Monte Carlo Study of Semiﬂexible Polymers

Monte Carlo Study of Semiflexible

Polymers

Inaugural-Dissertation zur Erlangung des Doktorgrades der

Fakultät für Mathematik und Physik der

Albert-Ludwigs-Universität Freiburg im Breisgau

vorgelegt von

Ganna Berezovska

2011

Dekan : Prof. Dr. Kay Königsmann

Leiter der Arbeit : Prof. Dr. Alexander Blumen

Referent : Prof. Dr. Alexander Blumen

Koreferent : Prof. Dr. Thomas Filk

Datum der mündlichen Prüfung : 29.11.2011

The results of this thesis are summarized in the

following papers:

• M. Dolgushev, G. Berezovska, and A. Blumen, “Cospectral polymers: Differentiation

via semiflexibility“, J. Chem. Phys. 133, 154905 (2010).

• M. Dolgushev, G. Berezovska, and A. Blumen, “Maximum entropy principle applied

to semiflexible ring polymers“, J. Chem. Phys., 135, 094901 (2011).

• M. Dolgushev, G. Berezovska, and A. Blumen, “Branched semiflexible polymers:

Theoretical and simulation aspects“, Macromol. Theory Simul., 20, 621 (2011).

• G. Berezovska, M. Dolgushev, and A. Blumen, “Semiflexibility Highlights the Poly-

mers’ Topology: Monte Carlo Studies“, submitted.

2

Contents

Introduction 5

1 Simulation Method and the Observables under Study 8

1.1 Simulation method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2 Physical observables under study . . . . . . . . . . . . . . . . . . . . . . . 13

1.2.1 Mean-square radius of gyration . . . . . . . . . . . . . . . . . . . . 14

1.2.2 g-ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2 Linear Chains and Stars 16

2.1 General properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.1.1 Radius of gyration . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.1.2 g-ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2 Local properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2.1 Interatomic distance . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2.2 Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3 Persistence length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.3.1 Theoretical background . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.3.2 Simulations: Persistence length extracted from exponential decay . 30

2.3.3 Simulations: Local persistence length due to Flory . . . . . . . . . . 37

3 Rings and Trefoils 42

3.1 General properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.1.1 Unknotted rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.1.2 Trefoils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.2 Local properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.2.1 Unknotted rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.2.2 Trefoils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4 Semiflexibility as a Way to Highlight the Polymers’ Topology 60

4.1 Linear chains and stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3

4.2 Cospectral polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.3 Rings and trefoils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.4 Rings and stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

Summary and Conclusions 71

Bibliography 74

Index of Abbreviations 78

4

Introduction

The properties of polymers are influenced both by their topology and also by the degree

of semiflexibility of their segments. During the last years these aspects were of active

interest in simulation studies. Exemplary, we recall here some studies, which are relevant

to the subject of this thesis: The role of topology for tree-like structures was addressed

in refs. [1–7] and for structures containing loops (having ring as the simplest object of this

category) in refs. [7–14]. Questions of semiflexibility were studied in refs. [15–21] applying

different simulation techniques.

Theoretical approaches based on a discrete chain model which account for branching

and semiflexibility also made progress during the last few years. Recent analytical exten-

sions of the generalized Gaussian structures picture (GGS) [22] to the case of semiflexible

branched polymers [7,23–25] and rings [7,8] give qualitative clues to the behavior of polymers.

However, numerical simulation techniques allow to investigate the role of the excluded

volume and thus, provide a more realistic approach. In this work we present our simu-

lation study of different topological structures by means of the bond fluctuation model

(BFM), [26,27] which allows to account both for branching and/or presence of loops and

also for semiflexible behavior. [6,7,28] For this we perform a generalization of the simulation

method combining semiflexibility and branching. To the best of our knowledge, in former

simulation studies in the framework of BFM semiflexibility was introduced in linear chains

only. [16–19,21]

We apply this simulation technique to some of those semiflexible objects, to which

the above mentioned analytical extensions of GGS are applicable. Namely, we investigate

chains and stars; the so-called cospectral structures, which are a special type of tree-like

polymers; and unknotted rings and trefoils. In the analysis of our results we focus on

the mean-square radius of gyration (as a general static property of polymers) and on the

bond-bond correlation function (as an example of local properties). In order to verify

the theoretical results we compare our simulation data with theoretical predictions based

on the extended GGS approach [23–25] and on the maximum-entropy principle (MEP). [7,8]

Moreover, a comparison of our Monte Carlo (MC) data with previous simulation studies

and with available scaling arguments is made. We also discuss the problem of a relevant

definition of persistence length under good solvent conditions; a question which recently

5

rose from several simulation studies showing that its standard definitions are incorrect for

certain systems. [1,2,29–31]

Now, a theoretical approach which was recently shown to be very well suitable for

the description of semiflexible tree-like polymers (STPs) as well as of semiflexible rings

is the MEP. [7] Being initially applied to semiflexible chains in a discrete [32] and in a

continuous [33] framework, it was implemented to STPs [24] and generalized to semiflexible

rings. [7,8] An important feature of the MEP method [34,35] is that it introduces constraints

only on adjacent bonds. Moreover, the treatment of STPs with MEP was shown to be

equivalent, and hence an alternative, to the recently developed approach for STPs [24] done

in the spirit of Bixon and Zwanzig. [36]

An example where the theoretical approaches mentioned above as well as the numerical

investigations turn out to be helpful is the study of so-called cospectral polymer (CP)

structures. [37,38] These structures are important, since in the GGS picture they have the

same Laplacian spectrum although being topologically different. Hence, in GGS, flexible

polymers whose structures are cospectral graphs are predicted to be indistinguishable

under the usual static and dynamical measurements. Taking into account that large

tree-like structures have (at least) one cospectral counterpart, [39,40] the problem becomes

of great importance. Our recent mathematical-analytical study [6] shows that when the

polymers are semiflexible one can distinguish between cospectral structures. In this work

we qualitatively confirm these theoretical predictions by our simulation results on the

smallest tree-like cospectral pair.

Discrete semiflexible rings provide another example of analytical results obtainable

through MEP. [7,8] It turns out that in the rigid limit, besides solutions pertaining to

unknotted rings, one obtains other solutions related to knotted rings. [7,8] To have a check

of the theory, MC simulations are very helpful, and particularly the BFM, which in its

standard form conserves the topology of the objects but also permits to turn off and on

the excluded volume interactions. [7,8]

In the framework of this thesis all presented simulation data (unless otherwise spec-

ified) were obtained by means of the BFM and its generalization, which accounts both

for branching and for semiflexibility. Now we would like to make a short overview of the

structure of this thesis:

• In Chapter 1 the simulation technique in the framework of BFM is recalled briefly. [26,27]

Furthermore, our extension of the BFM for semiflexible polymers with junctions of

functionalities f > 2 is described. The correspondence between stiffness parame-

ters for simulations and theory is discussed. Several physical observables, which are

studied in this work, are also presented.

• In Chapter 2 our simulation study of semiflexible chains and stars of functionality

6

f = 3 and f = 4 is presented. These structures are considered first, since we are

interested in the influence of a single branching point and the presence of additional

arms (treating chains as 2-arm stars) together with stiffness on general and local

static properties of semiflexible polymers. Simulation data for the mean-square

radius of gyration and the g-ratio are compared to the theoretical predictions based

on the STP-model [24] or, alternatively, on the MEP. [7,8] The interatomic distances

and bond-bond correlation functions along the star arms and close to the branching

points are studied and compared with previous simulations. At the end of the

chapter two commonly assumed definitions of persistence length are discussed in the

scope of their applicability to the good solvent conditions: The persistence length,

which is extracted from the exponential decay of the bond-bond correlation function

and the definition of the local persistence length usually addressed to Flory. [41] For

the latter a comparison to the renormalization group (RG) approach from ref. [42] is

performed. The scaling behavior of the bond-bond correlation function for flexible

chains is analyzed and compared with RG calculations from ref. [43]

Some results of this chapter are published in refs. [7,28]

• In Chapter 3 the verification of the theoretical results based on the MEP for rings is

performed. For unknotted rings theoretical predictions are compared quantitatively

with our simulation data, while for rings with one knot (known as trefoils) only a

qualitative comparison could be made. The reason for this lies in the analysis of

theoretical results: knotted topologies appear here only in a stiff limit which is not

accessible in simulations. The theoretical curves for semiflexible unknotted rings

and stiff trefoils are provided by Maxim Dolgushev. The results of this chapter are

published in refs. [7,8,28]

• Considering results from previous chapters we have noticed that under semiflexible

conditions the topology gets more pronounced. Thus, in Chapter 4 we analyze in

how far a semiflexible behavior enhances differences in properties of the classes of

polymers whose topological structure varies. We focus on three pairs of macromolec-

ular classes: stars vs. chains, unknotted rings vs. trefoils, and stars vs. unknotted

rings. Additionally, we also study the role of semiflexibility for differentiating be-

tween CPs which are one of the most interesting and beneficial objects in this scope.

For this we consider the smallest pair of tree-like cospectral structures. Our sim-

ulation data is compared to the theoretical curves provided by Maxim Dolgushev.

The results of this chapter are published in refs. [6,28]

7

Chapter 1

Simulation Method and the

Observables under Study

Having the aim to perform a simulation study of the interplay between branching and

semiflexibility and to validate the theoretical approaches for tree-like polymers [7,24] and

for rings [7,8] we need a well-established and readily manageable numerical method, in

which features of interest like branching and semiflexibility can be easily incorporated.

For this, being interested also in introducing realistic aspects such as excluded volume

(EV) interactions, we chose to perform three-dimensional coarse-grained lattice MC sim-

ulations, applying the bond fluctuation model (BFM). [26,27] Now, the BFM accounts for

excluded volume interactions and prohibits bonds from crossing each other and thus, it

conserves the topology of the systems under consideration. The BFM was extensively

applied to a great variety of problems. We recall studies involving polymer melts, [26,27]

the adsorption and conformations of tethered chains at substrates, [3,44–48] neutral and

charged dendrimers, [4,49,50] bottle-brushes, [1,2] and membranes. [51] In the BFM framework

semiflexibility was accounted for in studies involving chain polymers, such as the question

of the coil-globule transition, [16–18,21] the isotropic-nematic ordering, [19] tethered chain ad-

sorption on a surface [20] and also ring polymers [13,14] and bonded rings and catenates. [9,10]

In this thesis we extend the BFM to semiflexible polymers with branching points [6] and

apply the developed simulation scheme to semiflexible stars and to cospectral polymers. [6]

Moreover, we also consider semiflexible unknotted rings and trefoils.

1.1 Simulation method

We start our considerations by recalling the basic features of BFM. [26,27] In the BFM

each monomer of a coarse-grained polymer is represented by a cube of unit length on

the simple cubic lattice. The excluded volume property is introduced by requiring that

8

Chapter 1: Simulation Method and Observables under Study

y

z

x

d1

d6 d2

d3

d4

d5

Figure 1.1: Realization of a branched polymer in the bond-fluctuation model(BFM) (N = 7 elements are displayed). The bond vectors di are indicated by bluearrows; the violet arrows show allowed elementary moves, see text for details.

each lattice site belongs to one cube at most. Although the lengths of the bonds are

allowed to fluctuate, they have to belong to the set of lengths 2,√5,√6, 3 or

√10. All

spatial distances are measured in units of the lattice spacing. Altogether, the three-

dimensional BFM allows 108 different bond vectors and 87 different angles between them.

Applying the cubic point group operations to the set of vectors {(2, 0, 0), (2, 1, 0), (2, 1, 1),(2, 2, 1), (3, 0, 0), (3, 1, 0)} one obtains the complete set of allowed bond vectors.

Starting from an initial configuration one creates a new one through one local move:

First, one chooses at random a unit cube; then one attempts to move it randomly by a

unit length in one of the six lattice directions. The attempt is rejected if at least one of

the eight sites of the unit cube in the new position lands on an occupied site or if the

new bond does not belong to the allowed set. The restrictions on the bond lengths are

topology-preserving, since they prevent the crossing of segments. This feature makes the

method very well suitable for simulations of branched topologies, unknotted rings and

rings with one knot (so called trefoils), the objects which this work in devoted to.

One Monte Carlo step (MCS) is achieved when in average each bead has attempted

9

one trial move. The BFM scheme was originally applied to chain-like structures, but it

now encompasses a large number of polymer structures. [1,2,4,13,14,48,51] In Figure 1.1 we

represent schematically a branched polymer and some of the allowed BFM moves.

Since we want to model semiflexible objects we introduce an energy penalty for moves

which are allowed under the above scheme, but which involve an energy change of ∆U .

For this we use the Metropolis algorithm [52] to determine the transition probability w

w = min

[

1, exp

(

−∆U

kBT

)]

(1.1)

for accepting an allowed local move.

Now, to be in line with the theoretical calculations for semiflexible tree-like structures

based on MEP [24] and using the fact that the sum of the cosines of all the angles between

the bond vectors at each junction point i is bounded, [53] we assume the bending energy

Ui corresponding to the junction i with functionality fi to be:

Ui

kBT= Bi

fi2−∑

{(a,b)}

(−1)s cos θab

. (1.2)

Here Bi is the stiffness parameter corresponding to the junction i (Bi being 0 in the

flexible case), the sum {...} runs over all the distinct pairs (a, b) of bond vectors involving

junction i, and θab is the angle between the bond vectors a and b. The parameter s is

either 0 (for bond vectors in a head to tail configuration) or 1 otherwise.

The total energy of a configuration is the sum of energies of every junction point. But

being interested in the energy difference ∆U it is enough to calculate only the contributions

from the junctions which are affected by the trial motion. If for example a junction, say

i, experiences a trial move, then one has to take into account only contributions to the

energy change ∆U from all pairs of adjacent bonds of which at least one is attached to i.

Thus, using Equation (1.2), the energy difference ∆U to be used in Equation (1.1) is:

∆U

kBT= −

∑

[(a,b)]

(−1)sBj

(

cos θnewab − cos θoldab

)

. (1.3)

In Equation (1.3) the sum [...] runs over all distinct pairs of adjacent bond vectors (a, b),

of which at least one involves junction i, and j denotes the junction of the (a, b)-pair. The

θnewab (θoldab ) denote the new (old) angles.

A remark on the form of the potential given by Equation (1.2) is in order. Obviously,

in it the term fi/2 is irrelevant. It is introduced in order to have for fi = 2, in a head to

10


tail orientation of the bond vectors, from Equation (1.2):

Ui

kBT= Bi (1− cos θ) . (1.4)

With Bi ≡ B Equation (1.4) is one of the classical potentials used in simulations of

semiflexible chains [21,31] and of rings. [13,14]

A few remarks concerning the stiffness parameters used in simulations (Bi) and in

theoretical studies (ti[7]) are now required. In most of our simulations we will let the

semiflexibility parameter Bi to be the same for all junction types, Bi ≡ B, and we will

use B = 0 in the flexible case. For chains and stars in the semiflexible case parameter

B takes the values B = 1, 2, 3, 4, and 6. For rings and trefoils parameter B takes the

values B = 0, 3 and 6. Only for CP we will introduce different semiflexibility parameters

for the junctions with different functionalities. For the latter parameters Bi vary in the

range Bi = 0, 1, 2, . . . 10.

Given that in the theoretical framework the mean-square lengths of all bonds are equal,

〈l2b〉theory = l2, the theoretical stiffness parameter ti can be introduced through [7,8,24,28]

〈da · db〉 = (−1)s〈l2b 〉ti, (1.5)

where da and db are adjacent bond vectors connected through the bead i, s is the same

as in Equation (1.2) and√

〈l2b〉 is around 2.7 lattice units, which is a typical value for

the BFM. [2,14] The theoretical framework based on MEP is fully applicable to semiflexible

tree-like polymers and unknotted rings. Thus, in order to compare theoretical predictions

with simulations we associate Bi to ti for these objects. As the MEP allows a description

of trefoils only in a stiff limit, which is impossible to reach with our simulation method, we

cannot compare quantitatively our simulation data for trefoil to theoretical predictions.For chains and rings we consider homogeneous situations, in which all the bonds are

connected head to tail (see figure (a) of Figure 1.2 for the typical numbering of beads

and the orientation of bonds in a chain) and ti ≡ t for all i. The size of chains and

rings is described by the amount of beads N . Thus, the amount of bonds is Nb = N − 1

and Nb = N for chains and rings, respectively. In the case of stars the bonds have a

head to tail orientation in the arms and we will set ti ≡ t for them as well. Only the

bonds directly attached to the core have tail to tail orientations and, in principle, another

stiffness coefficient, namely qc = t/(f − 1), where f is the functionality of the core of the

star. The typical numbering of beads and the orientation of bonds in a star are presented

in figure (c) of Figure 1.2. In this thesis we consider stars of functionality f = 3 and 4

and molecular weight N = fn + 1, where n is the number of beads/bonds in each arm.

In this sense linear chains can also be viewed as “two-arm“ stars, f = 2, with arm length

n and N = 2n + 1. The corresponding orientation of bonds and the numbering of beads

11

Bt,

f = 2qtheoryc ,f = 3

qsimc ,f = 3

qtheoryc ,f = 4

qsimc ,f = 4

0 0.19 0.097 0.163 0.065 0.1411 0.41 0.203 0.280 0.135 0.2172 0.57 0.286 0.341 0.191 0.2533 0.68 0.343 0.375 0.228 0.2724 0.75 0.375 0.397 0.252 0.2846 0.84 0.420 0.424 0.280 0.298

Table 1.1: Theoretical stiffness parameters t for chains and stars, which corre-spond to the stiffness parameter B used in simulations, obtained from the simu-lations using Equation (1.5). The values for qtheoryc are obtained using equationqc = t/(f − 1) for the corresponding t and f values. qsimc are the values for meancosine in the branching point, obtained from the simulations, see text for details.

N = 16 N = 32 N = 64 N = 128 N = 256 N = 512

B = 0 0.159 0.176 0.182 0.185 0.185 0.186B = 3 0.620 0.653 0.663 0.667 0.670 0.670B = 6 0.764 0.807 0.816 0.819 0.821 0.821

Table 1.2: Theoretical stiffness parameters t for unknotted rings, which cor-respond to the stiffness parameters B used in simulations, obtained from thesimulations using Equation (1.5), see text for details.

for chains considered as stars are depicted in figure (b) of Figure 1.2. In certain cases,

considering deviations for stars from a chain behavior, it is reasonable to treat two arms

of a star as a backbone chain and the additional arms as side chains. The figure (d) of

Figure 1.2 shows the proper orientation of bonds in two selected star arms.

Based on Equation (1.5) one can now readily connect t to B. For chains and stars the

values of B and t related in this way are almost independent of the chain or of the arm

length and are presented in Table 1.1. Here the values of t follow from the MC data for

B = 0, 1, 2, 3, 4 and 6, as the mean cosine of the angles between two successive bond

vectors; around two-fifth of the bond vectors are excluded from the average, namely those

that are closest to the core or to the end(s). For a given t the values of theoretical stiffness

parameter in the branching point, qtheoryc , is calculated using the above expression for qc,

qtheoryc = t/(f − 1). qsimc is its simulation counterpart.

For unknotted rings the t-values for several N and for the values of the stiffness

parameter B = 0, 3 and 6 are presented in Table 1.2. One can notice that in the case of

rings the t-values depend on the ring length, which happens due to the closure condition. [8]

For sufficiently large ring lengths the values of t are almost constant.

12


d6

d1 d2

d3

d4

d5

a)

0

1

23

4 5

6

Chain N=7 Chain as star f=2, n=3

Star f=3, n=3 Star as chain N=7

d6

d1

d2d3

d4

d5

b)

0

4 5

61

2

3

d6

d1 d2

d3

d4

d5

d)

0

1

2 3

4 5

6

d6

d1

d2d3

d4

d5

d7

d8

d9

c)

9

8

7

01

2

3

4 5

6

Figure 1.2: Schematic picture of a chain with N = 7 (a); a chain from (a) whichis considered as a star with f = 2 and n = 3 (b); a star with f = 3 and n = 3(c); a star from (c) considered as a chain backbone of length N = 2n+ 1 with aside chain (d), see text for details. The parts of the structures which have to bere-oriented are shown in red.

1.2 Physical observables under study

In our study we focus mainly on the normalized mean-square radius of gyration 〈R2g〉/〈l2b〉

as an example of general properties of polymers and on the normalized bond-bond cor-

relation functions as an example of their local properties. In what follows we calculate

the normalized bond-bond correlation functions in two ways depending on the type of

comparison. For stars and chains we mainly compare this property obtained from our

MC data to previous simulation studies. For these objects we practically calculate the

13

mean cosine between the bonds di and dj , namely 〈di · dj/|di||dj|〉, the quantity which

is usually used in simulations. For the rings, having the aim to compare our simulation

data with recent theoretical results obtained using MEP, [7,8] we calculate the normalized

bond-bond correlation functions as 〈di · dj〉/〈l2b〉. This quantity is more relevant to the

way of introducing stiffness in theory (see Equation (1.5)).

Additionally, as a general property of stars, we also consider a so-called g-ratio which

gives a size of a star in terms of size of a linear chain.

1.2.1 Mean-square radius of gyration

One of the measures of the average chain dimension is the mean-square radius of gyration

〈R2g〉 (the ensemble average of the radius of gyration). This quantity depends strongly on

the molecular structure of the polymer chain. To determine 〈R2g〉 from the MC data we

use the following expression [54]

〈R2g〉 =

1

N

N∑

i=1

〈(ri −RC)2〉, (1.6)

where RC is the center of mass. Scaling theory [55] predicts that the mean-square radius

of gyration for stars under good solvent conditions scales with N in the same way as for

chains

〈R2g〉 ∼ N2ν , (1.7)

where ν = 0.588 is a critical exponent. [56] For ideal chains/stars the corresponding scaling

is 〈R2g〉 ∼ N having ν = 0.5. [54]

From the STP-model one has the following theoretical expression for 〈R2g〉

〈R2g〉 =

l2

N

N∑

k=2

1

λk

. (1.8)

Here l2 is the mean-square length of each bond and {λk} are the non-vanishing eigenvalues

of the matrix ASTP , which determines the set of Langevin equations in the semiflexible

case. [7,24] Alternatively, for stars and chains 〈R2g〉 can be obtained using explicit expressions

from ref. [7,57] Namely, for stars of functionality f the following expression can be used

〈R2g〉

l2=

f

N2

[

1 + t

1− t

n(n + 1)

2

(

n(f − 1) +n+ 2

3

)

+t

(1− t)2(

n+ 1− 2n2(f − 1))

+

2t2

(1− t)3

(

n + 1 + n(f − 1)(1− tn)− tn+1 − 1

t− 1

)

+t(tn+1 − 1)

(t− 1)3

(

tn+1 − 1

t− 1− 2(n+ 1)

)]

(1.9)

Here it was assumed that the stiffness parameter of the core is qc = t/(f−1), which implies

14


that qc = 0 for t = 0 and qc = 1/(f − 1) for t = 1. In this way, changes in t allow to reach

the flexible and the rigid limits for the branches and for the core simultaneously. [23,53] For

chains with f = 2 and n = (N − 1)/2, the above expression simplifies to

〈R2g〉

l2=

1

N

(

1 + t

1− t

(N − 1) (N + 1)

6− t(N − 1)

(1− t)2+

2t2

(1− t)3− 2t2

(1− t)41− tN

N

)

. (1.10)

1.2.2 g-ratio

A useful parameter which gives a size of a star in terms of the size of a chain is the g-ratio.

It is defined in the following way

g = limN→∞

gN , gN =〈R2

g〉f〈R2

g〉2, (1.11)

where 〈R2g〉f is the mean-square radius of gyration of a star with functionality f and 〈R2

g〉2is the mean-square radius of gyration of a chain with the same molecular weight N . In

the case of a Gaussian distribution of the intramolecular distances between the beads one

obtains the following value for g = g0[53,58]

g0 =3f − 2

f 2(1.12)

for a star of functionality f . In a rigid-rod limit the g-ratio takes the value g = gr =

4/f 2. [53]

For self-repelling stars the expression for the g-ratio was calculated using RG methods.

In the excluded volume limit one finds

g =3f − 2

f 2

(

1− ǫ

8

[

13 (f − 1) (f − 2)

2 (3f − 2)− 4 (f − 1) (3f − 5)

3f − 2ln 2 + ln f

])

, (1.13)

where ǫ = 4−d and d is the spatial dimension. [59] Apparently, Equation (1.12) is a special

case of this result. RG calculations show that the values of the g-ratio under good solvent

conditions are very close to the g-values in the Gaussian limit, see Equation (1.12). [60]

15

Chapter 2

Linear Chains and Stars

We start with an investigation of the role of a single branching point. For this we consider

stars of functionalities f = 3 and f = 4 and we compare the results with those for chains,

f = 2. In our BFM-simulations for stars we take n up to n = 50 and for chains we go with

N up to N = 320. The initial configuration of chains are straight lines. For the 3-arm

stars we allocate the arms along x+, y+ and z+ directions. The initial configuration of

4-arm stars is built by the x+ and y+ directions and the complete z axes. The size of

the simulation box varies with the size of the object considered and for the largest stars

or chains it contains 700 × 700 × 700 lattice units. We implement periodic boundary

conditions in x, y and z directions. Each object is equilibrated for some 109 MCS, after

which the conformations are saved every 1000 MCS. The averages are then taken over at

least 106 realizations.

2.1 General properties

2.1.1 Radius of gyration

Here we focus on the mean-square radius of gyration 〈R2g〉 as a static property of semiflex-

ible chains and stars. For chains and for stars with core functionalities f = 3 and f = 4

we compare now our simulation data obtained from the BFM with the theoretical results

developed in ref. [7,24,57] The simulation data for the radius of gyration are obtained using

Equation (1.6).

Figures 2.1, 2.2 and 2.3 display in double-logarithmic scales 〈R2g〉/〈l2b〉 as a function

of N for chains, f = 2, and for star polymers with f = 3 and f = 4, respectively.

The curves show the theoretical values calculated for stiffness parameter t corresponding

to given B (see Table 1.1). The agreement between simulation and theory is rather

qualitative; it gets better for larger B and for not too large N . With increasing N

the excluded volume interactions become more important; the same holds for smaller B

16

Chapter 2: Linear Chains and Stars

Figure 2.1: Double-logarithmic plot of 〈R2g〉/〈l2b〉 versus N for the linear chain

(f = 2). The symbols correspond to different stiffness parameters B and the linesdepict the theoretical results for different t obtained using Equation (1.10), seetext for details. The inset shows the semi-logarithmic plot of 〈R2

g〉/〈l2b〉 versus Bfor N = 21, 61 and 101.

values. The scaling exponents for B = 0 and B = 1 are about 1.2 (being 1.212 ± 0.004

and 1.193± 0.003 for f = 2; 1.221± 0.002 and 1.203± 0.003 for f = 3; and 1.217± 0.001

and 1.203± 0.003 for f = 4, respectively). The presence of excluded volume lets then the

simulation data deviate from the results of the STP-model, which does not account for

excluded volume interactions. Similar findings arise when comparing chains to stars: the

former feel the excluded volume less. These tendencies are also in qualitative agreement

with the simulation results in ref. [15], where freely-rotating chains and stars were studied.

17


Figure 2.2: Double-logarithmic plot of 〈R2g〉/〈l2b〉 versus N for the star (f = 3).

The symbols correspond to different stiffness parameters B and the lines depict thetheoretical results for different t obtained using Equation (1.9), see text for details.The inset shows the semi-logarithmic plot of 〈R2

g〉/〈l2b〉 versus B for N = 31, 91and 151.

2.1.2 g-ratio

Figure 2.4 shows plots for gN versus log10 N for regular stars with f = 3 (upper plot) and

f = 4 (lower plot) for different values of the stiffness parameter B. The symbols represent

the MC data. The curves show the theoretical values calculated from Equations (1.9)

(1.10) and (1.11) for corresponding values of the stiffness parameter t. The horizontal

lines correspond to the Gaussian limit, see Equation (1.12), according to which g0 = 7/9

and g0 = 5/8 for f = 3 and f = 4, respectively. [53] The simulation data for all values

of B is well above the rigid-rod limit, gr, which is gr = 4/9 and gr = 1/4 for f = 3

and f = 4, respectively. [53] Our results in Figure 2.4 are in good qualitative agreement

18


Figure 2.3: Double-logarithmic plot of 〈R2g〉/〈l2b〉 versus N for the star (f = 4).

The symbols correspond to different stiffness parameters B and the lines depict thetheoretical results for different t obtained using Equation (1.9), see text for details.The inset shows the semi-logarithmic plot of 〈R2

g〉/〈l2b〉 versus B for N = 41, 121and 201.

with previous MC simulations of regular semiflexible 3-arm stars of Ida et al. [15] As in the

latter work we observe downward deviations from the ideal chain curves for small values

of the stiffness parameter B. Similarly to ref. [15], with increasing the value of B our

simulation data become almost identical with those for the random coil limit: Namely, in

ref. [15] a sufficiently good agreement of gN for 3-arm stars with a corresponding ideal freely

rotating curve is obtained for t = 0.71 (that corresponds to the angle of 135◦ between

the consecutive bonds); our data both for stars with f = 3 and with f = 4 and stiffness

parameter B = 4 (t = 0.75) also coincides well with the random coil curve.

A comparison of the g-values from different authors is presented in Table 2.1. Here

the g-ratio for stars with f = 3 and f = 4 in a the random coil limit, [58] theoretical values

19


Figure 2.4: Dependence of gN versus log10N for stars f = 3 and f = 4. Thesymbols correspond to simulation data and lines depict theoretical results. Thehorizontal line shows the Gaussian chain limit, see text for details obtained usingEquations (1.9), (1.10) and (1.11), see text for details.

20


reference g, f = 3 g, f = 4[58] (Gaussian) 0.778 0.625[59] (1-order RG) 0.798 0.667[61] (2-order RG) 0.778 0.631this work (MC) 0.760 0.601[15] (MC) 0.760[62] (MC) 0.59[63] (MC) 0.766 0.611[64] (MC) 0.760 0.600[65] (MC) 0.75 0.61[66] (experiment) 0.790[67] (experiment) 0.610

Table 2.1: The values of g -ratio for stars f = 3 and f = 4 in the case ofGaussian statistics, RG calculations under good solvent conditions, obtained fromsimulations and experiments, see text for details.

calculated using the first- and second-order ǫ-expansion, [59,61] gN obtained from our MC

data for the largest flexible stars, values from other MC studies, [15,62–65] as well as exper-

imental results for regular MC f -arm polystyrenes in good solvents [66,67] are presented.

Our values of the g-ratio for stars with f = 3 and f = 4 are in good agreement with

the values obtained from other MC simulations but smaller then for ideal chains. The

theoretical values predicted in ref. [61] are larger then ours as the excluded volume is not

fully developed for the chain lengths we are dealing with.

2.2 Local properties

2.2.1 Interatomic distance

Now we turn to the comparison of local properties of chains and star polymers and study

the influence of single branching point on the local properties of stars in comparison to

chains. We continue our investigation in the spirit of Forni et al. [5], considering the internal

structure of the semiflexible chains and stars. As the influence of the branching point is

expected to be more prominent for large f , [5] we consider here only stars with f = 4. In

Figure 2.5 the normalized mean-square interatomic distance < r2ij > /(n < l2b >) between

the fixed number of bead i and bead j as a function of j/n is presented. Both chains,

f = 2, and stars, f = 4, have the arm length n = 50. The stiffness parameter B takes

values B = 0, 2, 4, and 6. The data for different B is differentiated by color. In each plot

of Figure 2.5 three groups of curves are presented depending on the locations of the fixed

21


Figure 2.5: Normalized interatomic distances < r2ij > /(n < l2b >) versus j/nfor chains, f = 2, with N = 101 (upper plot) and stars with f = 4 and N = 201(lower plot). Three possible locations of the fixed bead i are considered: i is a freeend of one arm, i = 0, and j approaches the free end of another arm, jmax = 2n,(©); i is the central bead of one arm, i = n/2, and j runs up to the centralbead of another arm, jmax = 3n/2, (�); i is at the core of a star, i = n, and japproaches the free end of the arm, jmax = 2n, (△), see text for details. Data fordifferent B-values is color-coded.

22


bead i: In the first group of curves shown in circles, i is at the free end of an arm and j

runs to another free end. In the second group of curves bead i is located at the central

bead of the arm and j runs over the branching point to the central bead of another arm.

In the third group bead i is located at the branching point of the star and j runs to the

free end of the star. We fix the location of the bead i on a certain arm, so for the first

two choices of i the data is averaged over the remaining star arms. In the case, when i is

set in the branching point of a star, the data is averaged over all arms.

In the case of chains with f = 2 (upper plot of Figure 2.5) we observe that the

interatomic distance < r2ij > /(n < l2b >) is smoothly increasing with the topological

separation between beads i and j. In the case of stars with f = 4 (lower plot of Figure 2.5)

we observe a cusp point when j is close to the branching point of the star. This behavior

of interatomic distance for stars with f = 4 qualitatively agrees with findings of Forni et

al. [5] for 12-arm stars. In our case this effect is weaker as the loss of correlation around

the branching point, which will be discussed in the next section, is smaller due to the

smaller functionality of the branching points.

2.2.2 Correlations

We continue our study of the influence of a single branching point (core) on the properties

of the macromolecule. Here we focus on the bond-bond correlation function around the

core of a star. Following the ideas of Forni et al. [5], we investigate the bond-bond corre-

lation function 〈di · ds/|di||ds|〉, when the position of one bond in a star, say di, is fixed

and the position of bond dj in a star varies approaching the branching point and then

running away from it along another arm. Having position of bond di fixed on a certain

star arm, the data for the star is averaged over the remaining arms.

Figure 2.6 shows the behavior of the bond-bond correlation function close to the

branching point for flexible stars (B = 0). Three different choices of the position of

the bond di in a star and n = 20 are considered: i = 6th, 8th and 10th bond from the

branching point (the orientation of the bonds in a star is shown in figure (d) of Figure 1.2).

The red dashed lines with symbols display our data for stars with f = 4, and the solid

red line corresponds to the linear chain of length N = 81, for which the bond at position

i = 10 from the central bead (counterpart of the branching point in the case of stars) is

fixed (the orientation of the bonds in a chain is shown in figure (a) of Figure 1.2). The

black dashed lines with symbols are the simulation data of Forni et al. [5] for the 12-arm

star (also having n = 20) with the following structure: Four bonds with a fixed tetrahedral

geometry are connected by their one end to the branching point; their other ends serve as

secondary branching points for three new arms each, which have no further geometrical

constraint. In the applied off-lattice MC procedure the central 5 beads of a star were kept

23


1 10s

0.01

0.1

1

<d i. d

s/|di||d

s|>

f=4, i=6f=4, i=8f=4, i=10f=2, N=81, i=10Forni: f=12, i=6Forni: f=12, i=8Forni: f=12, i=10

B=0

Figure 2.6: Double-logarithmic plot of bond-bond correlation function 〈di ·ds/|di||ds|〉 versus topological separation s for flexible star with f = 4 and n = 20(red dashed lines with symbols) and chain N = 81 (solid red line). The curves inblack display the previous off-lattice simulations results for 12-arm star of Forni etal. [5] The position of bond di in the star is kept fixed and dj passes the branchingpoint, see text for details.

fixed during the simulations. The same choices of the positions of bond di are considered

and denoted by different symbols.

A first point to notice is that the correlation between the bonds in a star gets larger in

comparison to a chain, when bond dj approaches the branching point but still belongs to

the same arm as the bond di. Moreover, for star with f = 4 the correlation almost does

not change with the topological separation when bond dj is one of the first three bonds

in the arm. A similar but more pronounced behavior was obtained by Forni et al. [5] for

12-arm stars. This effect shows that the star arms are strongly expanded in the vicinity

of branching points compared to the case of chains. This expansion is caused by the

excluded volume interactions due to the high congestion of beads close to the branching

point; and it gets stronger with increasing the functionality of the star.

While for chains the correlation decreases smoothly crossing the central bead, for

24


Stiffness Drop amplitude, f = 3 Drop amplitude, f = 4B = 0 0.3286 0.4967B = 1 0.3406 0.5046B = 2 0.3573 0.5159B = 3 0.3802 0.5255B = 4 0.3826 0.5278B = 6 0.3696 0.5200

Table 2.2: The lost of correlation in the branching point extracted from thedouble-logarithmic plot of 〈di · ds/|di||ds|〉 versus s for 3- and 4-arm star withn = 20 and different stiffness parameters B, see text for details.

stars we observe 2-fold behavior with an abrupt drop down when the bond dj passes the

branching point. This findings also qualitatively agree with the results obtained by Forni

et al. [5]. However, for the 12-arm stars the correlations drop down by almost one order of

magnitude just after j has passed the branching point. [5] In our case for star wuth f = 4

the drop is much smaller. After passing the branching point the correlation becomes small

and this indicates the loss of memory about the initial arm direction. The magnitudes

of the drop of correlation passing the branching point obtained from our MC data for

semiflexible stars with f = 3 and f = 4 are presented in Table 2.2. One can see that with

increasing the functionality f of the star the loss of correlation increases, a fact which

was also predicted by Forni et al. [5] This tendency is observed for all values of stiffness

parameter B.

After crossing the core the correlation decreases smoothly and the data obtained by

the two different simulation methods and for stars with different functionalities almost

coincide.

Finally, let us investigate the influence of semiflexibility on the bond-bond correlation

function close to the branching point of a star in detail. Our findings for 〈di ·ds/|di||ds|〉as a functions of s are shown in Figure 2.7. Here symbols represent the data for the stars

with f = 4 and n = 50. Different stiffness parameters are depicted by different colors.

As in the previous figure, the location of bond di in the star is kept fixed (i = 10 from

the branching point in the upper part of Figure 2.7 and i = 20 from the branching point

in the lower part) and bond j crosses the branching point (again the bonds in a star are

oriented as in figure (d) of Figure 1.2). The dot-dashed line shows the correlation of the

fixed bond di at position i = 5 from the branching point with j running along the star

arm towards its free end. The dashed line represents the bond-bond correlation function

for the intermediate region of the corresponding chain with N = 101. For the bond-bond

correlation function via the branching point the data is averaged over the three remaining

arms of the star and for the correlation within the same arm the average is taken over all

25


Figure 2.7: Double-logarithmic plot of the bond-bond correlation function 〈di ·ds/|di||ds|〉 versus topological separation s for flexible 4-arm star with n = 50 andchain N = 101. The symbols correspond to the bond-bond correlation for the stararound the core: The location of bond di is kept fixed (i = 10 in the upper plotand i = 20 in the lower one) and j passes the branching point. The dot-dashedline displays the correlations withing the arm: the fifth bond from the core is fixedand j goes to the free end of the star. The dashed line shows the correlations inthe linear chain, see text for details.

26


4 arms of the star.

The bond-bond correlation function of a star within the same arm (dot-dashed lines)

deviates upwards with respect to the bond-bond correlation function of a chain (dashed

lines) with increasing the topological separation between the bonds. Thus, the presence

of the branching point influences the correlation even if the distance between the bonds

is rather large. If the bond di, whose position in the star/chain is fixed, is located rather

far from the branching point (say i = 20) and dj approaches the branching point (curves

shown by symbols in the lower part of Figure 2.7), then the correlation decay initially

follows the curve for the corresponding chain (dashed lines). As dj comes closer to the

branching point, the correlation starts to increase in comparison to the one of a chain, as

was already observed for flexible stars in Figure 2.6. Here the effect is present for values of

B ≤ 3, showing that the arms are stretched more in the vicinity of a branching point. We

also want to mention, that this increasing of correlation and, thus, of stiffness of the star

arm near the branching point justifies the theoretical ideas implemented in the partially

stretched star model. [57,68] For large values of the stiffness parameter B, B = 4, 6, we do

not observe a stretching of the star arms near the branching point. The curve for stars

with dj approaching the branching point (symbols) starts to coincide rather well with the

dot-dashed lines, which depicts a correlation in the star arms with dj approaching the

free end of the arm. Thus, we can summarize that with increasing the stiffness parameter

B the role of the branching point in aligning the bonds around it decreases and starts to

be suppressed by the intrinsic chain stiffness.

Similarly to the flexible case studied in Figure 2.6, we observe a 2-fold behavior for the

bond vector correlation function crossing the branching point for all values of the stiffness

parameter B.

2.3 Persistence length

In this section we continue our study of bond-bond correlation functions, however in

another perspective. These quantities are often used for the definitions of persistence

length which is a characteristics of the flexibility of a chain. In what follows we touch

upon two definitions of persistence length:

1. Persistence length which is extracted from exponential decay of the bond-bond

correlation function with the distance between the bonds; [69]

2. The definition of a local persistence length due to Flory. [41]

Both these quantities have attracted much attention recently as their application for

the systems under good solvent conditions was found not to be justified. Recent studies

27


of linear and comb polymers in good solvents show that the standard picture of existence

of a constant persistence length lp is not applicable to systems with excluded volume in-

teractions. [1,2,42,43,70,71] Practically, this notion is valid for strictly ideal chains only. Active

current interest to the influence of excluded volume interactions on the persistence length

of polymers motivates us to reconsider the semiflexible chain where the semiflexibility is

not induced by the presence of side chains as in the case comb-polymers [1,2,70,71] but is

an internal feature of the chain. Moreover, in the case of stars we also investigate the

influence of branching point and presence of additional arms on the persistence length

definitions mentioned above.

2.3.1 Theoretical background

So let us consider definitions of persistence length separately.

1. Persistence length extracted from exponential decay.

It is known from textbooks that for the model obeying Gaussian statistics the correla-

tions between bonds decay exponentially with the contour distance between the bonds. [69]

Thus, persistence length lp of an ideal chain can be extracted from the exponential decay

of the bond-bond correlation function (mean cosine < cos θ(s) > between the bonds di

and dj) with the distance slb along the chain

< cos θ(s) >=<di · dj

|di||dj|>= exp

(

−slblp

)

, (2.1)

where di = ri−ri−1 is a bond vector with ri being the position of the ith monomer in space,

s = |i− j| is the topological separation between the bonds and lb = |di|. [1,69] Persistence

length defined in this way is a universal characteristic of the chains and asymptotically

becomes independent of the chain length.

However, recent simulation studies show that even for dilute θ-solutions, where poly-

mer chains were believed to behave like ideal, Equation (2.1) does not hold, as long-range

connectivity induced correlations of the bond vectors which are far away from each other

in the chain have to be taken into account. [31] Because of the latter the power law decay

of correlations

< cos θ(s) >∼ s−3

2 (2.2)

is observed instead of the exponential one. [31] The same power law decay was found in

melts, where the polymers were assumed to be ideal as well. [29,30]

In the case of good solvent conditions the bond-bond correlation function was predicted

to obey the following scaling relation

< cos θ(s) >∼ s−µ, µ = 2− 2ν (2.3)

28


where ν = 0.588 is the critical exponent in good solvent. [43] The power lower of Equa-

tion (2.3) with µ = 0.824 was recently also confirmed by simulations by Hsu et al. [2]

using PERM algorithm. [72] Furthermore, using renormalization group methods Schäfer et

al. [43] showed that if the chain length is finite the scaling relation of Equation (2.3) is also

strongly affected by the position of the considered bonds in the chain. To account on this

the crossover scaling function has been constructed. Particularly, the scaling function for

the bond vector correlation function was found in the form

h(s) ∼ s−µS̃ (p, q, z̃) , i < j (2.4)

where p =i

s, q =

Nb − j

sand z̃ contains the information about the microscopic structure

of the chain, so that z̃ → ∞ in the excluded volume limit. [43] The function S̃ could be

split up into the two multipliers

S̃ (p, q, z̃) = S̃∞ (z̃) S̄ (p, q, z̃) . (2.5)

In the excluded volume limit z̃ → ∞ the amplitude S̃∞ (z̃) was found either to have the

value S̃∞ (z̃) = 0.111 or S̃∞ (z̃) = 0.102 depending on whether the strict ǫ-expansion or

the approximate value of the fixed-point coupling constant was taken and the expression

for S̄ (p, q, z̃) was obtained analytically. [43]

2. Local persistence length due to Flory.

The alternative and more general definition of persistence length usually addressed to

Flory, [41] which gives it as a local quantity for each bond in a chain, also turns out not

to be suitable under good solvent conditions. Ideally, the persistence length should be

extracted from the plateau of the projection of the end-to-end chain vector Ree on each

unit bond vector jlp(j)

lb= 〈dj ·Ree

|dj |2〉, j = 1, . . . , Nb. (2.6)

For flexible chains under good solvent conditions the following approximation for the

latter quantity was obtained using renormalization group techniques: [42]

lp(j)

lb≈ c

(

j(Nb − j)

Nb

)2ν−1

(2.7)

where Nb is the amount of bonds in the chain and c is a nonuniversal constant. Here one

can easily see that lp(j) for the middle bond Nb/2 is now divergent with increasing chain

length Nb:[2,42]

lp(Nb/2) ∼ clb (Nb/4)2ν−1 → ∞, with Nb → ∞. (2.8)

Recent simulation studies of comb-polymers also confirm the difficulties in application of

29


the Flory definition under good solvent conditions. [1,2,70]

2.3.2 Simulations: Persistence length extracted from exponential

decay

Now we turn to the consideration of our simulation results for the persistence length

definitions presented above. First, we consider the decay of the bond-bond correlation

function (mean cosine between the bonds) with topological separation, investigating the

deviations from the exponential decay of Equation (2.1). In Figure 2.8 the simulation

data for 〈di · ds/|di||ds|〉 versus topological separation s, s = |i− j|, and i (j) is the ith

(jth) bond in a chain, are presented. The length of the chain is N = 101 and the stiffness

parameter B takes values B = 0, 1, 2, 3, 4, and 6. The chain is considered as star f = 2

(see figure (b) of Figure 1.2). The position of bond di is kept fixed at the first bond in

the arm (say i = 1) and the bond dj approaches the free end of the arm. The result is

averaged over the two arms of the star f = 2. In Figure 2.8 symbols correspond to the

simulation data and lines show the predicted scaling behavior according to Equation (2.1)

with lp/lb taken from Table 2.3. The latter were determined from the initial exponential

decay of 〈di ·ds/|di||ds|〉. One can see that for all values of B this standard fit is obviously

not applicable for the large distances between the bonds: The decay of the bond-bond

correlation function with increasing s becomes slower then the exponential one for large

values of s. For large value of stiffness parameter B and small s the fit gets better but also

slows down as the topological separation increases. This results are in a good agreement

with previous simulation results for simiflexible chains, [70] where authors considered larger

stiffness values but used another MC simulation method.

The inset of Figure 2.8 presents the dependence of lp/lb versus B for chains of length

N = 101. One has to note, that in the intrinsically flexible case, B = 0, the chains

possesses the persistence length lp/lb = 0.61. This is in agreement with the RG theoretical

predictions, [42] where self-repelling chains in good solvents are expected to have a nonzero

persistence length due to the excluded volume interactions. The previous simulation

studies of ref. [70] report about the value of lp/lb = 0.569 for flexible excluded volume

chains, which also agrees with our result.

Figure 2.9 shows analogical dependence for 〈di · ds/|di||ds|〉 versus topological sepa-

ration s, as in Figure 2.8, for chains of length N = 101 and for stars with functionality

f = 3 and f = 4 and arm length n = 50. Again chains are considered as stars (figure

(b) of Figure 1.2) and orientation of bonds in the star are as in figure (c) of Figure 1.2.

As in Figure 2.8, the position of bond di is fixed at the first bond in the star arm and j

runs towards the end of the arm. Te correlations are averaged over the amount of stars

arms. In Figure 2.9 three groups of curves, which are distinguished by color, correspond

30


Figure 2.8: Semi-logarithmic plot of 〈di ·ds/|di||ds|〉 versus s, where s = |i− j|is a topological separation between bonds i and j, for chains with N = 101. Thebonds in a chain are numbered as in case of two-arm star. The first bond in thearm (exp. i = 1) is kept fixed and the bond j approaches the free ends of the arm.The symbols correspond to the simulation data for different stiffness values B andlines depict the initial exponential decay exp (−slb/lp) with lp/lb from Table 2.3.The inset shows dependence of lp/lb in B for the linear chain taken from Table 2.3,see text for details.

to the stiffness parameters B = 0, 3 and 6. Different symbols correspond to different func-

tionalities of branching point: Circles stand for the chains, f = 2; triangles and crosses

depict for the stars with core functionalities f = 3 and f = 4, respectively. Now, for stars

we observe even more slow decay of the bond-bond correlation function in comparison

to chains: The larger the functionality f of the branching point, the more slowly decays

the bond-bond correlation function and, thus, the more memory remains in the system

about bonds’ orientation. This effect is more prominent for B = 0; whereas for large B,

say B = 6, gets suppressed by the local chain stiffness. From the Table 2.3 comparing

values of lp/lb for chains, f = 2, with the one for stars, we can assume that presence of

additional arms in the case of 3- and 4-arm stars do not effect significantly the initial

31


Blp/lb,f = 4

lp/lb,f = 3

lp/lb,f = 2

0 0.61 0.61 0.61

1 1.19 1.18 1.16

2 1.89 1.88 1.87

3 2.77 2.76 2.77

4 3.78 3.74 3.73

6 6.10 6.08 6.05

Table 2.3: The persistence length lp/lb for stars with f = 3, 4 and n = 50, andchains (f = 2) with N = 101 for different values of stiffness parameter B.

exponential decay. However, presence of branching points (the junction of functionality

f = 3 and f = 4) and additional arms strongly influences the bond-bond correlation func-

tion on rather large distances. This shows that influence of the excluded volume which

is stronger for stars with f = 3 and f = 4, then for chains, f = 2, affects the whole arm

length of the star.Following the ideas of ref. [31], in Figure 2.10 we plot 〈di ·ds/|di||ds|〉 versus topological

separation rescaled by persistence length, slb/lp. Here the data for stiffness parameters

B = 0, 1, 2, 3, 4, and 6 are presented and color-coded. Chains, f = 2, and stars, f = 3

and f = 4, are of the same size as in Figure 2.9 and the data are depicted using the same

symbols. Also positions of the bonds di and dj in the star arm are taken the same as in

Figure 2.9. Again the persistence lengths lb/lp of a chain (star arm) was determined from

the initial exponential decay of the curves (that holds better for more intrinsically stiff

polymers) and are taken from Table 2.3. More slow decreasing of correlations with rescaled

topological separation, slb/lp, for higher functionalities of branching point is highlighted

in this figure. Qualitatively this behavior of bond-bond correlation function in Figure 2.10

is very similar to the one obtained for dilute θ-solutions. [31] However, the scaling behavior

for the latter is vastly different and follows the scaling law of Equation (2.2). [31]

From our simulation data presented above we summarize that application of Equa-

tion (2.1) for the definition of persistence length is not applicable for the systems under

good solvent condition. Now we verify the theoretical predictions of power law decay

of the bond-bond correlation function for flexible chains under good solvent conditions

(Equation (2.3)). However one needs to remark, that predicted power law decay in Equa-

tion (2.3) holds in the infinite chain limit. [43] As it was already shown by RG theory [43]

and simulations, [2,5,43] free ends of a chain are more flexible and thus, strongly influence

the decay of correlation. Being interested in this work mainly in the study of effects of

semiflexibility on the properties of chains/star polymers we perform here simulations for

rather short chain/arm lengths for which these end effects are not negligible. Thus, in

32


0 10 20 30 40 50s

0.001

0.01

0.1

1<

d i. ds/|d

i| |d s|>

f=2, B=0f=2, B=3f=2, B=6f=3, B=0f=3, B=3f=3, B=6f=4, B=0f=4, B=3f=4, B=6

Figure 2.9: Semi-logarithmic plot of 〈di ·ds/|di||ds|〉 versus s, where s = |i− j|is a topological separation between bonds i and j, for chain with N = 101 andstars with f = 3 and f = 4, and n = 50. The bonds in a chain are numbered as incase of two-arm star. The first bond in the arm (for example i = 1) is kept fixedand the bond j approaches the free ends of the arm. The symbols correspond tothe simulation data for different core functionalities f , and different stiffness valuesB are color-coded: Black, blue and violet stand for B = 0, 3 and 6 respectively,see text for details.

order to verify Equation (2.3) and to decrease the loss of correlation due to the proximity

of the free ends, we fix the position of bond di close to the middle of the chain while

the bond dj runs up to the jmin = 5, the 5th bond from the closer chain end. For the

sake of comparison with the previous MC simulations of Forni et al. [5] we take i to be the

17th bond from the center of the chain (the bonds in the chain are oriented as in figure

(a) of Figure 1.2). We average the data over the chains conformations. The Figure 2.11

shows in the double-logarithmic plot the obtained dependence of the 〈di · ds/|di||ds|〉on topological separation s for flexible chains of different lengths. We observe the same

rapidly decreasing strongly curved function, predicted by Schäfer et al. [43] and reported in

33


0 20 40 60 80sl

b/l

p

0.0001

0.001

0.01

0.1

1<

d i. ds/|d

i| |d s|>

f=2, B=0f=2, B=1f=2, B=2f=2, B=3f=2, B=4f=2, B=6f=3, B=0f=3, B=1f=3, B=2f=3, B=3f=3, B=4 f=3, B=6f=4, B=0f=4, B=1f=4, B=2f=4, B=3f=4, B=4f=4, B=6 exp(-sl

b/l

p)

Figure 2.10: Semi-logarithmic plot of 〈di · ds/|di||ds|〉 versus slb/lp, wheres = |i − j| is a topological separation between bonds i and j, for chain withN = 101 and stars with f = 3 and f = 4, and n = 50. The bonds in a chainare numbered as in case of two-arm star. The first bond in the arm (exp. i = 1)is kept fixed and the bond j approaches the free ends of the arm. The symbolscorrespond to the simulation data for different core functionalities f , and differentstiffness values B are color-coded. The dashed line corresponds to the exponentialdecay exp (−slb/lp) with lp/lb from Table 2.3, see text for details.

the simulation studies. [2,5,63] The resulting scaling exponent in our study for the longest

chain N = 320 is µ = 0.892 which is larger then the theoretically predicted µ = 0.824,

but smaller then the 1.15 and 0.95 obtained in ref. [63] and ref. [5], respectively. The recent

studies show that in order to provide convincing evidence of the theoretically predicted

exponent µ = 0.824 one has to go into the simulation of significantly longer chains. [2] In

the latter study flexible chains of up to Nb = 6400 were investigated in order to obtain

exponent µ. For the shorter chains the effect of the chain ends causes the rapid decay

of the bond vector correlation function in the range s > Nb/10 in the double-logarithmic

plot [2] that decreases the value of the universal exponent ν, which in our case is ν = 0.554.

In Figure 2.12 the double-logarithmic plot of the bond-bond correlation function 〈di ·

34


0 0.25 0.5 0.75 1 1.25 1.5 1.75 2log

10s

-3

-2.5

-2

-1.5

-1

-0.5

0

log 10

[<d i. d

s/|di| |

d s|>]

N=81, i=24, jmin

=5N=101, i=34, j

min=5

N=151, i=59, jmin

=5N=257, i=109, j

min=5

N=320, i=143, jmin

=5

slope -0.892

f=2

Figure 2.11: Double-logarithmic plot of 〈di · ds/|di||ds|〉 versus s, where s =|i− j| is a topological separation between bonds i and j, for the flexible chains ofdifferent lengths. The beads are numbered starting for the free end. The bond iis kept fixed while j approaches the closer free end of the chain and its minimumvalue is jmin = 5. The violet line corresponds to the power law fitting s−µ withµ = 0.892, see text for details.

ds/|di||ds|〉 versus s for flexible chains, f = 2, and 4-arm stars, f = 4, is compared with

theoretically predicted RG corrections to scaling for chains [43] and with simulation data

of Forni et al. [5] for chains and 12-arm stars. To make a link with the latter work we take

chains of length N = 81 and stars with arm length n = 40. Again position of bond di

is kept fixed at the 17th bond from the center of the chain and bond dj approaches the

closer free end of the chain/ star arm. The bonds in chains are oriented as in figure (a)

of Figure 1.2. The orientation of bonds in stars is as in figure (c) of Figure 1.2.

Our MC data for chains and stars is depicted in Figure 2.12 by red empty circles and

green squares respectively. The data shown by black symbols presented MC data from

ref. [5] for chains and 12-arm stars. Our simulation data for chains with N = 81 (red

35


1 10s

0.001

0.01

0.1

<d i. d

s/|di| |

d s|>

Theory: f=2, N=81, i=24, jmin

=5

This work: f=2, N=81, i=24, jmin

=5

This work: rescaled simulation data for f=2, N=81This work: f=4, n=40, i=17, j

max=36

Forni et al.: f=2, N=81, i=24, jmin

=5

Forni et al.: f=2, N=473, i=138, jmin

=5

Forni et al.: f=12, n=40, i=17, jmax

=36

Figure 2.12: Double-logarithmic plot of 〈di · ds/|di||ds|〉 versus s, where s =|i−j| is a topological separation between bonds i and j for the flexible linear chainN = 81 (red empty circles) and 4-arm star f = 4 polymer with n = 40 within thesame arm (green squares). The bond i is kept fixed while j approaches the closerfree end of the chain/star arm and arrives at a four-bond distance from the freeend. The filled red circles correspond to the simulation data for the linear chaindivided by factor 1.8. The blue line is the theoretical prediction [43] for the chain;by the black symbols the previous simulation results [5] are denoted, see text fordetails.

empty circles) are in a good agreement with the previous off-lattice simulations of Forni

et al. [5] (black filled triangles). Analyzing data from ref. [5] for 12-arm stars and chains

(black filled triangles and black empty squares respectively) we observe the same tendency

of stronger correlation at large s for stars relative to the case of chains, as was observed

above in Figures 2.9 and 2.10. We also notice that bond-bond correlation function for

stars f = 4 (green squares) is smaller then for 12-arms stars from ref. [5] This is also in

agreement with previously mentioned effect of stronger excluded volume interaction for

stars with larger functionality of the branching point.

The blue curve in Figure 2.12 presents theoretical bond-bond correlation function for

36


chains obtained with RG approach. [43] For calculation of the theoretical scaling relation

from ref. [43] we took S̃∞ (z̃) = 0.102, [43] and the same chain length N = 81, as for the

simulations. The curve which corresponds to our simulation data for chains (red empty

circles) lies above the theoretical scaling relation (Equation (2.4)). Dividing the simulation

data by the factor 1.8 we can report about the good agreement with the theoretical

predictions [43] (the shifted data is represented by the filled red circles on the plot).

2.3.3 Simulations: Local persistence length due to Flory

In Equation (2.1) it was assumed that the stiffness is uniform along the chain length. But

this is not valid for the semiflexible objects studied here as in our case chain/arm ends

have more freedom to move in comparison to the segments which are in the middle. Thus

the persistence length can vary along the chain/arm. The definition of persistence length

due to Flory, Equation (2.6), accounts for these effects. In the latter persistence length is

defined locally as projection of the end-to-end vector of a chain on each segment vector:

for the bonds which are close to the chain free ends persistence length should decrease

due to their higher flexibility and for the middle part of the chain it is supposed to have a

plateau which is independent of the bond number. The height of the plateau defines the

persistence length of the chain.

As it was shown in ref. [2], this expectations hold rather well for the chains at the

θ-temperature where the persistence length is a constant for the interior part of the chain

and independent of the chain length Nb, while in a good solvent the value of persistence

length obtained from the plateau is divergent with increasing Nb (see Equation (2.8)).

N c ν

46 1.546±0.038 0.601± 0.005

81 1.559± 0.024 0.599± 0.003

121 1.563± 0.017 0.599± 0.002

257 1.641± 0.014 0.589± 0.001

Table 2.4: The fitting parameters obtained using Equation (2.7) for flexible chainsof different lengths, see text for details.

Figure 2.13 presents dependence of lp(j)/lb on j/Nb for flexible chains of length N =

46, 81, 121, and 257 (symbols). The solid lines display fitting using Equation (2.7). The

simulation data follows the theoretical predictions from ref. [42] rather accurately. The

good agreement with Equation (2.7) was shown also by similar but more extensive study

37


0 0.2 0.4 0.6 0.8 1j/N

b

1

1.5

2

2.5

3

3.5

l p(j)/

l b

N=46N=81N=121N=257

υ=0.601

υ=0.599υ=0.599

υ=0.589

Figure 2.13: The persistence length obtained using Equation (2.6) for flexiblechains of length N = 46, 81, 121, and 257. The simulation data for different chainlengths are shown by symbols and the solid lines show fitting using Equation (2.7),see text for details.

for flexible chains with lengths up to Nb = 6400 performed by Hsu et al. [2] using PERM

algorithm. [72] The reported corresponding fitting parameters obtained are c = 1.6888 and

ν = 0.5876. [2] The fitting parameters for c and ν obtained from our simulation data are

collected in Table 2.4. One can see that already for the chain length N = 257 our values

both for c and for ν are close to those of Hsu et al. [2] Moreover, from the data of Table 2.4

it is clearly seen that with increasing chain length the value of scaling exponents ν slightly

decrease, while the value of c increase. This qualitative dependence of c and ν on the chain

length agrees with the tendencies observed for comb-polymers in ref. [2]: While the value

of ν for the backbone length Nbb = 131 of comb-polymers is larger then the corresponding

scaling exponent for Nbb = 259, the parameter c for Nbb = 131 is smaller then c for

Nbb = 259. This dependence was observed for all side chain lengths considered. [2]

From the Figure 2.13 it is apparent that Flory definition for persistence length is not

38


0 20 40 60 80 100j

0

2

4

6

8

10

12lp(j)/l

bf=4, B=0f=4, B=1f=4, B=2f=4, B=3f=4, B=4f=4, B=6f=3, B=0f=3, B=1f=3, B=2f=3, B=3f=3, B=4f=3, B=6f=2, B=0f=2, B=1f=2, B=2f=2, B=3f=2, B=4f=2, B=6

Figure 2.14: Local persistence length lp(j)/lb versus number of the bond j forchains, f = 2, and stars with f = 3 and f = 4, and n = 50 shown by black, redand green symbols respectively, see text for details.

applicable for chains under good solvent conditions: With increasing chains length N we

observe the increasing of the height of plateau, the effect which qualitatively agrees with

Equation (2.8).

In the case of stars the situation turns out to be even more complicated. Figure 2.14

illustrates the dependence of lp(j)/lb versus j for the chains, f = 2, (black symbols) and

stars with f = 3 and f = 4 with n = 50 (red and green symbols respectively). Different

values of stiffness parameter B = 0, 1, 2, 3, 4 and 6 are depicted by different symbols.

As we are interested in the influence of branching points and presence of additional

arms on the local persistence length, we consider two arms of the star as a linear chain.

For this purpose the bonds are renumbered starting from one free end of the arm and

running up to the end of the other arm (see figure (d) of Figure 1.2). Instead of the

end-to-end vector of a chain, in the case of stars we take a vector which connects two last

39


beads of two star arms. Its normalized projection on each bond of selected two arms is

compared with local persistence length of a chain. One arm is always taken the same, so

that the data in Figure 2.14 is average over the remaining f − 1 arms: Thus, for each

curve of Figure 2.14 the data for j ≤ 50 are averaged over star conformations, and the

data for j > 50 are averaged over f − 1 arms and conformations.

For flexible stars lp(j)/lb increases toward the branching point (bonds j = 50 and

j = 51) relative to lp(j)/lb for flexible chains. Upward deviations are also observed close

to the arm ends, showing that influence of branching point and additional arms affects

all bonds in the star. It is in agreement with the results of Section 2.2.2 (see Figure 2.10

for example), where influence of excluded volume caused by higher steric congestion of

the monomers acts on bonds which are rather far from the branching point. For the same

reason the upward deviations of lp(j)/lb for the stars f = 4 is larger then for the case

f = 3.

Turning now to the influence of semiflexibility on the local persistence length of chains

we observe increasing rigidity for all parts of the chain with increasing bending penalty.

This is in a qualitative agreement with the MC simulations on the basis of Metropolis al-

gorithm [52] for the bead-spring model. [70] For semiflexible stars the situations turns out to

be much more complicated. For B > 0 we observe cavities in curves close to the branching

points. This happens against a background of an increase of lp(j)/lb towards the core of

the star. The cavity gets larger as B increases. We can explain this in the following way:

As stiffness increases, the value of stiffness in the branching point, qc, approaches the

value, which corresponds to the stiff limit in branching point which is characterized by

symmetrical distribution of the first bonds around the center of the star. [53] For star with

f = 3 qc takes the value qc = 0.5 which corresponds to angle θ = 120◦ between the first

bonds in the star arms. The corresponding values for stars with f = 4 are qc = 0.33 and

θ ∼ 109.47◦. While for flexible stars, B = 0, qsimc obtained from simulations (0.16 and

0.14 for stars with f = 3 and f = 4 respectively) do not differ much from t = 0.19 valid

in the central part of the chain/star arm, the difference between qsimc and t increases with

the increasing B: Exemplary for B = 6 t = 0.84 in the star arms and qsimc = 0.42 and

qsimc = 0.3 for f = 3 and f = 4, respectively, in the branching point (see Table 1.1). Thus

close to the branching point we observed an interplay between the intrinsic stiffness of

the arm described by parameter t and influence of the branching point, where the angles

between the bonds connected to it are characterized by qsimc . The role of this factor is

already noticeable for B = 1, where the same tendency of the projection increasing is

observed for all the bonds except the first two in the arms, and becomes more significant

for the larger B. For the large values of bending penalty the excluded volume starts to be

suppressed by the intrinsic stiffness and we observe a strong decay of the projection with

j approaching the core due to the smaller value of the average angle between the bonds

40


in the branching point in comparison to the average angle in the middle part of the arm.

41

Chapter 3

Rings and Trefoils

In the previous chapter we have considered semiflexible linear and branched polymers

which are the simplest examples of tree-like polymers. The theoretical treatment of these

objects, besides applying STP-model, [7,23,24] can be alternatively based on MEP. [7,24] How-

ever this method was also found to be suitable and well applicable to semiflexible ring

polymers. [7,8] Thus, being motivated by recent achievements in theoretical description of

semiflexible rings we decided to verify theoretical results and to perform simulation study

of these objects taking into account effects of excluded volume interactions which are

omitted in the theory. In this chapter we study semiflexible unknotted rings and rings

with one knot known as trefoils.

The simulation setup for unknotted rings and trefoils is the following.

Ring: The amount of monomers in the unknotted ring takes the values N = 16, 32,

64, 128, 256, 512 and the values of stiffness parameter are B = 0, 3, 6. The initial ring

configuration is taken to be a square with all bond lengths equal to 2 lattice units.

Trefoil: The length of trefoil takes the values 32, 64, 128, 256, 512 and stiffness

parameter takes the values B = 0, 3, 6. For the length N = 64 in order to investigate the

role of excluded volume on the bond-bond correlation function we also perform simulations

with switched off excluded volume interactions for beads. An example of the initial

configuration of trefoil is presented in Figure 3.1, where each bead stands for the lower

left front corner of the coarse-grained monomer and all bond lengths are equal to 2 lattice

units. For the purpose of generalization of the way the initial configuration for different

trefoil lengths is build, we distinguish three types of beads, depending on the part of the

initial configuration they belong to. The beads denoted by gray color correspond to the

”ring” part of the trefoil. This ”ring” is represented as a rectangle in xy plane with 2 sides

having 2k bonds and other 2 containing 2 bonds each. Thus, the amount of gray beads

is 2(2k+1)+ 2. Ten red beads correspond to the realization of the knot and this number

does not change with changing the length of the object. The blue beads represent the

last ”loop” of the trefoil which is built of 2 sides parallel to z axes and having m bonds

42

Chapter 3: Rings and Trefoils

yz

x

Figure 3.1: Initial configuration of knotted ring (trefoil) with N = 32 beads(each bead represents a BFM cube), see text for details.

each. This sides are connected by 3 bonds in the xy plane. The amount of beads in the

loop is 2m + 2. Thus the total amount of beads in trefoil is N = 4k + 2m+ 16. For the

reason of convenience we take the following dependence of k and m on N in the initial

configuration of trefoil: k = 3(N − 16)/16 and (N − 16)/8.

Both for ring and for trefoil the size of simulation box is 200×200×200 lattice units for

N = 16, 32, 64; 400×400×400 for N = 128 and 256; and 800×800×800 for N = 512. We

implement periodic boundary conditions in the x, y, and z directions. The configurations

were equilibrated between 109 and 3 × 109 MCS depending on the size of the structures

and stiffness: the data were averaged over between 106 to 2× 106 realizations. We store

conformations in intervals of 1000 MCS.

3.1 General properties

3.1.1 Unknotted rings

We start our consideration by investigating the normalized mean-square radius of gyration

of unknotted rings and of trefoils for the stiffness parameters B = 0, 3 and 6. The MC

results for 〈R2g〉/〈l2b〉 as a functions of N for unknotted rings are presented in Figure 3.2.

The values obtained from simulations are computed using Equation (1.6), and are pre-

sented through open circles for the unknotted rings with excluded volume interactions and

through solid triangles for the flexible unknotted rings with turned of excluded volume.

43


10 100 1000N

1

10

100

1000

<R

g2 >/<

l b2 >

B=0B=3B=6B=0, no EV

~N1.20

~N0.99

Unknotted ring

Figure 3.2: Double-logarithmic plot of 〈R2g〉/〈l2b〉 versus N for semiflexible un-

knotted rings for different values of stiffness parameter B, B = 0, 3 and 6. Theopen symbols stand for excluded volume rings, while filled triangles present our MCdata for flexible unknotted rings with switched off excluded volume interactions.The lines show the obtained power fitting, see text for details.

The solid and dashed lines in Figure 3.2 show fits to the simulation data by power laws

(Equation (1.7)), where the scaling exponents equal 1.20 ± 0.01 for the flexible unknot-

ted rings and 0.99 ± 0.01 for the rings without excluded volume interaction. The value

for excluded volume rings is in good agreement with the scaling exponents reported in

ref. [11], which are 1.176 for the unknotted rings, and with exponent 1.18 obtained from the

mean-square diameter in ref. [14] For small values of N with increasing stiffness parameter

B the power law gets larger then for the flexible unknotted ring. However, for large N

the excluded volume interactions become more important and exponent approaches the

value of the one for flexible ring.

In the next figure (Figure 3.3) we compare our MC data for unknotted rings with

excluded volume interaction to the theoretical predictions obtained using MEP by M.

Dolgushev. [7,8] The circles depict the same data for unknotted rings as in Figure 3.2

44


1 1.5 2 2.5 3log

10N

0

0.5

1

1.5

2

2.5

3lo

g 10[<

Rg2 >

/<l b2 >

]

B=0, theoryB=3, theoryB=6, theoryB=0, simulationsB=3, simulationsB=6, simulations

Figure 3.3: Double-logarithmic plot of 〈R2g〉/〈l2b〉 versus N for semiflexible un-

knotted rings. Circles stand for simulation results and crosses show the theoreticalpredictions obtained using MEP by M. Dolgushev, [8] see text for details.

and crosses stand for theoretical values of 〈R2g〉/〈l2b〉. The latter are evaluated based on

Equation (1.8) using t values from Table 1.2. For flexible unknotted rings, B = 0, we

observe upward deviations of MC data from its theoretical counterpart for all ring lengths.

For B = 3 and B = 6 and rather small N the agreement with theory is very good as

excluded volume interactions are suppressed by the local stiffness. For large N we again

observe upward deviations from theoretical results due to the increasing role of excluded

volume interactions.

3.1.2 Trefoils

Now we turn to investigation of the normalized mean-square radius of gyration for trefoils.

The MC data for trefoil for different values of stiffness parameter B, which are color-

coded, are shown in Figure 3.4. Again, the line shows fit with power law. The value

45


10 100 1000N

1

10

100

1000

<R

g2 >/<

l b2 >

B=0B=3B=6

~N1.27

Trefoil

Figure 3.4: Double-logarithmic plot of 〈R2g〉/〈l2b〉 versus N for semiflexible trefoils

for different values of stiffness parameter B, B = 0, 3 and 6. The line shows theobtained power fitting, see text for details.

of the obtained scaling exponent is 1.27 ± 0.01 which is in a very good agreement with

scaling exponent 1.266 from ref. [11] Both values for scaling exponent are somewhat larger

then the typically expected value of 2ν = 1.176. [11] One has to mention in this relation,

that extrapolation results of Mansfield et al. [11] show that for large N the same scaling

exponent (2ν = 1.176) is expected to be valid for all knot types. However, so far there is

no direct MC evidence for this universal exponent, while simulation data display a very

slow convergence. [11]

46


3.2 Local properties

3.2.1 Unknotted rings

Now we turn to the bond-bond correlation function for ring polymers. This quantity

allows direct comparison of the theoretical results with simulations for unknotted rings.

As the MEP applied to semiflexible rings reveals knotted topologies only in the stiff

limit, we can compare our simulation data for trefoils only qualitatively with theoretical

predictions for these objects.

We start our consideration from unknotted rings. The values of the theoretical stiffness

parameter t, which corresponds to the stiffness parameter B used in the simulations, are

presented in Table 1.2. One can notice immediately that for the case B = 0 we have a

significant value of t which is certainly expected due to the excluded volume interactions

which are strictly taken into account in BFM. Moreover, as it was already mentioned in

Section 1.2, the correlation t between successive bonds in the unknotted ring for the given

B depends on the length of the ring and for large N it becomes almost a constant.

In Figure 3.5 we display the dependence of the bond-bond correlation functions on the

distance between the segments. The correlation function is considered between the first

bond in the ring, d1, and successive bonds dj . Thus, the distance between the segments

varies between unity and N . The upper part shows the results for the flexible unknotted

rings, B = 0, which implies that angular interactions between the segments are absent.

The lower part of the figure shows again the data for flexible unknotted rings but for the

case when excluded volume interactions for beads were switched off in the BFM scheme.

In both figures we present data for N = 16, 32, 64, 128 and 256. The data for N = 512

are omitted here as is does not differ much from the case N = 256.

N = 8 N = 16 N = 32 N = 64 N = 128 N = 256

B = 0 0.101 0.172 0.193 0.200 0.203 0.204B = 3 0.428 0.530 0.560 0.570 0.576 0.578B = 6 0.574 0.721 0.751 0.761 0.765 0.767

Table 3.1: Effective stiffness parameters t′ of unknotted rings calculated from thesimulations. [8] Here each theoretical segment is considered as two BFM segments.

For unknotted rings under excluded volume conditions (upper part of Figure 3.5) the

correlations at short range are positive. With increasing the distance between the bonds so

that the bond j approaches the bond dN/2 the correlation becomes negative. This result

can be understood intuitively as for an unknotted ring the bonds which are opposite

to each other along the ring, i.e. bonds d1 and dN/2, are expected to have negative

correlation. For small N this effect is more pronounced while for large N the simulation

47


Figure 3.5: Bond-bond correlation functions for flexible rings of different lengths.The symbols show simulation data and the lines depict the correspond theoreticalpredictions obtained using MEP by M.Dolgushev. [8] In the upper figure the ex-cluded volume flexible ring is presented; in the lower figure the excluded volumeinteractions are turned off, see text for details.

48


Figure 3.6: Bond-bond correlation functions for semiflexible rings of differentlengths. The symbols show simulation data and the lines depict the correspondtheoretical predictions obtained using MEP by M.Dolgushev. [8] The upper andlower figures correspond to the stiffness parameters B = 3 and B = 6 respectively,see text for details.

49


Figure 3.7: New bond vectors d̃i are shown in green, taken as the sum of twotypical BFM bonds depicted by blue arrows, see text for details.

curve becomes flatter. We explain this as interplay between the closure condition for

rings, d1+d2+ · · ·+dN = 0, and excluded volume interactions: For short rings excluded

volume interactions are dominant, while for longer rings the closure condition plays the

main role. [7,8]

Apart from the simulation data, the upper part of Figure 3.5 contains also theoretical

results for the corresponding t-values from Table 1.2. [7,8] The color of the theoretical

curves is the same as for their simulation counterparts. For large N there is a qualitative

agreement between theory and simulations, however the agreement gets worse for small

N . One can assume that the role of closure condition is overestimated in the theory and

presence of excluded volume, which is not accounted for in the theoretical framework,

slows down decay of correlation. In what follows we show that the agreement between

theory and simulations gets better if we include the bending energy in the BFM scheme

having B 6= 0, and when we modify our interpretation of the basic unit bond in the BFM.

In the lower part of Figure 3.5 we show our simulation data for unknotted rings

with switched off excluded volume interaction. Here bond-bond correlation function is

independent of N for all ring lengths displayed (16, 32, 64, 128 and 256). The solid lines

50


0 0.2 0.4 0.6 0.8 1(j-1)/N

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

<d 1. d

j>/<

l b2 >

N=8N=16N=32N=64N=128

B=0

Figure 3.8: Bond-bond correlation function for flexible rings of different lengths.The symbols show simulation data and the lines depict the correspond theoreticalpredictions obtained using MEP by M.Dolgushev [8], see text for details.

present the corresponding theoretical flexible ring limit, t∞ = −1/(N − 1). [8] One has

to underline that in contrast to the ideal flexible chains, where t = 0, for the flexible

rings there is correlation between the neighboring bonds and one has to use t∞ instead of

t = 0. Having N → ∞ one gets t∞ → 0, which is a standard assumption for the stiffness

parameter t for flexible chains. [8] In Figure 3.5 we obtained a very good agreement between

theory and simulations and in this way we confirm that MEP applied to semiflexible rings

gives a correct flexible limit for rings.

In the next figure (Figure 3.6) we again present the bond-bond correlations functions of

unknotted rings but now with bending potential, Equation (1.4). The considered stiffness

parameters are B = 3 (upper part) and B = 6 (lower part of Figure 3.6). The lengths

of the rings are the same as for the flexible case discussed above. With increasing B the

bonds which are located oppositely along the ring (say d1 and dN/2) tend to orient more

and more antiparallel to each other. This effect is more pronounced for smaller N and

51


Figure 3.9: Bond-bond correlation function for semiflexible rings of differentlengths. The symbols show simulation data and the lines depict the correspondtheoretical predictions obtained using MEP by M.Dolgushev. [8] The upper andlower figures correspond to the stiffness parameters B = 3 and B = 6 respectively,see text for details.

52


0.001 0.01 0.1(1-(j-1)/N)(j-1)/N

1

10

100[<

d 1. dj>

/<l b2 >

]N2-

2v+

7.66

[(1-

(j-1

)/N

)(j-

1)/N

]2-2v

N=64N=128N=256N=512

0.185[(1-(j-1)/N)(j-1)/N]2-2v

B=0

Figure 3.10: Double-logarithmic plot for the bond-bond correlation functionrescaled according to the Equation (3.1). The symbols show simulation data forseveral ring lengths and the blue line depict the first term in Equation (3.1) withA = 0.185, see text for details.

washes out for long rings.

We also compare theoretical bond-bond correlation function shown by solid curves to

the simulation data in Figure 3.6. The corresponding t-values are as previously taken

from Table 1.2. The agreement gets better, especially for rather small or for very large

rings. It could be explain in the following way: For short rings local stiffness is dominating

and excluded volume interactions are getting more suppressed, while for very large rings

the closure condition provides a sufficiently good description of simulation data by theory.

The worse agreement for intermediate ring lengths may be due to the role of the segments’

thickness, which is quite high in the BFM.

In order to verify this idea we introduce new correspondence between the theoretical

segment and segment in simulations. Now each theoretical segment, say d̃a, is represented

by two BFM segments d̃a = d2a−1+d2a, see Figure 3.7. A new normalized nearest neighbor

bond-bond correlation functions 〈d̃i · d̃i+1〉/〈l̃2b〉 as a function of N for B = 0, 3 and 6 are

presented in Table 3.1. As previously we use these values as an estimate for t.

53


In Figures 3.8 and 3.9 we again confront the results from the simulations with the

theory. Figure 3.8 present the case of flexible unknotted ring (B = 0) and in Figure 3.9

the upper part presents the data for the stiffness parameters B = 3 and the lower part

corresponds to the case B = 6. The amount of new segments in the ring is now 8, 16,

32, 64 and 128. While for the flexible unknotted ring the agreement between theory and

simulations still remains only qualitative, for the semiflexible rings it becomes now very

satisfactory for the whole range of parameters involved. We would like to stress that

for the rings which are considered the correlation functions display a clear-cut negative

minimum around j = N/2, which is vastly different from the case of a linear chain where

the correlation functions are always positive. This pronounced minimum is a clear sign

of an unknotted ring, for which the angular potential is quite strong, the interpretation

being that of a rather flat geometrical arrangement of the segments, in which segments

at a distance of one half of the contour length have very evident antiparallel orientations.

So far we were comparing our MC data for unknotted rings with theoretical pre-

dictions based on the MEP. [7,8] However, the latter theory does not take into account

excluded volume interactions. In the next figure (Figure 3.10) we present scaling analysis

of the bond-bond correlation function for flexible self-avoiding unknotted rings proposed

in ref. [73] In the latter work the bond-bond correlation function was expected to obey the

following scaling relation

< d1 · dj/|d1||dj| > ∼ 1

N2−2ν

(

A/

[

j − 1

N

(

1− j − 1

N

)]2−2ν

−

B

[

j − 1

N

(

1− j − 1

N

)]2−2ν)

, (3.1)

where A and B are constants. In this way bond-bond correlation function decomposes

into a short range part, which is positive defined and a long range negative part. For small

j, such as j−1 ≪ N , Equation (3.1) leads to the decay of bond-bond correlation function

according to 1/|j − 1|2ν−2. In Figure 3.10 we perform estimations of the constants A and

B from our MC data following the way used in ref. [73] Namely, we compare the MC data

for j = 2 and j = N/2+1 for different N with Equation (3.1). We use the value ν = 0.588

for the critical exponent. The obtained scaling with constants A = 0.185 and B = 7.66

is presented in Figure 3.10. The plot reveals that the scaling function (Equation (3.1))

coincides rather well with our MC data. Using another MC technique, Baumgärtner [73]

found also very accurate agreement with Equation (3.1) taking the constants A = 0.142

and B = 5.7.

54


0 0.2 0.4 0.6 0.8 1(j-1)/N

-0.4

-0.2

0

0.2

0.4

0.6

0.8

<d 1. d

j>/<

l b2 >

B=0, without EVB=3, without EVB=6, without EVB=0, with EVB=3, with EVB=6, with EV

N=64

Figure 3.11: Bond-bond correlation function for trefoils N = 64 for differentstiffness parameters B obtained from BFM simulations with excluded volume in-teractions turned on (symbols) and turned off (lines), see text for details.

3.2.2 Trefoils

We now turn our attention to trefoils, i.e. to rings displaying one knot. In Figure 3.11

we present our simulation results for the bond-bond correlation functions for trefoils of

length N = 64, in which we vary the semiflexibility parameter B. The simulations

start with a knotted initial configuration, as exemplified in Figure 3.1. The results are

presented in Figure 3.11 through differently colored symbols. With increasing B the

curves display a double minimum, quite in line with theoretical expectations. Namely,

analyzing theoretical expression obtained in the framework of MEP one gets for quite stiff

trefoils the bond-bond correlation function presented in Figure 3.12. [7,8] Here the symbols

stand for N = 32, 64 and 128 and perfectly fit the solid curve displaying the bond-bond

correlation function for double-folded polygon, for which 〈d1·dj〉 = cos(4π(j−1)/N). Two

minima and one maximum lie at positions 0.25, 0.75 and 0.5 respectively. Now comparing

the simulation data from the Figure 3.11 with theoretical results of Figure 3.12 one can see

55


0.0 0.2 0.4 0.6 0.8 1.0

-1.0

-0.5

0.0

0.5

1.0

<d1

dj>/l2

(j-1)/N

Cos(4 (j-1)/N) N=32 N=64 N=128

Figure 3.12: Theoretical bond-bond correlations function for trefoils N = 32, 64and 128 close to the rigid limit for different lengths N (symbols) obtained by M.Dolgushev. [7,8] The solid line corresponds to the double-folded polygon, see textfor details.

that with increasing B the positions of the minima tend towards 0.25 and 0.75, denoting

an antiparallel arrangement of bonds, consistent with a doubly folded structure. In a

similar way, in Figure 3.11 a maximum develops at the position 0.5 with increasing B-

values, again in accordance with the doubly folded structure. As a consistency check we

have also performed BFM simulations by turning off the excluded volume interactions.

The results are presented in Figure 3.11 as continuous lines. Evidently, when the excluded

volume interactions are removed the bonds are permitted to cross each other, by which

the knot disappears. The final situation is the same as when starting from an unknotted

initial structure. Here we again remark for B = 0 the plateau-type behavior found in the

lower part of Figure 3.5.

The next two figures (Figure 3.13 and Figure 3.14) consider the bond-bond correlations

of trefoils of different length for B = 0, 3 and 6. The lengths of trefoil take values N = 32,

64, 128, 256 and 512 and are depicted using different colors of symbols. Comparing the

bond-bond correlations for different N within each of the pictures we see that for short

knotted rings a double minimum pattern is evident; the feature gets blurred when N

increases. Correspondingly, the pronounced maximum at 0.5 also gets flattened with

56


0 0.2 0.4 0.6 0.8 1(j-1)/N

-0.2

-0.1

0

0.1

0.2

0.3<

d 1. dj>

/<l b2 >

N=32N=64N=128N=256N=512

B=0

Figure 3.13: Bond-bond correlation functions for flexible trefoils of differentlengths. The symbols show simulation data for different trefoil lengths N , see textfor details.

increasing N . With increasing semiflexibility parameter B for given N the extrema are

getting more pronounced.

Another point to notice is that for flexible trefoil the bond-bond correlations between

successive bonds decrease with increasing N (see data for B = 0 of Table 3.2). This

is in contrast to the flexible unknotted rings for which bond-bond correlations between

neighboring bonds increase with N (see Table 1.2). We explain this by stronger local

correlations and excluded volume interactions due to the realization of a knot for small

trefoils. With increasing N the influence of a knot on the correlation of neighboring bonds

decreases and its value tends to the corresponding value for unknotted rings. This ex-

planation is also supported by recent extrapolation results of ref. [11] which predict knot

localization for sufficiently long self-avoiding walks (SAWs). The latter phenomenon con-

sists in the tendency for a knot to be localized in a certain part of the ring rather then to

be spread over it. [11] The effect takes place for sufficiently large rings. [11]

For semiflexible trefoils the influence of excluded volume due to the presence of a knot

is suppressed by the intrinsic stiffness of trefoil and the dependence of the bond-bond

57


Figure 3.14: Bond-bond correlation functions for semiflexible trefoils of differentlengths. The symbols show simulation data for different trefoil lengths N . Theupper and lower figures correspond to the stiffness parameters B = 3 and B = 6respectively, see text for details.

58


correlations on N becomes analogical to the one of unknotted ring (see data for B = 3

and B = 6 of Table 3.2).

N = 32 N = 64 N = 128 N = 256 N = 512

B = 0 0.254 0.222 0.202 0.193 0.190B = 3 0.631 0.663 0.669 0.670 0.671B = 6 0.751 0.804 0.817 0.820 0.821

Table 3.2: Stiffness parameters t of trefoils calculated from the simulations usingEquation (1.5), see text for details.

59

Chapter 4

Semiflexibility as a Way to Highlight

the Polymers’ Topology

Considering the bond-bond correlation functions in the previous chapter (see Sections 3.2.1

and 3.2.2) we have noticed that under semiflexible conditions the topology is getting more

pronounced. Exemplary, this can be clearly seen by comparison bond-bond correlation

functions for unknotted rings and trefoils: correlations for trefoils have a maximum which

manifests itself more for the large values of stiffness parameter. Moreover, possibility of

highlighting topology is not only of pure theoretical interest; for certain objects, such as

cospectral structures, [37,38] it turns to be a crucial problem. The peculiarity of these ob-

jects is that while being represented by different graphs, they are indistinguishable through

the macroscopic measurements. However, theoretical approach based on the MEP pro-

vides a possibility to differentiate between CP via introducing semiflexibility in them. [6]

Being motivated by recent achievements in the theory, we decided to perform simulation

study of CP and verify the theoretical predictions. In order to make a more extensive

investigation we study in detail the influence of the semiflexibility on other topologies as

well. For this, first we revisit our simulations results for chains and star polymers with

functionality f = 3 and f = 4, which have only one branching point. Then we verify

theoretical ideas for CP by considering the simplest tree-like CP pair. As a next step we

compare unknotted rings and trefoils and we end up this chapter by comparison of stars

with unknotted rings.

4.1 Linear chains and stars

We start our study of semiflexibility as a tool of highlighting topological structures of

polymers from comparing chains, f = 2, and stars with f = 3 and f = 4. For this we

replot the data from Section 2.1.1 for the mean-square radius of gyration. Figure 4.1

60

Chapter 4: Semiflexibility as a Way to Highlight the Polymers’ Topology

Figure 4.1: Double-logarithmic plots of 〈R2g〉/〈l2b〉 versus N for chains (f = 2)

and for stars (f = 3 and f = 4). The symbols indicate the simulation results andthe stiffness parameter is B = 0 in the upper part and B = 6 in the lower partof the figure. The lines depict the theoretical results [7] for the corresponding t,t = 0.19 and 0.84, see text for details.

61


presents in double-logarithmic scales 〈R2g〉/〈l2b〉 as a function of N for chains and for stars

for B = 0 (upper figure) and for B = 6 (lower figure). The theoretical curves are evaluated

based on Equation (1.8) or, alternatively, using Equations (1.9) and (1.10) taking t = 0.19

and t = 0.84 as stiffness parameters (the values for t which correspond to B are taken

from Table 1.1). Comparing the simulation data for 〈R2g〉/〈l2b〉 as a function of N one can

immediately see that 〈R2g〉f=2 > 〈R2

g〉f=3 > 〈R2g〉f=4 holds both for B = 0 and B = 6.

However, for the semiflexible case, B = 6, the differences between the gyration radii

for chains and for stars get much more pronounced. In line with this observation, for

B = 6 and increasing N the objects start to be more flexible and the distance between

the different curves is getting smaller.

4.2 Cospectral polymers

An example where the theoretical STP-approaches as well as the numerical investiga-

tions turn out to be helpful is the study of so-called cospectral polymers (CP). [37,38] Such

objects have encountered much interest in mathematical, [40,74–77] and chemical [78] litera-

ture. The attention to these structures is motivated by the fact, that in the GGS picture

they have the same Laplacian spectrum although being topologically different. Hence, in

GGS, flexible polymers, whose structures are cospectral graphs, are predicted to be indis-

tinguishable under the usual static and dynamical measurements (such as the radius of

gyration and the mechanical relaxation moduli). Taking into account that large tree-like

structures have (at least) one cospectral counterpart, [39,40] the problem becomes of much

importance. However, recent mathematical-analytical study applied to the smallest tree-

like cospectral pair shows that when these polymers are semiflexible one can distinguish

between cospectral structures [6]: Introducing semiflexibility in GGS picture leads to the

different spectra and thus to different macroscopic properties. Moreover, static proper-

ties were shown to be very well suitable for differentiation between topologies. [6] In this

section we verify theoretical predictions performing simulations for the smallest tree-like

cospectral pair.

An example of the smallest nontrivial tree-like CP-pair is displayed in Figure 4.2. Both

structures consist of N = 11 beads and have the same amount of junctions of certain

functionality fi, where i is the number of junction. Namely, there are two junctions of

functionality fi = 2, two junctions of functionality fi = 3, and one junction of functionality

fi = 4. Both structures, notwithstanding differences in topology, have the same eigenvalue

spectrum of their connectivity matrices. Introducing semiflexibility of junctions according

to Equation (1.5) leads to differences in eigenvalues for both structures and, thus, in their

static (radius of gyration < R2g >) and dynamic (loss modulus) properties. [6] However,

differences between the structures for the loss modulus were found to be much smaller

62


Structure 1

Structure 2Figure 4.2: The smallest pair of treelike CP, see the text for details.

then for the radius of gyration. [6] This makes the latter parameter more relevant for

experimental purposes. [6]

Let us start by recalling the theoretical results for CP displayed in Figure 4.2. Both

structures are tree-like and thus a theoretical study can be performed using generalization

of the GGS picture for semiflexible polymers. [7,24] The same results also follow from the

MEP. [7,24] These theoretical approaches (as well as our simulation method) allow to treat

each junction independently, the property which turns out to be important in this study.

So in theoretical description three semiflexibility parameters related to junction with

given functionality are introduced: Junctions with functionalities 2, 3 and 4 have stiffness

parameters t, q and p. Altogether three choices of parameters were considered. A first

63


0 0.2 0.4 0.6 0.8 1t

1.5

2

2.5

<R

g2 >/l

2

Structure 1, t=/ 0, q=t/2, p=t/3Structure 2, t=/ 0, q=t/2, p=t/3Structure 1, t=/ 0, q=0, p=t/3Structure 2, t=/ 0, q=0, p=t/3

0 0.1 0.2 0.3 0.4 0.5q

Structure 1, t=0, q=/ 0, p=0Structure 2, t=0, q=/ 0, p=0

Figure 4.3: Averaged square radius of gyration for different choices of the semi-flexibility parameters and for the CP-pair of Figure 4.2 obtained theoretically byM. Dolgushev. [6,7], see text for details.

choice of parameters lets q and p depend on t, namely q = t/2 and p = t/3. [6,7] In this way

both limiting cases of flexible and stiff junctions are satisfied: t = 0 implies that q = p = 0;

for t → 1 one has q → 1/2 and p → 1/3 and these are the maximal possible values of

stiffness allowed for corresponding functionalities. [53] In two other choices of parameters

the role of trifunctional junctions, for which fi = 3, was considered. They were treated

either as flexible, having q = 0, while all other junctions remained semiflexible, or only

trifunctional junctions were kept semiflexible with stiffness parameter q and junctions

with functionalities fi = 2 and fi = 4 were flexible, t = p = 0. [6,7]

The theoretical predictions for the normalized radius of gyration 〈R2g〉/〈l2b〉 for different

stiffness values for CP from Figure 4.2 are presented in Figure 4.3. Here solid and dashed

lines correspond to structure 1 and structure 2 respectively and the color stands for differ-

ent choices of stiffness parameters (t, q, p) discussed above: (t, t/2, t/3) in black, (t, 0, t/3)

in blue, and (0, q, 0) in green. All lines have the point t = q = 0 in common, which

corresponds to the parameter choice (0, 0, 0), i.e., to fully flexible situation. For non-zero

t and/or q both structures are not identical. The difference between the structures gets

larger with increasing t and/or q. However, not all parameter choices are best suited for

differentiation between topologies. From Figure 4.3 we can see that the largest difference

64


0 2 4 6 8 10B

1.75

2

2.25

2.5

2.75

<R

g2 >/<

l b2 >

S1: Bi=B

S2: Bi=B

S1: Bi=0 for f

i=3

S2: Bi=0 for f

i=3

S1: Bi=0 for f

i={2, 4}

S2: Bi=0 for f

i={2, 4}

Figure 4.4: The normalized averaged square radius of gyration, 〈R2g〉/〈l2b〉, plotted

as a function of the stiffness. Three parameter choices are considered: Bi = Bfor all i, in black; Bi = 0 for fi = 3 and Bi = B else, in blue; Bi = B for fi = 3and Bi = 0 else, in green. The results for structure 1 are indicated by circles andfor structure 2 by filled triangles, see the text for details.

is for the case when trifunctional junctions are kept flexible, namely the parameter choice

(t, 0, t/3), given in blue.

Another remarkable feature to notice is that different choices of stiffness parameters

lead to different relative location of curves displaying structure 1 and structure 2. While

for the parameter choices (t, t/2, t/3) and (t, 0, t/3) the 〈R2g〉-values for structures 1 are

larger then the corresponding values for structure 2, the opposite is true for the parameter

choice (0, q, 0): the 〈R2g〉-values for structures 2 lie now above those for structure 1. The

theoretical calculations for CP are provided by M. Dolgushev. [6,7]

After recalling theoretical expectations for the 〈R2g〉-values for the CP from Figure 4.2

lets turn to our simulation results for the mean-square radius of gyration for these objects.

Now, we can tune the semiflexible behavior at each junction through the parameters Bi

of Equation (1.2). To be close to the different parameter choices discussed above, we

take either the Bi to be constant for all junctions or we will set some Bi for particular

65


junctions to be zero, depending on their functionality. The mean-square radius of gyration

divided by 〈l2b〉, namely 〈R2g〉/〈l2b〉, both evaluated as described above through the BFM,

is presented in Figure 4.4. In the figure the results for structure 1 (S1) are denoted by

open circles and those for structure 2 (S2) by filled triangles. Using in each case the

5 × 106 realizations involved, the relative standard error is around 1.05 × 10−4 for 〈R2g〉

and around 0.4× 10−4 for 〈l2b 〉, giving a total relative standard error of around 1.5× 10−4

on 〈R2g〉/〈l2b〉, well below the size of the symbols used. As is evident, depending on the

Bi-parameters used, the results fall into three groups, which we color-coded. Black stands

for the cases in which the parameters Bi are the same for all junctions, blue denotes the

cases in which we set Bi = 0 only for trifunctional (fi = 3) junctions and green denotes

the cases in which only the trifunctional junctions are semiflexible (and hence Bi = 0

for fi = 2 and fi = 4). This choice of colors is in line with that used in Figure 4.3 to

differentiate the junctions according to their flexibility. On the other hand, varying the

parameters t, q and p in the STP-model or varying the Bi in BFM-simulations is not the

same thing, and thus the results of Figure 4.3 and of Figure 4.4 differ; what interests us

here is their qualitative agreement.

The first thing to note is that for full flexibility (all junctions having Bi = 0) the results

for structures 1 and 2 are indistinguishable: While this is no surprise for the analytical

GGS model, it is very noteworthy here, since in the BFM-simulations excluded volume

constrains are explicitly taken into account. We conclude that in what the value of 〈R2g〉

is concerned, short ranged excluded volume effects can hardly be seen.

Turning now to the role of the angular restrictions we find, similarly to the theoretical

results, that the 〈R2g〉/〈l2b〉-values are larger for structure 1 than for structure 2 when

all Bi are taken to be equal. The same holds in the case when only the trifunctional

junctions are allowed to be totally flexible. On the other hand, 〈R2g〉/〈l2b〉 is smaller for

structure 1 than for structure 2 when only the trifunctional junctions are semiflexible. In

this way everything is in line with theoretical results of Figure 4.3. To be somewhat more

quantitative: 〈R2g〉/〈l2b〉 for structure 1 is larger than 〈R2

g〉/〈l2b〉 for structure 2 when the

Bi are the same, Bi ≡ B for all junction types, the difference between the 〈R2g〉/〈l2b〉 being

(1.477± 0.007)% for B = 10. When only the trifunctional junctions are kept flexible and

for the other ones we set Bi ≡ B, the deviations become even larger, being (2.117±0.013)%

for B = 10. Finally, when only the trifunctional junctions are semiflexible and only for

them Bi ≡ B, 〈R2g〉/〈l2b〉 is smaller for structure 1 than for structure 2, the difference

between them being (1.033± 0.015)% for B = 10. All this agrees qualitatively well with

the theoretical results presented in Figure 4.3.

66


4.3 Rings and trefoils

In this subsection we study the influence of stiffness on unknotted rings and on trefoils.

Here we consider not only the normalized mean-square radius of gyration 〈R2g〉/〈l2b〉 but

also recall our results the bond-bond correlation functions from Sections 3.2.1 and 3.2.2

(Figures 3.5 and 3.6 for unknotted rings and Figures 3.13 and 3.14 for trefoils).

Starting with the local properties of unknotted rings and trefoils we draw attention to

the bond-bond correlation functions 〈di · dj〉. We compare the the data for flexible un-

knotted rings and trefoils (upper figure of Figure 3.5 and Figure 3.13) and for semiflexible

unknotted rings and trefoils with stiffness parameter B = 6 (lower parts of Figures 3.6

and 3.14). The symbols stand here for the simulation data; the lines in figures relative to

rings show theoretical results for unknotted rings obtained in refs. [7,8]

Comparing the simulation data for unknotted rings with those for trefoils we observe

that the data for rings show minima around 0.5, while for trefoils there are maxima around

0.5 and minima around 0.25 and around 0.75. With growing B these features get more

pronounced, both for the unknotted rings and for the trefoils. Hence,as we discussed

above, for quite stiff unknotted rings the bonds d1 and dN/2 are almost antiparallel,

whereas for trefoils that pair is parallel, the antiparallel pairs being, say, d1 and dN/4 as

well as dN/2 and d3N/4. Thus, accounting for the symmetry of the ring, 〈d1+k · dj+k〉 =〈d1 · dj〉, in quite stiff situations one minimum implies a steady rotation of consecutive

angles by a total amount of 2π, whereas for trefoils this amount is 4π. These findings

are intuitively clear and are, furthermore, supported by the theoretical analysis. [8] For

larger N and fixed B the polymers become more flexible and the extrema get washed out;

then it is more difficult to distinguish between unknotted rings and trefoils. This again

emphasizes the main idea of this section: Topological differences are better seen when

considering quite stiff objects.

We proceed further by comparing the mean-square radius of gyration of unknotted

rings and of trefoils for the stiffness parameters B = 0 and B = 6. For this the results

of Figures 3.2 and 3.4 are replotted in Figure 4.5. The values obtained from simulations

are computed using Equation (1.6), and are presented through open symbols for the

unknotted rings and through solid ones for the trefoils. The solid and dashed lines in

Figure 4.5 show fits to the simulation data by power laws with denoted exponents. Now,

as a function of N we always find that 〈R2g〉trefoil < 〈R2

g〉ring, given that the trefoil is a

more compact object than the corresponding unknotted ring. As in the case of chains

and stars discussed above, for moderate values of N the stiffness considerably increases

the differences in the 〈R2g〉/〈l2b〉 values between the unknotted rings and the trefoils. For

large N we get again a more flexible situation and the 〈R2g〉-values of the trefoils approach

the ones of the corresponding unknotted rings.

67


10 100 1000N

1

10

100

1000

<R

g2 >/<

l b2 >

ring; B=0ring; B=6trefoil; B=0trefoil; B=6

~N1.20

~N1.27

Figure 4.5: Double-logarithmic plots of 〈R2g〉/〈l2b〉 versus N for unknotted rings

(empty symbols) and for trefoil (filled symbols) for B = 0 and B = 6. The straightlines correspond to fitting with power laws with indicated exponents, see text fordetails.

4.4 Rings and stars

As a last example we compare the situations encountered for unknotted rings and for stars

of core functionalities f = 3 and f = 4. We again study the influence of stiffness on the

normalized mean-square radius of gyration 〈R2g〉/〈l2b〉. For fully flexible, phantom stars

and rings, Zimm et. al. found [58,79] 〈R2g〉star/〈R2

g〉chain = (3f − 2)/f 2, see Equation (32)

and (39) of ref., [79] and 〈R2g〉ring/〈R2

g〉chain = 1/2, as can be readily inferred from Equation

(52a) of ref. [58] For fixed N one has thus for flexible, phantom objects:

〈R2g〉ring < 〈R2

g〉f=4 < 〈R2g〉f=3. (4.1)

The situation changes when the structures get stiff. Thus, for rigid, maximally ex-

tended, unknotted rings and stars geometric arguments lead in the limit of very large N

68


Figure 4.6: Double-logarithmic plots of 〈R2g〉/〈l2b〉 versus N for unknotted rings

and for stars (f = 3 and f = 4). The symbols are obtained from simulationswith the stiffness coefficients being B = 0 (upper) and B = 6 (lower part of thefigure). The lines depict the theoretical results for the corresponding t, [7] see textfor details.

to 〈R2g〉f=4 : 〈R2

g〉ring : 〈R2g〉f=3 =

14: 3π2 : 4

9and hence to

〈R2g〉f=4 < 〈R2

g〉ring < 〈R2g〉f=3, (4.2)

69


i.e. to a change in the order of the radii of gyration. The estimation of values for 〈R2g〉

are provided by M. Dolgushev.

For a more realistic picture we again perform BFM-simulations and find a very satis-

factory agreement with the theory. To show this we plot in Figure 4.6 the gyration radii

both for stars and for unknotted rings for B = 0 (upper part) and B = 6 (lower part of

Figure 4.6). Here again the symbols correspond to the simulation data and the curves

are the theoretical results. For stars the values of 〈R2g〉/〈l2b〉 were obtained using Equa-

tion (1.8) with t = 0.19 and t = 0.84 for B = 0 and for B = 6, respectively. The values

of t for rings are taken from Table 1.2. For B = 0 we find for all N that Equation (4.1)

holds. Going to a more stiff situation, B = 6, the behavior changes, and for N . 64

the order of the radii of gyration is that of Equation (4.2). Thus, we indeed find in the

simulations a crossover between fully flexible and fully rigid situations; hence by varying

the stiffness coefficient one can pinpoint the underlying topologies even on a qualitative

level.

70

Summary and Conclusions

In this thesis different semiflexible topological structures under good solvent conditions

were studied by means of the bond fluctuation model (BFM). [26,27] In our investigation

we focused on the following topologies: simple branched structures such as stars with

functionality f = 3 and f = 4 of branching points and chains (the latter can be viewed

as two-arm stars), cospectral polymers (CP), [37,38] and simple loop structures such as un-

knotted rings and rings with one knot, known as trefoils. In order to perform simulations

for semiflexible branched structures mentioned above we introduced a generalization of

the BFM, which allows to account both for branching and for semiflexibility. While, to

the best of our knowledge, previous applications of the BFM to semiflexible structures

confined themselves to semiflexible chains and rings, this work presents a new possible

treatment of branching points together with semiflexibility in the framework BFM.

We confronted our findings from MC simulation studies, in which the BFM technique

was used, to the theoretical results obtained with maximum entropy principle (MEP). [24]

This allowed us to investigate the influence of semiflexibility under realistic conditions,

by also accounting for the excluded volume interactions which are not taken into account

in the theoretical studies mentioned above. We considered general properties of the topo-

logical structures mentioned above (such as mean-square radius of gyration) and of their

local properties (such as bond-bond correlation functions).

The mean-square radius of gyration we found to be in a good agreement with theoret-

ical predictions and with previous simulations for all structures under consideration. Our

MC data for the bond-bond correlation function for unknotted rings also confirms well

theoretical results based on MEP. [7,8] In the study of bond-bond correlation functions for

chains and stars we focused on the role of branching points and presence of additional

arms on this quantity. Our results for flexible chains and stars agree very well with previ-

ous theoretical renormalization group (RG) [43] and simulation studies of local properties

of such objects. [5]

Besides description of local chain/star properties, bond-bond correlation function is

usually related to the definition of persistence length. The latter is a measure of intrinsic

chain stiffness and is supposed to be extracted from the exponential decay of the bond-

bond correlation function. However, we found this definition completely unsuitable under

71

good solvent conditions. Instead of the exponential decay we observe a power law decay

of the bond-bond correlation function, 〈di · ds/|di||ds|〉 ∼ s−µ, with µ = 2 − 2ν. This

conclusion agrees well with the theoretical RG predictions [43] and recent simulation re-

sults. [1,2,70,71] The value of scaling exponent µ obtained from our MC data (µ = 0.892)

is somewhat larger then the theoretically predicted one (µ = 0.824). This happens due

to the end effects which are rather strong in our case, as the chain lengths are not long

enough.

We also studied applicability of definition of persistence length due to Flory, which

gives it as a local property and defines chain stiffness in a long chain limit, to systems under

good solvent conditions. Our MC simulations show that this definition does not provide

a universal characteristic for chain stiffness under good solvent conditions. This result

agrees with theoretical calculations based on RG methods. [42] We got a good quantitative

agreement between our MC data for local persistence length and approximate theoretical

expression for local persistence length obtained using RG methods. [42] The similar findings

were also reported in the simulation studies in refs. [1,2,70] For stars the situation gets even

more complicated as presence of branching points and additional arms affects dramatically

the local persistence length.

In this thesis we also address the question how changes in flexibility help in highlighting

the topology of polymers. For some structures, such as CP, this question becomes of

major importance since fully flexible, ideal CP are indistinguishable. [6] By comparisons

of stars vs. chains, of unknotted rings vs. trefoils and of stars vs. unknotted rings

we could investigate the influence of semiflexibility on the static properties of polymers,

namely on the mean-square radii of gyration and the bond-bond correlation functions.

For not-too-large N , both for stars and chains, and for unknotted rings and trefoils the

differences between the mean-square radii of gyration increase when the semiflexibility

parameter B gets larger. This allows a better differentiation between these polymers.

For unknotted rings and trefoils the differences in topologies manifest themselves even

stronger in their bond-bond correlation functions: With growing B the extrema in the

bond-bond correlation functions become more pronounced. Comparing different stars vs.

unknotted rings we observe qualitative changes: Our simulations for B = 0 support the

theoretical ordering 〈R2g〉ring < 〈R2

g〉f=4 < 〈R2g〉f=3 whereas for B = 6 and small N the

order is 〈R2g〉f=4 < 〈R2

g〉ring < 〈R2g〉f=3. Thus our simulation results are in good agreement

with the theoretical findings.

The MC investigations presented in this thesis have possible future perspectives. Ex-

emplary, further developments of the simulation method are needed in order to allow

description of branching points with higher functionalities. The problem lies in the de-

creasing mobility of the branching point with increasing functionality. Because of this

effect we limited ourselves to stars of functionality 3 and 4 only. Furthermore, having

72

conservation of topology as an intrinsic feature of the BFM allows its application to a

large variety of topological structures. Especially interesting and challenging would be a

more general and extensive studies of influence of semiflexibility on static and dynamic

properties of CP, while their theoretical treatment could be easily done in the framework

of MEP [7,8] or, alternatively, applying GGS picture. [7,24]

73

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Index of Abbreviations∗

GGS generalized Gaussian structure(s)BFM bond-fluctuation modelMEP maximum entropy principleMC Monte CarloSTP semiflexible treelike polymer(s)CP cospectral polymer(s)RG renormalization groupEV excluded volumeMCS Monte Carlo step(s)PERM pruned-enriched Rosenbluth methodS1 structure 1S2 structure 2

∗The abbreviations are given in the order of their appearance in the text

78

Acknowledgements

I would like to give my cordial acknowledgments to Prof. Dr. Alexander Blumen for

the continuous supervision and support during my PhD study. Special thanks to Maxim

Dolgushev for providing theoretical data. The Deutsche Forschungsgemeinschaft is ac-

knowledged for the financial support.

79

Documents

Monte Carlo Study of Semiﬂexible Polymers