicai with the

number of bonds IV, therefore fiom now on we wilI use N instead of L.

and

ln $ 2 ~ = In $N + In(8) [UA ($NI - ~ R ( @ N ) ] (5.37)

Starting with the values of Rf and A2,1 for short chains of the length Z the recursion

equations allow the calculation of R2 and A2 of a polymer chah of any length N = 2"1

using iterating equations (5.35) and (5.36). In the case of a negligible excluded

volume one starts with R: = Z2 and A2,1 = ~ W Z * (W is small) irnplying that QI zz 0.

For srna11 values of Q ~ , u A ( $ ~ ) - UR(@^) = 116 and therefore = &+N (from

eq. (5.37)). As larger values of qN are gradually reached in the iteration process,

VA(@) - UR(@) zz O. At a certain value of Ilrnr, the h e d point ~* (limw,, $ J ~ = @*)

is reached where vA(@*) - vR($') = O and both R2 and A2 have a cornmon scaling

exponent vR(@*) = uA(q8) = V. The long chah behavior of properties R2 and A2

near the fixed point is given by eqs. (1.8) and (1.9)

where the nonuniversal scaling amplitudes a ~ , bR, a ~ , and ba depend on the initial

conditions $l . By linearizing the recursion equations around +* value, the correction

to scaling exponent il can also be estimated from the formula

In order to use the recursion equations (5.35) and (5.36), the knomledge of uR(@)

and uA(@) is crucial. As eqs. (5.33) and (5.34) show, the functions uR(@) and va(@)

can be approximated using two-parameter results of eqs (5.25) and (5.26). However,

these functions are valid only for very small values of II, and for larger values of $ the

exponents acquire non-physical values. Following Nickel's work, we choose

instead of using eqs. (5.25) and (5.26). This is obviously a very crude approximation,

however it surprisingly well recovers al1 important features of both two-parameter

limit and self-avoiding walk limit of behavior of polymer in solvent. The parameter

c is a free parameter larger than 1.18687 by introduction of which we make sure

that the effective exponent y&b) is well defined over the whole range of physically

accessible values of @. This parameter also allows us to fix one of the three universal

quantities v, $*, or A. We chose to fLus the value of v to 0.3876. This gives us

c = 3.5527 and the other two universal quantities $* = 0.2338 and A = 0.4523 are

obtained by iteration of the equations (5.35) and (5.36). The iteration process starting

from different initial conditions gives different solutions to the OPRM as shown

on Figure 5.1. Individual curves represent the iteration solutions obtained by using

various initial conditions A2,1 and Rf to start the iteration process. If one starts from

a small value of @L that represents small excluded volume, one recovers the solution

close to the two-parameter model solution. Within the approximations used in the

method, the lower envelope of the curves is the two-parameter mode1 solution itself.

On the other hand, for greater initial value of the SAW model can be recovered.

Depending on what is the initial value of we can get approach to the asymptotic

value @* that is either from above or from below.

8The estirnate of the cntical exponent value obtained in this work as will be shown later.

Figure 5.1: Flow lines of QN for different initiai conditions as obtained by the

one-parameter recursion model. The flow lines are: (a) is the "near" two-parameter

solution, (b) the two-parameter solution and (c) SAW-like solution of the OPRhd.

5.3 Generalization of the one-parameter recursion

In principle, the recursion equations (5.35) and (5.36) determining properties of chains

of length 2N from those of chains of length N may include some other parameter in

addition to $. This new dimensionless parameter (say $), that might be related to

some other physical property of a polymer in a solution such a s the third virial coef-

ficient AJ or the chain stiffness, allows us to generalize the oneparameter recursion

model equation for (eq. (5.37)) to the continuous-N form of Wilson's renormaliza-

tion group equations

where the functions Z L ~ ( $ ~ , dN) and u 2 ( $ ~ , #N) do not depend on iV explicitlyg. The

complete model solution is obtained by solving these differential equations. This

might be a suggestion for further work, but at present we focus on the approximation

of eqs. (5.43) and (5.44) where only one parameter @ is used. This is what Nickel

assumed in OPRM [4]. In addition, eqs. (5.43) and (5.44) would not reproduce the

expected analytic 1/N-dependence (see eq. (1.7)) and for that reason one must include

explicit N dependence in the u functions. Then the renormalization group equation

where ül (N, $) = 3(vA(N, $) - vR(N, $)) as follows from eq. (5.37). Introduction

of the N-dependence is a natural step in the generalization of the OPRM. It is also

gSee the discussion of Wilson's renormdization group differential equations in the first section of

this chapter.

necessary to enable the reproduction of the Domb-Joyce (DJ) model results in the

limit @ + O exactly. The 1/N corrections in the limit of large excluded volume

are included through the choice of the fitting functions. Monte Carlo method (see

Appendix C.2) was used to estimate the averages (R;) and and the obtained

data were used to calculate UR and U A according to eqs. (5.31) and (5.32). This was

done for many different values of the Domb-Joyce excluded volume parameter w and

plotted versus 11. The details of the actual data generation and analysis are presented

in Chapters 6 and 7.

Chapter 6

The Domb-Joyce model

The most widely used lattice model of real polymer chains is the self-avoiding walk

(SAW) model in which excluded volume interactions of segments are rnodeled by

the infinite-range correlations between steps of a walk. These correlations are imple-

mented algorithmically as a self-avoidance check preventing a walker to visit the same

Iattice site twice during the walk'. In 1972 the generalization of the self-avoiding waik

model was suggested by Domb and Joyce [5]. This so called Domb-Joyce (DJ) model

allows the excluded volume effects to be varied. The DombJoyce lattice mode1 was

used in Our work and in this chapter it is described in greater detail. -- - --

'If the self-avoidance condition is to be violated in a Monte Carlo simulation, the lattice walk is

aborted or the algorithm returns to the previous configuration of the chain, depending on the type

of sampling used.

6.1 The mode1

The partition function of a polymer in solvent, defined by eq. (2.4), can be rewritten

into the following form

where P ( O ) is given by eqs. (2.7) and (2.8) and the integration goes over al1 possible

configurations of the chain. If we are looking for the simplest possible model in which

the repulsion forces can be varied, we just replace the exponential of eq. (6.1) by an

expression with the &-type effective pseudopotential as follows:

where X(fii) = 1 - exp(- w ( ~ ) ( $ ) / ~ ~ T ) and ,f3 is given by equation (2.29). Now, it

is straightforward to mi t e d o m a lattice version of the partition function z ( ~ ~ ) for

the DJ model as

where the parameter w = 1 - exp(-Woo/kBT) is the lattice analogue of the binary

cluster integral S . The lattice potential Wij is such that only the direct overlap of

segments labeled as i and j gives a nonzero interaction and al1 other relative positions

of the segments present zero contributions. Qi is the lattice position of the segment

i of the configuration Q and JAiAj is the Kronecker syrnbol with dAB = 1 for A 3 B

and zero otherwise. The sum goes over al1 possible lattice embeddings of a walk of

N steps and the product (representing the statistical weight of a configuration Q)

goes over al1 pairs of chah segments. As T increases (i.e., T -+ cm), w decreases to

zero and the effect of chain overlaps on its statistical weight diminishes. At w = O

the statistical weights of al1 configurations are equal. This is the random walk limit.

On the other hand, if T decreases (i.e., T -+ O) , w increases to one and the chains

with one or more overlaps have smaller statistical weights. At w = 1 their statisticd

weights are al1 equal to zero. This is the SAW limit. As we can see, the DJ model,

defined by the partition function of equation (6.3), smoothly interpolates between the

random and the self-avoiding walk models. There are two main advantages to using

the DJ model:

A number of numerical data (available from SAW simulations) can be readily

used for comparisons2 a t the SAW limit of the DJ mode1 (w = 1).

The detailed knowledge of random walk generating functions (see Appendix 3)

allows us to develop the perturbation theory expansion near the random walk

limit of the DJ model ( w = 0) and use these exact data as boundary conditions

for the numerical solution of the DJ model.

The partition function of eq. (6.3) can be recast into a form

where

and the constants

{QI count the number of configurations with k overlaps. Other physical properties such

as (R2) and (A2) given by eqs. (1.1), (1.2) and (1.3) can also be rewritten into the

DJ form. Explicitly,

?We include these tests at the end of the thesis in Chapter 8.

where

and the runs over al1 possible values of squares of the end-to-end vector. Similady,

the second vinal coefficient in the DJ model forrnalisrn is given by the formula

where

The factor 1/V from the definition of the second virial coefficient (see eq. (1.2)) was

eliminated by the fixing the origin of chain one. The symbol {Q, Q'} represents the

configurations of individual chains as well as their relative positions. If the two chains

do not overlap (Le., k' = 0) the contribution to the second virial coefficient is zero.

A2 can be rewritten (sirnilady to eq. (6.4) and (6.7)) as

where cF~ ccounts the nurnber of configurations in which the chains overlap at k

sites. In the small w limit A2 behaves as (AîqN(w)) = ~ W ( N + 1)2 which is the lattice

analogue of the two-parameter result AZVL = ~ w L ~ .

There are two limits in which exact results for the DJ model can be obtained.

One is the short chain limit in which exact enurnerations of short chains allow us

to cdculate &(w), (R&(w)) and (&N(W)) exactly for al1 values of parameter W.

The other limit is the Iimit of w + O in which perturbation series expansions in the

small parameter w can be evaluated. Outside these two regions approximate methods

have to be used. The standard method used for statistical estimates of averages of

physical quantities is the Monte Car10 method. In the following sections these three

approaches are briefly discussed.

6.2 Exact enurnerat ions

Since the complexity of exact enurnerations increases exponentially with N, the enu-

merations were performed (921 only up to N = 16 for CL: and oniy up to N = 8 for

C g coefficients. The typicai exact enurneration results for CN,k, cF~ and CS for

N = 4 are shown in Table 6.1. These data are later used as the boundary conditions

for the global numerical Ieast squares fit to the MC data. This will be discussed in

greater detail later.

Table 6.1: Exact counts for chain length of N = 4.

6.3 Perturbation expansions

In this section we will brïefiy explain how various properties3 are developed as power

series in smdl parameter W . The results for (R%(w)) and (A2,N(w)) will dso be

listed. Here we only briefly outline the basis of the method of generating functions;

for more detailed discussion see [89, 9014.

Instead of working with individual ZN values, one can define a new function

that simplifies the manipulation with the sums that appear in eq. (6.14). This is

the so called

recovered by

generating function. Once it is evaluated, coefficients ZN can be easily

a contour integral

The asyrnptotic expansions of G(x) for many different lattices are known; in our work

we need the results for a simple cubic lattice that has been given by Joyce [68]. The

leading N terms of the (RN) have been worked out before by Barrett and Domb [89]

and the two-parameter mode1 expansions were obtained (see eqs. (3.23) and (3.24)).

In the Iimit v + O the ZN(w) (see eq. (6.4)) can be rewritten as

where the counts in the square brackets, that are related to the terms wo, w 1 and

w2, are the total nurnber of configurations, the total number of overlaps and the total

number of pairs of overlaps, respectively. For easier evaluation of the sums such as - -- - - -

3We will use the partition function to outline the method.

4AAlso see Appendix B

Figure 6.1: Graphs in the first and second order of perturbation expansion.

those above (see eq. (6.14)), the generating function Gz(s, w ) for ZN(w) is defined

as follows

where G$)(z) is the i-th perturbation order of Gz(z, w). Let us briefly explain how

the sums in eq. (6.14) are evaluated. In the first order of w, the symbol bqiQj has a

contribution of 1 for al1 such configurations Q that i-th and j-th segments coincide

on the lattice, t herefore we can write

This can be represented by graph (a) in Figure 6.1 which consists of random walk

segment of total length nl and the return-to-origin segment of length nl. &, and

6n1 represent number of returns to origin and randorn walks, respectively. Functions

1/(1- x) and R(x) are the random walk and the return-to-origin generating functions

given by the formulas

P(4

R(4

where r, = &/6" represents the

ongin after m steps and c, = 1

probability of a random walker to reach the lattice

represents the probability of a random w d e r to

reach any point on the lattice. The second order of w, namely G ~ ) ( x ) , counts the

total number of pairs of overlaps. Here the ordering of contacts on the chain becomes

important. The second order contributions to Gz(x, w) are shown in Figure 6.1. The

individual terms of G ~ ) ( z ) correspond to graphs (c): (d), (e) and (f), reçpectively.

The individual orders of perturbation series of Gz(s, w) are

+ 3p2 (2) R~ (x) + p2 (2) S ( x ) (6.19)

Functions S ( x ) and S,(x) (see eq. (6.22) below) and other details are listed in A p

pendix B. Let us now concentrate on (R$(w)). In the limit w + O the numerator of

eq.(6.7) can be rewritten as

In order to evaluate the sums in eq. (6.20), one needs to keep track of where on the

lattice the particular lvdk ends. Similarly to eq. (6.15) one can define the generating

78

function for CR,N ( w ) as

where

G ~ ) ( X ) = X P ~ ( X )

G;' (X) = 2 x p 3 ( x ) R(X)

( x ) = 3 x 2 p 4 ( x ) R2 (x) + 2 x p 3 (x)

+ 3 x p 3 ( x ) R~ (x) + 3x P3 ( x ) S ( x ) + p 2 ( x ) Sq (x) (6.22)

Now the mean square end-to-end distance can be written using the generating

functions as follows

Similarly, the average second virial coefficient ( .42 ,H(~) ) can be written as

where the generating functions of various orders are

Let us now recall the definition of linear expansion factors c r ~ and c z l ~ given by equa-

tions (2 .41) and (2.42). In the DJ mode1 these will be defined as

and evaluated by adding the coefficients of xN separately in each order of w of equa-

tions (6.23) and (6 .24) , and then performing the Taylor expansion of the eqs. (6.26)

and (6.27) in W. One gets the perturbation series

in which the leading N-dependence leads to the two-parameter model. Eqs. (6.28)

and (6.29) were first derived by Barrett and Domb [89]. Two different approaches at

evaluating the coefficients in the perturbation series are possible. One approach is the

evalutation of k i ( N ) for short chains. For this evaluation one needs series expansions

such âs eq. (6.18) for al1 generating functions involved, i.e. R ( x ) , S ( x ) and S , (x ) .

Random walk generating functions have been investigated before [67] and the detailed

study of the properties of the simple cubic lattice generating function can be found

in [68] . This allows us to avoid approximations in the perturbation theory expansions.

Much of what we need in our calculations is listed in Appendix B. The evaluated

coefficients ki are listed in Table 6.2 for selected values of N . Another approach is

the evaluation of k i ( N ) in the long chain limit N + oo. To find the asymptotic

contributions to CN for each generating function R ( x ) , S ( x ) and S,(x) respectively

one has to transforrn them into analytic continuations about their singular points.

For " loose-packed" lattices such as the simple cubic lattice there are two dominant

singularities at the circle of convergence 1x1 = 1, namely x = f 1. Joyce used the

connection with Heun's differential equation and derived the analytic continuation

formula for R(x) about singular points [68]. If we perform the calculations we get the

following asymptotic forms for the coefficients k , ( N )

Table 6.2: Values of perturbation series coefficients of a&(w) and ( Y ~ , ~ ( W ) .

The coefficients of the leading orders of N are the well known coefficients of the two-

parameter expansion. The two-parameter excluded volume variable z is defined in

such a way that the factor ( 3 / 2 ~ ) ~ / ~ is absorbed into the z definition.

Next, we perform the transformation to the new effective exponent variables uR(+)

and UA($) exactly as described for the TPM in Section 5.2. First, we substitute the

asymptotic form of coefficients k[R) (N), kiR) (N) and k i A ) ( ~ ) from eqs. (6.30), (6.31)

and (6.32) into eqs. (6.28) and (6.29). Using the definition of effective exponents

(see eqs. (5.31) and (5.32)) we derive the series expansions in small excluded volume

parameter W. These are of the following form

There are no asymptotic terms of relative order 1 / 0 in coefficients of w in f iR (N, w)

and ûA ( N , w ) but there are terms of relative order l/n in w2 and higher orders of

W. These terms are eliminated by the transformation w -+ $, explanation of which

follows. After introducing the interpenetration function + ( N , w )

as a new variable5 and after inverting this expression with respect to w one c m get

U R and U A as functions of II, in the small II, limit6. These data for smail values of N

are listed in Table 6.3. The asymptotic behavior of the shape of the functions uR($)

and uA($) near the origin is

5by substituting k i (N) into eqs. (6.28) and (6.29)

6The first and second derivatives of VR($J) and first derivative of VA(@) at ~ = O can be evaluated.

Table 6.3: The derivatives of uR($) and uA($) for srnall iV values.

As can be seen, the asymptotic terms of the order 1 / n were eliminated from the

N-dependence of effective exponents by the use of the transformation of the form

U N ( $ ) = ûN(w($) ) . The transformation into the effective exponent variables itself is

not sufficient to eliminate the 1/f i dependence. The 1/N expansions of eqs. (6.36),

(6.37) and (6.38) are set up in such a way that the leading terms of order 1 and order

1/N are exact and the rest of the coefficients in the expansion are the optimal values

of the least squares fit. The least square fit is performed with respect to the exact

values listed in the Table 6.3 in such way that the values of N = 4, N = 6 and N = 8

are fitted exactly. The quality of this fit is graphically represented in Figure 6.2.

Figure 6.2: The residues of the fits compared to the exact data plotted for IV. In-

dividuai lines correspond to 10' x A ( ~ ~ $ ( ' ) ) (zero at N = 15), 105 x A(;~"(*))

(zero at N = 14) and 106 x A(~*,$*)) (zero at N = 20). Symbol A represents the

residue.

6.4 Monte Carlo method

For such values of u or N that neither the exact counts, nor the perturbation series

can be used one has to use approximate numerical methods of evaluating the ensemble

averages. The most widely used method is the Monte Carlo (MC) method. With the

MC method properties of chains of any length can be calculated but the answer

is only a statistical estirnate with finite error. In the MC method (for details see

Appendix C.2) ensemble averages are replaced by sample averages such as

where W I ( w ) = niCj(L - wbqiQj) is the weight of the configuration QI and the total

number of elements of the surn is much smaller than the size of the ensemble. This

is so called simple sampling where the configurations QI are chosen from the ensem-

ble at random. This approach is very inefficient because for most randomly chosen

configurations weight Wl(w) is very sma117 and this prevents the reliable evaluation

of averages from reasonably large samples. In the importance sampling, on the other

hand, one selects the configurations QI with a bias probability proportional to Wl(w).

The average mean square end-to-end distance is then evaluated by the formula

where each element I of the sample is drawn from the probability distribution W(w).

The evaluation of the R: is straightforward. For each configuration, the surn

of squares of the lattice positions of the N-th step8 is calculated using the formula

R c ( I ) = x$ + y; + z& and then averaged by eq. (6.40) where n is the total number

'This means that the real system would not spend significant amount of tirne in the region of

configuration space near Q 1.

8We place one end of the chah (labeled as O) to the origin of the lattice, Qo = (0,0,0).

of MC steps. For the second virial coefficient, however, the evaluation of the average

is little bit more involved. The definition of A2 is given by formula (6.9) the lattice

analogue of which is

The sum runs over al1 relative lattice positions FOo of origin of chain 2 with respect to

the fixed origin of chain 1. The total interaction energy U(FOo) represents the energy

between the two chains that depends on the configurations Q and Q' and also on the

relative position foo of the chains. Since for most of the relative positions the lattice

chains do not overlap at al1 (kl(Q, QI) = O), to improve the computational efficiency

Barrett suggested [87] a convenient way of calculating A2 using formula

where the sum runs over al1 pairs of chah segments i and j For each particular pair

i and j the algorithm overlaps the chains such that the lattice position of segment

i of chah 1 is identicai with the lattice position of segment j of chain 2, namely

Qi = Q; and the total number of overlaps k1 between c h a h 1 and 2 are counted.

This procedure requires the CPU time of order 0(N2). Since the evaluation of (A2)

is only approximate due to statistical estimates, we can afford to evaluate Ap only

approximately using the formula

Here i and j instead of running through al1 N + 1 segments of both chains as in

eq. (6.42), only m segmentsg of each chah are chosen a t random and from those

segments m2 pair overlaps of the chains are constructed to estimate A2.

9By choosing m much srnéder compared to fV + 1 we can significantly reduce the CPU required

to calcuiate the estimate of Az for a single MC step.

Figure 6.3: Residues of MC generated data compared to exact data for (Rf6(zu

t

As a check to whether our Monte Carlo algorithm is correct we compared the MC

data to the exact counts. This can be done for chah lengths of up to N = 16 (in case

of (Rc(w))) and for chahs up to N = 8 (in case of (A2,N(w))). In Figure 6.3 the plot

of residues of MC data compared to exact data for (R&(w)) is shown for al1 values

of W. The goodness of fit of the MC data set is 85.2%. In Figure 6.4 the residues of

( ~ 2 ~ (w))/( f w(N + 1)*) versus variable w are plotted. The goodness of fit of the MC

data set is 64.3% that suggests that the calculation of the averages is correct.

In the next chapter we will explain how the calculation of the universa1 quantities

was performed and also explain some details of the fitting procedure.

0.001 --

O. O005 U 0 O

N e 0.- I

I

4.001-- V

O 2 4 6 8 IO 12 14 16

counter for w

Figure 6.4: Residues of MC generated data compared to exact data for second virial

coefficient.

Chapter 7

Calculation of model properties

Scaling of global properties of a polymer with excluded volume interaction w in the

long chah limit is given by the renormalization group (see eq. (1.7)) as

where v is the critical exponent and Ai and A2 are corrections to scaling exponents.

Higher order corrections to scaling are of the form Those with

al1 ki = O are called analytic, othenvise they are called nonanalytic corrections to

scaling. The exact values of exponents Al and A2 are unknown, but previous studies

(e-g. [74, 581) suggest that Al = 0.5 and it is believed that A2 = 1. Much effort

has been spent to estimate the universal exponent v and the leading correction to

scaling exponent Ai with ever increasing precision. In this chapter we present details

of our work on the Domb-Joyce (DJ) Lattice rnodel [5] that we used to describe the

excluded volume problem of a polymer in solution and to extract universal properties

such as the leading critical exponent v and the correction to scaling exponent Ai

(referred to as A in the further discussion). Information on non-universal properties

such as the sc&g amplitudes a ~ ( w ) and aA(w) as well as the Ieading correction to

scding amplitudes b:)(w) and b!)(u) as also obtained. Monte Carlo simulations of

chains of different Lengths N ( N is the bond number) with varying excluded volume

parameter w were performed on a three-dimensiond simple cubic lattice using the

pivot algorithm [66]. The detaiIed asymptotic analysis of the DJ mode1 near zero

excluded volume (w z 0) was performed and the results were used in the fit of the

Monte Carlo data. Exact data available from direct enumerations of short chains for

al1 values of DJ parameter ur were also included in the final analysis.

7.1 Flow near the RG fixed point

In this work new variables, the effective exponents

were defined for both the second virial coefficient (Az) and the mean square end-to-

end distance (R2). They were plotted versus the interpenetration function given by

formula

for different chah lengths N.

Here are few reasons why we chose parameter -$ as an independent variable:

a In the long chah lirnit $ represents the dirnensionless amplitude ratio the

asymptotic value of which, according to the prediction of the renormdization

group theory, is a universal quantity. The value of $ reaches a k e d point of

the renormalization group for al1 vaIues of w in the interval O < w 5 1. The

plot therefore provides a convenient graphical representation of the crossover

behavior.

The function $ itself is not a universal function of the excluded volume param-

eter z as predicted by the two-parameter theory (TPT). This can be seen from

the flow pattern. For various values of w: $(w) may approach the fixed point

either from above or from below. By looking a t the plot, one can easily identify

which properties are universal and which are mode1 dependent.

The range of values of is finite compared to the infinite range of values of the

parameter t (the independent variable of the TPT).

According to Nickel's assumption, which he made to derive the one parameter

recursion mode1 [4], the interpenetration function Sis of eq. (1.10) is the only

relevant parameter of the excluded volume problem. This assumption works

well and was also used in this work except that we use QR.

a In the limit of $J + O the cornparison with the TPT can be easily made.

Let us look closer at the behavior of the scalinggiven by equations (7.1) and (7.2) near

the renormalization group fixed point ($', v*). By substituting the equations (7.1)

and (7.2) into (7.3) and (7.5) we get the following asymptotic behavior of UR and $

where

Similar equations can be written for VA. If the equation (7.7) is inverted and the

result is substitued into the equation (7.6), the following hnctional dependence near

the fixed point is obtained

where dl) = bt')(w)/bi)(w) is independent of w since the ratio of the correction to

scaling amplitudes is a universal quantity and higher order coefficients d2)(w) and

d3)(w) are dependent on w since they are obtained from $1 (tu) and b$)(w) as various

nonuniversal ratios. Therefore in the variables Av and A+ al1 the models belonging to

the same universality class approach the k e d point along the same line, in this work

called the leading correction-to-scaiing line, given by formula Au = c( ' ) A$. Higher

order corrections (either analytic or non-analytic) manifest thernselves as deviations

from this line near the h e d point. This allows us to find the deviations of the 0ow

lines from the leading correction-to-scaling line. By substituting the first order of

1/N from eq. (7.7) into eq. (7.10) it can be found that the deviations are on the

order of WA2. The leading corrections to scaling are thus eliminated by the use

of transformation equations (7.3), (7.4) and (7.5). This is analogous to elimination

of 1/a terms €rom uR(N, $) and vA(N, $) expressions (see eqs. (6.36), (6.37) and

(6.38)) in 2/i E O Iimit. This is the main advantage of this method.

Let us now look at this transformation from another perspective. In Figure 7.1 a

characteristic 0ow pattern is shown. For different values of the UJ parameter Au is

plotted versus A@. For any value of parameter w the flow is tomrds the fixed point

(represented by the dot in the center of the graph). Different corrections to scaling

for different values of w can be easily identified by various distances of flow lines from

the fixed point. Clearly, if one chooses the mode1 that corresponds to the w value

mith the largest leading correction to scaling (flow line furthest away from the fixed

point), the chah length required to reach the fixed point would be much greater than

that with small corrections to scaling. One faces the same problem when trying to

estimate the universal exponent v with a reasonable precision using the SAW mode1

only. The lengths of chains one needs to simulate are enormous (Li et al. [58] used

N = 80000) which makes the computations CPU-expensive. In Our method, on the

other hand, we use multiple values of the pararneter w and simultaneously analyze

data. This effectively eliminates the corrections to scaling and allows us to estimate

v , A and other universal quantities with greater precision.

Our method represents a new approach to the numerical solution of the excluded

volume problem. It also allows us to predict the values of ( A ~ & I ) ) and (R%(w))

for any chain length iV and any parameter W. The values of (RC(1)) and ( A 2 , ~ ( 1 ) )

predicted by Our method were compared to the results of self-avoiding walk (SAPV)

simulations available in the literature [58]. An excellent agreement was fourid for

both the mean square end-to-end distance and the second virial coefficient. Let us

now explain in greater detail how the universal quantities were calculated and also

make some general comments on how the fit of the effective exponents UR and UA was

performed.

7.2 Calculation of critical exponents and ot her uni-

versal quant it ies

7.2.1 Elementary analysis

By using the dynamic Monte Carlo (MC) method (for details see Chapter 6 and A p

pendix C.2) the estimates of (R2) and (A2) were obtained. The variances of the means

of these quantities, a2 ((R2)) and 02((A2)), as well as their covariances, a* ((R2), (A2)),

flow towards 6xed point

point

Figure 7.1: Typical renormalization group flow pattern.

were obtained based on the blocking' method [69] (for more details see Appendix D).

15 values of w were used in this work, namely w ={0.01, 0.03, 0.05, 0.07, 0.1, 0.15,

0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0). Chain lengths a t which MC simulations were

performed were N ={ 12, 16, 24, 32, 48, 64, 96, 128). Raw Monte Carlo data are

listed in Appendix F. By applying the effective exponent transformation to pairs of

(R2, Az) we obtained the data points v ~ , ~ ) and ($N, vRN) for every value of the

D J parameter w and for every chain length N . This data set is plotted in Figure 7.2.

Two sets of curves can be clearly seen. One set corresponds to the VA and the other

one to UR. The 8ow Iines, corresponding to the fixed values of w , are also depicted in

the figure. The particularly interesting one is the flow line near the left edge of the

graph (Le., the region of small $) which represents flow for the value of w = 0.01.

This flow line Erst approaches the random walk limit value of the critical exponent

V A = 2/3, but then (for longer chains) it is "repelled" from it and eventualiy reaches

the universal fixed point a t a value of UA x 0.588. I t was already mentioned in

Chapter 1 that there is an arnbiguity in the definition of N. It may represent either

the length of the chah or the number of monomers of the chain which is larger by

one than the chah length. In the limit of N + oo the definitions are equivalent,

but for a finite N, the quantities based on various definitions of 1V differ by ana-

lytic terms of the f o m N-'. Nevertheless the corresponding data sets are physically

equivalent. With this in mind we chose to transform the second virial coefficient using

IV 2 + (A2,N)(w) . By this transformation the data set becomes more compact.

'In this method variauces of "block dataJ7 coiiecting (i.e. averaging out) progressively larger

number of generated MC data are evaluated. In case of equilibrated çamples, the blocking method

is much more efficient compared to the traditional integrated autocorrelation time method because

a floating point operation on blocks of two MC values is required only in every other MC step and

the operations with blocks of larger sizes (Le. four, eight etc.) are progressively less frequent. It has

virtually no storage requirements and the analysis is straightforward.

Figure 7.2: Monte Carlo data of effective exponents VR and nu^ versus rl, (without

transformation). Set of curves with the cornmon value of 0.5 in the limit 11, + O is

the V R set.

This, hopehlly, can improve the fitting of the data2. Also in the random walk limit

d l the VA cuves originate from the random walk vdue of the exponent VA = 213. The

shape of UA curves becornes much like those of v ~ . The transformed data are plotted

in Figure 7.3. The effective exponents vR,&,bnr) and U ~ , ~ ( $ J ~ ) are plotted along with

their errors3. Each individual line originating at $ = O region corresponds to a dif-

ferent chain length N . Such lines are constructed simply by connecting the values of

the effective exponent of vN(w) for the same N. Every line vRTN(@) starts at a value

of 112 since this is the randorn walk limit of the critical exponent u and similarly al1

vAtN (q) lines originate at 213. One can also use another representation of the data

set, namely the plot of flow lines (see Figure 7.4). In this case vN(w) of the same w

are connected for different values of N. Irrespectively of the value of parameter w, all

flow lines approach the cornmon region in the (111, v ) space - the region of crîticality

- and eventually end up reaching the same point, i.e. the critical point (q*, v*). This

means that the polyrner chains with different excluded votume interactions share the

same critical exponerit v* in the long chain limit. This is the ba i s of the universality

hypothesis. The direction from which these flow Lines approach 11' is clearly not the

same for al1 the flow curves. For SAW mode1 (w = 1) and al1 models with w zz 1,

$N - @* is positive and for w rr O qN - $J* is negative. The value of w*, the excluded

volume strength at which the " two-parameter regime" changes to the " SAW regime"

is difficult to estimate at this point due to the fact that for chain lengths used in the

MC simulation the available flow lines stop far away from the critical point. Let us

now describe the procedure that we used to fit vN(+) and mention some problems

2This may not be the case since during the analysis we encountered the difficulties in fitting the

region of SAW. To further investigate the effect of transformation on the fit, one wodd have to

perform analogous fits for untransformed data set.

3The error bars are smaller than the width of the lines.

) v~ set

} VA set

Figure 7.3: Monte Car10 data of effective exponents v versus @ after the transforma-

tion was applied.

Figure 7.4: The flow lines constructed as connecting lines between successive Monte

Carlo data for v at the same value of W.

encountered dong the way.

Smooth N-curves (corresponding to a k e d chain length N) of Figure 7.3 suggest

that even some approxirnate graphical method could give us a reasonable estimate

of u*. If we merely connect the MC data points (as we did to plot the curves in

Figure 7.3) and zoom into the region of criticality (see Figure 7.5) quite a regular

pattern of U A and un lines can be observed. This pattern is almost perfect for UA lines

in a sense that the distance between an N-line and 2N-line is approximately half of

the distance between the (N/2)-Line and N-line. This suggests that the approach of

the v ~ , ~ ( $ J ) lines towards the limiting line UA (oo, .$) can be described reasonably well

by N-dependence of the form a + b / N . This is in agreement with the conclusion of

the previous section that the deviations of flow lines frorn the leading correction-to-

scaling line are of the order of N-l and/or of the order N-*z where A2 x 1. Indeed,

if we plot e.g. dLY(N) = ~ ~ ~ ~ ( O . 2 4 ) where X = R or A as a function of 1/N (see

Figure 7.6) we get alrnost a perfect straight Iine for d A ( N ) . The shape of d R ( N )

suggests that some higher powers of 1/1V are needed for a correct fitting in the 1/N

space. Finally, in Figure 7.7 the estimates of dopes of different N-curves at II, = 0.24

are shown.

Al1 the parameters needed to approxirnately reproduce the curves uR(oo, $1 and

uA (OO, $J) can be easily found and the asymptotic curves can be d r a i n in order to get

estimates of the critical exponent v*, the interpenetration function $*, the correction

to scaling exponent A and the correction to scaling amplitude ratio bA/bR. These

universal quantities are calculated Erom equations

Figure 7.5: Zoom into the region of criticality. Both N-curves as well as the flow lines

are plotted.

Figure 7.6: The plot of dx(N) = ~ ~ . ~ ( l t ' ) l * , ~ . ~ ~ where Ar = R or A.

Figure 7.7: The plot of s ( N ) = d@ ~ k 0 . 2 4 '

and the approximate results are: v* = 0.5876I0.0003, +* = O.232f 0.001, A x 0.51,

and bA/bR = -0.96. The errors are subjective errors; the enor of v* is about 5 times

larger than the largest MC error of u e f f .

7.2.2 Global linear fit

To obtain better and more precise estimates of Y*! the use of a rigorous procedure such

as a r n i n i r n ~ r n - ~ ~ fit is in order. For most of the data points UN) the error in qN

is much smaller compared to the error in VN. This allows us to make the assumption

that o(Qiv) = O which in tum linearizes the fitting procedure. Needless to Say that

by this assumption the fitting is greatly simplified and dlows us to quickly test the

adequacy of various fitting functions. After finding the optimal fitting parameters,

the limit N + oo is taken to obtain the asymptotic curves vR(cq @) and va (w, $).

This can formally be rvritten as

It has to be stressed that the curves v(oo, @) could be very sensitive to the fit

and consequently their behavior can be rather unpredictable. One reason for this

unpredictable behavior is that with many fitting parameters we get a good fit to the

data in the region where there are data points available whereas in the region with no

data points the function is unpredictable. Originally we used 10 values of w equally

spaced between 0.0 and 1.0. Later we found out that the spacing was not fine enough.

As we go to longer chah lengths, the flow (see Figure 7.4) "pulls" the variable dnr

closer to the fixed point thus leaving a large gap between zero and QN. In order to

get a finer spacing for data points in the region of srna11 + we perforrned more MC

simulations at w = 0.01, 0.03, 0.05, 0.07, 0.15 and added the data to the set. The

lack of MC data for longer chahs is another reason for large variability behavior of

v(m, $). In order to get a better estimate of the & ( N ) functions later we decided to

run more MC simulations to obtain two more data points a t N = 48 and N = 64.

Ideally, we would like to reprcsent each N-curve, sampled by 15 MC data points,

as a simple function of $. If we use a linear fitting procedure, the most probable (Le.

optimal) values of the fitting parameters are given by minimizing x2 ({c)) defined as

where

is the linear fitting function, n is the number of MC data points, m is the number

of functions and ci are the fitting coefficients. The errors of v are obtained by using

standard formulae for propagation of errors from variances cr2((R2)) and cr2((A2)).

Tt is also desirable to include as much information as possible about the DJ mode1

into the fit. The exact values of R2(w) for chain lengths up to N = 16 and A2(w)

values for up to N = 8 for any value of w can be obtained from direct enumerations.

By applying the effective exponent transformation, the exact curves LJ,~,~($) , vR,* ($) ,

LJRt6 ($) , and v ~ , ~ ($) can be obtained.

The question arises how to incorporate both the exact data and the MC data

into the fit. One could assign uniform error bars to exact v values and pretend they

were MC generated data. These artificial errors, however, would make it difficult

to interpret the X2 value especially in the global fit where a11 MC data are fitted

simultaneously. Another possibility is to force the fitting functions to "reproduce"

the exact functions v($). This "exact reproduction" has a reasonable meaning in

the context of a minimum deviation (MD) fit that can be defined by the following

relation

f M D ($) = min max 1 F (@) - zfxad ($1 1 { F } O<rlr<*rn,

In order to proceed in this way, the fitting coefficients have to be N-dependent. Let

us assume as an example that the coefficients & ( N ) have the following form4

where the functions l/NP represent the fitting functions in N-space. In order to

reproduce the MD fit the fitting function has to obey the relation

which translates into constraints for coefficients

To retain the desired asyrnptotic form of the l/N-fitting functions for large 1V as

welI as to satisfy the constraints imposed by MD fit for small N we rewrite the

equation (7.18) for the case of a single Nezact = 4 as

This relation defines the functions g j (N) and b ( N ) where gj are the terms that reduce

to zero for any NeZact, namely gj(Nesact) = O. The hi reproduce the MD values of

MD for al1 i. By the coefficients c Y D for any given N,,a,t, namely h,(NZad) = ci,Nelact

introducing N constraints directly into the fitting function, the use of the method

of Lagrange multipliers5 is avoided. Even with this reduction the total number of

parameters is around 50. - --

4For discussion on the expected fonn of N-dependence of the fitting coefficients see Section 7.1. 5Using method of Lagrange multipliers would only increase the number of parameters in the

problem.

Let us now describe how the actual fits were performed in a procedure we cal1 a

"global linear" fit. The exponents v~ and VA are treated as independent quantities

and each one is fitted separately using a function of the form

where { fi(x)} is a set of m, hinctions of variable6 s, g j ( N ) is set of mi functions of

N and hi(N) are previously defined functions of N . There is still more information

available on y($) that we can use. From the perturbation theory results near the

random walk limit (w = O) we know the values of dvA/&, duR/dx and dvi/dx2 a t

x = O for any short N as well as their asymptotic expansion for large N . Al1 this can

be included into the fit using the function FO defined as

mo

FO (x, N) = 1 c: ( N ) ff (x) i= 1

where { f:(x)) are functions of the form f ,"(x) h: 1, fi(x) zz x and fi(x) h: x2 near

x =V O and c:(N) define the exact asymptotic behavior of the derivatives of v a t zero,

Optimal parameters of the fit, {c i ) were obtained by rninimizing the following X2

function

where the summation goes over al1 Monte Carlo data V ~ J . We get separate X2-values

for V A and UR data sets. The results are shown in Tables 7.1 and 7.2. The first two

columns show number of x- and 1/N-functions used for the fit. The third column

presents powers of l / N used to generate gj (N) functions. Because of the constraints,

the total number of g j (N) functions is smaller than mg; for V A being mg - 1 and for v~

being mg - 3. The total number of fitting parameters and the degrees of freedom (#

=This is a @ variable that, for convenience, we chose to rescaie using transformation x =

so that $J* N corresponds to x* x 1 (see Appendix E).

d.f. = number of data - number of parameters) are listed in the fourth and the fifth

columns, respectively. The last two columns represent the x2 value and the goodness

of fit, respectively. The goodness of fit is defined as the probability that any random

set of data points would yield a larger value of x2 and is calculated using the formula

Table 7.1 shows that in the case of UR only 10 to 1 2 parameters are sufficient to fit

60 MC data with goodness of fit approximately at the level of 80%. However, in the

Table 7.1: Fits of UR data (total # of MC data 60 , N=12,16,24,32).

Table 7.2: Fits of UA data (total # of MC data 90 , N = 6,8,12,16,24,32).

g. of fit(%) X2 # d. f. # fit. p. mg powers of 1/N in g ( N )

case of UA where 70 parameter fit (for 90 MC data) gives only about 10% goodness of

fit, one can immediately recognize some problems with the Etting of UA data in the

form presented above. Problems arise due to the unexpected bend in the UA data near

the SAW lirnit as shown in Figure 7.8. This bend is so severe cornpared to the size

of error bars, that the obtained fit of UA for mf = 14 and m, = 6 as well as uA(oa, $)

are virtually useIess and cannot be used in Our final analysis. This is because so many

parameters of the fit make the fitting function unstable in the regions where there

are no data as was discussed before.

One possibility for overcoming this problem is to not include the MC data close

to the SAW region, namely w = 1.0 and w = 0.9, into the data set. This way a much

better fit is obtained as can be seen from Table 7.3. Now we can get reasonable fits

Table 7.3: Fits of UA data (total # of MC data 78 , N=6,8,12,16,24,32).

mf

10

10

11

11

of UA with as few as 44 parameters. By ignoring the SAW data we improve the fit

but we lose the possibility to predict long chain SAW quantities and compare them

to other SAW data from the literature. That would mean a waste of the SAW data

altogether.

After spending some time on this problem we realized that by adding the term

f (x)/(l + w;) into our fitting function we can improve our fits significantly. The

m,

3

4

4

5

powers of l / N in g ( N )

0117+

0,1,&2

O,l,$

0,1,!,2,3

# fit. p.

20

30

33

44

# d. f.

58

48

45

34

X*

184.84

50.42

47.25

28.68

Figure 7.8: Unusual behavior of the effective exponent variables u.4 near the SAW

limit.

fitting function (for va ody) is now

where the last term of F({c ) , x) is a function with the pole at x,, = - l / ~ + l .

This takes care of the "difficult-to-fit" behavior in the SAW region but the optimiza-

tion has to be done more carefully. We scanned the range of reasonable values of

kci to find an estimate of its optimal value and then used a nonlinear least squares

procedure to optimize i t even further. In Table 7.4 we compare the polynomial fit

(P) and the " polynomial with the pole" (PP) fit results. It is clear that adding one

more parameter k + l into the functional form (see eqn (7.26)) provides much better

fit than the one with the linear form of function (see eqn (7.16)) (compare column 3

and column 4 in Table 7.4). In Table 7.4 the values of optimized parameter k + l , the

calculated pole position and the maximum value of s (x,, = $ J ~ ~ ~ / & ) for different

N are also listed. In the process of fitting we have to make sure that xp > x,, for

any given N and also x&o) > x*. The fit would not be acceptable otherwise since

the pole position would fd l right into the middle of the data set.

7.2.3 Global nonlinear fit

In any of the fits mentioned above we did not consider the errors of the independent

variable $. We also ignored the correlation between (R2) and (A2) . Strictly speaking

that was not correct. In what follows the parameters obtained the way described

above are considered to be only the initial estimates of the "global nonlinear" fit.

From the definition of effective exponents (see eqs. (7.3) and (7.4)) the recurrence

formulas

Table 7.4: Improving X2 by adding one parameter. " P" represents the polynomial fit

and " PP" the " polynomial with the pole" fit.

can be derived.

If we know RN (w) and (w) and the effective exponents uR(N, q!~) and va ( N , +)

the estimates of G N ( w ) and &,2N(w) can be found. Including the correlations

02 ((R2), (A2 ) ) is straightforward so Ive simply define X* as

where

and a2(. . .) are the appropriate variances and covariances of logarithmic quantities

given by

In order to find optimal pararneters we have to solve the system of nonlinear equations.

This system of equations c m be Iinearized and solved iteratively to get a better

approximation to the exact solution. Linearized equations are of the form

where {do) ) are the estimates of fitting parameters at any given iteration step and

a! is the index numbering the equations. Our experience tells us that if we start

with the initial estimate obtained from the global linear fit, the rnethod converges

into minimum in about 10-15 iterations. In Tables 7.5 and 7.6 the results for the

nonlinear global fit are shown for ml = 8 and different choice of 1/N-fitting functions

for parameter h ( N ) . FOC a11 the other parameters c,(N) had the form

Ci (O) Ci

(2) ,-, (3) ci (4) c..(lV) = c i +-+-+-+- N N3/2 N5/2

where mg was equal to 5 (see Table 7.5) or G(N) was was expressed as

(2) ,-, q (O) ci (5) (0 ,-, (3) c,

q ( N ) = c , + -+- +-+-+- N N3/2 fV2 N5/2 N3

when mg was equal to 6 (see Table 7.6).

Total of 44 to 60 parameters were used to fit 210 MC data (120 values of UA and

90 values of uR). Later we found an error in this fit as well as in other fits listed in

Tables 7.5 and 7.6. This was caused by the bug in the program that fixed the value

of the second derivative of V R($) at $ = O to the half of the exact value. This

Table 7.5: Nonlinear fit of UR data (mJ = 5, mg = 5) and VA data (mJ = 8, mg = 5).

The last line represents the fit with the corrected constraints a t $J = O (see text).

m, for cg

6

5

4

3

6

5

4 - 3

4

minor bug did not propagate to the region of critical point so the values of universal

properties were not affected. The only difference was in the cornparison of our uL(z)

result to that of des Cloizeaux et al. for small values of z (see the next chapter). The

corrected fit is listed in the last line of Table 7.5. Its goodness of fit is 17.39% and

we can expect al1 other fits to be of similar quality. At this point the choice of the

fitting function is subjective, because as we can see, the goodness of fit of most of

the fits listed in Tables 7.5 and 7.6 is virtually the same. We chose the one with

17.39% goodness of fit (from Table 7.5) because it uses a fewer number of parameters

than any other fit. In this fit each fitting parameter c i (N) except cg (N) is given by

eqn (7.36) and the nonlinear parameter cg (N) is fitted to the form

powers of 1/N in g ( N )

(3,1,$2,53

o,l,$,$

o , l , t ,2

o,L$

o , ~ J , $ , ~

0,&1,&2

o,$l,!

O$

o,l ,$

# fit. p.

4 7

46

45

44

47

46

45

44

45

g. of fit(%)

5.73

6.37

6.81

0.30

5.93

6.18

6.06

5.83

17.39

# d. f.

163

164

165

166

163

164

165

166

165

x2

192.44

192.44

192.84

220.46

192.09

192.75

194.04

195.52

181.94

Table 7.6: Nonlinear fit of UR data (ml = 5, mg = 6) and VA data (mf = 8, mg = 6 ) .

mg for cg

6

5

4

3

6

5

4

The residues of InR$(w) and In A2,~(w) for this fit are shown in Figures 7.9 and 7.10.

In order to study the sensitivity of the optimal fit we compared it to three other fits

Mth mf = 8 and mg = 6; mf = 9 and mg = 5; rnj = 9 and mg = 6, respectively. Al1

of them use the same form (see eqn (7.38)) for k + l . The difference between Iimiting

estimates v(cq $) of mj = 8 and rnf = 9 fits is smaller than IO-' for the whole range

of values for mg = 5 and no more than For mg = 6. Since the error bars for

the long chains are on the order of 6 x 1 0 - ~ for UR and about 3 x IO-' for V A Our

choice mg = 5 gives the " enor" of fit I V ~ , ~ , = ~ ( O O , JI) - vA,mf=9 (m, Q) 1 smaller than

MC errors for large N. It is not so for m, = 6.

In the next two figures graphical representation of the final fit c m be seen. In

Figure 7.11 vR(N1$) and vA(N7 $) for several values of N are plotted and in Fig-

ure 7.12 the zoom into the region close to the fixed point is shown. Both figures show

also the uA(co, $1 and vR(oo,$) (the triangle near the left edge, see Figure 7.12).

These oo-curves give us values of v*, .Sr*, A and 6 A / b R . We can also reconstruct

powers of l / N in g ( N )

0 ~ l ~ $ ~ 2 ~ ~ ~ 3

O l l l f ,z1q

0717$2

0,h:

04,1,$,2,$

0,~1i7~72

01$717t

# fit. p.

60

59

58

57

6 O

59

58

# d. f.

150

151

152

153

150

151

152

x2

177.82

179.49

179.85

192.09

175.34

179.85

xp(co)<x*

g. of fit(%)

6.01

5.66

6.09

1.98

7.69

5 -45

r-4

Figure 7.9: Residues of In RL(w) for different values of W.

Figure 7.10: Residues of InA2,N(w) for difTerent values of W .

Figure 7.11: Fitted functiunal form of UR and u.4 for different values of N used in MC

simulation.

Figure 7.12: Zoom into region of criticality of v-fits.

the flow lines for any \due of the parameter W. These are shown in Figure 7.13 (a

global view) and Figure 7.14 (zoom into the vicinity of the fixed point). The line

in the Figure 7.14 crossing the x-axis a t about 0.996, making an angle of about 30

degrees with the limiting lines, and heading straight towards the fixed point is the

flow line corresponding to w* 2 0.3875 for which bA = bR = O. Figure 7.14 implies

that the Domb-Joyce model with w in the range 0.0 < w < 0.3875 exhibits a " tm-

parameter-Iike" behavior whereas the behavior cf model with 0.3875 < w 5 1.0 is

"SAW-Iike". In Table 7.7 we present the final estimates of the universal quantities

that we were able to determine with great precision from the fit of the data set. These

Table 7.7: Our results.

results will be discussed further in Chapter 8 and also compared to other results from

the literature. The first line of Table 7.7 represents our best fit obtained with the

correct constraints of the second derivative of vR(@). The rest of the Table 7.7 can

be considered a "sensitivity" test for various universal quantities. In the rest of this

chapter, instead, we d l focus on the determination of the nonuniversal properties of

the D J model, i.e. those that depend on the value of the DJ parameter W.

Figure 7.13: GIobal view of calculated flow. The recursion was carried to very high

values of N.

Figure 7.14: Zoom of the caiculated flow showiiig the region close to the critical point.

7.3 Calculation of non-universal quant it ies

One of the advantages of our method over other methods is an easy calculation

of model dependent scaling amplitudes aR(w), aa(w), bR(w) and bA(w). Once the

numerical solution of the DJ model has been obtained the evaluation of scding am-

plitudes is straightforward. Using the recurrence relation form of the solution we can

easily generate7 the values of R;(w) and AZVN(w) for any chain length N = 2 x 2" or

N = 3 x 2", starting from exact values of R$(w) and A 2 , ~ ( w ) for short c h a h (Le.

N = 4 or N = 6).

Scaling of any global observable X in the asymptotic limit is given by formula

where the w-dependent scaling amplitudes ax(w) and bx(w) can be found rather

easily using the recursion relations. We can rewrite eq. (7.39) into the form

and by generating XN(w) (where X represents either R2 or Aq) for increasing Ai we

obtain estimates of In ax(w). These approximate values of ln a , - (w) are approaching

a plateau (the true asymptotic value of ln ax(w)). When the corrections to scaling

terms N-A become srnaller than the numerical precision of a cornpute9 one can

assume that the value of lnax(w) is not contaminated by leading corrections to

scaling. The process of iteration has to be monitored, because after the plateau value

of ha&) has been reached the roundoff error sets in and the numerical stability is

destroyed. This process can be repeated for any valueg of u and thus the functions

'This is explained in greater detail in Chapter 8 where the cornparison between our prediction

of &SB,,, AZVNpSAW and MC resdts of Li et ai. [58] is done.

8For 30 digits of precision the chah length one needs to iterate to is approximately N = 2L87.

griot just for those w values that have been used for the MC data generation

aR(ut) and aA(w) can be obtained. The error of lnax can also be easily calculated.

The correlation rnatrix for the coefficients of the fit, namely u2(ci, c j ) obtained hom

a nonlinear optimization procedure is used to calculate the variance of lnax in the

Following way

The value of lnax(w) can later be used in a calculation of the leading correction to

scaling amplitude bx-(w) given by the formula

This formula, however, is not convenient for estimation of a2 (bx ) , because in that case

the rnatrix of derivatives LI2 ln ax/aciacj must be evaiuated. Instead, the equivalent

formula 1

b n ( 4 = (ln xN(w) - ln XNI2(w) - a ln(2)) i ~ " (7.43)

is used which is more convenient for error calculation. The formula for error cdcula-

tion corresponding to eq47.43) is

where

One can readily see that in this case there is no need for evaluation of derivatives

of Inax, which simplifies numerical calculations. In Table 7.8 we present the results

for ln aR(w), ln aA (w), bR(w) and bA(w) for selected values of w parameter along

with their statistical errors calculated in the described way. One c m notice that

bA/bR = cmst. as well as ln a~ - 3/2 ln a~ = const. because both quantities are

universal. The first one is the ratio of the correction to scaling amplitudes and the

second one is equal to the logarithm of the universal function 4(f )'12+*.

Table 7.8: Nonuniversal scaling amplitudes and their statistical errors.

Let us comment on a feature that we noticed in the course of calculation of nonuni-

versd amplitudes. From the dehition of the effective exponent (see eq. (5.30)) it is

clear that UN is analogous to the derivative of a function ln XN with respect to ln N.

The function XN is, however, defined only for discrete values of IV. The iteration

procedure that was used to generate X N values for N = 2 x 2* and N = 3 x 2" is

analogous to the process of integration

that can be used to find the function value y(xl) if y(xo) and the derivative yt(x)

over the entire interval are known. In our case, however, the underlying discretness

of v ~ ( @ ) as a function of N prevents us to get a unique answer for ln ax(w) since two

distinct sets of chain lengths N = 2 x 2" and N = 3 x 2" are used in recursion iteration.

Due to the discrete character of recursion relations there is a srnaIl difference between

the calculated values of ax(w) and bx(w) obtained from these two sets. Fortunately,

the difference is much smaller than the statistical error so that the estimates are

consistent. The non-universal scaling amplitudes are also presented in the graphical

form in Figure 7.15 where aR(w) and aA(w) are plotted versus w and aIso Figure 7.16

where bR(zu) and bA(w) are plotted versus W . From Figure 7.16 it can be determined

that the value of w where the functions bR(w) and bA(w) change signs is about

w* = 0.3875. The non-universal scaling amplitude b*(w) = bA(w) - 3/2bR(w)

(-0.9091 -3/2)bR(w) bas a negative value b@ = -2.4091 bR for any vahe of w srnalier

than w'. In this region of w values the DJ model asymptotically behaves as the two-

pararneter model since the interpenetration function .Sr approaches the asymptotic

value $* from belom. For a larger excluded volume parameter, namely w > w*, S,

approaches $' from above which is characteristic for the self-avoiding w d k model.

At w = wn the leading correction to scaling vanishes. This nonuniversal behavior of

long chain flexible polymer molecules in solvent follows naturally from the complete

numerical solution of the DJ model. In the next chapter the results obtained by our

rnethod both in the SAW limit and in the two-parameter Iimit will be compared to

other results available in the literature.

Figure 7.16: The plot of nonuniversal scaling amplitudes bR(w) and bA(w).

Chapter 8

Comparison wit h ot her studies

As we already mentioned before, the advantage of our numerical solution of the

Domb-Joyce model of the excluded volume problem is in the simultaneous use of the

whole range of models parametrized by the excluded volume parameter' w and thus

in its predictive power for al1 the models involved. In this chapter we focus on the

comparison of our results first to the results obtained from SAW model studies and

later to the TPM results.

8.1 Comparison with the SAW mode1

Let us compare our results to those predicted from SAW simulations of Li et al. [58].

In their work a very effective pivot aIgorithm was used allowing simulations of chains

of the total length of up to N = 80000 to be performed. To our knowledge, this is the

SAW simulation with longest chains reported to date and thus predictions based on

our method for very long N can be tested against their MC results. Another reason

for choosing their data for comparison is that they also calculated the second virial

lThe SAW model is the w = 1 limit and the two-parameter model (TPM) is the w + O b i t of

the D J model.

coefficient.

Let us first compare our estimates of the universal quaatities v*, A, @* and bA/bR

to those of Li et al. and other estimates available in the literature that were obtained

&om MC simulations of SAW-s on lattices. These results are presented in Table 8.1.

Our best estimate of u* is in agreement with that of Li et al. but the the estimates of

Li et al.

Rapaport

Madras

Table 8.1: Comparison to previous work.

range of N

80000

Eizenberg

this work

u* obtained by others that are bûsed on shorter chah simulations are systematicalIy

larger (see Table 8.1). This is caused, very likely, by the fact that when using a

single model to estimate u* the influence of the correction to scaling on value of v*

is great and one needs to go to much longer chains than those of Rapaport, Madras

and Eizenberg. Frorn Table 8.1 one can see that the shorter the simulated chah the

larger the estimated value of v. This suggests that the above reason for discrepancy is

correct. It is conceivable that the chain length of N = 80000 is sufficient for estimating

the leading universal quantities, the scaling exponent v and $*, using the SAW model

only. We know, that the parameters such as the correction to scaling exponent A and

the correction to scaling amplitude b are correlated a lot. In the fitting procedure

that uses a single model data (such as that of Li et al. ) it is therefore extremeIy

difficult to obtain a good estimate. The ratio b A / b R estimated by Li et al. is not in

130

2400

3000

u*

-5877 (6)

7168

128

.592 (2)

.592 (2)

A

.56( 3)

.5909 (3)

-58756 ( 5)

N

N

$*

.2322 ( 4)

N

.5295 ( 33)

b~ PR

-1.64(17)

,-Q

N

N

ru

P"

.23221 ( 11)

P4

-.go91 (233)

agreement with our result. It is very likely that this disagreement is caused by extreme

sensitivity of the fits when only data of SAW mode1 are available [58]. We believe

that our method where the corrections to scaling were eliminated by sirnultaneous

analysis of the data for many models and by a convenient choice of variables can be

considered to give a superior estimate of v* as well as of al1 other universal quantities

even using short chain lengths (i.e. only up to N = 128 employed in our approach).

Let us now turn to the non-universal quanitities such as the averages and

(A1,N)SAW. The values generated by our rnethod are compared to the data of Li et al.

Using an iteration procedure based on eqs. (5.35) and (5.36) the values of (R~(w))

and (A2,N(2~)) can be generated for any value of the excluded volume parameter w

and for any chain length N. The recurrence formulas given by eqs. (5.35), (5.36) and

(5.37) allow us to estimate quantities g N ( w ) and A ~ J ~ ( w ) if we know the values of

R$(w), A 2 , ~ (w) and also the functional form of both vR(N, g!~) and wA(N, $). Thus

if we start fiom exact SAW values R:(1) and A2,.,(1), by using eqs. (5.35) and (5.36),

we can generate both Ri(1) and A2,s(l) and continue on in this way to generate al1

values of R$(1) and AzVN(1) where N = 2 x 2". By starting from Rz(1) and &$(1)

we can generate al1 values of R%(I) and A2,N(1) where N = 3 x 2". The cornparison

of Li et al. to our data can be represented in a graphical way (see Figures 8.1 and 8.2)

where the plots of ratio of Li et al. data to Our data versus N are shown on a log-log

scale. MC values of R2 and A2 of Li et al. were, however, evaluated for such chah

lengths N that are not compatible with our values of N = 2 x 2" or N = 3 x 2".

This is not a problem since we can dways use a simple interpolation method to find

out what our prediction of RN(1) and A2,~(1) would be for N = NLi. Quadratic

interpolation was used to estirnate our values of both ln RL(1) and ln A 2 , ~ ( 1 ) as weil

as their approximate errors for N = NLZ. Figures 8.1 and 8.2 show the residues

of ln(R$)sAw and ln(Az,N)sAw, respectively. It is clear from the Figure 8.2 that

0.003

0.002 our MC data

N = 128

log, 1v

Figure 8.1: The cornparison of Li et al. (1995) bIn(R$) data to Our data.

Figure 8.2: The cornparison of Li et al. (1995) d ln(AzlN) data to our data.

the values of 1nAalN(l) for short c h a h are systematically lower than our predicted

values. We found the X 2 value to be about 70.12 for 35 degrees of freedom (the total

number of Li et al. data) which gives the goodness of fit at about 0.038% level. The

prediction for In R%(1) is almost perfect with 99.5% of goodness of fit.

To investigate the ln A 2 , ~ ( 1 ) prediction further, we decideci to do more simulations

for longer chains a t the SAW limit of the Domb-Joyce mode1 only. We performed extra

simulations of chains of lengths N= 192, 256, 384, 512, 768, 1024 and included these

data into the original data set. as optimized using the same fitting function with

the same number of parameters as described previously. The results are sumrnarized

in Table 8.2. The plot of residues of Li et al. data versus our predictions for ln A2,N(1)

g. of our fit(%)

# of S-4W data

"goodne~s~~ of prediction (%)

Table 8.2: Cornparison between results obtained from the "old" data set and fiom

the "SAW enhanced data set.

old data set

I l

for the enhanced data set is shown in Figure 8.3 and one can clearly see that by

including the SAW data for longer chains we get excelient overall prediction. The

"SAW enhanced" data set

17

short c h a h A2,N data of Li et al. are slightly lower t han our MC data. This can be

due to a statistical fluctuation. In order to be able to comment more on this matter

and/or properly compare Our predictions to MC data sets of Li e t ai., we would have

to know what the covariances cr2((R2): (A2) ) of the corresponding MC data were2.

The effect of "SAW enhanced" data set on the universal quantities u*, Q*, A and

bA/bR is a h shown in Table 8.2, We can see that both us and A obtained kom the

enhanced data set are within O of their old values and the universal amplitude ratios

did not change by more than 20 , Le. there is no significant change.

8.2 Cornparison with the two-parameter theory

The two-parameter mode1 limit of the recurrence relations (5.35) and (5.36) is ob-

tained by taking the small excluded volume limit (w + O) dong with the long chain

h i t (N + oo) and keeping the value of the product z - W N ' I ~ k e d . Our recursion

relations can be rewritten into the form

where T/J has to be determined repeatedly a t each iteration step using the fomuia

In the small z region of the two-parameter limit the predictions obtained from our re-

cursion relations (8. l), (8.2) and (8.3) are identical with those of TPM series. This is

because the numerical solution of the DJ mode1 in the small ui limit was constructed

based on the perturbation series that were shown to be identical to the TPM. There-

*The covariances were not induded in Li's et al. publication.

I our MC data

N = 1024

Figure 8.3: The comparison of Li et al. (1995) ~ 5 l n ( A ~ , ~ ) data to our own data after

including the SAW data of N = 192, 256, 384, 512, 768, 1024.

fore the agreement with al1 the two-parameter theory results in the region of a small

z is to be expected.

Let us first present the results of al1 the calculations based on the two-parameter

series expansions and the summation techniques (such as those of Muthukumar and

Nickel [74] and des Cloizeaux, Conte and Jannink [75]) as well as the results based on

eexpansions (Le Guillou and Zinn-Justin [84]) and n = O component field theory (Le

Guillou and Zinn-Justin [48]). Our result for the universal critical exponent u* agrees

very well with al1 the other results based on various calculation techniques presented

in the literature (see TabIe 8.3). Our value of the correction to scaling exponent 4

is larger than the rest of the values in Table 8.3. As we mentioned before, the SAW

1 this work 1 33756 ( 5) 1 -5295 ( 33) 1 - - - -- - --

Table 8.3: Comparison of Y* and A d u e s obtained by our method to those obtained

by different methods.

model is the w += 1 limit of the DJ model, whereas the TPM is its w + O limit. Both

models are therefore compIementary and thus the estimates of the universal quantities

are very likely to be biased. In the SAW model the bias is caused by restricting MC

simulations to chains of a finite length N and a subsequent estimation of the values of

Y* and A from those data. In the TPM, on the other hand, the summation techniques

applied to the finite series are also only of a limited validity. Estimates calculated

137

using TPM series are very likely to fail to reproduce the true values of ag(z) and

d ( z ) near the h e d point of the renormalization group and the values of v* and A

may be biased. Looking a t the values in Tables 8.1 and 8.3 one can note that while

the leading exponent is virtually the same for both the SAW model and the TPM,

the values of the correction to scding exponent obtained hom these two methods are

significantly different. It is interesting to note that our value of A is almost halfway

between the estimates from the SAW model and the TPM. It seems plausible, that

our estimated value of 4 is closer to the true value than any of the previous estimates.

Let us now compare graphically some TPM results mentioned before with our re-

sults obtained using the recursion relations (8.1), (8.2) and (8.3) in the two-parameter

limit. In Figure 8.4 the crossover behavior of the linear expansion factor a i ( z ) is plot-

ted versus the excluded volume variable 2. From the top to the bottom of the figure,

the results for &(z) are plotted in the following order

ln a&&) = In (0.572 + 0.428(1 + 6.232) Il2) ln ai,,, ( z ) = 0.1772 In (1 + 7.5242 + 11.062~)

2 ln ( Y ~ , ~ ~ ~ ~ ~ ~ ~ (z) = 0.17512 ln (1 f 7.6142 + 1 2 . 0 4 ~ ~ )

and the " modified" Flory formula

The subscript DB indicates the results of Domb and Barrett [88], YT those of Ya-

makawa [18], MN those of Muthukumar and Nickel [74]. The last equation for a;(z)

was obtained in this work3. As we can see, both curves obtained by Muthukurnar

3The hinction of eq. (8.7) is the expression simiiar in form to that of eq. (8.6). This hinction

approximates our exact recursion results very well both in z -+ O and t -+ oo limits.

Figure 8.4: The results obtained from various two-parameter theories (see text for

details) .

and Nickel and by us are, on the scale plotted in Figure 8.4, virtually identical and

thus more thorough cornparison is needed. This type of cornparison is presented in

Figure 8.5 where the result of des Cloizeaux et al. [75], the most precise TPM

result available in the literature, is also included. In this figure the residues of

2 lna%(z) - lnaR,,,si,(z) for MN 1741, dCCJ [75] and formula (8.7) are plotted

versus log., z. From analysis of result of des Cloiseaux et al. we know [92] (for qual-

itative picture, see e.g. FIG. 9 of [74]), that vanous constraints4 imposed on their

formula

do not affect the results for values of r smaller than 1. For larger values of z (Le.

log2(z) > O), however, the constraints can change des Cloizeaux answer significantly.

In this context one c m look at the Figure 8.5 as follows; for values of z < 1 it can

be considered a check of our work5 whereas, for values of z > 2 , this is a check of

des Cloizeaux et al. formula (8.9). In the region of z 5 1 the agreement is very

good, namely less that 0.02%. In the critical region our method of Monte Car10

renormalization group is more reIiable than any method of series analysis that is

using only a finite number of series terms. Clearly, for z 2 1, neither MN result nor

the des Cloizeaux TPM results are compatible6 with our recursion data as can be

4e.g. fixed d u e of u and/or A 5 0 ~ recursion results are represented by the base line of the graph. Our approximate formula

of MN type (see eq. 8.7) is the function with the smallest absolute value.

=The estimated critical exponents v of both methods are too high.

seen from the error bar7 for our method shown in the right side of the Figure 8.5.

The des Cloizeau result start to differ more significantly from our recursion values

of a i ( z ) at about log,(z) x O which is in agreement with the discussion above. Our

approximate formula (8.7) is even better than our exact recursion data in the region

of smdl values of z as can be seen from its series expansion

since it starts to differ from the exact series expansion of eq. (3.23) signihcantly only

with the coefficient of z5 term,

Another possible way of comparing our data with those in Literature is to evaluate

the constants in the TPM asymptotic formula

where the constants a, and b, are given by

In Figures 8.6 and 8.7 constants a,(w) and b,(w) are plotted for various values of W.

In the TPM limit ive get a, = Iimw,o a,(w) = 1.546(1) and bz = limw,o b,(w) =

0.122(2). This allows us to compare the TPM results of our analysis to other results

in the Literature. These asymptotic (z + oo) formulas for linear expansion factor c r ~

are

2 a i ( z ) = 1.546 (1 + 0.122 z-1.062 + . 9) (8.16)

?This error bar is the 1 u uncertainty in the t = oo amplitude l n a ~ ( 0 ) (see Table 7.8. The

uncertainty at finite z will be srnailer.

Figure 8.5: The relative error of TPM results of des Cloizeaux, Conte and Jannink

(1985) and Muthukurnar and Nickel (1987) compared to our exact recursion results

plotted versus log,(z). At log, z = O the functions from top axe dCCJ, our approxi-

mate formula (8.7) and MN, respectively. The function with less than 0.1% relative

error in the region of intermediate values of z corresponds to our approximate formula

given by eq. (8.7).

Figure 8.6: Plot of a,(zu) versus W.

Figure 8.7: Plot of b,(w) versus. W.

For better understanding of the qualitative relationship between the TPM and

our renormalization group method we inchde Table 8.4. In this table we present

the two-parameter mode1 data a i ( z ) and cri(z) for various discrete values of log,(z)

generated using Our recurrence formulas (8.l), (8.2) and (8.3). For every log2 (2) also

the values of the effective exponents UR, UA and the interpenetration function .Si are

also included.

In Figure 8.8 the linear expansion factor for the second vinal coefficient a i ( z ) is

compared to the results of others (see eq. (3.9)). From the bottom of the Figure 8.8

to the top, the functions

are the results of a differential equation approach of Yamakawa [17] and the semiem-

pirical procedure of Orofino and Flory [91] based on the smoothed density theory.

The uppermost curve is our function obtained by recursion equations (8.1) and (8.2).

So far there were no high-precision results similar to 1741 or [75] for a i ( z ) . The rea-

son is that the derivation of the two-parameter perturbation series for a>(z) is much

more complex task compared to the derivation of a%(z) and therefore the resu1ts are

known [74] only up to the second order in r . Similar to eq. (8.11) Ive can obtain an

asyrnptotic expression for second virial expansion factor aA(z) of the form

where the estimated values of constants in the limit of w + O are aiAl = 0.446(15)

and by) = -0.110(5). The resulting expression therefore is

this work

Figure 8.8: The results of various two-parameter theories (see text for details). Plot

of aA(z) versus z.

Based on the cornparison of ari(z) to other high precision results derived from the

series (3.23) and (3 .24) , we believe that the result for ai ( z ) obtained by our method

is very likely of a comparable accuracy and is the only high-precision result available.

Table 8.4: Relationship between the TPM values of linear expansion factors (r;(a),

cui(z) and the effective exponents VR, UA for various values of log,(r).

Chapter 9

Conclusions

In this work universal critical exponents v and A of the excIuded volume problem of a

polymer c h a h in solution have been calculated using a novel approach. The method

that has been used allowed us to calculate the scaling exponent v, the correction to

scaling exponent .A and other properties with a much better precision than that of

any other method reported up to date.

In Our approach we used the Domb-Joyce (DJ) mode1 to describe a linear flexible

polymer chain. In the MC simulation of the ensemble of DJ chains, the new chain

configuration was generated from the old one by the pivot algorithm. The Metropolis

sampling was applied to calculate the global averages based on the DJ weight factor.

The DJ model was chosen to represent a polymer chain since by varying of the

parameter w within the interval O < w 5 1 one c m get a continuous range of rnodeIs

that al1 belong to the same universality class and thus share the sarne universal

properties.

The standard numerical approach to the excluded volume problem is based on the

SAW lattice model. In the SAW model the presence of the corrections to scaling poses

many problems when the values of critical exponents are to be extracted from the

finite chah data. The use of MC data generated from different models (i.e. w is varied

between O and 1) and the use of the effective exponent transformation allowed us to

eliminate the corrections to scaling effects and thus evaluate exponents v and A with

a very good precision. Our estimates of the universal quantities are v = 0.58756(5),

Q* = 0.2322(1), A = 0.530(3) and bA/bR = -0.92(1). The errors in these universal

quantities were significantIy smaller than errors O bt ained by ot her methods described

in the literature. For the leading exponent v the error was up to 10 times smaller

compared to the error of the next most precise numericd estimate of u [58]. This is

a very good result in its own right, not to mention that the total CPU time used was

only about 136 days of a Silicon Graphics " Challenge -XL" computer' time compared

to effectively much longer time required by other methods (e.g. [58]). Other universal

quantities such as +* and bA/bR were also calculated with a very good precision. The

difference in our estimate of bA/bR and that of Li et al. can be attributed to high

correlation between the exponent A and the correction to scaling amplitude b when

fitting of a single model data is used.

The numerical solution of the DJ model presented in this work is a nonlinear fit of

MC data for Rc(w) and A2,~(w) . TWO constraints were built into the form of fitting

function. The first constraint represents the limit of short N in which the exact count

data were used to fix the fitting function using the minimum deviation fit. The second

constraint is the small Q limit constraint obtained from the perturbation expansion

in smdl W. The detailed knowledge of the randorn walk generating functions allowed

us to include al1 l / N corrections exactly in the limit of w + O. We were also able to

show that by using the effective exponent transformation, in the limit of small w, the

terms 1/m disappear. Similarly, the correction to scaling terms l/NA are expected

'with 150 MHz "MIF'S" R4400 Processor CPU-s

to disappear for any value of w as a result of the effective exponent transformation.

The numerical solution of the DJ model, presented in the form of the recursion

relations, allawed us to generate d l values of R$(w) and A 2 , N ( ~ ) for chains of length

N = 2 x 2" or N = 3 x 2" and for any value of DJ parameter W . This was used

both to evaluate the nonuniversal scaling amplitudes aR(w), aA(w), bR(w) and bA(w)

and to compare our results to other results available in the literature in two speciai

limits. One iimit is the SAW limit where the generated data R$(1) and A2,N(1) were

compared to the MC data of Li et al. [58] and were found to be in a very good

agreement.

In the two-parameter limit we compared our linear expansion factor a$(%) to the

TPM results of des Cloizeaux et al. [75] and also found a very good agreement for

2 5 1. However, due to a different asymptotic form of des Cloizeaux et al. and our

results, for iarger vdues of z, the agreement becomes worse. From Our numerical

solution of the DJ model the two-parameter solution was obtained in the limit w + O

and N + oo while z oc wlVL/* was fi~ed. This is the only numerical solution of the

TPM available in the literature. The direct approach to the numerical solution of the

TPM, where the simulation of models with small w is used for very long chains, is

time consuming. The large values of the correction to scaling amplitudes bR(w) and

bA(w) in the limit w + O do not a1low us to get near the fixed point fast and the

results are valid only over a very short interval of z values. The same drawback is

inherent to any finite series analysis (e.g. the work of des Cloizeax et al.). Our result

for a;(z), on the other hand, is exact (within the statistical and numerical precision)

for al1 values of the excluded volume variable z. Moreover, in our work, the result for

the linear expansion factor a i ( z ) is obtained "for free". This result is essentially new

since the high precision two-parameter series analysis of des Cloizeaux et al. type for

second virial coefficient is non-existent due to the complexity of series evaluation.

In order to compare the theoretical predictions to the experiment, MC data of

the radius of gyration instead of R2 data must be generated. In this work, however,

we concentrated on developing the method and comparing the results to the most

accurate theoretical results available in the literature. In this context the presented

method proves itself to be a new and powerful approach to the numerical solution

of the excluded volume problem and is likely to be of importance in other areas of

criticd phenornena.

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Appendix A

History of polymer science

When in 1833 Berzelius coined the term 'polymerization', the meaning of the word he

used was much the same as it is today. Since then it took one hundred years for sci-

entists to understand the structure of polymers. Nowadays it is a common knowledge

that polymers are held together by covalent forces. This so called macromolecular

hypothesis, however, did not get much attention until 1930's when Staudinger's pri-

mary valence viewpoint of polymer structure was accepted. Nevertheless, polymers

were manufactured long before there was a clear understanding of their molecular

structure or the origin of forces responsible for this structure. Goodyear's discovery

of vulcanization in 1839 and Hyatt's patented use of camphor with cellulose nitrate

in 1869 are but two initiating factors in the rise of industrial scale production of

polymer materials. Hyatt's invention, for exapmle, allowed to produce hard objects

with smooth surfaces. What else could be more practical use of his invention than

the billiard ball? This not only improved the quality of the game of billiard but more

importantly saved lives of thousands of elephants that would have been otherwise

slaugthered to cover the ever increasing demand for ivory.

Even long after scientists started to study polymers they believed that p01~vrners

were aggregates of small molecules held together by "secondary valence" forces. This

so called association hypothesis was favored over macromolecular hypothesis for many

years. There were many reasons why macromolecular concept did not get wider

acceptance earlier. Here are some of the reasons:

In 1861 Graham measured extremely small rates of diffusion of some substances in

solution and accordingly called them colloids to ernphasize their glueiike character.

This classification put a lot of bias into the reasearch in years to corne so that even if

experiments pointed out that rubber or cellulose were large molecules, scientists were

rehctant to accept it.

Raoult's discovery of more precise cryoscopic method for determining molecular

weights of substances in 1885 made the old diffusion rate measurements obsolete.

Still, many years later, the new method was not trusted because it waç assumed not

to be applicable to materials in a colloidal state. In 1914 Casperi entirely rejected his

osmotic pressure measuremens on dilute rubber solutions that gave him estimate of

about 10000 for molecular weight simply because he could not believe it.

a Another reason for not considering macromolecular hypothesis was the popularity

of the concept of the secondary association of molecules that prevailed at the end

of the nineteenth century. The coordination complexes and van der Waals forces

playing an important role in the association hypothesis attracted wide attention after

Ramsay and Shields provided a method to detect molecular association by using the

temperature coefficient of the molar surface energy.

The terrn polymerization applied to the coordination complexes has lost its original

meaning given by Berzelius and this confusion in terminology also played its part in

neglecting macromolecular hypothesis.

Gradually experirnental evidence pointed out that the macromolecular hypothesis

was correct and the association hypothesis was badly shaken by the following findings:

In 1922 Staudinger and Fritschi showed that after hydrogenation rubber retains

its polyrneric (colloidal) character. The secondary valence forces were thought to

require the presence of double bonds that would be destroyed by hydrogenation and

the substance would not be colloidai anymore.

a The 1925 work of Mayer and Mark on X-ray diffraction of crystalline cellulose was

best explained by using a concept of long chain molecules.

The sharp molecular weight distributions were difficult to explain on the associa.tion

basis.

a Carothers £rom the DuPont Company was interested in preparing polymeric molecules

of various structures using reactions of organic chemistry. In 1929 he succeeded by

carrying out a straightforward polycondensation reaction that could only lead to

macromolecules. This was the final blow to already dying association hypothesis.

After macromolecular hypothesis was accepted things started to move relatively

quickly. The theory of viscosity of polymer solutions advanced by Kuhn, Mark and

Guth in the early 1930's and later the statistical mechanical theory of rubber elas-

ticity presented by the same group were the first two successfull theories based on

the macromolecular hypothesis. In 1942 Flory and Huggins presented the thermo-

dynamic theory of polyrner solutions. Flory also introduced the theory of excluded

volume effect of polymer in solution along with the concept of O-solvent. Gradually

cornputers became available to the general scientific community and in 1954 Wall,

Hiller, Wheeler and Atchison started publishing a series of papers on statistical com-

putation of mean dimensions of macromolecules that paved the way for completely

new approach in polymer physics. More recently an important connection between

polymer physics and critical phenomena was made in 1972 by de Gennes. He formu-

lated the mathematical equivalence between n = O of the n-vector mode1 of critical

phenomena and the self-avoiding walk on a regular lattice. Despite the attention of

the best scientists in the field, many problems of poIymer science are not yet com-

pletely solved. One of them is the excluded volume problem of a polymer molecule

in a solution.

Appendix B

Generat ing funct ions

Let us consider the random walk (RW) on a simple subic lattice (d = 3). In the case

of Markovian RW where the next step is not influenced by any of the previous steps,

the probability P,(q that the walker is a t site Safter n stepsl satisfies the following

equation

where pj is the probability of stepping to the site j and Z, is the vector pointing

towards that site. The surn runs over the nearest neighbor sites.

Also xj pj = 1. The characteristic function R(k) is defined as a Fourier transform

of of P,(q by the formula

If one substitutes eq. (BA) into eq. (B.2) one gets the result

By definition &(k) = 1 (çince Po(s') = ~5~,,), so we c m write

'assuming it started at the origin s'= (0,0,0)

164

where 1 ~ ( k ) = - (COS ki + cos k2 + COS ka) 3

is the characteristic function for the single step walk on the simple cubic lattice.

Probability P,(q can be obtained as an inverse Fourier transform by the following

formula

AU the information about random walk of certain type on the regular lattice can be

conveniently collected into a compact form called the generating functzon

Taking eqs. (BA) and (B.6) into account one can write U(S, x) int othe form

The detailed knowledge of the behavior of hnction U(S, z) and especially its S = 6

where

P(k) = A - I

1 - %(cos k, + cos k2 + COS k3) called the return tu origin generatzng function is crucial in nurnerous physics applica-

tions such as spin-wave theory, spherical mode1 of ferromagnetism and the theory of

random walks [68]. A power series representation for R ( x ) can be obtained directly

expanding the integral in eq. (B.9) so that one gets2

00

R(z) = Q . ~ 2 n = f (ri / / [:(COS k, + cos k2 + cos h)] ln d3$) x2. n=l n=i 2 ~ ) -a

(B.11)

and the explicit expression derived by Joyce [68] in terms of terminating generalized

hypergeometric series is

where r(t) is the Gamma function and F is the hypergeometric function. By definition

eq. (B.7) coefficient rpn is the probability p2,(6) that the walker will return to its point

of origin after 271 steps3. The analytic continuation formula

derived by Joyce [68] is of considerable importance in the theory of simpe cubic lattice

random walks and in this work it was used to derive the asyrnptotic expansions of

R$ and A2,N in the limit w + O. Other important generating functions are

and

(B. 14)

*Expansion of eq. (B.ll) is directly related to the generating function F(x) given by the formula

R(x) = 1 + F(x)R(x ) . Coefficients of expansion F(x) = Cr=O fnxn represent the probabiiity

that the waker returns to the origin for the first tirne. The so c d e d escape probabiiity, given by

P,,,,, = 1 - F ( l ) = l /R( l ) , gives the probabiIity that the walker never returns to the origin. For

1D and 2D Iattices P..,,,. = O (the walker returns to the origin after certain number of steps).

3return to the origin sooner during the walk is not prohibited

Appendix C

Monte Carlo Method

Monte Carlo (MC) is a method of numerical analysis widely used in many areas

of physics. It was named after the famous casino in Monaco to reflect its random

character. MC is based on using (pseudo-)random numbers (see Section C.2). A

sequence of random numbers is generated by a computer and used to " wdk" through

the configuration space of the system. The statistical accuracy of MC is therefore

closely related to the CPU performance of a computer.

C.1 Monte Carlo rnethod

Ideas on which MC is based were known even before 1900's. Lord Kelvin, for example,

used random sampling, which consisted of drawing marked pieces of paper frorn a hat

in order to evaIuate some integrals in the kinetic theory of gases. MC methods,

however, became popular only after the first computers started to emerge in the early

1950's. In statistical mechanics the average of a physical quantity F is given by the

multiple integral

where the integration runs over the whole configuration space R and p(X) is the

probability of finding the system at the configuration X. The numerical evaluation of

multi-dimensional integrals is therefore necessary. For simplicity let us consider the

one-dimensional integral r 1

Numericd quadrature methods of estimating I often use equally spaced values of x k

to approximate I by 1 n

One can also view In as an average of f (x) over the intemal [O, 11 and choose xk in

eq. (C.3) randornly with uniforrn probability distribution over the interval. This is

the essence of a simple MC sampling of I . The uncertainty associated with estimation

where

is an unbiased estimator of the variance o f f . The fi decrease of or (see eq. (C.4)) is

typical of MC rnethod. The error of conventional quadrature method in d-dimensional

space is on the order of ~ ( n - ~ l ~ ) where rn is a small integer, and therefore for higher-

dimensional integrals the MC method is always superior to the quadrature method.

However, MC would still be useless for most problems of interest in physics if it was

not for variance-reduction techniques.

In order to illustrate variance reduction, let us rewrite the integral 1 into the form

This can be done provided we can find the inverse x(y) of a function y (x) = 1; w ( x ' ) h r .

The integral I can be approximated by

If the function w(x) behaves approximately z f (x) then ft(x) = f (x)/w(x) is a

smoother function than f (x) and < a/ which in turn reduces al. According to

eq. (C.7) the estimator of (F) (see eq. (C.l)) can be written as

Generally, in a multi-dimensional case it is impossible to find a function Y(X) with

Jacobian lûY (X)/ûXl = w (X) such that the inverse function X(Y) can be found.

Instead, one can think of X (Yk) simply as of a set of points Xk distributed wit h w (X)

distribution.

In statistical mechanics p(X) is the Boltzmann probability density function given

by formula p(X) oc exp(H(X)/kBT) where H ( X ) is the Harniltonian of the system.

The range of possible values of this function spans several orders of magnitude and

therefore it is convenient to choose w(X) = p ( X ) so that

Imagine that the evaluation of (F) would be based on the simpIe sampling such as

that used in eq. (C.3) in which Xk are chosen a t random. Most of the time the weight

factor exp(-H(X)/kBT) would be so small that the total contribution to (F) would

be negligible. Thus in the simple sampling technique the computer time is wasted

on regions of R of very little importance. Choosing w(X) = p(X) represents a more

efficient way of ( F ) estimation because Xk of eq. (C.9) are preferably from regions of

R where p ( X ) is large. This is why the method is called the "importance sampling"

method. However, in most cases of practical interest, generation of Xk directly from

p(X) is impossible. Fortunately, there is another method that can be used to generate

configuration points with the desired probability, called Metropolis sampling.

C.2 Metropolis sarnpling

This method uses a Markov chah defined by the transition probability W(Xi + Xf)

where Xi is the initial state and Xf the final state of the transition. If the probability

distribution is p ( X ) at any time t then its rate of change is

-- d p ( X ) - C [-p(X) W ( X -t XI) + p(X1) W(XI -, X)] (C. 10) dt {XfI

By choosing the proper form of the transition probability W a Markov process con-

verging to the desired equilibrium probabiiity peq(X) can be constructed. The detailed

balance condition

imposed on W is a sufficient condition for Markov process to reach the equilibrium

probability distribution peq(X) provided W is ergodic. Equation (C.11) shows that

on average there is an equal number of " jumps" from X to Xf to those from XI to X.

In the case of polyrner chain simulations where the statistical weight is (1 - w)#werLaPS

the Metropolis sampling can be algorithmically implemented as follows

c--- metropolis test

i f ( l p t . l e . l p ) go t o 20

cal1 rng(0)

if(yr.ge.(i-w)**(lpt-lp)) go to 28

20 continue

C--- update the chain

28 continue

return

where I p and lpt are number of overlaps within the chain of the old chain and the

new chain, respectively. If the uniformly generated random number y r is greater than

the ratio of weights of individual configurations, the new configuration is abandoned

and the algorithm returns to the old configuration of the c h a h Othemise the chain is

"updated" to the new configuration where the new configuration had been previously

generated by a very efficient pivot algorithm. In this algorithm a lattice site on the

polymer is chosen and the element of group of symmetries of a simple cubic lattice

is applied to the shorter part of the chain. The newly obtained configuration is then

tested for overlaps and the standard Metropolis test is applied (as discussed above).

The pivot algorithm has been well described in the literature [66], therefore we will

not get into more de tails. For generating uniformly distributed ( pseudo-) random

nurnbers we use the subroutine described in the next section.

C.3 Random number generator

Many random number generators have been suggested and used over the years (for

a thorough review see [93]). The generator used in our simulation [92] is the sum of

two linear congrueutid (modulo) generators with shuffling. Its FORTRAN code is

presented below.

subroutine rng(itst)

implicit integer (i-n)

implicit real*8 (a-h,o-z)

parameter(ax=2147483647/2147483648d0,bx=l6807+2101dO/2**28)

Save ix,r2

common/blk/rnx(2435),m,yr

if (itst) 2,15

c--- initialize random array

2 do 5 i=1,2435

xr=16807*xr-ax*idint(bx*xr) !(il

5 rnx(i)=xr

r2=rnx (2433)

ix=idint(2431*rnx(2435))+1

return

c--- generate random xr(nseed1 , yr

15 r=rnx(ix)

xr=16807*xr-ax*idint (bx*xr) ! ( 1 )

rnx(ix)=xr

ix=idint(243l*r)+l

r2=1664525*r2+1013904223/4294967296dO ! (2)

r2=r2-idint(r2)

yr=r+r2-idint(r+r2)

return

end

The generator denoted as (1) in the code is the simpIe multiplicative congruential

generator h s t proposed by Lewis et al. [94] and defined by

This generator has passed al1 known theoretical tests [95] and most importantly has

been used with success in the past. The other generator that riras used in our work

(denoted as (2)) is defined by

Ij+l = (1664525 Ij + 1013904223)rnod(2~~)

For more details, see [94].

Appendix D

Blocking met hod of calculat ing

standard errors

The sample attrition effect in the simulation of self-avoiding walks prevents one to

use the simple sampling Monte Carlo method to generate long walks in two and three

dimensions. Instead, other sampling methods such as the reptation, kink-jump or

the pivot algorithm have to be used. These dynamic sampIing methods eliminate

the attrition, however, they al1 have one serious drawback: the generated configu-

rations are not statistically independent and t herefore standard s tatistical analysis

does not apply. There are two alternative approaches to the estimation of the vari-

ance of the mean, namely the correlation function method and the blocking method.

Even though the blocking method was used in computer simulations before, it was

described by Flyvbjerg and Petersen [69] only recently using the real space renor-

malizarion group. In this appendix, comparison between the correlation function and

the blocking methods is presented and advantages of the latter are pointed out. The

description of the blocking method use is given.

Dynamic Monte Carlo simdation of a physical system provides a sequence of

highly correlated physical quantities {xi):= '=, . Their finite t ime average

is a fluctuating quantity the variance of which has to be estimated. It depends on

correlations between individual X i in the following way

where Yi j = ( s i x j ) - (xi) (xj) where the anguhr brackets denote the expectation

value (i. e. the ensemble average) given by the exact probability distribution p(x) as

(f) = J j (x )p (x )dx and c m be approximated as an average over a n i d a i t e number

of independent Monte Carlo runs. If invariance under a tirne translation is assumed1

we can write

where yt = (xoxt ) - (xO) (xt) and t = li - jl. In most cases the dynamically simulated

data are not independent (i. e. y, > 0) and therefore yO/n = 0 2 ( x ) / n is only the lower

estimate of the true a2(z). It would be easy to estimate a2($ if the values of yt were

readily available, but these have to be estimated hom the finite size data set too. A

standard estimator ct of yt is

1 n-t q=- C ( x i - z) (xi+, - z)

n - t +'=,

but one has to realize that this estimator is a biased. Flyvbjerg and Petersen [69]

suggest that if one can find T such that r « T « n where T is the Iargest correlation

time then a good estimator for a2(3c) is

'which is a good assumption as long as the simulated system is in equilibrium

175

Simila estimators have been used elsewhere with certain variations in the fom,

however they al1 contain the sum cT=, c,. Because of the computational effort involved

in calculating ct by using eq. D.4, the reduction of T to a value as mal1 as possible is

desirable, however, to ensure the accuracy of oz@), T has to be several times larger

than the correlation time T.

Blocking represents an alternative method of 02(2) estimation. As the name

suggests, the data are blocked into groups of progressively larger sizes by cdculating

the averages in the following way

where superscript k denotes the k-th level of blocking. It is easy to show that this

transformation of data set has two invariants, namely the mean

and the variance of the mean

Values of yt are affected by the transformation in the following way

1 (k) 1 (k) #+l) = - io + 571

2

and

1 (k) 1 (k) ( k ) yt(k+i) = -7*&l + p t + 4 7 2 t f l 4

(D. 10)

One can show that yt/n oc is the fuced point of the blocking transformation so that

by increasing the block sizes (k -t m) we expect the yo/n increases as we go on with

the blocking. In the limit of k + w the true value of 02(z) is reached. This can be

seen from Figures D.1 and D.2 where the MC data are treated in the way described

Figure D.l: The blocking method of estimating the statisticai errors of (R&(w)) for

various values of W.

above. Individual curves correspond to different d u e s of W. The horizontal axis

on both graphs represents Iog2(nb) where nb is the

one block. As we can see, by increasing the number

fi& increases until the limiting value is reached.

number of data " collected" into

of data in one block the value of

The true statistical errors of the

MC data that were used in our analysis are the plateau values of individual curves.

This is very simple to estimate. The main advantage of the method compared to

other methods of estimating statisticai errors is its efficiency.

Figure D.2: The blocking method of estimating the statistical errors of (A$,,(w)) for

various values of W.

Appendix E

Fitting functions

In this Appendix the choice of fitting functions is discussed. We also give the expres-

sion for fitting functions with the optimal values of coefficients and their errors. This

may be of some interest for those who want to investigate the renormalization group

recursion relations.

E.l The choice of fitting functions

If simple powers of @ (namely @'-') are used in the form of the fitting function (e.g.

the truncation of the Taylor series) the absolute error is proportional to the first

missing power of the truncated series. This error function (the absolute error) is very

small for small values of x but increases very fast with increasing x. In order to

distribute the error of the approximation more evenly over the entire interval, the so

called "equal ripple" (or minimax) approximations are used. To do that, Chebyshev

polynomidsl are usually employed. In our work we decided to use the set of fitting

functions that are not as efficient in distributing the error over the entire interval as

'These polynornials oscillate between values of +1 and -1 on the intenml -1 5 x 5 +l.

Chebyshev polynomials are, but they have another advantage. Each function fi (@)

of the set is related to the value (or derivative) of F($) a t one particular point of set

{$k) that has been chosen beforehand. The chosen points were distributed uniformly

over the interval O < -I/J < where & is the expected asymptotic value of the

interpenetration function. From

point $' to be approximately Ilo

that $* falls close to x* = 1 and

fitting functions is defined in the

preliminary fits we found the position of the fixed

= 0.232 and we rescaled 11-axis using x = so

thus Zk = ?,6k/?,60 are rational numbers. Our set of

following way

I

where {Zk ) lk=L ,m and {&)( t , l fm are t m sets defining positions of equally spaced

reference points and the derivatives of the functions in those points, respectively. This

restriction is very convenient because in the final f o m of the fitting function given

by eq. (7.16) each individual fitting parameter ci represents the value of either the

function F ( x ) or its derivative dF(x)/dx at one of the points defined by {i?k}lk=l,m.

Nso the error of the estimate of the function F(xk) is given directly by ~ ( 4 ) and

does not need to be evaluated from the matrix of the variances.

E.2 Theoptimalfit

In this section we give exact expressions for vR(N, x) and uA(N, x). The functions

that were used in the fit of vR(N, x) are

-I

The functions used in the fit of uA(N, x ) are

From these fitting functions, uR(N, x) and V A ( N , x) are constructed as follows

where $0 = 0.2328868 and various derivatives of v~ and VA as functions of N are

given by eqs. (6.36), (6.37) and (6.38).

Funetions gy), dependent on N, are of the following forrn

These fitted amplitude functions vanish at N = 4, 6 and 8 for X = R and also at

N = 4 for X = A and therefore they do not contribute to U R and va for these N

values. The k d amplitude functions h!X) are N-dependent functions of the form

These functions when multiplied by corresponding fRqi(x) or fAqi(x) as shown in

eqs. (E.4) and (E.5), reproduce exactly the functions u ~ , ~ ( x ) , vRl6(x) and vRl8(x) and

also vAl4 (x) . Finally the fitting parameters q and their optimal values are given in the Ta-

183

ble E.1-E.2. In the first two columns the linear (L) parameters that were obtained

by the "global linear" fit are tisted dong with their error bars. Columns three and

four contain the non-linear (N) parameters obtained using the "global nonlinex" fit

that were used in expressions (E.4) and (E.5). From eqs. (E.4) and (E.5) and the

definitions of set of fitting hinctions given by eqs. (E.2) and (E.3) one can recognize

that near the fixed point 11i rr x $* we have

4z(w Si)

(E. 10)

From these 5 numbers one can get 11' = 0.23321, u* = 0.58756, A = 0.532 and

bA/bR = -0.910, values that are very close to the exact solutions presented in the

text (see line (1) of Table 7.7 or line (5) of Table 8.1).

For convenience we also include the asymptotic functions vR(oo, I,!J) and z . q (00, $)

1 U~(OO, @) = - + 0.3983892349 $ - 0.1999578457$* - 0.4462249616 $3

2

+21.9130724001 $4 - 171.5308963434 111' + 526.9501834671 $6

From eqs. (5.25), (5.26) and (5.33), (5.34) we can obtain the predicted series expansion

for both UR and UA in terms of 11. The functions are

These resdts show that both the predicted value of the third derivative of ~ ~ ( $ 1 at

$ = O and the predicted value of the second derivative of ~ ~ ( $ 1 at $ = O are very

close to the exact two-parameter values (see eqs. (E.13) and (E.14)).

Table E. 1: Fitting parameters of linear and nonlinear fit.

-

Table E.2: Fit ting parameters of linear and nonlinear fit (continued) .

187

Appendix F

Monte Carlo data

In this appendix we present the mean square end-to-end distance and the second virid

coefficient data for a linear flexible chah calculated within the Domb-Joyce model.

The statistical errors a~ = a((RN(w))) and UA = o ( ( A ~ , ~ ( w ) ) ) are also listed where

applicable. The values of (R$(w)) and (A~J~(w) ) , where a~ or 0.4 are equd to zero,

are obtained by exact counts. Those with nonzero error bars were obtained by the

Monte Carlo method (see Appendix (2.2). These data might be of some interest to

those who would like to investigate the fits or other possibilities of analyzing the DJ

model results further.

Table F.l: Monte Carlo data for w = 0.01 and w = 0.03.

189

Table F.2: Monte Carlo data for w = 0.05 and w = 0.07.

190

Table F.3: Monte Carlo data for ul = 0.10 and w = 0.15.

Table F.4: Monte Car10 data for w = 0.20 and w = 0.30.

-- -




195

Table F.8: Monte Car10 data for w = 1.00.

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UNIVERSAL CRITICAL MOLECULE · scaling amplitudes bA/bR were also calculated with a very good precisiun. The results are u = 0.58756(5), A = 0.5310(33), ** = 0.23221(11) and bA/bR