Constitutive modeling of concentrated solutions of mainchain liquid crystalline polymers
Matveichuk, O.
DOI:
10.6100/IR750672
Published: 01/01/2013
Document Version
Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)
Please check the document version of this publication:
• A submitted manuscript is the author’s version of the article upon submission and before peer-review. There can be important differences
between the submitted version and the official published version of record. People interested in the research are advised to contact the
author for the final version of the publication, or visit the DOI to the publisher’s website.
• The final author version and the galley proof are versions of the publication after peer review.
• The final published version features the final layout of the paper including the volume, issue and page numbers.
Link to publication
Citation for published version (APA):
Matveichuk, O. (2013). Constitutive modeling of concentrated solutions of main-chain liquid crystalline polymers
Eindhoven: Technische Universiteit Eindhoven DOI: 10.6100/IR750672
General rights
Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners
and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.
• Users may download and print one copy of any publication from the public portal for the purpose of private study or research.
• You may not further distribute the material or use it for any profit-making activity or commercial gain
• You may freely distribute the URL identifying the publication in the public portal ?
Take down policy
If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately
and investigate your claim.
Download date: 16. Jun. 2017
Constitutive modeling of
concentrated solutions of
main-chain liquid crystalline
polymers
c
Copyright 2013
by Oleg Matveichuk, Eindhoven, The Netherlands.
All rights are reserved. No part of this publication may be reproduced, stored in a
retrieval 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 Aramid B.V.
A catalogue record is available from the Eindhoven University of Technology Library
ISBN: 978-90-386-3339-8
Constitutive modeling of
concentrated solutions of
main-chain liquid crystalline
polymers
PROEFSCHRIFT
ter verkrijging van de graad van doctor aan de
Technische Universiteit Eindhoven, op gezag van de
rector magnificus, prof.dr.ir. C.J. van Duijn, voor een
commissie aangewezen door het College
voor Promoties in het openbaar te verdedigen
op maandag 11 maart 2013 om 16.00 uur
door
Oleg Matveichuk
geboren te Odessa, Oekraı̈ne
Dit proefschrift is goedgekeurd door de promotor:
prof.dr. J.J.M. Slot
Contents
1
Introduction
1.1 Liquid crystals and liquid-crystalline polymers (LCPs) . . . .
1.2 Rod-like polymers and the Isotropic-Nematic phase transition
1.3 Hairpin defects in an LCP backbone . . . . . . . . . . . . . . .
1.4 Rheology of an LCP solution . . . . . . . . . . . . . . . . . . .
1.5 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Phase-space theory for LCPs
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . .
2.2 Model for a main-chain liquid crystalline polymer .
2.3 Hamiltonian for the ensemble of chains . . . . . . .
2.4 Smoluchowski equation . . . . . . . . . . . . . . . .
2.5 Stochastic differential equations for a polymer chain
2.6 Forces acting on a polymer chain . . . . . . . . . . .
2.7 Hairpins and entanglements . . . . . . . . . . . . . .
2.8 Dimensionless version of the evolution equations .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
1
. 1
. 4
. 7
. 10
. 15
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
17
17
18
20
23
27
30
36
40
3 Rouse-like model in the highly-ordered limit
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Elimination of fast variables . . . . . . . . . . . . . . . . . . .
3.3 Dynamics of a polymer chain . . . . . . . . . . . . . . . . . .
3.4 Normal modes expansion . . . . . . . . . . . . . . . . . . . .
3.5 Ensemble average behavior of the normal-mode coordinates
3.6 Evolution equation for the director . . . . . . . . . . . . . . .
3.7 Modified stress tensor . . . . . . . . . . . . . . . . . . . . . .
3.8 Uniaxial elongational flows . . . . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
47
47
48
53
56
58
59
61
62
4 Numerical simulations of semi-flexible LCPs
4.1 Introduction . . . . . . . . . . . . . . . . . .
4.2 Euler-Maruyama method . . . . . . . . . .
4.3 Values of the main parameters in the model
4.4 Algorithm . . . . . . . . . . . . . . . . . . .
4.5 Equilibrium properties of the system . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
71
71
73
76
80
84
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
vi
5
6
Contents
Rheology of entangled LCP solutions containing hairpins
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Linear rheology of an LCP solution . . . . . . . . . . . . . . . . . . . . . .
5.3 Evolution of the director in steady shear flow . . . . . . . . . . . . . . . .
5.4 Shear viscosity and the first normal stress difference in the steady shear
.
.
.
.
Conclusions
A Details of derivations
A.1 Evolution equation for the configurational distribution function
A.2 Evolution equation for the momentum-space averages . . . . .
A.3 Itô or Stratonovich interpretation . . . . . . . . . . . . . . . . . .
A.4 Derivation of the expression for θmax . . . . . . . . . . . . . . . .
A.5 Derivation of the expression for the stress tensor . . . . . . . . .
A.6 Normal modes expansion . . . . . . . . . . . . . . . . . . . . . .
A.7 Formal solution of a linear matrix differential equation . . . . .
A.8 Normal modes in the equilibrium state . . . . . . . . . . . . . . .
A.9 Free energy of the ensemble of chains . . . . . . . . . . . . . . .
87
87
88
95
100
107
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
111
111
112
115
117
119
121
123
125
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 benzoate 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 conventional 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 molecular 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 preserved 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 schematic picture of the ordering of molecules in the nematic phase is shown in fig. 1.1a. In
the smectic phases (from Greek ”σµηκτ ικoς” - 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 applications. 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., shortrange 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 dispersion 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
c
Kwolek [10]. PpPTA is commonly known by the brand names Twaron 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) solutions in general. Since 1965 the number of applications for yarns spun out of LCP solutions 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 process [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
b) Smectic A
a) Nematic
Figure 1.1: Schematic picture of molecular ordering in liquid crystalline mesophases.
a) sport equipment
c) tyre reinforcement
b) heat and cut protection
d) cable reinforcement
Figure 1.2: Applications of LCP.
e) body armor
Introduction
4
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 incorporated 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 further 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 persistence 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
l
effectively. If the ratio lpc ≪ 1, then the polymer is called flexible and it does not show the
ability to form a nematic phase. On the contrary, when
For the intermediate values of
lp
lc
lp
lc
≈ 1 the polymers are rod-like.
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 isotropicnematic (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 description 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
1
S = huui − I.
3
(1.1)
Here h·i 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 eigenvalue 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 mesophase 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.
Introduction
6
The tensor huui is symmetric by construction. This guarantees the possibility of its
diagonalization and the orthogonality of the eigenbasis, i.e.,
huui = λ1 nn + λ2 τ τ + λ3 ss
(1.2)
where n, τ , s form an orthonormal set of eigenvectors and λ1 , λ2 , λ3 are the corresponding eigenvalues associated with the tensor huui. 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 eigenvector 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 huui). Taking the dot product between
the director and the orientation vector of the rod and considering the average value of
its second Legendre polynomial hP2 i 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 hP2 i changes from 0 to 1 when the system transits from an isotropic
state to a perfectly aligned state. This gives
hP2 i =
3
(n · S · n) =
2
1
3
2
(n · u) −
2
2
.
(1.3)
The relation between hP2 i and λ1 is linear, namely
hP2 i =
3
1
λ − .
2 1 2
(1.4)
In the Doi model the interaction causing rods to align is treated by the nematic MaierSaupe potential. This potential originates from the anisotropic part of the long-range
attractive dispersion forces. The expression for the potential is
Un = −
allX
rods
m=1
1
H : um um − I ,
3
(1.5)
where um is the orientation of the m-th rod and H is the strength of the nematic potential. 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 = H0 S, 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 hP2 i on the inverse strength of the nematic field is depicted in fig. 1.4.
When HT is greater than approximately 0.15, hP2 i remains zero. But when HT becomes
0
0
slightly less than 0.15, hP2 i rapidly increases, reflecting the spontaneous alignment appearing in the system. This sharp growth of hP2 i 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 competing 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 restriction 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 completely flexible polymers. As the flexibility of the backbone increases more configurations 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-
Introduction
8
1.0
0.8
0.6
hP2 i
0.4
0.2
0.0
0.00
0.10
0.05
0.15
T
H0
LC
+
IS
T
IS
LC
Figure 1.4: Theoretical curve for the order parameter hP2 i as a function of the reduced inverse
nematic field strength for Doi’s rigid rod model with Maier-Saupe mean-field potential.
LC
+C
P
wt%
Figure 1.5: Part of a schematic phase diagram showing temperature T versus weight percentage
of the polymer present. There are three phases depicted in the diagram: an isotropic phase (IS), a
liquid crystalline phase (LC), and a crystalline polymer phase (CP).
1.3 Hairpin defects in an LCP backbone
a) non-hairpin state
9
b) hairpin state
Figure 1.6: Examples of possible configurations of the backbone of the chain: a) without a hairpin
defect, 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 predicts 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 associated 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 viscous. 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 solutions typically show viscoelastic behavior on macroscopic time scales, i.e., they have
much greater characteristic internal times than the low molecular weight liquid crystals. In many industrial applications, like fiber spinning, the deformation rates are
usually much higher than the inverse relaxation times of the LCPs. Therefore, to account for these internal relaxation processes becomes an important issue and the original Ericksen-Leslie-Parodi theory is not capable of capturing such processes. Furthermore, in equilibrium, a sample of LCP material usually contains a huge number of defects (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 relaxation 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 behavior. 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 unentangled 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 function 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 Maffettone 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 semiflexibility effects altogether. Some ways of treating the semi-flexibility of polymer backbones 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 approach 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 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 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 investigation 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 hairpins 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 steadystate 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 synchronization time, because this is the time required for the polymer chains to synchronize 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 therefore 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 observed. 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 maximum 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 oscillations 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
Region I
Region II
ln(η)
Region III
ln(γ̇)
Figure 1.7: Typical three-region plot for the steady-state viscosity as a function of the shear rate
for LCP solutions.
N1
Region I
Region III
0
γ̇
Region II
Figure 1.8: Schematic plot for the first normal stress difference as a function of shear rate for an
LCP 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 normal 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 experimental 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 tumbling 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 details 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 highlyaligned and unentangled chains containing hairpins. The general case and the nonlinear 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 evolution equations are derived from the basic principles of statistical mechanics. The
Smoluchowski equation for the ensemble of chains is derived from the Liouville equation. 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 explained. Then the equations are brought to a dimensionless form. The resulting dimensionless 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 unentangled highly aligned rod-spring chains containing hairpins. The evolution equation 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 EulerMaruyama 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 · 1023 mole−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 description 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 all N 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 BogoliubovBorn-Green-Kirkwood-Yvon hierarchy of equations (BBGKY hierarchy) [62–66]. Then
the BBGKY hierarchy is truncated. After truncation of the BBGKY hierarchy of equations 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 additional 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 systems. 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 mechanical 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 solution 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 −1 Nch blp2 > 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 similar 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 Gaussian 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 correlation 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 nematogens, 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 equipartition 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 involved 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-
Phase-space theory for LCPs
20
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
pri
J
φi
θi
pu i
pφ i
pθ i
mb
pb i
mass of a rod,
translational momentum of the i-th rod,
two-dimensional tensor of inertia of a rod,
azimuthal angle of the i-th rod,
inclination angle of the i-th rod,
generalized rotational momentum vector of the i-th rod,
azimuthal component of the generalized momentum of the i-th rod,
inclinational component of the generalized momentum of the i-th rod,
mass of a bead,
translational 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.
pu i = pφ i + pθ i
J=JI
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 cos φi i + sin θi sin φi j + cos θi k
(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
ui−1
bi−1
ui
bi
ri
ri+1
bi+1
O
Figure 2.1: Part of a chain near the i-th bead. Here ri ,ri+1 denote the positions of the adjoining
nematogens, bi−1 ,bi ,bi+1 denote the positions of the beads. O is the origin of the laboratory
reference frame.
z
pφ
u
θ
φ
pθ
pu
y
x
Figure 2.2: Illustration of the meaning of generalized momentum associated with the orientational degrees of freedom.
Phase-space theory for LCPs
22
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:
H
single
=
N
X
i=1
pu · J−1 · pui
p2ri
+ i
+ Urext
i
2mr
2
!
+
N
−1
X
i=1
p2bi
+ Ubexti
2mb
!
+ U intra
(2.2)
Here Urext
and Ubexti are the potential of the external field for the i-th rod and i-th bead
i
respectively. U intra is the intramolecular interaction potential, i.e., the potential of interactions between parts of the same chain. The Hamiltonian of an ensemble of chains is
given by the following expression.
H=
Nch
X
s=1
Hssingle + U inter
(2.3)
Here Hssingle is the Hamiltonian for the s-th chain and U inter is the intermolecular interaction 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)
∂H
∂
1
p
r =
=
∂t i
∂pri
mr ri
(2.4)
∂
∂H
1
p
b =
=
∂t i
∂pbi
mb b i
(2.5)
∂H
∂
u =
= J−1 · pui
∂t i
∂pui
(2.6)
∂H
∂
intra
inter
p =−
= Fext
ri + Fri + Fri ≡ Fri
∂t ri
∂ri
(2.7)
∂
∂H
intra
inter
= Fext
pb i = −
bi + Fbi + Fbi ≡ Fbi
∂t
∂bi
(2.8)
∂
∂H
intra
inter
= Fext
p = − (I − ui ui ) ·
ui + Fui + Fui ≡ Fui
∂t ui
∂ui
(2.9)
Here Fri , Fbi and Fui denote the total force acting on the i-th rod, the total force acting on the i-th bead and the total generalized angular force acting on the i-th rod respectively. Superscripts ext, intra and inter stand for contributions from external forces,
intramolecular interactions (between parts of the same chain) and intermolecular interactions (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
= −Lf
(2.10)
∂t
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
Nch
L=
N
N
−1
N
X X
X
X
1
∂
1
∂
∂
pusi · J−1 ·
prsi ·
+
pbsi ·
+
+
mr
∂rsi
mb
∂bsi i=1
∂usi
s=1
i=1
i=1
+
N
X
Frsi
i=1
N
−1
N
X
X
∂
∂
∂
Fbsi ·
Fusi ·
+
+
·
∂prsi
∂pbsi
∂pusi
i=1
i=1
!
(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 h·i
to denote an ensemble average.
Let B be a dynamical variable. Then from the Liouville equation (2.10) the general equation of change of B can be derived. It describes the evolution in time of the ensemble
average of the dynamical variable.
∂
hBi = hLBi
∂t
(2.12)
For instance, if we take the dynamical variable B to be
BΨ (y) =
Nch N
X
Y
s=1 i=1
δ (rsi − ri ) δ (usi − ui )
N
−1
Y
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.
Phase-space theory for LCPs
24
This step is described in Appendix A.1. We show here only the resulting equation.
X
N
−1
N
N
X
Jpri K
Jpbi K
∂ −1
∂
∂Ψ X ∂
·
Ψ +
· J · Jpui KΨ +
·
Ψ = 0 (2.15)
+
∂t i=1 ∂ri
mr
∂ui
∂bi
mb
i=1
i=1
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 averaged 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 momentumspace-averaged quantities: Jpri K, Jpui K and Jpbi K. 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 closure relations. Closure relations should give expressions for the momentum-space averaged quantities obtained in the n-th iteration in terms of quantities defined in lower
iterations. The equations for Jpri K, Jpui K and Jpbi K 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
(e)
(intra)
0 = F(b)
+ F(h)
ri + Fri + Fri
ri
(2.16)
(e)
(intra)
0 = F(b)
+ F(h)
ui + Fui + Fui
ui
(2.17)
(b)
(e)
(intra)
0 = Fbi + Fbi + Fbi
(h)
+ Fb i
(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].


N
p
−
m
v
K
J
p
−
m
v
X
rj
r
ri
r
∂ 
1 
·
Ψ +
Fr(b)
=
−
j
Ψ i=1 ∂ri
mr


N
−1
N
X
X
∂ −1
∂  J pbi − mb v prj − mr v K 
+
· J Jpui prj − mr v KΨ +
·
Ψ
∂ui
∂bi
mb
i=1
i=1

(2.19)
2.4 Smoluchowski equation
F(b)
uj
1
=−
Ψ
N
X
∂
·
∂ri
i=1
25
!
N
X
J pri − mr v puj K
∂ −1
Ψ +
· J Jpui puj KΨ +
mr
∂ui
i=1
!!
N
−1
X
J pb i − m b v pu j K
∂
+
·
Ψ
(2.20)
∂bi
mb
i=1



N
J pri − mr v pbj − mb v K
X
∂
1
(b)
·
Ψ +
Fbj = − 
Ψ i=1 ∂ri
mr


N
N
−1
J
p
−
m
v
p
−
m
v
K
X
X
b
b
b
b
∂
∂ 
i
j
+
· J−1 Jpui pbj − mb v KΨ +
·
Ψ 
∂ui
∂bi
mb
i=1
i=1
(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 − mr v pri − mr v K = mr T Iδij
Jpuj pui K = T Jδij
J pbj − mb v pbi − mb v K = mb T Iδij
(2.22)
(2.23)
(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 momentum space” approximation yields the following simplified expressions for the Brownian forces.
F(b)
rj = −T
∂
ln Ψ
∂rj
(2.25)
F(b)
uj = −T
∂
ln Ψ
∂uj
(2.26)
∂
ln Ψ
∂bj
(2.27)
(b)
Fbj = −T
Let us briefly recapitulate the procedure that we are following in order to get a closed
Phase-space theory for LCPs
26
equation for Ψ. We start from the evolution equation for the single-particle configurational distribution function (2.15). Then we derive similar equations (2.16)-(2.18) for
Jpri K, Jpui K, Jpbi K. These equations are again not closed. In order to make these equations closed we employ the ”equilibration in momentum space” approximation to compute the Brownian forces. The mesoscopic external forces and mesoscopic intramolecular forces allow direct computation because the microscopic external forces and the microscopic 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 twoparticle 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 interactions 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 systems ( [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 = −ζ ri · (Jṙi K − v)
(2.28)
F(h)
ui = −ζ ui · (Ju̇i K − (I − ui ui ) · ∇v · ui )
(2.29)
(h)
Fbj = −ζ bj · Jḃj K − v
(2.30)
Here ζ ri , ζ ui and ζ bi are 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 ζ ri , ζ bi and ζ ui on 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 approximation.
ζ ri = ζ Tri
ζ bi = ζ Tbi
ζ ui = 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 interactions. For a neutral polymer chain the long-range interactions are Van der Waals polarization 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 function after substituting the expressions for Jpri K, Jpui K, Jpbi K.
N
∂Ψ X ∂
(e)
(intra)
−1 ∂Ψ
+
·
v (ri ) + ζ −1
·
F
+
F
Ψ
−
T
ζ
·
+
ri
ri
ri
ri
∂t
∂ri
∂ri
i=1
N
X
∂
−1 ∂Ψ
(e)
(intra)
+
Ψ
−
T
ζ
·
·
(I − ui ui ) · ∇v (ri ) · ui + ζ −1
·
F
+
+
F
ui
ui
ui
ui
∂ui
∂ui
i=1
N
−1
X
∂
(e)
(intra)
−1
−1 ∂Ψ
= 0 (2.32)
·
v (bi ) + ζ bi · Fbi + Fbi
Ψ − T ζ bi ·
+
∂bi
∂bi
i=1
This Smoluchowski equation is the central equation describing the dynamics of a polymer chain in our model. This equation will be used in derivations of macroscopic properties 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 Itô or Stratonovich interpretations are also recapitulated in Appendix A.3.
In the previous section we have derived the evolution equation for a single-chain distribution function (2.32). In this section we formulate a system of SDEs that is equivalent
to (2.32).
Phase-space theory for LCPs
28
We rewrite terms containing the second order derivatives in the expression (2.32).
∂
∂Ψ
∂
∂ − 2T ∂ − 12 ∂
− 12
− 21
=
· ζ −1
·
·
ζ
·
ζ
·
ζ
·
ζ
=
(2.33)
·
Ψ
·
Ψ
ri
ri
ri
ri
ri
∂ri
∂ri
∂ri
∂ri
∂ri
∂ri
A similar derivation is made for ζ bi .
∂
∂ − 2T ∂
∂ − 12 ∂
∂Ψ
− 12
− 12
·
Ψ
=
·
Ψ
=
· ζ −1
·
·
ζ
·
ζ
·
ζ
·
ζ
bi
bi
bi
bi
bi
∂bi
∂bi
∂bi
∂bi
∂bi
∂bi
(2.34)
The set of possible values for ui forms a unit sphere. The set of possible values for the
derivatives u̇i forms the set of planes tangent to this unit sphere. Because we treat derivatives 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
(2.35)
ζ ui = ζrot (I − ui ui )
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.
3
2
A remark should be made about ζ −1
ui . The projector ζ ui acts from R to R . Therefore
ζ ui is the degenerated transformation. Then ζ −1
ui does not exist. In order to avoid this
−1
misunderstanding when writing ζ ui we actually mean the inverse of the operator ζ ui
restricted to the plane perpendicular to ui .
−1
ζ −1
ui = ζrot (I − ui ui )
(2.36)
Returning to equation (2.32), we transform the terms containing second order derivatives with respect to u.
∂
∂Ψ
∂Ψ
∂
−1
=
=
· ζ −1
·
·
ζ
(I
−
u
u
)
·
rot
i i
ui
∂ui
∂ui
∂ui
∂ui
∂
∂
−1
−1
=
· ζrot2 (I − ui ui ) ·
· ζrot2 (I − ui ui ) Ψ
=
∂ui
∂ui
1
∂
∂
− 21
2
· ζ−
·
ζ
·
Ψ
(2.37)
=
ui
∂ui
∂ui ui
2.5 Stochastic differential equations for a polymer chain
29
Employing results (2.33), (2.34), (2.37) in equation (2.32) gives
N
∂Ψ X ∂ (e)
(intra)
· v (ri ) + ζ −1
·
F
+
F
Ψ −
+
ri
ri
ri
∂t
∂ri
i=1
N
X
1
∂
∂ 12 − 12 − 12
2
−
· T ζ ri ·
· T ζ ri Ψ +
∂ri
∂ri
i=1
N
X
∂ (e)
(intra)
+
Ψ −
· (I − ui ui ) · ∇v (ri ) · ui + ζ −1
u i · Fu i + Fu i
∂ui
i=1
N
X
1
1
1
1
∂
∂
2
2 −2
· T 2 ζ−
·
T
−
·
ζ
Ψ
+
ui
ui
∂ui
∂ui
i=1
+
N
−1
X
i=1
∂ (e)
(intra)
Ψ −
· v (bi ) + ζ −1
·
F
+
F
bi
bi
bi
∂bi
N
−1
X
1 −1
∂
∂ 12 − 21 2
2
−
· T ζ bi ·
· T ζ bi Ψ
=0
∂bi
∂bi
i=1
(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 corresponding 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 quantities 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
dri
(e)
(intra)
r
= v (ri ) + ζ −1
+ ζ −1
(2.39)
ri · Fri + Fri
ri · fi
dt
dui
(e)
(intra)
u
(S)
= (I − ui ui ) · ∇v (ri ) · ui + ζ −1
+ ζ −1
(2.40)
ui · Fui + Fui
ui · fi
dt
dbj
(e)
(intra)
b
+ ζ −1
(2.41)
= v bj + ζ −1
·
F
+
F
(S)
bj · fj
bj
bj
bj
dt
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 , f u and f b 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
f r , f u and f b
r
fi (t) fir′ t′ = 2ζ ri T δii′ δ t − t′
(2.42)
u
fi (t) fiu′ t′ = 2ζ ui T δii′ δ t − t′
(2.43)
(S)
Phase-space theory for LCPs
30
D
fjb (t) fjb′ t′
E
= 2ζ bj T δjj ′ δ t − t′
(2.44)
These relations are the analogs of the formal property of the Wiener process
dW ′ dW
= 2δ t − t′
(t)
t
dt
dt
(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) supplemented 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 19581960 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) = −Un S :
I
uu −
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 X
i=1
ui ui −
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 − 3I . 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 − 3I .
The non-linearity is introduced by claiming the self-consistency condition: the mean of
uu − 3I computed with the nematic potential (2.46) should coincide with S. This selfconsistency 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 meanfield 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 Hütter at al. [75] the modification of the linear response theory for meanfield approximations is discussed. It turns out that introducing mean-field approximations 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.
(e)
(e)
In the equations (2.39), (2.40) and (2.41) the terms F(e)
ri , Fui and Fbj 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
chain
UMS
= −Un
F(e)
ui = −
N
X
i=1
S:
I
ui ui −
3
∂ chain
U
= 2Un S · ui
∂ui MS
(2.48)
(2.49)
(e)
The terms F(e)
ri and Fbj are equal to zero, because the nematic interaction contributes
only to the orientational degrees of freedom, but not to the translational degrees of freedom 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 potential 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) connectors [77]. He chose the following force-extension law for the FENE connector

k R

− oR·R if |R| ≤ lmax ,
1− 2
(2.50)
Fel (R) =
lmax


−∞
otherwise
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 Grün [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
(intra)
means the expressions for the intramolecular forces F(intra)
, F(intra)
and Fbi
will
ri
ui
follow from (2.50). In the region |R| ≤ lmax the potential for the FENE force law can be
constructed
2
ko lmax
R·R
Uel (R) =
ln 1 − 2
(2.51)
2
lmax
2.6 Forces acting on a polymer chain
33
Then the elastic energy of the rod-spring-bead chain is given by
ch
Uel
=
N
−1
X
i=1
X
N
1
1
Uel bi−1 − ri + lui
Uel bi − ri − lui +
2
2
i=2
(intra)
The expressions for the intramolecular forces F(intra)
, F(intra)
and Fbj
ri
ui
by differentiation of
ch
Uel
(2.52)
are obtained
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.
N
−1
X
U b (ui · ui+1 )
(2.53)
Ubend =
i=1
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 persistence 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 potential. 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 equations (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. Instead 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 comparable 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
Phase-space theory for LCPs
34
Bocquet et al. [80] computed the friction coefficients for a hard spherical particle submerged 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 system 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
ζ ri = ζ r (ui ) = ζk ui ui + ζ⊥ (I − ui ui )
(2.54)
ζ ui = ζ u (ui ) = ζrot (I − ui ui )
ζ bj = ζbead hj I
(2.55)
(2.56)
Here hj ≡ 21 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., ζk ≪ 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)
1
1
dri
r
+
r
−
+
F
+
= κ · ri + ζ −1
(u
)
·
F
lu
,
b
lu
,
b
i
i
el
i
el
r
dt
2 i i−1
2 i i
r
+ ζ −1
r (ui ) · fi
(S)
(S)
dui
= (I − ui ui ) ·
dt
−1
ζrot
! !
l
+
2
−1
−1 u
(2.59)
+κ · ui + 2ζrot
Un S · ui + ζrot
fi
lui
ri +
,b
2 i
Fel
dbj
−1
Fel
hj
= κ · bj + ζbead
dt
!
l
bj , rj + uj
2
l
− Fel
2
lui
ri −
,b
2 i−1
!
!!
l
bj , rj+1 − uj+1
+
2
−1
+ ζbead
hj fjb (2.60)
+ Fel
For i = 1 and i = N
(S)
(S)
dr1
dt
= κ · r1 +
du1
= (I − u1 u1 ) ·
dt
ζ −1
r
1
r
(u1 ) · Fel r1 + lu1 , b1 + ζ −1
r (u1 ) · f1
2
κ · u1 +
(2.58)
−1
ζrot
Fel
lu1
r1 +
, b1
2
!
(2.61)
l
+
2
+2Un S · u1 + f1u )) (2.62)
(S)
(S)
1
drN
r
−1
= κ · rN + ζ r (uN ) · Fel rN − luN , bN −1 + ζ −1
r (uN ) · fN
dt
2
duN
= (I − uN uN ) ·
dt
κ · uN −
−1
ζrot
Fel
luN
rN −
, bN −1
2
!
(2.63)
l
+
2
u
+2Un S · uN + fN
)) (2.64)
This system of SDEs represent the rod-spring-bead model presented in this thesis.
Phase-space theory for LCPs
36
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 formation 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 orientation 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 diagnosed 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
2
j ∈ {1, . . . , N − 1}
(2.65)
Defined in this way hj gives zero, when the consecutive rods point in the same direction, 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 question 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 with
a stiff backbone
z
θ
φ
y
x
UMS
Un
Figure 2.4: Spherical plot of the Maier-Saupe potential for the case when maximum eigenvalue
of huui is equal to 0.8
Phase-space theory for LCPs
38
a)
b)
Figure 2.5: Illustration of the role of the hairpin defects in the chain’s backbone in the formation
of entanglements: a) aligned chains without hairpins can not entangle, b) aligned chains can
entangle 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
rot
exp( UTn ), and it rapidly inbackbone. This first time-scale can be estimated as τnem
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 excluded 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 defects, 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 translational 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
s
2T
(2.66)
sin θmax =
(3λ − 1) Un
Here λ is the maximum eigenvalue of the matrix huui. The derivation of this formula is
given in Appendix A.4. For example, if UTn = 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, entanglements 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 entanglement 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 bj
become 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” hcrit
couples 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.
Phase-space theory for LCPs
40
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
Time of evolution
Characteristic time of process
Length of a rod
Force of elasticity of a spring
Elasticity coefficient of a spring
Strength of the nematic Maier-Saupe potential
Rod center mass position vector
Rod orientation vector
Bead center mass position vector
Orientation (order) tensor
Velocity gradient (Deformation rate)
Temperature
Friction tensor for a rod
Rotational friction for a rod
Friction coefficient for a bead
Symbol
t
τ
l
Fel
ko
Un
r
u
b
S
κ
T
ζ
ζrot
ζbead
SI units
s
s
m
N
N/m
J
m
m
s−1
J
N·s/m
J·s
N·s/m
Basic units
s
s
m
kg m s−2
kg s−2
kg m2 s−2
m
m
s−1
kg m2 s−2
kg s−1
kg m2 s−1
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 · 10−23 J/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 different. 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
l
t
t̃ =
τ
Un
Ũn =
T
s
τ −1 r
ζ 2 · f (t)
f̃ r̃ t̃ =
2T
s
τ − 12 u
f̃ u t̃ =
ζ f (t)
2T rot
s
τ − 12 b
ζ
f (t)
f̃ b̃ t̃ =
2T bead
b̃ =
r̃ =
b
l
κ̃ = κτ
F̃el (r̃) =
Fel (r)
ko l
ζ̃ r (u) =
ζ r (u)
ζk
ζ̃rot =
ζ̃bead =
ζrot
ζk l2
ζbead
ζk
Then the set of evolution equations for i ∈ {2, . . . , N − 1} and j ∈ {1, . . . , N − 1} becomes
(S) dr̃i =
−1
αtr
spr ζ̃ r
(ui ) ·
u
F̃ r̃i − i , b̃i−1 + F̃el
2
el
ui
r̃i + , b̃i
2
!!
− 21
+ κ̃ · r̃i dt̃ + αtr
dif f ζ̃ r
dt̃+
(ui ) · dWir̃
u
u
el
r̃i + i , b̃i − F̃el r̃i − i , b̃i−1 dt̃ +
(S) dui = (I − ui ui ) · αrot
spr F̃
2
2
rot
u
+κ̃ · ui dt̃ + αrot
nem S · ui dt̃ + αdif f dWi
(S) db̃j =
−1
αtr
spr ζ̃bead
hj
el
F̃
uj
b̃j , r̃j +
2
(S)
dr̃1 =
el
+ F̃
uj+1
b̃j , r̃j+1 −
2
!!
el
(u1 ) · F̃
u1
r̃1 + , b̃1
2
!
− 12
dt̃ + κ̃ · r̃1 dt + αtr
dif f ζ̃ r
(2.68)
dt̃+
− 21
b̃
+ κ̃ · b̃j dt̃ + αtr
dif f ζ̃bead hj dWj
For i = 1 and i = N
−1
αtr
spr ζ̃ r
!
(2.67)
(2.69)
(u1 ) · dW1r̃
(2.70)
Phase-space theory for LCPs
42
u
el
r̃1 + 1 , b̃1 dt̃ +
(S) du1 = (I − u1 u1 ) · αrot
spr F̃
2
rot
u
+κ̃ · u1 dt̃ + αrot
nem S · u1 dt̃ + αdif f dW1
(S) dr̃N =
−1
αtr
spr ζ̃ r
el
(uN ) · F̃
uN
, b̃N −1
r̃N −
2
!
(2.71)
dt̃+
+ κ̃ · r̃N dt̃ + αtr
dif f ζ̃
− 12
r̃
(uN ) · dWN
uN
el
,
b̃
r̃
−
(S) duN = (I − uN uN ) · −αrot
F̃
N −1 dt̃ +
N
spr
2
rot
u
+κ̃ · uN dt̃ + αrot
nem S · uN dt̃ + αdif f dWN
(2.72)
(2.73)
The elements dWr̃ ,dWu and dWb̃ are the increments of the 3D-Wiener process corresponding to the time-step dt̃. Expressions for the dimensionless groups in equations
(2.67)-(2.73) are given below
αtr
spr =
ko τ
ζk
αrot
spr =
αtr
dif f
v
u
u 2T τ
=t 2
ζk l
ko l2 τ
2ζrot
αrot
nem =
αrot
dif f =
s
2T τ
ζrot
2Un τ
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
tr
rot
rot
rot
by equating each of the coefficients αtr
spr , αdif f , αspr , αnem and αdif f 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 meaning. Such a parameter is the average relative amplitude of the spring length fluctuations
à 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., Ã ≪ l. Therefore, to express the relation between ko
and à we can simplify the expression (2.50) to a corresponding Hookean potential.
3
ko A2
= T
2
2
=>
ko =
3T
2
A
=
3T
Ã2 l2
(2.75)
2.8 Dimensionless version of the evolution equations
43
Description of the characteristic relaxation time
Symbol
Combination
ζk
ko
Relaxation time of translational motion of a rod due to the
presence of the springs
tr
τspr
Relaxation time of translational motion of a rod due to
thermal fluctuations
tr
τdif
f
Relaxation time of rotational motion of a rod due to the
presence of the springs
rot
τspr
Relaxation time of rotational motion of a rod due to the
nematic field
rot
τnem
ζrot
2Un
Relaxation time of rotational motion of a rod due to
thermal fluctuations
rot
τdif
f
ζrot
2T
ζk l2
2T
2ζrot
ko l2
Table 2.2: Time scales of the model
In the limit of a strong nematic potential, which is likely to occur for a highly concentrated solution of semi-flexible polymers, the characteristic relaxation time of the orientations 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.
rot
Therefore we choose the characteristic time τ to be τnem
. When expressions for ko and
rot
τnem (2.75) are substituted into (2.74) we get
αtr
spr =
3ζ̃rot
αrot
spr =
2
2Ũn Ã
αtr
dif f
=
s
ζ̃rot
Ũn
3
αrot
nem = 1
4Ũn Ã2
αrot
dif f
=
s
1
Ũn
(2.76)
Here ζ̃rot , Ũn and à are the parameters described in table 2.3.
From expressions (2.76) some qualitative conclusions can be already drawn. For example, the change of the concentration at the same temperature corresponds to the
change of strength of the nematic field Ũn . The increase of Ũn makes αrot
nem to be a domtr
rot
inant term, which means nematic ordering. The decrease of Ũn makes αtr
spr , αdif f , αspr
and αrot
dif f 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 ζ̃ r and ζ̃bead
determine 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
Phase-space theory for LCPs
44
Description
Symbol
Dimensionless nematic field strength
Ũn
Relative amplitude of the thermal fluctuations of the
spring
Ã
Relative maximum length of the springs
˜lz
Asymmetry of translational friction coefficient
ξtr
Ratio between rotational and translational frictions
ζ̃rot
Combination
Un
sT
3T
ko l2
lz
l
ζ⊥
ζk
ζrot
ζk 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 nondimensionalized by the the pressure of an ideal gas under equivalent conditions.
σ̃ =
V
σ
Nch T
(2.77)
Employing relation (2.75) we obtain the dimensionless expression for the stress tensor.
σ̃ = −
3
Ã2
"N
XD
i=1
F̃r̃i r̃i
E
#
−1 D
N
E
E NX
1 XD
F̃b̃i b̃i
F̃ui ui +
+
2 i=1
i=1
(2.78)
Here F̃ri , F̃ui and F̃bi are 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 nondimen2
sionalized by ko l. The rotational forces Fui are nondimensionalized by ko2l . The factor
1
l
2 appears because the radius of the sphere generated by a rotating rod of length l is 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 fiu may produce a non-isotropic contribution if S is anisotropic.
N
−1
X
i=1
hfiu ui i = −3N T S
(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 entangled 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 contour 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.
N
∂Ψ X ∂
(e)
(intra)
−1 ∂Ψ
+
·
v (ri ) + ζ −1
·
F
+
F
Ψ
−
T
ζ
·
+
ri
ri
ri
ri
∂t
∂ri
∂ri
i=1
N
X
∂
(e)
(intra)
−1 ∂Ψ
+
·
(I − ui ui ) · ∇v (ri ) · ui + ζ −1
·
F
+
F
Ψ
−
T
ζ
·
+
ui
ui
ui
ui
∂ui
∂ui
i=1
N
−1
X
∂
(e)
(intra)
−1 ∂Ψ
−1
+
Ψ − T ζ bi ·
·
v (bi ) + ζ bi · Fbi + Fbi
= 0 (3.1)
∂bi
∂bi
i=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 distribution function Ψ (r, u, b) can be represented by the product of the distribution function for the slow variables and the distribution function for the fast variables conditioned 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 variables 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 conditional 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 contribution to the stress tensor of 12 VT I 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
Y
i=1
1
1
δ bi − (ri + ri+1 ) + (ui − ui+1 )
2
4
(3.3)
The evolution equation for Φ (r, u, t) is obtained by substituting (3.3) into (3.2), substituting (3.2) into (3.1) and then integrating (3.1) with respect to beads variables. The last
Rouse-like model in the highly-ordered limit
50
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 replacing bi by 21 (ri + ri+1 ) + 41 (ui − ui+1 ). For the forces and the friction tensors we use
the expressions suggested in Chapter 2.
N
∂Φ X ∂
−1 ∂Φ
el
Φ
−
T
ζ
·
+
·
v (ri ) + ζ −1
·
F
+
ri
ri
ri
∂t
∂ri
∂ri
i=1
N
X
∂
−1
nem
el
−1 ∂Φ
·
(I − ui ui ) · ∇v (ri ) · ui + ζ ui · Fui + Fui Φ − T ζ ui ·
=0
+
∂ui
∂ui
i=1
(3.4)
el
Here Fel
ri and Fui are the elastic forces arising from the spring potential after elimination 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 Itô or Stratonovich interpretation does make any difference. But for the consistency we will stick to the Stratonovich
interpretation in this chapter. For i ∈ {1, . . . , N }
dri
el
−1
r
= κ · ri + ζ −1
r (ui ) · Fri + ζ r (ui ) · fi
dt
dui
−1 el
−1
−1 u
= (I − ui ui ) · κ · ri + ζrot
Fui + 2ζrot
Un S · ui + ζrot
fi
dt
(S)
(S)
(3.5)
(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 N 2 (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 variables u 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 represented 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.
exp − U(u,r)
T
φ (u; r) = R
U(u,r)
du
exp − T
(3.8)
Then we substitute (3.7) into (3.4) and the perform integration with respect to all orientational degrees of freedom u.
Z
N
∂ψ X ∂
∂ ln φ
el
φdu+
· ψv (ri ) + ψ ζ −1
·
F
+
−
T
ri
ri
∂t
∂ri
∂ri
i=1
Z
∂ψ
−1
= 0 (3.9)
−T ζ ri φdu ·
∂ri
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 represents the contribution from the orientational Brownian motion of the rods. We recall
the expression for the elastic force
Fel
ri = −
∂U
∂ri
(3.10)
and employ expression (3.8) to obtain
Fel
ri − T
∂ ln φ
∂
=
∂ri
∂ri
Z
U (u, r)
du
T ln exp −
T
(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 gradient of some potential. This potential can be seen as the free energy of a single chain, that
we denote by F ch
Z
U (u, r)
ch
du
(3.12)
F = −T ln exp −
T
Rouse-like model in the highly-ordered limit
52
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
R −1
ζ ri φdu. 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
Z
hui ui i = ui ui φdu ≈ 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 ui ui . This closure relation is used to estimate the integral for the averaged
mobility
Z
1
1
nn +
(I − nn)
(3.14)
ζ −1 (n) = ζ −1
ri φdu ≈
ζk
ζ⊥
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
N
X
∂
∂ψ
+
·
∂t
∂ri
i=1
ψv (ri ) + ψζ
−1
(n) ·
∂F ch
−
∂ri
!
− Tζ
−1
∂ψ
(n) ·
∂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 Itô or the Stratonovich interpretation. 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 mobility tensor ζ −1 (n) has no direct dependence on r, there is no difference whether we
choose the Itô or the Stratonovich interpretation in that case.
(S)
∂ri
∂ ch
F (r1 , ..., rN ) + fi (t)
= v (ri ) − ζ −1 (n) ·
∂t
∂ri
(3.16)
Here i ∈ {1, ..., N } and the Brownian force fi (t) satisfies the relations
hfi (t)i = 0
(3.17)
3.3 Dynamics of a polymer chain
53
fi (t)fi′ (t′ ) = 2δi,i′ ζ(n)T δ 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 entanglements. 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 =
N
1 X
r
N i=1 i
(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.
Rouse-like model in the highly-ordered limit
54
n
l
ci
ri
ri+1
H
Figure 3.1: The part of a chain in the neighborhood of the i-th nematogen. Here ci is the vector
connecting centers of the i-th and (i + 1)-st nematogen, O is the origin of the laboratory-fixed
frame 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
Z
U (u, r)
exp −
du
T
(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
a.
55
um
um+1
n
um
b.
um+1
Figure 3.2: a. The normal state. um and um+1 are pointing in the same direction. b. The
hairpin state. um and um+1 are pointing in the opposite directions.
hairpins has the following structure [87].
F
ch
N
−1
X
1
= w0
(r
− rm ) · K0 · (rm+1 − rm ) +
2 m=1 m+1
+
N
−1
X
1
w1
(r
− rm − ln) · K1 · (rm+1 − rm − ln)
2 m=1 m+1
(3.22)
Here w0 is the fraction of the hairpin states, and w1 = 1 − w0 . K0 = k0 I is the ”elasticity”
matrix for the hairpin state and K1 = k0 nn + k1 (I − nn) is the ”elasticity” matrix for
the normal state. The anisotropy is due to the nematic interaction of the nematogens.
k1 =
k0 H0
H0 + k0 l2
(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.
Rouse-like model in the highly-ordered limit
56
Using expression (3.22) for the free energy of the chain F ch , the gradients
readily obtained by the straightforward differentiation.
∂F ch
= − (K · (r2 − r1 ) − w1 lK1 · n)
∂rs
∂F ch
= −K · (rm+1 + rm−1 − 2rm )
∂rs
∂F ch
= − (K · (rN −1 − rN ) + w1 lK1 · n)
∂rs
(s = 1)
ch
∂F
∂rs
may be
(3.24)
(s ∈ {2, . . . , N − 1})
(s = N )
(3.25)
(3.26)
Here K denotes w0 K0 + w1 K1 .
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 ) − w1 lK1 · n + f1 (t)
(S) ζ (n)·(rs − κ · rs ) = K·(rm+1 + rm−1 − 2rm )+ fs (t)
(s = 1)
(3.28)
(s ∈ {2, ..., N − 1}) (3.29)
(S) ζ (n) · (rN − κ · rN ) = K · (rN −1 − rN ) + w1 lK1 · n + fN (t) (s = N )
(3.30)
In here the stochastic forces satisfy the following conditions.
hfs (t)i = 0
(3.31)
fs (t)fs′ (t′ ) = 2δs,s′ ζ(n)T δ 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 sequence 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 motions of the rod-spring chain. Because these equations are independent, the coordinates
qm are called normal modes.
(S) q̇m − κ · qm = −λm τ −1 · qm + αm w1 lτk n + vm (t)
(3.33)
−1
τ −1 (n) = ζ −1 (n) · K(n) = τk−1 nn + τ⊥
(I − nn)
(3.34)
vm (t) = ζ −1 (n) · hm (t)
(3.35)
hvm i = 0
vm (t)vm′ (t′ ) = 2ζ −1 (n)λm T δm,m′ δ t − t′
(3.36)
(3.37)
We now write these equations in a dimensionless form by the following scaling:
qm
l
t
t̃ =
τk
ṽm (t̃) =
q̃m =
τk
v (τ t̃)
l m k
κ̃ = κτk
τ̃ −1 = τ −1 τk
τ
τ= ⊥
τk
2T
Θ=
k0 l2
ζ̃
−1
= ζ −1 ζk
ζ
ζ= ⊥
ζk
The characteristic time scale used for non-dimensionalization is τk , which is the characteristic 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)
with
and
d
q̃p = − λp τ̃ −1 − κ̃ · q̃p + αp w1 n + ṽp t̃
dt
D
ṽp t̃ = 0
E
−1
ṽp t̃ ṽp′ t̃′ = ζ̃ Θλp δp,p′ δ t̃ − t̃′
(3.38)
(3.39)
(3.40)
If the director dynamics is prescribed, then these equations allow us to compute the q̃p .
From the internal dynamics of the polymer chain, the macroscopic properties and, in
Rouse-like model in the highly-ordered limit
58
particular, the stress tensor of the LCP solution can be deduced. Equation (3.38) is a
linear differential equation for q̃p and can be formally solved if κ̃ t̃ ,n t̃ and ṽp t̃ are
given.
3.5 Ensemble average behavior of the normal-mode coordinates
In the current section we present the solution of equation (3.38) and then deduce from
it the ensemble averages q̃p (t̃) and q̃p (t̃)q̃p (t̃) . The mathematical basis of this approach is described in Appendix A.7.
First introduce
Ap (t̃) ≡ − λp τ̃ −1 (t̃) − κ̃(t̃)
ap (t̃) ≡ αp w1 n(t̃) + ṽp (t̃)
(3.41)
(3.42)
and a matrix Mp (t) satisfying the initial-value problem:

 Ṁp (t̃) = Ap (t̃) · Mp (t̃)

(3.43)
Mp (0) = I
Then the solution of (3.38) is given by (A.79)
q̃p (t̃) = Mp (t̃) · q̃p (0) + Mp (t̃) ·
Zt̃
0
ds M−1
p (s) · ap (s)
(3.44)
Averaging (3.44) then gives
q̃p (t̃) = Mp (t̃) · q̃p (0) + αp w1 Mp (t̃) ·
Zt̃
0
ds M−1
p (s) · n(s)
(3.45)
By averaging the dyadic product of (3.44) with itself we get the expression for q̃p (t̃)q̃p (t̃) .
T
q̃p (t̃)q̃p (t̃) = q̃p (t̃) q̃p (t̃) + Mp (t̃) · Bp (t̃) · Mp (t̃)
(3.46)
3.6 Evolution equation for the director
59
where Bp (t̃) satisfies the following initial-value problem.

T
−1

−1
−1

 Ḃp (t̃) = Θλp Mp (t̃) · ζ̃ (t̃) · Mp
(3.47)


 B (0) = q̃ (0)q̃ (0) − q̃ (0) q̃ (0)
p
p
p
p
p
These three quantities q̃p (t̃) , q̃p (t̃)q̃p (t̃) and Bp (t̃) will be required to compute the
stresses.
Now let us see which q̃p (0) and q̃p (0)q̃p (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 q̃p (0) and q̃p (0)q̃p (0)
should be determined in equilibrium state. This means we have to derive expressions
for q̃p eq and q̃p q̃p eq from (3.45) and (3.46) in the limit t → ∞ with κ(t) = 0. In
Appendix A.8 the expressions for q̃p eq and q̃p q̃p eq are obtained. They are given by
(A.83) and (A.87). It follows that if the system was in equilibrium at moment t̃ = 0, then
Bp (0) = Θ τ̃ · ζ̃
−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 q̃p (t̃) and q̃p (t̃)q̃p (t̃) . However, the evolution of the director is itself determined by the evolution of the conformations of the chains, i.e., by the ensemble
averages q̃p (t̃) , q̃p (t̃)q̃p (t̃) possibly by even higher order moments. Therefore in the
next section we will focus on establishing the missing link between the evolution of n(t̃)
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 q̃p (t̃) ,
q̃p (t̃)q̃p (t̃) evolve, we can consider the system to be in an equilibrium state with respect 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
Rouse-like model in the highly-ordered limit
60
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
X
Fich
(3.49)
i=1
We will assume that all chains have an equal number of nematogens and Nch ≫ 1. Then
the law of large numbers allows us to write (3.49) as
E
D
Fsys = Nch F ch
(3.50)
It is convenient to make Fsys dimensionless via
F̃sys =
Fsys
Nch 21 k0 l2
(3.51)
Using (3.22) and results (A.95),(A.96),(A.97) from, Appendix A.9 then gives us
F̃sys = 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 constraint 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
= 2 (Λ + µI) · n − p
(3.54)
∂n
∂L
= n·n−1
(3.55)
∂µ
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
−1
(Λ + µI) · p
2
(3.57)
From Sylvester’s criterion [88] it follows that, provided the eigenvalues of Λ are positive, 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
q̃p (t̃)q̃p (t̃) and q̃p (t̃) . 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
σ̃ = −
3
Ã2
"N
XD
i=1
F̃r̃i r̃i
E
#
−1 D
N
E
E NX
1 XD
F̃b̃i b̃i
F̃ui ui +
+
2 i=1
i=1
(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
*
+
N
Nch X ∂F ch (t)
σ(t) =
ri (t)
(3.59)
V i=1
∂ri
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) = −Pid N I +
N −1 w α
2Pid X
K · qp (t) qp (t) − 1 m n qp (t)
Θ m=1
λm
(3.60)
Rouse-like model in the highly-ordered limit
62
where
Nchains T
(3.61)
V
The non-dimensionalization of the stress tensor is done in the same way as in expression
(A.50), i.e.,
σ
(3.62)
σ̃ =
Pid
Pid =
and thus
where
N −1 w1 αm 2 X
K̃ · q̃p (t̃)q̃p (t̃) −
n q̃p (t̃)
σ̃(t̃) = −N I +
Θ m=1
λm
(3.63)
K̃ = τ̃ −1 · ζ̃
(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 homogeneous 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 discussed 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 orientation 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 constant vector. The way to solve (3.38) is described in Appendix A.7. Then we use the
resulting expressions for q̃p (t̃) and q̃p (t̃)q̃p (t̃) 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) considerably.
Uniaxial elongational flow is characterized by one scalar function ǫ̃˙ (t) ≥ 0 and the
velocity gradient tensor has the following diagonal structure

˙ ǫ̃ t̃
0
0
κ̃ t̃ =  0
0 
− 21 ǫ̃˙ t̃
0
0
− 21 ǫ̃˙ t̃
(3.65)
Because of the absence of off-diagonal components in κ̃ t̃ 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.


σ̃k t̃
0
0
σ̃ t̃ = −p̃ t̃ I +  0
0 
σ̃⊥ t̃
0
0
σ̃⊥ t̃
(3.66)
The relevant rheological quantity is the so-called tensile stress σ̃T
σ̃T ≡ σ̃k − σ̃⊥
(3.67)
A non-zero tensile stress expresses the anisotropy of the stress tensor (3.66). For small
deformation rates, i.e., ǫ̃˙ (t) N 2 ≪ 1, the tensile stress can be represented by an integral
involving the response to the deformation from all previous moments of time
σ̃T t̃ =
Zt̃
−∞
Ẽ t̃ − t̃′ ǫ̃˙ t′ dt̃′
(3.68)
Substitution of (3.45) and (3.46) into (3.63) and the explicit evaluation of the matrices
Mp t̃ and Bp t̃ leads to an expression for the response function Ẽ t̃ . After shifting
Rouse-like model in the highly-ordered limit
64
the initial moment of time to −∞ the following expression results
Ẽ t̃ =
N
−1
X
p=1
!
2
2 αp w1
t̃
+
exp −λp t̃
2 exp −2λp t̃ + exp −2λp
τ
Θ
λp
(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 Ẽ t̃ . But before doing that we have to determine the values for
the parameters of the model τ , Θ, ζ, i.e.,
ζ=
τ=
ζ⊥
ζk
τ⊥
ζ kk
ζ
k0
= ⊥
= ⊥
τk
ζk k⊥
ζk w k + w
0 0
1
and
Θ=
2T
k0 l2
(3.70)
2
k0 l H0
2
k0 l +H0
(3.71)
(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 Ũn , Ã, ˜lz , ξtr , ζ̃rot . The detailed explanation 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 Ũn , Ã, l̃z , ξtr , ζ̃rot . The relation between the parameters is given by the following
relations
3 + 2Ũn Ã2
(3.73)
τ = ξtr
3w0 + 2Ũn Ã2
ζ = ξtr
Θ=
4 2
Ã
3
(3.74)
(3.75)
Taking à = 0.1, ξtr = 2 and Ũn = 25 the values Θ and ζ are computed straightforward
and the parameter τ is expressed as a function of the hairpin fraction w0 . The dependence of log10 Ẽ as function of hairpin fraction w0 and time t̃ 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 Ẽ. The character of this dependence is clearly shown in fig. 3.4. If the fraction 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
log10 Ẽ
−5
0
0.0
0.2
50
t̃
w0
0.4
100
Figure 3.3: 3D-Plot of log10 Ẽ versus hairpin fraction w0 and time t̃. The values of the parameters are à = 0.1, ξtr = 2, Ũn = 25 and N = 5.
10
Ẽ
0.1
w0 = 0.0
w0 = 0.1
0.001
w0 = 0.2
w0 = 0.3
10-5
w0 = 0.4
10
20
30
40
50
60
70
w0 = 0.5
t̃
Figure 3.4: Plot of Ẽ as function of time t̃ for different fractions of hairpins w0 . The values of
the parameters are à = 0.1, ξtr = 2, Ũn = 25 and N = 5.
Rouse-like model in the highly-ordered limit
66
storage and loss moduli are introduced.
′
Ẽ = ω̃
Z∞
Ẽ = ω̃
Z∞
Ẽ t̃ sin ω̃ t̃ dt̃ =
0
′′
Ẽ
φ̃
ω̃
!
φ̃
ω̃
!
0
Ẽ t̃ cos ω̃ t̃ dt̃ =
0
Z∞
Z∞
Ẽ
0
sin φ̃ dφ̃
(3.76)
cos φ̃ dφ̃
(3.77)
By substituting (3.69) into (3.76) and (3.77) and performing the integration we obtain
the explicit expression for Ẽ ′ and Ẽ ′′ .
′
Ẽ (ω̃) =
N
−1
X
p=1
′′
Ẽ (ω̃) =
N
−1
X
p=1
2ω̃ 2
τ 2 ω̃ 2
αp w1
λp
2
2
2 + 2 2
2 + Θ
ω̃ + 4λp
τ ω̃ + 4λp
4λp ω̃
αp w1
λp
2λp τ ω̃
2
2
2 + 2 2
2 + Θ
ω̃ + 4λp
τ ω̃ + 4λp
2
2
ω̃ 2
ω̃ 2 + λ2p
λp ω̃
ω̃ 2 + λ2p
!
!
(3.78)
(3.79)
According to (3.78) the storage modulus Ẽ ′ exhibits a quadratic dependence on ω̃ for
ω̃ → 0. From (3.79) follows that Ẽ ′′ has a linear dependence on ω̃ for ω̃ → 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 dependence 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 dependence on ω̃. 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 Ẽ ′ and the
loss modulus Ẽ ′′ are plotted as functions of the frequency ω̃ for different fractions of
hairpins.
Another quantity to be analyzed here is the zero-deformation rate elongational viscosity.
This quantity is determined as
Z∞
(3.81)
η̃E0 ≡ Ẽ t̃ dt̃
0
Substitution of the explicit expression for the response function (3.69) then gives
η̃E0 =
N
−1
X
p=0
1
λp
τ
2
1+ +
2 Θ
αp w1
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
200
100
50
Ẽ ′
20
Ẽ ′′
10
5
2
1
0.05
0.10
0.15
ω̃
0.20
0.25
0.30
(3.80)
Figure 3.5: Plot of the storage modulus Ẽ ′ (blue lines) and the loss modulus Ẽ ′′ (red lines)
as functions of the frequency ω̃ for different fractions of hairpins w0 . The upper curve in each
set corresponds to w0 = 0.0 and the lower curve corresponds to w0 = 0.5. The values of the
parameters are à = 0.1, ξtr = 2, Ũn = 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 η̃E0 on 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
η̃E0 ∼ N 3
(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, because during the non-dimensionalization procedure the viscosity is scaled by the comT
bination of Nch
V τk (units of pressure times units of time). But in real concentrated systems 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 example, if the number of nematogens per unit volume is kept constant, then the increase
Rouse-like model in the highly-ordered limit
68
1500
η̃E0
1000
500
0.1
0.2
0.3
0.4
0.5
w0
Figure 3.6: Plot of the elongation viscosity η̃E0 as 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 takes
values 2, 4, 8 and 16 respectively. The values of the parameters are à = 0.1, Ũn = 25 and
N = 5.
6
5
4
log10 η̃E0
3
2
1
0.5
1.0
1.5
log10 N
Figure 3.7: Plot of the elongation viscosity η̃E0 as 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 to
w0 = 0.2 and the upper curve to w0 = 0.4. The values of the parameters are à = 0.1, ξtr = 2
and Ũn = 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 number of chains per unit volume by the same factor of α. Therefore, the quantity ηE0 before
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 response 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 reflects 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 Ũn is above 20, because in the elasticity tensor K̃ given by (3.64) becomes
almost independent of Ũn .
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 solution containing hairpins that accounted for the possible entanglements between chains
with hairpin defects was formulated in terms of a set of stochastic differential equations (2.67)-(2.73). We recapitulate these equations here in a brief form by introducing
el
el
the shorthand notations F̃el
ri , F̃ui and F̃bj for the sums of generalized elastic forces associated with the corresponding degrees of freedom ri , ui , bj For i ∈ {1, . . . , N } and
j ∈ {1, . . . , N − 1} these equations read
−1
− 21
tr
el
(S) dr̃i = αtr
spr ζ̃ r (ui ) · F̃ri dt̃ + κ̃ · r̃i dt̃ + αdif f ζ̃ r
(ui ) · dWir̃
el
rot
rot
u
(S) dui = (I − ui ui ) · αrot
spr F̃ui dt̃ + αnem S · ui dt̃ + κ̃ · ui dt̃ + αdif f dWi
el
− 21
−1
tr
b̃
(S) db̃j = αtr
spr ζ̃bead hj F̃bj dt̃ + κ̃ · b̃j dt̃ + αdif f ζ̃bead hj dWj
(4.1)
(4.2)
(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 equations. 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.
Numerical simulations of semi-flexible LCPs
72
The main dimensionless parameters were also determined in Chapter 2. These five parameters 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.
αtr
spr =
3ζ̃rot
αrot
spr =
2
2Ũn Ã
αtr
dif f =
s
ζ̃rot
Ũn
3
αrot
nem = 1
2
4Ũn Ã
αrot
dif f =
s
1
Ũn
(4.4)
There are various reasons that make it prohibitively difficult to solve the system of equations (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 dynamics 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 microelectronics, 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 numerical 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 equilibrium 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 (Itô 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) ∈ Rn , 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)
t
The width of each interval is ∆t = nf . Then for the above described stochastic differential 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 integral equation. In order to obtain the discrete scheme (4.7) the approximations for the
integrals in (4.5) on each time-subinterval were made
tZi+1
ti
B (x (s) , s) dW (s) ≈ B (yi , ti ) · ∆Wi
(4.8)
ti+1
Z
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
Numerical simulations of semi-flexible LCPs
74
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 − ti
(y
− yi )
ti+1 − ti i+1
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 E x − y∆t = 0
∆t→0
(4.11)
where E (·) denotes the expected value.
The numerical scheme is weakly convergent if for any polynomial g(·)
lim E (g (x)) − E g y∆t = 0
∆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 functions 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 process 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 considered. The numerical scheme is weakly convergent of order γ if for any polynomial
g(·)
∆t
γ
(4.14)
E (g (x (t))) − E g y (t) ≤ βg (t) ∆t
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 21 [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
k
1 X ∆t E f y (t)
(4.15)
E (f (x (t))) ≈
k i=1
The error of such an estimation is of order √1k . Thus, the total error contains two contributions: 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 MonteCarlo estimator is
is
β
em
p
β
mc
k
, and the variance due to the deviation of f y∆t (·) from f (x(·))
for each realization. Thus, the sum of the variances is
Error =
β mc
β em
β mc
β em 2
+k
=
+
k
k
p
k
C
(4.16)
For a fixed computational cost, this function has a minimum at
k=
β mc C
2β em
13
p=
2β em C 2
β mc
! 31
(4.17)
2β
em
2
(β mc )
Computing each of the terms in (4.16) at the optimal point (4.17) gives values
C
em mc 2 13
β (β )
and
for the first and second term respectively. The first term is twice big4C
31
Numerical simulations of semi-flexible LCPs
76
ger than the second term. When performing the simulations we usually do not know
the values of β em and β ms a priori, but these quantities can be estimated from the simulations. 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 implementation, 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 80 o C).
Description
Symbol
Number of rods per chain
Combination
N
Relative amplitude of the thermal fluctuations of the
spring
Ã
Dimensionless nematic field strength
Ũn
Relative maximum length of the springs
˜lz
Asymmetry of translational friction coefficient
ξtr
Ratio between rotational and translational frictions
ζ̃rot
s
3T
ko l2
Un
T
lz
l
ζ⊥
ζk
ζrot
ζk 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 Ã. For this purpose we will need the data about the
contour length, persistence length and the transverse gyration radius of polymers.
g
Typical PpPTA polymers have an averaged molar mass µP pP T A ≈ 30000 mole
and a
contour length of lc ≈ 1600 Å [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 Å, Schaefgen et al [96] found it to be in the range
4.3 Values of the main parameters in the model
77
up to 240Å. Cotts et al [97] have published even higher values (up to 430Å), and Chu
and Ying [98] reported a value for the persistence length of 290Å. We will estimate the
persistence length to be about 290Å and we will take the length of a nematogen in the
model equal to this value, i.e., l = 290Å.
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
lcch = 2 (N − 1) A + N l
(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 experimentally determined from small angle neutron scattering (SANS) measurements. The values
were found to be rg k = 250Å for parallel gyration radius and rg ⊥ = 70Å for perpendicular 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 meanq
2
3A
each. The factor
2
3
occurs because we account here only for
ch
the transverse component of the spring. Then rg ⊥ is estimated from the following
expression
r
ch
4
rg ⊥ = A
(N − 1)
(4.19)
3
square displacement
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 .
l
N= c −
lp
v
√ u
2
3 rg ⊥ u
t lc − lp + 3 rg ⊥
lp
lp
4lp2
(4.20)
Substitution of lc = 1600Å, lp = 290Å, rg = 70Å 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 30.3Å.
Then
A
à =
≈ 0.10Å
(4.21)
lp
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
Numerical simulations of semi-flexible LCPs
78
suggested
hP2 i ≈ 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
hP2 i ≈ 1 −
T
ǫ (T )
(4.23)
helps us to recover the expression for the strength of the nematic field
1
ǫ (T )
=
T
0.22
Tni
T
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 545 K. From this data it follows that
for T = 353 K the value Tǫ ≈ 16.7. The strength of the nematic field Un defined in this
thesis is by the factor 23 greater, than ǫ. Therefore,
Ũn =
3ǫ
≈ 25
2T
(4.25)
We also adopt the value suggested in [11] for the rotational diffusivity D̄r = 3.3 · 104 s−1
at T = 80 o C. In [11] the value for D̄r was estimated from the following formula
−2
D̄r = βDr0 νL3
(1 − hP2 i)−2
(4.26)
where L is the rod length (290 Å), P2 ≈ 0.95 at rest for Tǫ = 16.7, ν is the number of rods
per unit volume at a given conditions (2.42 · 1025 m−3 ), β = 103 is a correction factor and
Dr0 is the rotational diffusivity coefficient for a dilute solution
D r0 =
3T ln
L
b
πηs L3
−γ
(4.27)
where Lb is the aspect ratio of the rod, b = 6.6 Å is the width of the rod, T is the temperature in Joules, ηs is the solvent viscosity (5.3 · 10−3 P a · s for sulfuric acid at 99%
concentration and at 80 o C), γ ≈ 0.8 is a correction term. Use of these values gives
D̄r = 3.3 · 104 s−1 . This result in [11] is followed by a very interesting discussion. D̄r
−2
has a very strong dependence on L, because of the factor νL3
in (4.26) and strong
dependence of Dr0 on 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 simulations 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
2πηs L
(4.28)
ζk =
ln Lb
ζ⊥ = 2ζk
(4.29)
ξtr = 2
(4.30)
ζ̃rot = 0.68
(4.31)
then
The rotational friction is estimated from the Einstein relation ζrot =
extensibility of the springs is taken to be ˜lz = 0.5.
T
.
D̄r
The maximum
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).
αtr
spr = 4.09
αrot
spr = 3.00
αtr
dif f = 0.165
αrot
nem = 1
αrot
dif f = 0.20
(4.32)
These values will be used in the simulations of the rheology of the LCP solution presented 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
tr
= 1.48 · 10−6 s
τspr
rot
= 2.02 · 10−6 s
τspr
tr
−5
τdif
s
f = 2.22 · 10
rot
τnem
= 6.06 · 10−7 s
rot
−5
τdif
s
f = 1.52 · 10
(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
tr
mode scales with the number of rods as N 2 , then the longest time scale is N 2 τspr
. 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 translarot
to scale the time t. That is why t̃ = 1 corresponds to
tional modes. We have chosen τnem
6.06·10−7 s for the considered solution of PpPTA in sulfuric acid, and κ̃ = 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
Numerical simulations of semi-flexible LCPs
80
was scaled by the quantity
Pid =
Nch T
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 kP a, then the partial pressure created by
the PpPTA chains can be estimated by determining the fraction of number of PpPTA
chains in the solution.
NP pP T A
Pid =
P
(4.35)
NP pP T A + NH2 SO4 atm
µH2 SO4 CP pP T A
NP pP T A
=
NH2 SO4
1 − CP pP T A µP pP T A
(4.36)
NP pP T A
1
≈
NH2 SO4
1250
(4.37)
g
is the molar mass of sulfuric acid, CP pP T A = 0.197 is the weight
where µH2 SO4 = 98 mole
kg
gives
fraction of the PpPTA chains. Substitution µP pP T A = 30 mole
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 P a
(4.38)
Thus, the unit of non-dimensionalized stress corresponds to approximately 81 P a 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 characteristic 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 necessary 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 numerical 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}
− 12
−1
el
tr
(S) dr̃i = αtr
spr ζ̃ r (ui ) · F̃ri dt̃ + κ̃ · r̃i dt̃ + αdif f ζ̃ r
(ui ) · dWir̃
el
rot
rot
u
(S) dui = (I − ui ui ) · αrot
spr F̃ui dt̃ + αnem S · ui dt̃ + κ̃ · ui dt̃ + αdif f dWi
el
− 12
tr
b̃
−1
(S) db̃j = αtr
spr ζ̃bead hj F̃bj dt̃ + κ̃ · b̃j dt̃ + αdif f ζ̃bead hj dWj
(4.39)
(4.40)
(4.41)
Before we formulate the equivalent Euler-Maruyama scheme for these equations we
have to derive the corresponding SDEs in Itô 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 u̇sp
i for
equation (4.40) turns out to be non-trivial
u̇i sp =
1 rot
1 rot 2
∂ rot
αdif f (1 − dimui ) ui
αdif f +
αdif f (I − ui ui ) ·
2
∂ui
2
(4.42)
The first term in this expression is obviously zero, because αrot
dif f 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 − ui ui ), 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 higherorder 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 renormalizing 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 spurious 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
−1
r̃k+1
= r̃ki + αtr
spr ζ̃ r
i
− 12
tr
k
uki · F̃el
uki · ∆Wir̃
k ∆t̃ + κ̃ · r̃i ∆t̃ + αdif f ζ̃ r
r
i
(4.43)
Numerical simulations of semi-flexible LCPs
82
el
uk+1
= uki + I − uki uki · αrot
+ αrot
Sk · uki + κ̃ · uki ∆t̃ + αrot
∆Wiu (4.44)
spr F̃uk
nem
dif
f
i
i
− 21
b̃
k
k
el
k
tr
−1
(4.45)
b̃k+1
= b̃kj + αtr
spr ζ̃bead hj F̃bk ∆t̃ + κ̃ · b̃j ∆t̃ + αdif f ζ̃bead hj ∆Wj
j
j
where ∆t̃ is the time-step of the iteration, and ∆Wir̃ , ∆Wiu , and ∆Wjb̃ are the increments of the 3D-Wiener process corresponding to the time-step ∆t̃, i.e. the components
of ∆Wir̃ , ∆Wiu , and ∆Wjb̃ are independent Gaussian-distributed random variables
with variance ∆t̃.
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 Mathematica 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 performed 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 iteration 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
Nch
1
1 X
uk uk − I
S =
Nch i=1 i i
3
k
(4.46)
Actually, the term − 13 I is omittedin the implemented code, because it drops out after
applying the projector I − uki uki . 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
≈ 0.03. In fact, the fluctuations are also
we keep Nch = 1000. Which gives √ 1
Nch
dependent on the values of the parameters, especially on the strength of the nematic
4.4 Algorithm
83
field Ũn . The increase of the strength of the nematic field leads to smaller fluctuations
of Sk . For Ũn = 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 timeintervals. 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
−1
el
term in equation (4.45) is αtr
∆t̃, because ζ̃bead → 0. In this limit equation
spr ζ̃bead F̃bk
j
(4.45) turns into a force balance
F̃el
(4.47)
k = 0
b
j
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 dramatically reduced. This formally means ζ̃bead → ∞ in (4.45). In that case the motion of the
bead reduces to affine motion
b̃k+1
= b̃kj + κ̃ · b̃kj ∆t̃
j
(4.48)
Formally, the large friction coefficient ζ̃bead corresponds to a respectively large characteristic 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 defect itself, as a kink in the chain’s backbone, is proportional to eŨn , which for Ũn = 25
rot
and τnem
= 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.
Numerical simulations of semi-flexible LCPs
84
4.5 Equilibrium properties of the system
In order to generate an ensemble of possible initial configurations of the chains the configurational 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 distribution
exp − U(x)
T
f (x) = R
(4.49)
exp − U(x)
dx
T
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 configurations 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 equilibration 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 results of simulations for N = 3, N = 5 with the rigid rod model. The self-consistency
equations for the rigid rod model [11] is
hP2 i =
R
exp
R
where Tǫ = 23 Ũn and P2 (x) =
expressed in terms of hP2 i
ǫ
T
hP2 i P2 (n · u) P2 (n · u) du
exp Tǫ hP2 i P2 (n · u) du
3 2
2x
(4.50)
− 12 . The maximum eigenvalue of huui can be
1 2
+ hP i
(4.51)
3 3 2
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
λmax =
4.5 Equilibrium properties of the system
85
Simulations for N = 5
Simulations for N = 3
Rigid Rod model N = 1
λmax
1.0
0.8
0.6
0.4
0.2
0.0
0.00
0.05
0.10
0.15
1
Ũn
Figure 4.1: The maximum eigenvalue λmax of huui as function of the inverse relative strength
of the nematic field for different number of rods per chain. Computed from both simulations and
the 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 Ũn constant, then the chain consisting of two rods has approximately twice bigger total nematic energy. If we keep the total nematic energy per chain
fixed, then the increase of N leads to the decrease of Ũn . Then the degree of orientational
ordering decreases for more flexible chains and starting from some N the nematic phase
does not occur. In other words, in this model if
nematic
Uper
chain lp
< 6.8
T
lc
(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 behavior 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-
Numerical simulations of semi-flexible LCPs
86
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 introduce 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 mirrorsymmetric with respect to a plane normal to the director. Therefore, the ensemble averNP
−1
hui · ui+1 i should be zero, i.e., hhi = 0.5. Sometimes, preparation of an enage N1
i=1
semble by the above equilibration procedure ends up with an h that deviates essentially
from 0.5. Because Ũn 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 hhi closest
to 0.5. Usually, the deviation does not exceed 0.03.
If hhi = 0.5, then the chain consisting of N rods and N − 1 flexible spacers should have
on average N 2−1 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 Nmethyl-pyrolidone molecules by means of quantum-chemical and molecular dynamics
simulations. N-methyl-pyrolidone was chosen in order to reduce the effect of protonation of amide groups in PpPTA. From these simulations it followed that the energy
difference between the cis-trans (folded and unfolded) configurations of the consecutkJ
ive amide groups is varying between 18 and 15 mole
depending on the dielectric permittivity of the solvent and on the degree of protonation of the amide groups. Using
kJ
and accounting for the number of amide units per chain
the energy penalty 15 mole
g
(32000 mole ) 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 flowaligning, 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 polymeric 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 dynamics 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 complexity 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 equations. 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 evolution 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 highlyaligned 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 Ũn ≈ 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 corresponding 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.
rot
ζrot
In the rod-spring-bead model we have nondimensionalized the time by τnem
= 2U
n
rot
(Table 2.2). In the highly-aligned limit this time-scale formally turns to zero τnem
→ 0,
and, therefore, can not be chosen for nondimensionalization. That is why we took anζ
other time-scale, namely, τk = kk defined by relation (A.71). Thus, for a correct compar0
ison of the results of these two models we have to account for this different scaling of
τk
time. The scaling factor is rot
. Taking into account k0 = k2o we derive the expression
τnem
for this factor in terms of the main dimensionless groups of the model
τk
rot
τnem
=
ζrot 2ζk
4 Ũn Ã2
=
2Un ko
3 ζ̃rot
(5.1)
τ
k
≈ 0.5. Moreover,
In particular, for ζ̃rot = 0.68, Ã = 0.1, Ũn = 25 this factor is rot
τnem
this factor also gives an indication of the ratio of the translational and rotational timescales. The lowest Rouse mode has the time-scale τk N 2 . In the highly-aligned model
rot
the separation of time-scales was justified by the assumption
3 ζ̃rot
≪1
4 Ũn Ã2 N 2
For N = 5 and
0.08.
τk
rot
τnem
τnem
2
τk N
≪ 1, i.e.,
(5.2)
= 0.5 the ratio on the left-hand side of equation (5.2) is equal to
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:
3 ζ̃rot
εtol = 0,
Ũn ≫ 1,
≪ 1.
(5.3)
4 Ũn Ã2 N 2
Although the analytical model for highly-ordered limit formulated in Chapter 3 is derived 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 ordering, 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
1
0
Γ = δe  0 − 12
0
0


0
0 
(5.4)
− 21
The rotational degrees of freedom are transformed slightly different, according to
unew = uold + (I − uold uold ) · 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.
Ẽ
⋄
Ñ1⋄ t̃
t̃ ≡
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 towards 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 timescale 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 timescale 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 numerical 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 consistent 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
t̃ = 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
100
Ẽ ⋄
10
1
Ẽ
0.1
0.01
2
4
6
8
10
t̃
Figure 5.1: Comparison between the elongational response function Ẽ ⋄ obtained in numerical
simulations and the elongational response function Ẽ 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
dt̃ = 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, t̃ < 1.
We also notice that for time region t̃ > 1 a slight difference (up to 10%) between the
responses Ẽ ⋄ and Ẽ 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 entanglements 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-
Rheology of entangled LCP solutions containing hairpins
92
εtol = 1.00
100
ε =
εtol =
0.05 tol 0.07
10
εtol =
0 .0
Ẽ
3
1
01
0. 0
= 0 .0
l
ε to ol =
εt
0.1
0.01
0
5
10
15
20
t̃
Figure 5.2: Elongational response function plotted versus time for different values of εtol . The
values of the parameters are: Ã = 0.1, ξtr = 2, Ũn = 25, l̃z = 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 increase 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 important 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 system. This is clearly seen in fig. 5.3. Decreasing the frequency corresponds to decreasing 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 entanglements 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 general, the existence of competing phenomena in the system leads to non-monotonicity
5.2 Linear rheology of an LCP solution
93
100
Ẽ ′
1
ε to
ε to
0.01
ε to
=
l
=
l
=
l
ε tol
ε to
=
=
l
0 .1
0 .0
0
7
0 .0
5
0 .0
3
0 .0
1
0.001
0.01
0.1
1
ω̃
Figure 5.3: Storage modulus plotted versus frequency for different values of εtol . The values of
the parameters are: Ã = 0.1, ξtr = 2, Ũn = 25, ˜lz = 0.5 and N = 5.
100.0
50.0
Ẽ ′′
10.0
5.0
ε to
ε to
ε to
=
l
=
l
1.0
0.5
ε tol
ε to
0 .1
=
l
=
0 .0
0
7
5
0 .0
3
0 .0
=
l
0.001
0 .0
1
0.005
0.010
0.050
0.100
0.500
1.000
ω̃
Figure 5.4: Loss modulus plotted versus frequency for different values of εtol . The values of the
parameters are: Ã = 0.1, ξtr = 2, Ũn = 25, ˜lz = 0.5 and N = 5.
Rheology of entangled LCP solutions containing hairpins
94
100
Ẽ ′
10
Ẽ ′′
1
0.1
0.001
0.005
0.010
0.050
0.100
0.500
1.000
ω
Figure 5.5: Storage Ẽ ′ and Ẽ ′′ moduli plotted versus frequency. The values of the parameters
are: εtol = 0.07, Ã = 0.1, ξtr = 2, Ũn = 25, ˜lz = 0.5 and N = 5
2 .104
Simulations
Interpolation
104
5000
η̃E0
2000
theory for w0 = 0
1000
500
theory for w0 = 0.5
200
100
0.00
0.02
0.04
0.06
0.08
0.10
εtol
Figure 5.6: Zero-elongation-rate viscosity η̃E0 plotted versus εtol . Dashed lines indicate the
values of the zero-elongation-rate viscosity of unentangled highly-ordered LCP solution with
w0 = 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 hairpins, 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, because 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 contribution 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 huui to characterize the orientational order in the system.
Rheology of entangled LCP solutions containing hairpins
96
Simulations
Interpolation
25
˜
γ̇
200
τ̃kayak
150
100
0.10
0.15
0.20
0.30
0.50
γ̇˜
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 à = 0.1, ξtr = 2,
Ũn = 25, l̃z = 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
γ̇˜ ≡ κ̃12 . Then we run the simulations with time-step dt̃ = 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 γ̇˜ < 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
˜ for the chosen values of the parameters. By a period we unγ̇
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
3.0
2.5
2.0
θL
1.5
1.0
0.5
0.0
0
5
10
15
20
γ̇˜
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 folded 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. However, 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 hairpin 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 < γ̇˜ < 0.65 the transition to a wagging type of behavior 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 γ̇˜ ≈ 0.7 flow-aligning type of behavior sets in. However, 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 < γ̇˜ < 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 huui is plotted for different 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 γ̇˜ −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 orientation 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 amplitude 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 following 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
0.8
λ1
λ2 0.6
λ3
0.4
0.2
125
250
375
500
t̃
Figure 5.9: Eigenvalues of the orientation tensor huui as a function of time in shear flow for
γ̇˜ = 0.2. Kayaking regime.
0.8
λ1
λ2
λ3
0.6
0.4
0.2
125
250
375
500
t̃
Figure 5.10: Eigenvalues of the orientation tensor huui as a function of time in shear flow for
γ̇˜ = 0.6. Wagging Regime.
0.8
λ1
λ2 0.6
λ3
0.4
0.2
50
100
150
200
t̃
Figure 5.11: Eigenvalues of the orientation tensor huui as a function of time in shear flow for
γ̇˜ = 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 important 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 γ̇˜ −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 predictions 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 γ̇˜ = 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 < γ̇˜ < 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
14
12
10
σ̃12
8
6
4
2
125
250
375
500
t̃
Figure 5.12: Shear stress as a function of time in shear flow for γ̇˜ = 0.2. Kayaking regime.
15
σ̃12
10
5
125
250
375
500
t̃
Figure 5.13: Shear stress as a function of time in shear flow for γ̇˜ = 0.6. Wagging regime.
10
8
σ̃12
6
4
2
50
100
150
200
t̃
Figure 5.14: Shear stress as a function of time in shear flow for γ̇˜ = 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
η̃sh
1
=
˜
γ̇ τ̃
t̃Z
0 +τ̃
t̃0
σ̃12 t̃ dt̃
(5.7)
where τ̃ 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 (γ̇) = η∞ +
η0 − η∞
1 + (τr γ̇)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 γ̇˜ > 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 γ̇˜ > 0.6. At the initial moment of time hhi = 0.56. But after
the flow is imposed hhi 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 γ̇˜ = 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 threeregion behavior. The transition from region I to Region II takes place at γ̇ ≈ 5 s−1 , and
the transition from Region II to Region III takes place at γ̇ ≈ 100 s−1 . The value of the
viscosity in the plateau region changes in the interval from 30 P a · s to 50 P a · s. In the
5.4 Shear viscosity and the first normal stress difference in the steady shear
103
Simulations
Interpolation
20.0
15.0
10.0
η̃sh
7.0
5.0
3.0
0.1
0.2
0.5
1.0
2.0
5.0
γ̇˜
Figure 5.15: Steady state shear viscosity plotted versus shear rate.
0.55
0.50
0.45
hhi
0.40
0.35
0.30
0
125
250
375
500
t̃
Figure 5.16: Average value of hairpin variable as a function of time in shear flow for γ̇˜ = 0.6.
Wagging regime.
15
10
Ñ1
5
0.2
0.4
0.6
0.8
1.0
γ̇˜
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 γ̇˜ ≈ 0.55. Comparing this result with the experimental data
rot
we can estimate τnem
that should be chosen in order to match the transition point from
rot
Region II to Region III. From this comparison it follows that τnem
≈ 5.5 · 10−3 s which
−1
suggests the value for rotational diffusion coefficient D̄r = 3.6s , 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 D̄r =
3.3·104 s−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 D̄r
can be easily several orders of magnitude. Thus, from our simulations follows a more
accurate way of estimating the rotational diffusivity D̄r by the transition point from
Region II to Region III.
Of course, we understand that in our model the change of D̄r 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 translational 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 .
rot
If we adopt τnem
≈ 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 η̃sh ≈ 9 P a · s. As was mentioned
earlier the value found in the experiments is in the range of 30 P a · s to 50 P a · 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 η̃sh ≈ 5 P a · 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 wagging 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 formulated in Chapter 3 if
ζ̃rot
3
4 Ũn Ã2 N 2
≪ 1 and εtol = 0.
• The presence of the hairpins in the unentangled LCP solution decreases the response 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 dynamic regimes of the director, the region with negative normal stress difference,
shear-thinning behavior.
• If possible entanglements between hairpins are accounted for, then both qualitative 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 composite 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 properties 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 follows 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 assumption 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
Conclusions
108
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
ζ̃rot
2
2
Ũn à N
≪ 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 substantially 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 agreement 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 calculated 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 homogeneous flows. However, for the practical purposes it is often needed to compute
non-homogeneous flow in particular geometries. The rod-spring-bead model presented 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 socalled poly-domain structure would be a natural extension [59]. The presented rodspring-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 initial 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 distribution 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.
∂
hBi = hLBi
∂t
(A.1)
If we choose the dynamical variable BΨ below
BΨ =
Nch N
X
Y
s=1 i=1
δ (rsi − ri ) δ (usi − ui )
N
−1
Y
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
Z
hBΨ i =
BΨ dX ≡ Ψ.
(A.3)
X
On the right hand side of the equation (A.1) we get hLBΨ i. The Liouville operator is
given by (2.11). We now employ the following identity for the delta-function
d
d
δ r − r′ = − ′ δ r − r′
dr
dr
(A.4)
Details of derivations
112
in order to get
LBΨ = −
Nch
N
−1
N
N
X
X
X
X
∂
1
∂
∂
1
pusi · J−1 ·
prsi ·
+
pbsi ·
+
+
m
∂r
m
∂b
∂u
r
si
b
si
si
s=1
i=1
i=1
i=1
!
N
N
−1
N
X
X
X
∂
∂
∂
Frsi ·
+
Fbsi ·
Fusi ·
BΨ (A.5)
+
+
∂prsi
∂pbsi
∂pusi
i=1
i=1
i=1
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
N
N
N
−1
−1
X
X
X
∂
∂
∂ −1
1
1
− hLBΨ i =
·
Jpri KΨ +
·
Jpbi KΨ +
· J · Jpui KΨ
∂ri
mr
∂bi
mb
∂ui
i=1
i=1
i=1
(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 singlechain configuration distribution function Ψ
X
N
−1
N
NX
Jpri K
Jpbi K
∂ −1
∂
∂Ψ X ∂
·
Ψ +
· J · Jpui KΨ +
·
Ψ = 0 (A.7)
+
∂t
∂ri
mr
∂ui
∂bi
mb
i=1
i=1
i=1
A.2
Evolution equation for the momentum-space averages
In order to obtain the evolution equation for the momentum-space averaged momenta
Jpri K, Jpbi K, Jpui K we have to consider the general equation of change for the appropriate 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
N
Y
X
s=1
i=1
δ (rsi − ri ) δ (usi − ui ) δ prsi − pri δ pusi − pui ×
×
BJpr
−mr vKΨ
j
BJpu
j
N
−1
Y
i=1
δ (bsi − bi ) δ pbsi − pbi
= prj − mr v Bψ
KΨ
= pu B ψ
!
(A.8)
(A.9)
(A.10)
A.2 Evolution equation for the momentum-space averages
BJpb
j
−mb vKΨ
= pb j − m b v B ψ
113
(A.11)
Here v is the velocity of a macroscopic sample of the solution.
D
D
BJpr
j
−mr vKΨ
E
= Jprj − mr vKΨ
(A.12)
N
N
−1
N
X
X
X
1
∂
1
∂
∂
LBJpr −mr vKΨ =
pusi · J−1 ·
prsi ·
+
pbsi ·
+
+
j
m
∂r
m
∂b
∂u
r
si
b
si
si
i=1
i=1
i=1
!
N
N
−1
N
X
X
X
∂
∂
∂
Frsi ·
Fbsi ·
Fusi ·
+
+
+
prj − mr v Bψ
(A.13)
∂prsi
∂pbsi
∂pusi
i=1
i=1
i=1
E
All terms in this expression are treated in the same way as expression (A.6) in the previous appendix, except for terms that contain ∂p∂r . These terms we integrate by parts.
j
In the end we get
N
−1
X
NX
∂ ∂ ∂ v·
v·
Jpbj − mb vKΨ +
Jprj − mr vKΨ +
Jprj − mr vKΨ =
∂t
∂ri
∂bi
i=1
i=1



N
N
X
∂  J pri − mr v prj − mr v K  X ∂ −1
= −
·
Ψ +
· J Jpui prj − mr v KΨ +
∂ri
mr
∂ui
i=1
i=1



N
−1
X
∂  J pbi − mb v prj − mr v K  (h)
(intra)
·
Ψ
+ Frj + F(e)
Ψ (A.14)
+
rj + Frj
∂bi
mb
i=1
Here we have to clarify the meaning of the new notations for hydrodynamic force F(h)
rj ,
(intra)
external force F(e)
.
rj and intramolecular force Frj
1 D inter E
Frj Bψ
Ψ
1 D ext E
F(e)
=
Frj Bψ
rj
Ψ
1 D intra E
F(intra)
=
Frj Bψ
rj
Ψ
F(h)
rj =
(A.15)
(A.16)
(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-
Details of derivations
114
ing the gradient-terms as Brownian forces.
F(b)
rj



N
1 X ∂  J pri − mr v prj − mr v K 
·
Ψ +
=−
Ψ i=1 ∂ri
mr
+
N
X
∂ −1
· J Jpui prj − mr v KΨ +
∂ui
i=1


N
−1
X
∂  J pbi − mb v prj − mr v K 
+
·
Ψ
∂bi
mb
i=1
(A.18)
Finally, we obtain the following equation
(h)
(e)
(intra)
F(b)
=0
rj + Frj + Frj + Frj
(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.
(h)
(e)
(intra)
F(b)
=0
u j + Fu j + Fu j + Fu j
(A.20)
Here
F(b)
uj
1
=−
Ψ
N
X
∂
·
∂r
i
i=1
!
N
X
J pri − mr v puj K
∂ −1
Ψ +
· J Jpui puj KΨ +
mr
∂ui
i=1
!!
N
−1
X
J pb i − m b v pu j K
∂
·
Ψ
(A.21)
+
∂bi
mb
i=1
1 D inter E
(A.22)
Fuj Bψ
Ψ
1 D ext E
(A.23)
Fuj Bψ
F(e)
uj =
Ψ
1 D intra E
(A.24)
F(intra)
=
Fuj Bψ
uj
Ψ
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(h)
uj =
(b)
(h)
(e)
(intra)
Fb j + Fb j + Fb j + Fb j
=0
(A.25)
A.3 Itô or Stratonovich interpretation
(b)
Fbj
115



N
1 X ∂  J pri − mr v pbj − mb v K 
·
Ψ +
=−
Ψ i=1 ∂ri
mr
+
N
X
∂ −1
· J Jpui pbj − mb v KΨ +
∂ui
i=1


N
−1
X
∂  J pbi − mb v pbj − mb v K 
·
Ψ
+
∂bi
mb
i=1
1 D inter E
Fbj Bψ
Ψ
1 D ext E
(e)
Fb j =
Fbj Bψ
Ψ
1 D intra E
(intra)
Fbj Bψ
Fb j
=
Ψ
(h)
Fb j =
(A.26)
(A.27)
(A.28)
(A.29)
Equations (A.19), (A.20), (A.25) are the evolution equations for Jprj K, Jpuj K and Jpbj K.
A.3 Itô or Stratonovich interpretation
In this section we will discuss the difference of the Itô 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) ∈ Rn 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 ) =
Zt
t0
where
Rt
t0
a (x(t), t) dt +
Zt
B (x(t), t) · dW(t)
(A.31)
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.
Details of derivations
116
Then the Stratonovich integral is the limit in probability of the integral sum
(S)
Zt
0
A(t) · dW(t) = lim
d→0
k−1
X
i=0
(A(ti ) + A(ti+1 ))
· (W(ti+1 ) − W(ti ))
2
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 Itô’s interpretation differs in the way the value
of the integrand is chosen in each subinterval of the partition.
(I)
Zt
0
A(t) · dW(t) = lim
d→0
k−1
X
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 Itô 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 Itô’s interpretation, namely, the use of
the value of the integrand in the beginning of the subsegment of the partition when constructing 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 Itô interpretation of the stochastic integral is the necessity to use a so-called ”Itô-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 Itô 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 Itô interpretation can be transformed into a corresponding Stratonovich integral. Consequently,
a stochastic differential equation in the Itô interpretation can be transformed into Stratonovich stochastic differential equation and vice versa. So the Itô SDE
(I) dxi = ai (x, t) dt +
m
X
j=1
Bij dWj
(A.32)
A.4 Derivation of the expression for θmax
117
has an equivalent Stratonovich stochastic differential equation given by

m
n X
m
X
X
∂
1
Bij dWj
Bik  dt +
(S) dxi = ai (x, t) −
Bjk
2 j=1
∂xj
j=1

(A.33)
k=1
The reverse transition is also possible. The Stratonovich SDE
(S) dxi = ai (x, t) dt +
m
X
Bij dWj
(A.34)
j=1
is equivalent to the following Itô stochastic differential equation

(I) dxi = ai (x, t) +
The term
1
2
n P
m
P
j=1 k=1
n m
1 XX
2
Bjk
j=1 k=1

m
X
∂
Bij dWj
Bik  dt +
∂xj
j=1
(A.35)
∂
Bik dt is usually called ”spurious drift” term. This term repBjk ∂x
j
resents the difference in definitions of the Itô 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 Itô 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 distribution function for quantity x at moment of time t. Then the Fokker-Planck equation
corresponding to the stochastic differential equation in the Itô interpretation (A.32) is
N
n
n
m
∂ψ X ∂
1 XXX ∂ ∂
(ai ψ) −
+
Bik Bjk ψ = 0
∂t
∂xi
2 i=1 j=1
∂xi ∂xj
i=1
(A.36)
k=1
If we take the stochastic differential equation in the Stratonovich interpretation (A.34)
then the corresponding Fokker-Planck equation is
N
n
n
m
∂ψ X ∂
1 XXX ∂
(ai ψ) −
+
∂t
∂xi
2 i=1 j=1
∂xi
i=1
k=1
∂
Bik
=0
Bjk ψ
∂xj
(A.37)
A.4 Derivation of the expression for θmax
We derive here the expression for θmax , which is the maximum angle that two oppositely aligned neighboring rods are allowed to deviate from the perfectly aligned state
Details of derivations
118
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 21 T .
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 corresponding 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.
λ−

S=
0
0

1
3
0
1−λ
2 −
0
1
3

0

0
1
1−λ
2 − 3
(A.38)
Here λ is the maximum eigenvalue of the matrix huui. Expression (A.38) follows from
the property of huui that
Tr huui = hu · ui = h1i = 1.
(A.39)
The trace of an operator is equal to the sum of its eigenvalues. The largest eigenvalue of
huui is λ. Because the other two eigenvalues are equal and their sum should be 1, they
are equal to 1−λ
2 .
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


sin θ cos φ
u =  sin θ sin φ 
cos θ
(A.40)
A.5 Derivation of the expression for the stress tensor
119
The nematic energy for one rod is
rod
UMS
= −Un S :
3λ − 1
1
I
= Un
sin2 θ − λ −
uu −
3
2
3
(A.41)
And the equation for θmax therefore becomes
Un
(3λ − 1)
sin2 θmax = T
2
(A.42)
and then a little manipulation yields
sin θmax =
s
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 expression [67].
(j)
Nch Nb D
E
X
1 X
(j)
σ=
(A.44)
F(j)
ri ri
V j=1 i=1
(j)
In this formula ri 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
(j)
within the volume of a sample V . Nch denotes the number of chains and Nb 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 configurational 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 }
Z
Ach [Ψ] = Ψ (x) (T ln (Ψ (x)) + U (x)) dx
(A.45)
Details of derivations
120
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 = Nch Ach .
The variation of the free energy under small instantaneous deformation δǫ is related to
the stress σ created.
δA = σµν δǫµν V
(A.46)
The variation of the free energy is determined by the variation of the distribution function.
Z
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 instantaneously the dominating terms in equation (2.32) are terms containing deformation rates.
Thus for this particular case equation (2.32) can be simplified.
N
N
−1
N
X
X
∂
∂
∂Ψ X ∂
·(κ · ri Ψ)+
·((I − ui ui ) · κ · ui Ψ)+
·(κ · bi Ψ) = 0 (A.48)
+
∂t i=1 ∂ri
∂u
∂b
i
i
i=1
i=1
Here N is the number of rods per chain. We use the relation δΨ =
δǫ = κδt to obtain
δΨ = −
∂Ψ
∂t δt
and the relation
N
N
N
−1
X
X
X
∂
∂
∂
·(δǫ · ri Ψ)−
·((I − ui ui ) · δǫ · ui Ψ)−
·(δǫ · bi Ψ) (A.49)
∂r
∂u
∂b
i
i
i
i=1
i=1
i=1
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.
#
"N
−1
N
NX
X
Nch X Fbi bi
Fui ui +
σ=−
Fri ri +
V
i=1
i=1
i=1
(A.50)
Here Fri , Fui and Fbi are 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 = −
∂
(T ln (Ψ (x)) + U (x))
∂ri
(A.51)
Fui = −
∂
(T ln (Ψ (x)) + U (x))
∂ui
(A.52)
Fbi = −
∂
(T ln (Ψ (x)) + U (x))
∂bi
(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
N ζ · (ṙc − κ · rc ) = fc (t)
In here fc (t) =
N
P
(A.54)
fs (t) with
s=1
hfc (t)i =
and
N
X
s=1
hfc (t)i = 0
(A.55)
N X
N
X
fc (t)fc (t′ ) =
fs (t)fs′ (t′ ) = 2N ζ (n) T δ t − t′
(A.56)
s=1 s′ =1
The equations for internal degrees of freedom became
(S) ζ · (ċ1 − κ · c1 ) = K · (−2c1 + c2 ) + w1 lk0 n + f2 − f1
(S) ζ · (ċs − κ · cs ) = K · (cs+1 − 2cs + cs−1 ) + fs+1 − fs
(s = 1) (A.57)
(s ∈ {2, ..., N − 2}) (A.58)
and
(S) ζ ·(ċN −1 − κ · cN −1 ) = K·(cN −2 − 2cN −1 )+w1 lk0 n+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}
N
−1
X
(S) ζ(n) · ċp − κ · cp = −K(n) ·
Ap,s cs + w1 lk0 n δp,1 + δp,N −1 + gp (t) (A.60)
s=1
with
gp (t) = 0
D
E
gp (t)gp′ (t′ ) = 2Ap,p′ ζ(n)T δ t − t′
(A.61)
(A.62)
By means of this change of variables we have separated the translational motion of the
Details of derivations
122
center of mass (A.54) from the internal motions of the chain (A.60). We see that the N −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
× (N − 1) can be diagonalized by the (N − 1) × (N − 1) matrix
qsize (N − 1)
πpm
2
Γm,p = N sin N , leading to
Jm,s = λm δm,s =
N
−1 N
−1
X
X
p=1 n=1
where λm = 4 sin2
πm
2N
Γm,p Ap,n Γ−1
n,s
(A.63)
are the eigenvalues of the Rouse matrix.
These considerations imply that the normal-mode coordinates qm are related to the connector vectors cs by the following relation
qm =
N
−1
X
Γm,s cs
(A.64)
s=1
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) · (q̇m − κ · qm ) = −λm K(n) · qm + αm w1 lk0 n + hm (t)
where
αm ≡
N
−1
X
p=1
Γm,p δp,1 + δp,N −1 =
and
hm (t) ≡
N
−1
X
r
πm 2
(1 − (−1)m ) sin
N
N
Γm,p gp (t)
(A.65)
(A.66)
(A.67)
p=1
with
hhm (t)i = 0
and
(A.68)
−1
−1 N
X
NX
Γm,p Γm′ ,p′ gm (t)gm′ (t′ )
hm (t)hm′ (t′ ) =
p=1 p′ =1
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)λm T δm,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 independent equations for the internal motions of the rod-spring chain. Because these equations
are independent, the coordinates qm are called normal modes.
(S) q̇m − κ · qm = −λm τ −1 · qm + αm w1 lτk n + vm (t)
(A.70)
−1
τ −1 (n) = ζ −1 (n) · K(n) = τk−1 nn + τ⊥
(I − nn)
(A.71)
vm (t) = ζ −1 (n) · hm (t)
(A.72)
hvm i = 0
vm (t)vm′ (t′ ) = 2ζ −1 (n)λm T δm,m′ δ t − t′
(A.73)
(A.74)
A.7 Formal solution of a linear matrix differential equation
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 timedependent vector in Rn solving the initial-value problem.

 ẋ(t) = A(t) · x(t) + c(t)

(A.75)
x(0) = x0
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 corresponding 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
ẋ(t) = A(t) · x(t).
Ṁ(t) · x0 = A(t) · M(t) · x0
This equation should be satisfied for any initial condition x0 . Thus we obtain an equation for M(t). Supplemented with the initial condition x(0) = M(0) · x0 = x0 , i.e.
Details of derivations
124
M(0) = I, we obtain an initial-value problem for M(t).

 Ṁ(t) = A(t) · M(t)

(A.76)
M(0) = I
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).

−1
 ċ(t) = M (t) · c(t)

(A.77)
c(0) = x0
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 +
Zt
0
ds M−1 (s) · c(s)
(A.78)
From (A.78) we readily obtain an expression for x(t).
x(t) = M(t) · x0 + M(t) ·
Zt
0
ds M−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 equation 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.

−1
−1
 Ṁ (t) = −M (t) · A(t)

M
−1
(A.80)
(0) = I
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 κ̃(t̃) = 0 and n does not change its direction in time. In this case system
(3.43) can be explicitly solved.
Then (3.45) gives
Mp (t̃) = exp −λp τ̃ −1 t̃
q̃p (t̃) = exp −λp τ̃ −1 t̃ · q̃p (0) +
Zt̃
0
In the limit t → ∞ this boils down to
q̃p
eq
=
ds exp −λp τ̃ −1 (t̃ − s) · αp w1 n
αp w1
n
λp
In order to get an expression for q̃p q̃p eq we can use the result (3.46).
T
q̃p q̃p eq = q̃p eq q̃p eq + lim Mp (t̃) · Bp (t̃) · Mp (t̃)
t̃→∞
(A.81)
(A.82)
(A.83)
(A.84)
Expression for Bp (t̃) can be found from (3.47) by direct integration, because in equilibrium the matrices Mp (t̃) are known and ζ̃ is a constant matrix. Thus, in equilibrium we
get
−1
1
Bp (t̃) = Θ τ̃ · ζ̃ · exp 2λp τ̃ −1 t̃ + Bp (0)
(A.85)
2
and hence
T
−1
1
lim Mp (t̃) · Bp (t̃) · Mp (t̃) = Θ τ̃ · ζ̃
(A.86)
2
t̃→∞
Finally, this leads to the following expression for q̃p q̃p eq
α2p w12
−1
1
q̃p q̃p eq =
2 nn + 2 Θ τ̃ · ζ̃
λp
(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 system of chains expressed in normal-mode coordinates. We start with (3.22) and the
Details of derivations
126
definition
(3.51). oFirst we transform from the set of coordinates {r̃1 , ..., r̃N } to the set
n
R̃c , c̃1 , ..., c̃N −1 , leading to
F̃sys =
N −1
1 X
[w hc̃ · K0 · c̃m i + w1 h(c̃m − n) · K1 · (c̃m − n)i]
k0 m=1 0 m
Then we rearrange the brackets and use the symmetry of the operator K1
F̃sys =
N −1
1 X
[(w0 K0 + w1 K1 ) : hc̃m c̃m i + w1 K1 : (nn − 2n hc̃m i)]
k0 m=1
Finally, using K0 = k0 I and K1 = k0 nn + k1 (I − nn) then gives
F̃sys = 2w1 (Λ : nn − p · n) + C
where
Λ=
1
2
NX
−1
k
1− 1
hc̃ c̃ i
k0 m=1 m m
p=
N
−1
X
m=1
and
C=
(A.88)
(A.89)
hc̃m i
(A.90)
NX
−1
k
w0 + w1 1
hc̃ · c̃ i + w1 (N − 1)
k0 m=1 m m
(A.91)
Now we want to express Λ, p and C in terms of the coordinates {q̃1 , q̃2 , ..., q̃N −1 }.
N
−1 D
X
E
b̃m b̃m =
m=1
N
−1 N
−1 N
−1 X
X
X
Γ−1
m=1 n=1 s=1
m,s
Γ−1
m,n
hq̃s q̃n i
We use Γ−1 = ΓT to get
N
−1 D
X
m=1
−1
E NX
hq̃n q̃n i
b̃m b̃m =
(A.92)
n=1
Similarly,
−1 N
−1 E NX
X
Γ−1
b̃m =
N
−1 D
X
m=1
m,n
m=1 n=1
where
βn =
N
−1
X
m=1
Γ−1
m,n
=
r
hq̃n i =
N
−1
X
n=1
βn hq̃n i
N −1
πnm 2 X
sin
N m=1
N
(A.93)
A.9 Free energy of the ensemble of chains
127
We use formula
N
X
sin(am) =
m=1
to get
1
sin
a sin
2
βn =
aN
2
sin
a (N + 1)
2
αn
λn
(A.94)
Thus we rewrite (A.89),(A.90),(A.91) in coordinates {q̃1 , q̃2 , ..., q̃N −1 }
Λ=
1
2
NX
−1
k
1− 1
hq̃ q̃ i
k0 m=1 m m
N
−1
X
αm
hq̃m i
λ
m=1 m
N −1
k1 X
hq̃ · q̃ i + w1 (N − 1)
C = w0 + w1
k0 m=1 m m
p=
(A.95)
(A.96)
(A.97)
These results (A.95),(A.96),(A.97) express the free energy (A.88) in terms of the coordinates {q̃1 , q̃2 , ..., q̃N −1 }.
128
Details of derivations
Bibliography
[1] F. Reinitzer. Beiträge zur Kenntnis des Cholesterins. Monatshefte für Chemie, 9:421–
441, 1888.
[2] O. Lehmann. Über Fließende Kristalle. Zeitschrift für Physikalische Chemie, 4:462–
472, 1889.
[3] P.G. de Gennes. Polymer Liquid Crystals. Academic Press, New York, 1982.
[4] L. Onsager. Anisotropic Solutions of Colloids. Physical Review, 62:558–829, 1942.
[5] L. Onsager. The effects of shape on the interaction of colloidal particles. Ann. N.
Y. Acad. Sci., 51:627–659, 1949.
[6] W. Maier and A. Saupe. A simple molecular theory of the nematic liquidcrystalline state. Zeitschrift für Naturforschung Teil A, 13:564–566, 1958.
[7] W. Maier A. Saupe. A simple molecular theory of the nematic liquid-crystalline
state. part i. Zeitschrift für Naturforschung Teil A, 14:882–889, July 1959.
[8] W. Maier A. Saupe. A simple molecular theory of the nematic liquid-crystalline
state. part ii. Zeitschrift für Naturforschung Teil A, 15:287, 1960.
[9] W.L. McMillan. Simple Molecular Model for the Smectic A Phase of Liquid Crystals. Physical Review A, 4(3):1238–1246, September 1971.
[10] J.R. Schaefgen S.L. Kwolek P.W. Morgan and L.W. Gulrich. Synthesis, anisotropic
solutions, and fibers pf poly(1,4-benzamide). Macromolecules, 10(6):1390–1396,
1977.
[11] S.J. Picken S. Zwaag and M.G. Northolt. Molecular and macroscopic orientational
order in aramid solutions: a model to explain the influence of some spinning
parameters on the modulus of aramid yarns. Polymer, 14(1):2998–3006, July 1992.
[12] P.J. Flory. Statistical Thermodynamics of Semi-Flexible Chain Molecules. Proc. R.
Soc. Lond. A, 234(1196):60–73, January 1956.
130
BIBLIOGRAPHY
[13] A.R. Khokhlov A.N. Semenov. Liquid-crystalline ordering in the solution of long
persistent chains. Physica A, 108:546–556, 1981.
[14] A.R. Khokhlov A.N. Semenov. Liquid-crystalline ordering in the solution of partially flexible macromolecules. Physica A, 112:605–614, 1982.
[15] J.D. Parsons. Nematic ordering in a system of rods. Physical Review A, 19(3):605–
614, March 1979.
[16] R.B. Meyer S.D. Lee. Computations of the phase equilibrium, elastic constants,
and viscosities of a hardrod nematic liquid crystal. The Journal of Chemical Physics,
84(6):3443–3448, March 1986.
[17] S.D. Lee. A numerical investigation of nematic ordering based on a simple
hardrod model. The Journal of Chemical Physics, 87(8):4972–4974, October 1987.
[18] C. Vega S. Lago. Isotropic-nematic transition of hard polar and nonpolar molecules. The Journal of Chemical Physics, 100(9):6727–6737, May 1994.
[19] D.C. Williamson G. Jackson. Liquid crystalline phase behavior in systems of hardsphere chains. The Journal of Chemical Physics, 108(24):10294–10302, June 1998.
[20] P. Bryk R. Roth. Bulk and inhomogeneous mixtures of hard rods and excludedvolume polymer: a density functional approach. Physical Review E, 71:011510–1–8,
2005.
[21] F. Gamez P.J. Merkling S. Lago. Parsons-Lee approach for oblate hard spherocylinders. Chemical Physics Letters, 494:45–49, 2005.
[22] D. Frenkel B.M. Mulder. The hard ellipsoid–of–revolution fluid. Molecular Physics,
55(5):1171–1192, 1985.
[23] H. Fynewever A. Yethiraj. Phase behavior of semiflexible tangent hard sphere
chains. The Journal of Chemical Physics, 108(4):1636–1644, January 1998.
[24] M. Doi. Molecular Dynamics and Rheological Properties of Concentrated Solutions of Rodlike Polymers in Isotropic and Liquid Crystalline Phases. Journal of
Polymer Science: Polymer Physics Edition, 19(2):229–243, February 1981.
[25] M. Doi and S.F. Edwards. The Theory of Polymer Dynamics. Clarendon Press, 1985.
[26] L.D. Landau. On the theory of phase transitions. Part I. Zh. Eksp. Teor. Fiz., 7:19,
1937.
[27] L.D. Landau. On the theory of phase transitions. Part II. Zh. Eksp. Teor. Fiz., 7:627,
1937.
[28] E.F. Gramsbergen L. Longa W.H. de Jeu. Landau theory of the Nematic–Isotropic
phase transition. Physics Reports (Review Section of Physics Letters), 135(4):195–257,
1986.
BIBLIOGRAPHY
131
[29] D. Long and Morse D.C. A rouse-like model of liquid crystalline polymer melts:
Director dynamics and linear viscoelasticity. Journal of Rheology, 42(1):49–92, January/February 2002.
[30] Y.-G. Tao. Kayaking and wagging of rigid rod-like colloids in shear flow. PhD Thesis,
Technische Universiteit Twente, May 2006.
[31] W. Wang M.C. Hund G. Wegner and G. Lieser. Partial phase diagrams of poly(pphenylene terephthalamide) copolymers in sulfuric acid. Polymer, 34(15):3247–
3251, 1993.
[32] M.H. Li A. Brulet P. Keller and J.P. Cotton. Liquid crystal polymers: evidence
of hairpin defects in nematic main chains, comparison with side chain polymers.
Journal of Molecular Structure, 383(1):11–18, 1996.
[33] S.J. Picken L. Noirez G.R. Luckhurst. Molecular conformation of a polyaramid in
nematic solution from small angle neutron scattering and comparison with theory.
J.Chem.Phys., 17:7612–7617, 1998.
[34] D.C. Morse. Viscoelasticity of concentrated isotropic solutions of semiflexible
polymers. 3. nonlinear rheology. Macromolecules, 32:5934–5943, 1999.
[35] H.P. Westerhof. On the Structure and Dissolution Properties of Poly(p-phenylene terephthalamide): Effect of Solvent Composition. PhD Thesis, Technische Universiteit
Delft, December 2009.
[36] X.J. Wang M. Warner. Discrete and continuum models of nematic polymers.
Journal of Physics A: Mathematical and General, 19:2831–2841, 1992.
[37] M. Warner D.R.M. Williams. Statics and dynamics of hairpins in worm-like mainchain nematic polymer liquid crystals. J. Phys. France, 51:317–339, 1990.
[38] J.M. Adamsa M. Warner. Hairpin rubber elasticity. European Physical Journal E:
Soft Matter and Biological Physics, 16:97–107, 2005.
[39] J.L. Ericksen. Theory of anisotropic fluids. Transactions of the society of rheology,
4:29–39, 1960.
[40] N. Kuzuu M. Doi. Constitutive equation for nematic liquid crystals under weak
velocity gradient derived from a molecular kinetic equation. Journal of the Physical
Society of Japan, 52:3486–3494, 1983.
[41] N. Kuzuu M. Doi. Constitutive equation for nematic liquid crystals under weak
velocity gradient derived from a molecular kinetic equation. II. leslie coefficients
for rodlike polymers. Journal of the Physical Society of Japan, 53:1031–1040, 1984.
[42] R.G. Larson. Arrested tumbling in shearing flows of liquid-crystal polymers. Macromolecules, 23:3983–3992, 1990.
132
BIBLIOGRAPHY
[43] R.G. Larson D.W. Mead. Toward a quantitative theory of the rheology of concentrated solutions of stiff polymers. Macromolecules, 29:1271–1285, 1991.
[44] P.L. Maffettone and G. Marrucci. Description of the liquid-crystalline phase of
rod-like polymers at high shear rates. Macromolecules, 22:4076–4082, 1989.
[45] F. Greco and G. Marrucci. The slightly bending-rod model of nematic polymers.
Molecular Crystals and Liquid Crystals, 266:1–8, 1995.
[46] F. Greco and G. Marrucci. The tumbling or flow–aligning nature of nematics
as predicted from the Slightly Bending Rod molecular model. Liquid Crystals,
22(1):11–16, 1997.
[47] F. Greco M. Minale G. Marrucci. Dynamics of stiff polymers with the slightlybending-rod model. Journal of Non–Newtonian Fluid Mechanics, 76:351–362, 1998.
[48] P.L. Maffettone and G. Marrucci. The nematic dumbbell model. Journal of Rheology, 36:1547–1561, 1992.
[49] A.N. Semenov. Rheological properties of a nematic solution of semiflexible macromolecules. Soviet Physics. Journal of Experimental and Theoretical Physics, 66:712–716,
1987.
[50] A. Subbotin. Dynamics of slightly flexible rods in the liquid-crystalline state. Macromolecules, 26:2562–2565, 1993.
[51] C.V. Chaubal L.G. Leal. Effects of Flexibility on Liquid Crystalline Polymer Behavior: The Nematic Broken Rod. Journal of Polymer Science Part B: Polymer Physics,
37:281–300, 1999.
[52] T. Asada and S. Onogi. Rheology and rheo-optics of polymer liquid crystals. Rheology, Plenum, NY (papers presented at Eighth International Congress on Rheology),
1:127–147, 1980.
[53] F. Greco and G. Marrucci. Flow behavior of liquid crystalline polymers. Advances
in Chemical Physics, 86:331–401, 1993.
[54] G. Kiss and R.S. Porter. Rheology of concentrated solutions of poly(gammabenzyl-glutamate). Journal of Polymer Science Part C: Polymer Symposia, 65(193),
1978.
[55] P. Moldenaers and J. Mewis. Transient behavior of liquid crystalline solutions of
poly(gamma-benzyl-glutamate). Journal of Rheology, 30:567–584, 1986.
[56] G. Kiss and R.S. Porter. Rheology of concentrated solutions of helical polypeptides. Journal of Polymer Science Polymer Physics Edition, 18:361–388, 1978.
[57] N. Grizziti S. Cavella and P. Cicarelli. Transient and steady-state rheology of a
liquid crystalline hydroxypropylcellulose solution. Journal of Rheology, 34:1293–
1310, 1990.
BIBLIOGRAPHY
133
[58] P.L. Maffettone and G. Marrucci. Nematic phase of rodlike polymers. i. prediction
of transient behavior at high shear rates. Journal of Rheology, 34:1217–1230, 1990.
[59] P.L. Maffettone and G. Marrucci. Nematic phase of rodlike polymers. i. polydomain predictions in the tumbling domain. Journal of Rheology, 34:1231–1244, 1990.
[60] M. Mortier P. Moldenaers and J. Mewis. Transient rheological behaviour of
poly-para-phenylenetherephthalamide (pppta) in sulfuric acid. Rheologica Acta,
35(1):57–68, January 1996.
[61] R.B. Bird C.F. Curtiss R.C. Armstrong and O. Hassager. Dynamics of polymeric
liquids. Volume 2. John Wiley & Sons, 1987.
[62] N.N. Bogoliubov. Kinetic Equations (in Russian). Journal of Experimental and Theoretical Physics, 16(8):691–702, 1946.
[63] N.N. Bogoliubov. Kinetic Equations. Journal of Physics USSR, 10(3):265–274, 1946.
[64] J. Yvon. La theorie statistique des fluides et l’equation d’etat (in French). Actual.
Sci. and Indust., 203, 1935.
[65] J.G. Kirkwood. The Statistical Mechanical Theory of Transport Processes I. General Theory. The Journal of Chemical Physics, 14(3):180–201, 1946.
[66] M. Born and H.S. Green. A General Kinetic Theory of Liquids I. The Molecular
Distribution Functions. Proc. Roy. Soc. A, 188:10–18, 1946.
[67] J.H. Irving and J.G. Kirkwood. The Statistical Mechanical Theory of Transport
Processes. IV. The Equations of Hydrodynamics. Journal of Chemical Physics,
18(6):817–829, June 1950.
[68] J.W. Gibbs. Elementary principles in statistical mechanics. Charles Scribner’s sons,
1902.
[69] Xin-Jiu Wang and Qi-Feng Zhou. Liquid Crystalline Polymers. World Scientific
Publishing Co., 2004.
[70] W.T. Coffey Yu.P. Kalmykov and J.T. Waldron. The Langevin Equation. World Scientific Publishing Co., 2004.
[71] L.C. Evans. An Introduction to Stochastic Differential Equations. Version 1.2. Department of Mathematics. UC Berkeley, 2006.
[72] N.G. van Kampen. Stochastic Processes in Physics and Chemistry. Elsevier, 2006.
[73] H. Risken. The Fokker-Planck Equation. Springer-Verlag Berlin Heidelberg, 1989.
[74] N.G. van Kampen. Itô Versus Stratonovich. Journal of Statistical Physics, 24(1):175–
187, 1981.
134
BIBLIOGRAPHY
[75] M. Hütter and H. C. Öttinger. Modification of linear response theory for meanfield approximations. Physical Review E, 54(3):2526–2530, 1996.
[76] R. Kubo M. Toda and N. Hashitsume. Nonequilibrium Statistical Mechanics. Statistical Physics II: 2nd ed. Springer, 1985.
[77] H.R. Warner Jr. Kinetic Theory and Rheology of Dilute Suspensions of Finitely Extendible Dumbbells. Industrial and Engineering Chemistry Fundamentals, 11(3):379–
386, 1972.
[78] W. Kuhn and F Grün. Relationships between elastic constants and stretching
double refraction of highly elastic substances. Kolloid-Z., 101:248–271, 1942.
[79] M. Herrchen H.C. Öttinger. The tumbling or flow-aligning nature of nematics
as predicted from the Slightly Bending Rod molecular model. Journal of NonNewtonian Fluid Mechanics, 68:17–42, 1997.
[80] L. Bocquet J. Hansen J. Piasecki. On the Brownian Motion of a Massive Sphere
Suspended in a Hard-Sphere Fluid. II. Molecular Dynamics Estimates of the Friction Coefficient. Journal of Statistical Physics, 76:527–548, 1994.
[81] P.E. Rouse. A Theory of the Linear Viscoelastic Properties of Dilute Solutions of
Coiling Polymers. Journal of Chemical Physics, 21(7):1272–1281, 1953.
[82] B.H. Zimm. Dynamics of Polymer Molecules in Dilute Solution: Viscoelasticity,
Flow Birefringence and Dielectric Loss. Journal of Chemical Physics, 24(2):269–278,
1956.
[83] D.C. Morse. Viscoelasticity of Concentrated Isotropic Solutions of Semiflexible
Polymers. 1. Model and Stress Tensor. Macromolecules, 31:7030–7043, 1998.
[84] D.C. Morse. Viscoelasticity of Concentrated Isotropic Solutions of Semiflexible
Polymers. 2. Linear Response. Macromolecules, 31:7044–7067, 1998.
[85] D.C. Morse. Viscoelasticity of Concentrated Isotropic Solutions of Semiflexible
Polymers. 3. Nonlinear Rheology. Macromolecules, 32:5934–5943, 1999.
[86] O. Matveichuk J.J.M. Slot. A Rouse-like model for highly ordered main-chain liquid crystalline polymers containing hairpins. CASA report, 10(73):1–24, December 2010.
[87] O. Matveichuk J.J.M. Slot. A rod-spring model for main-chain liquid crystalline
polymers containing hairpins. CASA report, 12(16):1–20, May 2012.
[88] C.D. Meyer. Matrix Analysis and Applied Linear Algebra. The Society for Industrial
and Applied Mathematics (SIAM), 2000.
[89] R. Avikainen. On irregular functionals of SDEs and the Euler scheme. Finance
Stoch., 13:381–401, 2009.
BIBLIOGRAPHY
135
[90] P.E. Kloeden. The Systematic Derivation of Higher Order Numerical Schemes for
Stochastic Differential Equations. Milan Journal of Mathematics, 70:187–207, 2002.
[91] K. Kubilius E. Platen. Rate of weak convergence of the euler approximation for
diffusion processes with jumps. Monte Carlo Methods Appl., 8(1):83–96, 2003.
[92] P. Glasserman N. Merener. Numerical solution of jump-diffusion LIBOR market
models. Finance Stoch., 7(1):1–27, 2003.
[93] E. Platen W.J. Runggaldier. A benchmark approach to portfolio optimization under partial information. Asia-Pacific Financial Markets, 14(1):25–43, 2007.
[94] P.E. Kloeden E. Platen. Numerical solution of stochastic differential equations. Applications of Mathematics. Berlin, New York: Springer-Verlag, 1992.
[95] P. Doty H. Bernoit. Light scattering from non-Gaussian chains. Journal of Physical
Chemistry, 57(9):958–963, 1953.
[96] J.R. Schaefgen V.S. Foldi F.M. Logullo V.H. Good L.W. Gulrich F.L. Killian.
Viscosity-molecular weight relationshios in stiff-chain aromatic polyamides. Polymer Preprints, 17:69–74, 1976.
[97] P.M. Cotts G.C. Berry. Studies on dilute solutions of rodlike macroions. ii: electrostatic effects. Journal of Polymer Science: Polymer Physics Edition, 21(8):1255–1274,
1983.
[98] B. Chu Q. Ying. Persistence length of poly(1,4-phenyleneterephthalamide) in concentrated sulfuric acid. Macromol. Chem., Rapid Commun., 5:785–791, 1984.
[99] S. Wolfram. A new kind of science. Wolfram Media, 2002.
[100] Z. Xuelong W. Jiwen X. Manwu L. Fengyu. High-Quality Pseudo-Random
Sequence Generator based on One-dimensional Extended Cellular Automata.
lnfoSecu04, pages 222–223, November 2004.
[101] V. Faraoni M. Grosso S. Crescitelli and P.L. Maffettone. The rigid-rod model for
nematic polymers: An analysis of the shear flow problem. Journal of Rheology,
43(3):829–843, 1999.
[102] S. Hess. Non-newtonian viscosity and normal pressure associated with the flow
alignment in macromolecular liquids. Physica A, 86:383–399, 1977.
Summary
Constitutive modeling of concentrated solutions of mainchain 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 semiflexible LCP solutions are used in the production of fibers with outstanding mechanical properties. Understanding of the relation between the processing of LCP solutions
and the ultimate properties of the fibers requires the development of constitutive models 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 formation of hairpins and their connection with entanglements in concentrated semi-flexible
LCP solutions are still of great interest. However, the introduction of these concepts increases 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 concentrated 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 function representing a single polymer chain immersed into the LCP solution. We also involve the concept of the mean-field approximation for the nematic interaction (MaierSaupe 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 corresponding 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 entangled 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 peculiar dynamics of the director: kayaking, wagging and flow-aligning. From these simulations 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 experimental 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 eigenschappen. Het begrijpen van de relatie tussen de verwerking van LCP oplossingen
en de uiteindelijke eigenschappen van deze vezels vereist het ontwikkelen van constitutieve 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 geconcentreerde oplossingen bestaande uit semiflexibele LCPs die dit soort hairpins bevatten.
Tevens verdisconteerd het model ook het feit dat met de vorming van hairpins het aantal 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 aannames 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 Stratonovich 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 omstrengelde 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 afschuifstroming 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 stroomrichting” (flow-aligning). Uit deze simulaties kan ook de stationaire afschuifviscositeit bepaald worden. Deze grootheid vertoont een plateau voor niet al te hoge afschuifsnelheden 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 afschuifsnelheden 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 Hütter, 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 atmosphere 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 fruitful 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 special. Thank you Patricio Rosen, Maria Ugryumova, Mirela Darau, Kundan Kumar, Sudhir 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 Mercuri, 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 appreciated 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 Marèse Wolfs-van de Hurk who helped me with wide variety of administrative 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, Alexander 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 started 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 ”Dimerization 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 Technology 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.