Constitutive modeling of concentrated solutions of main-chainliquid crystalline polymersCitation for published version (APA):Matveichuk, O. (2013). Constitutive modeling of concentrated solutions of main-chain liquid crystalline polymers.Technische Universiteit Eindhoven. https://doi.org/10.6100/IR750672

DOI:10.6100/IR750672

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Constitutive modeling ofconcentrated solutions of

main-chain liquid crystallinepolymers

Copyright c©2013 by Oleg Matveichuk, Eindhoven, The Netherlands.All rights are reserved. No part of this publication may be reproduced, stored in aretrieval system, or transmitted, in any form or by any means, electronic, mechanical,photocopying, recording or otherwise, without prior permission of the author.

The present work was part of a project in collaboration with and funded by Teijin Ara-mid B.V.

A catalogue record is available from the Eindhoven University of Technology LibraryISBN: 978-90-386-3339-8

Constitutive modeling ofconcentrated solutions of

main-chain liquid crystallinepolymers

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan deTechnische Universiteit Eindhoven, op gezag van derector magnificus, prof.dr.ir. C.J. van Duijn, voor een

commissie aangewezen door het Collegevoor Promoties in het openbaar te verdedigen

op maandag 11 maart 2013 om 16.00 uur

door

Oleg Matveichuk

geboren te Odessa, Oekraıne

Dit proefschrift is goedgekeurd door de promotor:

prof.dr. J.J.M. Slot

Contents

1 Introduction 1

1.1 Liquid crystals and liquid-crystalline polymers (LCPs) . . . . . . . . . . . 1

1.2 Rod-like polymers and the Isotropic-Nematic phase transition . . . . . . . 4

1.3 Hairpin defects in an LCP backbone . . . . . . . . . . . . . . . . . . . . . . 7

1.4 Rheology of an LCP solution . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.5 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2 Phase-space theory for LCPs 17

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2 Model for a main-chain liquid crystalline polymer . . . . . . . . . . . . . . 18

2.3 Hamiltonian for the ensemble of chains . . . . . . . . . . . . . . . . . . . . 20

2.4 Smoluchowski equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.5 Stochastic differential equations for a polymer chain . . . . . . . . . . . . . 27

2.6 Forces acting on a polymer chain . . . . . . . . . . . . . . . . . . . . . . . . 30

2.7 Hairpins and entanglements . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.8 Dimensionless version of the evolution equations . . . . . . . . . . . . . . 40

3 Rouse-like model in the highly-ordered limit 47

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.2 Elimination of fast variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.3 Dynamics of a polymer chain . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.4 Normal modes expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.5 Ensemble average behavior of the normal-mode coordinates . . . . . . . . 58

3.6 Evolution equation for the director . . . . . . . . . . . . . . . . . . . . . . . 59

3.7 Modified stress tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.8 Uniaxial elongational flows . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4 Numerical simulations of semi-flexible LCPs 71

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.2 Euler-Maruyama method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.3 Values of the main parameters in the model . . . . . . . . . . . . . . . . . . 76

4.4 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.5 Equilibrium properties of the system . . . . . . . . . . . . . . . . . . . . . . 84

vi Contents

5 Rheology of entangled LCP solutions containing hairpins 87

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.2 Linear rheology of an LCP solution . . . . . . . . . . . . . . . . . . . . . . . 88

5.3 Evolution of the director in steady shear flow . . . . . . . . . . . . . . . . . 95

5.4 Shear viscosity and the first normal stress difference in the steady shear . 100

6 Conclusions 107

A Details of derivations 111

A.1 Evolution equation for the configurational distribution function . . . . . . 111

A.2 Evolution equation for the momentum-space averages . . . . . . . . . . . 112

A.3 Ito or Stratonovich interpretation . . . . . . . . . . . . . . . . . . . . . . . . 115

A.4 Derivation of the expression for θmax . . . . . . . . . . . . . . . . . . . . . . 117

A.5 Derivation of the expression for the stress tensor . . . . . . . . . . . . . . . 119

A.6 Normal modes expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

A.7 Formal solution of a linear matrix differential equation . . . . . . . . . . . 123

A.8 Normal modes in the equilibrium state . . . . . . . . . . . . . . . . . . . . . 125

A.9 Free energy of the ensemble of chains . . . . . . . . . . . . . . . . . . . . . 125

Summary 137

Samenvatting 139

Acknowledgments 141

Curriculum vitae 143

Chapter 1

Introduction

1.1 Liquid crystals and liquid-crystalline polymers (LCPs)

The discovery of liquid crystals is usually associated with the work of the Austrian

botanist Friedrich Reinitzer [1]. In 1888 he observed that a material called cholesteryl

benzoate had two distinct melting points. After the solid sample of cholesteryl ben-

zoate melted it turned into a hazy liquid. But when Reinitzer continued increasing the

temperature the liquid became clear and transparent. Reinitzer continued the study of

this phenomenon in collaboration with the physicist Otto Lehmann [2]. This led to the

discovery of a new phase state of matter, called liquid crystal. This state of matter is

usually called mesophase (from Greek ”µǫσo” - middle, intermediate), because it is an

intermediate phase during the transition from solid crystal to an isotropic liquid. For

some systems even several different mesophases can be observed in the process of going

from solid crystal to an isotropic liquid.

The liquid crystal is a state of matter that has properties in between those of a conven-

tional liquid and those of a solid crystal [3]. Conventional liquids have only short-range

order on molecular scales whereas solid crystals exhibit long-range order on molecu-

lar scales as well. Liquid crystals typically show long-range order in some directions,

like solid crystals, and display short-range order in other directions, like conventional

liquids. Thus, liquid crystals exhibit anisotropic properties like solid crystals and show

the ability to flow like liquids.

Liquid crystals can be classified by the types of symmetries that are preserved. In the

nematic phase (from Greek ”νηµα” - thread) the orientation of the molecules is pre-

served across length-scales much greater than the size of a molecule, while the positions

2 Introduction

of the centers of mass of the molecules do not preserve any long-range order. A schem-

atic picture of the ordering of molecules in the nematic phase is shown in fig. 1.1a. In

the smectic phases (from Greek ”σµηκτικoς” - having power to clean, soap-like) not

only the orientation of the molecules is preserved, but in addition to that the centers of

mass of the molecules are ordered in 2D-layers. Fig. 1.1b depicts such a situation for a

so-called smectic A phase.

A detailed description of the different types of liquid-crystalline mesophases can be

found in the book by P.G. de Gennes and J. Prost [3]. In this thesis we shall deal with

the nematic mesophase.

Liquid crystal mesophases can be also classified by the driving force that induces the

mesophase. For example, if the mesophase is induced by the temperature, then the

liquid crystal is called thermotropic. If the liquid crystal occurs due to changes in the

amount of solvent, then it is called lyotropic (from Greek λυω - to solve ). In either case,

the temperature remains an important factor for the stability of the phase.

In the first part of the 20th century liquid crystals did not have a wide range of ap-

plications. However, the problem of explaining the existence of these liquid-crystalline

mesophases from a thermodynamic point of view attracted many scientists. Essential

pioneering contributions were made by Lars Onsager [4, 5]. Onsager suggested that a

liquid-crystalline mesophase may be formed due to excluded volume effects, i.e., short-

range interactions, when the molecules have a highly prolate form.

Later works by Wilhelm Maier and Alfred Saupe [6–8] have shown that a nematic

mesophase can originate from the anisotropic attractive part of the dispersion forces,

i.e., long-range interactions. Obviously, both excluded volume interactions and disper-

sion forces contribute simultaneously in most systems. In 1971 McMillan [9] suggested

a model accounting for both of these effects in order to explain the transition from a

smectic A phase to a nematic phase.

In the second part of the 20th century the increased use of polymeric materials and, in

particular, the search for lightweight and strong materials for the production of yarn led

to the discovery of poly-paraphenylene terephthalamide (PpPTA) in 1965 by Stephanie

Kwolek [10]. PpPTA is commonly known by the brand names Twaron c©and Kevlar

c©). PpPTA forms a nematic mesophase while in solution. The fibers spun out of this

solution demonstrated outstanding mechanical strength. This discovery led to a rapid

growth of interest to PpPTA solutions and to the liquid crystalline polymer (LCP) solu-

tions in general. Since 1965 the number of applications for yarns spun out of LCP solu-

tions increased enormously. Some of the applications are depicted in fig. 1.2. The

quality of the yarns spun out of the LCP solutions highly depend on the spinning pro-

cess [11]. That is why a good understanding of the mechanical response, i.e., of the

rheology of LCP solutions, is of great importance and use.

1.1 Liquid crystals and liquid-crystalline polymers (LCPs) 3

a) Nematic b) Smectic A

Figure 1.1: Schematic picture of molecular ordering in liquid crystalline mesophases.

a) sport equipment b) heat and cut protection

c) tyre reinforcement d) cable reinforcement e) body armor

Figure 1.2: Applications of LCP.

4 Introduction

a) Side-chain LCPs

b) Main-chain LCPs

Mesogen

Polymer backbone

Spacer

Figure 1.3: Schematic representation of side-chain and main-chain LCPs.

With respect to their structure, LCPs can be divided into two groups, namely, side-chain

LCPs and main-chain LCPs. Side-chain LCPs have three major structural components:

the backbone, the spacer, and the mesogen. The mesogen is the element that causes the

liquid crystalline behavior, i.e., causes the appearance of the mesophase. The spacers

are the elements that attach mesogens to the backbone. In contrast, the main-chain

polymers have a simpler structure. They are linear chains with a mesogenic units incor-

porated into the backbone. A polymer with a sufficiently stiff backbone is an example

of a main-chain LCP. In this case the backbone itself is the mesogenic unit. A schematic

structure of side-chain LCP and main-chain LCP is depicted in fig. 1.3. Since in fur-

ther chapters we will consider nematic main-chain LCP solutions, we will refer to the

mesogens by the name nematogens.

1.2 Rod-like polymers and the Isotropic-Nematic phase

transition

As mentioned at the end of the previous section, main-chain LCPs are polymers with

a sufficiently stiff backbone. The stiffness of the backbone can be characterized by the

ratio of the persistence length lp and the contour length lc of the polymer. The persist-

ence length lp is defined as the minimum length along the contour of the polymer’s

1.2 Rod-like polymers and the Isotropic-Nematic phase transition 5

backbone, over which the correlations in orientation of the backbone elements vanish

effectively. If the ratiolplc

≪ 1, then the polymer is called flexible and it does not show the

ability to form a nematic phase. On the contrary, whenlplc

≈ 1 the polymers are rod-like.

For the intermediate values oflplc

the polymers are called semi-flexible.

When dissolved in solution, at low concentrations rod-like polymers form an isotropic

phase. But when the concentration reaches a certain critical value, rod-like polymers

can spontaneously orient, thus forming a nematic phase. This is the famous isotropic-

nematic (I-N) phase transition. Since the discovery of the I-N transition, a considerable

amount of effort was spent in order to obtain a more precise description. Several models

have been proposed for this purpose. A large number of models appeared starting

from the Onsager model [5], the Maier-Saupe model [6–8], the Flory model [12], the

Khokhlov-Semenov model [13,14], the Parsons-Lee model [15–17], and more recent [18–

20].

Besides these analytical efforts, a variety of numerical simulations of this phenomenon

were performed. These simulations allowed for a more careful consideration of the

excluded volume effects and a more detailed treatment of the polymer shape. The de-

scription of the lyotropic LCPs was extended from rigid rods to hard sphero-cylinders

[16, 17, 21], hard ellipsoids [22], semi-flexible tangent hard sphere chains [23], etc.

One of the most studied models for the solution of rod-like polymers is the Doi rigid

rod model [24, 25]. The relative simplicity and effectiveness of this model made it a

good candidate to serve as a reference model. The problems addressed in rheology are

concerned with non-equilibrium states of the physical system. In order to deal with

this kind of problems the geometrical simplicity of the Doi model becomes even more

valuable in analytical approaches.

The description of the I-N transition in the Doi model [24, 25] is formulated in terms of

the order parameter. This follows the spirit of the Landau-De Gennes phase-transition

theory [26–28]. For the I-N transition the order parameter is deduced from the order

tensor S (also called the orientation tensor), which is defined as

S = 〈uu〉 − 1

3I. (1.1)

Here 〈·〉 denotes the averaging over the whole system and u is an orientation vector

of a rod. The order tensor is zero in the isotropic phase and gets the maximum eigen-

value 23 in the perfectly aligned state. It is important to stress that a vector quantity can

not adequately reflect the state of the system because it does not have the same set of

symmetries as the nematic mesophase. For example, mirroring of the nematic meso-

phase with respect to a plane, perpendicular to the direction of alignment, belongs to

a group of symmetries of the nematic mesophase, while a vector quantity is in general

not invariant with respect to such a transformation.

6 Introduction

The tensor 〈uu〉 is symmetric by construction. This guarantees the possibility of its

diagonalization and the orthogonality of the eigenbasis, i.e.,

〈uu〉 = λ1nn+ λ2ττ + λ3ss (1.2)

where n, τ , s form an orthonormal set of eigenvectors and λ1, λ2, λ3 are the correspond-

ing eigenvalues associated with the tensor 〈uu〉. By construction λ1, λ2, λ3 ∈ [0, 1] and

λ1 + λ2 + λ3 = 1.

For highly-aligned systems the concept of a pseudovector director is introduced [29].

This situation corresponds to the special case when λ1 → 1 and λ2, λ3 → 0. Then the ei-

genvector n in (1.2) is called the director and represents the direction of local orientation

of the polymers. In such a way expression (1.2) defines n as the director .

The scalar order parameter, i.e., the characteristic of the degree of alignment, can be

represented by λ1 (the maximum eigenvalue of 〈uu〉). Taking the dot product between

the director and the orientation vector of the rod and considering the average value of

its second Legendre polynomial 〈P2〉 is sometimes more convenient. This is the case

when the expansion into spherical harmonics of the distribution function is used. It is

also convenient that 〈P2〉 changes from 0 to 1 when the system transits from an isotropic

state to a perfectly aligned state. This gives

〈P2〉 =3

2(n · S · n) =

⟨

3

2(n · u)2 − 1

2

⟩

. (1.3)

The relation between 〈P2〉 and λ1 is linear, namely

〈P2〉 =3

2λ1 −

1

2. (1.4)

In the Doi model the interaction causing rods to align is treated by the nematic Maier-

Saupe potential. This potential originates from the anisotropic part of the long-range

attractive dispersion forces. The expression for the potential is

Un = −all rods∑

m=1

H :

(

umum − 1

3I

)

, (1.5)

where um is the orientation of the m-th rod and H is the strength of the nematic poten-

tial. Expression (1.5) does not look like an interaction potential, due to the fact that the

contribution from each um enters in an additive way. However, the nematic potential is

actually an interaction potential, because the strength of nematic potential H depends

on the configuration of all polymer molecules in the solution, i.e., computing H requires

the information about the correlations between orientations of different molecules to be

known. In practice, the so-called mean-field approximation is usually used. In the

1.3 Hairpin defects in an LCP backbone 7

mean-field approach one has H = H0S, where H0 is the scalar strength of the nematic

field.

When the strength of the nematic field reaches a critical value, the I-N phase transition

takes place. This is a transition of the first kind, since the reordering of the polymers in

the system leads to a rapid change in the system’s free energy. The dependence of the

order parameter 〈P2〉 on the inverse strength of the nematic field is depicted in fig. 1.4.

When TH0

is greater than approximately 0.15, 〈P2〉 remains zero. But when TH0

becomes

slightly less than 0.15, 〈P2〉 rapidly increases, reflecting the spontaneous alignment ap-

pearing in the system. This sharp growth of 〈P2〉 represents the I-N transition.

Typically there is a region of metastable states in the vicinity of the phase-transition

point for a phase-transition of the first kind. The boundaries of this region are called

spinodals. It is valid for the I-N transition as well. The problem of stability of metastable

states and the positions of spinodals is addressed in [30].

The strength of the nematic field is dependent on the temperature and on the weight

percentage of the polymer molecules in the solution. This dependence determines the

shape of the phase diagrams of the solution when depicted in terms of temperature

and concentration. Further information on this dependence can be found in a paper by

Picken [11]. Fig. 1.5 shows a schematic view of a phase diagram for an LCP solution.

More detailed and complete phase diagrams can vary from polymer to polymer, due to

the formation of specific phases, like crystal solvate or other ordered phases. A detailed

phase diagram for PpPTA in sulfuric acid, for example, can be found in [31].

1.3 Hairpin defects in an LCP backbone

The degree of ordering of the chains in the nematic phase is determined by two compet-

ing factors: the ordering due to the nematic interaction and the tendency to maximize

the entropy by undergoing thermal motion. The rigid rod model imposes a huge re-

striction on the configurations of the backbone that are allowed. In general, this leads

to a drastic loss of entropy. However, LCPs of sufficient length (molecular weight) are

semi-flexible and hence form an intermediate case between that of rigid rods and com-

pletely flexible polymers. As the flexibility of the backbone increases more configura-

tions of the backbone become possible while undergoing thermal motion. De Gennes

suggested in [3] that for semi-flexible LCPs the formation of so-called hairpins or kinks

takes place. Fig. 1.6 illustrates typical configurations of the backbone with and without

a hairpin defect. Hairpins are the defects where the chain executes abrupt reversal.

These defects of the backbone satisfy two conditions. They keep the nematic potential

low and increase the entropy of the system by allowing many more configurations of

the backbone. Both of these contributions lower the Helmholtz free energy of the sys-

8 Introduction

〈P2〉

TH0

0.00 0.05 0.10 0.150.0

0.2

0.4

0.6

0.8

1.0

Figure 1.4: Theoretical curve for the order parameter 〈P2〉 as a function of the reduced inversenematic field strength for Doi’s rigid rod model with Maier-Saupe mean-field potential.

T

wt%

IS

IS+

LC

LC

LC + CP

Figure 1.5: Part of a schematic phase diagram showing temperature T versus weight percentageof the polymer present. There are three phases depicted in the diagram: an isotropic phase (IS), aliquid crystalline phase (LC), and a crystalline polymer phase (CP).

1.3 Hairpin defects in an LCP backbone 9

a) non-hairpin state b) hairpin state

Figure 1.6: Examples of possible configurations of the backbone of the chain: a) without a hairpindefect, b) with a hairpin defect.

tem, which makes hairpins more likely to appear if the polymer backbone is flexible

enough. For a polymer with stiff backbone, the huge penalty of the folding up forbids

the formation of hairpins. Experimental evidence of the existence of hairpins in LCPs

was found by small angle neutron scattering techniques (SANS) [32, 33]. In systems

with strong nematic interactions, like in concentrated LCP solutions, the hairpin defects

can become long-living objects. Taking into account the capability of the chains that

contain hairpin defects to form additional entanglements we may expect an additional

contribution from the hairpins to the stress tensor. In fact, in 1999, David Morse showed

that for semi-flexible systems at sufficiently high shear rates the major contribution to

the stress comes from the hairpins that arise in highly deformed molecules [34]. For

small shear rates the formation of hairpins changes the profile of the response moduli

at low frequencies, though hairpins do not provide the major contribution to the stress

tensor. Experimental evidence that suggests that the presence of hairpins may cause

an increase of elastic behavior of the LCP solution is given in [35]. The capability of

hairpins to modify the rheological properties of semi-flexible polymeric systems is an

important phenomenon to study. Examples of such kind of studies can be found, for

instance, in the work of M. Warner et al [36–38].

In this thesis we study the formation of hairpins and their contribution to the rheology

10 Introduction

of LCP solutions. For the results on this topic the reader is referred to the Chapters 3 and

5 of this thesis.

1.4 Rheology of an LCP solution

One of the first successive rheological models for uniaxial nematic materials was the

Ericksen-Leslie-Parodi model [39]. In this model the stress tensor is suggested to have

a linear dependence on the deformation rate and on the angular velocity of the director

vector. From this assumption and employing the symmetries present in the nematic

phase, an expression for the stress tensor was obtained. The Ericksen-Leslie-Parodi pre-

dicts the existence of two types of behavior of nematic materials under shear: tumbling

behavior and flow-aligning behavior. The tumbling behavior is characterized by the

fact that the director does not have a steady orientation when the sample is steadily

sheared. The director keeps rotating in the plane of shear. This peculiar kind of motion

of the director is called tumbling. The flow-aligning behavior is characterized by the fact

that the director keeps a steady orientation under shear. The orientation of the director

lays in the plane of shear and constitutes an angle θ with the direction of the velocity

vector. This angle θ is called the Leslie angle and lies within the interval [0, π/4].

The Ericksen-Leslie-Parodi model implicitly assumes that the characteristic time associ-

ated with the deformation rate is much larger than the characteristic internal relaxation

times of the nematic phase. In other words, the response of the material is purely vis-

cous. This explains the success of the Ericksen-Leslie-Parodi theory for low molecular

weight liquid crystals, because the characteristic internal times for these materials are

typically very small when compared with macroscopic time scales. However, LCP solu-

tions typically show viscoelastic behavior on macroscopic time scales, i.e., they have

much greater characteristic internal times than the low molecular weight liquid crys-

tals. In many industrial applications, like fiber spinning, the deformation rates are

usually much higher than the inverse relaxation times of the LCPs. Therefore, to ac-

count for these internal relaxation processes becomes an important issue and the ori-

ginal Ericksen-Leslie-Parodi theory is not capable of capturing such processes. Further-

more, in equilibrium, a sample of LCP material usually contains a huge number of de-

fects (like disclinations). These defects give additional contributions to the stress tensor

(the so-called Frank elasticity), which causes deviations from the Ericksen-Leslie-Parodi

theory even for very small velocity gradients.

LCPs, as all high-molecular weight polymers, show many characteristic time scales in

their relaxation behavior. Short and large time scales are to be associated with the re-

laxation of small and large parts of the chains respectively. The largest relaxation time

is related to the relaxation of the chain as a whole and dominates its macroscopic beha-

vior. This fact justifies the success of approaches in which only the longest relaxation

1.4 Rheology of an LCP solution 11

time is taken into account [29]. That is why isotropic solutions of flexible polymers

can be adequately described by theories based on rather coarse representations of the

chain microstructure. Examples of such theories are the Rouse and Zimm models of un-

entangled polymer liquids and the reptation model of entangled polymer liquids. The

situation with LCP solutions, however, is much more complicated due to the fact that

the persistence length of the chains is not negligible compared to their contour length.

This leads to an anisotropic equilibrium state and a dependence of the distribution func-

tion on the nematic order parameter. Moreover, there exist several different classes of

LCPs having a different chain microstructure such as main-chain and side-chain LCPs.

Nevertheless, it is very useful to develop simplified models of LCP solution dynamics

that only depend on a few microstructural parameters. These models allow us to study

the effect of these parameters on the macroscopic properties of the LCP solutions in

detail.

The most heavily studied models for LCPs are the Doi rigid-rod model [24], the Kuzuu

and Doi model [40, 41] and the models of Larson [42], Larson and Mead [43], and Maf-

fettone and Marrucci [44]. All these models treat the polymer chain like a rigid rod

that performs rotational thermal motion in a mean nematic field, but neglect the semi-

flexibility effects altogether. Some ways of treating the semi-flexibility of polymer back-

bones were suggested by Marrucci et al by considering the so-called slightly bending

rod model [45–47]. In the slightly bending rod model the polymer backbone is allowed

to take the shape of an arc of a varying but small enough curvature. The opposite ap-

proach to treat semi-flexibility is the nematic dumbbell model [48]. In this model the

orientation of the polymer backbone is characterized by the end-to-end vector, though

the polymer chain is treated like a dumbbell as in the case of flexible polymers. All these

models treat systems of unentangled polymers and are also not capable of treating hair-

pins in a natural way. The models accounting for entanglements in semi-flexible LCP

systems were considered by Semenov [49] and Subbotin [50], but the role of hairpins in

these models is not clear. While Semenov was considering the case of a chain having

many hairpins per chain, Subbotin considered the opposite limit of a chain having no

hairpins.

One of the aims of the research presented in this thesis is concerned with the invest-

igation of the impact that hairpins have on the rheological properties. Therefore, the

coarse-grained model chosen for the polymer chain has to be capable of treating hair-

pins in a natural way. Examples of such models are the Nematic Broken Rod model [51]

or rod-spring model [29]. In other words, the model for a polymer chain should have

degrees of freedom that allows to distinguish between a ”normal state” and a ”hairpin

state”.

Solutions of LCP are typically shear-thinning materials. The steady-state viscosity for

these materials usually follows the three region curve proposed by Onogi and Asada

[52]. The schematic log-log plot for the steady-state viscosity as the function of the

12 Introduction

shear rate is shown in fig. 1.7. The shear-thinning behavior in region I is explained by

the poly-domain structure of the solutions of LCP [53]. In equilibrium LCP solutions

do not form a mono-domain with the same direction of ordering of the polymer chains

within. The largest relaxation time of such a poly-domain system is the time associated

with the evolution of the domain, because the relaxation of the domain happens at a

larger time scale than the relaxation of the single polymer. That is why domain structure

is the most sensitive to the deformation applied and starts to play a role at the smallest

deformation rates. When shear is applied, domains start to evolve. After the shear rate

reaches some value the contribution to the stress tensor from the poly-domain structure

becomes insignificant. This is reflected in the Asada-Onogi plot by region II. The steady-

state viscosity of the LCP solution in region II remains more or less constant.

The characteristic time scale associated with the transition from region II to the region

III is the time scale of nematic ordering. This time-scale is the characteristic time that

is required for the nematic potential to orient all the chains in the system in the same

direction if they were initially disordered. We will refer to this time as the nematic syn-

chronization time, because this is the time required for the polymer chains to synchron-

ize their orientation (up to some spread due to thermal fluctuations). If the shear rate

is smaller than the inverse synchronization time, then the chains keep rotating more

or less synchronously. Due to this synchronization, macroscopic quantities such as the

director also rotate, i.e., the LCP solution demonstrates tumbling behavior. If the shear

rate becomes greater than the inverse synchronization time, then the relative strength

of the nematic interaction is not large enough to keep the rotation of the polymer chains

in phase. The orientation of the different polymer chains gets out of phase and there-

fore the average orientation of the chains, i.e., director, does not follow any more the

orientation of the individual chains and keeps the constant orientation. In this case, the

orientation of the director is determined by the average time that an individual chain

spends in each angular sector. While rotating under steady shear, the polymer chain

spends most of the time approaching the plane which contains the shear velocity vector

and the vorticity pseudovector. This explains why the Leslie angle θ is always positive

and is below π4 . The transition from region II to region III is referred to as the ”dynamic

transition” and it describes the change from tumbling to flow-aligning behavior. The

slope of the curve for the steady state viscosity in the region III is about 0.4-0.5.

Between the tumbling region and the flow-aligning region another region can be ob-

served. It is called the transient region. The transient behavior is characterized by the

oscillatory behavior of the director. In this case, the director oscillates between the max-

imum and minimum deviations from the velocity direction, but does not perform full

2π turns like when tumbling. As the shear rate increases, the amplitude of the oscil-

lations of the director decreases towards zero giving rise to the flow-aligning type of

behavior.

Another peculiar fact in the rheology of the LCP solutions is concerned with the first

1.4 Rheology of an LCP solution 13

ln(η)

ln(γ)

Region I

Region II

Region III

Figure 1.7: Typical three-region plot for the steady-state viscosity as a function of the shear ratefor LCP solutions.

N1

γ

Region I

Region II

Region III

0

Figure 1.8: Schematic plot for the first normal stress difference as a function of shear rate for anLCP solution under steady shear.

14 Introduction

normal stress difference. The first normal stress difference is the quantity N1 := σxx −σyy , where σxx is the normal stress in the direction of the shear velocity and σyy is the

normal stress in the direction of the velocity gradient. For ordinary polymers the first

normal stress difference is positive for all shear rates. This positive sign means that

there is a compression in the direction of the velocity gradient. In fact, the positive

normal stress means that when the sample is sheared between two plates, the plates

are pushed apart with the force N1. In contrast to this typical behavior, the first nor-

mal stress difference for LCP solutions shows a region with negative first normal stress

difference. This peculiar behavior was first reported by Kiss and Porter [54]. The ex-

perimental evidences of the negative first normal stress difference were also provided

in [55–57]. The schematic plot for the first normal stress difference as function of the

shear rate is shown in fig. 1.8. In region I the first normal stress difference is reported

to be positive. In region II it changes sign to negative values and then transits back to

positive values in region III. The two-dimensional model developed by Marrucci and

Maffettone [44,58,59] predicts the region with negative first normal stress and the tum-

bling phenomenon which is followed by wagging and flow-aligning behavior at higher

shear rates. Larson [42] extended this model to a 3D-case.

Besides the steady state experiments mentioned above, a large amount of information

on the transient rheological properties of the LCP solutions is available [57, 60]. The

typical experiments performed for the transient flows are: start up flow, flow-reversal,

stepwise change of flow rate, rapid cessation of flow and also oscillatory flow (like Large

Amplitude Oscillatory Shear (LAOS)). The transient behavior will be discussed in de-

tails in Chapter 5.

In typical processing of LCP solutions the behavior of the LCP solution under both

shear and elongation flow is important. For example, in fiber spinning procedure LCP

solution is spun out of a spinneret. Inside the spinneret, shear flow plays an important

role, and outside the spinneret the fiber experiences elongational flow. Besides that, the

characteristic processing time can be of the same order or even smaller than the internal

relaxation time of the LCP solution. Therefore, both linear and non-linear rheological

properties of LCP solutions are important for the typical processing.

Despite the big progress on the study of the rheology of LCP solutions, there is a room

for further investigations. For example, the contribution of hairpins to the rheological

properties is still not well understood. In this thesis a model capable to account for the

presence hairpins is presented.

1.5 Thesis outline 15

1.5 Thesis outline

In this thesis a model for solutions of semi-flexible main-chain LCPs containing hairpins

is presented. Within this model the polymer chain is represented by a rod-spring-bead

chain. We perform an analytical study of the linear rheology for the limit of highly-

aligned and unentangled chains containing hairpins. The general case and the non-

linear rheology are studied via numerical simulations based on the Euler-Maruyama

method. We investigate the isotropic-nematic phase transition, the life-time distribution

function of the hairpins and some rheological properties, like viscosities, normal stress

differences, response moduli and the transient behavior. The ”dynamic transition” from

oscillatory orientational motion to the flow-aligning behavior is revisited. In particular,

a lot of attention is focused on the contributions from the hairpins to the rheology.

In Chapter 2 the mechanical model for the polymer chain is described and the evol-

ution equations are derived from the basic principles of statistical mechanics. The

Smoluchowski equation for the ensemble of chains is derived from the Liouville equa-

tion. We provide a detailed reasoning behind the closure relations that are used. Then

the obtained Smoluchowski equation is reformulated in terms of a system of stochastic

differential equations (SDEs) and the choice of the interpretation for these SDEs is ex-

plained. Then the equations are brought to a dimensionless form. The resulting dimen-

sionless groups of parameters are analyzed and the set of characteristic time scales that

are present in the system is described.

Chapter 3 is devoted to the analysis of the linear rheology for the limiting case of un-

entangled highly aligned rod-spring chains containing hairpins. The evolution equa-

tion for the director is derived. The response moduli, the first normal stress difference

and the steady-state viscosity are also obtained and their dependence on the fraction of

hairpins and other relevant parameters is described.

The final chapters are devoted to the numerical simulations performed using the Euler-

Maruyama method. In Chapter 4 we describe the algorithm and the choice for the values

of the parameters. The values of the parameters are chosen on the basis of experimental

data available for a solution of PpPTA in sulfuric acid of about 19.7 wt% [11, 35, 60].

Furthermore, in Chapter4 the results of the simulations for the equilibrium properties are

presented as well. In particular, the dependence of the degree of orientational ordering

as a function of the strength of the nematic potential and the number of rods in the

chain, i.e., the degree of flexibility of the chain, is examined.

The results of the simulations for the behavior of the LCP solution undergoing various

types of flows are presented in Chapter 5. First, the effect of ”hairpin entanglements” is

examined in the elongational flows. The contribution of the hairpins shifts the response

of the solution towards a more elastic behavior. This agrees with what was sugges-

16 Introduction

ted in [35]. Secondly, the results for the rheological behavior in steady shear flow are

presented. They include the steady-state viscosity and the first normal stress differences

as functions of the shear rate. Also the analysis of the orientational motion is done. The

transition from kayaking through wagging to flow-aligning is observed. The results

show qualitative agreement with both theory and experiments. Finally, we summarize

our findings.

Chapter 2

Phase-space theory for LCPs

2.1 Introduction

Macroscopic systems typically contain a large number of particles (Avogadro constant

is approximately 6.02 · 1023mole−1), making the use of the usual dynamical description

of mechanical systems inappropriate. Even if we were able to determine the motion of

each particle in the system it would be extremely difficult to imagine what all these data

would mean from a macroscopic point of view. Fortunately, such a detailed descrip-

tion is not needed. Macroscopic properties of a many-particle system can be deduced

from the principles of phase-space kinetic theory [61]. Within that approach the state

of the system is described by a probability distribution function in the phase-space of

the system, i.e., the space of all coordinates and momenta of the particles. Using this

distribution function macroscopic quantities can be obtained as ensemble averages of

combinations of dynamical variables.

This phase-space probability distribution function is sometimes called the N -particle

distribution function, because it gives the probability to find allN particles of the system

in a specified configuration with specified momenta. This function, however, contains

too much information to be convenient for practical use, except in very simple cases,

like the ideal gas. Therefore, a simplification is usually made. The Liouville equation

for the N -particle distribution function is reformulated into the equivalent Bogoliubov-

Born-Green-Kirkwood-Yvon hierarchy of equations (BBGKY hierarchy) [62–66]. Then

the BBGKY hierarchy is truncated. After truncation of the BBGKY hierarchy of equa-

tions the description typically passes from a N -particle distribution function to 1- and

2-particle distribution functions. This truncation step is not rigorous and requires addi-

tional assumptions that lead to closure relations. In other words this step can be seen

18 Phase-space theory for LCPs

as a modeling step, because depending on the assumptions made, different results are

obtained. On the other hand the phase-space theory approach is established and its

application to polymeric systems has been extensively studied. For example in [61, 67].

Bird et al. [61] has described the way to apply the BBGKY hierarchy to polymeric sys-

tems. In this chapter we apply the general procedure formulated by Bird et al. [61] to

a system consisting of chains of a particular structure. We describe a simplified mech-

anical model of a main-chain liquid crystalline polymer and formulate a Smoluchowski

equation for its time evolution.

2.2 Model for a main-chain liquid crystalline polymer

The aim of this work is a study of the rheological properties of concentrated solutions of

main-chain liquid crystalline polymers, like solutions of PpPTA in sulfuric acid. A solu-

tion of PpPTA in concentrated sulfuric acid forms a lyotropic nematic phase between 8

and 20 wt% [11]. The industrially relevant parameters are 19.6 wt% concentration and

a temperature of about T = 350K . At this concentration solutions of PpPTA in sulfuric

acid are concentrated solutions, because V −1Nchbl2p > 1. In this criterion V is a volume

of a solution sample, Nch is the number of chains inside this sample, b is the diameter

of the polymer chains, and lp is the persistence length of the polymer molecules. In the

original criterion formulated for Doi’s rigid rod model [25] the length of the rod instead

of lp was used.

The molecular weight of a PpPTA chain is about 30 kDalton. Under these conditions

the persistence length of a PpPTA-chain is some 5-10 times smaller than its contour

length [35]. Thus, the backbone of a polymer is stiff enough to form a nematic phase,

but not stiff enough to display rigid-rod dynamical behavior. We are confronted with a

concentrated solution of main-chain semi-flexible polymers. We expect therefore to find

essential deviations in the rheology of PpPTA solutions from the theoretical predictions

based on the rigid-rod model or the slightly bending rod model. These models, even

though successful for more stiff polymeric systems, are not capable of incorporating the

effects related to hairpin formation.

In order to study a polymeric solution we need to specify the model of a polymer chain.

Real polymers usually have lots of details on the length-scale of a monomer. It is a

very complicated problem to model polymers when taking these micro-scale details

into account. Fortunately, polymers with different chemical structure may show sim-

ilar behavior on a macro-scale. This suggests that lots of details of the ”shape” of the

polymer backbone are irrelevant for the macroscopic physical properties and can be

omitted. Consequently, we expect to get a good description of the polymer solution

even if we represent the polymers by some coarse-grained objects, provided that these

coarse-grained objects are chosen appropriately. Actual polymers are usually represen-

2.2 Model for a main-chain liquid crystalline polymer 19

ted by relatively simple mechanical chain configurations consisting of springs, rods and

beads. We will also use such a model.

Because the polymer chain is 5-10 times longer than its persistence length, the model of

a chain should be able to mimic the rigidity of the polymer backbone on a length-scale

smaller than the persistence length and the backbone’s flexibility on larger scales. For

this purpose we can adopt the rod-spring model with the length of rods taken to be of

the order of the persistence length. Such a model is used in [29], but for our purposes

we will need to modify it.

Firstly, we give a different interpretation to the elasticity of the springs. In [29] the

springs are considered as Gaussian ”entropic” springs. It is known that a Gaussian

spring can be seen as the limit of a freely jointed rod chain model, that consists of a

very large number of short rods for which the average end-to-end distance is much

smaller than the contour length. This suggests that the rod-spring model with Gaus-

sian springs is more applicable to the case of a heteropolymer, consisting of long stiff

nematogenic units and flexible spacers with a much shorter persistence length. In the

case of PpPTA the stiffness of the chain does not vary along the backbone. Here we

introduce springs between the nematogens in order to eliminate the orientational cor-

relation between neighboring nematogens. Modeling the polymer as a single rod leads

to a correlation of orientations of the ends of this rod. Such an artifact of the model is

inappropriate in our case. This correlation should vanish when the persistence length is

several times smaller than the contour length and there are no long range interactions

in the system. The elasticity of the springs is determined by the average contour length

fluctuations of a chain in equilibrium at a given temperature.

Secondly, we will introduce additional point-like beads in the middle of the neighboring

springs. If the mobility of these beads is much larger than the mobility of the nemato-

gens, then these beads do not give any contribution to the response functions and the

model reduces to a rod-spring model. The only change these beads give rise to will be

the modification of the ”ideal-gas” part of the stress tensor. This is due to the equiparti-

tion theorem, because we introduce additional degrees of freedom. On the other hand,

the mobility of these beads may depend on the concentration of the solution or on the

fraction of hairpins present. This can mimic the hindrance that the surrounding has on

the motion of a chain. If a polymer chain has developed a hairpin, then the probability

that it will get stuck with the surrounding chains increases. This effect will be reflected

by the decrease in mobility of the bead that is closest to the hairpin. In order to make

this idea more clear we stress that although the mobility of the beads is usually not in-

volved in the expression for the stress tensor explicitly, the actual values of the stress

tensor depend on the mobility of the beads. This is due to the fact that the evolution of

chains is affected by bead’s mobility.

Having specified the structure of the chain model, we have now to choose an appropri-

20 Phase-space theory for LCPs

ate parametrization. Let each chain consist of N nematogens. Then the configuration of

a chain can be specified by the following set of variables:

{r1, r2, ..., rN} positions of the centers of mass of the rods,

{u1,u2, ...,uN} unit vectors specifying the orientation of the rods,

{b1,b2, ...,bN−1} positions of the beads.

This choice of variables is clarified in fig. 2.1.

2.3 Hamiltonian for the ensemble of chains

For each polymer chain we introduce the following set of variables in order to write

down the Hamiltonian of the system.

mr mass of a rod,

pritranslational momentum of the i-th rod,

J two-dimensional tensor of inertia of a rod,

φi azimuthal angle of the i-th rod,

θi inclination angle of the i-th rod,

puigeneralized rotational momentum vector of the i-th rod,

pφiazimuthal component of the generalized momentum of the i-th rod,

pθi inclinational component of the generalized momentum of the i-th rod,

mb mass of a bead,

pbitranslational momentum of the i-th bead

It is important to notice that objects pui, pφi

, pθi and J are defined on a sphere, and

therefore are 2-dimensional. Illustration of this remark is shown in fig. 2.2.

pui= pφi

+ pθi

J = J I

Here J is the moment of inertia of a rod when rotated along an axis perpendicular to

the rod, and I is the two-dimensional identity matrix. In a Cartesian coordinate system

the orientation vector of the i-th rod can be expressed by the following standard linear

combination of Cartesian basis vectors i,j,k.

ui = sin θi cosφii+ sin θi sinφij+ cos θik (2.1)

Following the phase-space kinetic theory [61] we have to formulate the Hamiltonian

for an ensemble of chains. This Hamiltonian will be used to derive the Smoluchowski

2.3 Hamiltonian for the ensemble of chains 21

O

bi−1 ribi ri+1

bi+1

ui−1 ui

Figure 2.1: Part of a chain near the i-th bead. Here ri,ri+1 denote the positions of the adjoiningnematogens, bi−1,bi,bi+1 denote the positions of the beads. O is the origin of the laboratoryreference frame.

x

y

z

upθ

pφ

pu

φ

θ

Figure 2.2: Illustration of the meaning of generalized momentum associated with the orienta-tional degrees of freedom.

22 Phase-space theory for LCPs

equation for the evolution of a chain. For the chain described in previous section, the

Hamiltonian for a single chain is given by the following expression:

Hsingle =

N∑

i=1

(

p2ri

2mr

+pui

· J−1 · pui

2+ U ext

ri

)

+

N−1∑

i=1

(

p2bi

2mb

+ U extbi

)

+ U intra (2.2)

Here U extri

and U extbi

are the potential of the external field for the i-th rod and i-th bead

respectively. U intra is the intramolecular interaction potential, i.e., the potential of inter-

actions between parts of the same chain. The Hamiltonian of an ensemble of chains is

given by the following expression.

H =

Nch∑

s=1

Hsingles + U inter (2.3)

Here Hsingles is the Hamiltonian for the s-th chain and U inter is the intermolecular inter-

action potential. The intermolecular interaction potential is the potential of interaction

between parts of different chains.

Having introduced the Hamiltonian for the system we can write down the equations of

motion (Hamilton’s equations)

∂

∂tri =

∂H∂pri

=1

mr

pri(2.4)

∂

∂tbi =

∂H∂pbi

=1

mb

pbi(2.5)

∂

∂tui =

∂H∂pui

= J−1 · pui

(2.6)

∂

∂tpri

= −∂H∂ri

= Fextri

+ Fintrari

+ Finterri

≡ Fri(2.7)

∂

∂tpbi

= − ∂H∂bi

= Fextbi

+ Fintrabi

+ Finterbi

≡ Fbi(2.8)

∂

∂tpui

= − (I− uiui) ·∂H∂ui

= Fextui

+ Fintraui

+ Finterui

≡ Fui(2.9)

Here Fri, Fbi

and Fuidenote the total force acting on the i-th rod, the total force act-

ing on the i-th bead and the total generalized angular force acting on the i-th rod re-

spectively. Superscripts ext, intra and inter stand for contributions from external forces,

intramolecular interactions (between parts of the same chain) and intermolecular inter-

actions (between different chains) respectively.

These Hamiltonian equations are used to construct the Liouville operator, which is the

key-object for the general equation of change.

2.4 Smoluchowski equation 23

2.4 Smoluchowski equation

One of the central equations in statistical mechanics is the Liouville equation. It was first

published in its modern form by Gibbs in 1902 [68]. This equation describes the time

evolution of the phase space distribution function and expresses the conservation of

the normalization of the distribution function. Liouville’s theorem states that the phase

space distribution function is constant along any trajectory in the phase space. If the

phase space distribution function is denoted by f then the Liouville equation is given

by∂f

∂t= −Lf (2.10)

where L is the Liouville operator. For the ensemble of chains obeying Hamiltonian

equations (2.4)-(2.9) the Liouville operator is given by the following expression

L =

Nch∑

s=1

(

N∑

i=1

1

mr

prsi· ∂

∂rsi+

N−1∑

i=1

1

mb

pbsi· ∂

∂bsi+

N∑

i=1

pusi· J−1 · ∂

∂usi+

+N∑

i=1

Frsi· ∂

∂prsi

+N−1∑

i=1

Fbsi· ∂

∂pbsi

+N∑

i=1

Fusi· ∂

∂pusi

)

(2.11)

The physical properties of our macroscopic polymeric system are ensemble averages of

certain dynamical variables, i.e., functions of the phase space variables. We will use 〈·〉to denote an ensemble average.

LetB be a dynamical variable. Then from the Liouville equation (2.10) the general equa-

tion of change of B can be derived. It describes the evolution in time of the ensemble

average of the dynamical variable.

∂

∂t〈B〉 = 〈LB〉 (2.12)

For instance, if we take the dynamical variable B to be

BΨ (y) =

Nch∑

s=1

N∏

i=1

δ (rsi − ri) δ (usi − ui)

N−1∏

j=1

δ(

bsj − bj)

(2.13)

where y denotes the set of variables related to a single chain

y ≡ {r1, . . . , rN ,u1, . . . ,uN ,b1, . . . ,bN−1} (2.14)

then the ensemble average of BΨ becomes the single-chain configurational distribution

function. Consequently, when BΨ is substituted into the general equation of change

(2.12) we obtain the evolution equation for the configurational distribution function Ψ.

24 Phase-space theory for LCPs

This step is described in Appendix A.1. We show here only the resulting equation.

∂Ψ

∂t+

N∑

i=1

∂

∂ri·(

JpriK

mr

Ψ

)

+

N∑

i=1

∂

∂ui·(

J−1 · Jpui

KΨ)

+

N−1∑

i=1

∂

∂bi·(

JpbiK

mb

Ψ

)

= 0 (2.15)

The double-brackets J...K are used to denote a momentum space average. Equation

(2.15) expresses the conservation of normalization for the single-chain configurational

distribution function Ψ. Terms involving the double-brackets are momentum-space av-

eraged fluxes of probability density for the corresponding degrees of freedom. In order

to extract any results from (2.15) we need to specify the evolution of the momentum-

space-averaged quantities: JpriK, Jpui

K and JpbiK. Equations for these quantities can be

obtained in a similar way from the general equation of change (2.12). The dynamical

variable B should be chosen differently for this case. It should be BΨ multiplied by the

corresponding momentum variable. Equations obtained in this way will contain new

momentum-space averaged quantities, such as pairs of momenta. For these quantities

we will need to find a similar set of equations. This is the well known BBGKY-hierarchy

of equations. At some point we have to stop this iterative procedure by specifying clos-

ure relations. Closure relations should give expressions for the momentum-space av-

eraged quantities obtained in the n-th iteration in terms of quantities defined in lower

iterations. The equations for JpriK, Jpui

K and JpbiK are derived in Appendix A.2. At

time scales larger than the characteristic time of equilibration in momentum space these

equations become force balance equations and are given by

0 = F(b)ri

+ F(e)ri

+ F(intra)ri

+ F(h)ri

(2.16)

0 = F(b)ui

+ F(e)ui

+ F(intra)ui

+ F(h)ui

(2.17)

0 = F(b)bi

+ F(e)bi

+ F(intra)bi

+ F(h)bi

(2.18)

We will refer to these forces as mesoscopic forces. Superscripts (e),(b),(intra),(h) in these

equations refer to the kind of forces such as external, Brownian, intramolecular and

hydrodynamic forces respectively. Expressions defining these quantities are given in

Appendix A.2. As an example we will give the resulting expressions for the Brownian

forces. Although, at this stage it is not clearly seen why these expressions represent

Brownian forces, we follow the terminology suggested by Bird et al. [61].

F(b)rj

= − 1

Ψ

N∑

i=1

∂

∂ri·

J(

pri−mrv

)

(

prj−mrv

)

K

mr

Ψ

+

+N∑

i=1

∂

∂ui·(

J−1Jpui

(

prj−mrv

)

KΨ)

+N−1∑

i=1

∂

∂bi·

J(

pbi−mbv

)

(

prj−mrv

)

K

mb

Ψ

(2.19)

2.4 Smoluchowski equation 25

F(b)uj

= − 1

Ψ

(

N∑

i=1

∂

∂ri·(

J(

pri−mrv

)

pujK

mr

Ψ

)

+

N∑

i=1

∂

∂ui·(

J−1Jpui

pujKΨ)

+

+

N−1∑

i=1

∂

∂bi·(

J(

pbi−mbv

)

pujK

mb

Ψ

))

(2.20)

F(b)bj

= − 1

Ψ

N∑

i=1

∂

∂ri·

J(

pri−mrv

)

(

pbj−mbv

)

K

mr

Ψ

+

+

N∑

i=1

∂

∂ui·(

J−1Jpui

(

pbj−mbv

)

KΨ)

+

N−1∑

i=1

∂

∂bi·

J(

pbi−mbv

)

(

pbj−mbv

)

K

mb

Ψ

(2.21)

In these expressions v stands for the velocity of the center of mass of the whole ensemble

of chains.

All expressions for the Brownian forces contain momentum-space averages of pairs of

momenta. On the time scale larger than the relaxation in momentum space the so-called

”equilibration in momentum space” closure relation can be used. This closure relation

is a consequence of the equipartition theorem. In our case this means that we take

J(

prj−mrv

)

(

pri−mrv

)

K = mrT Iδij (2.22)

Jpujpui

K = TJδij (2.23)

J(

pbj−mbv

)

(

pbi−mbv

)

K = mbT Iδij (2.24)

We omit Boltzmann’s constant and measure temperature directly in Joules. All the

cross-combinations of pairs of momenta are equal to zero. The ”equilibration in mo-

mentum space” approximation yields the following simplified expressions for the Brow-

nian forces.

F(b)rj

= −T ∂

∂rjlnΨ (2.25)

F(b)uj

= −T ∂

∂ujlnΨ (2.26)

F(b)bj

= −T ∂

∂bjlnΨ (2.27)

Let us briefly recapitulate the procedure that we are following in order to get a closed

26 Phase-space theory for LCPs

equation for Ψ. We start from the evolution equation for the single-particle configur-

ational distribution function (2.15). Then we derive similar equations (2.16)-(2.18) for

JpriK, Jpui

K, JpbiK. These equations are again not closed. In order to make these equa-

tions closed we employ the ”equilibration in momentum space” approximation to com-

pute the Brownian forces. The mesoscopic external forces and mesoscopic intramolecu-

lar forces allow direct computation because the microscopic external forces and the mi-

croscopic intramolecular forces are dependent only on the variables related to a single

polymer chain. Difficulties occur with the hydrodynamic forces. These mesoscopic

forces arise from the microscopic intermolecular interactions. In order to compute the

hydrodynamic force, the N -particle configuration distribution function is, in general,

required. Even upon the assumption of pairwise interactions and vanishing of higher

than two-particle correlations, the hydrodynamic force can not be computed directly.

We still need a two-chain distribution function. The evolution equation for the two-

particle distribution function will contain new momentum space-averaged quantities,

which will require new evolution equations. At some point we have to terminate this

sequence. We can split the inter-particle interactions in two parts: short-range interac-

tions and long-range interactions. For the treatment of the short-range interactions the

so-called ”modified Stokes’ law empiricism” approximation is usually very efficient. It

is described in the book by Bird et al. [61] and is very successful for many polymeric sys-

tems ( [29], Rouse model, Zimm model). Actually, any theory assuming a linear relation

between the hydrodynamic force and the corresponding velocity can be regarded as a

theory with a ”modified Stokes’ law empiricism”. In our case this means introducing

the following set of relations.

F(h)ri

= −ζri· (JriK − v) (2.28)

F(h)ui

= −ζui· (JuiK − (I− uiui) · ∇v · ui) (2.29)

F(h)bj

= −ζbj·(

JbjK − v)

(2.30)

Here ζri, ζui

and ζbiare friction coefficients for translational motion of the i-th rod,

rotational motion of the i-th rod and translational motions of the j-th bead respectively.

We assume that the properties of the medium surrounding a polymer chain on the

length-scale of the order of the persistence length is homogeneous. This implies the

independence of the friction tensors ζri, ζbi

and ζuion the actual position of the rod or

the bead in space, i.e., on the variables ri and bi. We also assume the friction tensors

to be symmetric. This statement is true for an object with axial symmetry moving in a

homogeneous isotropic medium. For the system with nematic ordering this is an ap-

proximation.

ζri= ζ

Tri

ζbi= ζ

Tbi

ζui= ζ

Tui

(2.31)

The superscript T indicates the transposed tensor here.

2.5 Stochastic differential equations for a polymer chain 27

The contribution of the long-range interactions depends on the type of these interac-

tions. For a neutral polymer chain the long-range interactions are Van der Waals po-

larization forces. It was shown by Maier and Saupe [7] that these interactions can be

described in a mean-field way by a Maier-Saupe potential (1.5). This approximation

allows to reduce the description of the N -chain system to a single-chain distribution

function and to avoid the necessity to use a two-chain distribution function. Technically,

this means that the contribution of the long-range forces can be treated as an external

potential, although in a mean-field way. The treatment of the nematic phase using the

Maier-Saupe potential is a very common approach and is already described in many

books, for example in [69].

To conclude this section we recapitulate the evolution equation for the distribution func-

tion after substituting the expressions for JpriK, Jpui

K, JpbiK.

∂Ψ

∂t+

N∑

i=1

∂

∂ri·(

(

v (ri) + ζ−1ri

·(

F(e)ri

+ F(intra)ri

))

Ψ− Tζ−1ri

· ∂Ψ∂ri

)

+

+N∑

i=1

∂

∂ui·(

(

(I− uiui) · ∇v (ri) · ui + ζ−1ui

·(

F(e)ui

+ F(intra)ui

))

Ψ− Tζ−1ui

· ∂Ψ∂ui

)

+

+

N−1∑

i=1

∂

∂bi·(

(

v (bi) + ζ−1bi

·(

F(e)bi

+ F(intra)bi

))

Ψ− Tζ−1bi

· ∂Ψ∂bi

)

= 0 (2.32)

This Smoluchowski equation is the central equation describing the dynamics of a poly-

mer chain in our model. This equation will be used in derivations of macroscopic prop-

erties of the LCP solution.

2.5 Stochastic differential equations for a polymer chain

It is widely known [70–73] that the description of a stochastic process by means of a

partial differential equation (Smoluchowski equation, Fokker-Planck equation) for the

probability distribution function is equivalent to the description by the corresponding

stochastic differential equation (SDE). Since an SDE can be interpreted in a different

way it is important to specify the interpretation of the SDE when formulating the SDE.

A good discussion on this issue is given by van Kampen [74]. Expressions for equivalent

SDEs in Ito or Stratonovich interpretations are also recapitulated in Appendix A.3.

In the previous section we have derived the evolution equation for a single-chain distri-

bution function (2.32). In this section we formulate a system of SDEs that is equivalent

to (2.32).

28 Phase-space theory for LCPs

We rewrite terms containing the second order derivatives in the expression (2.32).

∂

∂ri·(

ζ−1ri

· ∂Ψ∂ri

)

=∂

∂ri·(

ζ− 1

2ri

· ∂

∂ri·(

ζ− T

2ri

Ψ)

)

=∂

∂ri·(

ζ− 1

2ri

· ∂

∂ri·(

ζ− 1

2ri

Ψ)

)

(2.33)

A similar derivation is made for ζbi.

∂

∂bi·(

ζ−1bi

· ∂Ψ∂bi

)

=∂

∂bi·(

ζ− 1

2

bi· ∂

∂bi·(

ζ− T

2

biΨ)

)

=∂

∂bi·(

ζ− 1

2

bi· ∂

∂bi·(

ζ− 1

2

biΨ)

)

(2.34)

The set of possible values for ui forms a unit sphere. The set of possible values for the

derivatives ui forms the set of planes tangent to this unit sphere. Because we treat deriv-

atives with respect to ui as unconstrained derivatives, we have to project the result onto

the plane perpendicular to ui. We will incorporate this projection operation into the

rotational friction tensor. Then the expression for the rotational friction tensor should

be given by the projector

ζui= ζrot (I− uiui) (2.35)

Here ζrot is the rotational friction coefficient. This coefficient can depend on the local

state of ordering of the surrounding medium, but it is independent of ui. In isotropic

media this coefficient is a constant.

A remark should be made about ζ−1ui

. The projector ζuiacts from R

3 to R2. Therefore

ζuiis the degenerated transformation. Then ζ

−1ui

does not exist. In order to avoid this

misunderstanding when writing ζ−1ui

we actually mean the inverse of the operator ζui

restricted to the plane perpendicular to ui.

ζ−1ui

= ζ−1rot (I− uiui) (2.36)

Returning to equation (2.32), we transform the terms containing second order derivat-

ives with respect to u.

∂

∂ui·(

ζ−1ui

· ∂Ψ∂ui

)

=∂

∂ui·(

ζ−1rot (I− uiui) ·

∂Ψ

∂ui

)

=

=∂

∂ui·(

ζ− 1

2rot (I− uiui) ·

(

∂

∂ui· ζ−

12

rot (I− uiui)Ψ

))

=

=∂

∂ui·(

ζ− 1

2ui

·(

∂

∂ui· ζ− 1

2ui

Ψ

))

(2.37)

2.5 Stochastic differential equations for a polymer chain 29

Employing results (2.33), (2.34), (2.37) in equation (2.32) gives

∂Ψ

∂t+

N∑

i=1

∂

∂ri·((

v (ri) + ζ−1ri

·(

F(e)ri

+ F(intra)ri

))

Ψ)

−

−N∑

i=1

∂

∂ri·(

T12 ζ

− 12

ri· ∂

∂ri·(

T12 ζ

− 12

riΨ)

)

+

+N∑

i=1

∂

∂ui·((

(I− uiui) · ∇v (ri) · ui + ζ−1ui

·(

F(e)ui

+ F(intra)ui

))

Ψ)

−

−N∑

i=1

∂

∂ui·(

T12 ζ

− 12

ui·(

∂

∂ui· T

12 ζ

− 12

uiΨ

))

+

+

N−1∑

i=1

∂

∂bi·((

v (bi) + ζ−1bi

·(

F(e)bi

+ F(intra)bi

))

Ψ)

−

−N−1∑

i=1

∂

∂bi·(

T12 ζ

− 12

bi· ∂

∂bi·(

T12 ζ

− 12

biΨ)

)

= 0 (2.38)

In order to write a system of corresponding SDEs we compare (2.38) with (A.37) and

(A.36). Equation (2.38) has the same structure as (A.37). This means that the corres-

ponding SDEs in the Stratonovich interpretation should have the structure of (A.34).

But when performing this transformation we have to take into account that the quantit-

ies t and x in equation (A.34) are dimensionless, which does not hold for (2.38). The set

of corresponding SDEs in the Stratonovich interpretation is therefore

(S)dridt

= v (ri) + ζ−1ri

·(

F(e)ri

+ F(intra)ri

)

+ ζ−1ri

· fri (2.39)

(S)duidt

= (I− uiui) · ∇v (ri) · ui + ζ−1ui

·(

F(e)ui

+ F(intra)ui

)

+ ζ−1ui

· fui (2.40)

(S)dbjdt

= v(

bj)

+ ζ−1bj

·(

F(e)bj

+ F(intra)bj

)

+ ζ−1bj

· fbj (2.41)

The first of these equations describes the translational motion of the rods, the second

equation describes the rotational motion of the rods, and the third equation describes

the translational motion of the beads. Indices i and j are from the following sets i ∈{1, ..., N}, j ∈ {1, ..., N − 1}. Symbols f

r, fu and fb denote the ”white-noise” forces

acting on translational and rotational degrees of freedom of the rods, and translational

degrees of freedom of the beads respectively. The following relations should hold for

fr, fu and f

b

⟨

fri (t) f

r

i′

(

t′)⟩

= 2ζriTδii′δ

(

t− t′)

(2.42)⟨

fui (t) fu

i′

(

t′)⟩

= 2ζuiTδ

ii′δ(

t− t′)

(2.43)

30 Phase-space theory for LCPs

⟨

fbj (t) fb

j′

(

t′)

⟩

= 2ζbjTδjj′δ

(

t− t′)

(2.44)

These relations are the analogs of the formal property of the Wiener process

⟨

dW

dt(t)dW

dt

(

t′)

⟩

= 2δ(

t− t′)

(2.45)

The factors in front of delta-function in (2.42)-(2.44) are due to the fact that the original

Smoluchowski equation (2.38) is not dimensionless.

2.6 Forces acting on a polymer chain

In the previous section we have derived the set of SDEs (2.39), (2.40) and (2.41) supple-

mented with (2.42), (2.43) and (2.44). So far we have not specified the expressions for

the forces and friction tensors. In this sections we will introduce the explicit expressions

for forces and friction tensors.

The formation of the nematic phase is due to the so-called nematic interaction. In 1958-

1960 Wilhelm Maier and Alfred Saupe developed a mean-field model to describe the

nematic-isotropic transition by introducing a continuous long-range nematic potential

( [6–8]). According to their theory the nematic potential emerges as the result of the

attractive dispersion forces. They assumed the molecules to be rigid and symmetric

around the molecular axis. The potential consists of an isotropic and an anisotropic

part, but only the anisotropic part gives birth to the nematic phase. The anisotropic part

of this potential is usually called a nematic potential or Maier-Saupe potential when the

nematogenic systems are considered. In the mean-field approach the nematic potential

for one rod reduces to the expression

UMS (u) = −UnS :

(

uu− I

3

)

(2.46)

This relation gives the energy a rod with orientation u would acquire due to nematic

interaction with the surrounding medium. Here Un is the strength of the nematic field,

S is the order tensor (orientation tensor). The order tensor is symmetric by definition.

S ≡N∑

i=1

⟨

uiui −I

3

⟩

(2.47)

The mean-field approach exploits the idea of replacing the two-particle interaction by

the single-particle interaction with the homogenized surrounding field. The properties

of the surrounding field are estimated as the mean of the corresponding quantity. In

case of the Maier-Saupe potential the field quantity is the order tensor S. It reflects the

2.6 Forces acting on a polymer chain 31

average state of orientation of the surrounding rods. The local quantity corresponding

to S is uu − I3 . The advantage of this approach is the ability to treat a two-particle

interaction potential as the external field which is quasi-linear with respect to uu − I3 .

The non-linearity is introduced by claiming the self-consistency condition: the mean of

uu − I3 computed with the nematic potential (2.46) should coincide with S. This self-

consistency condition is given by the definition (2.47).

The mean-field approach is an approximation. It works well when three-particle or

higher order correlations are not essential, which is expected in dilute systems. The

success of this approach depends on the ratio between the mean part of the local field

and the fluctuating part of the local field. The smaller the fluctuating part of the local

field, the more accurate results the mean-field approach gives. The larger the number of

objects giving essential contribution to the local field, the smaller the relative deviation

of the local field from its mean value will be. This justifies the success of the mean-

field approach for concentrated systems. Applied to the concentrated liquid-crystalline

polymeric solutions this approach turned out to be successful not only in the pioneering

works by Maier and Saupe [6], [7], [8], but also in many later works [11], [29], [47], [46].

In an article by Hutter at al. [75] the modification of the linear response theory for mean-

field approximations is discussed. It turns out that introducing mean-field approxim-

ations leads to a necessity of modifying the Green-Kubo relations [76]. However, as

it is shown in [75], if the mean-field is introduced in a way which is frequently used in

plasma physics (modifying only the interaction potential, but not the mobility matrices),

then the Green-Kubo relations experience only a slight modification. Due to the fact that

we employ the mean-field approximations only for the nematic potential, we expect that

in our case the Green-Kubo relations also experience only a slight modification.

In the equations (2.39), (2.40) and (2.41) the terms F(e)ri

, F(e)ui

and F(e)bj

are the external

field forces. Though we are considering concentrated LCP solutions without external

fields, the nematic interaction can be treated as an external field in the mean-field case.

The nematic energy per chain is

U chainMS = −UnN∑

i=1

S :

(

uiui −I

3

)

(2.48)

F(e)ui

= − ∂

∂uiU chainMS = 2UnS · ui (2.49)

The terms F(e)ri

and F(e)bj

are equal to zero, because the nematic interaction contributes

only to the orientational degrees of freedom, but not to the translational degrees of free-

dom like ri or bi.

The next type of forces present in the system are the intramolecular forces. These are

the forces acting between parts of the same molecule. In our case these forces are the

32 Phase-space theory for LCPs

elasticity forces of the springs connecting consecutive rods and beads. It is a classical

result [25] that for small perturbations of the chain a good approximation for the poten-

tial of the springs is the quadratic Hookean potential. The advantage of this choice is

the linear expression for the elastic forces. The disadvantage of this choice is revealed

when a shear or elongation flow is considered. When the shear rate or elongation rate

exceeds a certain limit the springs start to extend without bound and the model fails

to predict the real behavior of the solution. This problem arises because the Hookean

potential does not grow fast enough. This problem is not crucial for flows with small

deformation rates, but for high shear rates a steeper potential should be used. In 1972

H.R. Warner introduced the concept of finitely extendible nonlinear elastic (FENE) con-

nectors [77]. He chose the following force-extension law for the FENE connector

Fel (R) =

− koR(

1− R·R

l2max

) if |R| ≤ lmax,

−∞ otherwise

(2.50)

where ko is a spring constant and lmax is the maximum extension length of the spring.

R is the end to end connector vector for the spring. We will also use the version of the

function Fel with two arguments Fel (rend, rstart). We can relate this function to (2.50)

by putting R = rend − rstart.

It is difficult to explain why the force-extension law should have exactly the structure

of (2.50). This force law was originally derived by Kuhn and Grun [78] and represents

the effective elastic force exerted by a big number of ”phantom” rods connected by

completely free joints. The potential derived for the freely-jointed chain consisting of

many ”phantom” rods is a very appropriate model for the flexible polymers. However,

for semi-flexible polymers this argument is not valid any more, because the semi-flexible

chain can not be treated as a freely-jointed chain on the scales of the persistence length.

That is why we just quote H.R. Warner: ”The choice of this relationship is quite arbitrary

since innumerable equations can be proposed which predict Hookean spring behavior for small

extensions (|R|/lmax < 0.2) and yet have some length beyond which the connector cannot

be stretched.”. Warner applied this concept to a dumbbell model, but later this idea

was implemented in many other models. Also different variations of FENE force law

appeared (like FENE-P, FENE-CP). A good comparison of these three related force laws

is discussed in [79].

In our work we are going to consider a rod-spring-bead model with FENE-chains. This

means the expressions for the intramolecular forces F(intra)ri

, F(intra)ui

and F(intra)bi

will

follow from (2.50). In the region |R| ≤ lmax the potential for the FENE force law can be

constructed

Uel (R) =kol

2max

2ln

(

1− R ·Rl2max

)

(2.51)

2.6 Forces acting on a polymer chain 33

Then the elastic energy of the rod-spring-bead chain is given by

U chel =

N−1∑

i=1

Uel

(

bi − ri −1

2lui

)

+

N∑

i=2

Uel

(

bi−1 − ri +1

2lui

)

(2.52)

The expressions for the intramolecular forces F(intra)ri

, F(intra)ui

and F(intra)bj

are obtained

by differentiation of U chel with respect to a corresponding variable ri, ui, or bj .

At this point we should make a remark regarding the bending potential between the

consecutive rods.

Ubend =

N−1∑

i=1

U b (ui · ui+1) (2.53)

The reason why we do not introduce the bending energy between the consecutive rods

is the following. We want to associate the length of the rod in the model with the per-

sistence length of the polymer. But the orientations of the two consecutive rods becomes

correlated if we introduce a bending energy. Thus, the resulting persistence length of

the chain in the model becomes longer. Since the persistence length of the chain in

the model should coincide with the experimental values for the persistence lengths, the

length of each rod should be then shorter than the persistence length, i.e., the chain

should be represented by a larger number of rods. This makes the numerical scheme

more costly and requires some justification for the choice of the particular bending po-

tential. Since we choose the length of the rods to be equal to the persistence length, the

additional introduction of the bending potential would be inconsistent.

So far we have discussed the intramolecular forces and, partly, the intermolecular forces

(the nematic potential). As mentioned above, the nematic potential originates from the

anisotropic part of the dispersion forces and this is the contribution from the long-range

interactions. The short-range interactions should be also taken into account. In equa-

tions (2.39), (2.40) and (2.41) they are not present explicitly. Within this approach the

short-range interactions are incorporated in the expression of the friction forces. In-

stead of writing a potential for short-range repulsion between chains we have to specify

the expressions for the friction tensor.

For a spherical Brownian particle submerged into a fluid consisting of much smaller

molecules the hydrodynamic approach is applicable, therefore the Stokes’ law may be

used for the friction forces. However, when the size of the Brownian particle is com-

parable to the size of the surrounding molecules, a microscopic, statistical approach

becomes necessary. This has been the object of a large body of work, starting with the

pioneering paper of Kirkwood [65]. He showed that the friction coefficient is given by

a Green-Kubo formula, in terms of the time integral of the autocorrelation function of

the instantaneous total force exerted by the surrounding molecules on the test particle.

On the basis of his work friction tensors can be computed with high accuracy. In 1994

34 Phase-space theory for LCPs

Bocquet et al. [80] computed the friction coefficients for a hard spherical particle sub-

merged into a fluid consisting of the hard spherical particles as a function of the ratio of

the radii of the test particle and the fluid particles. It is a bit surprising, but it was found

that when a stick boundary condition is assumed, then the friction coefficient obtained

is very close to the one predicted by the Stokes law. The problem is that even for a sys-

tem with a simple geometry the computation of the friction coefficient becomes already

a very complex problem and requires a numerical approach.

The direct computation of the friction tensors for a rod or a bead submerged into a

fluid consisting of rod-spring-bead chains is expected to be a problem by itself. That is

why for our purposes we will adopt the expressions for friction tensors derived from

hydrodynamic consideration and modify them in order to reflect the peculiarities of the

system consisting of many rod-spring-bead chains. Therefore

ζri= ζr (ui) = ζ‖uiui + ζ⊥ (I− uiui) (2.54)

ζui= ζu (ui) = ζrot (I− uiui) (2.55)

ζbj= ζbead

(

hj)

I (2.56)

Here hj ≡ 12

(

1− uj · uj+1

)

is a ”hairpin variable”. We will discuss this quantity in detail

in the next section. Expression (2.54) reflects the fact that a rod can move much easier

along itself, than perpendicular to itself, i.e., ζ‖ ≪ ζ⊥. Expression (2.56) postulates an

isotropic friction tensor for the beads. The scalar coefficient ζbead(

hj)

is not constant,

but depends on whether there is a hairpin on the chain near the j-th bead. The idea

is to increase the friction tensor for a bead when the hairpin is formed on the chain.

This mimics the ability of hairpins to create topological constraints for the motion of the

chain when trapped with another hairpin or chain.

In concentrated LCP solutions the hydrodynamic interaction can be neglected [25]. That

is why the hydrodynamic interaction is omitted in expressions for the friction tensors

(2.54)-(2.56).

The last quantity in equations (2.39), (2.40) and (2.41) that we did not discuss yet is v.

It is the velocity field which the LCP solution is subjected to. We are going to consider

the behavior of the LCP solution under homogeneous deformations. Then v can be

represented by the velocity gradient tensor κ. κ is also called the deformation rate

tensor.

v (r) = κ · r (2.57)

To conclude this section we rewrite equations the (2.39), (2.40), (2.41) employing all the

2.6 Forces acting on a polymer chain 35

remarks made in this section. For i ∈ {2, . . . , N − 1} and j ∈ {1, . . . , N − 1}

(S)dridt

= κ · ri + ζ−1r (ui) ·

(

Fel

(

ri −1

2lui,bi−1

)

+ Fel

(

ri +1

2lui,bi

))

+

+ ζ−1r (ui) · fri (2.58)

(S)duidt

= (I− uiui) ·(

ζ−1rot

(

Fel

(

ri +lui

2,bi

)

l

2− Fel

(

ri −lui

2,bi−1

)

l

2

)

+

+κ · ui + 2ζ−1rotUnS · ui + ζ−1

rotfui

)

(2.59)

(S)dbjdt

= κ · bj + ζ−1bead

(

hj)

(

Fel

(

bj , rj +l

2uj

)

+ Fel

(

bj , rj+1 −l

2uj+1

))

+

+ ζ−1bead

(

hj)

fbj (2.60)

For i = 1 and i = N

(S)dr1dt

= κ · r1 + ζ−1r (u1) · Fel

(

r1 +1

2lu1,b1

)

+ ζ−1r (u1) · f

r1 (2.61)

(S)du1

dt= (I− u1u1) ·

(

κ · u1 + ζ−1rot

(

Fel

(

r1 +lu1

2,b1

)

l

2+

+2UnS · u1 + fu1 )) (2.62)

(S)drNdt

= κ · rN + ζ−1r (uN ) · Fel

(

rN − 1

2luN ,bN−1

)

+ ζ−1r (uN ) · frN (2.63)

(S)duNdt

= (I− uNuN ) ·(

κ · uN − ζ−1rot

(

Fel

(

rN −luN

2,bN−1

)

l

2+

+2UnS · uN + fuN )) (2.64)

This system of SDEs represent the rod-spring-bead model presented in this thesis.

36 Phase-space theory for LCPs

2.7 Hairpins and entanglements

The concept of hairpins was proposed by de Gennes [3]. He suggested that the loss of

entropy due to the high order in the nematic phase formed by the solution of polymers

with a stiff backbone will be partly recovered by sudden flips of the backbone, called

hairpins or kinks. The illustration of this concept is shown in the fig. 2.3. The dynamical

properties of a hairpin in a worm-like main-chain polymer were studied by Williams

and Warner [37]. In that article they also estimate the energy associated with the forma-

tion of a hairpin. Depending on the ratio between the strength of the nematic potential

and the stiffness of the polymer backbone the average number of the hairpins per chain

differs. In fig. 2.4, the spherical plot of the Maier-Saupe potential for a single nematogen

(rod) is depicted. Angles θ and φ are the inclination and the azimuthal angles of the ori-

entation vector u. It is clearly seen that the potential has a minimum not only for θ = 0,

but also for the opposite direction, where θ = π. Between these two orientations there

is an energy barrier of the order of Un. If this energy barrier is much higher then the

energy of the elastic deformation of the backbone, then once a hairpin is created it will

fluctuate for a relatively long time around a new local equilibrium. That is why hairpins

are also called sometimes ”trapped deformations”. The first experimental evidence of

the existence of hairpins in solutions of PpPTA in sulfuric acid was obtained by neutron

scattering in the work by Picken et al. [33].

In our rod-spring-bead model for a polymer chain the hairpin configuration can be dia-

gnosed by comparing the orientations of the neighboring rods. If the angle between the

consecutive ui is close to π, then the chain has a hairpin at that position. Therefore, the

variable hj is defined in the following way

hj ≡1−

(

uj · uj+1

)

2j ∈ {1, . . . , N − 1} (2.65)

Defined in this way hj gives zero, when the consecutive rods point in the same direc-

tion, and hj = 1 when the chain folds and the consecutive rods point in the opposite

direction, i.e., the rods are in a ”hairpin state”. For all other configurations the value of

hj will be between zero and one. The number of consecutive pairs of rods is equal to

the number of beads. That is why we will associate the variable h with the bead, that

is placed on the string between the corresponding rods. Due to the thermal motion the

orientations of the rods fluctuate and, thus, hi is almost never 1. This raises the ques-

tion about the proper estimation of the amplitude of fluctuations that are not destroying

the ”hairpin state”. Let us introduce the quantity hcrit, which clearly defines a ”hairpin

state”. If hj > hcrit, then we consider the part of the chain around the j-th bead in a

”hairpin state”. It is also convenient to use quantity εtol = 1 − hcrit, which we will call

”the tolerance to hairpins”.

The choice of this tolerance εtol associates some time-scale with the ”hairpin state”, be-

2.7 Hairpins and entanglements 37

a). Typical configuration

b). Hairpin configuration

Figure 2.3: Difference between the typical and the hairpin configuration of a polymer chain witha stiff backbone

x

y

z

θ

φ

UMS

Un

Figure 2.4: Spherical plot of the Maier-Saupe potential for the case when maximum eigenvalueof 〈uu〉 is equal to 0.8

38 Phase-space theory for LCPs

a)

b)

Figure 2.5: Illustration of the role of the hairpin defects in the chain’s backbone in the formationof entanglements: a) aligned chains without hairpins can not entangle, b) aligned chains canentangle by the backbone segments that are close to the hairpin defects

cause a smaller value of εtol correspond to a smaller lifetime of the ”hairpin state”. What

time-scale should be associated with such a ”hairpin state”?

There are several time-scales associated with a hairpin in the presented model. One

time-scale is associated with the process of folding and unfolding of the backbone of

the chain. This time-scale determines the lifetime of the hairpin defect in the chain’s

backbone. This first time-scale can be estimated as τrotnem exp(Un

T), and it rapidly in-

creases with an increase of Un. However, there is another time-scale associated with

the hairpins. This is the characteristic time-scale of the disentanglement of the chains.

Why do we associate the processes of entanglement and disentanglement of the chains

with the hairpins? In order to answer this question let us imagine a system of highly

aligned rigid rods, i.e., having no hairpins. The rods hinder each other because of ex-

cluded volume effects, but they can not entangle from each other, because they are all

more or less parallel. However, if some of the rods are folded, i.e., contain hairpin de-

fects, then different rods containing hairpins can entangle with each other at the hairpin

defect. A sketch of such configurations is shown in fig. 2.5. This figure illustrates why

the hairpin defects are so important for the entanglements in highly-aligned systems.

The disentanglement of such an entanglement can take place due to different processes.

In the first place chains can disentangle without destroying hairpin defects due to trans-

lational thermal motion. Secondly, if one of the hairpin defects unfolds due to thermal

motion, then the entanglement disappears. Thirdly, the imposed strong flow field can

destroy hairpin defects causing disentanglement of the chains. Each of these processes

will have characteristic time-scale. The process happening at the smallest time-scale,

2.7 Hairpins and entanglements 39

i.e., having the fastest rate, determines the average lifetime of the entanglement. In our

model we will associate the choice of hcrit and, consequently, εtol with the lifetime of

entanglements.

The lifetime of entanglements is dictated by thermal motion. If we estimate the hcrit on

the basis of the thermal fluctuations, then the corresponding maximum deviation angle

θmax is

sin θmax =

√

2T

(3λ− 1)Un(2.66)

Here λ is the maximum eigenvalue of the matrix 〈uu〉. The derivation of this formula is

given in Appendix A.4. For example, if Un

T= 25 and λ = 0.95, then θmax ≈ 0.2◦ and,

consequently, hcrit = 0.99 or εtol = 1 − hcrit = 0.01. If the expression on the right hand

side of the equation (2.66) becomes larger than 1, then the notion of ”hairpin state” is

not defined and it does not make sense to distinguish the ”hairpin state” from other

possible configurations.

Let us emphasize that in the system consisting of aligned semi-flexible chains, entan-

glements can be created only between the chains containing hairpins. We can conclude

this due to the orientational ordering of the system. That is why in concentrated nematic

LCP solutions the concepts of hairpins and entanglements are closely connected. In the

presented bead-rod-spring model the connection between hairpins and entanglements

is reflected by changing the mobility of the j-th bead when hj changes. For hj = 1

the mobility of the j-th bead is low, because the probability for the chain to get an en-

tanglement around bj is high. As hj decreases the mobility of the j-th bead increases.

At some characteristic value hcrit the probability for the chain to get entangled at bjbecome low and, therefore, the mobility of the j-th bead should approach the mobility

of the unentangled bead. In this way the criterion for defining a ”hairpin state” hcritcouples the possible formation of entanglements with hairpins.

Though, the stress tensor at each moment of time is determined by configuration of

the ensemble, and the mobility does not enter the stress tensor explicitly, the change of

mobility affects the evolution of the system. If the mobility is changed at some moment

of time, then the configurations that a chain will take at the next moment of time will

also change. Therefore, after the mobility is changed the stress will alter from the stress

that would be exhibited by the system if the mobility stays constant. For example, if

the system undergoes shear and the mobility of beads is low, then the part of the chain

in-between two beads is stretched a lot (because beads are mostly performing affine

motion), i.e., chains acquire conformations that result in a high value of the stress tensor.

This is the mechanism how the hairpins contribute to the stress tensor in the presented

model.

40 Phase-space theory for LCPs

2.8 Dimensionless version of the evolution equations

This section is devoted to the transformation of the equations (2.58)-(2.64) and (2.42)-

(2.44) to their dimensionless versions. The units of the quantities involved in these

equations are shown at the table 2.1.

Physical Quantity Symbol SI units Basic unitsTime of evolution t s sCharacteristic time of process τ s sLength of a rod l m m

Force of elasticity of a spring Fel N kg m s−2

Elasticity coefficient of a spring ko N/m kg s−2

Strength of the nematic Maier-Saupe potential Un J kg m2 s−2

Rod center mass position vector r m mRod orientation vector u - -Bead center mass position vector b m mOrientation (order) tensor S - -

Velocity gradient (Deformation rate) κ s−1 s−1

Temperature T J kg m2 s−2

Friction tensor for a rod ζ N·s/m kg s−1

Rotational friction for a rod ζrot J·s kg m2 s−1

Friction coefficient for a bead ζbead N·s/m kg s−1

Table 2.1: Units of the physical quantities

We should make a remark here about the units of temperature depicted in this table.

In SI the temperature has the unit Kelvin. This leads to the use of Boltzmann constant

k = 1.38 · 10−23J/K , which transforms Kelvin into Joule. Both Kelvin and Joule are

units of energy. To avoid ambiguity in energy units and the usage of non-fundamental

coefficient k we represent temperature in Joule.

Another remark is related to the quantity τ placed in the second line of table 2.1, called

a characteristic time. During the nondimensionalization procedure the time is scaled by

some characteristic time of the system. But the model described by equations (2.58)-

(2.64) and (2.42)-(2.44) has several characteristic times. For example the characteristic

relaxation times for a translational and rotational motion of a rod are in general differ-

ent. This already raises the question which characteristic time is more appropriate for

scaling t. That is why we do not specify an exact meaning of this parameter τ yet.

We apply the following transformation to get the dimensionless version of equations

2.8 Dimensionless version of the evolution equations 41

(2.58)-(2.64) with (2.42)-(2.44).

r =r

lb =

b

l

t =t

τκ = κτ

Un =Un

TFel (r) =

Fel (r)

kol

fr(

t)

=

√

τ

2Tζ− 1

2 · fr (t) ζr (u) =ζr (u)

ζ‖

fu(

t)

=

√

τ

2Tζ− 1

2rot f

u (t) ζrot =ζrot

ζ‖l2

fb(

t)

=

√

τ

2Tζ− 1

2

beadfb (t) ζbead =

ζbead

ζ‖

Then the set of evolution equations for i ∈ {2, . . . , N − 1} and j ∈ {1, . . . , N − 1} be-

comes

(S) dri = αtrspr ζ−1

r (ui) ·(

Fel(

ri −ui

2, bi−1

)

+ Fel

(

ri +ui

2, bi

))

dt+

+ κ · ridt+ αtrdiff ζ− 1

2

r (ui) · dWri (2.67)

(S) dui = (I− uiui) ·(

αrotspr

(

Fel(

ri +ui

2, bi

)

− Fel(

ri −ui

2, bi−1

))

dt +

+κ · uidt+ αrotnemS · uidt+ αrotdiffdWui

)

(2.68)

(S) dbj = αtrspr ζ−1bead

(

hj)

(

Fel

(

bj , rj +uj

2

)

+ Fel

(

bj , rj+1 −uj+1

2

))

dt+

+ κ · bjdt+ αtrdiff ζ− 1

2

bead

(

hj)

dWbj (2.69)

For i = 1 and i = N

(S) dr1 = αtrspr ζ−1

r (u1) · Fel(

r1 +u1

2, b1

)

dt + κ · r1dt + αtrdiff ζ− 1

2

r (u1) · dWr1

(2.70)

42 Phase-space theory for LCPs

(S) du1 = (I− u1u1) ·(

αrotsprFel(

r1 +u1

2, b1

)

dt +

+κ · u1dt+ αrotnemS · u1dt+ αrotdiffdWu1

)

(2.71)

(S) drN = αtrsprζ−1

r (uN ) · Fel(

rN −uN

2, bN−1

)

dt+

+ κ · rNdt+ αtrdiff ζ− 1

2 (uN ) · dWrN (2.72)

(S) duN = (I− uNuN ) ·(

−αrotsprFel(

rN − uN

2, bN−1

)

dt +

+κ · uNdt+ αrotnemS · uNdt+ αrotdiffdWuN

)

(2.73)

The elements dWr,dWu and dWb are the increments of the 3D-Wiener process cor-

responding to the time-step dt. Expressions for the dimensionless groups in equations

(2.67)-(2.73) are given below

αtrspr =koτ

ζ‖αrotspr =

kol2τ

2ζrotαrotnem =

2Unτ

ζrot

αtrdiff =

√

√

√

√

2Tτ

ζ‖l2 αrotdiff =

√

2Tτ

ζrot(2.74)

We did not yet specify what time scale to take for the characteristic time τ . There are

five characteristic time scales in the system. These time scales can easily be identified

by equating each of the coefficients αtrspr, αtrdiff , αrotspr , αrotnem and αrotdiff to 1. The results

are given in table 2.2

The physical role of the elasticity coefficient ko in the formula (2.50) is well known. The

bigger ko is, the stiffer the spring represented by this potential is. However, it is more

convenient to express ko in terms of another parameter with a clear geometrical mean-

ing. Such a parameter is the average relative amplitude of the spring length fluctuations

A at temperature T in the equilibrium state. For this we will again use the equipartition

theorem. As mentioned in the previous chapter the springs are introduced to represent

the semi-flexibility of the chain, but it would not be relevant to allow the contour length

of the chain to extend much, i.e., A ≪ l. Therefore, to express the relation between koand A we can simplify the expression (2.50) to a corresponding Hookean potential.

koA2

2=

3

2T => ko =

3T

A2 =3T

A2l2(2.75)

2.8 Dimensionless version of the evolution equations 43

Description of the characteristic relaxation time Symbol Combination

Relaxation time of translational motion of a rod due to thepresence of the springs

τ trsprζ‖ko

Relaxation time of translational motion of a rod due tothermal fluctuations

τ trdiffζ‖l

2

2T

Relaxation time of rotational motion of a rod due to thepresence of the springs

τrotspr

2ζrot

kol2

Relaxation time of rotational motion of a rod due to thenematic field

τrotnem

ζrot2Un

Relaxation time of rotational motion of a rod due tothermal fluctuations

τrotdiff

ζrot2T

Table 2.2: Time scales of the model

In the limit of a strong nematic potential, which is likely to occur for a highly concen-

trated solution of semi-flexible polymers, the characteristic relaxation time of the orient-

ations of the rods is expected to be very small. If this is the shortest time scale, then it

would be convenient to consider other relaxation processes in terms of this time scale.

Therefore we choose the characteristic time τ to be τrotnem. When expressions for ko and

τrotnem (2.75) are substituted into (2.74) we get

αtrspr =3ζrot

2UnA2 αrotspr =

3

4UnA2 αrotnem = 1

αtrdiff =

√

ζrot

Unαrotdiff =

√

1

Un(2.76)

Here ζrot, Un and A are the parameters described in table 2.3.

From expressions (2.76) some qualitative conclusions can be already drawn. For ex-

ample, the change of the concentration at the same temperature corresponds to the

change of strength of the nematic field Un. The increase of Un makes αrotnem to be a dom-

inant term, which means nematic ordering. The decrease of Un makes αtrspr, αtrdiff , αrotspr

and αrotdiff grow. At some point these coefficients become much greater than 1, which

means that the nematic ordering should break down due to thermal motion.

Four dimensionless coefficients from (2.76) and two dimensionless frictions ζr and ζbeaddetermine the evolution of the system. In the end we gather all the major parameters

of the model in table 2.3. The change of these parameters leads to an essential change

of the behavior of the system. Vice versa: if we change all the parameters of the system

44 Phase-space theory for LCPs

Description Symbol Combination

Dimensionless nematic field strength UnUnT

Relative amplitude of the thermal fluctuations of thespring

A

√

3T

kol2

Relative maximum length of the springs lzlzl

Asymmetry of translational friction coefficient ξtrζ⊥ζ‖

Ratio between rotational and translational frictions ζrotζrot

ζ‖l2

Table 2.3: Major parameters of the model

in such a way that the major parameters are kept fixed then the outcomes will be only

re-scaled, but a qualitative change in the results does not occur.

We also present here the expression for the stress tensor in the dimensionless form for

the presented rod-spring-bead model. Details of the derivation of the expression for the

stress tensor can be found in Appendix A.5. The stress tensor will be nondimensional-

ized by the the pressure of an ideal gas under equivalent conditions.

σ =V

NchTσ (2.77)

Employing relation (2.75) we obtain the dimensionless expression for the stress tensor.

σ = − 3

A2

[

N∑

i=1

⟨

Friri

⟩

+1

2

N∑

i=1

⟨

Fuiui

⟩

+

N−1∑

i=1

⟨

Fbibi

⟩

]

(2.78)

Here Fri, Fui

and Fbiare the dimensionless total forces associated with the variables

ri, ui and bi respectively. These forces originate from the corresponding gradients of

the expression (T ln (Ψ (x)) + U (x)). The translational forces Fri, Fbi

are nondimen-

sionalized by kol. The rotational forces Fuiare nondimensionalized by kol

2

2 . The factor12 appears because the radius of the sphere generated by a rotating rod of length l is l

2 .

Expression (2.78) will be used in further chapters to compute the stress tensor. For

the needs of rheology usually it is enough to compute the so-called excessive stress.

The excessive stress is the deviation of the stress from the stress in the equilibrium.

The contribution from the translational Brownian forces is isotropic and independent

of whether the system is in equilibrium or in a non-equilibrium state. Therefore these

2.8 Dimensionless version of the evolution equations 45

terms do not contribute to the excessive stress tensor. However, the rotational Brownian

forces fui may produce a non-isotropic contribution if S is anisotropic.

N−1∑

i=1

〈fui ui〉 = −3NTS (2.79)

This fact will be used in the numerical simulations in Chapter 4 and Chapter 5.

To conclude this chapter we briefly recapitulate what was done till this point. We have

formulated the system of stochastic differential equations with multiplicative noise to

describe the concentrated LCP solution. This system of equations is capable of taking

into account the processes of creation and destruction of hairpins. Besides that, it is

capable of treating the coupling between the hairpins and entanglements. This system

has been nondimensionalized and the set of relevant dimensionless groups has been

identified. The resulting system of equations (2.67)-(2.73) is going to be used in the

numerical simulations. In the next chapter the analytical study of the case of highly

ordered nematic phase will be performed.

46 Phase-space theory for LCPs

Chapter 3

Rouse-like model in the

highly-ordered limit

3.1 Introduction

In this chapter we present the analytical results for the linear rheology of unentangled

main-chain semi-flexible LCPs containing hairpins. We reduce the model described in

Chapter 2 to the limit of a strong nematic field, i.e., high degree of orientational ordering.

The computational power of modern computers allows one to simulate the rheology of

systems consisting of ensembles of quite sophisticated mechanical chains with various

types of interactions. But the importance of the analytical study of simplified models

can not be undervalued. The analytical formulae for the response functions depend

explicitly on the major parameters of the model. This makes the analytical formulae

very convenient for the understanding of the role of the various factors represented

by these corresponding parameters. Moreover, these analytical results can be helpful

for the development of fast computational algorithms. For range of parameters the

computationally costly blocks can be adequately replaced by the approximate analytical

results.

The viscoelasticity of isotropic liquids of flexible polymers can be adequately described

by analytical models that contain only few parameters, for example, by Rouse [81] and

Zimm [82] models for unentangled polymer liquids and the reptation model for en-

tangled polymer liquids. The case of LCP solutions, however, is more difficult. There

are additional complications with LCP liquids in comparison to systems consisting of

flexible polymers. These complications are the anisotropy of the equilibrium state, the

48 Rouse-like model in the highly-ordered limit

dependence of the distribution of the configurations of the LCP chain on the strength of

the nematic interaction, the details of the chain structure (like the ratio between the con-

tour length and the persistence length), the capability of formation of several different

liquid-crystalline phases. Because of all these complications the molecular model for an

LCP chain is neither expected to be as simple or as general as the model used to describe

a flexible polymer. Therefore, existing molecular models for LCP systems usually only

have a few parameters and make use of very simple mechanical models for the chains,

like the Doi rigid-rod model and its numerous extensions [24, 40–44].

Depending on the particular semi-flexible polymeric system and the aim of the study

different limiting cases can be considered. For example, in works [83–85] by Morse

the focus is made on the understanding of the role of entanglements in concentrated

solutions of semi-flexible polymers. Though the solutions are concentrated it is assumed

that that the phase state is isotropic, i.e., the chain is flexible enough not to form a liquid

crystalline phase even at high concentrations. In the present chapter we focus on the

opposite limit. At high concentrations the nematic field is stronger and, therefore, the

orientational ordering is high. When the chains are highly aligned the entanglements

between the chains mostly occur between the chains containing hairpin defects of the

backbone. If the fraction of hairpins is small, then the contribution of the entanglements

to the stress tensor is not significant. This question will be addressed in the next two

chapters. A similar model, but without hairpins, was studied in [29]. In this chapter we

address the question on how the linear response of the solution of unentangled chains

changes when the chains develop hairpins.

3.2 Elimination of fast variables

We begin this section with rewriting equation (2.32) derived in Chapter 2.

∂Ψ

∂t+

N∑

i=1

∂

∂ri·(

(

v (ri) + ζ−1ri

·(

F(e)ri

+ F(intra)ri

))

Ψ− Tζ−1ri

· ∂Ψ∂ri

)

+

+

N∑

i=1

∂

∂ui·(

(

(I− uiui) · ∇v (ri) · ui + ζ−1ui

·(

F(e)ui

+ F(intra)ui

))

Ψ− Tζ−1ui

· ∂Ψ∂ui

)

+

+N−1∑

i=1

∂

∂bi·(

(

v (bi) + ζ−1bi

·(

F(e)bi

+ F(intra)bi

))

Ψ− Tζ−1bi

· ∂Ψ∂bi

)

= 0 (3.1)

When formulating the model of the polymer chain we introduced the beads in order

to mimic the entanglements between hairpins by changing the mobility of the beads

depending on whether the particular bead is located within an unfolded segment of

3.2 Elimination of fast variables 49

the chain (with ui · ui+1 > 0) or within a folded segment (with ui · ui+1 > 0). When

the corresponding segment of the chain is not in a folded state the probability for an

entanglement to occur is small for a highly-aligned system and, therefore, the mobility

of the corresponding bead should be much higher than the mobility of the rods. In this

case the bead gives a negligible contribution to the extra stress tensor compared to the

contributions from the other degrees of freedom. In other words, to model a solution

of highly-aligned unentangled LCPs, these artificial beads are not needed and can be

eliminated as fast variables (because they have the smallest associated relaxation time

in the system). Then variables{

r1, . . . , rNch

}

and{

u1, . . . ,uNch

}

can be considered as

slow variables for unentangled LCP solutions.

For processes occurring on the time-scale of the slow variables, the configurational dis-

tribution function Ψ(r,u,b) can be represented by the product of the distribution func-

tion for the slow variables and the distribution function for the fast variables condi-

tioned upon the values of the slow variables.

Ψ(r,u,b, t) = Φ (r,u, t)χ (b; r,u) (3.2)

In this expression the arguments r,u or b denote the whole set of corresponding vari-

ables associated with a polymer chain. Φ (r,u, t) is the distribution function for the

slow variables at time t and χ (b; r,u) is the conditional distribution function giving the

probability to find the position of the beads b when the slow variables {r,u} have fixed

value. On time-scales relevant for the slow variables the system is in partial equilibrium,

because the degrees of freedom associated with fast variables have enough time to relax

towards equilibrium. That is why the conditional distribution function χ (b; r,u) does

not depend on time. Moreover, this suggests that the best approximation for the condi-

tional distribution function χ (b; r,u) is the Gibbs distribution where {r,u} are treated

as parameters of the potential. For example, in this case of the artificial beads, such an

approximation will lead to a modification of the elastic potential and will give a contri-

bution to the stress tensor of 12TVI per bead (according to the equipartition theorem). But

when these beads were introduced to mimic the entanglements, a remark was made that

this contribution is non-physical and only the excessive part of the beads’ contribution

to stress tensor will be taken. Therefore the right way to eliminate these artificial beads

for unentangled systems is not by taking the Gibbs distribution function for χ (b; r,u).

To eliminate this isotropic contribution we also have to omit the thermal fluctuations

of the beads around their temporary equilibrium point. This suggests the following

expression for the conditional distribution function

χ (b; r,u) =

N−1∏

i=1

δ

(

bi −1

2(ri + ri+1) +

1

4(ui − ui+1)

)

(3.3)

The evolution equation for Φ (r,u, t) is obtained by substituting (3.3) into (3.2), substi-

tuting (3.2) into (3.1) and then integrating (3.1) with respect to beads variables. The last

50 Rouse-like model in the highly-ordered limit

set of terms in equation (3.1) vanishes upon integration, because these terms contain the

divergence with respect to the beads’ variables. Integration of the remaining terms can

be easily performed by employing the properties of the delta-function, i.e., just repla-

cing bi by 12 (ri + ri+1) +

14 (ui − ui+1). For the forces and the friction tensors we use

the expressions suggested in Chapter 2.

∂Φ

∂t+

N∑

i=1

∂

∂ri·(

(

v (ri) + ζ−1ri

· Felri)

Φ− Tζ−1ri

· ∂Φ∂ri

)

+

+

N∑

i=1

∂

∂ui·(

(

(I− uiui) · ∇v (ri) · ui + ζ−1ui

·(

Fnemui

+ Felui

))

Φ− Tζ−1ui

· ∂Φ∂ui

)

= 0

(3.4)

Here Felri

and Felui

are the elastic forces arising from the spring potential after elimina-

tion of the bi’s. This equation describes the rod-spring chain and is equivalent to the

following set of SDEs. For these equations the choice of Ito or Stratonovich interpreta-

tion does make any difference. But for the consistency we will stick to the Stratonovich

interpretation in this chapter. For i ∈ {1, . . . , N}

(S)dridt

= κ · ri + ζ−1r (ui) ·Felri + ζ

−1r (ui) · fri (3.5)

(S)duidt

= (I− uiui) ·(

κ · ri + ζ−1rotF

elui

+ 2ζ−1rotUnS · ui + ζ−1

rotfui

)

(3.6)

Depending on the system parameters, the equilibration time-scales of the translational

and rotational variables can be different. The equilibration of the chain’s backbone is

caused by two processes: the orientational alignment of the rods due to the nematic

field and the relaxation of the springs towards equilibrium by motions of the centers of

mass of the rods. Let us denote the equilibration time scale associated with translational

degrees of freedom to be τr and the equilibration time scale associated with rotational

degrees of freedom to be τu. If τr ≪ τu, then the translations of the rods are relaxing

towards equilibrium much faster than the rods are aligning due to the nematic field.

The opposite limit is τr ≫ τu. In this case the alignment of the rods’ happens much

faster than the relaxation of the springs. This assumption seems to be in agreement also

with the assumption of a strong nematic field, because the increase of the strength of

the nematic field leads to a reduction of the time needed for the alignment to happen.

Moreover, for chains consisting of many rods τr is increasing proportionally to N2 (like

the largest Rouse time), while τu is the local time-scale of a rod independent of the

number of rods constituting a chain. In this chapter we consider the second case. The

orientational variablesu are considered to be fast variables and the positions of the rods’

centers of mass are considered to be slow variables.

3.2 Elimination of fast variables 51

In order to separate slow and fast variables we follow the same scheme as described

above for the elimination of the beads. First the distribution function Φ (r,u, t) is rep-

resented as the product of the distribution function for the slow variables ψ (r, t) and

the distribution function φ (u; r) for the orientational degrees of freedom conditioned

upon the positions of the rods’ centers of mass.

Φ (r,u, t) = ψ (r, t)φ (u; r) (3.7)

For φ (u; r) the Gibbs distribution function is taken, where slow variables r are treated

as parameters.

φ (u; r) =exp

(

−U(u,r)T

)

∫

exp(

−U(u,r)T

)

du(3.8)

Then we substitute (3.7) into (3.4) and the perform integration with respect to all orient-

ational degrees of freedom u.

∂ψ

∂t+

N∑

i=1

∂

∂ri·(

ψv (ri) + ψ

∫

ζ−1ri

·(

Felri− T

∂ lnφ

∂ri

)

φdu+

−T∫

ζ−1riφdu · ∂ψ

∂ri

)

= 0 (3.9)

In order to obtain a closed evolution equation for the slow variable distribution function

ψ (r, t) we need to compute the integrals with respect to u (the integration is performed

over the unit sphere). The first integral in expression (3.9) can be seen as the total force

averaged with respect to the orientational degrees of freedom. The first term in the

integrand represents the contribution from the elastic forces and the second term rep-

resents the contribution from the orientational Brownian motion of the rods. We recall

the expression for the elastic force

Felri

= −∂U∂ri

(3.10)

and employ expression (3.8) to obtain

Felri− T

∂ lnφ

∂ri=

∂

∂ri

(

T ln

∫

exp

(

−U (u, r)

T

)

du

)

(3.11)

This relation has a simple interpretation. The sum of the elastic force and the Brownian

force originating from the orientational motion of the rods can be replaced by the gradi-

ent of some potential. This potential can be seen as the free energy of a single chain, that

we denote by F ch

F ch = −T ln

∫

exp

(

−U (u, r)

T

)

du (3.12)

52 Rouse-like model in the highly-ordered limit

The expression on the right hand-side of the last equality is clearly independent of the

orientational degrees of freedom u. Consequently the expression on the left hand-side

is also independent of u and, therefore, can be taken out from the integral in (3.9). Then

the only integral left to be computed in (3.9) is the expression for the averaged mobility∫

ζ−1riφdu. We will denote this averaged mobility by ζ

−1. In the highly-aligned limit

the rods are mostly oriented along or opposite to the director vector n (defined in (1.2)).

Mathematically this implies the following closure relation

〈uiui〉 =∫

uiuiφdu ≈ nn (3.13)

The validity of this closure relation is independent of the number of hairpins present in

the system. For a system with high orientational ordering, the occurrence of a hairpin

leads to a change of the sign of the corresponding ui, but this does not affect the diadic

product uiui. This closure relation is used to estimate the integral for the averaged

mobility

ζ−1 (n) =

∫

ζ−1riφdu ≈ 1

ζ‖nn+

1

ζ⊥(I− nn) (3.14)

By writing the argument of ζ−1 we stress the fact that the averaged mobility tensor

depends on the director n.

Finally, we combine (3.9), (3.11), (3.12), (3.13) and (3.14) to obtain the following closed

evolution equation for the slow variables r

∂ψ

∂t+

N∑

i=1

∂

∂ri·(

ψv (ri) + ψζ−1 (n) ·(

−∂Fch

∂ri

)

− Tζ−1 (n) · ∂ψ∂ri

)

= 0 (3.15)

This equation is of a diffusion-type and, therefore, can be represented by an equivalent

system of stochastic differential equations. In Chapter 2 this connection was described.

First of all we have to specify whether we choose the Ito or the Stratonovich interpret-

ation. The mobility matrix ζ−1 (n) is time-dependent, because the director can be a

function of time. But ζ−1 (n) does not depend on the positions of the centers of the

nematogens r. Thus, the spurious drift terms are zero, because they involve gradients

of the mobility tensor with respect to the variables r. In other words, because the mo-

bility tensor ζ−1 (n) has no direct dependence on r, there is no difference whether we

choose the Ito or the Stratonovich interpretation in that case.

(S)∂ri∂t

= v (ri)− ζ−1 (n) · ∂

∂riF ch (r1, ..., rN ) + fi(t) (3.16)

Here i ∈ {1, ..., N} and the Brownian force fi(t) satisfies the relations

〈fi(t)〉 = 0 (3.17)

3.3 Dynamics of a polymer chain 53

⟨

fi(t)fi′ (t′)⟩

= 2δi,i′ζ(n)Tδ(

t− t′)

(3.18)

Equations (3.16) combined with (3.17), (3.18) and (3.14) are the central equations of the

model presented in this chapter for the highly-ordered LCP solution without entangle-

ments. In the other sections we shall present the explicit expression for F ch (r1, . . . , rN ),

introduce the normal mode expansion for the dynamics of the chain, and derive the

expression for the stress tensor in terms of the normal-mode coordinates. In addition,

the equation for evolution of the director will be obtained. Finally, the influence of the

hairpins on the response functions will be studied.

3.3 Dynamics of a polymer chain

In the system of unentangled polymer chains the center of mass of each polymer chain

undergoes unperturbed Brownian motion. This unperturbed Brownian motion gives

an isotropic contribution to the stress tensor, which is also sometimes called ideal gas

contribution. This part of the stress tensor is usually not of interest to the rheology,

because it is not caused by the imposed flow and remains when the system reaches

equilibrium. Usually, in rheological problems the so-called excessive stress tensor or

some of its components are studied. The excessive stress is the deviation of the stress

tensor from its equilibrium value when the system is perturbed. This motivates us to

change the set of variables describing the conformation of the chain. Because the motion

of the centers of mass of the polymer chains in an unentangled system do not give

contributions to the excessive stress tensor, it is convenient to separate the motion of the

chains’ center of mass from the set of internal motions of the chain.

The motion of the chain with respect to laboratory reference frame is described by the

original set of variables {r1, r2, . . . , rN}. They are the vectors connecting the origin of

some chosen laboratory reference frame with the centers of nematogens. In fig. 3.1 this

origin is denoted by O. Using the transformation

ci ≡ ri+1 − ri (i ∈ {1, . . . , N − 1}) (3.19)

in combination with the center of mass position rc

rc =1

N

N∑

i=1

ri (3.20)

we can switch to the set of variables {rc, c1, c2, . . . , cN−1}. Here rc is pointing from the

origin O to the chain’s center of mass. The vector ci is connecting the centers of the i-th

and (i+ 1)-st nematogen.

54 Rouse-like model in the highly-ordered limit

H

ci

l

n

ri

ri+1

Figure 3.1: The part of a chain in the neighborhood of the i-th nematogen. Here ci is the vectorconnecting centers of the i-th and (i + 1)-st nematogen, O is the origin of the laboratory-fixedframe of reference, ri is the vector connecting the origin O with the center of the i-th nematogen,n is the director.

The most convenient choice of the coordinates is usually dictated by the form of the

equations that are describing the system. We start with the set {r1, r2, . . . , rN}, because

of the very transparent meaning of these vectors in it, and later we will change to the

normal-mode coordinates to bring this system of equations in its simplest form.

Equation (3.16) contains the gradient of the single chain free energy F ch (r1, ..., rN ). In

order to compute this gradient explicitly we need to compute the integral

− T ln

∫

exp

(

−U (u, r)

T

)

du (3.21)

In a paper by Morse [29] it was suggested that in the highly-ordered limit the integration

over the unit sphere can be replaced by integration over a tangent plane, because the

integrant is decaying rapidly to zero for orientations that substantially deviate from

the director. In a similar way the above integral was computed for the highly-aligned

system containing hairpins. In [86] this was done for continuous chains and in [87]

for discrete rod-spring chains with quadratic potential for the springs. Thus, we can

use the result of computing the integral (3.21) obtained in [87] for discrete rod-spring

chains. The free energy of a single chain in the highly-ordered regime in the presence of

3.3 Dynamics of a polymer chain 55

n

a.

b.

um

um

um+1

um+1

Figure 3.2: a. The normal state. um and um+1 are pointing in the same direction. b. Thehairpin state. um and um+1 are pointing in the opposite directions.

hairpins has the following structure [87].

F ch =1

2w0

N−1∑

m=1

(rm+1 − rm) ·K0 · (rm+1 − rm) +

+1

2w1

N−1∑

m=1

(rm+1 − rm − ln) ·K1 · (rm+1 − rm − ln) (3.22)

Herew0 is the fraction of the hairpin states, andw1 = 1−w0. K0 = k0I is the ”elasticity”

matrix for the hairpin state and K1 = k0nn + k1 (I− nn) is the ”elasticity” matrix for

the normal state. The anisotropy is due to the nematic interaction of the nematogens.

k1 =k0H0

H0 + k0l2 (3.23)

One remark should be made at this point. The elasticity coefficient of the spring k0 is

twice smaller than the coefficient that was used in previous chapters. Each of the springs

connecting the consecutive nematogens after the elimination of beads are consist of two

springs connected in series. Therefore the elasticity of the resulting springs is twice

smaller than the elasticity of the original springs. The schematic picture of the hairpins

in the highly-ordered limit is shown in fig. 3.2. After the fast variables are eliminated,

the model is incapable of describing the transition from the normal state to the hairpin

state smoothly, because the rotational degrees of freedom are eliminated.

56 Rouse-like model in the highly-ordered limit

Using expression (3.22) for the free energy of the chain F ch, the gradients ∂Fch

∂rsmay be

readily obtained by the straightforward differentiation.

∂F ch

∂rs= − (K · (r2 − r1)− w1lK1 · n) (s = 1) (3.24)

∂F ch

∂rs= −K · (rm+1 + rm−1 − 2rm) (s ∈ {2, . . . , N − 1}) (3.25)

∂F ch

∂rs= − (K · (rN−1 − rN ) + w1lK1 · n) (s = N) (3.26)

Here K denotes w0K0 + w1K1.

In most cases the flow can be considered to be uniform on the scale of the polymer

chain. For a uniform flow the velocity at any point r can be written as κ · r + vorigin.

Now vorigin may be eliminated by a proper choice of the origin. Hence

v (rs, t) = κ · rs (3.27)

Combining (3.16),(3.27),(3.24),(3.25),(3.26) then leads to the following set of equations

governing the motion of the chain

(S) ζ (n) · (r1 − κ · r1) = K · (r2 − r1)−w1lK1 ·n+ f1 (t) (s = 1) (3.28)

(S) ζ (n) ·(rs − κ · rs) = K ·(rm+1 + rm−1 − 2rm)+fs (t) (s ∈ {2, ..., N−1}) (3.29)

(S) ζ (n) · (rN − κ · rN ) = K · (rN−1 − rN )+w1lK1 ·n+ fN (t) (s = N) (3.30)

In here the stochastic forces satisfy the following conditions.

〈fs (t)〉 = 0 (3.31)

⟨

fs(t)fs′(t′)⟩

= 2δs,s′ζ(n)Tδ(

t− t′)

(3.32)

We end this section with the following conclusion. The dynamics of a rod-spring chain

in the presence of a strong nematic field on the slow time-scales is described by the

system of stochastic differential equations (3.28), (3.29),(3.30) supplemented by (3.31)

and (3.32).

3.4 Normal modes expansion

In the previous section we have given the system of stochastic differential equations

describing the dynamics of the nematogens’ centers of mass of a polymer chain. This

3.4 Normal modes expansion 57

system consists of N coupled equations. In this section we are going to decouple those

equations by changing to a normal modes description. This is achieved through a se-

quence of coordinate changes. The detail of this procedure are described in Appendix

A.6. In the end we obtain the set of N − 1 independent equations for the internal mo-

tions of the rod-spring chain. Because these equations are independent, the coordinates

qm are called normal modes.

(S) qm − κ · qm = −λmτ−1 · qm + αmw1lτ‖n+ vm(t) (3.33)

τ−1(n) = ζ

−1(n) ·K(n) = τ−1‖ nn+ τ−1

⊥ (I− nn) (3.34)

vm(t) = ζ−1(n) · hm(t) (3.35)

〈vm〉 = 0 (3.36)⟨

vm(t)vm′(t′)⟩

= 2ζ−1(n)λmTδm,m′δ(

t− t′)

(3.37)

We now write these equations in a dimensionless form by the following scaling:

qm =qm

lvm(t) =

τ‖lvm(τ‖ t)

t =t

τ‖κ = κτ‖

τ−1 = τ

−1τ‖ ζ−1

= ζ−1ζ‖

τ =τ⊥τ‖

ζ =ζ⊥ζ‖

Θ =2T

k0l2

The characteristic time scale used for non-dimensionalization is τ‖, which is the char-

acteristic relaxation time for the spring when perturbed along the director. Clearly, the

time scale of the rotation of the nematogens in the nematic field is not present in the

system any more.

The dimensionless evolution equations are then

(S)d

dtqp = −

(

λpτ−1 − κ

)

· qp + αpw1n+ vp(

t)

(3.38)

with⟨

vp(

t)⟩

= 0 (3.39)

and⟨

vp(

t)

vp′

(

t′)

⟩

= ζ−1

Θλpδp,p′δ(

t− t′)

(3.40)

If the director dynamics is prescribed, then these equations allow us to compute the qp.

From the internal dynamics of the polymer chain, the macroscopic properties and, in

58 Rouse-like model in the highly-ordered limit

particular, the stress tensor of the LCP solution can be deduced. Equation (3.38) is a

linear differential equation for qp and can be formally solved if κ(

t)

,n(

t)

and vp(

t)

are

given.

3.5 Ensemble average behavior of the normal-mode co-

ordinates

In the current section we present the solution of equation (3.38) and then deduce from

it the ensemble averages⟨

qp(t)⟩

and⟨

qp(t)qp(t)⟩

. The mathematical basis of this ap-

proach is described in Appendix A.7.

First introduce

Ap(t) ≡ −(

λpτ−1(t)− κ(t)

)

(3.41)

ap(t) ≡ αpw1n(t) + vp(t) (3.42)

and a matrix Mp(t) satisfying the initial-value problem:

Mp(t) = Ap(t) ·Mp(t)

Mp(0) = I

(3.43)

Then the solution of (3.38) is given by (A.79)

qp(t) = Mp(t) · qp(0) +Mp(t) ·t∫

0

dsM−1p (s) · ap(s) (3.44)

Averaging (3.44) then gives

⟨

qp(t)⟩

= Mp(t) ·⟨

qp(0)⟩

+ αpw1Mp(t) ·t∫

0

dsM−1p (s) · n(s) (3.45)

By averaging the dyadic product of (3.44) with itself we get the expression for⟨

qp(t)qp(t)⟩

.

⟨

qp(t)qp(t)⟩

=⟨

qp(t)⟩ ⟨

qp(t)⟩

+Mp(t) ·Bp(t) ·(

Mp(t))T

(3.46)

3.6 Evolution equation for the director 59

where Bp(t) satisfies the following initial-value problem.

Bp(t) = ΘλpM−1p (t) · ζ−1

(t) ·(

M−1p

)T

Bp(0) =⟨

qp(0)qp(0)⟩

−⟨

qp(0)⟩ ⟨

qp(0)⟩

(3.47)

These three quantities⟨

qp(t)⟩

,⟨

qp(t)qp(t)⟩

and Bp(t) will be required to compute the

stresses.

Now let us see which⟨

qp(0)⟩

and⟨

qp(0)qp(0)⟩

are worthwhile to consider. We will be

interested in those cases where the polymer solution is at rest before t = 0 and then

subjected to some deformation. Thus, the initial conditions⟨

qp(0)⟩

and⟨

qp(0)qp(0)⟩

should be determined in equilibrium state. This means we have to derive expressions

for⟨

qp⟩

eqand

⟨

qpqp⟩

eqfrom (3.45) and (3.46) in the limit t → ∞ with κ(t) = 0. In

Appendix A.8 the expressions for⟨

qp⟩

eqand

⟨

qpqp⟩

eqare obtained. They are given by

(A.83) and (A.87). It follows that if the system was in equilibrium at moment t = 0, then

Bp(0) = Θ τ · ζ−1(3.48)

Before we end this section it is clear that if the time-evolution of the director is known,

then equations (3.43),(3.45),(3.46) and (3.47) allow us to determine the evolution of the

ensemble averages⟨

qp(t)⟩

and⟨

qp(t)qp(t)⟩

. However, the evolution of the director is it-

self determined by the evolution of the conformations of the chains, i.e., by the ensemble

averages⟨

qp(t)⟩

,⟨

qp(t)qp(t)⟩

possibly by even higher order moments. Therefore in the

next section we will focus on establishing the missing link between the evolution of n(t)

and the average conformations of the chains.

3.6 Evolution equation for the director

In this section we will establish the equation for the evolution of the director. This will

be achieved by minimization of the Helmholtz free energy of the chains with respect to

the fast variables. In the highly-ordered limit considered in this chapter the relaxation

of the orientational degrees of freedom happens much faster then the relaxation of the

translational degrees of freedom. This means that on the time-scale on which⟨

qp(t)⟩

,⟨

qp(t)qp(t)⟩

evolve, we can consider the system to be in an equilibrium state with re-

spect to the orientation of the nematogens. In the highly-ordered limit the orientation

of the nematogens is determined by the director. This leads us to the conclusion that

the director should minimize the total free energy of the chains. In an earlier section

60 Rouse-like model in the highly-ordered limit

we have already given the free energy (3.22) of a single chain in the mean nematic field

caused by the surrounding molecules. The total free energy is a sum of the free energy

over all the chains plus the interaction term. For unentangled chains only the nematic

interaction remains, which is already accounted for in this single chain free energy via

the mean-field potential. Thus, the sum of the free energy (3.22) over all chains gives

the free energy of the whole system.

Fsys ≡Nch∑

i=1

F chi (3.49)

We will assume that all chains have an equal number of nematogens andNch ≫ 1. Then

the law of large numbers allows us to write (3.49) as

Fsys = Nch

⟨

F ch⟩

(3.50)

It is convenient to make Fsys dimensionless via

Fsys =Fsys

Nch12k0l

2 (3.51)

Using (3.22) and results (A.95),(A.96),(A.97) from, Appendix A.9 then gives us

Fsys = 2w1 (Λ : nn− p · n) + C (3.52)

The problem is to minimize (3.52) with respect to n for given Λ, p, C, upon the con-

straint that n · n = 1. We use the method of Lagrange multipliers in order to find a

minimizer. First we construct a Lagrange function

L = Λ : nn− p · n+ µ (n · n− 1) (3.53)

Here µ is the Lagrange multiplier. The factor 2w1 and constant C are omitted because

they do not influence the minimum. Now we look for a global minimum of L in terms

of n and µ, i.e.,∂L

∂n= 2 (Λ+ µI) · n− p (3.54)

∂L

∂µ= n · n− 1 (3.55)

Combining (3.54) and (3.55) and employing the symmetry of Λ then gives the following

equation for µ

(Λ+ µI)−2 : pp = 4 (3.56)

3.7 Modified stress tensor 61

For a given µ the director n follows from (3.54), i.e.,

n =1

2(Λ+ µI)

−1 · p (3.57)

From Sylvester’s criterion [88] it follows that, provided the eigenvalues of Λ are posit-

ive, expression (3.57) gives the minimum we were looking for.

The evolution of the ensemble averages depends on the evolution of the director. The

orientation of the director, on the other hand, is determined by the ensemble averages⟨

qp(t)qp(t)⟩

and⟨

qp(t)⟩

. This means equations (3.45), (3.46) and (3.56), (3.57) should be

solved simultaneously.

3.7 Modified stress tensor

At the end of Chapter 2 the expression (A.50) for the stress tensor was derived. This

expression, however, can not be used directly in the highly aligned limit. The reason is

that it contains ensemble averaged combinations of both fast u and slow r variables

σ = − 3

A2

[

N∑

i=1

⟨

Friri

⟩

+1

2

N∑

i=1

⟨

Fuiui

⟩

+N−1∑

i=1

⟨

Fbibi

⟩

]

(3.58)

This complication can be avoided by rederiving the expression for the stress tensor. We

use again the method of virtual work with a minute modification. After eliminating

the fast variables the evolution equation (A.50) is reduced to the evolution equation for

the slow variables (3.15). In addition the potential should be modified. If the original

potential of the polymer chain was U , then the modified potential or the free energy

of the polymer chain F ch is given by (3.12). Therefore, we need to use (3.15) and 3.12

when deriving modified expression for the stress tensor. The result is very similar to the

Kramers-Kirkwood formula

σ(t) =NchV

N∑

i=1

⟨

∂F ch(t)

∂riri(t)

⟩

(3.59)

Substituting the expressions (3.24), (3.25), (3.26) for the gradients of F ch into (3.59) and

transforming to normal modes yields the following explicit expression for the stress

tensor in terms of the averages⟨

qp (t)qp (t)⟩

,⟨

qp (t)⟩

σ(t) = −PidNI+2PidΘ

N−1∑

m=1

(

K ·⟨

qp (t)qp (t)⟩

− w1αmλm

n⟨

qp (t)⟩

)

(3.60)

62 Rouse-like model in the highly-ordered limit

where

Pid =NchainsT

V(3.61)

The non-dimensionalization of the stress tensor is done in the same way as in expression

(A.50), i.e.,

σ =σ

Pid(3.62)

and thus

σ(t) = −NI+2

Θ

N−1∑

m=1

(

K ·⟨

qp(t)qp(t)⟩

− w1αmλm

n⟨

qp(t)⟩

)

(3.63)

where

K = τ−1 · ζ (3.64)

Equation (3.63) expresses the stress tensor in terms of the averages of the normal-mode

coordinates. We are going to use this expression to analyze the stress in simple homo-

geneous flows. For example, the next section is devoted to uniaxial elongational flows.

A final remark about the stress tensor should be made. The total stress tensor for a

polymer solution consists of two contributions. The first contribution, which is dis-

cussed here (3.63), is that from the polymer chains. The second contribution is from the

solvent molecules. This contribution is very important in dilute or semi-dilute systems.

In concentrated solutions, however, this contribution is small. This is why we did not

include this contribution in the derivation of the stress tensor.

3.8 Uniaxial elongational flows

In the previous sections we have described the model for a highly-ordered nematic LCP

solution. We have introduced the normal modes expansion for the internal motions

of the polymer chains and derived the stochastic differential equations (3.38) for the

evolution of the normal-mode coordinates supplemented with conditions imposed on

the stochastic forces (3.39), (3.40). The orientation of the director is determined from the

minimization of the free energy of the system (3.52). This procedure indirectly gives

the evolution of the director because terms like Λ and p in (3.52) are changing with

time. Finally, we derived the expression for the stress tensor (3.63). After all these

preparatory steps are done we can analyze the behavior of the stress tensor when the

system is subjected to the imposed flow field. We start the analysis when the uniaxial

elongational flow is imposed.

The evolution of the director is dependent on the evolution of the ensemble averages

of the normal-mode coordinates. But the evolution of the normal-mode coordinates

is dependent on the orientation of the director, because it is involved in the equation

3.8 Uniaxial elongational flows 63

(3.38). This coupling brings complications to the analytical analysis of the evolution of

the system.

If an elongational flow is imposed and the direction of extension coincides with the ori-

entation of the director, then the orientation of the director does not change. This can

be shown in two steps. First we solve (3.38) by treating the director vector as a con-

stant vector. The way to solve (3.38) is described in Appendix A.7. Then we use the

resulting expressions for⟨

qp(t)⟩

and⟨

qp(t)qp(t)⟩

in the minimization problem where

they are substituted in the explicit expressions for Λ and p in (3.52). The result we then

obtain from the minimization procedure is that the director’s orientation coincides with

the direction of elongation. Thus, in case of uniaxial elongation flow the minimization

problem becomes trivial, which simplifies the analysis of the equation (3.38) consider-

ably.

Uniaxial elongational flow is characterized by one scalar function ˙ǫ (t) ≥ 0 and the

velocity gradient tensor has the following diagonal structure

κ(

t)

=

˙ǫ(

t)

0 0

0 − 12˙ǫ(

t)

0

0 0 − 12˙ǫ(

t)

(3.65)

Because of the absence of off-diagonal components in κ(

t)

and the rotational symmetry

with respect to the first axis, the expression for the stress tensor of the system subjected

to uniaxial elongation flow has also a simple diagonal structure.

σ(

t)

= −p(

t)

I +

σ‖(

t)

0 0

0 σ⊥(

t)

0

0 0 σ⊥(

t)

(3.66)

The relevant rheological quantity is the so-called tensile stress σT

σT ≡ σ‖ − σ⊥ (3.67)

A non-zero tensile stress expresses the anisotropy of the stress tensor (3.66). For small

deformation rates, i.e., ˙ǫ (t)N2 ≪ 1, the tensile stress can be represented by an integral

involving the response to the deformation from all previous moments of time

σT(

t)

=

t∫

−∞

E(

t− t′)

˙ǫ(

t′)

dt′ (3.68)

Substitution of (3.45) and (3.46) into (3.63) and the explicit evaluation of the matrices

Mp

(

t)

and Bp

(

t)

leads to an expression for the response function E(

t)

. After shifting

64 Rouse-like model in the highly-ordered limit

the initial moment of time to −∞ the following expression results

E(

t)

=

N−1∑

p=1

(

2 exp(

−2λpt)

+ exp

(

−2λpt

τ

)

+2

Θ

(

αpw1

λp

)2

exp(

−λpt)

)

(3.69)

The response function is represented by a sum of decaying exponents with different

characteristic time scales. For N = 5 there are 12 contributions. Therefore, it is conveni-

ent to show plots of E(

t)

. But before doing that we have to determine the values for

the parameters of the model τ , Θ, ζ, i.e.,

ζ =ζ⊥ζ‖

(3.70)

τ =τ⊥τ‖

=ζ⊥ζ‖

k‖k⊥

=ζ⊥ζ‖

k0

w0k0 + w1k0l

2H0

k0l2+H0

(3.71)

and

Θ =2T

k0l2 (3.72)

We do this in the following way. At the end of Chapter 2 the key-parameters of the the

full model are listed in the table 2.3. They are Un, A, lz, ξtr, ζrot. The detailed explan-

ation how these values were extracted from the experimental data or were estimated

theoretically for PpPTA in sulfuric acid will be given in Chapter 4. Here we establish

the relation between the parameters of the original model and the parameters of the

reduced model, and then we compute the values of τ , Θ, ζ on the basis of the values

for Un, A, lz, ξtr , ζrot. The relation between the parameters is given by the following

relations

τ = ξtr3 + 2UnA

2

3w0 + 2UnA2 (3.73)

ζ = ξtr (3.74)

Θ =4

3A2 (3.75)

Taking A = 0.1, ξtr = 2 and Un = 25 the values Θ and ζ are computed straightforward

and the parameter τ is expressed as a function of the hairpin fraction w0. The depend-

ence of log10 E as function of hairpin fraction w0 and time t is shown in fig. 3.3. We

see that an increase of the hairpins fraction w0 leads to a faster decrease of the response

function E. The character of this dependence is clearly shown in fig. 3.4. If the frac-

tion of hairpins is small, then the response curve has two distinct regions with different

slopes. But as the fraction of hairpins increases the slope of the curve in the second

region becomes steeper and the decay goes faster.

The model clearly demonstrates viscoelastic type of response with different time scales

involved. In order to separate the viscous part of response from the elastic part the

3.8 Uniaxial elongational flows 65

0

−5

0.0

0.2

0.4

100

50

0

log10 E

w0

t

Figure 3.3: 3D-Plot of log10 E versus hairpin fraction w0 and time t. The values of the para-meters are A = 0.1, ξtr = 2, Un = 25 and N = 5.

10 20 30 40 50 60 70

10-5

0.001

0.1

10

E

t

w0 = 0.0

w0 = 0.1

w0 = 0.2

w0 = 0.3

w0 = 0.4

w0 = 0.5

Figure 3.4: Plot of E as function of time t for different fractions of hairpins w0. The values ofthe parameters are A = 0.1, ξtr = 2, Un = 25 and N = 5.

66 Rouse-like model in the highly-ordered limit

storage and loss moduli are introduced.

E′ = ω

∞∫

0

E(

t)

sin(

ωt)

dt =

∞∫

0

E

(

φ

ω

)

sin(

φ)

dφ (3.76)

E′′ = ω

∞∫

0

E(

t)

cos(

ωt)

dt =

∞∫

0

E

(

φ

ω

)

cos(

φ)

dφ (3.77)

By substituting (3.69) into (3.76) and (3.77) and performing the integration we obtain

the explicit expression for E′ and E′′.

E′ (ω) =N−1∑

p=1

(

2ω2

ω2 + 4λ2p+

τ2ω2

τ2ω2 + 4λ2p+

2

Θ

(

αpw1

λp

)2ω2

ω2 + λ2p

)

(3.78)

E′′ (ω) =N−1∑

p=1

(

4λpω

ω2 + 4λ2p+

2λpτω

τ2ω2 + 4λ2p+

2

Θ

(

αpw1

λp

)2 λpω

ω2 + λ2p

)

(3.79)

According to (3.78) the storage modulus E′ exhibits a quadratic dependence on ω for

ω → 0. From (3.79) follows that E′′ has a linear dependence on ω for ω → 0. These two

facts are very general. They follow from the fact that on time scales much larger than

the internal time scales of the system the purely viscous stress should have a linear de-

pendence on the deformation rate and the purely elastic response should have a linear

dependence on the deformation itself. After performing Fourier transformation these

limiting types of behavior result in the linear and correspondingly quadratic depend-

ence on ω. This suggests that the expressions (3.78) and (3.79) exhibit a qualitatively

correct behavior in the low-frequency limit. In fig. 3.5 the storage modulus E′ and the

loss modulus E′′ are plotted as functions of the frequency ω for different fractions of

hairpins.

Another quantity to be analyzed here is the zero-deformation rate elongational viscosity.

This quantity is determined as

ηE0≡

∞∫

0

E(

t)

dt (3.81)

Substitution of the explicit expression for the response function (3.69) then gives

ηE0=

N−1∑

p=0

1

λp

(

1 +τ

2+

2

Θ

(

αpw1

λp

)2)

(3.82)

We stress that τ here is not an independent parameter, but it is expressed in terms of

the other parameters of the model by formula (3.73). Taking this fact into account we

plot the set of curves for the elongational viscosity as function of the fraction of hairpins

3.8 Uniaxial elongational flows 67

0.05 0.10 0.15 0.20 0.25 0.30

1

2

5

10

20

50

100

200

E′

E′′

ω (3.80)

Figure 3.5: Plot of the storage modulus E′ (blue lines) and the loss modulus E′′ (red lines)as functions of the frequency ω for different fractions of hairpins w0. The upper curve in eachset corresponds to w0 = 0.0 and the lower curve corresponds to w0 = 0.5. The values of theparameters are A = 0.1, ξtr = 2, Un = 25 and N = 5.

in fig. 3.6. Different curves correspond to a different values of the asymmetry of the

friction tensor ξtr. For this set of curves the anisotropy of the friction tensor ranges from

2 to 16. But the change of this parameter by one order of magnitude results in changing

the elongational viscosity not more than by 30%. The dependence of the elongational

viscosity on the fraction of hairpins is much stronger. While the fraction of hairpins

varies between 0.0 and 0.5 the elongational viscosity changes by a factor of 3. The

dependence of the reduced elongational viscosity ηE0on the number of rods per chain

N is shown in fig. 3.7. In the log-log plot this dependence is almost linear with a

slope around 3.0. This suggests that the dependence of the dimensionless elongational

viscosity on the number of nematogens follows a power-law

ηE0∼ N3 (3.83)

The dimensionless zero-shear rate viscosity demonstrates an almost a cubic dependence

on the number of nematogens per chain, i.e., on the molecular weight of the polymer.

This result should be interpreted correctly when compared to experimental data, be-

cause during the non-dimensionalization procedure the viscosity is scaled by the com-

bination of NchTV

τ‖ (units of pressure times units of time). But in real concentrated sys-

tems the increase of the molecular mass of the polymer chains leads to a decrease in the

number of polymer molecules per unit volume due to excluded volume effects. For ex-

ample, if the number of nematogens per unit volume is kept constant, then the increase

68 Rouse-like model in the highly-ordered limit

0.1 0.2 0.3 0.4 0.5

500

1000

1500

ηE0

w0

Figure 3.6: Plot of the elongation viscosity ηE0as function of the fraction of hairpins w0 for

different values of ξtr. Starting from the lower curve to the upper curve the value of ξtr takesvalues 2, 4, 8 and 16 respectively. The values of the parameters are A = 0.1, Un = 25 andN = 5.

0.5 1.0 1.5

1

2

3

4

5

6

log10 ηE0

log10N

Figure 3.7: Plot of the elongation viscosity ηE0as function of the number of rods per chain N

for different fraction of hairpins w0. The lower curve corresponds to w0 = 0.0, the middle one tow0 = 0.2 and the upper curve to w0 = 0.4. The values of the parameters are A = 0.1, ξtr = 2and Un = 25.

3.8 Uniaxial elongational flows 69

of the number of nematogens per chain by a factor of α leads to the decrease of the num-

ber of chains per unit volume by the same factor of α. Therefore, the quantity ηE0before

non-dimensionalization shows a quadratic dependence on the number of nematogens.

The general conclusion can be made from the analysis of the dependencies of the re-

sponse functions, the response moduli and the elongational viscosity on the fraction of

hairpins present in the system. For the meaningful values of the parameters the increase

of the fraction of hairpins leads to a decrease of the response. The response becomes

weaker and decays faster if compared to the system without hairpins. This result re-

flects how the response of a single chain is affected by the presence of hairpins, because

this model do not treat entanglements that might occur between the chains containing

hairpins.

Another conclusion that can be made is the weak dependence of the response function

on the anisotropy of the friction tensor ξtr when it changes in the range (0, 20). Also the

results demonstrate a weak dependence on the strength of the nematic field when the

the strength Un is above 20, because in the elasticity tensor K given by (3.64) becomes

almost independent of Un.

70 Rouse-like model in the highly-ordered limit

Chapter 4

Numerical simulations of

semi-flexible LCPs

4.1 Introduction

In Chapter 2 the model for a concentrated liquid crystalline semi-flexible polymer solu-

tion containing hairpins that accounted for the possible entanglements between chains

with hairpin defects was formulated in terms of a set of stochastic differential equa-

tions (2.67)-(2.73). We recapitulate these equations here in a brief form by introducing

the shorthand notations Felri

, Feluiand F

elbj

for the sums of generalized elastic forces as-

sociated with the corresponding degrees of freedom ri, ui, bj For i ∈ {1, . . . , N} and

j ∈ {1, . . . , N − 1} these equations read

(S) dri = αtrspr ζ−1

r (ui) · Felridt+ κ · ridt+ αtrdiff ζ− 1

2

r (ui) · dWri (4.1)

(S) dui = (I− uiui) ·(

αrotsprFeluidt+ αrotnemS · uidt+ κ · uidt+ αrotdiffdW

ui

)

(4.2)

(S) dbj = αtrspr ζ−1bead

(

hj)

Felbjdt+ κ · bjdt+ αtrdiff ζ

− 12

bead

(

hj)

dWbj (4.3)

The stochastic differential equation (4.2) contains a multiplicative noise term. That is

why the correct interpretation of the stochastic integral is important for these equa-

tions. When these equations were derived from the Smoluchowski equation (2.32) it

was shown that the equation (4.2) had to be interpreted in the Stratonovich way. The

equations (4.1) and (4.3) are not affected by the choice of the interpretation of the noise

term.

72 Numerical simulations of semi-flexible LCPs

The main dimensionless parameters were also determined in Chapter 2. These five para-

meters are given in table (2.3). The various α’s occurring in equations (4.1), (4.2), and

(4.3) are computed from the main parameters by the following formulas.

αtrspr =3ζrot

2UnA2 αrotspr =

3

4UnA2 αrotnem = 1

αtrdiff =

√

ζrot

Unαrotdiff =

√

1

Un(4.4)

There are various reasons that make it prohibitively difficult to solve the system of equa-

tions (4.1), (4.2), and (4.3) analytically. These arise from the non-linearity of the elastic

forces, the coupling between the translational and rotational motion, the complex dy-

namics of the orientation tensor in shear flow, the necessity to account for creation and

destruction of the hairpin defects etc. This is the reason why we decided to solve the

system of equations (4.1), (4.2), and (4.3) by a numerical method.

Numerical treatment of stochastic differential equations is a dynamically evolving field

[89], [90], [91]. Besides the traditional applications in physics, chemistry and microelec-

tronics, stochastic differential equations play an important role in biology, epidemiology

and, of course, in financial mathematics [92], [93]. The analysis and description of many

different numerical schemes for solving stochastic differential equations can be found

in the book by Kloeden and Platen [94]. One of the simplest and most studied numer-

ical methods for stochastic differential equations is the stochastic Euler method, also

called Euler-Maruyama method. This method can be seen as a kind of extension of the

of the well-known Euler method used for ordinary differential equations to stochastic

differential equation. In this thesis we use the Euler-Maruyama method for solving the

equations (4.1), (4.2) and (4.3).

In the remainder of this chapter we will first describe the Euler-Maruyama algorithm.

Then we estimate meaningful values for the main parameters. Then we will explain

some peculiarities connected with the implementation of this numerical scheme for

equations (4.1), (4.2), and (4.3). We conclude the chapter by computing some equilib-

rium properties of the system, such as the degree of orientational ordering of the system

depending on the strength of the nematic potential and the distribution of the lifetime

of the hairpins.

4.2 Euler-Maruyama method 73

4.2 Euler-Maruyama method

Let us consider the n-dimensional stochastic differential equation (Ito interpretation)

(I) dx(t) = a (x, t) dt+B (x, t) · dW (4.5)

x(0) = x0

Here t ∈ R, x ∈ Rn, W is the m-dimensional Wiener process, a (x, t) ∈ R

n, B (x, t) is

the matrix of size n×m, x0 is the initial value of x(t).

In the Euler-Maruyama scheme [94] the discrete analog of the equation (4.5) on the time

interval[

0, tf]

is constructed in the following way. First of all, the time-interval[

0, tf]

is partitioned into n equal time subintervals.

0 = t0 < t1 < · · · < tn−1 < tn = tf (4.6)

The width of each interval is ∆t =tfn

. Then for the above described stochastic differen-

tial equation (4.5) the scheme has the form

yi+1 = yi + a (yi, ti)∆t+B (yi, ti) ·∆Wi (4.7)

where ∆Wi = W (ti+1)−W (ti), i ∈ [0, . . . , n− 1] and y0 = x0. If the solution of (4.5)

is denoted as x (t), then the Markov chain y generated by a sequence of equations (4.7)

is called the Euler-Maruyama approximation of x (t).

In Chapter ?? it was explained why the equation (4.5) should be interpreted as an in-

tegral equation. In order to obtain the discrete scheme (4.7) the approximations for the

integrals in (4.5) on each time-subinterval were made

ti+1∫

ti

B (x (s) , s)dW (s) ≈ B (yi, ti) ·∆Wi (4.8)

ti+1∫

ti

a (x (s) , s) ds ≈ a (yi, ti)∆t (4.9)

Due to the fact that the numerical scheme is an approximation of the original problem,

two important questions have to be answered before the numerical scheme is used for

simulations. In what sense should the numerical approximation y converge to the exact

solution x as the number of subsegments in the partition of the time interval increases?

And what is the rate of this convergence? For the Euler-Maruyama scheme the answer

to this question is already known, but we need to introduce a new notation to explain

74 Numerical simulations of semi-flexible LCPs

it.

For each time-step ∆t the numerical scheme (4.7) generates a Markov chain. On the

basis of this Markov chain we define a process y∆t (·) by linear interpolation, i.e.

y∆t (t) = yi +

t− titi+1 − ti

(yi+1 − yi) if t ∈ [ti, ti+1) (4.10)

The process y∆t (·) is the result of a simulation with time-step ∆t. In order for the

numerical scheme to be useful the decrease of the time-step ∆t should lead to a better

approximation of x (·) by the numerical solution y∆t (·), i.e. y∆t (·) should converge to

x (·) as ∆t → 0. Because the convergence of one process to another can be defined in

different ways, the convergence of the numerical scheme can also be defined in different

ways. Basically, there are two types of convergence of the numerical scheme: weak and

strong [94].

The numerical scheme is strongly convergent if

lim∆t→0

E

(∣

∣

∣x− y

∆t∣

∣

∣

)

= 0 (4.11)

where E (·) denotes the expected value.

The numerical scheme is weakly convergent if for any polynomial g(·)

lim∆t→0

∣

∣

∣E (g (x))− E

(

g(

y∆t))∣

∣

∣ = 0 (4.12)

In words it says that the strong convergence guarantees that the algorithm almost surely

reproduces the realizations of the stochastic process x(·) correctly (up to some small

deviation) if the time-step ∆t is small enough. If the numerical scheme is capable of

approximating the expected values of the random process that is f (x(·)), where f is

some smooth enough function, then the numerical scheme is weakly convergent. For

our purposes weak convergence is enough, because we are interested in the evolution

of macroscopic quantities, which are the expected values of functions of the realizations

of the stochastic process describing the evolution of the polymer chain.

The Euler-Maruyama scheme is both strongly and weakly convergent [94] if the func-

tions a and B are smooth enough. For example, sufficient conditions are that a and B

should be four times continuously differentiable with bounded derivatives. But if the

sufficient conditions do not hold, the numerical scheme can still be convergent for some

particular stochastic differential equations. For example, it might happen that the pro-

cess x(·) almost never approaches the singular points of a and/or B, which makes the

scheme still convergent in that particular case.

Another important question concerns the convergence order, that characterizes the rate

4.2 Euler-Maruyama method 75

of convergence. The numerical scheme is strongly convergent of order γ if

E

(∣

∣

∣x (t)− y∆t (t)

∣

∣

∣

)

≤ β (t)∆tγ (4.13)

Here β (·) depends on time and on the type of stochastic differential equation that is con-

sidered. The numerical scheme is weakly convergent of order γ if for any polynomial

g(·)∣

∣

∣E (g (x (t)))− E

(

g(

y∆t (t)

))∣

∣

∣ ≤ βg (t)∆tγ (4.14)

Here βg (·) is depends on time, on the polynomial g(·) and again on the type of stochastic

differential equation that is considered.

In other words, if the convergence order of the numerical scheme is γ, then decreasing

the time-step ∆t by a factor of p leads to a decrease of the approximation error by factor

of pγ . It is known that Euler-Maruyama scheme is weakly convergent of order 1 and

strongly convergent of order 12 [94].

In practice, when the expected values of f (x(·)) are needed, the Euler-Maruyama scheme

is used in combination with a Monte-Carlo method. First, by using the Euler-Maruyama

scheme a bunch of k realizations of y∆t(·) is computed. Then the required average is

estimated by

E (f (x (t))) ≈ 1

k

k∑

i=1

E

(

f(

y∆t(t)

))

(4.15)

The error of such an estimation is of order 1√k

. Thus, the total error contains two contri-

butions: one from the Monte-Carlo estimator (4.15) and one from the Euler-Maruyama

scheme (4.7). The computational cost C of the numerical simulation of k realizations in

a fixed time-interval with p steps in each of them is C = kp. The variance of the Monte-

Carlo estimator is βmc

k, and the variance due to the deviation of f

(

y∆t(·)

)

from f (x(·))is β

em

pfor each realization. Thus, the sum of the variances is

Error =βmc

k+ k

βem

p=βmc

k+βem

Ck2 (4.16)

For a fixed computational cost, this function has a minimum at

k =

(

βmcC

2βem

)13

p =

(

2βemC2

βmc

)13

(4.17)

Computing each of the terms in (4.16) at the optimal point (4.17) gives values

(

2βem(βmc)

2

C

)13

and

(

βem(βmc)

2

4C

)13

for the first and second term respectively. The first term is twice big-

76 Numerical simulations of semi-flexible LCPs

ger than the second term. When performing the simulations we usually do not know

the values of βem and βms a priori, but these quantities can be estimated from the simu-

lations. The general recommendation for the optimal use of computational power is to

keep both errors of the same order.

Solving equations (4.1), (4.2), and (4.3) requires a small modification of the traditional

Euler-Maruyama scheme. But before we give a detailed description of the implementa-

tion, let us first estimate the values of the main parameters a some realistic system. This

will provide us with some intuition about the meaningful order of magnitude of these

parameters.

4.3 Values of the main parameters in the model

In the Chapter 2 five dimensionless main parameters that affect the qualitative behavior

of the system were discussed. We show these parameters in table 4.1 again. In this

section we are going to estimate values of these parameters for a concentrated solution

of PpPTA in sulfuric acid for industrially relevant conditions ( 19.8wt% PpPTA at 80oC).

Description Symbol Combination

Number of rods per chain N

Relative amplitude of the thermal fluctuations of thespring

A

√

3T

kol2

Dimensionless nematic field strength UnUnT

Relative maximum length of the springs lzlzl

Asymmetry of translational friction coefficient ξtrζ⊥ζ‖

Ratio between rotational and translational frictions ζrotζrot

ζ‖l2

Table 4.1: Major parameters of the model

We start with estimating the number of rods per chain N and the relative amplitude of

the thermal fluctuations of springs A. For this purpose we will need the data about the

contour length, persistence length and the transverse gyration radius of polymers.

Typical PpPTA polymers have an averaged molar mass µPpPTA ≈ 30000 gmole

and a

contour length of lc ≈ 1600A [35]. The data on the persistence length of PpPTA reported

by different researchers for the same conditions shows quite a big spread. Benoit et

al [95] reported it to be about 175 A, Schaefgen et al [96] found it to be in the range

4.3 Values of the main parameters in the model 77

up to 240A. Cotts et al [97] have published even higher values (up to 430A), and Chu

and Ying [98] reported a value for the persistence length of 290A. We will estimate the

persistence length to be about 290A and we will take the length of a nematogen in the

model equal to this value, i.e., l = 290A.

The number of rods and the root-mean-square fluctuation of the springs is estimated

on the basis of the contour length lc of the chain and the perpendicular gyration radius⟨

rg⟩

⊥. Let us define the contour length of the rod-spring-bead chain to be the sum of

lengths of the springs plus the sum of lengths of the rods. If we estimate the length of the

spring by it’s root-mean-square fluctuation due to thermal motions, then the expression

for the thermal motion is

lchc = 2 (N − 1)A+Nl (4.18)

The amplitude A is estimated from the perpendicular radius of gyration⟨

rg⟩

⊥. In an

article by Picken et al [33] the gyration radii for PpPTA in sulfuric acid were experiment-

ally determined from small angle neutron scattering (SANS) measurements. The values

were found to be⟨

rg⟩

‖ = 250A for parallel gyration radius and⟨

rg⟩

⊥ = 70A for perpen-

dicular gyration radius. If we assume that the transverse size of the chain is due to the

springs, then the radius of gyration of the rod-spring-bead chain can be computed as a

mean-square displacement of a random walk consisting of 2 (N − 1) steps with mean-

square displacement√

23A each. The factor 2

3 occurs because we account here only for

the transverse component of the spring. Then⟨

rg⟩ch

⊥ is estimated from the following

expression⟨

rg⟩ch

⊥ = A

√

4

3(N − 1) (4.19)

We substitute A from (4.19) into (4.18) and we derive the expression for N in terms of

the transverse gyration radius⟨

rg⟩

⊥, the contour length lc and the persistence length lp.

N =lclp

−√3⟨

rg⟩

⊥lp

√

√

√

√

lc − lplp

+3⟨

rg⟩2

⊥4l2p

(4.20)

Substitution of lc = 1600A, lp = 290A,⟨

rg⟩

= 70A gives the value 4.62 for the number

of rods. We round it off to 5. Thus, in simulations we shall put N = 5. Then we use

(4.19) to compute the mean-square fluctuations of the springs. It turns out to be 30.3A.

Then

A =A

lp≈ 0.10A (4.21)

The estimation of the strength of the nematic field is made on the basis of the model

and data suggested by Picken et al [11]. In this article the strength of the Maier-Saupe

potential is denoted ǫ and for the concentrated regimes far above from the concentration

of nematic to isotropic transition the approximate formula for PpPTA in sulfuric acid is

78 Numerical simulations of semi-flexible LCPs

suggested

〈P2〉 ≈ 1− 0.22

(

T

Tni

)3

(4.22)

where Tni is the temperature of the nematic-isotropic transition. Relating this formula

to it’s analog in Maier-Saupe model

〈P2〉 ≈ 1− T

ǫ (T )(4.23)

helps us to recover the expression for the strength of the nematic field

ǫ (T )

T=

1

0.22

(

TniT

)3

(4.24)

In the same article [11] the extrapolated value of Tni for the PpPTA solution in sulfuric

acid at 19.8 wt% is suggested to be approximately 545K . From this data it follows that

for T = 353K the value ǫT

≈ 16.7. The strength of the nematic field Un defined in this

thesis is by the factor 32 greater, than ǫ. Therefore,

Un =3

2

ǫ

T≈ 25 (4.25)

We also adopt the value suggested in [11] for the rotational diffusivity Dr = 3.3 · 104s−1

at T = 80 oC. In [11] the value for Dr was estimated from the following formula

Dr = βDr0

(

νL3)−2

(1− 〈P2〉)−2 (4.26)

where L is the rod length (290 A), P2 ≈ 0.95 at rest for ǫT= 16.7, ν is the number of rods

per unit volume at a given conditions (2.42 ·1025m−3), β = 103 is a correction factor and

Dr0is the rotational diffusivity coefficient for a dilute solution

Dr0=

3T(

ln(

Lb

)

− γ)

πηsL3 (4.27)

where Lb

is the aspect ratio of the rod, b = 6.6 A is the width of the rod, T is the tem-

perature in Joules, ηs is the solvent viscosity (5.3 · 10−3 Pa · s for sulfuric acid at 99%

concentration and at 80 oC), γ ≈ 0.8 is a correction term. Use of these values gives

Dr = 3.3 · 104s−1. This result in [11] is followed by a very interesting discussion. Dr

has a very strong dependence on L, because of the factor(

νL3)−2

in (4.26) and strong

dependence of Dr0on L in (4.27). This discussion is concluded with expressing doubts

about the reported value. We will keep it in mind when comparing the results of simu-

lations with the experimental data, but so far we are going to use the suggested value.

4.3 Values of the main parameters in the model 79

If we estimate the translational friction coefficients on the basis of Doi model for dilute

solutions

ζ‖ =2πηsL

ln(

Lb

) (4.28)

ζ⊥ = 2ζ‖ (4.29)

then

ξtr = 2 (4.30)

ζrot = 0.68 (4.31)

The rotational friction is estimated from the Einstein relation ζrot =TDr

. The maximum

extensibility of the springs is taken to be lz = 0.5.

We have estimated all the main dimensionless parameters of the rod-spring-bead model

for the LCP solution. The corresponding values of α-coefficients can be now directly

computed from the formulas (2.76).

αtrspr = 4.09 αrotspr = 3.00 αrotnem = 1

αtrdiff = 0.165 αrotdiff = 0.20 (4.32)

These values will be used in the simulations of the rheology of the LCP solution presen-

ted in the next chapter.

To get an idea of the ratio of time scales present in the system we also compute the

various characteristic times from table 2.2

τ trspr = 1.48 · 10−6 s τrotspr = 2.02 · 10−6 s τrotnem = 6.06 · 10−7 s

τ trdiff = 2.22 · 10−5 s τrotdiff = 1.52 · 10−5 s (4.33)

The fastest time scale is the time scale of the orientational ordering of the nematogens

due to the nematic field. If we recall that the characteristic time of the lowest Rouse

mode scales with the number of rods as N2, then the longest time scale is N2τ trspr . Thus,

the qualitative picture of the relaxation of the polymer chain coincides with the picture

suggested for the model in the highly-ordered limit. If the chain is perturbed from the

equilibrium configuration, then, firstly, the alignment of the nematogens takes place.

Secondly, the different internal modes decay and, finally, the relaxation of the backbone

as a whole occurs. The relaxation of the backbone corresponds to the lowest transla-

tional modes. We have chosen τrotnem to scale the time t. That is why t = 1 corresponds to

6.06 ·10−7s for the considered solution of PpPTA in sulfuric acid, and κ = 1 corresponds

to a deformation rate of 1.65 · 106 s−1.

One more quantity should be estimated before finishing this section. The stress tensor

80 Numerical simulations of semi-flexible LCPs

was scaled by the quantity

Pid =NchT

V(4.34)

in the non-dimensionalization procedure.

When the solution of PpPTA in sulfuric acid is surrounded with the atmosphere having

standard atmospheric pressure Patm = 101.3 kPa, then the partial pressure created by

the PpPTA chains can be estimated by determining the fraction of number of PpPTA

chains in the solution.

Pid =NPpPTA

NPpPTA +NH2SO4

Patm (4.35)

NPpPTANH2SO4

=µH2SO4

1− CPpPTA

CPpPTAµPpPTA

(4.36)

where µH2SO4= 98 g

moleis the molar mass of sulfuric acid,CPpPTA = 0.197 is the weight

fraction of the PpPTA chains. Substitution µPpPTA = 30 kgmole

gives

NPpPTANH2SO4

≈ 1

1250(4.37)

This result shows that there are 1250 molecules of solvent per one polymer chain in this

system. Employing this ratio in (4.35) we obtain the estimation of the partial pressure

of polymer chains.

Pid ≈ 81 Pa (4.38)

Thus, the unit of non-dimensionalized stress corresponds to approximately 81 Pa for

this system.

We conclude this section with a brief summary. On the basis of the experimental data

available in the literature we have estimated the main dimensionless groups, the char-

acteristic time-scales present in the system and the values of the coefficients appearing

in the evolution equations (4.1), (4.2), and (4.3). Finally, we estimated the value of the

pressure used for non-dimensionalization of the stress. All this preparatory work is ne-

cessary for performing the simulations and for relating the results of these simulations

to available experimental data.

4.4 Algorithm

In an earlier section we described the Euler-Maruyama method. However, this numer-

ical scheme has to be modified to be applied to (4.1), (4.2), and (4.3). We repeat these

4.4 Algorithm 81

equations here. For i ∈ {1, . . . , N} and j ∈ {1, . . . , N − 1}

(S) dri = αtrspr ζ−1

r (ui) · Felridt+ κ · ridt+ αtrdiff ζ− 1

2

r (ui) · dWri (4.39)

(S) dui = (I− uiui) ·(

αrotsprFeluidt+ αrotnemS · uidt+ κ · uidt+ αrotdiffdW

ui

)

(4.40)

(S) dbj = αtrspr ζ−1bead

(

hj)

Felbjdt+ κ · bjdt+ αtrdiff ζ

− 12

bead

(

hj)

dWbj (4.41)

Before we formulate the equivalent Euler-Maruyama scheme for these equations we

have to derive the corresponding SDEs in Ito interpretation. This step was explained in

Appendix A.3 and explicitly stated in equation (A.35). As it was said in the beginning

of this chapter equations (4.39) and (4.41) are independent of the interpretation, i.e.,

the spurious drift term is zero for these equations. But the spurious drift term uspi for

equation (4.40) turns out to be non-trivial

uisp =

1

2αrotdiff (I− uiui) ·

∂

∂uiαrotdiff +

1

2

(

αrotdiff

)2

(1− dimui)ui (4.42)

The first term in this expression is obviously zero, because αrotdiff is a constant coefficient.

If the rotational friction coefficient would be dependent on ui, then the first term would

contribute. The second term originates from the projection operator (I− uiui), which

projects 3D-vectors onto the plane tangent to the unit sphere at the point ui. This term

accounts for the change of the tangent plane when moving over sphere.

If we do not take the radial spurious drift term (4.42) into account, then the discrete

analogs of ui will become non-unit vectors in a relatively short time-interval. The radial

spurious drift term (4.42) corrects for this and increases this time-interval. However,

due to round-off errors that are always present in numerical simulations and as higher-

order corrections are disregarded in the Euler-Maryama scheme, the discrete analogs

of ui will become anyway non-unit vectors at some point. This can be cured by renor-

malizing of the orientational degrees of freedom after some number of iterations. We

will renormalize after each iteration. In this way we can forget about the radial spuri-

ous drift, because the renormalization does the same job. The good part is that this

renormalization procedure does not accumulate errors in the magnitude of orientation

vectors.

For convenience let us denote the discrete analogs of the dynamical variables in (4.39),

(4.40), and (4.41) by the superscript k to the corresponding continuous variable x. The

value of the index k indicates the iteration step. Then the discrete analogs of equations

(4.39), (4.40), and (4.41) become

rk+1i = r

ki + αtrspr ζ

−1

r

(

uki

)

· Felrki∆t+ κ · rki∆t+ αtrdiff ζ

− 12

r

(

uki

)

·∆Wri (4.43)

82 Numerical simulations of semi-flexible LCPs

uk+1i = u

ki +(

I− uki u

ki

)

·((

αrotsprFel

uki+ αrotnemS

k · uki + κ · uki)

∆t+ αrotdiff∆Wui

)

(4.44)

bk+1j = b

kj + αtrspr ζ

−1bead

(

hkj

)

Fel

bkj∆t+ κ · bkj∆t+ αtrdiff ζ

− 12

bead

(

hkj

)

∆Wbj (4.45)

where ∆t is the time-step of the iteration, and ∆Wri , ∆W

ui , and ∆W

bj are the incre-

ments of the 3D-Wiener process corresponding to the time-step ∆t, i.e. the components

of ∆Wri , ∆W

ui , and ∆W

bj are independent Gaussian-distributed random variables

with variance ∆t.

At this point is important to make a remark about the algorithm that is used for the

pseudo-random number generation. The code is implemented in Mathematica 8.0.

Starting from version 6.0 the standard pseudo-random number generator used is the

so-called extended cellular automaton generator [99, 100]. This generator is considered

good by the ratio of quality and time consumption. The standard functions in Math-

ematica transform a uniform distribution into a Gaussian distribution without spoiling

the statistical properties of the pseudo-random numbers that are generated. Though

this algorithm of pseudo-random number generation is relatively fast, the necessity

to generate (9N − 3)Nch pseudo-random numbers at each time-step slows down the

Euler-Maryama scheme noticeably compared to the Euler scheme for the ”equivalent”

ordinary differential equations.

In the standard Euler-Maruyama scheme simulation of each realization can be per-

formed independently, because the functions a (x (t) , t) and B (x (t) , t) in (4.7) depend

only on the current value of x and on time t. In our case equation (4.44) contains the

tensor S, and the discrete version (4.44) contains the value of this tensor at a current it-

eration step (moment of time). But this tensor is itself an ensemble average, i.e., in order

to simulate the evolution of a single realization of the chain’s behavior the configuration

of the whole ensemble is required. This connection adequately reflects and arises from

the mean-field assumption for the Maier-Saupe potential. This assumption couples the

evolution of a single realization of the chain’s behavior to the state of the ensemble and

vice versa. In the numerical scheme we treat this complication in a following way. We

generate an ensemble of Nch chains. Using the state of the ensemble at k-th step, Sk is

computed by the next expression

Sk =

1

Nch

Nch∑

i=1

uki u

ki −

1

3I (4.46)

Actually, the term − 13I is omitted in the implemented code, because it drops out after

applying the projector(

I− uki u

ki

)

. The ensemble simulated at once has to be large

enough in order to make the fluctuations of Sk as small as possible. In our simulations

we keep Nch = 1000. Which gives 1√Nch

≈ 0.03. In fact, the fluctuations are also

dependent on the values of the parameters, especially on the strength of the nematic

4.4 Algorithm 83

field Un. The increase of the strength of the nematic field leads to smaller fluctuations

of Sk. For Un = 25, the fluctuations of Sk hardly exceed 1%.

The variety of different time scales present in the system causes another difficulty in the

numerical simulations. If we have two SDEs describing processes occurring on different

time-scales, then treating both of them simultaneously becomes a very costly affair. In

order to resolve processes occurring on a small time-scale we have to keep the time-step

small, but processes occurring on a large time-scale require the simulation of long time-

intervals. For some systems with clear separation of time-scales different optimization

schemes can be implemented. In our case the problems are caused by the changing

mobility of the beads. The original idea was simple. The beads in a ”non-hairpin state”

have to have a very high mobility in order to give no contribution to the excessive stress.

And vice versa, the mobility of beads in ”hairpin states” needs to be reduced in order

to mimic possible entanglements between chains. On the other hand a high mobility

corresponds to a small time-scale, and vice versa. Thus, both time-scales (the longest

and the shortest) are incorporated in equation (4.45). The trick to separate these time

scales works as follows. If the bead is in a ”non-hairpin state”, then the dominating

term in equation (4.45) is αtrspr ζ−1beadF

el

bkj∆t, because ζbead → 0. In this limit equation

(4.45) turns into a force balance

Fel

bkj= 0 (4.47)

plus some fluctuations around equilibrium point. In fact, we do not care about those

fluctuations, as beads with a high mobility almost give no contribution to the stress

tensor.

In the opposite limit, when the bead mimics the entanglement, its mobility is dramatic-

ally reduced. This formally means ζbead → ∞ in (4.45). In that case the motion of the

bead reduces to affine motion

bk+1j = b

kj + κ · bkj∆t (4.48)

Formally, the large friction coefficient ζbead corresponds to a respectively large charac-

teristic time. But the lifetime of such beads can not exceed the lifetime of a ”hairpin

state”. Thus, the tolerance for defining a ”hairpin state” should be chosen in such a way

that the lifetime of a ”hairpin state” coincides with the lifetime of an entanglement. This

tolerance, thus, represents the lifetime of entanglements. The lifetime of the hairpin de-

fect itself, as a kink in the chain’s backbone, is proportional to eUn , which for Un = 25

and τrotnem = 6.06 · 10−7 s may be of the order of 104 s. This is much larger than the other

time-scales.

Up to now we only have described the iteration scheme for the time evolution. But we

also have to specify the procedure for generating the initial state of the system. This

problem turns out to be non-trivial. The description of this procedure will be given in

the next section.

84 Numerical simulations of semi-flexible LCPs

4.5 Equilibrium properties of the system

In order to generate an ensemble of possible initial configurations of the chains the con-

figurational distribution function is needed. Using this function the ensemble of chains

with appropriate dispersion and higher moments can be generated. From statistical

mechanics it is well known, that for the system in equilibrium the Gibbs distribution

should hold. For a configurational distribution function it reduces to a Boltzmann dis-

tribution

f(x) =exp

(

−U(x)T

)

∫

exp(

−U(x)T

)

dx(4.49)

where x stands for all the variables determining the configuration of the system, U (x) is

the potential energy of the system. Unfortunately, this approach is not applicable in our

case. The nematic potential is expressed in terms of the orientation tensor Sk. However,

to compute Sk we need an already generated ensemble. Of course we could estimate

the Sk on the basis of the rigid rod model, but a priori we do not have arguments why

the equilibrium state of the ensemble of rigid rods should coincide with the equilibrium

state of the ensemble of rod-spring-bead chains. That is why we take another route.

We generate the ensemble of chains with uniformly distributed orientations of the rods.

Then we let the system evolve. After large enough number of steps the system evolves

towards the equilibrium state. The ensemble averages reach steady values and do not

change any more. When the equilibrium state is reached the ensemble of chain config-

urations is saved. The results of such simulations are depicted in fig. 4.1. The typical

interval of time computed ranges between 100 and 600 units of time. As the strength

of the nematic field approaches the critical value of about 6.8 the time needed for equi-

libration increases. We have performed these simulations with a time-step 0.005 for the

ensemble of 1000 chains. In our simulations we have compared the outcomes for N

varying from 1 to 5. They all collapse to the same line. In fig. 4.1 we compare the res-

ults of simulations for N = 3, N = 5 with the rigid rod model. The self-consistency

equations for the rigid rod model [11] is

〈P2〉 =∫

exp(

ǫT〈P2〉P2 (n · u)

)

P2 (n · u) du∫

exp(

ǫT〈P2〉P2 (n · u)

)

du(4.50)

where ǫT

= 23 Un and P2 (x) = 3

2x2 − 1

2 . The maximum eigenvalue of 〈uu〉 can be

expressed in terms of 〈P2〉λmax =

1

3+

2

3〈P2〉 (4.51)

Though, the self-consistency equation for the system of rods coupled with FENE-springs

does not coincide with the self-consistency equation (4.50), the deviations in the degree

of ordering are negligible for the two models for the chosen set of parameters. At first

4.5 Equilibrium properties of the system 85

λmax

1Un

Simulations for N = 5Simulations for N = 3Rigid Rod model N = 11.0

0.8

0.6

0.4

0.2

0.00.00 0.05 0.10 0.15

Figure 4.1: The maximum eigenvalue λmax of 〈uu〉 as function of the inverse relative strengthof the nematic field for different number of rods per chain. Computed from both simulations andthe analytical result (4.50).

sight this result might seem counterintuitive, since the chain consisting of two rigid rods

connected by a flexible spacer seems more flexible than the single rigid rod. However,

if we keep the parameter Un constant, then the chain consisting of two rods has approx-

imately twice bigger total nematic energy. If we keep the total nematic energy per chain

fixed, then the increase ofN leads to the decrease of Un. Then the degree of orientational

ordering decreases for more flexible chains and starting from someN the nematic phase

does not occur. In other words, in this model if

Unematicper chain

T

lplc< 6.8 (4.52)

then the polymer is too flexible and does not show the nematic phase.

For N = 1 this model reduces to a rigid rod model. Thus, we may also consider these

equilibration-simulations as some tests of the algorithm before we switch to the be-

havior of the system in flow. For example, from these equilibration simulations we

concluded that the beads do not affect the equilibrium properties.

We perform also tests in which the tolerance for ”hairpin state” creation was varied. In

(2.65) we have defined the hairpin variable h. For the tolerance it is more convenient to

take εtol = 1 − h. Then εtol = 0 corresponds to the absence of entanglements between

hairpins. As expected the equilibrium properties are not affected by the change of εtol.

However, the increase of εtol leads to an increase of the relaxation time and for example

for εtol > 0.2 the relaxation time becomes too long to be simulated.

Since the relaxation starts from a completely disordered isotropic state, the final ori-

86 Numerical simulations of semi-flexible LCPs

entation of the director in the equilibrated nematic phase is random. This is not very

convenient for further use, that is why, after the equilibration has taken place, we rotate

the whole ensemble to make the director parallel to Ox-axis. Ensembles generated in

such a way will be used in the next chapter as initial conditions.

One more remark regarding the hairpin defects should be made. Since we did not intro-

duce the bending energy for subsequent pairs of rods, both normal and folded states of

the chain’s backbone are equally favorable. Moreover, the nematic potential is mirror-

symmetric with respect to a plane normal to the director. Therefore, the ensemble aver-

age 1N

N−1∑

i=1

〈ui · ui+1〉 should be zero, i.e., 〈h〉 = 0.5. Sometimes, preparation of an en-

semble by the above equilibration procedure ends up with an h that deviates essentially

from 0.5. Because Un is large it takes an enormous time to come to true equilibrium.

The least biased way to treat this problem is to make several runs of the equilibration

procedure. Among the list of generated ensembles we choose the one with 〈h〉 closest

to 0.5. Usually, the deviation does not exceed 0.03.

If 〈h〉 = 0.5, then the chain consisting of N rods and N − 1 flexible spacers should have

on average N−12 consecutive pairs for which ui ·ui+1 > 0 and the same number of folded

pairs for which ui ·ui+1 < 0. This suggests that for N = 5 the average number of folded

pairs with ui · ui+1 < 0 per chain is about 2 in this model.

In 2009 Westerhof [35] has modeled a part of the PpPTA segment surrounded by N-

methyl-pyrolidone molecules by means of quantum-chemical and molecular dynamics

simulations. N-methyl-pyrolidone was chosen in order to reduce the effect of proton-

ation of amide groups in PpPTA. From these simulations it followed that the energy

difference between the cis-trans (folded and unfolded) configurations of the consecut-

ive amide groups is varying between 18 and 15 kJmole

depending on the dielectric per-

mittivity of the solvent and on the degree of protonation of the amide groups. Using

the energy penalty 15 kJmole

and accounting for the number of amide units per chain

(32000 gmole

) the number of 0.6 hairpins per chain comes out. However, from Picken’s

article [33] based on small angle neutron scattering experiments on PpPTA in sulfuric

acid the average number of hairpins per chain is estimated as 1.5, which is much closer

to our estimate.

Chapter 5

Rheology of entangled LCP

solutions containing hairpins

5.1 Introduction

LCP solutions exhibit unusual rheological behavior in shear flow, such as negative first

normal stress differences. Moreover, coupling between the flow and the orientational

ordering in the system gives rise to peculiar dynamics of the director, such as flow-

aligning, wagging or tumbling. The fascinating dynamics of LCP solutions combined

with their relevance for the industrial applications attracted a lot of interest to these

systems, resulting in a sequence of models treating the LCP solutions with increasing

accuracy. For example, the molecular models based on Doi’s rigid-rod model [24,40–44],

though capturing many aspects of the dynamical behavior of the system, neglected the

flexibility of the polymer chain altogether. However, many industrially relevant poly-

meric systems are semi-flexible. Two opposite approaches were used in the earliest

molecular models accounting for semi-flexibility of the polymer backbone. Either by

weakening the rigidity of the rod by considering the so-called slightly bending rod

model [45–47], or by introducing an additional nematic ordering into the dumbbell

model [48]. All these models treat systems of unentangled polymers and are also not

capable of treating hairpins in a natural way. The models accounting for entanglements

in semi-flexible LCP systems were considered by Semenov [49] and Subbotin [50], but

the role of hairpins in these models is not clearly evident. Aspects of statics and dynam-

ics of the hairpin defects were studied by M. Warner et al in [37]. He also investigated

their contribution to the rubber elasticity of nematic elastomers [38].

Besides theoretical and experimental methods of investigation, numerical simulations

88 Rheology of entangled LCP solutions containing hairpins

have become an increasingly valuable tool for understanding the dynamics of complex

systems. A good example of such approach can be found in [30], where a very detailed

and extended research of the rod-like colloidal systems is presented. Due to the com-

plexity of the model formulated in this thesis we also apply numerical methods to study

the properties of the dynamical model.

In this thesis we have presented a model for a semi-flexible main-chain nematic LCP

solution containing hairpins and possible entanglements between the chains containing

hairpin defects. In Chapter 2 we showed which assumptions for the closure relations

should be adopted to derive this model from the general phase-space kinetic theory.

Then we reformulated this model in terms of a system of stochastic differential equa-

tions. In specific limits this model reduces to other well-known models. For example,

if N = 1, then we obtain Doi’s rigid rod model. For N = 2 and ko → ∞ it reduces to

the Broken-Rod model [51]. A less trivial limit of high nematic ordering and time-scale

separation was considered in Chapter 3. In this limit the model can be reduced to the

Rouse-like rod-spring model containing hairpins. In case when hairpins are disregarded

it further reduces to the model suggested by Long and Morse [29].

Finally, in Chapter 4 the numerical scheme used for solving (2.67)-(2.73) was developed

and the choice of the values for parameters was explained.

In this chapter we present the results of the simulations when the solution is subject

to various homogeneous flows. We are particularly interested in the changes of the

rheological properties due to the presence of hairpins.

5.2 Linear rheology of an LCP solution

We start to examine the rheological properties of the model formulated by the evol-

ution equations (2.67)-(2.73) by considering the uniaxial elongational flow. As it was

mentioned in Chapter 3, uniaxial extensional flow is relatively simple to analyze. As the

direction of extension coincides with the orientation of the director, the dynamics of the

director is trivial. The director just preserves its orientation, as the chains continue to

elongate and align in the direction of extension. Moreover, in Chapter 3 we managed to

derive analytical expressions for the response function, response moduli and the zero

elongation rate viscosity as a function of the fraction of hairpins present for a highly-

aligned unentangled nematic LCP solution. Now we can use these analytical results

to test the numerical scheme in the limiting case, that the nematic field is assumed to

be strong enough to have a high orientational order, and to keep the time-scales of the

orientational and translational degrees of freedom well separated. This suggests, that

a proper numerical scheme should reproduce the theoretical predictions if we satisfy

these conditions and use the corresponding values for the parameters. In the previ-

5.2 Linear rheology of an LCP solution 89

ous chapter the value of the relative strength of the nematic potential was estimated

as Un ≈ 25. This high value guarantees the high degree of orientational alignment.

However, in the rod-spring-bead model (2.67)-(2.73) entanglements can be accounted

for. Entanglements are modeled by a change of the mobility of a bead when the corres-

ponding chain segment is in a ”hairpin state”.

The nondimensionalization of the time in the rod-spring-bead model (2.67)-(2.73) and in

the analytical model for the highly-aligned limit is done by different characteristic times.

In the rod-spring-bead model we have nondimensionalized the time by τrotnem = ζrot2Un

(Table 2.2). In the highly-aligned limit this time-scale formally turns to zero τrotnem → 0,

and, therefore, can not be chosen for nondimensionalization. That is why we took an-

other time-scale, namely, τ‖ =ζ‖k0

defined by relation (A.71). Thus, for a correct compar-

ison of the results of these two models we have to account for this different scaling of

time. The scaling factor isτ‖

τrotnem

. Taking into account k0 = ko2 we derive the expression

for this factor in terms of the main dimensionless groups of the model

τ‖

τrotnem

=ζrot2Un

2ζ‖ko

=4

3

UnA2

ζrot(5.1)

In particular, for ζrot = 0.68, A = 0.1, Un = 25 this factor isτ‖

τrotnem

≈ 0.5. Moreover,

this factor also gives an indication of the ratio of the translational and rotational time-

scales. The lowest Rouse mode has the time-scale τ‖N2. In the highly-aligned model

the separation of time-scales was justified by the assumption τrotnem

τ‖N2 ≪ 1, i.e.,

3

4

ζrot

UnA2N2 ≪ 1 (5.2)

For N = 5 andτ‖

τrotnem

= 0.5 the ratio on the left-hand side of equation (5.2) is equal to

0.08.

To summarize this discussion about the relation between the numerical rod-spring-bead

model and the analytical model in the highly-aligned limit we can formulate conditions

that guarantee their equivalent behavior. The parameters should satisfy the following

conditions:

εtol = 0, Un ≫ 1,3

4

ζrot

UnA2N2 ≪ 1. (5.3)

Although the analytical model for highly-ordered limit formulated in Chapter 3 is de-

rived from the rod-spring-bead model formulated by (2.67)-(2.73), these models are not

equivalent. The equivalence is achieved only if the assumptions of high nematic order-

ing, separation of time-scales, and absence of entanglements are satisfied. This is true

only if conditions (5.3) hold.

90 Rheology of entangled LCP solutions containing hairpins

Next, we explain some subtleties connected with performing simulations. First, we

take the ”equilibrated” ensemble of chains prepared according to the ”equilibration”

procedure, described in Chapter 4. Secondly, we apply an instantaneous deformation to

the system. This is achieved by applying the matrix I+ Γ to all translational degrees of

freedom. This matrix Γ is traceless and given by

Γ = δe

1 0 0

0 − 12 0

0 0 − 12

(5.4)

The rotational degrees of freedom are transformed slightly different, according to

unew = uold + (I− uolduold) · Γ · uold (5.5)

as they should only rotate. After this transformation the orientation vectors unew are

renormalized again. Finally, we use this ensemble as initial condition in the simulation

of the relaxation process. From this relaxation process we obtain the stress tensor as a

function of time. This information is enough to compute the response function.

E⋄ (t)

≡ N⋄1

(

t)

δe(5.6)

However, the stress measured in the simulations, just after the deformation, exceeds the

stress predicted by the analytical model. Nevertheless, the measured stress relaxes to-

wards the predicted stress on the time-scale of 0.1 unit of time. This time-scale is about

400 times larger, than the time-step. Therefore, this overestimation of the stress can

not be caused by a numerical artifact. The explanation of this discrepancy between the

measured and the predicted stress lies in the underestimation of the response function

in the highly-ordered limit on the time-scale of the fast variables. The procedure of time-

scale separation and elimination of the fast variables in the analytical model correctly

accounts for the total contribution of the fast variables to the stress tensor on the time-

scale of the slow variables. However, on the time-scale associated with the fast variables

the contributions from the fast variables to the stress tensor are neglected. Decreasing

the time-scale of the fast variables in comparison with time-scale of the slow variables,

causes the decreasing of the relative error in the the stress tensor. However, the numer-

ical scheme accounts for these contributions and, therefore, the measured stress exceeds

the predicted stress on the time-scale of these fast variables. That is why, for a consist-

ent comparison of the numerical outcomes with the theoretical predictions we start to

record the response function not from the initial moment of time, but from the moment

t = 0.1, when the contribution from the fast variables has already essentially died out.

The parameter δe in the initial instantaneous deformation is fixed to a value 0.1. This

parameter should not be too big, because then we will obtain the non-linear response

of the system as result of nonlinearity of the FENE-springs. On the other hand this

5.2 Linear rheology of an LCP solution 91

2 4 6 8 10

0.01

0.1

1

10

100

E⋄

E

t

Figure 5.1: Comparison between the elongational response function E⋄ obtained in numericalsimulations and the elongational response function E derived in the highly-ordered limit.

parameter should not be too small, because then the ratio of the response to the noise is

decreasing. For a maximum relative extensibility of the springs of lz = 0.5, a time-step

dt = 0.0025 and an ensemble of 1000 chains, the choice δe = 0.1 gives reasonable results.

Both measures and predicted results for the response function are plotted in fig. 5.1. The

elongational response function obtained from both methods are almost identical, t < 1.

We also notice that for time region t > 1 a slight difference (up to 10%) between the

responses E⋄ and E exists. This slight difference remains also in simulations with a

smaller time-step or for larger ensembles. This slight deviation is caused by the fact that

the ratio of the time-scales of fast and slow variables is finite in the simulations, while in

the theory this ratio is infinite. We also compare the elongational zero-deformation-rate

viscosity for the two models. Clearly, if the response functions almost coincide, then

the values for the viscosities also coincide. Namely, the theoretical value for w0 = 0.5

is 180 and the result from the simulations is 210 ± 30. The spread in the numerical

result is mainly caused by contributions from the fast variables in the beginning of the

relaxation process. This result and the equilibrium results (Chapter 4) can be considered

as validation tests of the numerical scheme.

Now, let us consider the results of another set of simulations. From the discussion

described above we concluded that the parameter εtol is related to the lifetime of the

entanglements between hairpins and, consequently, to the contributions of these entan-

glements to the stress tensor. In order to understand the role of these entanglements we

performed a set of similar simulations, but for different values of εtol. From each of these

simulations we obtained the response function, response moduli and zero-deformation-

92 Rheology of entangled LCP solutions containing hairpins

0 5 10 15 20

0.01

0.1

1

10

100

E

tεtol =

0.00

εtol =

0.01

εtol = 0.03

εtol = 0.05εtol = 0.07

εtol = 1.00

Figure 5.2: Elongational response function plotted versus time for different values of εtol. The

values of the parameters are: A = 0.1, ξtr = 2, Un = 25, lz = 0.5 and N = 5.

rate viscosity. The results are plotted in fig. 5.2-5.6. From fig. 5.2 it follows that an in-

crease of εtol, i.e., the lifetime of the entanglements between hairpins, leads to a slower

decay of the response. For εtol < 0.01 the response of the system is not very sensitive

to our choice of εtol. However, starting from εtol = 0.01 this factor start to play an im-

portant role, which is in agreement with the criterion (2.66) for the relevant choice of

tolerance εtol.

A slower decay of the response function implies an increase in the elasticity of the sys-

tem. This is clearly seen in fig. 5.3. Decreasing the frequency corresponds to decreas-

ing of the elastic response. However, as the lifetime of the entanglements increases

a plateau-region appears and extends more and more in the region of low frequencies.

Clearly, these results show that the viscoelastic properties of the LCP solution are highly

affected by the entanglements between hairpins, especially when the lifetime of these

entanglements is longer than the characteristic timescale of the internal modes of chains.

Fig. 5.6 shows how the zero-elongation-rate viscosity grows with εtol. From this plot a

very interesting conclusion can be drawn. On one hand, an increase of the number of

hairpins in the system leads to an increase in the number of entanglements. More en-

tanglements lead to an increase of the viscosity. However, as was shown in Chapter 3 in

an unentangled system hairpins play a different role, because they reduce the response

of the individual chains. Thus, we have two competing factors: the increase of response

due to entanglement and the decrease of the response from individual chains. In gen-

eral, the existence of competing phenomena in the system leads to non-monotonicity

5.2 Linear rheology of an LCP solution 93

0.001 0.01 0.1 1

0.01

1

100

E′

ω

ε tol=0.1

0

ε tol=0.0

7

ε tol=0.0

5

ε tol=0.0

3

ε tol=0.0

1

Figure 5.3: Storage modulus plotted versus frequency for different values of εtol. The values of

the parameters are: A = 0.1, ξtr = 2, Un = 25, lz = 0.5 and N = 5.

0.001 0.005 0.010 0.050 0.100 0.500 1.000

0.5

1.0

5.0

10.0

50.0

100.0

E′′

ω

ε tol=0.1

0

ε tol=0.0

7

ε tol=0.0

5

ε tol=0.0

3

ε tol=0.0

1

Figure 5.4: Loss modulus plotted versus frequency for different values of εtol. The values of the

parameters are: A = 0.1, ξtr = 2, Un = 25, lz = 0.5 and N = 5.

94 Rheology of entangled LCP solutions containing hairpins

0.001 0.005 0.010 0.050 0.100 0.500 1.000

0.1

1

10

100

E′

E′′

ω

Figure 5.5: Storage E′ and E′′ moduli plotted versus frequency. The values of the parameters

are: εtol = 0.07, A = 0.1, ξtr = 2, Un = 25, lz = 0.5 and N = 5

0.00 0.02 0.04 0.06 0.08 0.10

100

200

500

1000

2000

5000

104

2 104.

ηE0

εtol

SimulationsInterpolation

theory for w0 = 0

theory for w0 = 0.5

Figure 5.6: Zero-elongation-rate viscosity ηE0plotted versus εtol. Dashed lines indicate the

values of the zero-elongation-rate viscosity of unentangled highly-ordered LCP solution withw0 = 0 and w0 = 0.5.

5.3 Evolution of the director in steady shear flow 95

of some properties of the system. In our case it means the following. Let us consider

a concentrated solution in the nematic state with a persistence length of the order of

the contour length. The polymer chains in such a solution almost do not contain hair-

pins, because of the high penalty for bending. Because the system is highly-ordered

and contains only few hairpins, the number of entanglements should be small. Then

the assumptions made in the model formulated in Chapter 3 hold for this LCP solution.

According to the analysis of Chapter 3, the viscosity of the system without hairpins is

higher, than of the system containing hairpins. Therefore, if we consider a sequence of

nematic highly-ordered LCP solutions with a gradually decreasing persistence length,

then the viscosity will change in a non-monotonic way. While the fraction of hairpins

is small, the role of entanglements is negligible, and, therefore, the viscosity decays, be-

cause of the decrease of the average response of the individual chains. However, when

the number of hairpins increases sufficiently to make the entanglements between the

chains a sufficiently common event the viscosity starts to grow rapidly and the contribu-

tion from the entanglements starts to dominate. For example, in fig. 5.6 the dashed lines

indicate the zero-elongation-rate viscosity for unentangled highly-ordered LCP solution

for w0 = 0 and for w0 = 0.5. And for εtol = 0.03 the value for the zero-elongation-rate

viscosity of the entangled LCP solution with hairpins coincides with the result for the

unentangled highly-ordered nematic LCP solution without hairpins. Thus, εtol = 0.03

correspond to a dynamic equilibrium between the individual response of the chains and

the response due to entanglements for this particular choice of parameters.

From this discussion we conclude that the role of hairpins can be different depending

on the system. The presence of hairpins causes a weaker response in systems containing

hairpins compared to systems without hairpins. But for systems containing a significant

number of hairpins, such that entanglements between chains are quite common, the

presence of hairpins increases the response.

5.3 Evolution of the director in steady shear flow

The present section is devoted to the behavior of an LCP solution described by (2.67)-

(2.73) in shear flow. It was shown in many papers [30, 42–44, 46, 101] that the response

of the LCP solution to shear flow shows several peculiar features. For example: the

first normal stress difference has a region with negative values, the solution experiences

shear-thinning behavior, the director undergoes periodic motions while the system is

experiencing steady shear flow at low shear rates, and flow-aligning behavior of the

director at high shear rates. In this section we examine whether the rod-spring-bead

model with entanglements formulated by the set of equations (2.67)-(2.73) is capable to

showing the same type of behavior as reported in the previous studies for the rigid rod

model. Like in many other studies we will use the orientation tensor 〈uu〉 to character-

ize the orientational order in the system.

96 Rheology of entangled LCP solutions containing hairpins

������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������

0.10 0.500.20 0.300.15

100

200

150τkayak

˜γ

Simulations

Interpolation 25˜γ

Figure 5.7: Plot of the kayaking period as a function of shear rate.

In this series of simulations the values of the parameters are fixed to A = 0.1, ξtr = 2,

Un = 25, lz = 0.5,N = 5 and εtol = 0.01. The simulations are organized in the following

way. Using the equilibration procedure described in the previous chapter we prepare

the ensemble of chains that corresponds to the equilibrium state of the system. Then the

ensemble is rotated in order to make the director coincide with the Ox-axis. Then we

impose a steady shear flow. The direction of velocity will coincide with the Ox-axis, the

direction of the velocity gradient with the Oy-axis, and the direction of vorticity with

the Oz-axis. The only non-zero component of the velocity gradient tensor is denoted by˜γ ≡ κ12. Then we run the simulations with time-step dt = 0.0025. The interval of time

needed for the system to reach the steady state is different for different values of the

shear rate. For small shear rates ˜γ < 0.55 the system shows tumbling behavior initially.

After many (more than 5) periods of rotation the rotation goes out of the xOy-plane

giving rise to so-called kayaking behavior of the director. This is in agreement with the

predictions of Faraoni [101] and with the results of the simulations by Tao in [30].

In fig. 5.7 the period of kayaking as function of shear rate is plotted on a log-log scale.

We see that the results of simulations fall on a straight line in this log-log plot and can

be well interpolated by 25˜γ

for the chosen values of the parameters. By a period we un-

derstand here the time that the director takes to rotate over π radians, i.e., a half turn in

the xOy plane. Though in the presented model the rods are infinitely thin, the period

of rotation turns out to be a finite value. This is due to the fact that chains have a finite

perpendicular gyration radius due to springs and the spread in the orientation of the

rods. The aspect ratio of the chain configurations varies in this system from about 0.05

5.3 Evolution of the director in steady shear flow 97

0 5 10 15 20

0.0

0.5

1.0

1.5

2.0

2.5

3.0

θL

˜γ

Figure 5.8: Leslie angle (in degrees) plotted versus shear rate.

for a completely unfolded chain without hairpin defects to about 0.2 for completely fol-

ded chains containing 4 hairpins per chain. In the models treating unentangled chains

consisting of rods of a given aspect ratio the period of overall rotation coincides with

the period that an individual rod would have when subjected to the shear flow. How-

ever, in our case different chains in the ensemble will have different aspect ratios that

change during flow not only by stretching, but also by creation and destruction of hair-

pin defects. Thus, the observed period of director rotation is some kind of compromise

between all chains about the favorable period of rotation. Nevertheless, fig. 5.7 shows,

that the inverse proportionality between the period of kayaking and the shear rate still

holds.

For shear rates in the region 0.55 < ˜γ < 0.65 the transition to a wagging type of be-

havior occurs. In this region the director oscillates within a range of angles, but does

not perform a rotation over π. As will be shown in the next section the first normal

stress is found to be negative in this region, which is in agreement with a previous

work [30, 42, 44].

As the shear rate increases, the amplitude of the oscillations of the director in the xOy

plane decreases, and starting from ˜γ ≈ 0.7 flow-aligning type of behavior sets in. How-

ever, we could not exactly determine the location of the transition from wagging to

flow-aligning. Due to numerical errors and the finite size of the ensemble the director

in the flow-aligning case was also fluctuating and we could not distinguish whether

these fluctuations were real or were numerical artifacts. By computing the average ori-

98 Rheology of entangled LCP solutions containing hairpins

entation of the director during the long time interval the so-called Leslie angle θL was

found for different shear rates. The Leslie angle is the angle between the Ox-axis and

the projection of the director onto xOy plane in the flow-aligning case. The results are

plotted in fig. 5.8. In this plot the Leslie angle is plotted in degrees. From this figure we

conclude that θL is demonstrating a non-monotonic type of behavior. First, θL increases

from zero to about 2.7◦ in the region of shear rates 0.8 < ˜γ < 5, then θL decreases

to zero asymptotically as the shear rate goes to infinity. The existence of a maximum

for the Leslie angle was predicted by Marrucci and Maffettone [44] and also found in

simulations reported in [30].

In fig. 5.9, 5.10 and 5.11 the evolution of the eigenvalues of 〈uu〉 is plotted for dif-

ferent shear rates. These plots give information about the processes happening in the

ensemble subject to shear flow. For small shear rates the maximum eigenvalue λ1 of the

orientation tensor does not differ much from its equilibrium value. Clearly, this is due

to the nematic field, which is strong enough to synchronize the orientations of different

chains on the time-scale of the deformation ˜γ−1. Chains spend the major part of the

time in approaching the plane of shear. In fig. 5.9 these time intervals are represented

by the regions of the blue line having small slope. Once the plane of shear is crossed by

the chain, the shear flow forces the chain to rotate quickly out of this plane. Different

chains cross the plane of shear within some spread of time, due to the spread in orient-

ation caused by thermal motion. Due to the fact that rotation of the chain after crossing

the plane of shear is highly accelerated the above described spread in moments that the

plane is crossed causes an increase in the spread in orientation of the chains. This is

represented by the intervals where λ1 rapidly decreases in fig. 5.9. Then, as the chains

again approach the plane of shear the spread in orientation decreases again, which is

represented by the intervals where λ1 rapidly increases. For small deformation rates

the amplitude of the oscillations of λ1 is also small, because the nematic field penalizes

an increase in spread in orientations. However, when the shear rate increases, the amp-

litude of the oscillations of λ1 increases and λ1 drops to a lower value. But the main

feature of this dynamic regime is that the majority of the chains are still pointing along

the director. Starting from some shear rate, the nematic field is not capable any more

to slow down chains which have already crossed the plane of shear to such an extent

that the rest of the chains have enough time to move in phase with the faster chain. In

other words, the time of crossing the plane of shear by the whole ensemble becomes of

the same order as the characteristic time of an individual chain to traverse the fourth

quadrant (after crossing the plane of shear). In that case we will observe the follow-

ing behavior of the director. As the majority of the chains approach the plane of shear,

the director also will approache the plane of shear. But as a fraction of chains passes

the plane of shear, the chains rapidly flip over and start to approach the plain of shear

again. The director does not follow this fraction of chains, because the majority of the

chains are still approaching the plane of shear. When the fraction of chains that have

flipped over approaches the plane of shear again the director will increase the angle

with the plane of shear. According to observations this wagging motion of the director

5.3 Evolution of the director in steady shear flow 99

125 250 375 500

0.2

0.4

0.6

0.8

λ1λ2λ3

t

Figure 5.9: Eigenvalues of the orientation tensor 〈uu〉 as a function of time in shear flow for˜γ = 0.2. Kayaking regime.

125 250 375 500

0.2

0.4

0.6

0.8

λ1λ2λ3

t

Figure 5.10: Eigenvalues of the orientation tensor 〈uu〉 as a function of time in shear flow for˜γ = 0.6. Wagging Regime.

50 100 150 200

0.2

0.4

0.6

0.8

λ1λ2λ3

t

Figure 5.11: Eigenvalues of the orientation tensor 〈uu〉 as a function of time in shear flow for˜γ = 1.0. Flow-aligning regime.

100 Rheology of entangled LCP solutions containing hairpins

is non-damping, i.e., can be seen an as auto-oscillator. It is known that a very import-

ant component needed for creation of auto-oscillations is positive feedback. In the LCP

solution the role of feedback is introduced through the nematic field. For example, if

a considerable fraction of the chains have crossed the plane of shear, then through the

nematic interaction they speed up the crossing process for the rest of the chains which

are already very close to the plane of shear, thus increasing the number of chains which

cross simultaneously. And vice versa, when this fraction of chains approaches the plain

of shear, chains that are already closer to the plain of shear are pulled back from the plain

of shear. Thus, for some ratio of shear flow and nematic field the number of chains per

unit time that crosses the plane of shear starts to oscillate. This causes the oscillation of

the director, which we interpret as the wagging type of behavior.

For even higher shear rates the time-scale of deformations ˜γ−1 becomes small compared

to the time-scale of the synchronization of the orientation of the chains due to nematic

interactions. Therefore, the rotation of each chain becomes uncoupled from the rotation

of the other chains, i.e., the chains are rotating out of phase with each other. Obviously,

each chain spends the majority of time approaching the plain of shear. That is why, the

director is oriented at a small angle with the direction of flow. In fig. 5.11 we see that

the eigenvalues of the orientation tensor do not oscillate any more.

In this section we have examined the behavior of the model in shear flow. The system

shows a periodic kayaking motion for small shear rates. The kayaking period is found

to be inversely proportional to the shear rate, which is in agreement with the predic-

tions in earlier studies of the classical models [42]. For the flow-aligning case the Leslie

angle as function of shear rate was investigated. The non-monotonic dependence with

a maximum at about ˜γ = 5 was found. This result is also in agreement with earlier

studies.

However the region of wagging behavior it found to be rather narrow 0.55 < ˜γ < 0.65

compared to the standard rigid rod models for the chosen value of the nematic field

strength. Most probably, this is caused by the hairpins. The presence of the hairpins

changes the aspect ratios of different chains, and, thus, introduces an additional spread

in the characteristic timescales of rotation for individual chains in the flow field.

5.4 Shear viscosity and the first normal stress difference

in the steady shear

In this section we present the results of simulations for the rheological properties of the

system in a steady shear flow. Fig. 5.12, 5.13, and 5.14 show the plots for the shear

component of the stress tensor as function of time for the startup shear experiment and

5.4 Shear viscosity and the first normal stress difference in the steady shear 101

125 250 375 500

2

4

6

8

10

12

14

σ12

t

Figure 5.12: Shear stress as a function of time in shear flow for ˜γ = 0.2. Kayaking regime.

125 250 375 500

5

10

15

σ12

t

Figure 5.13: Shear stress as a function of time in shear flow for ˜γ = 0.6. Wagging regime.

50 100 150 200

2

4

6

8

10

σ12

t

Figure 5.14: Shear stress as a function of time in shear flow for ˜γ = 1.0. Flow-aligning regime.

102 Rheology of entangled LCP solutions containing hairpins

for the steady shear flows at different shear rates. In fig. 5.13 and 5.14 we clearly see an

overshoot and a couple of successive oscillations in the initial period of about 100 time

units. Then this transient behavior is followed by steady oscillations of the stress, like in

the wagging or kayaking regime, or by a slightly fluctuating value of the stress tensor,

like in the flow-aligning regime. After the transient behavior is passed, we compute the

steady state shear viscosity using

ηsh =1˜γτ

t0+τ∫

t0

σ12(

t)

dt (5.7)

where τ is either the period of the oscillations in the kayaking or wagging regime, or the

time interval sufficient to average out the fluctuations in the flow-aligning regime. The

results are plotted in fig. 5.15. In this plot we clearly see that the model demonstrates

shear thinning behavior. For small shear rates (in the kayaking regime) the viscosity

changes slightly. For higher shear rates the viscosity rapidly decreases with a slope of

about −0.45 in a Log-Log plot. This behavior is in agreement with the Asada-Onogi

plot (fig. 1.7), and corresponds to Region II and Region III.

For many shear-thinning liquids the formula suggested by Hess [102] is valid

η (γ) = η∞ +η0 − η∞1 + (τrγ)

2 (5.8)

where η0 is the zero shear-rate viscosity, η∞ is the viscosity at very high shear rates, and

τr is the rotational relaxation time. The red line in fig. 5.15 is given by (5.8) with the

following choice of parameters: η∞ = 3.8, η0 = 20, τr = 1.5.

At shear rates ˜γ > 0.6 the flow becomes strong enough to create or destroy hairpins. In

fig. 5.16 the average value of the hairpin variable h (defined in 2.65) is plotted versus

time for a shear rate ˜γ > 0.6. At the initial moment of time 〈h〉 = 0.56. But after

the flow is imposed 〈h〉 drops to 0.3. The chains are becoming more stretched by the

imposed flow and the hairpin defects are destroyed by this stretching. This process also

contributes to the shear-thinning behavior of the solution. Namely, the decrease of the

number of hairpin defects causes the decrease in the number of entanglements between

the chains. As we observe in fig. 5.16 this process is adequately described by our model.

The transition from Region II to region Region III takes place at about ˜γ = 0.55, which

corresponds to a dynamic transition from kayaking to wagging. We compare this result

with the experimental data available for PpPTA in sulfuric acid. According to Mortier

et al [60] the steady-state shear viscosity for PpPTA in sulfuric acid exhibits the three-

region behavior. The transition from region I to Region II takes place at γ ≈ 5 s−1, and

the transition from Region II to Region III takes place at γ ≈ 100 s−1. The value of the

viscosity in the plateau region changes in the interval from 30 Pa · s to 50 Pa · s. In the

5.4 Shear viscosity and the first normal stress difference in the steady shear 103

0.1 0.2 0.5 1.0 2.0 5.0

10.0

5.0

20.0

3.0

15.0

7.0

ηsh

˜γ

SimulationsInterpolation

Figure 5.15: Steady state shear viscosity plotted versus shear rate.

0 125 250 375 500

0.30

0.35

0.40

0.45

0.50

0.55

〈h〉

t

Figure 5.16: Average value of hairpin variable as a function of time in shear flow for ˜γ = 0.6.Wagging regime.

�����

����

�����

�

�

�

�

�

�

�

�

� �

�

�

�

0.2 0.4 0.6 0.8 1.0

5

10

15

�

N1

˜γ

Figure 5.17: First normal stress difference averaged over a period as a function of shear rate.

104 Rheology of entangled LCP solutions containing hairpins

shear-thinning region the slope is reported to be −0.4, which is in good agreement with

the results of our simulations. According to our simulations the transition from region

II to region III takes place at ˜γ ≈ 0.55. Comparing this result with the experimental data

we can estimate τrotnem that should be chosen in order to match the transition point from

Region II to Region III. From this comparison it follows that τrotnem ≈ 5.5 · 10−3 s which

suggests the value for rotational diffusion coefficient Dr = 3.6s−1, which is four orders

of magnitude smaller than the value reported in [11].

In the discussion that followed the result for the rotational diffusion coefficient Dr =

3.3·104s−1 in [11] it was explained that due to a very strong dependence of the rotational

friction coefficient on the choice of the persistence length L the error in estimation of Dr

can be easily several orders of magnitude. Thus, from our simulations follows a more

accurate way of estimating the rotational diffusivity Dr by the transition point from

Region II to Region III.

Of course, we understand that in our model the change of Dr changes the hierarchy of

the time-scales, and not only scales the result. However, most probably the hierarchy of

the time-scales does not change. In the previous chapter we have estimated the trans-

lational friction coefficients for the rods on the basis of formulae for a dilute solution.

Definitely, we have underestimated these coefficients by several orders of magnitude.

Thus, if we increase the rotational friction by factor of 104 and account for the increase

by several orders of magnitude of the translational friction coefficient in a concentrated

solution compared to a dilute solution, then the ratio between the translational friction

coefficient and the rotational friction coefficient changes much less than by factor 104.

If we adopt τrotnem ≈ 5.5 · 10−3 s and keep the dimensionless parameters of the model the

same, then the value for the viscosity in Region II is ηsh ≈ 9 Pa · s. As was mentioned

earlier the value found in the experiments is in the range of 30 Pa · s to 50 Pa · s. In

the simulations presented in this section we have used εtol = 0.01, as it follows from

criterion (2.66). Of course, in the criterion based on the dimensional considerations a

dimensionless factor was omitted, which is, as turned out of importance. The increase

of the role of entanglements between hairpins, as was shown in fig. 5.6 leads to a rapid

increase of the response and, clearly, of the viscosities. For example, according to fig. 5.6

the change of εtol from 0.01 to εtol = 0.04 changes the elongational viscosity by factor of

4. In simulations with εtol = 0 and for the same values of the dimensionless parameters

the computed viscosity in Region II is ηsh ≈ 5 Pa · s, while the position of the transition

from Region II to Region III is hardly changed. Thus, for shear flow we observe the

same tendency, as for elongational flow. To match the experimental data we have to use

εtol ≈ 0.04. This is a good result, because we have estimated εtol = 0.01 on the basis

of dimension considerations without introducing additional tuning coefficients. And

in the end we have obtained the result that differs from the experimental data only by

factor 4.

5.4 Shear viscosity and the first normal stress difference in the steady shear 105

The strong dependence of the response functions on the tolerance εtol, which reflects

the lifetime of the entanglements, occurs only when the average number of hairpins per

chain exceeds 1. In our simulations for the chain consisting of two rods (N = 2) the

excessive stress tensor almost did not change with the change of εtol. This is a trivial

result. If the chain contains only one hairpin and it entangles, then this entanglement

hinders primarily the motion of the center of mass of the chain. In our model the chain

behaves as if it is pinned to the background at a point. However, the main contribution

to the excessive stress tensor is due to the internal modes of the chain, which are almost

not perturbed by this this entanglement. The situation changes when the number of

the chains containing two hairpins or more becomes significant. In this case, a lot of

chains can become entangled at two points. Consequently, the segment of the chain

between these two entanglements can not relax easily, unless one of the entanglements

disappears. In this way hairpins start to hinder the relaxation of the internal modes

of the chains. This gives rise to a slower decay of the response functions and to larger

values of the viscosities, as we see from the presented in this chapter simulations.

Finally, we plot the first normal stress difference versus shear rate in fig.5.17. The first

normal stress changes sign two times. This is in agreement with the theoretical results

in [42], [43], [44], [46] and the simulations in [101], [30]. The first normal stress clearly

shows three regions: in the kayaking region of the shear rates it is positive, in the wag-

ging region it is negative, and then it again becomes positive in flow-aligning regime.

From the simulations described in this chapter we can draw the following conclusions:

• The numerical scheme described in Chapter 4 reproduces the predictions of the

model for highly-ordered unentangled LCP solutions containing hairpins formu-

lated in Chapter 3 if 34

ζrot

UnA2N

2 ≪ 1 and εtol = 0.

• The presence of the hairpins in the unentangled LCP solution decreases the re-

sponse functions.

• The entanglements between hairpins start to play an important role only when

there are on average more than one hairpin per chain.

• The rod-spring-bead model formulated in Chapter 2 is capable to reproduce the

main features of the rheology of the nematic LCP solutions, such as several dy-

namic regimes of the director, the region with negative normal stress difference,

shear-thinning behavior.

• If possible entanglements between hairpins are accounted for, then both qualit-

ative as well as quantitative agreement between experiments and theory can be

obtained.

106 Rheology of entangled LCP solutions containing hairpins

Chapter 6

Conclusions

In this thesis we have addressed a set of problems originating from the processing of

LCP solutions. The fibers spun from LCP solutions have demonstrated extraordinary

characteristics. These characteristics are the reason why such fibers are used in compos-

ite materials, sport equipment, body armory, thermal-insulating cloth, etc. However,

complex rheological behavior of industrially relevant LCP solutions makes control of

LCP processing difficult. The need for a better understanding of the rheological prop-

erties of the semi-flexible LCP solutions, and, in particular, the role of hairpins present

in the LCP solution was the main focus of this thesis. Both numerical and analytical

techniques were combined in order to achieve this goal. The following results were

obtained in this thesis:

• We have formulated the rod-spring-bead model for concentrated LCP solutions,

which is capable of treating semi-flexibility of chain and the presence of hairpins.

This model was expressed in two ways: by a Smoluchowski equation and by a set

of stochastic differential equations in Chapter 2.

• In Chapter 3 we have formulated a reduced model for LCP solutions, which fol-

lows from the rod-spring-bead model in the limit of the high nematic ordering,

separation of time-scales of rotational and translational motion, and the assump-

tion about the absence of entanglements between the chains. In this limit the

model was investigated analytically, in particular regarding its behavior in linear

rheology. The influence of hairpin defects of the chain’s backbone was examined

by considering the response moduli for different number of hairpins present. It

turned out that the increase of the number of hairpin defects in the system causes

the reduction of the response functions for such an unentangled LCP solution.

• In Chapter 4 we have formulated the modified Euler-Maruyama algorithm and

108 Conclusions

presented the results of simulating equilibrium properties of the rod-spring-bead

model formulated in Chapter 2.

• Numerical results of the rod-spring-bead model in homogeneous flows by means

of the modified Euler-Maruyama algorithm described in Chapter 4 were presented

in Chapter 5. The results are the following:

– From a series of numerical simulations of the degree of orientational ordering

in the equilibrium state the increase of the flexibility of the chain’s backbone

leads to a rapid reduction of the degree of orientational ordering if the total

nematic energy per chain is kept constant.

– The numerical scheme reproduces the results of the reduced analytical model

formulated in Chapter 3, if ζrot

UnA2N

2 ≪ 1 and εtol = 0.

– When entanglements between chains containing hairpins are accounted for

the relaxation time of the response function will increase, in particular a more

elastic response is observed.

– The contribution from the entanglements between hairpins to the stress tensor

becomes important when the average number of hairpins per chain substan-

tially exceeds 1. In many of the simulations presented in Chapter 5 this value

was close to 2. For this case it was shown that the response functions are

very sensitive in this case to the choice of εtol, which reflects the lifetime of

entanglements between hairpin defects.

– It follows from the numerical simulations of the behavior of the chain in a

steady shear flow that the well known dynamic transitions from kayaking

through wagging to flow-aligning are observed. Besides that the first normal

stress difference shows a region with negative values. The kayaking period is

found to be inversely proportional to the shear rate, which is in good agree-

ment with the results of previous works [30].

All parameters involved in the rod-spring-bead model presented in this thesis have a

clear physical meaning and can be either directly measured in experiments or calcu-

lated from the experimental data. In Chapter 4 we have estimated the values of these

parameters on the basis of available experimental data for the industrially relevant LCP

solution consisting of PpPTA dissolved in sulfuric acid.

In science there is always space for further developments and improvements. This

holds for this thesis as well. The model presented in this thesis works well for ho-

mogeneous flows. However, for the practical purposes it is often needed to compute

non-homogeneous flow in particular geometries. The rod-spring-bead model presen-

ted in this thesis is very computationally costly and, therefore, requires to be further

coarse-grained to be efficiently applied for these kind of problems.

109

Besides further coarse-graining of the presented model, an incorporating of the so-

called poly-domain structure would be a natural extension [59]. The presented rod-

spring-bead model reproduces the Region II and Region III of the Asada-Onogi plot

correctly. However, it is known that the shear-thinning behavior in Region I is caused

by poly-domain structure [45]. In our model the order tensor is assumed to be constant

at all points of the system, i.e., the system is mono-domain. This is also the reason why

we do not present the results for start-up flow. In mono-domain approach at the ini-

tial moment of time we have to specify the orientation of the director. The results of

simulations depend on this choice of the initial orientation of the director. However,

in experiments different parts of the sample have different orientation of the director

and, therefore, the results of start-up simulations should not be directly compared to

the experiments.

110 Conclusions

Appendix A

Details of derivations

A.1 Evolution equation for the configurational distribu-

tion function

We derive the equation for the evolution of the single-particle distribution function Ψ

in this section. The derivation is based on the general equation of change.

∂

∂t〈B〉 = 〈LB〉 (A.1)

If we choose the dynamical variable BΨ below

BΨ =

Nch∑

s=1

N∏

i=1

δ (rsi − ri) δ (usi − ui)

N−1∏

i=1

δ (bsi − bi) , (A.2)

then the general equation of change will give the evolution equation for the single-chain

distribution function. In order to make the nomenclature more compact we denote the

whole phase-space of the system by X . Infinitesimal parts of phase-space are denoted

by dX and we write

〈BΨ〉 =∫

X

BΨdX ≡ Ψ. (A.3)

On the right hand side of the equation (A.1) we get 〈LBΨ〉. The Liouville operator is

given by (2.11). We now employ the following identity for the delta-function

d

drδ(

r− r′) = − d

dr′δ(

r− r′) (A.4)

112 Details of derivations

in order to get

LBΨ = −Nch∑

s=1

(

N∑

i=1

1

mr

prsi· ∂

∂rsi+

N−1∑

i=1

1

mb

pbsi· ∂

∂bsi+

N∑

i=1

pusi· J−1 · ∂

∂usi+

+

N∑

i=1

Frsi· ∂

∂prsi

+

N−1∑

i=1

Fbsi· ∂

∂pbsi

+

N∑

i=1

Fusi· ∂

∂pusi

)

BΨ (A.5)

The last three contributions of the Liouville operator turn out to be zero because BΨ

does not depend on the momenta. Ensemble averaging of LBΨ then gives

−〈LBΨ〉 =N∑

i=1

∂

∂ri·(

1

mr

JpriKΨ

)

+

N−1∑

i=1

∂

∂bi·(

1

mb

JpbiKΨ

)

+

N−1∑

i=1

∂

∂ui·(

J−1 · Jpui

KΨ)

(A.6)

The quantities involving double brackets J·K are momentum space averaged quantities.

By combining (A.1), (A.2) and (A.6) we obtain the evolution equation for the single-

chain configuration distribution function Ψ

∂Ψ

∂t+

N∑

i=1

∂

∂ri·(

JpriK

mr

Ψ

)

+

N∑

i=1

∂

∂ui·(

J−1 · Jpui

KΨ)

+

N−1∑

i=1

∂

∂bi·(

JpbiK

mb

Ψ

)

= 0 (A.7)

A.2 Evolution equation for the momentum-space averages

In order to obtain the evolution equation for the momentum-space averaged momenta

JpriK, Jpbi

K, JpuiK we have to consider the general equation of change for the appropri-

ate dynamic quantity. We denote the single-particle phase-space distribution function

by ψ. This distribution function gives the probability density to find a single polymer

at a given point in the single-chain phase space. It follows that Bψ will generate ψ after

being averaged with respect to the ensemble.

Bψ =

Nch∑

s=1

(

N∏

i=1

δ (rsi − ri) δ (usi − ui) δ(

prsi− pri

)

δ(

pusi− pui

)

×

×N−1∏

i=1

δ (bsi − bi) δ(

pbsi− pbi

)

)

(A.8)

BJprj−mrvKΨ =

(

prj−mrv

)

Bψ (A.9)

BJpujKΨ = puBψ (A.10)

A.2 Evolution equation for the momentum-space averages 113

BJpbj−mbvKΨ =

(

pbj−mbv

)

Bψ (A.11)

Here v is the velocity of a macroscopic sample of the solution.

⟨

BJprj−mrvKΨ

⟩

= Jprj−mrvKΨ (A.12)

⟨

LBJprj−mrvKΨ

⟩

=

(

N∑

i=1

1

mr

prsi· ∂

∂rsi+

N−1∑

i=1

1

mb

pbsi· ∂

∂bsi+

N∑

i=1

pusi· J−1 · ∂

∂usi+

+N∑

i=1

Frsi· ∂

∂prsi

+N−1∑

i=1

Fbsi· ∂

∂pbsi

+N∑

i=1

Fusi· ∂

∂pusi

)

(

prj−mrv

)

Bψ (A.13)

All terms in this expression are treated in the same way as expression (A.6) in the pre-

vious appendix, except for terms that contain ∂∂prj

. These terms we integrate by parts.

In the end we get

∂

∂t

(

Jpbj−mbvKΨ

)

+

N∑

i=1

v · ∂

∂ri

(

Jprj−mrvKΨ

)

+

N−1∑

i=1

v · ∂

∂bi

(

Jprj−mrvKΨ

)

=

= −

N∑

i=1

∂

∂ri·

J(

pri−mrv

)

(

prj−mrv

)

K

mr

Ψ

+

N∑

i=1

∂

∂ui·(

J−1Jpui

(

prj−mrv

)

KΨ)

+

+

N−1∑

i=1

∂

∂bi·

J(

pbi−mbv

)

(

prj−mrv

)

K

mb

Ψ

+(

F(h)rj

+ F(e)rj

+ F(intra)rj

)

Ψ (A.14)

Here we have to clarify the meaning of the new notations for hydrodynamic force F(h)rj

,

external force F(e)rj

and intramolecular force F(intra)rj

.

F(h)rj

=1

Ψ

⟨

Finterrj

Bψ

⟩

(A.15)

F(e)rj

=1

Ψ

⟨

FextrjBψ

⟩

(A.16)

F(intra)rj

=1

Ψ

⟨

Fintrarj

Bψ

⟩

(A.17)

The terms on the left-hand side of (A.14) can be interpreted as ”acceleration terms” and

those on the right-hand side as force terms. For processes occuring on time scales much

larger than the time-scale of relaxation in momentum space ”acceleration terms” can be

omitted. In that case equation (A.14) converts to a force balance equation.

It becomes more evident that the right-hand side of (A.14) is a force balance after defin-

114 Details of derivations

ing the gradient-terms as Brownian forces.

F(b)rj

= − 1

Ψ

N∑

i=1

∂

∂ri·

J(

pri−mrv

)

(

prj−mrv

)

K

mr

Ψ

+

+

N∑

i=1

∂

∂ui·(

J−1Jpui

(

prj−mrv

)

KΨ)

+

+

N−1∑

i=1

∂

∂bi·

J(

pbi−mbv

)

(

prj−mrv

)

K

mb

Ψ

(A.18)

Finally, we obtain the following equation

F(b)rj

+ F(h)rj

+ F(e)rj

+ F(intra)rj

= 0 (A.19)

Applying the general equation of change (2.12) to the dynamical quantity (A.10) we

obtain in a completely similar way the force balance for the orientations of the rods.

F(b)uj

+ F(h)uj

+ F(e)uj

+ F(intra)uj

= 0 (A.20)

Here

F(b)uj

= − 1

Ψ

(

N∑

i=1

∂

∂ri·(

J(

pri−mrv

)

pujK

mr

Ψ

)

+

N∑

i=1

∂

∂ui·(

J−1Jpui

pujKΨ)

+

+

N−1∑

i=1

∂

∂bi·(

J(

pbi−mbv

)

pujK

mb

Ψ

))

(A.21)

F(h)uj

=1

Ψ

⟨

Finteruj

Bψ

⟩

(A.22)

F(e)uj

=1

Ψ

⟨

FextujBψ

⟩

(A.23)

F(intra)uj

=1

Ψ

⟨

Fintrauj

Bψ

⟩

(A.24)

Finally we apply the general equation of change (2.12) to the dynamical quantity (A.11)

to get the force balance equation for the positions of the beads.

F(b)bj

+ F(h)bj

+ F(e)bj

+ F(intra)bj

= 0 (A.25)

A.3 Ito or Stratonovich interpretation 115

F(b)bj

= − 1

Ψ

N∑

i=1

∂

∂ri·

J(

pri−mrv

)

(

pbj−mbv

)

K

mr

Ψ

+

+

N∑

i=1

∂

∂ui·(

J−1Jpui

(

pbj−mbv

)

KΨ)

+

+

N−1∑

i=1

∂

∂bi·

J(

pbi−mbv

)

(

pbj−mbv

)

K

mb

Ψ

(A.26)

F(h)bj

=1

Ψ

⟨

Finterbj

Bψ

⟩

(A.27)

F(e)bj

=1

Ψ

⟨

FextbjBψ

⟩

(A.28)

F(intra)bj

=1

Ψ

⟨

Fintrabj

Bψ

⟩

(A.29)

Equations (A.19), (A.20), (A.25) are the evolution equations for JprjK, Jpuj

K and JpbjK.

A.3 Ito or Stratonovich interpretation

In this section we will discuss the difference of the Ito and Stratonovich integral in detail.

Let us consider the n-dimensional SDE

dx = a (x, t) dt+B (x, t) · dW (A.30)

Here t ∈ R, x ∈ Rn, W is the m-dimensional Wiener process, a (x, t) ∈ R

n and B (x, t)

is a matrix of size n×m.

Due to the non-differentiability of the Wiener process the SDE should be understood in

the sense of the corresponding integral equation. Equation (A.30) means

x(t)− x(t0) =

t∫

t0

a (x(t), t) dt+

t∫

t0

B (x(t), t) · dW(t) (A.31)

wheret∫

t0

B (x(t), t) · dW(t) is a stochastic integral. The Stratonovich integral can be

defined in a manner similar to the Riemann integral, i.e., as a limit of Riemann sums.

116 Details of derivations

Then the Stratonovich integral is the limit in probability of the integral sum

(S)

t∫

0

A(t) · dW(t) = limd→0

k−1∑

i=0

(A(ti) +A(ti+1))

2· (W(ti+1)−W(ti))

Here d is the diameter of the mesh of the partition 0 = t0 < t1 < . . . < tk < t of [0, t],

and A is a matrix-valued function of time.

The definition of the stochastic integral in Ito’s interpretation differs in the way the value

of the integrand is chosen in each subinterval of the partition.

(I)

t∫

0

A(t) · dW(t) = limd→0

k−1∑

i=0

A(ti) · (W(ti+1)−W(ti))

Here d is again the diameter of the mesh of the partition 0 = t0 < t1 < . . . < tk < t of

[0, t].

Each of these definitions of the stochastic integral has its advantages, which determine

the fields of application of a particular interpretation. The Stratonovich interpretation

of the stochastic integral is more convenient in physics, because it allows the use of the

same ”calculus” as the usual Riemann-Stieltjes integral. The Ito interpretation is more

convenient in numerical applications [94] and in mathematical studies. This is due to the

non-anticipating structure of the integral sums in Ito’s interpretation, namely, the use of

the value of the integrand in the beginning of the subsegment of the partition when con-

structing the sum for the integral. It simplifies the studies of the stochastic differential

equations, because it allows to use explicit numerical schemes, like the Euler-Maruyama

method. The disadvantage of the Ito interpretation of the stochastic integral is the ne-

cessity to use a so-called ”Ito-calculus”, which is non-intuitive for those more used to

the rules of traditional calculus.

It was mentioned in the previous section that the relation between Ito and Stratonovich

interpretations of the stochastic integral was extensively studied [70], [72], [73]. Here

we briefly recapitulate the results of this study. A stochastic integral in the Ito inter-

pretation can be transformed into a corresponding Stratonovich integral. Consequently,

a stochastic differential equation in the Ito interpretation can be transformed into Stra-

tonovich stochastic differential equation and vice versa. So the Ito SDE

(I) dxi = ai (x, t) dt+

m∑

j=1

BijdWj (A.32)

A.4 Derivation of the expression for θmax 117

has an equivalent Stratonovich stochastic differential equation given by

(S) dxi =

ai (x, t)−1

2

n∑

j=1

m∑

k=1

Bjk∂

∂xjBik

dt+m∑

j=1

BijdWj (A.33)

The reverse transition is also possible. The Stratonovich SDE

(S) dxi = ai (x, t) dt+m∑

j=1

BijdWj (A.34)

is equivalent to the following Ito stochastic differential equation

(I) dxi =

ai (x, t) +1

2

n∑

j=1

m∑

k=1

Bjk∂

∂xjBik

dt+

m∑

j=1

BijdWj (A.35)

The term 12

n∑

j=1

m∑

k=1

Bjk∂∂xj

Bikdt is usually called ”spurious drift” term. This term rep-

resents the difference in definitions of the Ito and the Stratonovich stochastic integral.

From the structure of this term we see that if Bik is a constant, then the ”spurious drift”

term is equal to zero. In this trivial case both the Ito and the Stratonovich interpretations

of the stochastic differential equation are equivalent.

We also recapitulate here the relation between the stochastic differential equation and

the corresponding Fokker-Planck equation. Let ψ(x, t) be the probability density distri-

bution function for quantity x at moment of time t. Then the Fokker-Planck equation

corresponding to the stochastic differential equation in the Ito interpretation (A.32) is

∂ψ

∂t+

N∑

i=1

∂

∂xi(aiψ)−

1

2

n∑

i=1

n∑

j=1

m∑

k=1

∂

∂xi

∂

∂xj

(

BikBjkψ)

= 0 (A.36)

If we take the stochastic differential equation in the Stratonovich interpretation (A.34)

then the corresponding Fokker-Planck equation is

∂ψ

∂t+

N∑

i=1

∂

∂xi(aiψ)−

1

2

n∑

i=1

n∑

j=1

m∑

k=1

∂

∂xi

(

Bik∂

∂xj

(

Bjkψ)

)

= 0 (A.37)

A.4 Derivation of the expression for θmax

We derive here the expression for θmax, which is the maximum angle that two oppos-

itely aligned neighboring rods are allowed to deviate from the perfectly aligned state

118 Details of derivations

to still be considered in a hairpin state. According to the definition the expression for

θmax follows from the equality between the average thermal energy associated with the

rotational motion of a rod and the extra nematic energy, which this rod would acquire

when deviated by θmax from the direction of the most favorable orientation of the rods.

According to the equipartition theorem the average energy of the thermal motion is 12T .

Because rotational motion of a rod has two degrees of freedom the average energy of

the rotational motion of a rod is T .

Expression (2.48) determines the nematic potential. To use it in this particular case let

us do some manipulation with the orientation tensor S. By definition S is a symmetric

tensor, and consequently the corresponding matrix is also symmetric. It is known that

a symmetric matrix can be always diagonalized and the eigenbasis for such a matrix is

an orthogonal basis. We choose the cartesian coordinate system in such a way that the

Oz axis coincides with the direction on the eigenvector with the maximum eigenvalue.

Systems with nematic ordering have cylindrical symmetry around the direction corres-

ponding to the maximum eigenvalue of matrix S,because this is the direction of the most

preferable orientation of the rods. Therefore two other eigenvalues of S are equal, or in

other words, the eigenspace corresponding to this eigenvalue is two-dimensional. This

means that we can choose the axes Ox and Oy along any two perpendicular directions

in the plane perpendicular to Oz. In this coordinate system the matrix S is diagonal.

S =

λ− 13 0 0

0 1−λ2 − 1

3 0

0 0 1−λ2 − 1

3

(A.38)

Here λ is the maximum eigenvalue of the matrix 〈uu〉. Expression (A.38) follows from

the property of 〈uu〉 that

Tr 〈uu〉 = 〈u · u〉 = 〈1〉 = 1. (A.39)

The trace of an operator is equal to the sum of its eigenvalues. The largest eigenvalue of

〈uu〉 is λ. Because the other two eigenvalues are equal and their sum should be 1, they

are equal to 1−λ2 .

The vector u is a unit vector. Therefore it is convenient to express the coordinates of

u in terms of spherical coordinates: the inclination angle θ and the azimuthal angle φ.

The angle θ is the angle between the vector u and Oz axis. The angle φ is related to the

rotation in the xOy plane. Then the cartesian coordinates for u are

u =

sin θ cosφ

sin θ sinφ

cos θ

(A.40)

A.5 Derivation of the expression for the stress tensor 119

The nematic energy for one rod is

U rodMS = −UnS :

(

uu− I

3

)

= Un

(

3λ− 1

2sin2 θ −

(

λ− 1

3

))

(A.41)

And the equation for θmax therefore becomes

Un(3λ− 1)

2sin2 θmax = T (A.42)

and then a little manipulation yields

sin θmax =

√

2T

(3λ− 1)Un(A.43)

This is the expression for θmax that is used in the definition of the hairpin state.

A.5 Derivation of the expression for the stress tensor

For models treating polymer chains as bead-spring chains the contribution from the

polymer chains to the stress tensor is given by the famous Kramers-Kirkwood expres-

sion [67].

σ =1

V

Nch∑

j=1

N(j)b∑

i=1

⟨

F(j)ri

r(j)i

⟩

(A.44)

In this formula r(j)i is the position of the i-th bead of the j-th chain and F

(j)ri

is the

force exerted on this bead. The summation is performed over all beads and all chains

within the volume of a sample V . Nch denotes the number of chains and N(j)b denotes

the number of beads constituting the j-th chain. However, for our model we have to

modify the expression for the stress tensor, because the orientational degrees of freedom

also give contributions to the stress tensor. One of the ways to get the new expression

is by the method of virtual work [25]. The idea of this method is to relate the change of the

Helmholtz free energy of the sample to the stress under an instantaneous deformation

δǫ.

The contribution to the Helmholtz free energy Ach from each chain for the system of

chains, described by equation (2.32), can be expressed in terms of the single-chain con-

figurational distribution function Ψ. For convenience we will denote the whole set of

variables related to a single chain by x. x = {r1, . . . , rN ,u1, . . . ,uN ,b1, . . . ,bN−1}

Ach [Ψ] =

∫

Ψ(x) (T ln (Ψ (x)) + U (x)) dx (A.45)

120 Details of derivations

The total free energy of the polymer chains is given by the sum of contributions from

all chains in the sample V . If all the chains have identical structure, then the total free

energy is just A = NchAch.

The variation of the free energy under small instantaneous deformation δǫ is related to

the stress σ created.

δA = σµνδǫµνV (A.46)

The variation of the free energy is determined by the variation of the distribution func-

tion.

δA =

∫

δΨ(x) (T + T ln (Ψ (x)) + U (x)) dx (A.47)

The variation of the free energy upon the instantaneous deformation can be estimated

from the evolution equation (2.32). Because the deformation δǫ is performed instantan-

eously the dominating terms in equation (2.32) are terms containing deformation rates.

Thus for this particular case equation (2.32) can be simplified.

∂Ψ

∂t+

N∑

i=1

∂

∂ri·(κ · riΨ)+

N∑

i=1

∂

∂ui·((I− uiui) · κ · uiΨ)+

N−1∑

i=1

∂

∂bi·(κ · biΨ) = 0 (A.48)

Here N is the number of rods per chain. We use the relation δΨ = ∂Ψ∂tδt and the relation

δǫ = κδt to obtain

δΨ = −N∑

i=1

∂

∂ri·(δǫ · riΨ)−

N∑

i=1

∂

∂ui·((I− uiui) · δǫ · uiΨ)−

N−1∑

i=1

∂

∂bi·(δǫ · biΨ) (A.49)

Then we substitute (A.49) to (A.47) and perform integration by parts. Employing (A.46)

yields the expression for the stress originating from the polymer chains.

σ = −NchV

[

N∑

i=1

⟨

Friri⟩

+

N∑

i=1

⟨

Fuiui⟩

+

N−1∑

i=1

⟨

Fbibi⟩

]

(A.50)

Here Fri, Fui

and Fbiare the total generalized forces associated with the variables ri,

ui and bi respectively. These forces are given by the corresponding gradients of the

expression (T ln (Ψ (x)) + U (x)). The part of the forces originating from the gradients

of T ln (Ψ (x)) are the Brownian forces.

Fri= − ∂

∂ri(T ln (Ψ (x)) + U (x)) (A.51)

Fui= − ∂

∂ui(T ln (Ψ (x)) + U (x)) (A.52)

Fbi= − ∂

∂bi(T ln (Ψ (x)) + U (x)) (A.53)

A.6 Normal modes expansion 121

A.6 Normal modes expansion

We derive here the evolution equations for the normal coordinates of the rod-spring

chain.

First, we separate the translational motion of the chain’s center of mass from the internal

motions. This is done by switching to the set of variables {rc, c1, ..., cN−1}. The equation

for rc is obtained by taking the sum of equations for all r1, r2, ...rN and the equation for

cs is obtained by taking the difference of the equations for rs+1 and rs. This leads to

Nζ · (rc − κ · rc) = fc(t) (A.54)

In here fc(t) =N∑

s=1fs(t) with

〈fc (t)〉 =N∑

s=1

〈fc (t)〉 = 0 (A.55)

and⟨

fc(t)fc(t′)⟩

=

N∑

s=1

N∑

s′=1

⟨

fs(t)fs′ (t′)⟩

= 2Nζ (n) Tδ(

t− t′)

(A.56)

The equations for internal degrees of freedom became

(S) ζ · (c1 − κ · c1) = K · (−2c1 + c2)+w1lk0n+ f2− f1 (s = 1) (A.57)

(S) ζ ·(cs − κ · cs) = K ·(cs+1 − 2cs + cs−1)+ fs+1− fs (s ∈ {2, ..., N−2}) (A.58)

and

(S)ζ ·(cN−1 − κ · cN−1) = K·(cN−2 − 2cN−1)+w1lk0n+fN−fN−1 (s = N−1) (A.59)

We now introduce the stochastic forces gs = fs+1 − fs for s ∈ {1, 2, ..., N − 1} and the

Rouse matrix Aps = 2δp,s − δp+1,s − δp,s+1. Then equations (A.57),(A.58) and (A.59) can

be written in a general form for p ∈ {1, 2, ..., N − 1}

(S) ζ(n) ·(

cp − κ · cp)

= −K(n) ·N−1∑

s=1

Ap,scs + w1lk0n(

δp,1 + δp,N−1

)

+ gp(t) (A.60)

with⟨

gp(t)⟩

= 0 (A.61)⟨

gp(t)gp′(t′)⟩

= 2Ap,p′ζ(n)Tδ(

t− t′)

(A.62)

By means of this change of variables we have separated the translational motion of the

122 Details of derivations

center of mass (A.54) from the internal motions of the chain (A.60). We see that theN−1

equations describing the internal motions are coupled. The problem of decoupling these

equations boils down to diagonalization of the Rouse matrix. It is known that the Rouse

matrix of size (N − 1) × (N − 1) can be diagonalized by the (N − 1) × (N − 1) matrix

Γm,p =√

2Nsin(

πpmN

)

, leading to

Jm,s = λmδm,s =

N−1∑

p=1

N−1∑

n=1

Γm,pAp,n

(

Γ−1)

n,s(A.63)

where λm = 4 sin2(

πm2N

)

are the eigenvalues of the Rouse matrix.

These considerations imply that the normal-mode coordinates qm are related to the con-

nector vectors cs by the following relation

qm =N−1∑

s=1

Γm,scs (A.64)

The equations for the coordinates qm are derived from (A.60) by substituting cs in terms

of qm, then multiplying by Γm,p followed by summing over p. This gives for m ∈{1, ..., N − 1}

(S) ζ(n) · (qm − κ · qm) = −λmK(n) · qm + αmw1lk0n+ hm(t) (A.65)

where

αm ≡N−1∑

p=1

Γm,p(

δp,1 + δp,N−1

)

=

√

2

N(1− (−1)m) sin

(πm

N

)

(A.66)

and

hm(t) ≡N−1∑

p=1

Γm,pgp(t) (A.67)

with

〈hm(t)〉 = 0 (A.68)

and⟨

hm(t)hm

′(t′)⟩

=N−1∑

p=1

N−1∑

p′=1

Γm,pΓm′,p

′

⟨

gm(t)gm′(t′)⟩

With (A.62) and the properties of the symmetric and orthogonal matrix Γ the following

expression for⟨

hm(t)hm

′(t′)⟩

is derived.

⟨

hm(t)hm

′(t′)⟩

= 2ζ(n)λmTδm,m′δ(

t− t′)

(A.69)

A.7 Formal solution of a linear matrix differential equation 123

Finally, by applying ζ−1 (n) to both sides of (A.65) we obtain the set of N − 1 independ-

ent equations for the internal motions of the rod-spring chain. Because these equations

are independent, the coordinates qm are called normal modes.

(S) qm − κ · qm = −λmτ−1 · qm + αmw1lτ‖n+ vm(t) (A.70)

τ−1(n) = ζ

−1(n) ·K(n) = τ−1‖ nn+ τ−1

⊥ (I− nn) (A.71)

vm(t) = ζ−1(n) · hm(t) (A.72)

〈vm〉 = 0 (A.73)⟨

vm(t)vm′(t′)⟩

= 2ζ−1(n)λmTδm,m′δ(

t− t′)

(A.74)

A.7 Formal solution of a linear matrix differential equa-

tion

In this appendix a formal method to solve a system of linear inhomogeneous first order

differential equations with time-dependent coefficients is described. Let x be a time-

dependent vector in Rn solving the initial-value problem.

x(t) = A(t) · x(t) + c(t)

x(0) = x0

(A.75)

Here A is a time-dependent matrix of size n×n and c is a given time-dependent vector

in Rn.

Our aim is to derive the solution of the initial-value problem (A.75). In order to do

this we will first solve a corresponding homogeneous equation, and then we use the

modified method of variation of the constant. Our guess for the solution of the corres-

ponding homogeneous equation is x (t) = M (t) ·x0, where M (t) is an unknown matrix

to be found. Equation for M(t) can be established by direct substitution of x(t) into

x(t) = A(t) · x(t).M(t) · x0 = A(t) ·M(t) · x0

This equation should be satisfied for any initial condition x0. Thus we obtain an equa-

tion for M(t). Supplemented with the initial condition x(0) = M(0) · x0 = x0, i.e.

124 Details of derivations

M(0) = I, we obtain an initial-value problem for M(t).

M(t) = A(t) ·M(t)

M(0) = I

(A.76)

Now we turn back to the problem (A.75) with inhomogeneous equation. We will look

for a solution in the form x(t) = M(t) · c(t), where c(0) = x0 and M(t) is described by

(A.76). The direct substitution of the suggested expression for x(t) into (A.75) brings us

to an initial-value problem for c(t).

c(t) = M−1(t) · c(t)

c(0) = x0

(A.77)

If the matrix M−1(t) is known and the vector c(t) is given, then the problem (A.77) can

be integrated straightforward by

c(t) = x0 +

t∫

0

dsM−1(s) · c(s) (A.78)

From (A.78) we readily obtain an expression for x(t).

x(t) = M(t) · x0 +M(t) ·t∫

0

dsM−1(s) · c(s) (A.79)

This expression contains both M and M−1. Sometimes it is more convenient to solve an

evolution equation for M−1 instead of inverting M for every moment of time. The equa-

tion for M−1 is easily derived from (A.76) and the use of the definition of the inverse

matrix M−1(t) ·M(t) = I.

M−1(t) = −M

−1(t) ·A(t)

M−1(0) = I

(A.80)

We conclude this appendix giving the solution of the problem (A.75) in terms of (A.79),

(A.76), and (A.80).

A.8 Normal modes in the equilibrium state 125

A.8 Normal modes in the equilibrium state

In equilibrium κ(t) = 0 and n does not change its direction in time. In this case system

(3.43) can be explicitly solved.

Mp(t) = exp(

−λpτ−1t)

(A.81)

Then (3.45) gives

⟨

qp(t)⟩

= exp(

−λpτ−1t)

·⟨

qp(0)⟩

+

t∫

0

ds exp(

−λpτ−1(t− s))

· αpw1n (A.82)

In the limit t→ ∞ this boils down to

⟨

qp⟩

eq=αpw1

λpn (A.83)

In order to get an expression for⟨

qpqp⟩

eqwe can use the result (3.46).

⟨

qpqp⟩

eq=⟨

qp⟩

eq

⟨

qp⟩

eq+ limt→∞

Mp(t) ·Bp(t) ·(

Mp(t))T

(A.84)

Expression for Bp(t) can be found from (3.47) by direct integration, because in equilib-

rium the matrices Mp(t) are known and ζ is a constant matrix. Thus, in equilibrium we

get

Bp(t) =1

2Θ τ · ζ−1 · exp

(

2λpτ−1t)

+Bp(0) (A.85)

and hence

limt→∞

Mp(t) ·Bp(t) ·(

Mp(t))T

=1

2Θ τ · ζ−1

(A.86)

Finally, this leads to the following expression for⟨

qpqp⟩

eq

⟨

qpqp⟩

eq=α2pw

21

λ2pnn+

1

2Θ τ · ζ−1

(A.87)

a result that is in agreement with the equipartition theorem.

A.9 Free energy of the ensemble of chains

In this section we derive the formula for the dimensionless free energy of the sys-

tem of chains expressed in normal-mode coordinates. We start with (3.22) and the

126 Details of derivations

definition (3.51). First we transform from the set of coordinates {r1, ..., rN} to the set{

Rc, c1, ..., cN−1

}

, leading to

Fsys =1

k0

N−1∑

m=1

[w0 〈cm ·K0 · cm〉+ w1 〈(cm − n) ·K1 · (cm − n)〉]

Then we rearrange the brackets and use the symmetry of the operator K1

Fsys =1

k0

N−1∑

m=1

[(w0K0 + w1K1) : 〈cmcm〉+ w1K1 : (nn− 2n 〈cm〉)]

Finally, using K0 = k0I and K1 = k0nn+ k1(I− nn) then gives

Fsys = 2w1 (Λ : nn− p · n) + C (A.88)

where

Λ =1

2

(

1− k1k0

)N−1∑

m=1

〈cmcm〉 (A.89)

p =

N−1∑

m=1

〈cm〉 (A.90)

and

C =

(

w0 + w1

k1k0

)N−1∑

m=1

〈cm · cm〉+ w1 (N − 1) (A.91)

Now we want to express Λ, p and C in terms of the coordinates {q1, q2, ..., qN−1}.

N−1∑

m=1

⟨

bmbm

⟩

=

N−1∑

m=1

N−1∑

n=1

N−1∑

s=1

(

Γ−1)

m,s

(

Γ−1)

m,n〈qsqn〉

We use Γ−1 = ΓT to getN−1∑

m=1

⟨

bmbm

⟩

=N−1∑

n=1

〈qnqn〉 (A.92)

Similarly,N−1∑

m=1

⟨

bm

⟩

=

N−1∑

m=1

N−1∑

n=1

(

Γ−1)

m,n〈qn〉 =

N−1∑

n=1

βn 〈qn〉 (A.93)

where

βn =

N−1∑

m=1

Γ−1m,n =

√

2

N

N−1∑

m=1

sin(πnm

N

)

A.9 Free energy of the ensemble of chains 127

We use formula

N∑

m=1

sin(am) =1

sin(

a2

) sin

(

aN

2

)

sin

(

a (N + 1)

2

)

to get

βn =αnλn

(A.94)

Thus we rewrite (A.89),(A.90),(A.91) in coordinates {q1, q2, ..., qN−1}

Λ =1

2

(

1− k1k0

)N−1∑

m=1

〈qmqm〉 (A.95)

p =

N−1∑

m=1

αmλm

〈qm〉 (A.96)

C =

(

w0 + w1

k1k0

)N−1∑

m=1

〈qm · qm〉+ w1 (N − 1) (A.97)

These results (A.95),(A.96),(A.97) express the free energy (A.88) in terms of the coordin-

ates {q1, q2, ..., qN−1}.

128 Details of derivations

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Summary

Constitutive modeling of concentrated solutions of main-

chain liquid crystalline polymers

Processing of concentrated semi-flexible LCP (liquid-crystalline polymer) solutions is

very important from an industrial point of view. In particular, concentrated semi-

flexible LCP solutions are used in the production of fibers with outstanding mechan-

ical properties. Understanding of the relation between the processing of LCP solutions

and the ultimate properties of the fibers requires the development of constitutive mod-

els for LCP solutions. Despite the fact that processing of LCP solutions started more

than 50 years ago and a lot of research was devoted to the constitutive modeling of

these systems, there are still questions to be answered. Some aspects, such as the form-

ation of hairpins and their connection with entanglements in concentrated semi-flexible

LCP solutions are still of great interest. However, the introduction of these concepts in-

creases the complexity of the constitutive model and increases the difficulty of studying

the rheological properties of LCP solutions.

The present thesis deals with the development of the constitutive model for concen-

trated semi-flexible LCP solutions containing hairpins. The model also accounts for the

fact that the formation of the hairpins increases the number of entanglements between

the chains. The present work combines both, numerical and analytical techniques. We

start by formulating the coarse-grained mechanical model for semi-flexible polymer

molecule. Then we employ the methods of phase-space kinetic theory for deriving the

evolution equation (the Smoluchowski equation) for a single-chain distribution func-

tion representing a single polymer chain immersed into the LCP solution. We also in-

volve the concept of the mean-field approximation for the nematic interaction (Maier-

Saupe potential) to eliminate the two-chain distribution function. Though the methods

of phase space theory are extensively studied, we describe this derivation in detail in

order to show explicitly the assumptions involved in this derivation.

138 Summary

Next, we reformulate the obtained Smoluchowski equation in terms of the correspond-

ing system of stochastic differential equations (SDEs) in the Stratonovich interpretation.

This system of SDEs is used for the analytical study of the unentangled highly-ordered

concentrated semi-flexible LCP solution containing hairpins in elongational flow. Our

results for this limit indicate that for unentangled LCP solutions the presence of hairpins

reduces the response functions.

We have further developed a numerical code for solving the system of SDEs for the en-

tangled concentrated semi-flexible LCP solutions containing hairpins. Firstly, this code

is tested by reproducing the equilibrium properties of the LCP solutions. Secondly,

the results of simulations for the elongation flow are compared with the theoretical

prediction for unentangled LCP solution. It turns out that results of simulations for

unentangled LCP solutions reproduce the theoretical predictions correctly. However,

from the simulations for entangled LCPs it follows that the entanglements increase the

response functions and become important when the average number of hairpins per

chain becomes greater than 1.

In addition, we perform simulations of the behavior of the LCP solution under shear.

Our model is capable of reproducing the well known dynamical transition and a pecu-

liar dynamics of the director: kayaking, wagging and flow-aligning. From these simu-

lations we also obtain the steady-state shear viscosity, which demonstrates the plateau

for intermediate shear rates and the shear-thinning behavior for high shear rates. This

is qualitatively in agreement with Asada-Onogi plot for typical LCP solutions. Besides

that, our results of simulations for high and medium shear-rates agree with the experi-

mental data by order of magnitude.

Samenvatting

Het verwerken van geconcentreerde oplossingen bestaande uit semiflexibele vloeibaar

kristallijne polymeren (LCP) is industrieel zeer relevant. Deze geconcentreerde LCP

oplossingen worden met name verwerkt tot vezels met uitstekende mechanische ei-

genschappen. Het begrijpen van de relatie tussen de verwerking van LCP oplossingen

en de uiteindelijke eigenschappen van deze vezels vereist het ontwikkelen van consti-

tutieve modellen voor de reologie van LCP oplossingen. Ondanks het feit dat deze LCP

oplossingen al meer dan 50 jaar worden verwerkt en er al veel onderzoek verricht is

aan het modelleren van het constitutieve gedrag van dit soort systemen, zijn er nog veel

vragen onbeantwoord. Sommige aspecten, zoals de vorming van ”hairpin” defecten en

hun verband met ”omstrengelingen” (entanglements) in geconcentreerde semiflexibele

LCP oplossingen, zijn nog steeds onvoldoende begrepen. Echter, het verdisconteren

van dit soort fenomenen in een constitutieve model voor de reologische eigenschappen

van LCP oplossingen is niet eenvoudig en vergroot de complexiteit van zo’n model en

verdere analyse ervan.

Dit proefschrift gaat over de ontwikkeling van een constitutief model voor geconcent-

reerde oplossingen bestaande uit semiflexibele LCPs die dit soort hairpins bevatten.

Tevens verdisconteerd het model ook het feit dat met de vorming van hairpins het aan-

tal entanglements tussen de polymeerketens toe zal nemen. De aard van het onderzoek

in dit proefschrift is een combinatie van numeriek en analytisch werk. Het begint met

het formuleren van een grofkorrelig mechanisch model van zo’n semiflexibel polymeer

molecuul.

Vervolgens worden m.b.v. methoden uit de zogeheten faseruimte kinetische theorie

de evolutie vergelijking (Smoluchowski vergelijking) voor de verdelingsfunctie van een

enkele polymeerketen in zo’n LCP oplossing afgeleid. Daarnaast wordt een gemiddelde

veld benadering (Maier-Saupe potentiaal) gentroduceerd om de nematische interactie

te beschrijven tussen ketenden en om het gebruik van twee-keten verdelingsfuncties

te vermijden. Hoewel de methoden van de faseruimte kinetische theorie uitgebreid

beschreven zijn in de literatuur, wordt deze afleiding in detail gegeven om alle aan-

names die bij deze afleiding worden gebruikt expliciet weer te geven. Vervolgens wordt

140 Samenvatting

de verkregen Smoluchowski vergelijking geformuleerd in termen van een equivalent

stelsel van stochastische differentiaalvergelijkingen (SDEs) waarbij de zogeheten Stra-

tonovich interpretatie wordt gehanteerd. M.b.v. dit stelsel van SDEs wordt allereerst

een rekstroming geanalyseerd van een oplossing van niet-omstrengelde semiflexibele

LCPs die hairpins bevatten. De resultaten van deze theoretische analyse geven aan dat

in deze limiet van niet-omstrengelde LCP ketens, de aanwezigheid van hairpins leidt

tot een verlaging van de reologische responsefuncties.

Verder is er een numerieke code ontwikkeld voor de tijdsintegratie van het stelsel van

SDEs in het algemenere geval van geconcentreerde oplossingen van semiflexibele LCPs

waarin hairpin-bevattende ketens met elkaar omstrengeld kunnen raken. Deze code

wordt allereerst getest door te laten zien dat deze de evenwichtseigenschappen van dit

soort LCP oplossingen goed kan voorspellen. Vervolgens worden de resultaten van

simulaties van een rekstroming in de limiet van niet-omstrengelde ketens vergeleken

met de resultaten van de eerder genoemde theoretische analyse. Het blijkt dat beide

resultaten goed met elkaar overeenkomen. Uit de resultaten van simulaties van om-

strengelde LCP oplossingen volgt echter dat entanglements tot een verhoging van de

reologische responsefuncties leiden. Dit effect treedt op zodra het gemiddelde aantal

hairpins per keten groter wordt dan 1.

Eveneens worden simulaties van het gedrag in afschuiving gepresenteerd. In een af-

schuifstroming vertoont het model niet alleen de bekende dynamische overgang in

dit soort systemen, maar worden ook de diverse dynamische regimes van de director

teruggevonden zoals ”kajakken”, ”waggelen” (wagging) en ”oplijning in de stroom-

richting” (flow-aligning). Uit deze simulaties kan ook de stationaire afschuifviscos-

iteit bepaald worden. Deze grootheid vertoont een plateau voor niet al te hoge af-

schuifsnelheden en laat afschuifverdunnend gedrag zien bij hoge afschuifsnelheden.

De grafiek stemt kwalitatief overeen met de bekende Asada-Onogi plot voor typische

LCP oplossingen. Afgezien daarvan, blijken de resultaten voor hoge en medium af-

schuifsnelheden qua orde van grootte overeen te komen met experimentele data.

Acknowledgments

During these four years as a PhD researcher I met a lot of colleagues and new friends

who made my PhD-life an enjoyable experience. Thanks to all of you for the positive

and important role you have played for me to finish this thesis.

First and foremost, I would like to thank my promotor prof. Han Slot for giving me the

opportunity to participate in such an interesting and challenging project at the CASA

group of the Eindhoven University of Technology, and also for his guidance, stimulating

support during the past four years and careful reviewing of this thesis. I also would

like to thank him for the comfortable work atmosphere, and the opportunity to visit

conferences and summer schools where I have met many people from the field of my

research.

I would like to express my sincere gratitude to the members of the promotion committee

including prof. Markus Hutter, prof. Paul van der Schoot, prof. Jaap Molenaar and

prof. Wim Briels, together with my supervisor prof. Han Slot, and the members of

the extended promotion committee including prof. Mark Peletier and prof. Stephen

Picken. I would like to thank them for the time devoted to reading my thesis and their

willingness to evaluate my work .

I am grateful for the support of my research from the Teijin company, and, especially, to

Hans Meerman and Erik Westerhof for many fruitful discussions.

What made these four years especially bright and enjoyable was the great working at-

mosphere within CASA group. I would like to express my gratitude to Lena Filatova

for helping me with translating the summary of this thesis into Dutch. I also thank dr.

Adrian Muntean, dr. Sorin Pop, prof. Jan de Graaf and prof. Mark Peletier for fruit-

ful conversations. I would like to thank my current and former colleagues for many

social events we have shared together, all of which made my life in Eindhoven spe-

cial. Thank you Patricio Rosen, Maria Ugryumova, Mirela Darau, Kundan Kumar, Sud-

hir Srivastava, Michiel Renger, Maxim Pisarenco, Valeriu Savcenco, Erwin Vondenhoff,

Martien Oppeneer, Volha Shchetnikava, Iason Zisis, Steffen Arnrich, Tasnim Fatima,

142 Acknowledgments

Andriy Hlod, Maria Rudnaya, Evgeniya Balmashnova, Bas van der Linden, Carlo Mer-

curi, Patrick van Meurs, Antonino Simone, Mayla Bruso, Giovanni Bonaschi Akshay

Iyer, Hans Groot, Qingzhi (Darcy) Hou, Roxana Ionutiu, Godwin Kakuba, Ali Etaati,

Agnieszka Lutowska, Corien Prins, Jan Willem Knopper, Mark van Kraaij, Shona Yu,

Fan Yabin, Rostyslav Polyuga, Badr Kaoui, Arpan Ghosh, David Bourne, Lucia Scardia,

Nicodemus Banagaaya and many more. I thank my office mates Paticio Rosen, Volha

Shchetnikava, Martien Oppeneer, Evgeniya Balmashnova, Arpan Ghosh, Andriy Hlod

and Tasnim Fatima for always keeping a nice and productive atmosphere in the office.

The scientific discussions with Patricio, combined with his perfect sense of humor, were

always a pleasure for me and he was always willing to help me with all kind of matters.

Our discussions with Iason, especially on explaining some Greek terms, were an ap-

preciated input for my work. I also want to thank Andriy Hlod and Erwin Vonderhoff

for the role they played in the integration of all PhD researchers and postdocs in CASA

when I joined the group. I am grateful for the help and assistance given to me by Enna

van Dijk and Marese Wolfs-van de Hurk who helped me with wide variety of admin-

istrative issues. Furthermore, I am thankful to Alexander Zimin, Gleb Pavlenko, Lena

Filatova and Patricio Rosen for providing me additional computational power in the

latest stages of my PhD research. I am also very grateful to my paranymphs Tamerlan

Saidov and Patricio Rosen who agreed to be with me during the defense ceremony.

I am infinitely grateful to my teachers of physics Pavel Viktor, Valery Koleboshin, Vadim

Manakin and Vladimir Kulinskiy for inspiring my interest in physics.

I highly appreciate and value the moral support of my friends in Odessa who made

me feel at home every time I visited my home-city Odessa, even if the visits were not

frequent. Thank you Alexey Kunitskiy, Gleb Pavlenko, Valentin Munitsa, Grygoriy

Fuchedzhy, Yuriy Skvortsov, Andrey Sokolov, Yuriy Turbovets, Irina Soloviova, Alex-

ander Syvorotka, Eugene Britavskiy and many more.

Family support is always very important for achieving any result. It was important for

me to feel this support from parents and from my sister Liudmila Matveichuk. Last,

but certainly not least, I would like to thank my dearest wife Lena Filatova who is very

important for me and who supported me every day as long as I remember myself. It is

to you that I dedicate this thesis.

Oleg Matveichuk

Eindhoven, March 2013.

Curriculum vitae

Oleg Igorevich Matveichuk was born on 14th October 1985 in Odessa, USSR. In 2003

he started his studies in I.I. Mechnikov Odessa National University (ONU), where he

completed a bachelor program in Theoretical Physics with excellence. In 2007 Oleg star-

ted a Master degree program in Theoretical Physics in I.I. Mechnikov Odessa National

University. In January 2008 he was awarded Pinchuk scholarship for the work ”Dimer-

ization in water vapor in the vicinity of the critical point”. He graduated with excellence

(Red Diploma) in July 2008. He wrote his master’s thesis entitled “Rectilinear diameter

as a sensitive characteristic of the phase equilibrium”, under the supervision of prof.dr.

V.L. Kulinskii.

From 2008 to 2012, he worked as a PhD researcher in the Eindhoven University of Tech-

nology in the Centre for Analysis Scientific Computing and Applications (CASA), under

the supervision of prof.dr. J.J.M. Slot. The results of this research are presented in this

dissertation.