Binary Mixtures of Rod–like
Colloids:
Mesoscopic Equilibrium Theory
and
Shear Induced Instabilities
vorgelegt von:
Master of Science (Physics)
Rodrigo Lugo-Fr´
ıas
aus
Pachuca, Hidalgo, Mexiko
Von der Fakult¨at II – Mathematik und Naturwissenschaften
der Technischen Universit¨at Berlin
zur Erlangung des akademischen Grades
Doktor der Naturwissenschaften
Dr. rer. nat.
genehmigte Dissertation
Promotionsausschuss:
Vorsitzender: Prof. Dr. Michael Lehmann
1. Gutachterin: Prof. Dr. Sabine H. L. Klapp
2. Gutachter: PD Dr. Andreas Menzel
Tag der wissenschaftlichen Aussprache: 28. Juli 2016
Berlin 2016
I
Declaration of Authorship
Hiermit erkl¨are ich, dass ich die vorliegende Arbeit selbstst¨andig und eigenh¨andig sowie
ohne unerlaubte fremde Hilfe und ausschliesslich unter Verwendung der aufgef¨uhrten
Quellen und Hilfsmittel angefertigt habe.
Berlin, den
Rodrigo Lugo-Fr´ıas
III
Publications
Most of this work is based on the publications listed below. Texts are partially rewritten
and some extensions are introduced. Contributions from co-authors presented in this
work are indicated in the text.
Binary mixtures of rod-like colloids under shear: microscopically-based equi-
librium theory and order-parameter dynamics, R. Lugo-Frias and S. H. L. Klapp, J.
Phys.: Condens. Matter,28, 244022, (2016)
This paper is concerned with the dynamics of a binary mixture of rod-like, repulsive
colloidal particles driven out of equilibrium by means of a steady shear flow (Couette
geometry). To this end we first derive, starting from a microscopic density functional in
ParsonsLee approximation, a mesoscopic free energy functional whose main variables are
the orientational order parameter tensors. Based on this mesoscopic functional we then
explore the stability of isotropic and nematic equilibrium phases in terms of composition
and rod lengths. Second, by combining the equilibrium theory with the DoiHess approach
for the order parameter dynamics under shear, we investigate the orientational dynamics
of binary mixtures for a range of shear rates and coupling parameters. We find a variety
of dynamical states, including synchronized oscillatory states of the two components,
but also symmetry breaking behavior where the components display different in-plane
oscillatory states.
Shear banding in nematogenic fluids with oscillating orientational dynamics,
R. Lugo-Frias, H. Reinken and S. H. L. Klapp, submitted European Physical Journal E.
(2016)
We investigate the occurrence of shear banding in nematogenic fluids under pla-
nar Couette flow, based on mesoscopic dynamical equations for the orientational order
parameter and the shear stress. We focus on parameter values where the sheared homo-
geneous system exhibits regular oscillatory orientational dynamics, whereas the equilib-
rium system is either isotropic (albeit close to the isotropic–nematic transition) or deep
in its nematic phase. The numerical calculations are restricted to spatial variations in
shear gradient direction. We find several new types of shear banded states characterized
by regions with regular oscillatory orientational dynamics. In all cases shear banding is
accompanied by a non–monotonicity of the flow curve of the homogeneous system; how-
ever, only in the case of the initially isotropic system this curve has the typical S–like
shape. We also analyze the influence of different orientational boundary conditions and
of the spatial correlation length.
V
Abstract
In this work we investigate the shear induced instabilities occurring in colloidal binary
mixtures of rigid anisotropic particles. To this end, we derive a theoretical framework
including equilibrium and non–equilibrium methods.
The equilibrium description is derived on the basis of microscopic density functionals
in the Parsons–Lee and Ramakrishnan–Youssuf approximations. The result is a meso-
scopic free energy given in terms of a set of orientational order parameter tensors (one
for each component of the mixture) and their gradients. In this regard, our theory may
be seen as an extension of the Landau–de Gennes theory to mixtures. However, our
theory incorporates microscopic information, particularly the aspect ratios and number
densities characterizing the components of the mixture. Specializing to binary mixtures
of rod–like particles, these functionals are then used to explore the stability of isotropic
and nematic equilibrium phases in terms of composition and rod lengths.
We extend the description to consider non-equilibrium phenomena by combining our
equilibrium theory with the theory of irreversible processes. Thus, extending the Doi–
Hess approach to the study of binary mixtures under shear. As a result, we obtain a set of
hydrodynamic equations which take into account the competition between flow–induced
effects on the alignment and the relaxation of the entire system towards equilibrium.
Based on these hydrodynamic equations we turn to the study of shear induced
phenomena under planar Couette flow geometry. These investigations are divided in two
parts. First, we focus on a one–component system and study the underlying relation
existing between shear and stress. We observed that disregarding the equilibrium (un–
sheared) state of the system (close to the isotropic–nematic transition or deep in its
nematic phase), we find several new types of shear banded states characterized by regions
with regular oscillatory orientational dynamics.
Finally we concentrate on binary mixtures. Here, after exploring a vast range of pa-
rameters, we observe several dynamical states appearing in the system. These dynamical
states include the synchronization of of in– and out–of–the shear plane oscillatory states
of the two components. Interestingly, for a specific set of parameters, we observe the ap-
pearance of symmetry breaking behavior where both components of the mixture display
different in–phase oscillatory states.
VII
Zusammenfassung
In dieser Arbeit untersuchen wir scherinduzierte Instabilit¨aten, welche in bin¨aren Mis-
chung von starren anisotropen Teilchen auftreten. Hierzu leiten wir eine theoretische
Beschreibung mit Gleichgewichts- und Nichtgleichgewichtsmethoden her.
Die Gleichgewichtsbeschreibung wird auf der Basis von einer mikroskopische Dichte-
funktionaltheorie mit Hilfe von den N¨aherungen von Parsons–Lee und Ramakrishnan–
Youssuf hergeleitet. Dieser Ansatz f¨uhrt zu einem Ausdruck f¨ur die mesoskopischen
freien Energie als Funktion von tensoriellen Ordnungsparametern (einer f¨ur jede Mis-
chungskomponente) sowie deren Gradienten. Hierdurch kann unsere Arbeit als Erweiter-
ung der Landau-de GennesTheorie zur Beschreibung von Mischungen angesehen werden.
Allerdings ber¨ucksichtigt unsere Theorie die Abh¨angigkeit von mikroskopische Gr¨ossen,
insbesondere dem Verh¨altnis von Teilchenl¨angen und -durchmesseren, sowie die Teilchen-
dichten, welche die Mischungskomponenten charakterisieren. F¨ur den Spezialfall von
bin¨are Mischungen von st¨abchenf¨ormigen Teilchen verwenden wir das resultierende freie
Energiefunktional, um die Stabilit¨at von isotropen und nematischen Gleichgewichtsphasen
in Bezug auf ihre Zusammensetzung und St¨abchenl¨angen zu untersuchen.
Zur Beschreibung von Nichtgleichgewichtsprozessen verbinden wir die Gleichgewicht-
stheorie mit der Theorie irreversibler Prozesse. Dies entspricht einer Erweiterung der
Doi–Hess–Theorie zur Beschreibung von bin¨aren Mischungen unter Scherung. Wir er-
halten ein System aus hydrodynamischen Gleichungen, welche den Wettstreit zwischen
str¨omungsinduzierten Effekten und der Relaxation des gesamten Systems hin zum Gle-
ichgewicht beschreibt.
Auf Basis dieser hydrodynamischen Gleichungen widmen wir uns der Untersuchung
von scherinduzierten Ph¨anomenen in einer ebenen Couette-Geometrie. Diese Unter-
suchungen bestehen aus zwei Teilen. Zuerst betrachten wir ein einkomponentiges System
und die zugrundeliegende Beziehung zwischen Scherrate und Spannung. Ausgehend von
unterschiedlichen Gleichgewichtszust¨anden (nahe des isotrop–nematischen Phasen¨uber-
gangs und tief in der nematischen Phase) finden wir in den gescherten Systemen r¨aum-
lich inhomogene Strukturen, sogenannte ”shear bands”. Diese sind durch Bereiche mit
regul¨aren oszillatorischen Ausrichtungsdynamiken charakterisiert.
Anschliessend konzentrieren wir uns auf gescherte bin¨are Mischungen. Hierbei iden-
tifizieren wir nach der Untersuchung eines grossen Parameterbereichs verschiedene dy-
namische Zust¨ande, welche sowohl synchronisierte Oszillationen innerhalb als auch ausser-
halb der Scherebene beeinhalten. Bemerkenswert ist insbesondere die Beobachtung eines
symmetriebrechenden Verhaltens f¨ur bestimmte Parameterwerte. Hierbei weisen die bei-
den Mischungskomponenten verschiedene, aber phasengleiche oszillatorische Zust¨ande
auf.
Contents
Title page
Abstract V
Zusammenfassung VII
Contents IX
1 Introduction 1
1.1 Colloidal systems of rigid rod–like particles ................ 1
1.2 Equilibrium behavior of mixtures ..................... 2
1.3 Non–equilibrium behavior under shear .................. 4
1.4 Motivation, goals and scope of the thesis ................ 7
1.4.1 Outline ............................... 7
2 Phenomenological description 9
2.1 Measure of orientational order ...................... 9
2.1.1 Scalar order parameter ...................... 10
2.1.2 Tensor order parameter ...................... 11
2.1.3 Some properties of the second rank alignment tensor ...... 13
2.2 The isotropic–nematic phase transition .................. 14
2.2.1 Lyotropic energy .......................... 14
2.2.2 Elastic energy ........................... 16
2.2.3 Surface energy ........................... 18
3 Microscopical description 19
3.1 Systems of rigid anisotropic particles ................... 19
3.1.1 Pair interaction potential ..................... 20
3.1.2 Onsager’s second virial theory .................. 22
3.2 Density functional theory ......................... 23
3.2.1 Excess free energy ......................... 25
3.2.2 Density functional theories for systems of hard rods ....... 26
IX
X CONTENTS
3.3 Density functional theories for mixtures .................. 30
3.3.1 Many–fluid Parsons–Lee theory .................. 31
3.3.2 Second–order perturbation theory for mixtures .......... 32
4 Bridging microscopic and mesoscopic theories 35
4.1 DFT based Q–tensor theory for mixtures ................. 35
4.2 Homogeneous systems ........................... 36
4.2.1 Ideal contribution ......................... 36
4.2.2 Excess contribution ........................ 36
4.2.3 Full free energy .......................... 37
4.3 Inhomogeneous systems .......................... 39
4.3.1 Ideal contribution ......................... 40
4.3.2 Excess contribution ........................ 41
4.3.3 Gradient expansion of the direct correlation function ...... 43
4.3.4 Full free energy .......................... 46
4.4 Comparison between mesoscopic free energies .............. 47
4.5 Summary .................................. 48
5 Application to binary mixtures 51
5.1 Binary mixtures of hard spherocylinders ................. 51
5.1.1 Orientational free energy ..................... 52
5.2 Isotropic–Nematic transition ....................... 55
5.2.1 Single component system ..................... 56
5.2.2 Binary mixture ........................... 57
5.2.3 Scaled free energy ......................... 59
5.3 Elastic energy ............................... 63
5.3.1 Oseen–Frank theory ........................ 64
5.4 Summary .................................. 65
6 Dynamic equations and constitutive relations for binary mixtures 67
6.1 General remarks .............................. 67
6.2 Extension to binary mixtures ....................... 68
6.2.1 Gibbs fundamental relation .................... 69
6.2.2 Conservation laws and balance equation ............. 70
6.2.3 Entropy production and phenomenological equations ...... 71
6.2.4 Complete dynamic equations ................... 72
6.2.5 Equation for the stress tensor ................... 73
6.3 Relaxation of the alignment ........................ 74
6.3.1 Vanishing cross–coupling of the entropy production ....... 75
6.3.2 Final dynamic equations ..................... 77
6.4 Simple Couette geometry ......................... 78
6.4.1 Scaled dynamic equations ..................... 78
CONTENTS XI
6.4.2 Scaled momentum balance equation ............... 79
6.5 Explicit equations of motion ....................... 80
6.6 Summary .................................. 82
7 Shear induced instabilities in binary mixtures 83
7.1 General Remarks .............................. 83
7.1.1 Dynamical state diagram ..................... 84
7.2 Shear banding with oscillating orientational dynamics .......... 85
7.2.1 Numerical calculations and boundary conditions ......... 85
7.2.2 Homogeneous solutions ...................... 87
7.2.3 Spatiotemporal behavior and shear banding ........... 90
7.3 Binary mixtures of rod–like colloids .................... 94
7.4 Orientational dynamics of binary mixtures ................ 95
7.4.1 Variation of the tumbling parameter and the shear rate ..... 95
7.4.2 Variation of the cross coupling and the shear rate ........ 98
7.5 Summary .................................. 99
8 Concluding remarks 101
8.1 Summary .................................. 101
8.2 Outlook and further investigations .................... 103
Appendices 105
A Basic tensor concepts 107
B Spherical harmonics and their tensor representation 115
Bibliography 121
Acknowledgements 133
Chapter 1
Introduction
This work contains many things which are new and
interesting. Unfortunately, everything that is new is not
interesting, and everything which is interesting, is not new.
–Lev Davidovich Landau
Fluids composed of orientable particles form the basis for a range of high–end
technologies such as drug delivery, immunodetection and the design of new func-
tional materials. In general, these devices are composed of colloidal suspensions,
where anisotropic particles (in the micrometer range) are suspended in liquids.
Typically, applications require stable suspensions which are often obtained at very
high solute concentrations. However, in practice, these systems are highly poly-
disperse (i.e., composed of various shapes and sizes) and this may have strong
effects in the stability of the system. Thus, in order to tune key properties of such
systems, external fields, like shear and electro–magnetic fields, are usually applied.
The goal of this chapter is to introduce the reader to the study of mixtures of
rigid anisotropic particles in and out of equilibrium. To this end, we address some
recent experimental and theoretical developments of the behavior of such systems.
In particular, we discuss the underlying equilibrium phenomenology and the effect
of shear on the structure and rheological signature of these fluids. Finally, at the
end of the chapter, we list the motivation, goals and an outline of this thesis.
1.1 Colloidal systems of rigid rod–like particles
A colloidal suspension is a system where microscopic particles, ranging from nano– to
micrometer dimensions, are dispersed in a solvent composed of atomic–sized molecules.
In this thesis, we focus on the study of mixtures of rigid anisotropic particles embedded
in aqueous solutions.
1
2 Chapter 1. Introduction
Figure 1.1: (a) Example of single component dispersions of rod–like, disk–like and ellip-
soidal particles in aqueous solutions (see Ref. [8]). (b) Transmission electron microscopy
image of a mixture of magnetic platelets embedded in a nematic host (see Ref. [1]).
(c) Scanning electron microscopy image of a polydisperse solution of ”matchstick” rod–
like particles (see Ref. [2]).
To be more specific, we deal with particles having simple morphologies characterized
by their aspect ratio, i.e., the ratio between their length and diameter. Examples include
magnetic platelets embedded in nematic hosts [1] and polydisperse ”matchstick” rod–like
particles [2] (see Fig. 1.1). The versatility of these systems has allowed their successful
application in high–end technologies, e.g., panels and photographic devices [3,4], smart
windows [5] and even in drug delivery and immunodetection [6,7].
To ensure an effective performance, almost all devices depend on the optimization of
key material properties, e.g., rotational viscosity and dielectric anisotropy. Consequently
mixtures of colloidal rod–like particles are designed to be thermodynamically stable at
very high concentrations [9]. However, because of the complex orientational behavior
and high sensitivity to external perturbations, this is often a difficult task. Thus, it is
often necessary to apply external fields such as shear (see e.g. [10]) and electro–magnetic
fields (see e.g. [11]) to tune the performance of the fluid.
1.2 Equilibrium behavior of mixtures
Already in thermodynamic equilibrium, mixtures of rigid anisotropic particles display
a very rich phase behavior typically dominated by repulsive interactions arising from
the particle’s shape. In general, these fluids exhibit two kinds of long range ordering:
ordering of the centers of mass and ordering of the orientations of the particles. In
the absence of translational ordering, as the density increases, systems with little long–
range orientational order undergo a phase transition from the isotropic (orientationally
disordered) to the nematic (orientationally ordered) phase.
1.2. Equilibrium behavior of mixtures 3
Figure 1.2: Phase separation of binary mixtures of thin and thick rod–like particles.
(a) Highly fractionated isotropic–nematic coexistence of thin and thick fd–viruses [16].
Here Ndenotes the location of the nematic fluid (rich in thick particles) whereas I
denotes the isotropic fluid (rich in thin particles) (b) Snapshot of computer simula-
tions of binary mixtures of thick and thin hard spherocylinders [17]. Here the domain
is splitted into isotropic rich long–spherocylinders in the front and isotropic rich short–
spherocylinders in the back.
A prominent example of the complicated equilibrium behavior of true polydisperse
fluids is the experimental study of clay rods in Ref. [12]. Here they measure the collective
reorientation of the particles by looking at changes of the optical properties of the
sample, e.g. birefringence and light transmittance. However, recent studies focus on
binary mixtures as the first step towards understanding polydisperse fluids.
In fact, recent investigations of binary mixtures of rod–like particles show that these
systems display a very rich, entropy–driven phase behavior, including phase separation,
demixing and order–disorder phase transitions [13,14,15,16,17,18] (see Fig. 1.2).
Noteworthy are the phase diagrams of short and long molecules obtained by Monte
Carlo simulations [14], and the experimental phase diagrams of binary mixtures of thin
and thick fd–viruses [16].
From a theoretical standpoint, such phase behavior may be studied using two differ-
ent levels of description: a phenomenological (mesoscopic) description and a particle–
based (microscopic) description.
The phenomenological approach is based on a very simple idea: In orientationally
ordered phases the particles are, on average, aligned along a common direction; in this
case the system is said to be in a nematic state, and the local direction of alignment is
given in terms of the nematic director ˆ
n=ˆ
n(r, t)[19]. In the same way, it is possible
to describe the spatial distortions of the nematic director using a more general theory
which includes spatial derivatives of ˆ
n[20,21]. Further, the degree of orientational
4 Chapter 1. Introduction
order is then measured by defining a suitable set of order parameters. In the scope of
the Q–tensor theory, one can define a mesoscopic order parameter in terms of the one–
body orientational distribution function f(ˆu)(where ˆucontains the set of Euler angles
describing the particle’s orientation [19,22]). The resulting order parameter, Q, is a
second–rank tensor which corresponds to the lowest non–vanishing moment of f(ˆu)and
it can be measured in experiments looking at the birefringence and fluorescence of the
sample [19,23]. In terms of Qand its spatial derivatives, the phase behavior of the system
is described using the phenomenological Landau–de Gennes (LdG) theory [19]. However,
this description lacks quantitative measurement of the microscopic characteristics of the
system.
The microscopical description studies the equilibrium behavior of the system in terms
of all degrees of freedom of each particle. In certain approximations, this has been
achieved successfully, e.g., for needle–like particles in the low density limit [24]. More
generally, one can employ the framework of classical density functional theory (DFT) [25,
26], whose main idea is to minimize a (free energy) functional of the singlet density
ρ(r,ˆu)where rand ˆuaccount for translational and rotational degrees of freedom,
respectively [27,28].
For one component systems several studies have been devoted to construct free
energy functionals for homogeneous states of anisotropic fluids. Their common goal is to
describe the isotropic–nematic phase transition [27,29] or the distortions of the nematic
phase [30,31,32]. Considering binary mixtures, the effects of shape (and in some cases
flexibility) have been accounted for in various theoretical studies (see e.g. [33,34] and
references therein). Of special importance for the present study are the DFT description
of spatial homogeneous phases of mixtures of Malijevsky et al. [35] and the Phase–Field–
Crystal Model for liquid crystals of Wittkowski et al. [36]. These theoretical descriptions
are derived, respectively, as a generalization of the ideas of Parsons [37] and Lee [38]
for pure component systems, and an extension of a second–order perturbation theory in
the Ramakrishnan-Youssuf approximation [39,40,41].
1.3 Non–equilibrium behavior under shear
While the equilibrium behavior of mixtures is already challenging by itself, a further level
of complexity is reached when these systems are considered under shear [42,43,44].
Shear flow is a fundamental example of driving a soft–matter system out of equilib-
rium [45]. From a fundamental perspective, the combination of shear and the complex
morphologies of the particles can generate a variety of instabilities in the system. Promi-
nent examples are the shear induced shift of the isotropic–nematic transition and the
stabilization of the nematic phase [46,47], the appearance of orientational oscillatory
states [48,49,50], rheochaos [51,52] and the occurrence of spatially inhomogeneous
instabilities like shear banding [53,54]. Thus, investigating the impact of shear is crucial
for understanding the system’s rheological behavior.
1.3. Non–equilibrium behavior under shear 5
Figure 1.3: (a) Order parameter P2as a function of time showing the dynamics of the
system in an orientational oscillatory states (adapted form Ref. [60]). (b) The path of
the nematic director in the unit sphere in the wagging oscillatory state [55].
Here, we are interested in the study of two different types of shear induced phenomena
occurring in fluids of rigid anisotropic particles: shear induced oscillatory states and shear
banding instabilities. In general, these non–equilibrium phenomena are characterized by
a time–dependent collective behavior of the orientable particles. However, modeling
these collective shear induced phenomena on a microscopic level is still a challenge.
Thus, one typically employs particle based computer simulations [55,56] or non–linear
mesoscopic theories for the dynamics of the orientational order parameter Q[57,23,
58,59]. The later assumes that the nematic director ˆ
n(r, t)is effectively coupled with
the hydrodynamic flow field v(r, t).
Shear induced oscillatory states: Already for homogeneous systems, particles in a
shear flow can display spontaneous time–dependent oscillations. A prime example is the
wagging motion (characterized by the alternation of the nematic director around zero
along the shear direction) which has been observed in experiments [60] as well as in
particle–resolved computer simulations [55] (see Fig. 1.3).
A well–established path to explore the orientational dynamics under shear are the
mesoscopic equations describing the evolution of Qin time. Within the Doi–Hess ap-
proach [57,23], the time dependence of Qhinges on a competition between flow–induced
perturbations, and a contribution accounting for the relaxation of the system towards
equilibrium (stemming from the functional derivative of the phenomenological LdG free
energy [19]). Depending on the initial equilibrium (un–sheared) state of the system, these
nonlinear equations uncover a huge variety of complex dynamical behavior characterized
by different (regular) oscillatory states [48,49,50], chaotic behavior [51] and complex
spatiotemporal patterns [61,62,63].
6 Chapter 1. Introduction
Figure 1.4: (First row) Flow birefringence experiments showing shear banding structures
on a CTAB (cetyltrimethyammoniumbromide) solution in a Couette cell upon increasing
shear rate from (a) ˙γ≃3s
−1to (b) ˙γ≃16s
−1(taken from Ref. [70]). (Second row)
(c) Flow curve of a shear banding fluid. (d) Velocity profile in a shear banded flow (taken
from Ref. [71]).
Shear banding instabilities: The manifestation of shear induced orientational dy-
namics must have a direct impact on the rheological properties of the system. Thus, the
emerging oscillatory states may be associated with a non–Newtonian behavior of the
viscosity. In fact, when the applied shear rate goes beyond a critical value, the homo-
geneous system becomes unstable and separates into macroscopic bands with different
local shear rates (or stresses) (see Fig. 1.4(d)). This phenomenon is often referred to as
shear banding and was observed (for the first time) in worm–like micellar solutions by
Rehage and Hoffmann [64,65] (see e.g. Refs. [66,67,68] for recent reviews). More re-
cently, Chen et al and Decruppe et al have shown that this phenomenon can be observed
in rod–like colloidal dispersions [69] and liquid–crystalline polymers [70] (see Figs. 1.4(a)
and 1.4(b)). In all mentioned cases, the flow induced reorganization of the fluid’s mi-
crostructure leads to a non–monotonic relation between shear stress and shear rate, the
so–called flow curve of the fluid (see Fig. 1.4(c)).
Theoretically, shear banding (and the related vorticity banding [72]) has been studied
mainly via continuum models. A standard theory is the diffusive Johnson-Segelman (DJS)
model [73,74] in which the viscosity decreases with the stress (shear thinning) and
the shear bands are formed along the gradient direction. In contrast, recent theoretical
1.4. Motivation, goals and scope of the thesis 7
studies [52,75] have used the Doi–Hess model, by choosing a suitable parameter range,
to study the emergence irregular dynamical shear banding corresponding to rheochaos.
The choice of the Doi–Hess model over the DJS model is because the first one depends
on Q. Thus, the study of spatiotemporal complex dynamics and spatial inhomogeneities,
at the same time, is accessible.
1.4 Motivation, goals and scope of the thesis
The purpose of this thesis is to investigate the collective behavior of binary mixtures of
rod–like colloids under shear. The focus of our investigations is to study the nature of
the shear induced oscillatory states arising in such systems with respect to the mixture’s
components, as well as the possible occurrence of shear banding instabilities and the
related rheological behavior.
Specifically, we aim to develop novel techniques based on quantities that are directly
accessible to experiments, e.g., the orientational order parameter tensor Q. The start-
ing point of our investigations is related to the complex equilibrium phase behavior of
binary mixtures. In fact, as shown for one–component fluids, a deep understanding of
the underlying equilibrium behavior of the system leads to straightforward investigations
of the non–equilibrium setup. Thus, the first goal of our approach, is to construct an
expression for the equilibrium free energy functional capable of describing the thermody-
namic phase behavior of the mixture. To this end we use techniques from classical DFT
to develop a mesoscopic functional given in terms of the orientational order parameter
tensors, Qαwhere α=A, B (one for each component of the mixture).
Second, we couple the Q–tensors to the hydrodynamic flow field v(r, t). For this
purpose, the free energy functional resulting from the equilibrium study is used as an
input to extend the well–known Doi–Hess theory to the study of binary mixtures. Finally,
by expanding the tensorial system of equations in terms of an appropriate tensor basis,
we analyze the resulting equations, exploring a vast range of range of parameters.
1.4.1 Outline
In order to cover the content listed above, this work is organized as follows: First, in
Chapters 2and 3we introduce the theoretical background necessary to understand the
phenomenological and microscopic descriptions of rod–like systems in equilibrium.
We start our own investigations in Chapter 4. There we use the multicomponent
Parsons–Lee theory [35] and a second–order perturbation theory for mixtures as a start-
ing point to construct an approximate free energy functional given in terms of the
tensorial order parameters, Qα, and its gradients, ∇Qα, where α= 1,··· , n labels the
ncomponents of the mixture.
Further, in Chapter 5we concentrate on the study of binary mixtures in equilibrium.
Special attention is given to the semi–dilute regime in Section 5.2, where we investigate
8 Chapter 1. Introduction
the stability of isotropic and nematic states (for fixed densities). In Section 5.3 we briefly
discuss the mesoscopic free energy density of distortion for binary mixtures.
In Chapter 6, we combine our equilibrium theory with the theory of irreversible
processes in order to extend the Doi-Hess theory to binary mixtures of rigid anisotropic
particles. In Section 6.5, upon expanding the alignment tensors and the stress in terms
of an appropriate tensor basis, we obtain a set of coupled differential equations for the
independent components of Qαand T.
In Chapter 7, we present numerical results of the shear banding in one component
systems and shear induced orientational dynamics of binary mixtures. This work then
finishes in Chapter 8with concluding remarks and an outlook. Some technical details
are given in the Appendices Aand B.
Chapter 2
Phenomenological description
Fluids consisting of anisotropic particles exhibit two kinds of long range ordering:
ordering of the centers of mass and ordering of the orientations of the particles.
In a system characterized by little to no long–range orientational order, as the
density is gradually increased, the ordering of the molecules becomes larger. This
behavior constitutes the general picture of the isotropic–nematic phase transition.
In classic liquid crystal theories, the phenomenology of this transition is de-
scribed in terms of a set of appropriate order parameters. The aim of this chapter
is to introduce the tensor order parameter Qand the ideas behind the well es-
tablished phenomenological Landau–de Gennes theory. This theory is a standard
premise used in the study of suspensions of rod–like (disc–like) particles and liquid
crystals. 1
2.1 Measure of orientational order
In the absence of long range positional ordering the particles are, on average, aligned
along a common direction. In this case the system is in a nematic state, where the
local direction of alignment is defined by the nematic director. In different regions of
the fluid, this director changes from point to point and may also depend on time, thus,
ˆ
n=ˆ
n(r, t). When both, positional and orientational order vanish, the system is in an
isotropic fluid phase. A sketch of this behavior is shown in Fig. (2.1). The director ˆ
nis
a local measure of the average molecular orientation of the particles in the fluid.
A single particle has its own orientation which is specified by the unit vector ˆ
u. In
general, depending on the complexity of the particles ˆ
uis a complicated vector dependent
1This chapter is based on four publications: Landau theory of nematic–isotropic phase transition by
E. F. Gramsbergen et al [76], The physics of liquid crystals by P. G. de–Gennes [19], Introduction to
Q–tensor theory by N. J. Mottram et al [77] and Tensors for physics by S. Hess [22].
9
10 Chapter 2. Phenomenological description
Figure 2.1: Rod–like particles in the (a) isotropic fluid phase and (b) nematic phase. In
the isotropic phase there is not an average orientation of the particles, whereas, in the
nematic phase the average orientation is given by the nematic director ˆ
n.
on Euler angles and presents no symmetries. However, for uniaxial particles the head–
to–tail symmetry is conserved and ˆ
u=−ˆ
u. For these systems, a measure of how spread
out is the particle’s orientation with respect to the nematic director is given by the angle
θp, defined by the relation
cos θp=ˆ
u·ˆ
n.(2.1)
In real systems it is impossible to track the amount of orientational order of each
particle with respect to the director. Hence the statistical state of the fluid is char-
acterized by a probability distribution function in terms of the molecular orientations,
f=f(ˆ
u,r,t). In terms of f, the probability to find a molecule in [r,r+dr]with an
orientation [ˆ
u,ˆ
u+dˆ
u]at a given time tis f(ˆ
u,r,t)drdˆ
u. The orientational distribution
function satisfies the normalization condition
S2
f(ˆ
u,r,t)dˆ
u=1,(2.2)
where f(ˆ
u,r,t)is integrated over the an arbitrary surface S2, corresponding to the solid
angle integral. As a last remark, one should note that considering all the molecules in a
volume V, the mean value of f(ˆ
u,r,t)defines the direction of the nematic director ˆ
n.
2.1.1 Scalar order parameter
The usual measure of the amount of order in fluids of anisotropic particles is the scalar
order parameter Sintroduced by A. Saupe and W. Maier in 1959 [78]. Like in any order
parameter theory, a common requirement is that it satisfies the following: it should
vanish in the more symmetric (less ordered) phase and it must be different from zero in
the less symmetric (more ordered) phase [76].
2.1. Measure of orientational order 11
According to this principle, Saupe and Meier defined a scalar order parameter S
as a weighted average in terms of the second order Legendre polynomial. This order
parameter is given by
S=
S2
P2(ˆ
u·ˆ
n)f(ˆ
u)dˆ
u.(2.3)
When the material has all its molecules exactly aligned with the director, cos2θp=1
and therefore S=1. In contrast, when all molecules are randomly oriented in a plane
perpendicular to the director cos2θp=0and thus S=−1/2. In the isotropic liquid
phase the molecules are randomly oriented, f(ˆ
u)=1/4πand as a result S=0. When
greater accuracy is required it is usual to define an order parameter in terms of higher
order polynomials of the Legendre expansion [77].
In general the nematic state is not only defined by a single nematic director. In fact,
in some materials there are two or three axis of complete rotational symmetry. For such
materials a set of perpendicular axes {ˆ
l,ˆ
m,ˆ
n}may be defined. The uniaxial or biaxial
macroscopic nature of the system should not be confused with the geometry of the
particles. In principle uniaxial particles (e.g. rods or discs) may be arranged into biaxial
phases [19,77].
There are some theories able to describe biaxial nematic phases using scalar order
parameters. These theories generalize the ideas of Maier and Saupe and often involve
a representation of rotational invariants in terms of Euler angles. The disadvantage of
such theories is that in some specific cases they may present some multi–valuedness
problems [77]. Therefore there is a need of a further generalization of the scalar order
parameter Sin Eq. (2.3).
2.1.2 Tensor order parameter
A natural extension from a scalar order parameter is a tensor one. This arbitrary function
should be a non–zero rank Cartesian tensor 2containing all the information of the
rotational and reflective symmetries of the nematic phase. In contrast to the scalar order
parameter, the tensor order parameter is not defined in terms of a set of perpendicular
axes, but instead it is a quantity defined in terms of the moments of the orientational
distribution function.
A proper construction of the tensor order parameter involves the evaluation of the
average of a hierarchy of dyadic products ˆ
u(k)3over fiso =1/4πand f(ˆ
u). In general,
2For background information about this topic please refer to the Appendix A.
3Here we consider the k–order dyadic product of ˆ
uwith itself given by the relation
ˆ
u(k)=ˆ
u⊗ˆ
u⊗···⊗ ˆ
u
k−times
and ˆ
u(k)∈R3⊗R3⊗···⊗R3
k−times
.
12 Chapter 2. Phenomenological description
these moments are defined by the relation
ˆ
u(k)=
S2
ˆ
u(k)f(ˆ
u)dˆ
u.(2.4)
With respect to the isotropic orientational distribution function fiso the first two mo-
ments are
ˆ
uiso =0 and ˆ
uˆ
uiso =1
3δ. (2.5)
Note that the first moment, and in fact all odd moments of the distribution vanish since
f(ˆ
u)=f(−ˆ
u). Therefore the hierarchy of dyadic products ˆ
u(k)is restricted for values
of k=2nwhere nis an integer.
In analogy to S, the new tensor order parameter Q(k)should be a function of a
deviation of the full tensor ˆ
u(k)with respect to the isotropic phase, i.e.
ˆ
u(k)−ˆ
u(k)iso .(2.6)
In terms of ˆ
uˆ
uusing Eq. (2.5) it follows
ˆ
uˆ
u−ˆ
uˆ
uiso =ˆ
uˆ
u−1
3δ=ˆ
uˆ
u,(2.7)
where xstands for the symmetric traceless part of the tensor x. The first non–vanishing
moment which distinguishes the anisotropic distribution from the isotropic one is the
second rank tensor order parameter Q(2) =Q[22]. This quantity is defined as an
ensemble average of the deviation ˆ
uˆ
uover the orientational distribution function f(ˆ
u),
i.e.
Q(r,t)=15
2ˆ
uˆ
u.(2.8)
The translational dependency of the second rank order parameter Q(2) =Q(2)(r,t)
is implicit since ˆ
u=ˆ
u(r)[19,22]. In most phenomenological theories, all physical
quantities depending on the orientation ˆ
uof the particles can be described in terms of
the order parameter Qalso known as the alignment tensor [63]. In Eq. (2.8) the factor
15/2is a convention [23].
The construction of the second rank alignment tensor can be generalized in a very
direct manner in terms of ˆ
u(k)so that
Q(k)(r,t)=(2k+ 1)!!
k!ˆ
u(k),(2.9)
is the k–th order alignment tensor [22]. As mentioned before, for systems involving
particles presenting head–to–tail symmetry it is sufficient to consider tensors with even
values of k. In the following we will focus only on the second rank alignment tensor Q
in Eq. (2.8) since it is in terms of this quantity that the phenomenological Landau–de
Gennes theory is given.
2.1. Measure of orientational order 13
2.1.3 Some properties of the second rank alignment tensor
By definition Qis a 3×3symmetric traceless matrix. It has five independent components
where three of them may be fixed to the molecular reference frame and the other two can
be used to describe the average orientational order. In the molecular reference frame, Q
is written as a linear combination of the principal eigenvectors (parallel to the molecule’s
principal axes) l,mand n,i.e.
Q(r,t)=μ1ll +μ2mm +μ3nn .(2.10)
Here μ1,μ
2and μ3are the eigenvalues of the matrix satisfying μ1+μ2+μ3=0. In terms
of these eigenvalues it is possible to characterize different states of order. The following
distinction should be made: if all eigenvalues are zero the system presents an isotropic
state; if two of these eigenvalues are equal then the system is in a uniaxial nematic state;
finally, if all eigenvalues are different the system presents a biaxial nematic state.
For the special case of uniaxial phases, the order parameter can be found contracting
Qand the principal direction n,i.e.,nn :Q. Hence
Q(r,t)=3
2μ3nn (2.11)
where μ3is the largest eigenvalue in Eq. (2.10). The equilibrium value μ3is proportional
to the Maier–Saupe order parameter via μ3=10/3S[22]. Using the definition of the
Maier–Saupe scalar order parameter we find that the bounds of μ3are
−5
6≤μ3≤10
3.(2.12)
For biaxial nematic states the description is more complicated. In this case the
full tensor Qshould be contracted as well with the tensors ll and mm. However,
due to the symmetries of the tensor, the three principal values are not independent.
A proper description of the biaxial states of the system can be done in terms of the
rotational invariants of Q. By definition, a second rank symmetric traceless tensor has
three rotational invariants: the trace, the norm and the determinant. Using the Einstein
convention these are:
I1=Qμμ ,
I2=QμνQμν ,(2.13)
I3=QμνQνλQλμ .
Since Qis traceless I1=0. In terms of I2and I3a suitable measure of the biaxiality is
b, the biaxiality parameter, given by [79]
b2=1−I2
3
I3
2
.(2.14)
With this parameter one realizes that in the case of uniaxial alignment b=0and in the
case of biaxial alignment b=1.
14 Chapter 2. Phenomenological description
2.2 The isotropic–nematic phase transition
Normally depending on the temperature, concentration or the applied electric or mag-
netic field these systems show a variety of phases. Therefore the total free energy F
of the material may include terms such as: elastic energy due to any distortion of the
structure of the material; lyotropic energy which dictates the preferred phase of the
material; and, surface energy terms representing the interaction between the boundaries
of the system and the particles at the surface [77]. Therefore, the total energy of the
system is:
F=Felastic +Flyotropic +Fsurface .(2.15)
In terms of the energy densities Fe,F
land Fs, Eq. (2.15) may be written as
F=
V
(Fe+Fl)dr+
S2
Fsds , (2.16)
where the energy densities depend on the tensor order parameter Q.
2.2.1 Lyotropic energy
The lyotropic energy density Flis the function which dictates the state the fluid would be,
e.g. uniaxial order, biaxial nematic or isotropic fluid phase [77]. At low concentration this
free energy density should have a minimum energy in the isotropic state Q=0whereas
at high concentration there should be minima at three uniaxial states, i.e. the states
where any two of the eigenvalues of Qare equal. The simplest form of such function is
a truncated Taylor expansion about Q=0which up to 4–th order reads [19,77],
Fl=1
2AQ
μνQμν −1
3BQ
μνQνλQλμ +1
4C(QμνQμν )2.(2.17)
It is immediate to recognize that this expansion is given in terms of the scalar rotational
invariants In[see Eqs. (2.13)] of the tensor order parameter Q. In fact, a more general
expression of Flis constructed in terms of a multiplication of the scalar rotational
invariants as shown in Ref. [76]. However, for this study we will focus in the classical
Landau–de Gennes expression in (2.17).
The numerical coefficients in Eq. (2.17) are introduced for future convenience, how-
ever, the most important information is given by the coefficients A, B and C. These
coefficients may or may not depend on the concentration of the system. For simplicity
here we assume that
A(c)=A01−c
c∗,(2.18)
and A0,B and Care positive dimensionless coefficients that may be related to molecular
properties of the system [19,57,23,80]. In Eq. (2.18) the characteristic concentration
2.2. The isotropic–nematic phase transition 15
c∗is a model parameter known as the pseudo–critical concentration [22]. Before going
any further, it is important that to take into account the following remarks [76]:
1. There is no linear term in Q. This allows the possibility of an isotropic phase.
2. Odd terms of order three are allowed. This causes the isotropic–nematic phase
transition to be of first order.
3. The isotropic–nematic phase transition takes place in the neighborhood of A=0.
Therefore the density dependence of the free energy density of the system is
contained in Aalone.
The minimum of the Landau–de Gennes free energy density
Considering only uniaxial nematic phases inserting the uniaxial nematic order parameter
given by Eq. (2.11) in Eq. (2.17) the lyotropic energy becomes
Funi
l=9
8A(c)μ2
3−9
8Bμ3
3+81
64Cμ4
3.(2.19)
Equation (2.19) has stationary points when the second derivative vanishes, i.e. dF uni
l/dμ3=
0, thus
μ0
3=0,(2.20)
μ±
3=1
3CB±B2−4AC.(2.21)
For the system it would be energetically favorable to lie in one of the minima of Funi
l.
By calculating the second derivative (d2Funi
l/dμ2
3)and comparing the energies of each
solution one finds that:
•At μ0
3the isotropic state is globally stable for A>2B2
9C, metastable for 0<A<
2B2
9Cand unstable for A<0.
•At μ+
3the nematic state is globally stable for A<2B2
9C, metastable for 2B2
9C<
A<B2
4Cand not defined for A>B2
4C.
•At μ−
3the nematic state has negative (non–physical) values and is usually dis-
carded.
There are three important values of A:B2
4Cthe low concentration c†where the nematic
state disappears; 2B2
9Cthe concentration cIN at which the energy of the isotropic and
nematic are exactly equal; A=0the high concentration c∗where the isotropic state
loses stability. Using the ansatz in Eq. (2.19) these critical concentrations are:
c†=1−B2
4Cc∗and cIN =1−2B2
9Cc∗.(2.22)
16 Chapter 2. Phenomenological description
−1−0.5 0 0.5 1 1.5 2
−1.0
−0.5
0.0
0.5
µ3
Funi
t
c†
cIN
c∗
Figure 2.2: Free energy density Funi
tas a function of the scalar order parameter µ3for
the concentrations c†, cIN and c∗, for µIN
3= 0.4.
In Fig. (2.2) the Landau–de Gennes free energy density in Eq. (2.19) is shown as a
function of the scalar order parameter µ3for the special concentrations c†, cIN and c∗
with µIN
3= 0.4.
It is important to say that by construction the Landau–de Gennes free energy density
does not restrict the order parameter to be within its bounds [see Eq. (2.12)]. Thus it
is common to introduce an ansatz such that the magnitude of the alignment tensor is
bounded. In Ref. [63] an amended free energy density is given in terms of a maximum
value for the magnitude of Q. However, the differences between these ansatz and the
Landau–de Gennes free energy are small and it is convenient to maintain the original
one [63].
2.2.2 Elastic energy
The most general theory of curvature-elasticity of uniaxial fluids is due to F. C. Frank
extending the seminal work of C. W. Oseen from 1933 [20]. In his work of 1958 On
the theory of liquid crystals Frank derives a phenomenological elastic energy density in
terms of the nematic director ˆ
n[21] . For non–chiral particles the Frank–Oseen elastic
free energy density is:
Fe=K11
2(∇· ˆ
n)2+K22
2(ˆ
n·∇׈
n)2+K33
2(ˆ
n×∇׈
n).(2.23)
2.2. The isotropic–nematic phase transition 17
Figure 2.3: Splay, twist and bend deformations of the nematic director ˆ
n.
This elastic energy takes into account 3different types of distortion that can occur in the
system: splay, twist and bend [see Fig. (2.3)]. The Frank constants are dependent on the
particular fluid being described and can be determined by experimental measurements
of the structure factor of the material [19].
The Landau–de Gennes free energy density can also be extended to a description
of spatially inhomogeneous alignment only by including combinations of gradients, di-
vergence and curl of the order parameter Q[19]. The contribution due to distortion in
space of the alignment tensor is such that any gradients in Qwould lead to an increase
of the elastic energy density Fe[77].
However, in order to write a phenomenological description of these phenomena in
terms of Qwe have to take into account that given a fixed distortion in space of
Qthe elastic energy must remain unchanged under translations and rotations of the
material. Such restrictions imply that not all combinations of derivatives of Qin space
are allowed [77]. A common expression of the phenomenological elastic energy density
is:
Fe=L1
2(∇λQµν)2+L2
2(∇λQµλ)(∇γQµγ),(2.24)
where the elastic coefficient L1and L2are phenomenologic elastic constants. A more
general elastic free energy density contains terms proportional to ∇λQµγ∇γQµλ and
cubic in the alignment tensor [19], but in the following they will not be considered.
The elastic parameters L1and L2are related to the Frank elastic constants Kij
18 Chapter 2. Phenomenological description
in (2.23)by[81]
L1=1
6S2
eq
(K33 −K11 +3K22),(2.25)
L2=1
S2
eq
(K11 −K22),(2.26)
where Seq is the equilibrium uniaxial order parameter of the fluid [81]. In terms of elastic
constant L1order parameter fluctuations are correlated over a distance of order
ξ=L1
A(c),(2.27)
where A(c)is given in Eq. (2.19). In normal lyotropic fluids this correlation length is at
most a few tens of nanometers even close the isotropic–nematic transition [82].
2.2.3 Surface energy
In experiments surfaces can be treated in different ways. Usually the surface undergoes a
special type of chemical or physical modification such that there is a preferred direction
of alignment near the boundaries [83,84,85,86]. Associated with this preferred direction
of alignment directly at the border is the amount of order in a vicinity close to it. These
restrictions imply that the surface energy density Fshas minimum at the preferred state.
If Sis the surface in contact with the fluid
Fs=Fs(Q|S).(2.28)
Sometimes the variables near the boundaries are forced to move out of the minima of Fs
because the bulk of the fluid is in some alternate state. Finally the competition between
the surface energy and the bulk energy reaches a balance such that the total energy of
the system in Eq. (2.15) is minimized.
The most commonly obtained types of anchoring are characterized by a particular
anchoring direction. One these is the so called weak anchoring condition. In this type
of anchoring Fshas a single minimum at the point where the magnitude of Q|Sis
reached. Another common type of anchoring is the strong (infinite) anchoring, which
corresponds to the usual Dirichlet boundary condition at the surface. Finally, another
common anchoring condition is the planar anchoring condition. In this situation the
physical boundaries are prepared such that the directors of the fluid lie parallel to the
substrate [84].
Chapter 3
Microscopical description
In this chapter we revisit the basic concepts of standard statistical physics start-
ing with the Hamiltonian of a one component system and the pair interaction po-
tential between uniaxial rigid molecules. We continue introducing the main ideas
behind the variational principle known as classical density functional theory (DFT)
and familiarize with the theories developed (within this formalism) for systems of
one–component non–spherical bodies. Further, we briefly introduce the density
functional theories for mixtures of non–spherical particles. We focus mainly on
two theories: the Many–fluid Parsons–Lee theory [35] which is focused on spatially
homogeneous one–body densities; and an extension of the Ramakrishnan–Youssuf
approximation [39] (Second–order perturbation theory) for mixtures which focus
on spatially inhomogeneous one–body densities.
3.1 Systems of rigid anisotropic particles
Consider a system of Nidentical uniaxial anisotropic rigid particles filling a simply
connected volume V⊂R3. Due to their anisotropy the complete set of variables neces-
sary to describe the k–th particle in Vis: {rk,ˆ
uk}. This set accounts for translational
and rotational degrees of freedom. Here rkdenotes the center of mass position and
ˆ
uk=ˆ
uk({φk,θ
k})is a unit vector dependent on the set of Euler angles that fix the
particle’s orientation.
In absence of external fields, the Hamiltonian of the system, H({rk,ˆ
uk}),isasum
of kinetic energy Kand interaction energy Hgiven by
K=
N
i=1
p2
i
2m+1
2Iω2
iand H=U({rk,ˆ
uk}),(3.1)
19
20 Chapter 3. Microscopical description
where piis the linear momenta of particles of mass m,Ithe moment of inertia, ωi
the angular velocity of rotation and U({rk,ˆ
uk})is the total inter molecular potential
energy [87]. Assuming that the interactions occur solely by pairs U({rk,ˆ
uk})is given
by
U({rk,ˆ
uk})=1
2
N
i=1
N
j=1
U(ri−rj,ˆ
ui,ˆ
uj)(3.2)
where U(rij,ˆ
ui,ˆ
uj)is the pair interaction potential between the particle iand the
particle jat positions riand rjhaving orientations ˆ
uiand ˆ
uj, respectively.
In accordance with classical statistical mechanics in the canonical ensemble, the
configurational partition function is
Q=1
N!
N
k=1
V
drk
S2
dˆ
ukexp ⎛
⎝−1
2
N
i=1
N
j=1
U(rij,ˆ
ui,ˆ
uj)
kBT⎞
⎠,(3.3)
where S2is the unit sphere and kBis Boltzmann’s constant. Given the configurational
partition function, Eq. (3.3), the conformational ensemble average of an observable A
is given by
A=1
Q
N
k=1
V
drk
S2
dˆ
ukAexp ⎛
⎝−1
2
N
i=1
N
j=1
U(rij,ˆ
ui,ˆ
uj)
kBT⎞
⎠.(3.4)
In statistical mechanics, the properties of a system are obtained after the partition
function (3.4). However, in most cases, the configurational integral, Q, and the confor-
mational ensemble average of an observable, A, cannot be calculated analytically for a
given pair potential U(rij,ˆ
ui,ˆ
uj). Nevertheless, it is possible to write these functions in
the form of series expansions. In fact, there are several ways to do this and the simplest
is one is the Mayer cluster expansion [87,88,89]. This expansion [90] consists of a power
series of the configurational partition function (3.3) around a system of non-interacting
particles. Here, the two particle function
fij(rij,ˆ
ui,ˆ
uj)=exp−U(rij,ˆ
ui,ˆ
uj)
kBT−1,(3.5)
is introduced, and in terms of this function the cluster expansion is performed. For
homogeneous rigid anisotropic fluids the Mayer function plays a key role in the context
of Onsager’s theory [24] and Density Functional Theory [29].
3.1.1 Pair interaction potential
The structure and thermodynamic properties of a system in equilibrium are determined
by inter molecular forces. These forces, or rather their related interaction potentials, can
3.1. Systems of rigid anisotropic particles 21
Figure 3.1: (a) Interaction between two anisotropic particles. Here rij denotes the center
of mass distance between the particles and d(ˆ
ui,ˆ
uj,ˆ
uij)is the orientational dependent
distance of closest approach. (b) Schematic representation of the inter particle potential
energy U(rij)and Mayer function fij (rij )of a couple of hard anisotropic particles with
fixed orientation. Since the orientation of the molecules is fixed the distance of closest
approach between the particles is d.
be obtained, e.g., from ab initio calculations based on quantum mechanics [91]. However
even for simple fluids these calculations are complex and computationally demanding. For
this reason theorists resort to model these interactions using suitable analytic functions
determined by a number of parameters.
Restricting ourselves to apolar and uniaxial rigid molecules, the pair interaction po-
tential must have the following regularities [87]
U(rij,ˆ
ui,ˆ
uj)=U(−rij,ˆ
ui,ˆ
uj)=U(rij,−ˆ
ui,ˆ
uj)=U(rij,ˆ
ui,−ˆ
uj),(3.6)
where rij =ri−rj. For this reason, the average orientation of the particles present the
so called head–tail symmetry. Some examples of pair interaction potentials suitable to
describe such anisotropic fluids are generalizations of the Gay-Berne potential [92] and
excluded volume interactions [24].
The simplest model is one of hard core interactions. Hard particle systems combine
conceptual simplicity and computational tractability, qualities that make them accessible
to theoretical studies. For these systems the pair potential is given by
U(rij,ˆ
ui,ˆ
uj)=∞r<d(ˆ
ui,ˆ
uj,ˆ
uij)
0otherwise ,(3.7)
where d(ˆ
ui,ˆ
uj,ˆ
uij)is the orientational dependent distance of closest approach between
the particles, and rij =rij ˆ
uij [see Fig. 3.1(a)]1. For a system of rigid anisotropic
1The simplest hard core pair potential model is the one of a hard–sphere liquid where the particles
have isotropic shape and drepresents the effective diameter of the particle.
22 Chapter 3. Microscopical description
particles the Mayer function (3.5) is given by
fij(rij,ˆ
ui,ˆ
uj)=−1r<d(ˆ
ui,ˆ
uj,ˆ
uij)
0otherwise .(3.8)
In Fig. 3.1(b) we show schematically the inter particle potential energy U(rij )and Mayer
function fij(rij)for a couple of hard anisometric particles with fixed orientation.
Due to the singular nature of the hard particle potential U(rij,ˆ
ui,ˆ
uj)the tempera-
ture Tdoes not appear on the right hand side of Eq. (3.8), consequently, hard particle
fluids are often called athermal. In turn, the configurational energy depends only on the
density and is determined by the non–overlapping configurations in space.
3.1.2 Onsager’s second virial theory
Historically the first attempt to describe the structural and thermodynamical properties
of anisotropic fluids from a microscopic point of view was made by Lars Onsager in
1949. In his seminal work The effects of shape on the interaction of colloidal particles
he considered a low density gas of needle–like particles and wrote down the orientational
free energy introducing a much more efficient way of using the virial expansion [24]. To
this end, he did a cluster expansion of the free energy following the work of Mayer [90].
This expansion reads
F
NkBT=c(T)+lnρ−1−β1
2ρ−β2
3ρ2+··· ,(3.9)
where c(T)is a constant that depends only on the temperature and βnare the irreducible
cluster integrals 2, given in terms of Mayer functions as
β1=1
V
S2
f12 dˆ
u1dˆ
u2and β2=1
2V
S2
f12f23f31 dˆ
u1dˆ
u2dˆ
u2.(3.10)
For hard–core potentials −β1determines the excluded volume between particles. Indeed,
excluded volume effects can account for most ordering transitions in simple liquids and
liquid crystals [93].
For a pair of hard–spheres of radius rthe excluded volume is obviously a sphere of
radius 2r. However, the determination of an exact expression of the excluded volume
between non–spherical particles is not an easy task. In fact, this assignment normally
includes concepts from analytic geometry and integral geometry3. Onsager himself de-
termined the pair–excluded volume of hard–spherocylinders (see Ref. [24]). Hard sphero-
cylinders are cylinders of length lcaped at both ends with hemispheres whose diameters
2For a modern and didactic introduction to the Mayer cluster integrals and its formalism we refer the
reader to Chapter 3 in the book of Hansen and McDonald Theory of simple liquids, Ref. [88].
3A very nice introduction to the topic is given in the book: Theory of anisotropic colloidal solutions
by A. Isihara, Ref. [94].
3.2. Density functional theory 23
are the same as the diameter of the cylinder d. They are characterized by their aspect
ratio κ=l/d and possess a particle volume of v=(π/12)(3κ+2)d3. The excluded
volume between a pair of spherocylinders (i, j)is
Vexc(γ)=π
4(κid3
i+κjd3
j)+π
4didj(di+dj) sin γ+π
4(κidid2
j+κjdjd2
i)|cos γ|
+κiκjdidj(di+dj) sin γ+(κidi+κjdj)didjE(sin γ),(3.11)
where E(sin γ)denotes the complete elliptic integral of the second kind and γ=
arccos (ˆ
ui·ˆ
uj). A modern derivation in the framework of general expressions for gen-
eralized hard sphero–zonotopes can be found in Ref. [93].
The irreducible cluster integrals (3.10) are related to the well known virial coefficients
Bnby Bn+1 =−n/(n+1)βn[82]. Omitting all constants that are independent of density
Eq. (3.10) can be rewritten as
F
NkBT=lnρ+ρB2+1
2ρ2B3+··· .(3.12)
For highly elongated spherocylinders (κ→∞)Onsager showed that the contribution to
Eq. (3.12) of all virial coefficients beyond the second goes to zero. As it is confirmed by
numerous simulations, in this limit all higher order terms vanish and may be neglected
for all finite densities ρB2κ[82].
Up to this point we have assumed that the density of the system is constant. However,
we should allow the possibility of an orientational ordered phase with a lower free energy.
To this end the density of the system should be a one–particle density distribution of the
form ρ=ρ(r,ˆ
u). Although this approach is made in a very clear fashion by Onsager in
Ref. [24] we will turn to a much more modern description. For this purpose we use the
density functional description for fluids which is described in the following sections.
3.2 Density functional theory
In its original form, the density functional formalism was developed in 1964 to deal with
a gas of interacting electrons in an external potential. Hohenberg and Kohn [95] derived
a variational principle to determine the ground state of this quantum mechanical system.
One year later this theoretical approach was generalized to non-zero temperatures by
Mermin [96].
In Soft Matter theory this formalism was introduced by R. Evans in The nature
of the liquid-vapor interface and other topics in the statistical mechanics of nonuni-
form, classical fluids in 1979 [25,26]. The following summary of the theory is based on
the above mentioned publications as well as in the review articles Density Functional
Theories of Hard Particle Systems by P. Tarazona et al [28] and Density functional the-
ory of inhomogeneous classical fluids: recent developments and new perspectives by H.
L¨owen [97].
24 Chapter 3. Microscopical description
In short, density functional theory (DFT) consists on a variational principle of the
grand canonical free energy functional Ω(T,μ,[ρ(r,ˆu)]). Here Tis the temperature of
the system and μis its chemical potential [96,25]. Within this formalism, the basic
variables describing the microscopic degrees of freedom of a many–body fluid are the
one–particle densities ρ(r,ˆu). Here ρ(r,ˆu)dV is the average number of particles in an
infinitesimal volume dV located at position rwith orientation ˆu. In terms of {rk,ˆuk}
the one–particle density is defined as
ρ(r,ˆ
u)≡N
k=1
δ(r−rk)δ(ˆu−ˆuk),(3.13)
where · · · denotes the conformational average in Eq. (3.4). In the DFT the structural
an thermodynamic properties of the system will be described in terms of ρ(r,ˆ
u).The
one–particle density distribution ρ(r,ˆ
u)gives the probability of finding a particle at
position rwith orientation ˆu.
At fixed Tand μ,Ω(T,μ,[ρ(r,ˆu)]), the functional Ω([ρ(r,ˆu)]) becomes minimal
for the equilibrium one–particle density ρ0(r,ˆ
u), i.e.,
δΩ[ρ]
δρ ρ=ρ0
=0,Ω[ρ0(r,ˆ
u)] = Ω .(3.14)
Here, the value of the functional at the equilibrium density Ω0is the real equilibrium
grand canonical free energy. Moreover, the functional Ω(T,μ,[ρ(r,ˆu)]) can be decom-
posed as
Ω[ρ]=F[ρ]+
V
dr
S2
dˆ
uρ(r,ˆu))(Vext(r,ˆu))−μ),(3.15)
where Vext(r,ˆu))is the external potential acting on the particles. Normally, this external
potential is in charge of describing the system boundaries and external fields (gravita-
tional, electro–magnetic, etc.).
The free energy functional F[ρ]appearing in Eq. (3.15) may be splitted in an ideal
contribution, Fid[ρ], plus an excess free energy, Fexc[ρ], that accounts for the particles
interaction. In kBTunits F[ρ]is written as
βF[ρ]=βFid[ρ]+βFexc[ρ],(3.16)
where β=1/kBT. The ideal free energy functional is known exactly [29] and reads
βFid[ρ]=
V
S2
ρ(r,ˆu){ln (Λ3ρ(r,ˆu)) −1}dˆudr,(3.17)
where Λ2=h2/2πmkBTis the thermal de Broglie wavelength with mthe particle’s
mass and includes the translational and rotational contributions of the kinetic energy. If
the system is an ideal gas, Fexc[ρ]vanishes. For a system of non-vanishing pair interaction
the excess part is not known in general and one has to rely on approximations.
3.2. Density functional theory 25
3.2.1 Excess free energy
The effects of particle interactions are introduced through the excess free energy func-
tional Fexc[ρ]. For its construction it is useful to introduce an effective one–body potential
known as the direct correlation function. The direct correlation function c(1) is related
to the excess free energy functional via
c(1)[ρ(r,ˆ
u)] ≡−βδFexc[ρ]
δρ .(3.18)
In fact, c(1)[ρ(r,ˆ
u)] is only the first member of a hierarchy of correlation functions
generated by Fexc[ρ]. Higher order functions are obtained by differentiation
c(n)[ρ]=−βδnFexc[ρ]
δρ1···δρn
=−βδnFexc[ρ]
δρn···δρ1
,(3.19)
where ρi=ρ(ri,ˆ
ui). Of particular interest is the second derivative of this hierarchy, the
so–called direct correlation function
c(2)(r12,ˆ
u1,ˆ
u2)=−βδ2Fexc[ρ(r,ˆ
u)]
δρ(r1,ˆ
u1)δρ(r2,ˆ
u2).(3.20)
The direct correlation function serves as a measure of the structure and the properties of
the fluid [98] and is related to the total pair correlation function which can be measured
using diffraction experiments and computer simulations.
In terms of the direct correlation function the total pair correlation function h(r12,ˆ
u1,ˆ
u2)
is given by4:
h(r12,ˆ
u1,ˆ
u2)=c(2)(r12,ˆ
u1,ˆ
u2)+ρ0
V
dr3c(2)(r13,ˆ
u1,ˆ
u2)h(r32,ˆ
u3,ˆ
u2)ˆ
u3,
(3.21)
and it measures the total effect of a molecule 1on a molecule 2at a separation r12 and
with orientations ˆ
u1and ˆ
u2. Equation (3.21) is a generalization of the Ornstein–Zernike
equation to non–spherical molecules for a homogeneous fluid [99]. This expression indi-
cates that the total pair correlation between two particles consists of two contributions:
the direct correlation function between them and an indirect correlation arising because
of their interactions with the remaining particles in the system[91]. In general, the knowl-
edge of the direct correlation function allows us to calculate many fluid properties, for
example, the isothermal compressibility.
Approximations work on different levels. The usual virial expansion of Onsager can
be obtained directly from Eqs. (3.20) and (3.21) doing a cluster expansion of the pair
correlation function in a system with uniform density ρ0, i.e.,
−c(2)(r12,ˆ
u1,ˆ
u2;ρ0)=f12 +ρ0f12
V
S2
f23f31 dˆudˆudrdr+··· ,(3.22)
4For an inhomogeneous fluid ρ0will depend on r3and ˆ
u3and cannot be taken outside the integral.
26 Chapter 3. Microscopical description
where fij =fij(rij,ˆ
ui,ˆ
uj)is the corresponding Mayer function. This choice is asymptot-
ically exact in the dilute limit [24,82]. Another choice is a the mean field approximation
which is asymptotically exact at very high densities for bounded potentials [100]. A per-
turbative treatment out of the homogeneous liquid which needs the direct correlation
function as an input is the Ramakrishnan–Youssuf (RY) approximation [39]. It is a func-
tional Taylor expansion of Fexc[ρ]in terms of the density difference Δρ=ρ(r,ˆ
u)−ρ0
where ρ0is the homogeneous density in the isotropic phase. It follows from Eqs. (3.20)
and (3.21)that
−c(2)(r12,ˆ
u1,ˆ
u2;ρ0)=βδ2Fexc[ρ(r,ˆ
u)]
δρ(r1,ˆ
u1)δρ(r2,ˆ
u2)ρ0
,(3.23)
with the second functional derivative of Fexc[ρ]evaluated at ρ(r,ˆ
u)=ρ0. Therefore,
βFexc[{ρ}]≃−
1
2
V
S2
c(2)(r−r;ˆu,ˆu)Δρ(r,ˆu)Δρ(r,ˆu)dˆudˆudrdr.(3.24)
For hard spheres, the RY approximation leads to freezing and was the first demonstration
that freezing can be described within DFT. There are a couple of non–perturbative
functionals based on Rosenfeld’s fundamental measure theory [101]. These are more
accurate approximations of the excluded energy of hard convex bodies and can be found
in Refs. [102,103].
3.2.2 Density functional theories for systems of hard rods
In this section we summarize the most important results regarding isotropic and nematic
phases in the DFT formalism. We discuss different theories for systems of hard rods
using the smoothed density approximation of Poniewierski and Holyst [29] as a starting
point.
The smoothed density approximation in its original form is a DFT of a homogeneous
hard–sphere system (see for example Refs. [27,104]). For systems of hard rods, it relies
on the definition of a weight function which is related to the direct correlation function.
The ideal gas contribution to the free energy is given as usual [see Eq. (3.17)] by
βFid[ρ]=
V
S2
ρ(r,ˆu){ln (Λ3ρ(r,ˆu)) −1}dˆudr,(3.25)
where ρ(r,ˆu)is the one–particle density function. On the other hand, within this ap-
proximation the excess free energy of the system is
Fexc[ρ]=
V
ρ(r)Δψ(¯ρ(r)dr(3.26)
3.2. Density functional theory 27
where ρ(r)and ¯ρ(r)are the physical and smoothed densities and Δψis the excess free
energy per particle of the reference fluid, which, in this case, is a homogeneous and
isotropic fluid of hard rods. The first nontrivial step beyond the SDA corresponds to a
relation between ρand ¯ρ. To this end Poniewierski et al used the following expression
¯ρ(r,ˆ
u)=
V
S2
w(r−r,ˆ
u,ˆ
u)f(r,ˆ
u)f(r,ˆ
u)drdˆ
udˆ
u.(3.27)
Here f(r,ˆ
u)=ρ(r,ˆ
u)/ρ(r)is the angular distribution function also known as orien-
tational distribution function and w(r−r,ˆ
u,ˆ
u)is a weight function satisfying the
normalization condition
V
S2
w(r−r,ˆ
u,ˆ
u)drdˆ
udˆ
u=(4π)2.(3.28)
For the isotropic fluid the orientational distribution function is given by f=1/dˆ
u=
1/4π. In general, the weight function wshould be calculated via the direct correlation
function of a homogeneous and isotropic fluids of hard rods. However any analytic form
of c(2) is in general unknown. Instead of making approximations on the level of c(2)
Poniewierski and Holyst did some approximations on the weight function itself.
Onsager’s approximation
Onsager’s second virial theory can be obtained directly from the SDA assuming that
w(r−r,ˆ
u,ˆ
u)=−f12(r−r,ˆ
u,ˆ
u)
2B2
,(3.29)
together with
βΔψ(ρ)=ρB2+O(3) .(3.30)
Here f12 is the Mayer function for hard rods and B2is the second virial coefficient given
by
B2=1
2
V
S2
f12(r−r,ˆ
u,ˆ
u)dˆ
udˆ
u.(3.31)
For uniform fluids (ρ=ρ(r)) the Mayer function only depends on the orientational
degrees of freedom and determines the volume of exclusion between the particles. In this
limit the second virial coefficient is
B2=1
2ρV
S2
Vexc(ˆ
u,ˆ
u)f(ˆ
u)f(ˆ
u)dˆ
udˆ
u,(3.32)
28 Chapter 3. Microscopical description
where Vis the volume of the system5. As we can see, introducing the simple ansatz (3.28)
in the DFT formalism develops into the well known Onsager approximation [see Eq. (3.12)].
Finally the excess contribution to the free energy is
βFexc
V=1
2ρ2
S2
Vexc(ˆ
u,ˆ
u)f(ˆ
u)f(ˆ
u)dˆ
udˆ
u.(3.33)
One can obtain the isotropic and nematic coexistence densities directly from the total
free energy Eqs. (3.25) and (3.32). To this end one needs to do functional minimization
of the total free energy with respect to f(ˆ
u). After this procedure one gets the integral
equation
f(θ)=
exp −8ρ∗
π1
−1d(cos θ)K(θ, θ)f(θ)
1
−1d(cos θ) exp −8ρ∗
π1
−1d(cos θ)K(θ, θ)f(θ),(3.34)
where ρ∗=ρ∗BI
2is a dimensionless density scaled with respect to the value of B2for
the isotropic fluid and the kernel function K(θ,θ)is
K(θ,θ)=2π
0
dφ1−(cos θcos θcos φ+ sin θsin θ)2,(3.35)
Equation (3.32) can be solved for f(θ)using an iterative procedure at a fixed ρ∗. The
resulting function should be inserted back into the total free energy and after using the
Maxwell construction (see Ref. [105]) obtain the coexistence densities [28,106]. The
main drawback of this theory is that it is limited to very long aspect ratios and it is only
valid in the dilute regime (very low densities).
Parsons–Lee Approximation
For finite values of κfurther virial coefficients should be included in order to obtain
with increased accuracy the I−Ncoexistence densities. In the original treatment in
Ref. [29] the ansatz for Δψ[see Eq. (3.30)] has an additional contribution which takes
into account the packing between the particles. This inclusion corresponds to the virial
expansion of Parsons and Lee [37,38,107]. Parsons and Lee proposed an excess free
energy density given by
βFexc[f(ˆ
u)]
V=ρGHS(η)B2[f(ˆ
u)]
BHS
2
.(3.36)
5The excluded volume between the particles 1 and 2 is related to the contact distance (see Eq. (3.7))
via
Vexc (ˆ
u1,ˆ
u2)=1
3
S2
d(ˆ
u1,ˆ
u2,ˆ
u12)dˆ
u12 .
3.2. Density functional theory 29
where BHS
2=4vis the second virial coefficient of the hard sphere model with a molecular
volume vequal to the volume of the hard rod. The prefactor GHS(η)is the compress-
ibility factor of the system and corresponds to the usual Carnahan–Sterling free energy
per particle of the hard sphere fluid. The compressibility factor GHS(η)is given by
GHS(η)=η(4 −3η)
(1 −η)2,(3.37)
with η=ρv the packing fraction. From construction, this approximation recovers the
second virial low–density limit when ρ→0. For a system of spherocylinders with aspect
ratios κ≃5−8, this simple approach is very accurate. As it is proven by Bolhuis et al
in Ref. [108] using Gibbs-ensemble simulations.
Density-functional theory of distortion
The distortions of the nematic director can be formulated within the DFT formalism. A
first approximation was done by A. Poniewierski and J. Stecki in 1979–1982 [40,41].
Later, in a series of studies between 1985 and 1987, Y. Singh and K. Singh derived
an excess free energy of the aligned nematic liquids in terms of the direct correlation
functions of the isotropic fluids [30,31,32].
In both approximations the Frank elastic constants [see Eq. (2.23)] are expressed
in terms of the direct correlation function c(2)(r12,ˆ
u1,ˆ
u2)and the one–particle density
distribution function. These studies are based on the assumption that the one–particle
density is such that
ρ(r,ˆ
u)=ρ0(ˆ
n(r)·ˆ
u),(3.38)
where ˆ
nis the nematic director specifying the axis of symmetry of the uniaxial homoge-
neous sample. In terms of the director field and the one–particle density the free energy
of deformation is [40]:
βFdef[ρ]=
V
Aαμ(r)∇αnμdr+1
2
V
[Mαβμν (r)+Hαβμν(r)] ×∇
αnμ∇βnνdr
+1
2
S2
Bαβμ(r)∇αnμdσβ.(3.39)
where the coefficients Aαμ,B
αβμ,M
αβμν , and Hαβμν are functions of rdefined as
integrals of a hierarchy of direct correlation functions integrated over ρ0(ˆ
n(r)·ˆ
u), (see
Ref. [40]).
Finally, in terms of the direct correlation function of the undistorted nematic state
30 Chapter 3. Microscopical description
c(2)(r12,ˆ
u1,ˆ
u2;ρ0)in Eq. (3.24) the Frank elastic coefficients are
K1=1
2
V
S2
(r12)2
xc(2)(r12;ˆu,ˆu;ρ0)ρ0(ˆ
n·ˆ
u)ρ0(ˆ
n·ˆ
u)ˆuxˆu
xdˆ
udˆ
udr,(3.40)
K2=1
2
V
S2
(r12)2
xc(2)(r12;ˆu,ˆu;ρ0)ρ0(ˆ
n·ˆ
u)ρ0(ˆ
n·ˆ
u)ˆuyˆu
ydˆ
udˆ
udr,(3.41)
K3=1
2
V
S2
(r12)2
zc(2)(r12;ˆu,ˆu;ρ0)ρ0(ˆ
n·ˆ
u)ρ0(ˆ
n·ˆ
u)ˆuyˆu
ydˆ
udˆ
udr,(3.42)
where (r12)idenotes the magnitude of the i–th component of the vector r12 =r−r.
To calculate these constants for a model nematic fluid we must know the direct corre-
lation function and the orientational distribution function calculated in the undistorted
homogeneous nematic [40,41].
3.3 Density functional theories for mixtures
For an n–component mixture of non spherical hard–body particles, the total number of
molecules of all species is
N=
n
α=1
Nα.
where αlabels the components of the mixture. In the scope of DFT, there is a one–
body density ραfor each component of the fluid and it is such that ρα(r,ˆu)dV is the
average number of α–molecules in an infinitesimal volume dV located at position rwith
orientation ˆu. In terms of {rα
k,ˆuα
k}the one–body density is
ρα(r,ˆu)≡Nα
k=1
δ(r−rα
k)δ(ˆu−ˆuα
k),(3.43)
where · · · denotes the conformational average. Since, the density functional approach
focuses on the Helmholtz free energy Fas a unique functional of these one–body densi-
ties ραthen F[ρα]. The free energy functional, F[ρα], is splitted in an ideal contribution,
Fid[ρα], plus an excess free energy, Fex[ρα], that accounts for the particles’ interaction.
In β=1/kBTunits F[ρα]is written as
βF[ρα]=βFid[ρα]+βFex[ρα].(3.44)
3.3. Density functional theories for mixtures 31
Ideal contribution
A straightforward generalization of the single component system (see Ref. [29]) of the
ideal free energy for the n–component mixture reads
βFid[ρα]=
n
α=1
V
S2
ρα(r,ˆu){ln (Λ3
αρα(r,ˆu)) −1}dˆudr,(3.45)
where Λαis the thermal de Broglie wavelength of component αof mass mαand is given
by
Λα=h2
2πmαkBT1/2
.
Excess contribution
As stated previously, the excess free energy functional Fex[ρα]is unknown and therefore
one has to rely on several approximations. In the following we present two approximations
of the one–body density of each component αwhich lead to two very distinct density
functionals. First we consider spatially homogeneous densities where the dependency
of ραon the translational degrees of freedom is disregarded. Later we assume that the
dependence of the one–body densities on the translational degrees of freedom is inherited
through ˆ
uwith the ansatz ˆ
u=ˆ
u(r).
3.3.1 Many–fluid Parsons–Lee theory
Here we focus on spatially homogeneous densities such that
ρα(r,ˆu)=ραfα(ˆu),(3.46)
where ρα=Nα/V is the number density of species α(with Vbeing the volume) and
fα(ˆ
u)is the orientational distribution function (ODF), which is normalized such that
dˆ
ufα(ˆ
u)=1. This ansatz allows us to describe isotropic and nematic phases of the
system.
With this specification, the ideal part of the free energy functional [see Eq. (3.45)]
can be written as
βFH
id
V[ρα]=
n
α=1
ραln(ραΛ3
α)−1+σ[fα],(3.47)
where σ[fα]is the orientational entropy given by
σ[fα]=
S2
dˆ
ufα(ˆ
u) ln(4πfα(ˆ
u)),(3.48)
32 Chapter 3. Microscopical description
The quantity σis zero in the isotropic state, where fα(u)=f0=(4π)−1and becomes
positive in the nematic state. It thus reflects the loss of orientational entropy due to
ordering.
Regarding the excess part of the free energy, we here employ a generalization of the
so–called Parsons–Lee theory [37,38] to mixtures, as suggested by Malijevsky et al. [35].
This approach is derived by starting from the virial expression for the pressure, which
involves (apart from the singlet densities) the spatial derivative of the pair potential,
Uαβ and the pair correlation function gαβ. Noting that Uαβ can be re–expressed in
terms of a hard–sphere (HS) potential with scaled distance rij/dαβ(ˆ
ui,ˆ
uj,uij), the
same assumption is made for the correlation function, i.e., gαβ =gHS
αβ (rij/dαβ)(note
that this is an approximation). By integrating over the total density ρ=αρα(and
noting that ∂UHS
αβ (y)/∂y is proportional to a delta function in space), one obtains the
excess free energy
βFH
ex
V[ρα]=1
2
n
α=1
n
β=1
ραρβGαβ
S2
dˆ
ui
S2
dˆ
ujVexc
αβ (ˆ
ui,ˆ
uj)fα(ˆ
ui)fβ(ˆ
uj).(3.49)
In (3.49), Vexc
αβ is the excluded volume between particles iand j, which is related to the
contact distance via
Vexc
αβ (ˆ
ui,ˆ
uj)=1
3
S2
dˆ
uij d3
αβ(ˆ
ui,ˆ
uj,ˆ
uij).(3.50)
The quantities Gαβ are averages (in terms of the total density) of the contact values of
the HS pair correlation functions, gHS
αβ (1+), in the considered density range [35].
As discussed extensively in Ref. [35], the free energy given in (3.49) reduces to
the familiar second–virial theory of Onsager [24] in the limit of low densities. Then,
one has gHS
αβ (1+)→1, and therefore Gαβ =1for all combinations α, β. Although
the Onsager theory provides an intuitive approach regarding the effects related to the
geometry of the particles, it is known to become inaccurate for particles of finite length
at the densities close to orientational phase transitions [108,109]. In this respect the
generalized Parsons–Lee approach provides a significant improvement, as demonstrated
by comparison with Monte Carlo simulations of hard Gaussian molecules with different
elongations and concentrations [14].
3.3.2 Second–order perturbation theory for mixtures
To our account the excess free energy functional of a mixture of hard–body particles
in terms of spatially inhomogeneous one–body densities has not been addressed in lit-
erature. To this end we assume that the one–body density of the αcomponent of the
mixture can be written as:
ρα(r,ˆu)=ραfα(ˆu(r)) .(3.51)
3.3. Density functional theories for mixtures 33
This ansatz of the one–body densities is similar to the one in Ref. [40,41], however,
the spatial dependence of the orientational degrees of freedom is not given in terms of
the nematic director but in terms of full vector ˆ
u. This allows a description of the fluid
in terms of the space–fixed frame of reference instead of the usual director dependent
frame of reference. With this assumption the ideal contribution of the free energy [see
Eq. (3.45)] is simply given by:
βFI
id[ρα]=
n
α=1
V
S2
ρα(ˆ
u(r)){ln (Λ3
αρα(ˆ
u(r)) −1}dˆudr,(3.52)
where Λαis the thermal de Broglie wavelength of component α.
Since our interest is to study spatially homogeneous nematic phases a first choice of
the excess free energy functional may be an extension of Onsager’s second virial theory
for mixtures or a mean field approximation. However, inspired on previous results for
the one component system [39,40,41,30,31,32] we use a second order perturbation
theory of the full excess free energy functional, Fex[ρα]. This perturbation resolves on an
extension of the well known Ramakrishnan–Youssuf [39] to a mixture of n–components
which is:
βFI
ex =−1
2
n
α=1
n
β=1
V
S2
c(2)
αβ (r−r;ˆu,ˆu)Δρα(r,ˆu)Δρβ(r,ˆu)dˆudˆudrdr.
(3.53)
Here Δρα(r,ˆu)=ρα(r,ˆu)−¯ραwith the mean number density ¯ρα=ραf0and the
direct correlation function c(2)
αβ (r−r;ˆu,ˆu)calculated in the isotropic phase. As it is
discussed previously, with this assumption Uαβ(rij,ˆui,ˆuj)is a smoothly varying long
range function of the distance and orientation [110,36,39].
Chapter 4
Bridging microscopic and
mesoscopic theories
The first goal of our investigations is to construct an expression of the equilib-
rium free energy functional capable of describing the thermodynamic phase behav-
ior of a mixture. In addition, this functional should contain microscopic features of
the system such as, number density, length and diameter of the particles, etc. In
this chapter we take on the task of bridging the microscopic and phenomenological
descriptions introduced in the preceding chapters. We address this issue for both,
spatially homogeneous and inhomogeneous systems, starting from the previously
discussed density functional theories for mixtures (Many–fluid Parsons–Lee theory
and the Ramakrishnan-Youssuf approximation for mixtures).
4.1 DFT based Q–tensor theory for mixtures
The aim of this chapter is to rewrite the free energy functionals consisting of ideal and
excess contributions [see Eqs. (3.47)–(3.53)] in terms of the k–th order alignment tensors
Qα
(k)(one for each component of the mixture) where α=1,··· ,n[see Chapter 2]. These
tensors are defined in terms of the k–fold product ˆ
u(k)as [22]
Qα
(k)=(2k+ 1)!!
k!ˆ
u(k)α
.(4.1)
where · · · αdenotes the configurational ensemble average of ˆ
u(k)with respect to the
orientational distribution function fα(ˆu). For most systems of rigid particles, it is suf-
ficient to consider tensors with even kdue to the head–to–tail symmetry. Within this
subset, the tensors of rank k=2and k=4are of particular importance since they may
be related to the birefringence and fluorescence of the sample [19,23].
35
36 Chapter 4. Bridging microscopic and mesoscopic theories
4.2 Homogeneous systems
First we will discuss the construction of the Q-tensor functional for homogeneous systems
based on the publication Binary mixtures of rod–like colloids under shear: microscopically–
based equilibrium theory and order–parameter dynamics which was written in collabora-
tion with S. H. L. Klapp [111].
4.2.1 Ideal contribution
The first task is to rewrite the orientational entropy, given in (3.48). Here we follow
earlier approaches [23,112], where the one–particle ODF was expressed as fα(ˆ
u)=
f0(1 + ψα(ˆ
u)), with ψ(ˆ
u)being a small deviation from the isotropic state, f0=1/4π.
Inserting this ansatz into (3.48) and performing a Taylor expansion of the logarithm
yields [23,113]
σ[fα]=
∞
n=2
(−1)n
n(n−1)ψn
α(ˆ
u)0,(4.2)
where ···0denotes the orientational average evaluated with f0. Due to normalization
of the ODF the term n=1is zero and thus it is disregarded. Next, one assumes that the
function ψ(ˆ
u)may be expanded in terms of the tensor quantities ((2k+ 1)!!/k!) ˆ
u(k)
[see (4.1)], where the prefactors are determined by their orientational averages, Qα
(k).
Following [23,113], we restrict this expansion to terms involving k=2and k=4.
Inserting the resulting ansatz for ψαinto (4.2) and retaining terms up to fourth power
in Qα
(2) and up to second power in Qα
(4), one obtains
σ[Qα
(2),Qα
(4)]=1
2Qα
(2) Qα
(2) −√30
21 Qα
(2) ·Qα
(2) Qα
(2) +5
28 Qα
(2) Qα
(2)2
+1
2Qα
(4) Qα
(4) +O(Qα
(4))2.(4.3)
In (4.3), the product A(k)·B(k)yields a tensor of rank k[see Appendix A].
4.2.2 Excess contribution
We now turn to the reformulation of the excess free energy as functions of the Qα
(k).To
our knowledge, before our publication [111], this problem has not been addressed in the
literature so far.
As a starting point we expand the excluded volume of the hard–core particles ap-
pearing in Eq. (3.49) in terms of the Legendre polynomials (following Refs. [35,114]
and references therein),
Vexc
αβ (ˆ
ui,ˆ
uj)=
∞
k=0
Vαβ
kPk(cos γ),(4.4)
4.2. Homogeneous systems 37
where γis the angle between ˆ
uiand ˆ
uj(see Appendix B) and the coefficients Vαβ
kare
calculated using the relation
Vαβ
k=2k+1
21
−1
dcos γPk(cos γ)Vexc
αβ (γ).(4.5)
Next, we replace Pkin (4.4) by using the addition theorem for spherical harmonics [99]
(see Appendix B), yielding
Vexc
αβ (ˆ
ui,ˆ
uj)=
∞
k=0
4πV αβ
k
2k+1
k
m=−k
Y∗
km(ˆ
ui)Ykm(ˆ
uj).(4.6)
The sum on the right side of Eq. (4.6) can be rewritten as a full scalar product of the
two tensors ˆ
u(k)[99]. One obtains
Vexc
αβ (ˆ
ui,ˆ
uj)=
∞
k=0
Vαβ
k
(2k+ 1)!!
(2k+1)k!ˆ
u(k)
iˆ
u(k)
j.(4.7)
Inserting (4.7) into (3.49), noting that the angular integrals (weighted by the ODFs)
in (3.49) lead to orientational averages, and using the definition of the tensor order
parameter [see (4.1)] one finds
βFH
exc
V=1
2
n
α=1
n
β=1
ραρβGαβ
∞
k=0
Vαβ
k
2k+1Qα
(k)Qβ
(k).(4.8)
Equation (4.8) provides the excess free energy as a function of order parameter tensors.
It still depends on the microscopic parameters of the mixture, that is, the geometry of
the particles (defining the excluded volume) and the densities.
4.2.3 Full free energy
We now turn to the full free energy containing ideal and excess parts, FH=FH
id +FH
ex.
Restricting to terms involving 2nd– and 4th– rank tensors we have
βFH
id [Q]
V=
n
α=1 Fα
00 +ρασQα
(2),Qα
(4),(4.9)
where Fα
00 =ρα(ln(ραΛ3
α)−1),σis given in (4.3), and the notation [Q]on the left side
indicates the dependence of the (ideal) free energy on all four order parameter tensors
(k=2,4,α=1,··· ,n). Further, we obtain from Eq. (4.8)
βFH
ex[Q]
V=1
2
n
α=1
n
β=1
ραρβGαβ Vαβ
0+Vαβ
2
5(Qα
(2) Qβ
(2))+Vαβ
4
9(Qα
(4) Qβ
(4)).
(4.10)
38 Chapter 4. Bridging microscopic and mesoscopic theories
These expressions can be further simplified by using a ”closure” relation expressing the
4th–rank tensor with the one of second rank. For uniaxial systems characterized by a
(nematic) director n[115,116], there exists the exact relation S2
2Q(4) =S4Q(2)Q(2) ,
where S2=P2(ˆ
u·ˆ
n)and S4=P4(ˆ
u·ˆ
n). In strongly aligned systems one has
S4≈S2
2≈1. In that limit, the closure thus takes the form Q(4) ≡Q(2)Q(2) . This
choice is the simplest and most widely used approximation without requiring further
assumptions in the ODF. In this direction, one could have used the Bingham closure
which is suitable for rotational flows but loses precision for simple shear flow [117].
Since we can now express all quantities in terms of the second–rank tensor, we ease
notation and set Qα=Qα
(2) (α=1,···,n). Further, we rewrite the notation for the
tensor products appearing in Eq. (4.10) into the form commonly used for second–rank
tensors, that is QαQα=Qα:Qα=Tr(Q·Q)=QμνQμν . Adding ideal and excess
contributions of the free energy, we obtain
ΦH[Q]=
βFH[Q]
V
ΦH[Q]=
n
α=1 Fα
0+Aα(Qα:Qα)−BαTr(Qα·Qα·Qα)+Cα(Qα:Qα)2
+1
2
n
α=1
n
β=1 Fαβ
0+Aαβ(Qα:Qβ)+Cαβ(Qα:Qβ)2,(4.11)
where the coefficients are functions of the number densities, the factors Gαβ and the
coefficients of the excluded volume. Explicitly, they are given by
Fα
0=ραln(ραΛ3
α)−1+1
2ρ2
αGααVαα
0,(4.12)
Aα=1
2ρα+1
10ρ2
αGααVαα
2,(4.13)
Bα=√30
21 ρα,(4.14)
Cα=19
28ρα+1
18ρ2
αGααVαα
4,(4.15)
Fαβ
0=ραρβGαβVαβ
0,(4.16)
Aαβ =1
5ραρβGαβVαβ
2,(4.17)
Cαβ =1
9ραρβGαβVαβ
4.(4.18)
As seen from Eq. (4.11), the mesoscopic free energy derived here has the standard
Landau–de Gennes (LdG) form [19] in the sense that it involves an expansion into order
parameters. Contrary to standard LdG theory, however, the coefficients are related to
microscopic properties of the system. We also note that the free energy does not depend
4.3. Inhomogeneous systems 39
explicitly on the temperature, which reflects the hard–core character of the interactions.
Consequently, the ordering behavior of the resulting ”lyotropic” system is essentially
controlled by the concentration of the system.
4.3 Inhomogeneous systems
In the previous Subsection (4.2) we took on the task of writing the mesoscopic free energy
functional of the mixture in terms of tensor order parameters of second and fourth rank.
This was done using an expansion of the excluded volume between anisotropic particles
in terms of the spherical harmonic functions. Here we propose a different but at the
same time complementary approach based on a particular ansatz for the one–particle
density ρα(r,ˆu).
As in Eq. (3.51) in the following we assume that the one–particle density is of the
form
ρα(r,ˆu)=ραfα(ˆu(r)) ,(4.19)
where ρ=N/V is the number density and f(ˆu)is the normalized orientational distri-
bution function with ˆu=ˆu(r)) [see Chapter 3]. Following Ref. [112] assuming simple
angle dependence of the ODF viz f(ˆu)=f0(1 + ϕ(ˆu)) we employ the ansatz
ρα(r,ˆu)=ραf0(1 + ϕα(ˆu)) (4.20)
=ραf0 1+15
2ˆuˆu:Qα!,(4.21)
where ϕ(ˆu)describes a small deviation from f0corresponding to anisotropic states. In
expressions (4.20–4.21) the explicit dependence of ˆuand Qμν with the space dependent
vector ris not written to avoid any confusion.
Equation (4.20) implies that the one–particle density is comparable with the mean
number density of the fluid in the isotropic phase given by ¯ρ=ρf0. Inserting Eq. (4.21)
in the density difference Δρα(r,ˆu)=ρα(r,ˆu)−¯ραgiven by the RY approximation [see
Eq. (3.53)] results in
Δρα(r,ˆu)=ραf0ϕα(ˆu),
=ραf015
2ˆuˆu:Qα.(4.22)
With the above expressions of the one–particle density and the difference Δρα(r,ˆu),
Eqs. (4.21–4.22), we will proceed writing the ideal (3.52) and excess (3.53) contributions
of the free energy functional in terms of the spatially inhomogeneous second rank tensor
order parameters Qα(r).
40 Chapter 4. Bridging microscopic and mesoscopic theories
4.3.1 Ideal contribution
The first assignment is to write the ideal contribution of the free energy (3.52) in terms
of Qα(r). Using the initial assumption for inhomogeneous systems Eq. (4.19), the ideal
free energy functional in Eq. (3.52) is given by 1
βFI
id[ρα]=
n
α=1
V"ραln(ραΛ3
α)−1+σ[fα]#dr,(4.23)
where σ[fα]is the inhomogeneous orientational entropy given by
σ[fα]=
S2
dˆ
ufα(ˆ
u) ln(4πfα(ˆ
u)) .(4.24)
Inserting the ansatz (4.20)inEq.(4.24) results in
σ[fα]=f0
S2
(1 + ϕα(ˆu)) ln((1 + ϕα(ˆu)))dˆu.(4.25)
For small non–zero deviations we expand the integrand in (4.25) up to fourth order, thus
(1 + ϕα(ˆu)) ln((1 + ϕα(ˆu))) ≈ϕα(ˆu)+1
2ϕα(ˆu)2−1
6ϕα(ˆu)3+1
12ϕα(ˆu)4+O(ϕ5
α),
(4.26)
yielding to the inhomogeneous orientational entropy
σ[fα]=f0
S2
ϕα(ˆu)+1
2ϕα(ˆu)2−1
6ϕα(ˆu)3+1
12ϕα(ˆu)4dˆu.(4.27)
Due to normalization of the ODFs the first integral in (4.27) must vanish. Further,
inserting the explicit ansatz (4.21) in each term of the right side of (4.27) results in
σ[Q(r)] = 1
2(Qα:Qα)−√30
21 Tr (Qα·Qα·Qα)+ 5
28 (Qα:Qα)2,(4.28)
Evidently the above derivation is consistent with the one for homogeneous systems [see
Eq. (4.3)]. Finally, inserting Eq. (4.28) in Eq. (4.23) results in
βFI
id[Q(r)] =
V
$Φα
0+Φ
id[Q(r)]%dr.(4.29)
1Note that the form of the ideal contribution to the free energy is similar to the one for spatially
homogeneous phases [see Eq. (3.47)] with the difference that here the integration over the volume Vis
in general more complicated because of the implicit dependence ˆu(r).
4.3. Inhomogeneous systems 41
where Φα
0=αραln(ραΛ3
α)−1and Φid[Q(r)] is the ideal contribution of the meso-
scopic free energy density (per unit volume)
Φid[Q(r)] =
n
α=1 $1
2ρα(Qα:Qα)−1
710
3ραTr (Qα·Qα·Qα)+ 5
28ρα(Qα:Qα)2%.
(4.30)
4.3.2 Excess contribution
Now we turn to the reformulation of the excess free energy as function of Q(r).To
our knowledge this has not been done systematically from a DFT basis, however, some
attempts have been reported following Taylor expansions of the one–particle densities
and moments of the Mayer function [118] similar to the generalized DFT for weakly
twisted director fields [119].
Here we use the extension to mixtures of the RY approximation [see Eq. (3.53)].
Inserting the ansatz given for the density difference in (4.22) into Eq. (3.53) and after
some rearrangement yields2
βFI
ex[Q(r)] = −1
2
n
α=1
n
β=1
V
dr
V
drρα
4πρβ
4π15
2Qα
μν(r)Qβ
μν(r)
×
S2
dˆu
S2
dˆuc(2)
αβ (r−r;ˆu,ˆu)ˆuμˆuνˆu
μˆu
ν.
(4.31)
Now, the task is to rewrite the integrals over the unit sphere, i.e.,
Intˆuˆu=
S2
dˆu
S2
dˆuc(2)
αβ (r−r;ˆu,ˆu)ˆuμˆuνˆu
μˆu
ν,(4.32)
in terms of the tensor quantities Qα. Here it is important to mention, once again, that
according to the RY approximation [39] the direct correlation function is calculated in
the isotropic phase.
Referred to an arbitrary space–fixed reference frame the direct correlation function
c(2)
αβ (r−r;ˆu,ˆu)can be expanded in terms of the spherical invariants as (see e.g.
Refs. [88,99])
c(2)
αβ (r−r;ˆu,ˆu)=
l1l2l
c(2)
αβ (l1l2l;r−r)Φ(l1l2l;ˆu,ˆu,ˆux),(4.33)
2For simplicity in the following expressions Einstein’s index convention will be used.
42 Chapter 4. Bridging microscopic and mesoscopic theories
where ˆuxis the unit vector pointing in the direction of the vector difference x=r−r
and c(2)
αβ (lil2l;r−r)are the harmonic expansion coefficients of the direct correlation
function
c(2)
αβ (l1l2l;x)=(2l1+ 1)(2l2+1)
4π(2l+1)
S2
dˆu
S2
dˆuc(2)
αβ (r−r;ˆu,ˆu)Φ(l1l2l;ˆu,ˆu,ˆux).
(4.34)
Here, the rotational invariants Φ(l1l2l;ˆu,ˆu,ˆux)are given by
Φ(l1l2l;ˆu,ˆu,ˆux)=
m1m2m
C(l1l2l;m1m2m)Yl1m1(ˆu)Yl2m2(ˆu)Y∗
lm(ˆux),(4.35)
where Ylm are the spherical harmonics and C(l1l2l;m1m2m)are the Clebsch–Gordan
coefficients [see Appendix B]3. Inserting the series expansion (4.33) in Eq. (4.32) yields
Intˆuˆu=
S2
dˆu
S2
dˆu
l1l2l
c(2)
αβ (l1l2l;x)Φ(l1l2l;ˆu,ˆu,ˆux)ˆuμˆuνˆu
μˆu
ν.
(4.36)
Since the Cartesian product ˆuμˆuνis isomorphic to the l=2spherical harmonics
[see Appendix B], i.e.,
ˆuˆum=8π
15 Y2m(ˆu),(4.37)
then we find after substitution of the isomorphism (4.37) in Eq. (4.36)
Intˆuˆu=8π
15
S2
dˆu
S2
dˆu
l1l2l
c(2)
αβ (l1l2l;x)Φ(l1l2l;ˆu,ˆu,ˆux)Y2m1(ˆu)Y2m2(ˆu).
(4.38)
Considering that the rotational invariants Φ(l1l2l;ˆu,ˆu,ˆux)are given in terms of spher-
ical harmonics [see Eq. (4.35)] and since [see Appendix B]
S2
Y∗
l1m1(ˆu)Yl2m2(ˆu)dˆu=δl1l2δm1m2,and Y∗
l1m1(ˆu)=(−1)−m1Yl1−m1(ˆu),
3The notation in Eqs. (4.33) and (4.35) represents explicitly the following [see Appendix B]:
l1l2l
→
∞
l1=0
∞
l2=0
|l1+l2|
l=|l1−l2|
and
m1m2m
→
l1
m1=−l1
l2
m2=−l2
l
m=−l
δm,m1+m2.
4.3. Inhomogeneous systems 43
the integral over ˆuand ˆuin Eq. (4.38) reduces to
Intˆuˆu=8π
15
4
l=0
l
m=−l
c(2)(22l;x)C(22l;00m)Y∗
lm(ˆux).(4.39)
Equation (4.39) defines the reduced direct correlation function (independent of ˆuand
ˆu)as
¯c(2)
αβ(r−r)=
4
l=0
l
m=−l
c(2)
αβ(22l;x)C(22l;00m)Y∗
lm(ˆux).(4.40)
Finally, writing Eq. (4.31) in terms of the reduced direct correlation function (4.39)
yields
βFI
ex[Q(r)] = −1
2
n
α=1
n
β=1
V
dr
V
drραρβ
4π¯c(2)(r−r)Qα
μν(r)Qβ
μν(r).(4.41)
4.3.3 Gradient expansion of the direct correlation function
As mentioned in Chapter 3,c(2)
αβ(r−r)is related to the pair interaction between particles.
To gain an insight of the properties of the system and compare them with experimental
results (often obtained by small–angle X–ray or neutron scattering see e.g. Ref. [120]) it is
useful to write the direct correlation function in the Fourier space (see e.g. Ref. [88,89]).
Here, we follow Ref. [121], to write a gradient expansion of the pair correlation
function. First we focus on the integral over rappearing on the right hand side of
Eq. (4.41),
J(r)=
V
dr¯c(2)
αβ(r−r)Qβ
μν(r).(4.42)
Taking the Fourier transform of Eq. (4.42),
F"J(r)#=F$
V
dr¯c(2)
αβ(r−r)Qβ
μν(r)%,(4.43)
and after using the convolution theorem (see e.g. Ref. [122]) from Eq. (4.43) it follows
&
J(k)= 1
(2π)3/2'
¯c(2)
αβ(k)'
Qβ
μν(k).(4.44)
where kis the wave number and the notation &
f(k)represents the Fourier transform of
the real valued function f(r). Second, in the small wave number approximation (k≈0)
44 Chapter 4. Bridging microscopic and mesoscopic theories
we expand '
¯c(2)
αβ(k)in the series
'
¯c(2)
αβ(k)=Mαβ
0+Mαβ
2|k|2+Mαβ
4|k|4+O(|k|6),(4.45)
where Mαβ
nare expansion coefficients related to the moments of the direct correlation
function in the Fourier space [89]. Inserting (4.45) back into Eq. (4.44) we obtain
&
J(k)= 1
(2π)3/2Mαβ
0+Mαβ
2|k|2+Mαβ
4|k|4'
Qβ
μν(k).(4.46)
Back in real space the Fourier transform (4.46) corresponds to the product between the
gradient expansion of the direct correlation function and the tensor Qβ
μν,i.e.
J(r)=
V
drMαβ
0−M
αβ
2∇2+Mαβ
4∇4δ(r−r)Qβ
μν(r),(4.47)
where the operator ∇is taken with respect to the rcoordinates. Finally, upon integration
of (4.47) with respect to rone obtains
J(r)=Mαβ
0−M
αβ
2∇2+Mαβ
4∇4Qβ
μν(r),(4.48)
where the coefficients Mαβ
nare the moments of the expansion coefficients of the direct
correlation function given by
Mαβ
n=1
(2π)3/2
V
dx¯c(2)
αβ(x)|x|n.(4.49)
In this analysis the properties of the mixture are parametrized only by the moments Mαβ
0,
Mαβ
2and Mαβ
4. In general, these moments depend on the thermodynamic properties
of the mixture expressed in terms of the mean number densities ρα. For example, the
k=0term is related to the liquid phase isothermal compressibility whereas the k=2
and k=4terms are usually related with the bulk modulus of the isotropic system and
the Frank elastic constants [121].
Inserting Eq. (4.48) into the excess free energy contribution Eq. (4.41) yields:
βFI
ex[Q(r)] = −1
2
n
α=1
n
β=1
ραρβ
4π
V
drQα
μν(r)Mαβ
0−M
αβ
2∇2+Mαβ
4∇4Qβ
μν(r).
(4.50)
Equation (4.50) may be interpreted as the excess free energy cost due to the spatial
inhomogeneities of the mixture. However, as it is written it does not measure properly
4.3. Inhomogeneous systems 45
how the inhomogeneities of the tensor quantities Qαand Qβare correlated. To this end
it is useful to re–write Eq. (4.50) in terms of gradients of the alignment tensors Qαand
Qβ.
Focusing on the integral
−
V
drQα
μν(r)Mαβ
2∇2−M
αβ
4∇4Qβ
μν(r),(4.51)
we note that the ∇2and ∇4operators are acting (only) on Qβ. However, both may be
rewritten in terms of the derivatives of the product Qα:Qβ.
Focusing on second order derivatives and using Green’s tensor identity [see Ap-
pendix A]
∇2(Qα
μνQβ
μν)=Qα
μν(∇2Qβ
μν)+2(∇λQα
μν)(∇λQβ
μν)(4.52)
+(∇λQα
λν)(∇μQβ
μν)+Qα
μν(μγδ∇γ(δκλ∇κQβ
λν)) ,
equation (4.51) after integration by parts yields (where we used the property of the
Levi–Civita symbol μγδδκλ =δμκδγλ −δμλδκγ )
−
V
drQα
μν(r)Mαβ
2∇2Qβ
μν(r)= (4.53)
=
V
drMαβ
2$∇λQα
μν∇λQβ
μν+∇νQα
μν∇λQβ
μλ+∇νQα
μλ∇λQβ
μν%.
Finally, the excess free energy contribution Eq. (4.41) in terms of gradients of Qαand
Qβis
βFI
ex[Q(r)] = −
V
Φex[Q(r)] dr,(4.54)
where Φex[Q(r)] is the excess contribution of the mesoscopic free energy density (per
unit volume) which in the tensor notation is
Φex =1
2
n
α=1
n
β=1
ραρβ
4π"Mαβ
0Qα:Qβ+Mαβ
2∇Qα∇Qβ(4.55)
+Mαβ
2∇·Qα·∇·Qβ+Mαβ
2∇×Qα:∇×Qβ#.
In general each gradient term on the right hand side of Eq. (4.55) contributes differently
to the excess free energy. However, this expression indicates that in our approximation,
all distortions of the fluid (splay, twist, bend and saddle–splay) contribute equally to the
energy cost of deformation. Taking this into account Eq. (4.55) may be rewritten as
Φex =1
2
n
α=1
n
β=1
ραρβ
4π"Mαβ
0Qα:Qβ+Mαβ
2∇Qα∇Qβ#.(4.56)
46 Chapter 4. Bridging microscopic and mesoscopic theories
4.3.4 Full free energy
As in Subsection (4.2), in the following we write the full free energy given by the sum
of ideal and excess parts FI=FI
id +FI
ex. Adding ideal (4.29) and excess (4.54) contri-
butions we get
βFI[Q(r)] =
V
$Φα
0+Φ
id[Q(r)] −Φex[Q(r)]%dr,
where Φα
0,Φid[Q(r)] and Φex[Q(r)] are the ideal (4.30) and excess (4.56) contributions
of the mesoscopic free energy density (per unit volume), respectively.
In terms of the alignment tensors the mesoscopic free energy density of the system:
Φ[Q(r)] =
n
α=1 Φα
0+Aα(Qα:Qα)−BαTr (Qα·Qα·Qα)+Cα(Qα:Qα)2
+1
2
n
α=1
n
β=1 AαβQα:Qβ+Cαβ∇Qα∇Qβ.(4.57)
where the coefficients Φα
0,Aα,Bα,Cα,Aαβ and Cαβ are given by
Φα
0=ραln(ραΛ3
α)−1,(4.58)
Aα=1
2ρα,(4.59)
Bα=√30
21 ρα,(4.60)
Cα=5
28ρα,(4.61)
Aαβ =ραρβ
4πMαβ
0,(4.62)
Cαβ =ραρβ
4πMαβ
2.(4.63)
and the moments of the direct correlation function ¯c(2)
αβ(r−r)in Eq. (4.49) are given
by
Mαβ
n=1
(2π)3/2
V
dx¯c(2)
αβ(x)|x|n.
As in the case of homogeneous systems, the mesoscopic free energy density derived for
inhomogeneous systems (4.57) has the standard de Gennes form of the elastic energy [19]
in terms of an expansion in terms of the order parameters and its gradients [see Chap-
ter 2]. The main difference with the de Gennes’s approximation and our approach is that
terms with third order derivatives vanish. It is important to note that the microscopical
4.4. Comparison between mesoscopic free energies 47
information entering in Eq. (4.57) through the expansion coefficients is closely related to
the intrinsic properties of the fluid namely the number density and the moments of the
direct correlation function [see Eq. (4.49)]. We have to remark that similar expressions
of these moments are given in various previous studies in Refs. [110,36,123,124] based
on a particle–fixed frame description (parallel to ˆez). Both descriptions are equivalent,
however, with our approach the connection with simulation analysis and experiments
may be simpler.
4.4 Comparison between mesoscopic free energies
To conclude this chapter it is necessary to discuss and establish the connection between
the full mesoscopic free energy density for homogeneous systems, Eq. (4.11)], and the
one corresponding to inhomogeneous systems, Eq. (4.57). This can be done in straight-
forwardly focusing only on the part of the free energy which depends on the orientational
order parameters Qα. With this specialization the mesoscopic free energy densities (per
unit volume), Φ[Q]are:
Homogeneous systems:
ΦH[Q]=
n
α=1 Aα(Qα:Qα)−BαTr(Qα·Qα·Qα)+Cα(Qα:Qα)2(4.64)
+1
2
n
α=1
n
β=1 Aαβ(Qα:Qβ)+Cαβ(Qα:Qβ)2,
Inhomogeneous systems:
ΦI[Q]=
n
α=1 Aα(Qα:Qα)−B
αTr (Qα·Qα·Qα)+Cα(Qα:Qα)2(4.65)
+1
2
n
α=1
n
β=1 AαβQα:Qβ+Cαβ∇Qα∇Qβ,
where the coefficients appearing in Eqs. (4.64) and (4.65) are listed (for convenience)
on Table 4.1. Clearly, the first point of convergence between both expressions is that for
each independent component αterms of second, third and fourth terms of Qαappear.
This confirms that both derivations result in a Landau–like expression of the single
component system.
The coefficients appearing in Eqs. (4.64) and (4.65) depend on the number densities
of each components of the mixture and on different microscopic details. On the one hand,
for homogeneous systems the dependency of the excluded volume coefficients implies
short–range structural properties of the fluid. On the other hand, for inhomogeneous
systems the microscopic dependence on the direct correlation function of the isotropic
48 Chapter 4. Bridging microscopic and mesoscopic theories
Homogeneous systems Inhomogeneous systems
Eq. (4.64) Eq. (4.65)
Qα:Qα1
2ρα+1
10 ρ2
αGααVαα
2
1
2ρα
Tr (Qα·Qα·Qα)√30
21 ρα√30
21 ρα
(Qα:Qα)219
28 ρα+1
18 ρ2
αGααVαα
4
5
28 ρα
Qα:Qβ1
5ραρβGαβVαβ
2
ραρβ
4πMαβ
0
Qα:Qβ21
9ραρβGαβVαβ
4—–
∇Qα∇Qβ—– ραρβ
4πMαβ
2
Table 4.1: Comparison of the coefficients appearing on the final forms of the mesoscopic
free energy densities for homogeneous (4.11) and inhomogeneous systems (4.57).
system implies an measure of the mean interaction of the particles. This feature is what
makes both descriptions, albeit independent, complementary.
Regarding the coupling terms between component αand component βwe see that
the derivation of the mesoscopic free energy for homogeneous systems includes and
extra term of the type Qα:Qβ2which is evidently due to the detailed expansion
of the excluded volume interaction up 4–th rank tensors (see Section 3.3.1). On the
other hand, this term is not present on the derivation for inhomogeneous systems due
to the simple angle dependence assumption introduced in Eq. (4.20). However, what
is recovered by this assumption are terms of gradients of the alignment tensors Qα.
Therefore we conclude that both derivations do not exclude each other but are indeed
complementary and, to certain extent, reciprocal.
4.5 Summary
In this chapter we established a bridge between density functional theory and meso-
scopic Q–tensor theory. To create this link we proposed two different, but at the same
time complementary strategies. First, based on the many fluid Parson-Lee theory, we
performed an expansion of the free energy functional in powers of the Q–tensors of
each species. This procedure yields a spatially homogeneous Landau–like expression for
4.5. Summary 49
the orientational part of the free energy. On the other hand, starting with an ansatz of
the one–particle densities (which assumes small deviations from the isotropic state), we
expressed the second order perturbation theory (Ramakrishnan–Youssuf approximation)
in terms of powers of Qand its gradients. The resulting expressions for the free energy
density are:
Homogeneous systems:
ΦH[Q]=
n
α=1 Aα(Qα:Qα)−BαTr(Qα·Qα·Qα)+Cα(Qα:Qα)2
+1
2
n
α=1
n
β=1 Aαβ(Qα:Qβ)+Cαβ(Qα:Qβ)2,
Inhomogeneous systems:
ΦI[Q]=
n
α=1 Aα(Qα:Qα)−B
αTr (Qα·Qα·Qα)+Cα(Qα:Qα)2
+1
2
n
α=1
n
β=1 AαβQα:Qβ+Cαβ∇Qα∇Qβ,
where the coefficients appearing in these expressions are functions of the microscopic
properties of the system, namely, the number density, aspect ratio and, in the case of
spatially inhomogeneous systems, moments of the inter–component direct correlation
function.
The combination of microscopic (DFT) and mesoscopic descriptions makes our ap-
proach a part of the theories aiming to contribute to a scale–bridging characterization of
complex colloidal mixtures. However, the biggest restriction of our approach is that we
focused on the orientational distribution function alone. This enhances a clear qualitative
difference with classical density functional theories for fluids which describe a variety of
density driven transitions (such as demixing). In the future, it would be interesting to un-
derstand to what extent the above–mentioned expressions are preempted (or modified)
by coupling to density fluctuations.
Chapter 5
Application to binary mixtures
Athermal mixtures of non spherical particles are widely used in functional ma-
terials. These include liquid crystal displays, adaptive optic devices and switchable
windows. Typically these mixtures are characterized by their polydispersity, i.e.,
non uniform range of lengths and diameters. The lowest form of polydispersity is
constituted by binary mixtures. In this chapter we aim to connect our theoretical
approach with real systems.
There are a number of studies focusing on the equilibrium behavior of binary
mixtures [13,14,15,16,17,18]. However, for our theoretical investigations we
focused on two of them: particle based simulations [14] and experimental mea-
surements [16]. First, based on a standard Monte Carlo simulations in Ref. [14]
the behavior of a binary mixture of Hard Gaussian particles is studied. In this
study, they observed the spontaneous formation of orientationally ordered phases
in systems of short and long molecules. On the other hand, the first experimental
observation of such phenomena is due to Purdy et al [16]. Here, for a system
composed of thin semi flexible fd–viruses and thick PEG coated fd–viruses, they
observed that as the concentration of the system is higher, a vast variety of coexist-
ing phases (isotropic–nematic, isotropic–nematic–nematic and nematic–nematic)
appear. Interestingly, for sufficiently low concentrations there is a clear separation
of an isotropic (thin–rich) and nematic (thick–rich) phases. As we can see both
studies describe systems of rigid rod–like particles with very different geometry
which develops in different quantitative results. However, the overall qualitative
behavior of both, simulations and experiments, is very similar.
5.1 Binary mixtures of hard spherocylinders
Since the derivation presented in Chapter 4can be applied to binary mixtures of any type
of rigid anisotropic particles (as long as they are uniaxial and have head–tail symmetry),
51
52 Chapter 5. Application to binary mixtures
Figure 5.1: (a) Binary mixture of A(short) and B(long) molecules (in this case hard
spherocylinders) having the molecular orientations ˆ
uiand ˆ
uj. (b) Hard spherocylinders
are cylinder cape at both sides with hemispheres characterized by two geometrical pa-
rameters: their length lαand their diameter dαwhich defines their aspect ratio, i.e.,
κα=lα/dα.
to make a connection with real systems, in the following we focus on systems of long
and short hard spherocylinders labeled Aand Bfor convenience [see Fig. (5.1)].
Hard spherocylinders are characterized by two geometrical parameters, their length
lαand their diameter dα, defining their aspect ratio κα=lα/dα. The resulting particle
volume is
vα=π
12(3κα+ 2)d3
α.
Between a couple of spherocylinders the excluded volume Vexc(γ)[defined in (3.50)] is
given by
Vexc(γ) = π
4(καd3
α+κβd3
β) + π
4dαdβ(dα+dβ) sin γ+π
4(καdαd2
β+κβdβd2
α)|cos γ|
+κακβdαdβ(dα+dβ) sin γ+ (καdα+κβdβ)dαdβE(sin γ),(5.1)
where E(sin γ)denotes the complete elliptic integral of the second kind. Within the
original Onsager theory, only terms up to linear order in sin γare considered; that is, the
terms involving |cos γ|and E(sin γ)are neglected. In the present work we keep the full
expression (5.1).
5.1.1 Orientational free energy
To calculate the resulting expansion coefficients Vαβ
kwhich are defined in (4.5) and
which determine the coefficients appearing in the free energy [see (4.13–4.15)], we derive
the following analytic expressions for the coefficients Vαβ
k,k= 0,2,4, appearing in the
expansion of Vexc(γ)in Legendre polynomials [see equations (4.4–4.5)]. To this end we
5.1. Binary mixtures of hard spherocylinders 53
expand the functions |cos γ|and E(sin γ)in Eq. (5.18) up to fourth order in sin γ,
yielding
|cos γ|=(1−sin γ)1/2=1−1
2sin2γ−1
8sin4γ+O(6) ,(5.2)
and
E(sin γ)=π
2−π
8sin2γ−3π
128 sin4γ+O(6) .(5.3)
Neglecting the term O(6), the excluded volume becomes
VO(4)
exc (γ)=
π
4c1+π
4c2sin γ+π
4c31−1
2sin2γ−1
8sin4γ
+c4π
2−π
8sin2γ−3π
128 sin4γ+c5sin γ, (5.4)
where the constants cidepend on the aspect ratio and diameters of the particles and
are
c1=καd3
α+κβd3
β,c
2=dαdβ(dα+dβ,
c3=καdαd2
β+κβdβd2
α,c
4=(καdα+κβdβ)dαdβ
c5=κακβdαdβ(dα+dβ).(5.5)
We note that the coefficients c3and c4vanish within the original Onsager approach [24].
Starting from Eq. (5.4), we can now employ Eq. (4.5) to calculate the expansion
coefficients of interest (k=0,2,4):
Vαβ
k=2k+1
2π
0
dγ V O(4)
exc (γ)Pk(cos γ) sin γ. (5.6)
Using P0(cos γ)=1and performing the angular integral, we obtain the zero–order
coefficient
Vαβ
0=π
4c1+π2
16c2+3π
20 c3+97π
240 c4+π
4c5.(5.7)
To calculate Vαβ
2we use that P2(cos γ)=1
2(3 cos2γ−1). Since
π
0
dγ P2(cos γ) sin γ=0,(5.8)
all linear functions of sin γdo not contribute to the coefficient. However, powers of sin γ
do contribute since
π
0
dγ sin γP
2(cos γ) sin γ=−π
16 ,(5.9)
π
0
dγ sin2γP
2(cos γ) sin γ=−4
15 ,(5.10)
π
0
dγ sin4γP
2(cos γ) sin γ=−32
105 .(5.11)
54 Chapter 5. Application to binary mixtures
Inserting these results into (5.6) with k=2we obtain
Vαβ
2=−5π2
128c2+3π
28 c3+17π
168 c4−5π
32 c5.(5.12)
The coefficients Vαβ
4involve the Legendre polynomial P4(cos γ)=1
8(35 cos4γ−30 cos2γ+
3). We find that integrals involving linear and cubic terms in sin γvanish whereas the
other integrals give non-zero results. Specifically,
π
0
dγ sin γP
4(cos γ) sin γ=−π
128 ,(5.13)
π
0
dγ sin4γP
4(cos γ) sin γ=16
315 .(5.14)
With this we finally obtain
Vαβ
4=−9π2
1024c2−π
140c3−3π
560c4−9π
256c5.(5.15)
It is worth noting that, since the aspect ratio and the diameter are always positive, the
coefficients cimust be positive as well. It follows that Vαβ
4is always negative, whereas
the sign of Vαβ
2depends on the combination of sizes. This is different from standard
Onsager theory where c3=c4=0and thus, Vαβ
2is always negative.
Focusing only on the binary mixture (where the components are labeled Aand B)
and on the part of the free energy which depends on the orientational order parameters,
that is,
βFor
V=βF[Q]/V −(FA
0+FB
0+FAB
0),(5.16)
where the terms on the right side are defined in equations (4.12 and (4.13), respectively.
We further specialize on situations where the hard spherocylinders have equal diameters,
i.e., dA=dB=d, but different aspect ratios, κA=κB. This allows to introduce a
dimensionless form of the orientational free energy, For =βFord3/V , and the reduced
densities ρ∗
α=ραd3. With these assumptions, the orientational free energy of a binary
mixture of hard spherocylinders becomes
For =For[ρ∗,κ;Q]
=
α=A,B "Aα(Qα:Qα)−BαTr(Qα·Qα·Qα)+Cα(Qα:Qα)2#
−AAB(QA:QB)−CAB(QA:QB)2,(5.17)
5.2. Isotropic–Nematic transition 55
where the coefficients Aα,Bα,ρα,AAB and CAB, depend on {ρ∗
α},{κα}and are
Aα=1
2ρ∗
α−1
10ρ∗2
α5π2
64 −5π
12 κα+5π
16 κ2
αGαα ,(5.18)
Bα=√30
21 ρ∗
α(5.19)
Cα=19
28ρ∗
α−1
18ρ∗2
α9π2
512 +π
40κα+9π
128κ2
αGαα ,(5.20)
AAB =1
5ρ∗
Aρ∗
B5π2
64 −5π
24 (κA+κB)+5π
32 κAκBGAB ,(5.21)
CAB =1
9ρ∗
Aρ∗
B9π2
512 +π
80(κA+κB)+ 9π
256κAκBGAB .(5.22)
An explicit expression for the quantities Gαβ in equations (5.18–5.22) results from a
generalization of the Carnahan–Starling theory for HS systems to mixtures [125,126],
yielding [35]
Gαβ =4−3(ρ∗
ανα+ρ∗
βνβ)
4(1 −ρ∗
ανα−ρ∗
βνβ)2,(5.23)
where να=vα/d3.
5.2 Isotropic–Nematic transition
As a background for our later discussion of mixtures under shear we first investigate the
isotropic–nematic transition in equilibrium. Specifically, we are interested in the stability
of the two phases in dependence of the densities and aspect ratios.
As mentioned in detail in Chapter 2, in equilibrium one expects a uniaxial ne-
matic phase described by a single director, n, characterizing the preferred alignment
of the rods [82,127].For uniaxial alignment, the second rank tensor reduces to Qα=
(3
2qαnn , where qαis the eigenvalue related to n. An important remark is that the
products appearing in (5.17) become
Tr(Qα·Qβ)=qαqβand Tr(Qα·Qα·Qα)=qα3/√6.
By definition of the second rank tensor [see Eq. (4.1) with k=2] the eigenvalues qα
are related to the well known Maier–Saupe order parameter S2via qα=√5S2[22].
As a further simplification, we focus on the semi–dilute regime corresponding to low
densities. Physically, this implies that the particles have few contacts [128] and that
long–range hydrodynamic interactions are negligible [129]. On a more formal level, the
low–density limit implies that the contact values of the pair correlation functions tend
to one, and the same holds for their density averages.
56 Chapter 5. Application to binary mixtures
Mathematically, this limit corresponds to Gαβ →1. Still, even for this simplification,
the mesoscopic theory we develop provides reliable results (in comparison to particle–
based simulations [108]) in the range κ=20–40 (as we will demonstrate below for the
case of a one–component system). Although the choice Gαβ =1corresponds to that
in the original Onsager theory [24], the present orientational free energy functional is
different due to the treatment of the coefficients in the excluded volume.
5.2.1 Single component system
To compare our approach against literature results, we initially consider the case of a
one–component system. The orientational free energy then reduces to the well–known
Landau free energy [19]
For =A(ρ∗,κ)q2−B(ρ∗)
√6q3+C(ρ∗,κ)q4,(5.24)
where the coefficients now depend on the aspect ratio and the density. From Eq. (5.24),
we find the three stationary solutions (determined by dFor/dq =0)
q0=0,(5.25)
q±=√6B
16C 1±1−64AC
3B2!.(5.26)
As usual, the isotropic phase becomes globally unstable (i.e., d2For/dq2|q0<0) when
the second–order coefficient A(ρ∗,κ)changes sign from positive and negative. In terms
of the density, this implies that within the stable or meta stable isotropic phase,
ρ∗<55π2
64 −5π
12 κ+5π
16 κ2−1
.(5.27)
Directly at A=0, the nematic state already exists as a meta stable state. The cor-
responding value of the order parameter follows from Eq. (5.26)asqc=√6B/8C.
Further, from the second derivative evaluated at q+one finds that the nematic state
becomes globally unstable (d2For/dq2|q±<0) when A>3B2/64C. The free energies
For(q0)and For(q+)become equal at
AIN =B2
24C,(5.28)
The resulting order parameter at isotropic–nematic coexistence is given by
q0=√6B
12C.(5.29)
This simple analysis provides straightforward relations between the density, ρ∗, and the
aspect ratio, κ. In Fig. 5.2 we visualize the stability ranges following from the above
5.2. Isotropic–Nematic transition 57
20 25 30 35 40 45 50 55 60
0
0.2
0.4
0.6
0.8
1
1.2
1.4·10−
2
Nematic
Isotropic
20.220.420.620.8
1.26
1.28
1.3
1.32
1.34 ·10−2
58.258.458.658.8
1.2
1.4
1.6
1.8·10−3
κ
ρ∗
Figure 5.2: Isotropic–nematic transition of a system of hard spherocylinders with aspect
ratios between 20 and 60. The (red) squares correspond to the lower limit of the density
ρ∗given by inequality (5.30), while (green) down–triangles are the numerical solutions
of the relation A=3B2/64C. For comparison, the (blue) circles and up–triangles
correspond to data for the isotropic–nematic phase coexistence from Gibbs ensemble
simulations of Bolhuis et al, see [108].
analysis where the axes are given by the density and the aspect ratio. In Fig. 5.2, the
(red) line represents the boundary of (meta–)stability of the isotropic state, determined
by the maximal density fulfilling the inequality (5.27), as function of κ. On the other
hand, the (green) line indicates the solutions of the relation A=3B2/64Cfor a given
value of κ. This line can be seen as the upper boundary of the stable isotropic state.
For smaller aspect ratios, we compare the stability limits with the coexistence densi-
ties of isotropic and nematic phases (plotted in (blue) •and ) obtained by Bolhuis et
al in Ref. [108]. We find that two sets of data agree reasonably well. Analytically, in the
limit of very large aspect ratios, κ>80, our theoretical description is consistent with
Onsager’s second virial theory [24]. However, for smaller aspect ratios (10 <κ<20) the
analytic calculations are very dissimilar, as we can see when they are compared to the
Gibbs ensemble simulations [see Ref. [108]]. A very important remark is that the orien-
tational free energy Eq. (5.24) can, by definition, not predict the coexistence densities,
since we neglected the densities ραas order parameters.
5.2.2 Binary mixture
Depending on the overall density and concentrations, binary mixtures of particles with
different aspect ratios can exhibit a very rich phase behavior [16,33]. As mentioned
58 Chapter 5. Application to binary mixtures
00.10.20.30.40.50.60.70.80.91
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2·10−
2
xB
ρ∗
Nematic
Isotropic
(a)
(a) κA=20,κ
B=25.
00.10.20.30.40.50.60.70.80.91
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2·10−
2
xB
ρ∗
Nematic
Isotropic
(b)
(b) κA=24,κ
B=25.
Figure 5.3: State diagrams for a mixture of long and short hard spherocylinders with
uniaxial ordering. The nematic ordering is marked with (•), whereas (×)denotes an
isotropic state. In the limit where xB=0these diagrams coincide with the one compo-
nent results portrayed in Fig, (5.2).
before, in order to calculate full phase diagrams, including first–order coexistence regions
in density space, one would have to consider the full free energy given in equations (3.47–
3.50). Here we restricted ourselves to computing orientational state diagrams of the
binary mixture by minimizing the orientational part of the free energy, For, with respect
to the uniaxial order parameters qAand qB. The state diagrams hereby presented are
evaluated in the plane spanned by the total density ρ∗=Nd3/V and concentration
xα=Nα/N(such that ρ∗
α=Nd3xα/V =ρ∗xαis the density of component α). We
focus on two mixtures characterized by aspect ratios also occurring in real systems [16].
The methodology is the following. Using a standard Newton–Raphson minimization
algorithm, we search for values of qAand qBcorresponding to minima of For . We then
evaluate the orientational free energy at these points and determine the minimum value,
min{For}. In the case that min{For}=0the isotropic phase is stable whereas if
min{For} =0a nematic state exists. The result of this procedure is shown in Fig. 5.3.
In both diagrams, Figs. 5.3.(a) and (5.3.(b), we see that for relatively low densities,
ρ∗∼0.005, the isotropic state is stable for all concentrations. As the total density of the
system increases, the isotropic phase becomes unstable with respect to the nematic, and
for sufficiently high densities, ρ∗∼0.020, the system is in the nematic state irrespective
of the concentration.
5.2. Isotropic–Nematic transition 59
As expected, the mole fraction of long spherocylinders plays an important role for
the location of the transition. This is observed by focusing on a constant density value,
ρ∗
∼0.010 in Fig. 5.3.(b). For xB=0.2the nematic phase does not exist, but as xB
increases, xB=0.8, the nematic state appears. In the limiting case where xB=0the
results coincide with the one component system presented in Fig. 5.2.
Clearly, it would be very interesting to compare the equilibrium behavior found here
to that of real systems. However, to our knowledge, experimental results have only be
reported on binary mixtures of fd–viruses with equal length and different diameters [16].
This is different from the mixture of rods with different lengths and equal diameters
considered in Fig. 5.3. One could apply the present theory also to the case of hard
spherocylinders of equal length and different diameters only by re-scaling the number
densities by the length (instead of the diameter) of the particles and recalculating the
excluded volume coefficients in equations (5.21–5.25).
5.2.3 Scaled free energy
Here we make emphasis on the case of different length with the final motivation of
analyze and characterize of the non–equilibrium behavior of rod–disk mixtures such as
(models of) blood [44]. However, for the subsequent analysis of sheared systems it is
convenient to re–write Eq. (5.17) into a dimensionless form using an appropriate scaling
of the free energy. In principle, different scalings are possible. Here we follow the strategy
used in earlier studies [57,80,23,22] of one–component systems.
To develop a proper scaling, let us introduce the scaled alignment tensors, ˜
Qαand
the scaled free energy, For
ref , given by
˜
Qα=Qα
qα
0
and ˜
For =For
For
ref
(5.30)
where qα
0is the coexistence value of the uniaxial order parameter of the corresponding
one–component system given in Eq. (5.29), and For
ref =2CAB qA
0qB
02. In the latter
expression, CAB is the coupling coefficient of the fourth–order term (QA:QB)2of the
free energy (5.17).
To simplify the expression for the scaled free energy resulting from inserting Eq. (5.30)
into Eq. (5.17), we make the following assumptions. First, we set the coefficients Bα
and Cα[see equations (5.19) and (5.20), respectively] equal to their (positive) values
at I–N coexistence of the one–component system, that is,
Bα=Bα(ρ∗
αIN )and Cα=Cα(ρ∗
αIN ,κ
α),(5.31)
where the densities ρ∗
αIN are calculated by solving Eq. (5.31). Thus, we henceforth
neglect changes of Bαand Cαwith ρα. We further suppose that the fourth–order
60 Chapter 5. Application to binary mixtures
coefficient of the one–component system, CAand CB, are related to CAB via
CAB =CAqA
0
qB
02
=CBqB
0
qA
02
.(5.32)
Equation (5.32) implies that for given values of καand ρ∗
α,|C2
AB −CACB|=0. To test
this assumption for a representative example, we consider a binary mixture in the nematic
state (ρ∗=0.002) where 30% of the particles have an aspect ratio κA=20and the
others have an aspect ratio κB=25. In this case we find |C2
AB −CACB|≈3.79×10−7,
which makes our ansatz plausible.
With the simplifications (5.31–5.32), the scaled version of the orientational free
energy [see Eq. (5.17)] becomes
˜
For[˜
QA,˜
QB]=
B
α=A"Θα
2(˜
Qα:˜
Qα)−√6Tr(˜
Qα·˜
Qα·˜
Qα)+ 1
2(˜
Qα:˜
Qα)2#
−ΘAB
2(˜
QA:˜
QB)−1
2(˜
QA:˜
QB)2.(5.33)
In Eq. (5.33), the coefficients Θαof the second–orders term are given by
Θα=24
Aα(ρ∗
α,κ
α)Cα(ρ∗
αIN ,κ
α)
Bα(ρ∗
αIN )2=Aα(ρ∗
α,κ
α)
AIN(ρ∗
αIN ,κ
α),(5.34)
which shows the explicit dependence of Θαon the composition (that is, on ρα) and
κ. The remaining coefficient ΘAB is a positive quantity which is depends not on the
composition, but only on the aspect ratios. Explicitly,
ΘAB =
95π2
64 −5π
24 (κA+κB)+5π
32 κAκB
59π2
512 +π
80 (κA+κB)+ 9π
256 κAκBqA
0qB
0
.(5.35)
In the limiting case of the one–component system, the scaled free energy (5.33) reduces
to the corresponding expression given in [57,80,23,22].
Analysis of the scaled free energy
To better understand the properties of the scaled free energy we consider, first, the
explicit density dependence of the coefficients Θαappearing in front of the quadratic
powers of ˜
Qα[see Eq. (5.34)]. As an illustration, we show in 5.4 numerical results for
an equimolar binary mixture and three values of the aspect ratio (note that each Θα
depends only on κα).
In Fig. 5.4, one observes the same qualitative behavior irrespective of the actual
value of κ: the coefficient Θαfirst increases with density and then reaches a maximum,
after which it monotonically decreases and changes sign from positive to negative. This
change of sign is expected (within the Landau picture) for a system displaying a phase
transition. The corresponding density is the smaller, the larger κ.
5.2. Isotropic–Nematic transition 61
00.511.522.5
·10−2
−2
−1
0
1
ρ∗
Θi
κ=20
κ=24
κ=25
Figure 5.4: Relation of Θαversus the reduced density, given by Eq. (5.34), for equimolar
binary mixtures (xA=xB=0.5) and different aspect ratio. The coexistence densities
ρ∗
αIN are given by the roots of Eq. (5.31).
To examine the relation between the sign of Θαand the mixture’s stability, we
consider the case of uniaxial order. In this case 5.33 reduces to
˜
For
uni =
B
α=A
{Θα
2˜q2
α−˜q3
α+1
2˜q4
α}−ΘAB
2˜qA˜qB−1
2˜q2
A˜q2
B,(5.36)
where ˜qα=qα/qα
0. The stability of a minimum ˜
qcan be determined using the Hessian
matrix [122,130] given by
H(˜qA,˜qB)=⎛
⎝
ΘA−6˜qA+6˜q2
A−˜q2
B−ΘAB
2−2˜qA˜qB
−ΘAB
2−2˜qA˜qBΘB−6˜qB+6˜q2
B−˜q2
A
⎞
⎠(5.37)
evaluated of the minimum. We now focus on the stability of the isotropic state, ˜qt=
(0,0). Stability then requires that the two eigenvalues or, alternatively, the two diagonal
elements Θαand the determinant
det[H(˜qt)] = ΘAΘB−Θ2
AB
4>0,(5.38)
are positive (where we recall that ΘAB is a positive quantity). An illustration of the
regions of stability of the isotropic phase in the ΘA–ΘBplane at fixed ΘAB is portrayed
in Fig. 5.5.
62 Chapter 5. Application to binary mixtures
(a)
-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
ΘA
ΘB
Unstable
Isotropic
(b)
-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
ΘA
ΘB
Unstable
Isotropic
Figure 5.5: Stability regions of the isotropic state (˜qt=(0,0)) according to (5.38). The
values of ΘAB are (a) ΘAB =0.50 and (b) ΘAB =1.50.
From Fig. 5.5 we notice that ˜
qtis locally stable for ΘA,ΘB0.5and locally
unstable for ΘA,ΘB0.5. This areas are marked by the white striped and solid blue
portions, respectively. The center solid green area denotes values where (5.38) is negative
and thus ˜
qtis locally meta stable. The larger ΘAB , the smaller is the region where the
isotropic state is a minimum of the free energy.
It seems straightforward to repeat the same analysis for free energy minima cor-
responding to nematic states (qA=0,qB=0). However, in that context we face a
problem already discussed in the context of one–component systems [63,61], which con-
cerns the bounds of possible values. This can be seen as follows: The order parameters
qAand qBare related to the Maier–Saupe order parameter S2by qα=√5S2[22].
Since S2is defined within the range −1/2≤S2≤1. The bounds of the correspond-
ing scaled order parameters, qAand qB, involve the factor 1/qα
0. Using Eq. (5.32) and
the values of the coexisting densities ρ∗
αIN given by the roots of Eq. (5.31), we find that
the scaled order parameters are defined in the range
−1.30 qα2.60 .(5.39)
However, free minimization of the scaled free energy Eq. (5.33) does not automatically
respect these bounds; rather it yields, for a range of parameters, nematic minima outside
the allowed domain. To solve this problem, one could consider an ”amended” version
of the free energy, similar to what has been done for one–component systems [63,61].
However, this would go beyond the scope of the present work.
Given the simplifications involved in our scaling of the free energy, another question
that arises is to which extent the stability conditions for the isotropic phase found here
5.3. Elastic energy 63
Fig. 5.3.(a) Fig. 5.3.(b) Eq. (5.38)
ρ∗ΘαΘαdet[H(˜
qt)]
(10−2)κA=20 κB=25 κA=24 κB=25
0.50 0.599 0.825 0.783 0.825 + Isotropic
1.00 0.929 0.973 0.995 0.973 + Isotropic
1.75 0.918 -0.073 0.239 -0.073 - Unstable
2.00 0.779 -0.759 -0.298 -0.759 - Unstable
ΘAB =1.203 ΘAB =1.223
Table 5.1: Numerical values of the constants ΘA,ΘBand ΘAB for equimolar binary
mixtures (xA=xB=0.5). This values correspond to different points in the phase
diagrams in figures 5.3(a–b) and are calculated using Eq. (5.34) and Eq. (5.35).
are consistent with our earlier results obtained by minimization of the unscaled free
energy [see Fig. 5.3]. To solve this, in Table (5.1) we summarize a simple evaluation
of the quantities Θαfor several state points in the diagrams plotted in Fig. 5.3, which
shows consistency of unscaled and scaled free energy.
5.3 Elastic energy
In this section we turn to the elastic energy of the binary mixture. As mentioned at
the end of Chapter 4the difference between the mesoscopic free energy for homoge-
neous systems (4.64) and the one for inhomogeneous systems (4.65) is that in the later
terms of gradients of the alignment tensor Qare taken into account. These additional
terms correspond to the so–called elastic free energy or distortion free energy, For
e[∇Q].
Focusing on this contribution we have
For
e=DAA∇QA∇QA+DBB∇QB∇QB+DAB∇QA∇QB,(5.40)
where the Dαβ coefficients are given by
DAA =ρ2
A
4πMAA
2,DBB =ρ2
B
4πMBB
2and DAB =ρAρB
4πMAB
2.(5.41)
and Mαβ
2are the moments of the direct pair correlation function [see Chapter 4]:
Mαβ
2=1
(2π)3/2
V
dx¯c(2)
αβ(x)|x|2.
64 Chapter 5. Application to binary mixtures
5.3.1 Oseen–Frank theory
As mentioned in Chapter 2, the Landau–de Gennes theory [19] couples phenomenological
elastic coefficients Lwith the well known Frank elastic constants Kij. In the one–
constant approximation [19] this relation is given by
L=1
2S2
eq
K, (5.42)
where Seq is the equilibrium uniaxial order parameter of the single component fluid [81].
In analogy with this we propose that
Dαβ =1
4qα
0qβ
0
Kαβ ,(5.43)
where qα
0is the coexistence value of the uniaxial order parameter in Eq. (5.29). Substi-
tution of Eq. (5.41) in Eq. (5.43) leads to
Kαβ =ραρβ
4π
qα
0qβ
0
(2π)3/2
V
dx¯c(2)
αβ(x)|x|2,(5.44)
Doing so, we obtained the Frank elastic constants Kαβ of the binary mixture in the
one–constant approximation. Comparing to previous studies (see e.g. Refs. [41,32] and
references therein) the aforementioned elastic constants are given in terms of the direct
pair correlation function measured in the isotropic phase and the coexistence values
of the uniaxial order parameter of the one component fluid, instead of the direct pair
correlation function of the undistorted nematic phase [41,32]. As it is, Eq. (5.44)isan
approximation that relates weak distortions of the nematic phase to the the isotropic
fluid.
As a further simplification, as before, we may assume that the diameter of the
molecules is such that dα=dβ=d, we may write the dimensionless elastic energy For
e
in terms of the reduced densities ρ∗
α=ραd3viz:
For
e=ξ2
AA
2∇QA∇QA+ξ2
BB
2∇QB∇QB+ξ2
AB
2∇QA∇QB,(5.45)
where ξαβ are the elastic correlation length coefficients given in terms of the Frank
elastic constants as
ξ2
αβ =Kαβ
2qα
0qβ
0d3.(5.46)
In general ξαβ is given in terms of the pitch of the cholesteric which determines the
typical length scale associated to the helical periodicity. The pitch length is normally on
the order of hundreds or thousands times the particle length. However, a unified picture
which links microscopic and macroscopic chirality is still to be achieved since it is clear
that the relation between pitch and elastic constants depends on both the single-particle
properties and the thermodynamic state of the system [131].
5.4. Summary 65
5.4 Summary
In this chapter we developed a theory suitable for the description of isotropic and nematic
phases of binary mixtures of hard spherocylinders in thermodynamic equilibrium. We
started from the general expressions of the mesoscopic theory obtained in Chapter 4and
specialize our description to the semi–dilute regime reaching similar expressions to the
well–known Landau–de Gennes theory. The resulting expression for the orientational free
energy is then:
For=
B
α=A
"Aα(Qα:Qα)−BαTr(Qα·Qα·Qα)+Cα(Qα:Qα)2#
−AAB (QA:QB)−CAB (QA:QB)2,
where the coefficients Aα,Bα,ρα,AAB and CAB are
Aα=1
2ρ∗
α−1
10ρ∗2
α5π2
64 −5π
12 κα+5π
16 κ2
αGαα ,
Bα=√30
21 ρ∗
α
Cα=19
28ρ∗
α−1
18ρ∗2
α9π2
512 +π
40κα+9π
128κ2
αGαα ,
AAB =1
5ρ∗
Aρ∗
B5π2
64 −5π
24 (κA+κB)+5π
32 κAκBGAB ,
CAB =1
9ρ∗
Aρ∗
B9π2
512 +π
80(κA+κB)+ 9π
256κAκBGAB .
To validate our theory, we further analyze mixtures characterized by molecules hav-
ing equal diameters and different length as in Ref. [14] (opposed to the experimental
realizations of ”thick–thin” mixtures of fd–viruses having equal length and different
diameters [16]). Our numerical calculations show that even for the one–component sys-
tem, there is an reasonable agreement with the coexistence densities obtained by Gibbs
ensemble simulations [108]. In case of binary mixtures we did not resolve a full phase
diagram of the system. However, we calculate a state diagram which can be use to give
some predictions of the overall behavior of the mixture.
At the end of this chapter we briefly discuss the Oseen–Frank elastic theory and
its relation with the mesoscopic free energy density of distortion for a binary mixture.
We show how after a simple assumption it is possible to write an expression similar to
the well–known Frank elastic constants in terms of the direct correlation function. The
resulting expression is given in of gradients of Qviz
For
e[∇Q]=ξ2
AA
2∇QA∇QA+ξ2
BB
2∇QB∇QB+ξ2
AB
2∇QA∇QB,
66 Chapter 5. Application to binary mixtures
where ξαβ are the elastic correlation lengths
ξ2
αβ =Kαβ
2qα
0qβ
0d3where Kαβ =ρ∗
αρ∗
β
4π
qα
0qβ
0
(2π)3/2
V
dx¯c(2)
αβ(x)|x|2
where Kαβ are the Frank elastic coefficients of the weakly distorted nematic phase. As
in the case of homogeneous systems, this relation may be used to further character-
ize the material and to provide a first approximation to link between microscopic and
macroscopic chirality. However, we should point out that at this level of simplification
the material is only defined by a small number of quantities which may not be enough
to fully parametrize any material.
Chapter 6
Dynamic equations and
constitutive relations for binary
mixtures
The non–equilibrium phenomena associated with the molecular orientation of
anisotropic fluids is coupled with the exchange of mass, energy and momen-
tum. However, due to its molecular structure, a proper description of their dy-
namic properties requires a measure of the rotational degrees of freedom of the
molecules. Such description can be done in different levels: in terms of single–
particle models where each particle is described by the set {ri(t), φi(t)}(Jef-
fery orbits [132]); in terms of probability distribution functions p(r, φ, t)(Fokker–
Planck equation [80,133], Dynamical Density Functional theory [100,134]); or, in
terms of the evolution of macroscopic variables, like the nematic director n(r, t)
(Ericksen–Leslie theory [135,136]) or the alignment tensor Q(r, t)(Doi–Hess
theory [57,137]). In this chapter we aim to extend the later description for bi-
nary mixtures employing our equilibrium theory and the basic concepts of linear
irreversible thermodynamics.
6.1 General remarks
The Doi–Hess theory [57,80] of irreversible processes in liquid crystals is based on
the local equilibrium hypothesis. This hypothesis states that all equilibrium thermo-
dynamic relations are valid for the thermodynamic variables assigned to an elemental
volume [138]. Thus, individual molecules are ignored and the fluid consists of continu-
ous matter. Hence, all thermodynamic variables like temperature, internal energy Uand
67
68 Chapter 6. Dynamic equations and constitutive relations for binary mixtures
entropy Sare given in terms of field variables,i.e.,
T=T(r, t),U(t)=V
u(r,t)dV , S(t)=V
s(r,t)dV , (6.1)
where is the mass density of the fluid and the field variables u(r,t),s(r,t)are the
specific internal energy and entropy, respectively. Consequently, the thermodynamics of
irreversible processes is a continuum theory treating the state parameters of the system
as field variables [105,139].
For fluids with internal degrees of freedom the general assumption of the Doi–Hess
theory [57,80] is that the thermodynamic functions, such as specific internal energy,
specific volume and specific entropy, depend on the alignment. Further, these quantities
may be decomposed in two parts: one that is independent on the alignment (0) and the
other which vanishes in the absence of it (a). Thus,
u=u0+ua(Q),
−1=−1
0+−1
a(Q),s=s0+sa(Q).(6.2)
Under this assumption the Gibbs fundamental relation [57]is
Tds=du +pd−1−∂g
a(Q)
∂QdQ,(6.3)
where ga(Q)is the specific Gibbs free energy associated with the alignment
ga=ua+p−1
a−Tsa.(6.4)
Commonly, gais associated with the alignment tensor and its gradients [57],
ga=ga(Q,∇Q).(6.5)
6.2 Extension to binary mixtures
In order to extend the treatment of irreversible thermodynamics to a binary mixture
several modifications of the standard theory are needed 1. First, we assume that we
have an homogeneously mixed system, i.e. every element of volume has the same mass
density =m/V where mis the mass of the mixture.
Firstly, in the standard theory [57,23], the Gibbs free energy of alignment is given
by a dimensionless form of the sum of the Landau–de Gennes potential (2.17) with the
energy of distortion (2.23) [see Chapter 2] with a proportionality factor. For the binary
mixture we assume that ga(Q,∇Q)is given in terms of the free energy density functional
For[Q,∇Q]=For [Q]+For
e[∇Q],
ga(Q,∇Q)=kBT
mFor[Q,∇Q].(6.6)
1For an alternative derivation, please refer to Henning Reinken’s Master’s thesis [71].
6.2. Extension to binary mixtures 69
where For and For
eare given by
For[Q]=
B
α=A
"Aα(Qα:Qα)−BαTr(Qα·Qα·Qα)+C(Qα:Qα)2#
−AAB (QA:QB)−CAB (QA:QB)2,(6.7)
For
e[∇Q]=ξ2
AA
2∇QA∇QA+ξ2
BB
2∇QB∇QB,(6.8)
where Aα,Bα,ρα,AAB and CAB are given by Eqs. (5.18)–(5.22) in the semi–dilute
regime [see Chapter 5Sect. 5.2] and ξ2
αα by Eq. (5.46). Notice that in Eq. (6.8) terms
carrying the coupling ∇QA∇QBare not taken into account. This might corresponds
to weak distortions of the coexisting nematic state. This assumption is similar to the one
adopted in Ref. [133] which is based in the mean field potential approximation. Thus,
the coupling between component Aand component Bis only carried out through terms
containing the products (QA:QB)appearing in Eq. (6.7).
Finally, from Eq. (6.6), taking into account the concentration xα(μαchemical po-
tential) of the components in the mixture, the Gibbs fundamental relation (6.3) for the
binary mixture is
Tds=du +pd−1+
B
α=A
μαdxα−kBT
m
B
α=A
∂For[Q,∇Q]
∂QαdQα.(6.9)
6.2.1 Gibbs fundamental relation
In the non–equilibrium situation, the thermodynamic variables are functions of time and
thus the Gibbs fundamental relation becomes [57]
Tds
dt =du
dt +pd−1
dt +
B
α=A
μα
dxα
dt −kBT
m
B
α=A
∂For
∂Qα
dQα
dt .(6.10)
The quantities dA/dt are known as thermodynamic fluxes and measure the rate of
change of Adue to the different transport processes occurring in the system. To link
transport processes with thermodynamic fluxes a set of constitutive equations is needed.
For example, disregarding spatial inhomogeneities, one of the simplest (non–trivial) con-
stitutive equations for the alignment tensors is
dQα
dt =−ταQα,(6.11)
where ταis the relaxation time of the alignment of the system towards equilibrium. This
equation describes the simple exponential relaxation of the alignment of component
αin the absence of orienting fields or flows. However, Eq. (6.11) is an oversimplified
assumption and in reality it may be more complicated depending on the effects of flow
or external fields.
70 Chapter 6. Dynamic equations and constitutive relations for binary mixtures
6.2.2 Conservation laws and balance equation
From the standpoint of the theory of irreversible processes, the constitutive equations
are obtained from local conservation laws, particularly conservation of mass, momentum,
internal energy and angular momentum). In isothermal conditions the conservation of
mass, momentum and internal energy for the mixture are:
dxα
dt +∇·Jα=0,(6.12)
dv
dt −∇·Ttot =0,(6.13)
du
dt −Ttot :∇v=0.(6.14)
As it is common in continuum theories, d/dt denotes the material derivative given by
d
dt ≡∂
∂t +v·∇.(6.15)
In Eqs. (6.12)–(6.14)v(r,t)is the flow velocity, Ttot is the total stress tensor and Jα
is the diffusion flux of the component αwith concentration xα.
A key quantity used for the study of the hydrodynamic properties of the mixture is
the stress tensor Ttot. The total stress tensor can be written as
Ttot =−pα+Tasy +T,(6.16)
where the first term represents the (isotropic) hydrostatic pressure and the two other
terms describe flow–induced effects [57]. The antisymmetric part of the stress tensor,
Tasy, relates the angular momentum of the particles with the velocity field v. For liquid
crystals (and other colloidal systems) the angular relaxation times are much smaller than
the collective orientational order relaxation rate [140] and therefore Tasy is usually set
to zero. In fact, in Newtonian flow regimes Tasy naturally vanishes [22,63]. However,
as we will see in the following section, the remaining contribution of the stress tensor
Tdoes not vanish and in fact is associated with the alignment tensors Qα.
The time dependence of the alignment tensors Qαis described by balance equations
which couple the dynamics of the alignment with the velocity field v. Following the
ansatz of Hess–Pereira [57,58] and disregarding non–convective flow of alignment we
assume
dQα
dt =2Ω·Qα+2σΓ·Qα+δQα
δt irr
,(6.17)
where Ω=(1/2)(∇vT−∇v)and Γ=(1/2)(∇vT+∇v)are the flow vorticity and
deformation, respectively. In Eq. (6.17), (δQα/δt)irr represents the production and decay
of alignment due to irreversible processes [57,23,58].
6.2. Extension to binary mixtures 71
Finally, for the theory of non–equilibrium thermodynamics the so–called balance
equation for the entropy plays a central role. This equation reads:
ds
dt +∇·Js=δs
δtirr ≥0.(6.18)
Equation (6.18) expresses the fact that the entropy of a volume element changes in
time for two reasons: it changes because entropy flows into the volume element with a
rate Js; and because there is an entropy source due to irreversible phenomena inside the
volume element, (δs/δt)irr. The entropy production, (δs/δt)irr, is always non–negative
(Second law of thermodynamics) [139].
6.2.3 Entropy production and phenomenological equations
In order to find an explicit form of the entropy production in the volume element due
to the hydrodynamic interactions and the decay or production of alignment we have to
insert Eqs. (6.12)–(6.18) into the Gibbs fundamental relation (6.8). This yields
Tδs
δtirr
= T−2
mkBTσ
B
α=A
Qα·Ψα!:Γ−
mkBT
B
α=A
Ψα:δQα
δt irr
,
(6.19)
where the second rank tensor Ψαis given by [see Eqs. (6.7) and (6.8)]. The entropy
production in (6.19) takes into account two processes which correspond to dissipative
processes: viscous flow and production (decay) of alignment. Moreover, each of these
terms contains a thermodynamic force–flux pair. In general, the choice of which quantity
is a thermodynamic flux or force depends on their spatial and temporal symmetry. Con-
sidering that Γand (δQα/δt)irr are not independent under time reversal it is convention
to choose them as thermodynamic forces [58].
In the linear approximation [139] fluxes are proportional to the thermodynamic forces
and thus, the entropy production (6.19) is a bi–linear expression in the fluxes and ther-
modynamic forces. Within this approximation we write the relation between fluxes and
forces for the binary mixture as:
−ΨA=τAδQA
δt irr
+τAB δQB
δt irr
+√2τApΓ,(6.20)
−ΨB=τBδQB
δt irr
+τAB δQA
δt irr
+√2τBpΓ,(6.21)
T−2
mkBTσ
B
α=A
Qα·Ψα
=√2
mkBTτpA δQB
δt irr
+τpB δQA
δt irr
+√2τpΓ.(6.22)
72 Chapter 6. Dynamic equations and constitutive relations for binary mixtures
In Eqs. (6.20)–(6.22) the phenomenological coefficients τxand have units of time. It is
possible to write Eqs. (6.20)–(6.22) in a somewhat more transparent form as:
⎛
⎜
⎜
⎜
⎜
⎜
⎝
−ΨA
−ΨB
T−2
mkBTσB
α=AQα·Ψα
⎞
⎟
⎟
⎟
⎟
⎟
⎠
=
⎛
⎜
⎜
⎜
⎜
⎝
τAτAB τAp
τBA τBτBp
τpA τpB τp
⎞
⎟
⎟
⎟
⎟
⎠
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
δQA
δt irr
δQB
δt irr
2
mkBTΓ
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
,
(6.23)
where we identify Onsager’s matrix of phenomenological coefficients Lwhich is
L=
⎛
⎜
⎜
⎜
⎜
⎝
τAτAB τAp
τBA τBτBp
τpA τpB τp
⎞
⎟
⎟
⎟
⎟
⎠
.(6.24)
According to Onsager’s theorem Lis a symmetric matrix [141] and thus the cross
coupling coefficients obey the so–called Onsager’s reciprocal relations
τAp =τpA,τ
Bp =τpB and τAB =τBA .(6.25)
Since the production of entropy is always a non–negative quantity, a by–product of
Onsager’s theorem is that Lshould always be positive semi–definite matrix [141]. Thus
the following relations exist between the coefficients of the phenomenological laws in
Eqs. (6.20)–(6.22):
τA>0,τ
B>0,τ
p>0,(6.26)
τAτB>τ
2
Ap ,τ
Bτp>τ
2
Bp ,τ
AτB>τ
2
AB ,(6.27)
τAτBτp+2τABτApτBp >τ
Aτ2
Bp +τBτ2
Ap +τpτ2
AB .(6.28)
As expected making τB=τAB =τBp =0results in the original phenomenological laws
set by the standard Doi–Hess theory [57].
6.2.4 Complete dynamic equations
Using the subtraction method, the set of equations for the thermodynamic fluxes of
alignment (6.20)and(6.21) can be written into
δQA
δt irr
=πAΨA+πABΨB+πAp√2Γ,(6.29)
δQB
δt irr
=πABΨA+πBΨB+πBp√2Γ,(6.30)
6.2. Extension to binary mixtures 73
where the coupling coefficients πxhave units of time−1and are given in terms of the
relaxation times τxviz
πA=−τB
τAτB−τ2
AB
,π
AB =τAB
τAτB−τ2
AB
,π
B=−τA
τAτB−τ2
AB
,(6.31)
πAp =τABτBp −τApτB
τAτB−τ2
AB
,π
Bp =τABτAp −τBpτA
τAτB−τ2
AB
.(6.32)
Finally, inserting Eqs. (6.29) and (6.30) into Eq. (6.17) results into the equations
that describe the dynamical behavior of a binary mixture in the presence of a velocity
field v(r,t), which are,
dQA
dt =2Ω·QA+2σΓ·QA+πAp√2Γ+πAΨA+πABΨB,(6.33)
dQB
dt =2Ω·QB+2σΓ·QB+πBp√2Γ+πABΨA+πBΨB.(6.34)
where Ψαtakes into account the relaxation of the alignment towards equilibrium via
ΨA=∂For[Q]
∂QA−ξ2
AA∇2QA,(6.35)
ΨB=∂For[Q]
∂QB−ξ2
BB∇2QB.(6.36)
6.2.5 Equation for the stress tensor
Before further discussion and applications of the dynamic equations of the alignment
tensors [Eqs. (6.33) and (6.36)] some implications of the phenomenological laws (6.20)–
(6.22) have to view reviewed. Particularly the viscous flow of the binary mixture.
Inserting Eqs. (6.29) and (6.30) into the phenomenological law of the stress ten-
sor (6.22) results into the relation
T=2ηisoΓ+Tal .(6.37)
According to Eq. (6.37), Tsplits into a Newtonian contribution (already present in
fluids with vanishing orientational order) and a contribution depending explicitly on the
alignment. The Newtonian viscosity ηiso is modified viz
ηiso =
mkBTτp 1−τ2
ApτB+τ2
BpτA−2τABτApτBp
τAτBτp−τ2
ABτp!.(6.38)
The first term of Eq. (6.38) corresponds to the viscosity of the mixture when QAand
QBand consequently ΨA(ΨB)completely vanish. The second term characterizes the
74 Chapter 6. Dynamic equations and constitutive relations for binary mixtures
coupling between the stress tensor and the alignment when QAand QBare non–zero.
Lastly, the contribution depending explicitly on QAand QBis given by
Tal =
mkBT √2
B
α=A
παpΨα+2σ
B
α=A
Qα·Ψα!,(6.39)
where the coefficients παp are given by Eq. (6.32). Equation (6.39) is known as the stress
tensor of alignment and is this contribution which allows us to study the hydrodynamic
properties of the fluid.
Finally, to measure the hydrodynamic effects arising in the binary mixture due to the
alignment QAand QBthe momentum balance equation reads
dv
dt −∇·T=0,(6.40)
6.3 Relaxation of the alignment
In the absence of flow (v=0) and disregarding spatial inhomogeneities Eqs. (6.33)–
(6.34) read
dQA
dt =πA
∂For[Q]
∂QA+πAB
∂For[Q]
∂QB,(6.41)
dQB
dt =πAB
∂For[Q]
∂QA+πB
∂For[Q]
∂QB.(6.42)
Notice that (6.41)–(6.42) are non–linear relaxation equations since For[Q]as given
in (6.7) is non–linear with respect to QAand QB. The full relaxation of the order
parameters is governed by Eqs. (6.41) and (6.42). Though this non linear system can
solved in general, only some special features are discussed in the following. In particular,
we study the linear decay of the alignment.
If For[Q]is replaced by its linear approximation, Eqs. (6.41)–(6.42) reduce to the
system of coupled linear equations
dQA
dt =πA(AAQA−AABQB)+πAB(ABQB−AABQA),(6.43)
dQB
dt =πAB(AAQA−AABQB)+πB(ABQB−AABQA),(6.44)
which can be written as the system
˙
Q=MQ,(6.45)
where Q=(QA,QB)Tand the matrix Mis given by
M=⎛
⎝
πAAA−πABAAB πABAB−πAAAB
πABAA−πBAAB πBAB−πABAAB⎞
⎠.(6.46)
6.3. Relaxation of the alignment 75
10 15 20 25 30 35 40 45 50
0
0.2
0.4
0.6
0.8
1
(a)
time (seconds)
Order parameter
qA
qB
10 15 20 25 30 35 40 45 50
−2
−1
0
1
2
(b)
time (seconds)
Order parameter
qA
qB
Figure 6.1: Relaxation towards equilibrium of the binary mixture given in Eq. (6.51):
(a) Non coupled system AAB =0, (b) Coupled system AAB =0.
This approximation is applicable for small alignment and low concentrations. Since the
relaxation times τxand for small concentrations the coefficients Axare always posi-
tive, then Mis a positive definite matrix. Hence, Eq. (6.45) shows that the alignment
of the binary mixture relaxes to zero. This feature is common in liquid crystals with
concentrations above the isotropic–nematic coexistence characteristic value [19].
6.3.1 Vanishing cross–coupling of the entropy production
When the system relaxes towards its equilibrium value it is possible to track the relaxation
of the full alignment tensors QAand QBby only looking at the uniaxial alignment with
a constant director [57]. The coupling between the true relaxation times in the full non–
linear equations (6.41)) and (6.42)) is more complicated and requires further analysis
by its own. In order to simplify the analysis, in the following we will assume vanishing
cross–coupling.
The assumption of vanishing cross–coupling implies that the production of entropy
of component Aand the production of entropy of component Bare completely inde-
pendent. Thus, the relaxation time τAB vanishes and consequently πAB =0. Under this
approximations Eq. (6.45) reduces to
⎛
⎝
dtqA(t)
dtqB(t)
⎞
⎠=⎛
⎝
−τ−1
AAAτ−1
AAAB
τ−1
BAAB −τ−1
BAB
⎞
⎠⎛
⎝
qA(t)
qB(t)
⎞
⎠,(6.47)
where qA(t)and qB(t)are the uniaxial order parameters of the corresponding alignment
tensors and dtdenotes the material derivative (6.15). The quantities τAA−1
Aand τBA−1
B
76 Chapter 6. Dynamic equations and constitutive relations for binary mixtures
are known as the true relaxation time of alignment of the single component system [80].
Thus, the true relaxation times of alignment for the binary mixture are τAA−1
AB and
τBA−1
AB
In general, the solutions of the system (6.47) have the form
q(t) = C1x1exp(λ1t) + C2x2exp(λ2t)(6.48)
where Ci(with i= 1,2) are constants that depend on the initial conditions, and xiand
λiare the eigenvectors and eigenvalues of M. The simplest solution of (6.47) is obtained
when the relaxation towards equilibrium of Ais independent of the relaxation of B,i.e.
there is no inter–component coupling, thus, AAB = 0 [see Fig. 6.1.(a)]. However, the
real solution is obtained after solving the eigenvalue equation
det |M−λα|= 0 .(6.49)
Before solving Eq. (6.49) it is important to make some general remarks on the
phenomenological relaxation times τAand τB. In Ref. [80] the single particle relaxation
times ταare
τα=8πηR3
g
6kBT,(6.50)
where ηis the viscosity of the solvent (usually water for colloidal suspensions) and Rg
is the radius of gyration of the rigid molecule. Considering fd–viruses with a radius of
gyration ∼11nm embedded in a aqueous solution at room temperature (η≈0.79mPa·s
at 300K) [142,143] we get τfd–virus ≈10−2−10−3s.
Considering a binary mixture of short and long fd–viruses with aspect ratios κA= 20
and κB= 25 and phenomenological relaxation times τA= 7.5×10−3and τb= 2.5×10−3
in the isotropic state [see Fig. 5.3], Eq. (6.47) reads:
dtqA(t)
dtqB(t)
=
−0.215 −1.286
−0.428 −0.069
qA(t)
qB(t)
,(6.51)
Given an initial value different form zero, the analytic solutions of (6.51) may be obtained
straightforwardly from Eq. (6.47). The behavior of these solutions in time is showed in
Fig. 6.1.(b). As expected, even for the simplest inter–component coupling the mixture
relaxes towards the values qA=qB= 0. Here, the alignment presents a regular oscillatory
decay of the type sin(t) exp(−t)showing that even for a simple linear approximation
the coupling coefficient AAB plays a very important role.
In the same manner, it is possible to study the relaxation of uniaxial order parameters
towards its equilibrium nematic values. This task is in general more complicated since
(at least) terms of second order in qAand qB(i.e. q2
α(t)) should be included. Thus we
6.3. Relaxation of the alignment 77
have to consider the full system
dQA
dt =−1
τA
∂For[Q]
∂QA,(6.52)
dQB
dt =−1
τB
∂For[Q]
∂QB.(6.53)
Trivially, for the uncoupled system (AAB =CAB =0), the uniaxial order parameters
qAand qBmust decay uniformly [23] at a rate τ−1
αAαto the equilibrium value of the
corresponding one–component system (qc
Aand qc
B) [see above Eq. (5.28) in Sec. 5.2.1].
However, depending on the degree of nematicity of the system the dynamic behavior
may be sensitive to the values of Aα,Bαand Cαor even to the initial conditions of the
system. On the other hand, when the coupling coefficients does not vanish, i.e. AAB =0
and CAB =0, this analysis is even more complex and further studies in this direction
are needed.
In this work we assume that in the nematic state both components of the mixture
decay uniformly at a same rate τref . As it is the case for the one component system [80]
this rate is supposed to be modulated by the ratio of the coefficients Bαand Cα
τref =τA
24CA
B2
A
=τB
24CB
B2
B
.(6.54)
The ratio B2
α/24Cαcorresponds to the isotropic–nematic coexistence value of Aα[see
Eq. (5.28)]. Equation (6.54) indicates that the solutions of Eqs. (6.52) and (6.53) decay
uniformly at the same rate towards the value of qAand qBin the stationary state.
6.3.2 Final dynamic equations
Under the vanishing cross–coupling approximation (πAB =0), Eqs. (6.33) and (6.34)
simplify to the set of equations:
dQA
dt =2Ω·QA+2σΓ·QA−√2τAp
τA
Γ−1
τA
∂For[Q]
∂QA+ξ2
AA∇2QA,(6.55)
dQB
dt =2Ω·QB+2σΓ·QB−√2τBp
τB
Γ−1
τB
∂For[Q]
∂QB+ξ2
BB∇2QB.(6.56)
where Ω=1
2(∇vT−∇v)and Γ=1
2(∇vT+∇v)describe the effect of flow vorticity and
its deformation on the average alignment of the particles, respectively. The ratio τip/τα
in the third term of Eqs. (6.55–6.56) measures the impact of external perturbations. It
defines the tumbling parameter λαa quantity related to the aspect ratio καviz
λα=−ταp
τα
=3
5
κ2
α−1
κ2
α+1.(6.57)
For spherical particles λ=0, whereas for prolate and oblate particles, λ>0and λ<0,
respectively.
78 Chapter 6. Dynamic equations and constitutive relations for binary mixtures
Figure 6.2: Sketch of a binary mixture in a planar Couette flow. Here, the mixture of
rod–like molecules is enclosed between two infinite parallel plates at y=±Lmoving
along the x–axis with velocities vx=±L˙γ.
6.4 Simple Couette geometry
In the following we specialize to a planar Couette flow (see Fig. 6.2) where the fluid
is confined between two infinitely extended, parallel plates (separated by a distance 2L
along the y–direction) moving in opposite directions. The flow profile (for a Newtonian
fluid) is then given by v(r)= ˙γyˆ
ex, with ˙γthe shear rate.
6.4.1 Scaled dynamic equations
Reintroducing the scaled alignment tensors ˜
Qαand the scaled free energy according to
Eq, (5.30) and the definition of λαin Eq. (6.54), equations (6.55) and (6.56) become
d˜
QA
d˜
t=2
˜
˙γ˜
Ω·˜
QA+2σ˜
˙γ˜
Γ·˜
QA−˜
Φ
A+√2˜
λA˜
˙γ˜
Γ+˜
ξ2
A˜
∇2˜
QA,(6.58)
d˜
QB
d˜
t=2
˜
˙γ˜
Ω·˜
QB+2σ˜
˙γ˜
Γ·˜
QB−˜
Φ
B+√2˜
λB˜
˙γ˜
Γ+˜
ξ2
B˜
∇2˜
QB,(6.59)
where ˜
t=t/tref and ˜
˙γ=˙γtref are the re-scaled time and shear rate, ˜
∇2=∇2/L2,
and the reference time tref is given by
tref =τα(qα
0)2
For
ref
.(6.60)
The scaled vorticity and deformation tensors are ˜
Ω=(1/2)(ˆexˆey−ˆeyˆex)and ˜
Γ=
(1/2)(ˆexˆey+ˆeyˆex), respectively. Further, the scaled tumbling parameters and correlation
6.4. Simple Couette geometry 79
lengths are
˜
λα=λα
qα
0
=3
5
1
qα
0
κ2
α−1
κ2
α+1,˜
ξ2=(qα
0)2
L2For
ref
ξ2
ii .(6.61)
An example of typical values of the scaled tumbling parameters, we consider systems
involving the PEG–coated fd–virus, like the ones used in the experiments of thick and thin
colloidal rods of Purdy et al. [16]. For these systems, the Maier-Saupe order parameter
at coexistence is S0=P2≈0.5−0.8[143] . Using qα
0=√5S0[22] one obtains
˜
λα∼0.43 −0.69. Finally, the derivative of the scaled free energy [see Eq. (5.33)] is
given by
˜
Φ
α=∂˜
For(˜
QA,˜
QB)
∂˜
Qα,
˜
Φ
α=Θ
α˜
Qα−2√6˜
Qα·˜
Qα+2(˜
Qα:˜
Qα)·˜
Qα−1
2ΘAB ˜
Qβ−(˜
QA:˜
QB)·˜
Qβ.
(6.62)
6.4.2 Scaled momentum balance equation
Using the above definitions it is possible to write the scaled momentum balance equation
d˜
v
d˜
t=1
β˜
∇·˜
T,(6.63)
where we introduced the definition of the scaled stress tensor T=m−1For
ref ˜
Tand the
additional parameter βis defined as
β=˙γL2m
kBTFor
ref tref
.(6.64)
This coefficient, or rather the ratio between βand the scaled viscosity ˜ηiso =ηiso˜
˙γ
defines the Reynolds number of the solvent
Re =β
˜ηiso
=˙γL2ρ
ηiso
.(6.65)
Experiments of shear–induced instabilities are typically performed in the low Reynolds
number limit (Stokesian limit), Re 1[144,145]. In this limit the momentum balance
equation (6.63) reduces to
˜
∇·˜
T=0.(6.66)
We note that due to the time dependence of ˜
Q(t), the total stress ˜
Tgenerally depends on
time, however, at each time the total stress has to fulfill Eq. (6.66). The resulting velocity
profile may deviate from the linear profile assumed initially. This feature is essential for
80 Chapter 6. Dynamic equations and constitutive relations for binary mixtures
the description of spatial symmetry–breaking such as shear banding. Finally, the scaled
contribution to the stress tensor is
˜
T=−˜pI+2˜ηiso ˜
Γ+˜
Tal (6.67)
where ˜p=p/m−1For
ref and the contribution to the stress tensor due to the changes in
the alignment of the binary mixture is
˜
Tal =
B
α=A√2˜
λα˜
Φα−√2˜
λα˜
ξ2
α˜
∇2˜
Qα+2σ˜
Qα·˜
Φα−2σ˜
ξ2˜
Qα·˜
Φα.(6.68)
Remarks: In the following, we drop the tilde (∼)on all variables appearing in Eqs. (6.55–
6.68). Thus, all quantities are identified with the same symbols as originally unless vague-
ness could arise. Following previous studies of one–component systems [146,147] we set
σ=0since this parameter has minor effect on the dynamics of the system for planar
Couette flow geometry.
6.5 Explicit equations of motion
All together, the set of equations (6.58)–(6.68) represent the spatiotemporal evolution
of second rank tensors. In fact, this set of equations can not be solved analytically
beyond the uniaxial approximation (introduced to study the relaxation of the alignment
in Sect. 6.3). Therefore, in order to study the full non–linear evolution of the alignment
and the stress, Qαand Tmay be expanded in terms of a standard tensor basis Biwith
five independent elements [see Appendix A]. In this basis
Qα=
4
i=0
qα
iBiand T=
4
i=0
TiBi(6.69)
where the coefficients xi(where x=q,T) are determined using the relation xi=X:
Bi[79].
Using the basis (6.69) in the Stokesian limit (Re 1), Eqs. (6.58), (6.59) and (6.66)
reduce to
dQA
dt =ξ2
A∇2QA+2˙γΩ·QA+√2λΓ−Φ
A,(6.70)
dQB
dt =ξ2
B∇2QB+2˙γΩ·QB+√2λΓ−Φ
B,(6.71)
∇·T=0,(6.72)
where Φ
αis the derivative of the one component orientational free energy [see Eq. (6.62)]
Φ
α=Θ
αQα−2√6Qα·Qα+2(Qα:Qα)·Qα−1
2ΘABQβ−(QA:QB)·Qβ.
(6.73)
6.5. Explicit equations of motion 81
Further, we assume that the spatial variation of Qand T, is restricted to a one–
dimensional investigation along the y–axis (the direction of the shear gradient [see
Fig. 6.2]). This assumption reduces the degrees of freedom of the solution and implies
that the possibility of banding in vorticity (z–) direction is excluded.
Using the tensor basis (6.69), Eq. (6.70) transforms into the set of equations:
dqα
0
dt =−(φα
0+ϕβ
0)+ξ2∂2qα
0
∂y2,
dqα
1
dt =−(φα
1+ϕβ
1)+˙γ∂v
∂yqα
2+ξ2∂2qα
1
∂y2,
dqα
2
dt =−(φα
2+ϕβ
2)−˙γ∂v
∂yqα
1+ξ2∂qα
2
∂y2+˙γλ∂v
∂y ,(6.74)
dqα
3
dt =−(φα
3+ϕβ
3)+1
2˙γ∂v
∂yqα
4+ξ2∂2qα
3
∂y2,,
dqα
4
dt =−(φα
4+ϕβ
4)−1
2˙γ∂v
∂yqα
3+ξ2∂2qα
4
∂y2.
Here the quantities φα
idescribe the relaxation of the particles towards equilibrium and
are
φα
0=qα
0Θi−3qα
0+2q2
α+3(qα2
1+qα2
2)−3
2(qα2
3+qα2
4),
φα
1=qα
1Θi+6qα
0+2q2
α−3
2√3(qα2
3−qα2
4),
φα
2=qα
2Θi+6qα
0+2q2
α−3√3qα
3qα
4,(6.75)
φα
3=qα
3Θi−3qα
0+2q2
α−3√3(qα
1qα
3+qα
2qα
4),
φα
4=qα
4Θi−3qα
0+2q2
α+3
√3(qα
1qα
4−qα
2qα
3).
while the quantities ϕβ
idisplay the inter–component interaction within the mixture and
are
ϕβ
i=−qβ
i(ΘAB +q2
AB),(6.76)
where i=0,···,4. In equations (6.74)–(6.75), q2
α=qα2
iand q2
AB =qα
iqβ
i. For
one–component systems, it can be shown by general arguments [50] (and has been
demonstrated numerically [48,49]) that the structure of equations (6.74)–(6.75) leads
to various types of oscillatory solutions as long as that the shear rate ˙γ=0.
Finally, the divergence of the stress tensor in Eq. (6.72) is now rewritten as
∂T2
∂y =√2ηiso ˙γ∂2v
∂y2+
B
α=A √2λα
∂(φα
2+ϕβ
2)
∂y −√2λαξ2
α
∂3qα
2
∂y3!.(6.77)
The surviving component of the stress tensor T2refers to the in–shear plane stress (Txy)
exerted on the system due to the velocity field.
82 Chapter 6. Dynamic equations and constitutive relations for binary mixtures
6.6 Summary
In this chapter we present a natural extension of the theory of non–equilibrium alignment
phenomena to a system composed of a binary mixture of rigid anisotropic particles. We
achieved this combining the equilibrium theory developed in Chapter 5with the theory
of irreversible processes for liquid crystals.
Following standard calculations of the theory of linear irreversible thermodynamics
we obtained a set of phenomenological laws which relate the hydrodynamic behavior
of the fluid and the thermodynamic fluxes and forces of alignment. Further, employing
Onsager’s theorem we get a complete set of hydrodynamic equations where the alignment
tensors QAand QBare coupled with the flow velocity v. The final hydrodynamic
equations for the mixture are:
dQA
dt = 2 Ω·QA+ 2σΓ·QA+πAp√2Γ+πAΨA+πABΨB,
dQB
dt = 2 Ω·QB+ 2σΓ·QB+πBp√2Γ+πABΨA+πBΨB.
where πxare related to phenomenological times and the terms Ψα(given by the deriva-
tives of the scaled free energy densities derived in Chapter 5) account for the relaxation
of the alignment towards equilibrium. Here Ω=1
2(∇vT−∇v)and Γ=1
2(∇vT+∇v)
describe the effect of flow vorticity and its deformation on the average alignment of the
particles, respectively.
We analyze these equations in the limit when the entropy production of component
Aand Bare completely independent and study the relaxation of the mixture towards
equilibrium in the absence of flow. We show that even with the simplest inter–component
coupling (entering through the free energy of alignment) the dynamics of the decay of
alignment are very complicated. Thus we conclude that the vanishing cross–coupling
approximation is suitable as a first approach to study the full dynamics of the system.
Finally, specializing to a planar Couette shear flow geometry, we introduce the scaled
alignment tensors QAand QBand re–write the hydrodynamic equations (for the align-
ment and the stress tensors) into a dimensionless form which are used in the subsequent
chapter to analyze the shear–induced instabilities.
Chapter 7
Shear induced instabilities and
oscillatory states in binary mixtures
In this chapter we study the dynamic behavior of binary mixtures under shear.
However, to provide a systematic approach we subdivide this chapter in two parts:
On the first part, we review the well–known oscillatory orientational dynamics of an
homogeneous single component system [49]. However, we expand this analysis to
study the spatiotemporal behavior of the alignment order parameter and its impact
on the underlying relations between stress and shear rate. On the second part we
turn to the analysis of the orientational dynamics of a spatially homogeneous
binary mixture. This chapter includes the main results of the publications: Binary
mixtures of rod-like colloids under shear: microscopically-based equilibrium theory
and order–parameter dynamics [111] and Shear banding in nematogenic fluids with
oscillating orientational dynamics [148].
7.1 General Remarks
The shear induced behavior of nematogenic fluids has been intensely studied for the
one component system on the basis of equations (6.74)–(6.75) (see e.g. [149,49,50]).
In these studies it has been proven that nematogenic fluids present a wide variety of
orientational dynamics depending on the initial equilibrium state (isotropic or nematic).
The dynamics of the full alignment tensor are characterized by the dynamics of the
corresponding nematic director. Thus, in average, the particles in the fluid perform a
variety of shear induced oscillatory states. The different types of regular oscillatory states
are: wagging (W), tumbling (T), kayaking–wagging (KW), kayaking–tumbling (KT), and
(flow–)alignment states (A). Moreover, under certain conditions chaotic dynamics may
be present [48,51,49,150]. All these states are characterized by the time–dependent
83
84 Chapter 7. Shear induced instabilities in binary mixtures
Within the shear plane Out of the shear plane
Alignment
Tumbling
motion
Wagging Kayak
Tumbling
Kayak
Wagging
Figure 7.1: Sketch of the different types shear induced regular oscillatory states. The
blue arrows show the direction in which the Couette walls are moving whereas the green
arrows indicate the motion of the nematic director.
behavior Q(t)(and conversely of the coefficients qi(t)) and it is possible to set one
particular oscillatory state depending on the shear rate and the tumbling parameters
[see Eqs. (6.74)–(6.75)].
Excluding chaotic dynamics, the regular time–dependent states can be classified in
two groups: states lying within the shear plane (A, T, W) and states lying out of the shear
plane (KT, KW). Trivially, the A state corresponds to a fixed point of the system (6.74)–
(6.75) and therefore the dynamics of the qi(t)coefficients are ”frozen” as a function of
time. In contrast, the regular states W and T are characterized by full in–plane rotations
of nematic director and q3(t)=q4(t)=0. Finally, the out of the shear plane oscillatory
states (KW, KT) are named after their in–shear plane counterparts with the difference
that here qi(t)=0∀i∈[0,1,2,3,4].
7.1.1 Dynamical state diagram
All the different steady oscillatory states can be summarized in a dynamical state dia-
gram [49]. In Fig. 7.2, we show the dynamical state diagram spanned in the plane ˙γ−λ
for a one component system characterized by Θ=−1.0. The colored areas are com-
puted via direct numerical integration and each one corresponds to different dynamical
behavior: A =Alignment, W =Wagging, T =Tumbling, KW =Kayak–Wagging and
KT =Kayak–Tumbling (see Fig. 7.1). We should note that because of our choice of
scaling (see Chapter 6) the values of λdiffer from those in earlier studies (e.g. [49]) by
a constant factor.
In earlier studies [51,50] investigating similar values of Θ, the authors reported the
occurrence of an important region characterized by irregular and even chaotic motion
of the nematic director. This chaotic region is located around the point where the KT,
KW and A states meet. Here we do not detect such region because our algorithm does
not resolve Lyapunov exponents.
The boundaries between the colored regions in Fig. 7.2 correspond to different types
of dynamical bifurcations. The simplest one is a (supercritical) Hopf bifurcations occur-
7.2. Shear banding with oscillating orientational dynamics 85
0.5 0.7 0.9 1.1 1.3
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
λ
˙γ
A
KT
KT
KW
WT
KT
Figure 7.2: Dynamical state diagram for a one component system characterized by
Θ = −1.0. The colored areas are computed via direct numerical integration and each
one corresponds to different dynamical behavior: A = Alignment, W = Wagging, T =
Tumbling, KW = Kayak–Wagging and KT = Kayak–Tumbling.
ring at the boundary between alignment (A) and wagging (W). In Ref. [50] a complete
continuation analysis is presented which reveals the complexity of the dynamical behavior
already in the one component case (where one has already five dynamical variables).
In order to study the shear induced instabilities occurring in binary mixtures, it
is necessary to provide a systematic approach containing a deep analysis of the one
component case. Thus, in the following section (Section 7.2) we investigate systems
composed of only one component, i.e., in the limit ρb→0. Further, in Section 7.3 we
continue our analysis with the investigations of binary mixtures of rod–like colloids
7.2 Shear banding with oscillating orientational dynamics
In recent years, several theoretical approximations have explored the spatial inhomo-
geneities of the alignment Qand the stress T, yielding shear banding between different
steady states. In some cases, the unsheared system is close to the isotropic–nematic tran-
sition [59] and in others the dynamics of the sheared homogeneous system is irregular
or chaotic [52,75]. In this section, our purpose is to extend the work in Refs. [52,75] to
parameter values where the homogeneous system exhibits regular oscillatory dynamics.
7.2.1 Numerical calculations and boundary conditions
For the study of inhomogeneous systems the gradient terms appearing in (6.74)–(6.77)
should be discretized. To this end we use a central finite difference scheme of fourth order
86 Chapter 7. Shear induced instabilities in binary mixtures
Figure 7.3: Sketch of the boundary conditions applied to the tensor Qat the plates:
(a) Isotropic [see Eq. (7.1)] (b) Planar nematic [Eq. (7.2)] (c) Vertical nematic [Eq. (7.3)]
and (d) Planar degenerate [Eq. (7.4)].
where asymmetric stencils (using only available grid points) are implemented [151]. The
calculations are initialized with values of q0,··· ,q
4matching the boundary conditions,
with a small random perturbation. The steady configurations of the system are found
by monitoring the evolution of q0,··· ,q
4and T2, disregarding transient initial behavior.
The resulting dynamical states are characterized employing an algorithm that recognizes
the periodicity as in the previous section. For further details, please refer to Henning
Reinken’s Master’s thesis [71].
Regarding the boundary conditions at the plates (y=±L), we assume strong an-
choring conditions, i.e., the value of Qat the plates is constant in time, but may have
different symmetries. From an experimental point of view, strong anchoring conditions
are possible by chemical or mechanical treatment of the plates [83,84,85,86]. Here we
focus on the following idealized cases:
Q
y=L
y=−L
=0 (Isotropic) ,(7.1)
Q
y=L
y=−L
=3
2˜μeq
3ˆ
exˆ
ex(Planar nematic) ,(7.2)
Q
y=L
y=−L
=3
2˜μeq
3ˆ
eyˆ
ey(Vertical nematic) ,(7.3)
Q
y=L
y=−L
=−3
2˜μeq
3ˆ
eyˆ
ey(Planar degenerate) ,(7.4)
where ˜μeq
3is the equilibrium value of the alignment tensor in the nematic phase. Equa-
tions (7.1) and (7.4) describe disordered states where the particle orientations are dis-
tributed, either in all three directions (”isotropic”) or within the plane of the plates, i.e.,
7.2. Shear banding with oscillating orientational dynamics 87
in the x–zplane (”planar degenerate”) (see Fig. 7.3). The other two boundary con-
ditions correspond to nematic states, with the director lying either in the plane of the
plates (”planar nematic”) or along the gradient (y–) direction (”vertical nematic”) (see
Fig. 7.3). For the velocity field usual no–slip boundary conditions, vx(y=±1) = ±1(in
reduced units), are implemented [63].
7.2.2 Homogeneous solutions
Here we consider spatially homogeneous systems far away from the boundaries in the limit
of infinite plate separation, i.e.,L→∞. In this limit, the correlation length appearing in
Eq. (6.77) is zero and thus, all gradient terms vanish. In particular, the stress T2takes
the form
T2(t)=√2ηiso ˙γ+√2λΦ2.(7.5)
Homogeneous systems sheared from the isotropic state
We start considering systems whose equilibrium state is isotropic (Θ=1.20). Increasing
the shear rate from zero, the system develops a paranematic (PN) state1; this is illus-
trated in Fig. 7.4(a). The behavior upon further increase of ˙γdepends on the value of
the chosen tumbling parameter λ(a measure of the aspect ratio).
For λ0.62 one observes a transition from paranematic to shear–aligned (A) state.
The PN–A transition is accompanied by a narrow region of bistability (not visible in
Fig. 7.4) where the degree of ordering of the system is highly dependent on the initial
condition. For smaller values of the tumbling parameter (λ0.62) the Astate is
unstable and the system develops wagging (W) motion. Overall, the behavior found in
the present calculations agrees qualitatively with that reported in ref. [50], however, the
quantitative data for the boundary lines differ due to the different scaling of the free
energy.
For each of the parameter sets ( ˙γ,λ)we calculated the stress T2(in the W state,
averaging T2(t)over one period of time). Remarkably, we observe that different param-
eter sets may lead to the same value of T2. To illustrate this point, Fig. 7.4(a) includes
dashed (blue) lines and solid (green) lines corresponding to two constant values of T2.
Moreover, there are several regions of λwhere T2assumes the same value for different
shear rates. For example, at λ=0.55 there are three values of ˙γwith T2=0.6, and for
λ≥0.7one finds two solutions with T2=0.4.
Due to this multivalued behavior, it is interesting to consider the corresponding flow
curves, T2(˙γ). The results for λ=0.55 and λ=1.25 are plotted in Figs. (7.4(b)
and (7.4(c), respectively. The flow curve for λ=0.55 [see Fig. 7.4(b)] displays a region
with a negative slope between ˙γ≈0.357 and ˙γ≈0.365. Within this region the ho-
mogeneous flow is mechanically unstable, and as one might expect, the system forms a
1The paranematic state PN is similar to the AA state but it is characterized by weak nematic order.
88 Chapter 7. Shear induced instabilities in binary mixtures
0.5 0.7 0.9 1.1 1.3
0.10
0.20
0.30
0.40
0.50
0.60
λ
˙
γA
W
PN
T2= 0.6
T2= 0.4
(a)
PN
W
0.3 0.35 0.4
0.5
0.6
0.7
˙γ
T2
(b) λ= 0.55
PN
A
0.0 0.1 0.2
0.2
0.4
0.6
˙γ
T2
(c) λ= 1.25
Figure 7.4: a) State diagram in the plane spanned by tumbling parameter (λ) and shear
rate ( ˙γ) at Θ = 1.20 (isotropic equilibrium system). The dotted gray lines indicate
the boundaries between three different states: paranematic (PN), shear–aligned (A) and
wagging (W). The dashed (blue) and solid (green) lines connect points with constant
stress T2= 0.4and T2= 0.6, respectively. (Second row) Homogeneous flow curves
T2(˙γ)for (b) λ= 0.55 and (c) λ= 1.25.
spatially inhomogeneous, shear banded state. We also note that the shear rate ˙γ≈0.359
characterized by T2= 0.6and dT2/d˙γ < 0agrees roughly with the corresponding point
on the boundary line PN–W in Fig. 7.4(a). This indicates that the orientational tran-
sition from the PN state to the (oscillatory) W state, on the one side, and the shear
banding instability, on the other side, are closely correlated.
In contrast, for λ= 1.25 [see Fig. 7.4(c)] the flow curve does not display a region with
negative slope. Rather one observes a discontinuity and, associated with this, hysteretic
behavior. Upon increase of ˙γfrom the small values, i.e., from the paranematic (PN)
state, the systems discontinuously ”jumps” to the aligned (A) state only at ˙γ≈0.14
[which is above the upper blue dashed line in Fig. 7.4(a)]. However, decreasing ˙γstarting
from the large shear rates characterizing the A state, the jump occurs at the much smaller
shear rate ˙γ≈0.06. As we will show at the end of this section, the formation of shear
bands in this case (λ= 1.25) strongly depends on the boundary conditions.
Homogeneous systems sheared from the nematic state
We now turn to systems deep in the nematic phase. Specifically, we set Θ = −0.25 in
Eq. (6.75). The dashed (blue) and solid (green) lines in Fig. 7.5(a) indicate parameter
7.2. Shear banding with oscillating orientational dynamics 89
0.5 0.7 0.9 1.1 1.3
0.10
0.20
0.30
0.40
0.50
0.60
λ
˙
γ
AWT
KT
KT
KW
T2= 0.6
T2= 0.4
(a)
KT
T
4 5 6 7
4
6
8
10
12
˙γ
T2
(b) λ= 0.55
KT
A
1 2 3 4 5
2
4
6
8
10
12
˙γ
T2
(c) λ= 1.25
Figure 7.5: (a) State diagram at Θ = −0.25. The dotted gray lines indicate the bound-
aries between the different dynamical states: wagging (W), tumbling (T), kayaking–
tumbling (KT), kayaking–wagging (KW), and shear–alignment (A). The dashed (blue)
and solid (green) lines connect points with constant stress T2= 7.0and T2= 9.0, respec-
tively. (Second row) Homogeneous flow curves T2(˙γ)for (b) λ= 0.55 and (c) λ= 1.25.
sets at which the orientational dynamics yield the constant stress–values T2= 9.0and
T2= 7.0, respectively. In the first case, the line provides a unique relation in the sense
that an increase of ˙γat fixed λyields only one crossing with this line. This is different
for the case T2= 7.0where, depending on λ, one or two crossings can be observed.
Exemplary flow curves for two values of the tumbling parameter are shown in the right
hand side of Fig. 7.5. At λ≈0.55 [see Fig. 7.5(b)], where each of the constant–pressure
lines is crossed only once, one observes a monotonic increase of T2with ˙γ. In particular,
there is no discontinuity even at ˙γ≈6.5, where the underlying orientational dynamics
changes from the KT to the T states. In fact, a systematic bifurcation analysis (such as
the one in Ref. [50]) would reveal a bistable region characterized by the presence of (at
least) two attractors between the pure KT and the pure T state. In such a situation, the
stress T2would not be uniquely defined. This aspect certainly deserves more attention
in a future study.
In contrast, in the case λ= 1.25 showed in Fig. 7.5(a), an increase of ˙γfrom
small values yields two crossings with the constant–stress curve T2= 7.0. As seen from
Fig. 7.5(c), the flow curve exhibits a pronounced discontinuity related to the transforma-
tion of the (out–of–plane) oscillating KT state into the shear–aligned (A) steady state
at ˙γ≈3.67. One also recognizes a strong dependence on initial conditions (hysteresis),
similar to the situation discussed in Fig. 7.4(c).
90 Chapter 7. Shear induced instabilities in binary mixtures
0 50 100 150 200
−1.0
−0.5
0.0
0.5
1.0
t/tref
y
L
(a)
0 50 100 150 200
−1.0
−0.5
0.0
0.5
1.0
t/tref
y
L
(b)
0 50 100 150 200
−1.0
−0.5
0.0
0.5
1.0
t/tref
y
L
(c)
0.0
0.2
0.4
0.6
0.8
1.0
||Q||
Figure 7.6: Spatiotemporal behavior of the norm of Qat ˙γ= 3.65,λ= 1.25 and
different correlation lengths (a) ξ2= 10−5, (b) ξ2= 10−4and (c) ξ2= 10−3. The
equilibrium state ( ˙γ= 0) is isotropic (Θ = 1.20).
7.2.3 Spatiotemporal behavior and shear banding
In the preceding discussion we found indications of the formation of inhomogeneous
states in both, systems sheared from the isotropic and systems sheared from the nematic
phase. Now, we turn to the analysis of the corresponding systems (characterized by
certain values of the tumbling parameter) calculating the full spatiotemporal behavior
of the Qand the resulting shear stress T2. To this end we have solved numerically
Eqs. (6.74)–(6.77) using the methodology described in Sect. 7.2.1.
Our discussion in this section is divided into two parts: First, we focus on the cor-
relation length ξ, which appears as a prefactor of the gradient term in the orientational
free energy density [see Eqs. (6.74)–(6.75)]; later we turn our attention to the boundary
condition for Qat the plates [see Eqs. (7.1)–(7.4)].
Impact of the correlation length
Initially isotropic system We first consider the system at Θ = 1.20 and λ= 0.55
[see Fig. 7.4(b)] focusing on a shear rate ˙γ= 0.365. In Figs. 7.6(a)–(c) we show the
spatiotemporal evolution of ||Q|| (the norm of Q–tensor) at three different values of
the correlation length. Since the equilibrium state is isotropic, we choose the boundary
conditions according to Eq. (7.1), thus the boundaries do not support any orientational
ordering.
As seen from Fig. 7.6(a) the system forms spatiotemporal structures with locally
large values of ||Q|| already at the smallest correlation length considered, ξ= 10−5. The
observed pattern is rather ”loose” with its width changing in time. We can also observe
that, within the inhomogeneous regions, ||Q|| is oscillating in time. The oscillations in
these regions are consistent with a wagging (W) state and outside ||Q|| takes values
typical of a paranematic (PN) state. This behavior is expected since the value of ˙γ
considered in Fig. 7.6 is very close to the boundary between the PN and W state (see
Fig. 7.4).
Upon increasing the correlation length, ξ, we observe from Figs. 7.6(b) and 7.6(c)
7.2. Shear banding with oscillating orientational dynamics 91
0.35 0.40 0.45
−1.0
−0.5
0.0
0.5
1.0
˙γ
(
y
)
y
L
Applied shear rate
Local shear rate
(a)
0.32 0.34 0.36 0.38 0.40
0.56
0.58
0.60
0.62
0
.
64
˙γ
T2
ξ2=10
−3
ξ2=10
−4
ξ2=10
−5
(b)
Figure 7.7: (a) Local shear rate within the banded state of the initially isotropic system
(Θ=1.20,λ=0.55, average (applied) shear rate ˙γ=0.365). (b) Inhomogeneous flow
curves at different correlation lengths. The symbols (red), ×(blue) and ◦(green)
correspond to ξ2=10
−5,ξ2=10
−4and ξ2=10
−3, respectively. As a reference the
homogeneous flow curve is included (black dashed line).
that the regions characterized by W motion become more defined, in terms of the shape
and oscillatory motion of the order parameter. At the same time the interface between
the W and PN region broadens. As a consequence, the W oscillations are transferred to
some extent into the outer region, however, with a very small amplitude.
A further illustration of the presence of shear bands is plotted in Fig. 7.7(a), where
we present a snapshot of the local shear rate, ˙γ(y), for the system at ξ2=10
−3. Here,
we see that the band in the middle of the system has a significantly higher shear rate
than the PN state close to the boundaries. On the other hand, Fig. 7.7(b) shows the
flow curves obtained for the inhomogeneous (initially isotropic) systems for different
correlation length. We obtained these curves by calculating the mean value of T2in-
creasing gradually from lower to larger values of ˙γ(see Ref. [152]). As a reference, the
corresponding homogeneous flow curve [see Fig. 7.4(b)] is included.
Interestingly, the value of T2corresponding to the banded state is essentially inde-
pendent of ξ; in other words, the value of T2appears to be unique. This observation is
consistent with previous studies on the basis of both, a Q–tensor model [59] and the
DJS model [74]. We further observe from Fig. 7.7(b) that there is a slight dependence on
the value of T2on ξat high shear rates beyond the banded state. This is an effect stem-
ming from the inhomogeneities induced by the confining walls: the larger ξ, the larger
is the extent of these inhomogeneities into the bulk–like region between the plates. In
fact, excluding these regions from the calculations the results for T2completely agree
for different ξ.
92 Chapter 7. Shear induced instabilities in binary mixtures
0 50 100 150 200
−1.0
−0.5
0.0
0.5
1.0
t/tref
y
L
(a)
0 50 100 150 200
−1.0
−0.5
0.0
0.5
1.0
t/tref
y
L
(b)
0 50 100 150 200
−1.0
−0.5
0.0
0.5
1.0
t/tref
y
L
(c)
0.85
0.90
0.95
1.00
1.05
||Q||
˜µeq
3
Figure 7.8: Spatiotemporal behavior of the norm of Qat ˙γ= 3.65,λ= 1.25 and
different correlation lengths (a) ξ2= 10−5, (b) ξ2= 10−4and (c) ξ2= 10−3. The
equilibrium state ( ˙γ= 0) is nematic (Θ = −0.25).
Initially nematic system We now turn to the system at Θ = −0.25 and λ= 1.25,
where the homogeneous flow curve [see Fig. 7.5(c)] is discontinuous. Results for the
norm of Qas function of space and time are shown in Fig. 7.8, where we assumed
equivalent planar nematic boundary conditions [see Eq. (7.2)], but different correlation
lengths.
In all exemplary cases showed in Fig. 7.8, there is a a clear spatial separation of the
system into an inner band, where the orientational behavior corresponds to the kayak–
tumbling (KT) state, and an outer region where the system is in a shear–aligned (A)
state. Upon increase of ξthe width of the KT band widens, while the oscillations within
the band become more and more regular.
Corresponding results for the local shear rate and the inhomogeneous flow curves are
given in Fig. 7.9. Compared to the initially isotropic system, we see from Fig. 7.9(a) that
the oscillatory (KT) band is characterized by a smaller shear rate than the regions close
to the boundaries. A further difference comes up when we consider in Fig. 7.9(b) the
values of the stress plateau in the inhomogeneous flow curve. Here we find a dependence
on the correlation length; that is, the value of T2at the plateau increases with ξ. This
contrasts with our corresponding results for the initially isotropic system.
Role of the boundary conditions
In this final subsection we will address the role of the boundary conditions [see eqs. (7.1)–
(7.4)]. For the following analysis we set the correlation length to a constant value of
ξ2= 10−5. However, higher values yield very similar results. In Fig. 7.10 we present
numerical results for the resulting flow curves of the inhomogeneous systems previously
discussed.
As before, we first consider systems sheared from the isotropic state at λ= 0.55
where clear shear banding instabilities are found even for isotropic boundary conditions
[see figs. 7.6 and 7.7]. As indicated by the flow curves in Fig. 7.10(a), similar behavior
occurs for other (including nematic) boundary conditions. Indeed, all of the systems
7.2. Shear banding with oscillating orientational dynamics 93
23456
−1.0
−0.5
0.0
0.5
1.0
˙γ
(
y
)
y
L
Applied shear rate
Local shear rate
(a)
12345
2
4
6
8
10
12
˙γ
T2
ξ2=10
−3
ξ2=10
−4
ξ2=10
−5
(b)
Figure 7.9: (a) Local shear rate within the banded state of the initially nematic system
(Θ=−0.25,λ=1.25, average (applied) shear rate ˙γ=3.65). (b) Inhomogeneous flow
curves at different correlation lengths. The symbols (red), ×(blue) and ◦(green)
correspond to ξ2=10
−5,ξ2=10
−4and ξ2=10
−3, respectively. As a reference the
homogeneous flow curve is included (black dashed line).
form bands (within a range of shear rates ˙γ≈0.355 –˙γ=0.385) with wagging–like
oscillations within PN regimes at the plates. Outside the banding region, the systems
are characterized by the same value of T2.
We observe that the value of the ”selected” stress within the banding region, Tsel
2≈
0.59, is essentially independent of the boundary conditions. Specifically, for the two types
of isotropic boundary conditions, as well as for nematic ordering within the plane of the
plates, the system stays in the homogeneous PN state for all ˙γup to the maximum of the
flow curve. In contrast, the system with vertical alignment at the plates, i.e., alignment
in the shear gradient (y–) direction, forms bands once the value of Tsel
2is reached (at
˙γ≈0.34). In this sense, the vertical nematic ordering favors the occurrence of the W
state characterized by oscillations in the gradient direction. For completeness we also
note that upon decreasing ˙γfrom high values, the system goes without hysteresis into
the banded state (with the same stress), irrespective of the boundary conditions.
In the initially nematic case [see Fig. 7.10(b)], upon increasing ˙γfrom lower values
all systems, irrespective of boundary conditions, display shear band formation at shear
rates in the range ˙γ≈3.0–˙γ≈4.5. After this critical values they break up into a band
with kayaking–tumbling dynamics (i.e., oscillations out of the shear plane) surrounded
by regions of shear–alignment at the boundaries. The stress T2characterizing the banded
state seems to be unique, and the boundary conditions only affect the onset of shear
banding (upon starting from the low–shear rate branch, where the system is in the KT
state). Moreover, from Fig. 7.10(b) we see that the onset of banding is ”delayed” when
94 Chapter 7. Shear induced instabilities in binary mixtures
0.32 0.34 0.36 0.38 0.40
0.56
0.58
0.60
0.62
0.64
˙γ
T2
Isotropic
Vertical nematic
Planar degenerate
Planar nematic
(a)
12345
2
4
6
8
10
12
˙γ
T2
Isotropic
Vertical nematic
Planar degenerate
Planar nematic
(b)
Figure 7.10: Influence of the boundary conditions on the inhomogeneous flow curves for
(a) initially isotropic systems at Θ = 1.20,λ= 0.55 and (b) initially nematic systems
at Θ = −0.25,λ= 1.25. The correlation length is set to ξ2= 10−5. As a reference the
homogeneous flow curves have been included.
planar degenerate boundaries are used. This is because these boundary conditions seem
to support the KT state. Interestingly, the behavior upon decreasing the shear rate from
the aligned state is different: in that case, all systems stay in the aligned state until the
lower end of the high–shear rate branch is reached; then they directly jump into the KT
state without an intermediate shear banded state.
Finally, both systems considered in Fig. 7.10 display shear banding irrespective of
the detailed nature of the boundary conditions, if the shear rate is increased from low
values. The boundary conditions then only influence the ”critical” shear rate at which
the homogeneous state observed at small ˙γbreaks up into bands. On the contrary, shear
banding upon decreasing ˙γfrom high values is only seen in the initially isotropic system.
The behavior in the initially nematic system thus depends on the initial conditions,
similar to what has been found in the DJS model [153].
7.3 Binary mixtures of rod–like colloids
After the analysis of the spatiotemporal behavior, in this section we turn to the analysis
of binary mixtures. In the following we assume spatially homogeneous systems where the
boundaries of the system do not play a role. Thus, we numerically solve Eqs (6.74)–(6.77)
in the limit where L→ ∞.
To perform the numerical integration we employ a standard Runge-Kutta algorithm
with adaptive step size control [151]. As in the case of the one component system,
the starting values of the tensor components qα
0,··· , qα
4(where α=A, B) are chosen
randomly in the interval [0,1], using a uniform distribution. We note that due to the high
non–linearity of the problem at hand the choice of initial values and the overall number
of time–steps required to obtain reliable results (disregarding transient initial behavior)
7.4. Orientational dynamics of binary mixtures 95
strongly depends on the chosen parameters. Therefore, all numerical integrations have
been repeated several times. Finally, to characterize the resulting dynamical states, we
use an algorithm that recognizes the periodicity and sign change of the calculated time–
dependent order parameters qα
i(t)(see e.g. Refs. [149,49,146]).
7.4 Orientational dynamics of binary mixtures
In the following we investigate sheared systems whose equilibrium state ( ˙γ= 0) is
nematic. Specifically, we set ΘA= ΘB=−0.25. According to the stability analysis in
Chapter 5, in this case the isotropic phase is unstable irrespective to the value of ΘAB.
The section is divided into two parts. First, we discuss the effect of varying the
shear rate, ˙γ, and the tumbling parameter of species B, λB, while λAis fixed. We note
that in real experiments, variation of the tumbling parameter is somewhat difficult since
each value of this parameter corresponds to a different aspect ratio (and thus, particle
type), see Eq. (6.74). However, in the framework of our dynamical equations λB(A)is
the crucial parameter measuring the impact of the shear induced perturbation, thus it
seems worth to explore its role. Specifically, we set λA= 1.2and consider a range of
values λB> λA, implying that the B–particles have larger aspect ratios. The second
part of the section addresses the role of the ”cross–coupling” parameter ΘAB.
7.4.1 Variation of the tumbling parameter and the shear rate
The numerical integration leads to dynamical state diagrams for each of the two species
which we present in Fig. 7.11. As in Fig. 7.2, each colored region corresponds to a
different dynamical state characterized by a specific behavior of the respective nematic
director, related to the largest eigenvalue of the alignment tensor QA(B). Specifically,
we observe wagging states (W), tumbling states (T), kayaking–wagging states (KW),
kayaking–tumbling states (KT), and (flow–)alignment states (A). The boundaries be-
tween these colored regions correspond to different types of dynamical bifurcations and
an analysis for the mixture would have been beyond the scope for this work. However,
we note that the shape of each state diagram in Fig. 7.11 is qualitatively similar to that
of one component systems (see Fig. 7.2). As in the one component system, we expect
that a chaotic or irregular also occurs in the mixture (for both species) at values of ˙γ,
λBwhere the regions A, KW and KT meet.
While the overall behavior of the two species is very similar, there are interesting
differences for specific values of ˙γand λB. For example, consider the shear rate ˙γ= 7.5
and λB= 3.0. At these parameters, the long rods are in the W state while the short rods
are in a tumbling (T) state. In other words, both directors display oscillatory behavior
in the shear plane, but with different characteristics: Tumbling is characterized by full
in–plane rotations of nematic director, whereas wagging just implies finite back–and–
forth–motion in angular space.
96 Chapter 7. Shear induced instabilities in binary mixtures
A
T
W
KW
KT
(a)
2.0 2.5 3.0 3.5 4.0 4.5 5.0
5.0
6.0
7.0
8.0
9.0
10.0
λB
˙γ
Short Particles
(
A
)
(b)
2.0 2.5 3.0 3.5 4.0 4.5 5.0
λB
Long Particles
(
B
)
Figure 7.11: Dynamical state diagram for a binary mixtures characterized by ΘA=
−0.25,ΘB=−0.25,ΘAB =1.50, and λA=1.2. Parts a) and b) illustrate the behavior
of the short and long particles, respectively. Color bar on the right side: A =Alignment,
W=Wagging, T =Tumbling, KW =Kayak–Wagging, KT =Kayak–Tumbling.
The tumbling and wagging states are characterized by in shear plane periodic motion
of the nematic director, but the T state is characterized by full rotations of nematic
director. Therefore, it is of interest to compare the angle between both nematic directors.
Since in the T and W states qα
3=qα
4=0, in terms of the components qα
0,qα
1,qα
2,ϑ(t)
the angle between directors is given by
ϑ(t) = arccos ⎛
⎝
qA
jqB
j
(qA
jqA
j(qB
jqB
j
⎞
⎠,(7.6)
where j=0,1,2and we used the Einstein convention. The angle between nematic
directors is showed in Fig. 7.12(a) where we choose a constant shear rate ˙γ=7.5and
three different values of λB.
As it can be seen form Fig. 7.12(a), at the largest λBconsidered, both species are
in a shear aligned state. For single–component systems, it is well known that the shear
aligned state is characterized by a finite angle between the (stationary) director and shear
direction, and that this ”flow angle” depends on the tumbling parameter. Therefore, one
would expect the two directors of the mixture to have different (stationary) flow angles
with respect to the shear direction, and consequently, enclose a finite angle in between.
This is exactly what we see from the corresponding curve in Fig. 7.12(a). At λB=3.25,
both species are in the tumbling state (see Fig. 7.12), with the directors displaying
7.4. Orientational dynamics of binary mixtures 97
0 5 10 15 20 25 30 35 40 45 50
15
30
45
60
75
90
time (s)
ϑ(t)(degrees)
λA=1.20
λB=2.75
λB=3.25
λB=5.00
(a)
0 5 10 15 20 25 30 35 40 45 50
0.2
0.4
0.6
0.8
1
time (s)
drel(t)
λ
A
=1.20
λ
B
=2.75
λ
B
=3.25
λ
B
=5.00
(b)
Figure 7.12: (a) Angle between the nematic directors as function of time at ˙γ=7.5and
three values of λB. The green solid line () corresponds to shear–aligned states of both
species (λB=5.0), whereas the blue line () corresponds to synchronized tumbling
states (λB=2.75). Finally, the red line (•) represents a case where the A–species (B–
species) is in a tumbling (wagging) state (λB=2.75). (b) Relative alignment of the
nematic directors as a function of time at ˙γ=7.5. The dashed black lines and the
solid lines correspond to the behavior of the A–species (λA=1.20) and B–species (with
varying λB), respectively. The () symbol correspond to shear–aligned states of both
components (λB=5.0) whereas the () correspond to synchronized tumbling states
(λB=3.25). Finally, (•) represents the coexisting T/W state (λB=3.75).
full rotations. Still, we observe from Fig. 7.12(a) that the angle between the tumbling
directors is close to zero at all times considered. We interpret this behavior as (nearly)
synchronous tumbling. At λB=2.75, the behavior is markedly different: Here we observe
a periodic variation of the angle between the directors, reaching values up to nearly 90
degree. This reflects the simultaneous appearance of tumbling (species A) and wagging
(species B).
To further compare the in–plane oscillations of the nematic directors in the different
regions of the state diagram (see Fig. 7.11) we also look at their relative amplitudes.
To this end we introduce a quantity that compares, for each component of the mixture,
the instantaneous magnitude |Qα|(t)with its maximum value max{|Qα|} (we recall at
this point that the tensors considered here are not normalized to one). In terms of the
components qi
0,qi
1,qi
2, the relative alignment is therefore given by
dA(B)
rel (t)= (qA(B)
jqA(B)
j
max$(qA(B)
jqA(B)
j%,(7.7)
where j=0,1,2and we used the Einstein convention. This quantity is shown in
98 Chapter 7. Shear induced instabilities in binary mixtures
A
T
W
KW
KT
(a)
0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4
5.0
6.0
7.0
8.0
9.0
10.0
ΘAB
˙γ
Short Particles
(
A
)
(b)
0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4
ΘAB
Long Particles
(
B
)
Figure 7.13: Dynamical state diagram in the plane spanned by shear rate and cross
coupling parameter for a binary mixtures characterized by ΘA=−0.25,ΘB=−0.25,
λA=1.2and λB=3.0. Parts a) and b) illustrate the behavior of the short and long
particles, respectively.
Fig. 7.12(b). As expected, in the shear aligned state (λB=5.0) the relative alignment
of both components does not oscillate in time. Further, when both nematic directors
are tumbling (i.e., perform full rotations, here at λB=2.75), the oscillations are in
phase and also their magnitude is comparable. In contrast, in the coexisting T/W state
(λB=3.25), there is a marked difference between the amplitude of oscillations. Still,
the oscillations are in phase. Therefore, we conclude that both, the T state and the
coexisting T/W state are characterized by synchronous motion.
7.4.2 Variation of the cross coupling and the shear rate
In view of the variety of dynamical states at fixed cross coupling parameter ΘAB it
is interesting to further explore the impact of this parameter. To this end we present
in Fig. 7.13 the state diagrams obtained at fixed tumbling parameters λA=1.20 and
λB=3.0, in the plane spanned by ΘAB and ˙γ.At ˙γ=0(equilibrium), all systems
considered are in the (meta–)stable nematic phase.
At small shear rates ˙γ5.0, both species are in the out–of–plane KT state for all
cross coupling parameters considered. At larger shear rates the out–of–plane oscillations
then transform into in–plane oscillations of W or T type. As indicated from Fig. 7.13,
an increase of ΘAB shifts this boundary between KT and T or W towards larger shear
rates for both components of the mixture. A further effect becomes apparent when we
consider a fixed, high shear rate, e.g., ˙γ=8.0. Here, an increase of ΘAB can induce
7.5. Summary 99
profound changes of the dynamical state, such as a change from W to T to KT. Along
this way, there a large regions where the two components display different dynamical
behavior (coexistence of T and W), similar to what we have discussed before.
Finally, we notice the small ”islands” of KT regions within the T state (A–species)
or W–state (B–species). This indicates a bistability of the solutions which may be inter-
preted as a coexistence of in–shear plane and out–of–shear plane oscillations (as already
mentioned in [50]). In fact, as we have previously showed for the one component system,
any type of bistability may translate into spatially separated dynamical coexistence of
different states (shear banding).
7.5 Summary
We present the shear induced instabilities occurring in systems of rod–like particles. We
divide our discussion in two parts: the study of shear banding in one component systems
and the analysis of oscillatory states in spatially homogeneous binary mixtures.
For one component systems, we focused on instabilities along the gradient direc-
tion. We saw that systems with different initial equilibrium states (isotropic or nematic),
tumbling parameters (i.e., aspect ratios), and boundary conditions show complex shear
banding behavior. Interestingly, the shear bands appearing in these systems are accom-
panied by oscillatory orientational motion in certain regions of space. We observe that
band formation strongly depends on the orientational boundary conditions applied to
the system. In this sense certain boundaries may enable or inhibit the formation of shear
bands.
Similar to what occurs in one component systems, for a binary mixture we found
a variety of oscillatory dynamical states of the nematic directors, either with in–plane
or with out–of–plane symmetry. However, given an specific set of tumbling parameters,
we observe symmetry breaking behavior for intermediate and high values of the shear
rate. This behavior is characterized by the coexistence of simultaneous appearance of
tumbling motion (short particles) and wagging motion (long particles). However, albeit
the actual motion is different, there is an overall synchronization of the system.
We also observed that an additional control parameter of this oscillatory behavior is
the coupling interaction ΘAB. We show that by increasing ΘAB at fixed shear rate, one
can find transitions from in–plane oscillations to out–of–plane oscillations (kayaking–
tumbling).
Chapter 8
Concluding remarks
8.1 Summary
In this thesis we investigated mixtures of rigid anisotropic colloidal particles under shear.
In particular, we focused on binary mixtures of rod–like particles and developed a theo-
retical framework capable to describe their equilibrium behavior and the shear induced
rheological response.
In Chapter 1, we give a general introduction to colloidal mixtures of rigid rod–like
particles and addressed their basic equilibrium and shear induced behavior. Further, in
Chapters 2and 3, we continue this introductory remarks by presenting the phenomeno-
logical and microscopical descriptions of rod–like systems in equilibrium.
We begin our investigations in Chapter 4where we cover the first goal of our research:
to bridge microscopic and phenomenological theories. There, on the basis of the density
functional theory, we construct a free energy functional given in terms of mesoscopic
variables (the second rank alignment tensors and their derivatives). In addition, this
functional contains both microscopic features of the system, namely, number density
and aspect ratio of the particles. The combination of microscopic (DFT) and mesoscopic
descriptions makes our approach a part of the theories contributing to a scale–bridging
characterization of colloidal mixtures.
To validate our approach, we focus on binary mixtures characterized by molecules
having equal diameters and different length in Chapter 5. Compared to previous experi-
ments and particle based simulations our calculations show a reasonable agreement. In
fact, the calculated state diagrams can be used to make predictions of the overall be-
havior of the mixture. At the end of this chapter, we discussed the relation between our
theory and the Oseen–Frank elastic theory. In this regard, we obtained expressions for
the Frank elastic constants (in the one constant approximation) in terms of the direct
correlation function for the mixture. However, we should point out that at this level of
simplification the material is only defined by a small number of quantities which may
not be enough to fully parametrize any material.
101
102 Chapter 8. Concluding remarks
We note that the biggest restriction of our equilibrium investigations (in general) is
that we have focused on the orientational part of the free energy alone. Therefore, the
stability analysis done in Chapter 5does not give any information about phase transitions
(or coupled phase transitions) involving the number densities as order parameters. This
would clearly be an interesting extension of our work.
In Chapter 6, we presented a natural extension of the Doi–Hess description for binary
mixtures. We achieved this by combining our equilibrium theory with the concepts of
linear irreversible thermodynamics. By virtue of this, we got a pair of dynamic equations
of motion which relate the alignment tensors of the particles (one for each component)
with the hydrodynamic effects (through the stress tensor and the velocity field). Further,
specializing to a planar Couette flow geometry and by expanding these tensor equations
into an appropriate basis, we obtained an 11-dimensional coupled system of partial
differential equations.
Finally, in Chapter 7, we focused on the shear induced instabilities occurring in
binary mixtures. We divided our investigations in two parts: First, we analyzed the shear
banding instabilities occurring in one component systems, and then, we focused on the
shear induced orientational dynamics of binary mixtures.
In the case of shear banding instabilities, we focused on instabilities along the gradi-
ent direction and studied systems whose (un–sheared) equilibrium states were (stable)
isotropic or nematic. We studied various tumbling parameters (aspect ratios) and dif-
ferent anchoring conditions. Our results reveal the appearance of shear banded states in
the gradient direction accompanied by regions with oscillatory dynamics. This feature
is not seen in other models (such as the non–local DJS model) and it is due to the
coupling between alignment tensor and shear stress. Further, contrary to earlier stud-
ies [52,75], we focused on parameters where the homogeneous systems exhibit regular
oscillatory dynamics, and as a result the shear bands are characterized by regions of
regular oscillatory orientational dynamics.
We noticed that the band formation is strongly dependent on two factors: the corre-
lation length and the orientational boundary conditions. We observed that by changing
these features the shear banded states (appearing in the middle of the sample see e.g.
Fig. 7.8) become more defined in terms of width and regularity of the dynamics. This
implies that the bands can be tuned, in principle, only by changing one of these parame-
ters. Since the spatial correlation length is a property of the fluid, the obvious candidates
are the boundary conditions. In fact, our observations prove that certain boundary con-
ditions can support or hinder the formation of shear bands depending on the anchoring
close to the plates.
We then turn to the analysis of binary mixtures. In this case, we showed that in a
binary mixture deep in the nematic state there is a variety of synchronized oscillatory
states of the nematic directors, either with in–plane or with out–of–plane symmetry.
Strikingly, given a specific set of tumbling parameters (aspect ratios) in the mixture, we
observe an area where two different in–plane oscillatory states coexist: tumbling motion
for short particles and wagging motion for long particles. Interestingly, even though the
8.2. Outlook and further investigations 103
actual motion of the components is different, the oscillatory motion is still in synchrony.
Additionally, we observed that as the coupling between the particles is increased, at
a fixed shear rate, one can find transitions from in–plane oscillations to out–of–plane
oscillations (kayaking–tumbling). However, these transitions are highly bistable and this
can translate into an even more convoluted spatiotemporal behavior. At this point, it
would be very interesting to extend the present analysis to spatially inhomogeneous
systems.
8.2 Outlook and further investigations
Concerning the future development of our research, there are several directions that could
be followed to extend our work; both for equilibrium and non–equilibrium phenomena.
Regarding the equilibrium theory, it would be very interesting to understand to what
extent the mesoscopic free energy functionals are modified by coupling to density fluctu-
ations. Therefore, the next logical step would be to improve our theory in order to take
into account full spatial segregation. An idea is to the ansatz for the one body density:
ρ(r,ˆ
u)=(r)f(ˆ
u),
where (r)is an inhomogeneous density and f(ˆ
u(r)) is the usual orientational distri-
bution function. In fact, with our investigations we prove that the ODF (under certain
approximations) can always be written in terms of Q–tensors, therefore the prospective
free energy functional should be:
F[ρ]→F[(r),Q(r)] .
Doing so, we could encounter some challenges. For example, because rand ˆuare
not independent variables, even the normalization of ρ(r,ˆ
u)is not obvious. Another
interesting complication arising from this ansatz has to do with a more general issue, the
exact coupling between (r)and the complete set of Q(r)tensors. This is a problem that
has its roots on the closure approximation. Indeed changing the underlying assumption
of the theory will necessarily change the way we cut the series expansion in the excess
free energy [see Chapter 4].
Clearly, it would be interesting to extend the non–equilibrium investigations as well.
First, the obvious candidate is to extend the analysis for the binary mixture towards
spatially inhomogeneous systems. In fact, we ran some preliminary calculations for the
homogeneous flow curves and the related spatiotemporal behavior. These calculations
show that the region where the in–plane oscillatory states coexist is very unstable. The
nature of these instabilities is not clear, thus, further analysis should be done in this
direction.
Regarding the one component systems, it would also be interesting to study the
shear induced instabilities in the direction perpendicular to the gradient. In fact, recent
104 Chapter 8. Concluding remarks
publications [67,68] show that fd–virus suspensions exhibit vorticity banding (where
regularly stacked bands are formed along the vorticity direction). In this way it could be
interesting to see if the Doi–Hess model is able to reproduce these phenomena.
To close this discussion, for industrial and technological applications it would be
interesting to impose an additional control to the system in order to have a well–defined
rheological response. One idea is to tune the system introducing an additional feedback.
In fact, recent experimental studies [154,155] use oscillatory rheometers to control the
behavior of the viscosity and stress of the system. In view of this, an extension of our
work would be to develop a theoretical approach that reproduces such behavior. In fact,
earlier studies [156] showed that in the scope of the Doi–Hess theory, a feedback control
strategy may lead to the stabilization of different dynamical states. The scheme used in
these investigations is based on a single time delay τin the alignment tensor, such that
Q=Q(t−τ). However, considering our results for spatially inhomogeneous systems,
an alternative approach would be to introduce a similar scheme to the stress tensor, i.e.,
T=T(t−τ). This proposition would model the action of the oscillatory rheometer and
would make it easier to compare the theoretical results with experimental measurements.
Appendices
105
Appendix A
Basic tensor concepts
Introduction
Tensor calculus came into prominence with the development of Einstein’s general theory
of relativity in 1916. Since then it can be seen as a universal language in mathematical
physics [157]. Tensors not only enable us to write general equations in a very compact
manner, but they also serve as a guide in the selection of physical laws by indicating
invariance with respect to the transformation of coordinates.
The aim of this appendix is to serve as a short mathematical introduction to the
use of Cartesian tensors and tensor calculus and to provide the necessary identities and
relationships needed in the main text.
In physics it is usual to deal with real variables as functions represented by Cartesian
coordinates of a point in a plane. If we are dealing with a quantity of three variables
then the ordinary Euclidean space of three dimensions may be used. If the number of
variables exceeds three, the geometrical representation is more complicated and a space
of more than three dimensions is needed. For example, in the case of a rigid body in the
Euclidean space, additional variables (Euler angles) are needed to describe its rotations
around the fixed molecular axes.
In general, a tensor is a multidimensional array of numerical values. The rank (order)
of a tensor is the dimensionality of the array needed to represent it. For example, a scalar
is a zero rank tensor, whereas a linear map represented by a matrix is a 2nd order tensor.
A more abstract definition is that of tensors being multilinear maps taking an element of
a vector space into another vector space. Tensors do not change in any coordinate system
and its components only depend on the relating transformed coordinate systems [158].
Some examples of tensor quantities commonly used by physicists are:
•Scalar a∈R:0–order tensor
•Vector v∈Rn:1–order tensor
•Matrix M∈Rn×Rn:2–order tensor
107
108 Chapter A. Basic tensor concepts
•Levi–Civita symbols ijk ∈Rn×Rn×Rn:3–order tensor
A.1 General definitions and notations
Consider two right-handed coordinate systems Vand Vin R3. The components of a
vector v∈Vare denoted by vμwhere μ=1,2,3,orμ=x, y, z.Ifvis the image of
vin Vthen
v
ν=Dμνvμ
where Dis an orthogonal matrix describing the rotation V→V. The analogous trans-
formation for a 2nd rank tensor Aμν is
A
λγ =DλμDγνAμν ,
where Dkj =Djk. This transformation between Cartesian tensors can be extended to k
order tensors directly via [22]:
A
μ1μ2···μk=Dμ1ν1Dμ2ν2···DμkνkAμ1μ2···μk.
For many applications it is necessary to manipulate tensors in different ways. The
multiplication of a tensor with a real number ameans the multiplications of all its
elements by this number, i.e. aAμ1μ2···μk. On the other hand, the addition of two tensors
can be done only when both have the same rank [22].
A.1.1 Scalar product
An important tensor manipulation is that of the tensor product. Between two vectors,
the usual scalar product is
a·b=aμbμ,
where repeated indices are implicitly summed over (Einstein’s convention). However, for
second rank tensors the maximum contraction is done via the Frobenius inner product
defined by,
A:B=AμνBμν .
The most general contraction between two krank tensors always results in a scalar and
it is denoted by ,
A(k)B(k)=Aμ1μ2···μkBμ1μ2···μk.
A.1.2 Tensor product
The cross product between two vectors in Rnresults in a new vector in the same space,
i.e.
∀a,b∈Rn⇒a×b=c∈Rn.
A.2. Decomposition of tensors 109
However, the outer product, also known as tensor or dyadic product, between these pair
of vectors results into a matrix, i.e.
∀a,b∈Rn⇒a⊗b=C∈Rn×n.
In index notation the µν element of the product a⊗bis
Cµν = (abT)µν =aµbνwhere µ, ν = 1,2,··· , n .
A tensor of rank kcan be constructed as a hierarchy of outer products between a
set of vectors. Let {a,b,···k}be a set of vectors in Rnthen, in terms of this set, the
tensor A(k)of order kis
A(k)=a⊗b⊗···⊗kindex notation Aµ1µ2···µk=aµ1bµ2···kµk.
A.2 Decomposition of tensors
The decomposition of tensors is unique. For example, a second rank tensor, having 9
components can be decomposed into a sum of three tensors, one of each with 1, 3, and
5 linearly independent components [113]. The irreducible parts of a second rank tensor
Aare: an isotropic tensor Aiso; an antisymmetric tensor Aasy and a symmetric traceless
tensor A. Since Aiso = (TrA)I/3,
A=1
2TrAI+Aasy +A.(A.1)
Tensors are classified by their behavior under rotations. Therefore, we can learn
more about their representation by considering a decomposition of them in terms of the
rotation group SO(3). A Cartesian tensor of rank khas 3kcomponents and can be
decomposed into a linear combination of its irreducible parts, each of which spans one
of the irreducible representations of SO(3) [113].
For an arbitrary Cartesian tensor of rank k,A(k), the symmetric traceless part,
A(k), is the only irreducible representation which is not reduced into a lower rank
tensor [22].The number of independent components of an irreducible tensor of rank kis
2k+ 1 .
A.2.1 Isotropic tensors
The symmetric traceless part of a tensor A(k)can be written as a tensor contraction with
another tensor which does not change its form when the Cartesian coordinate system is
replaced by a rotated one, an isotropic tensor. Special cases are:
110 Chapter A. Basic tensor concepts
•Second rank (Kronecker delta):
δμν =1μ=ν
0μ=ν.
•Third rank (Levi–Civita symbol):
λμν =⎧
⎪
⎨
⎪
⎩
1λμν = 123; 312; 231
−1λμν = 213; 132; 321
0otherwise
.
In terms of this tensors, the antisymmetric and isotropic parts of a second rank tensor
Aare:
Aasy =−1
2λμνγηνAμν ,(A.2)
Aiso =1
3δμηδλνAμν .(A.3)
A combination of second and third rank isotropic tensors can be constructed to give the
projector of the symmetric traceless part:
Δ(2) =Δ
μλην =1
2(δμηδλν +δμνδλη)−1
3δμλδην .(A.4)
This identity is such that for a arbitrary second rank tensor A:
A=Δ(2) :A.(A.5)
A generalization of the projection tensor Δ(2) is straightforward noticing that: the
isotropic tensor of rank 2kshould be symmetric and traceless in the first kindices and
in the last kindices. This projection tensor is denoted by Δ(k)and is such that when
applied to an arbitrary tensor A(k)of rank kgives [113]:
A(k)=Δ(k)A(k).(A.6)
A.3 Tensor operations
Restricting to second rank tensor quantities, in this section we list some basic identities
and results of tensor analysis. For a more detailed discussion around this topic we refer the
reader to Tensor Calculus by J. L. Synge et al [157]andTensor Analysis and Elementary
Differential Geometry for Physicists and Engineers by H. Nguyen-Sch¨afer et al [158].
A.3. Tensor operations 111
A.3.1 First order derivatives
Gradient
The partial differentiation with respect to Cartesian components of the position vector
ris frequently denoted by the operator
∇μ=∂
∂rμ
.(A.7)
The application of this operator to a 2nd rank tensor results in a 2+1 rank tensor. The
gradient of a second rank tensor Ais
∇A=∇λAμν =Bλμν =Q(3) .(A.8)
A useful identity in terms of the Kronecker delta is
∇μrν=δμν .(A.9)
Divergence
The divergence of a tensor of rank kresults on a tensor of rank k−1. The divergence
of a second rank tensor is denoted by
∇·A=∇μAμν =Bν,(A.10)
where Bμis a vector.
Curl
The curl can operate on any tensor of rank k(higher than one) and results in a tensor
of the same rank. For example, the curl of a second rank tensor is denoted
∇×A=λμν∇μAνκ .(A.11)
A.3.2 Integration theorems
Gauss’ theorem
Let Sbe an open surface bounding a region Ωof volume V. Given the unit normal
vector ˆ
n(pointing outwards) and A(x)a tensor field, then:
V
A·∇dV =/
S
A·ˆ
ndS . (A.12)
112 Chapter A. Basic tensor concepts
Stoke’s theorem:
Let Sbe an open surface bounded by a closed curve C. Given the unit normal vector ˆ
n
(pointing outwards) and A(x)a tensor field, then:
/
C
A·dx=
S
(A×∇)·ˆ
ndS . (A.13)
A.3.3 Second order derivatives
Laplacian
The Laplacian refers to the special second derivative of a rank ktensor resulting on a
tensor of rank k. It is the result of taking the divergence of a gradient and is denoted by
∇2A=∇·∇A=∇λ∇λAμν .(A.14)
An important identity related to the second order derivative ∇2Ais:
∇2A=∇(∇·A)+∇×(∇×A).(A.15)
Theorem
For a rank (k=2) tensor field Athe following identities for the second order derivative
hold:
∇×(∇A)=0 (A∇)×∇=0 (A.16)
∇·(∇×A)=0 (A×∇)·∇=0 (A.17)
given that Ais smooth enough. This theorem can be reformulated in the following way:
•The gradient of a tensor field is curl–free, i.e., its curl vanishes.
•The curl of a tensor field is divergence–free, i.e., its divergence vanishes.
This theorem may be proved applying Gauss’ and Stoke’s theorems to the tensor products
(A∇) and (A×∇)[
157].
Green’s identity for tensors
∇2(A:B)=(∇2B):A+2(∇A)(∇B)+B:(∇(∇·A)) −B:(∇×(∇×A)) .
(A.18)
The fourth order derivative of a tensor field is often referred as ”the Laplacian of the
Laplacian” and is denoted by
∇4A=∇2∇2A.
The fourth order derivative of a tensor field is often used to stabilize an otherwise
divergent partial differential equation [130].
A.4. Tensor basis 113
A.3.4 Derivative with respect to second rank tensors
Given a scalar differentiable function f(A)which maps the matrix A∈Mn×mto R,
the derivative of fwith respect to Ais
C=∂f(A)
∂A=⎛
⎜
⎝
∂f
∂A11 ··· ∂f
∂An1
.
.
.....
.
.
∂f
∂A1n··· ∂f
∂Anm
⎞
⎟
⎠.(A.19)
where C∈Mn×m. Some higher order derivatives referenced in the main text are:
∂
∂ATr(Al)=l(Al−1)T,(A.20)
∂
∂ATr(ATBA)=BA +BTA,(A.21)
∂
∂ATr(ABAT)=ABT+AB .(A.22)
All of this identities may be extended according to the specific type of matrix operator.
Further information can be found in The Matrix Cookbook by K. B. Petersen et al [159].
A.4 Tensor basis
A symmetric traceless tensor Qhas five independent components. Thus, it can be written
in terms of these components using a standard orthonormal tensor basis [79]:
Q=
4
i=0
qiBi,where B0=3
2ˆ
ezˆ
ez,
B1=1
2(ˆ
exˆ
ex−ˆ
eyˆ
ey),B2=√2ˆ
exˆ
ey,(A.23)
B3=√2ˆ
exˆ
ezB4=√2ˆ
eyˆ
ez.
This tensor basis has the property Bi:Bj=δij, where δij is the Kronecker symbol.
The qicomponents are given by the contraction qi=Q:Bi. In matrix notation the
114 Chapter A. Basic tensor concepts
basis tensors are
B0=1
√6
−100
0−1 0
002
,B1=1
√2
100
0−1 0
000
,
B2=1
√2
010
100
000
,B3=1
√2
001
000
100
,(A.24)
B4=1
√2
000
001
010
.
Explicitly the qicomponents are related to the Cartesian components qij by
q0=−1
√6(qxx +qyy −2qzz), q1=1
√2(qxx −qyy),
q2=1
√2qxy , q3=1
√2qxz ,(A.25)
q4=1
√2qyz .
These basis tensors have been used in a wide variety of tensor equations including
flow alignment of liquid crystals [23,79] and the non–linear behavior of gases, simple
liquids and molecular fluids [80,160,161]. An equivalent set of basis tensors is used to
study the Landau theory of blue phases and cholesterics [162,163].
Appendix B
Spherical harmonics and their
tensor representation
The rotation group SO(3) is very relevant in fluid theory because it is used to repre-
sent physical quantities which are invariant under rotations, like tensors [see App. A],
intermolecular potentials or pair correlation functions [99].
In this appendix we provide a compendium of the properties of the spherical har-
monics and its relation with symmetric traceless tensors. For proofs of these relations we
refer to standard references in group theory and its representations (e.g. Introduction to
group theory with applications by G. Burns [164]).
B.1 Spherical harmonics
The spherical harmonic functions Yμν(ˆ
u)employed in this study are
Yμν(ˆ
u)≡Yμν(θ, φ)=(−1)ν0(2μ+ 1)(μ−ν)!
4π(μ+ν)! 11/2
Pμν(cos θ)eiνφ ,(B.1)
where ν=0,··· ,μ and μ=0,1,2,···. Here Pμν(ˆ
u·ˆ
u)refers to the associated
Legendre functions (for a short overview see Ref. [130]).The unit vector ˆ
uis such that
ˆ
u= (sin θcos φ, sin θsin φ, cos θ). This definition may be extended to ν<0using the
identity
Yμ−ν=(−1)νY∗
μν ,(B.2)
where Y∗
μν denotes the complex conjugate of Yμν. The spherical harmonics are orthog-
onalized functions such that:
S2
Y∗
μν(ˆ
u)Yμν(ˆ
u)dˆ
u=δμμδνν.(B.3)
115
116 Chapter B. Spherical harmonics and their tensor representation
The integral over S2denotes the solid angle integral where cos θ∈[−1,1] and φ∈
[0,2π].
B.1.1 Addition theorem
Spherical harmonics having two different directors ˆ
uand ˆ
ucan be related through the
addition theorem, which is:
∞
ν=0
Y∗
μν(ˆ
u)Yμν(ˆ
u)=2μ+1
4πPμ(ˆ
u·ˆ
u),(B.4)
where ˆ
u·ˆ
u=cosγ and γis the angle between ˆ
uand ˆ
u. Here Pk(ˆ
u·ˆ
u)is the k–th
term of the Legendre polynomials.
B.1.2 Product rule
The multiplication of a pair of spherical harmonics results in
Yμ1ν1(ˆ
u)Yμ2ν2(ˆ
u)=
μν (2μ1+ 1)(2μ2+1)
4π(2μ+1) 1
2C(μ1μ2μ; 000)×
C(μ1μ2μ;ν1ν2ν)Yμν(ˆ
u),(B.5)
where C(μ1μ2μ;ν1ν2ν)is a Clebsch–Gordan coefficient (see Section B.2) . For a full
definition of these coefficients and their use in mathematical physics we refer to Refs. [99,
113].
B.1.3 Integrals
Since the spherical harmonics are orthogonal functions [see Eq. (B.3)] the following
integration rules are satisfied:
S2
Yμν(ˆ
u)dˆ
u=(4π)1/2δμ0δν0,(B.6)
S2
Yμν(ˆ
u)Y∗
μν(ˆ
u)dˆ
u=δμμδνν.(B.7)
B.1.4 Series expansion
The spherical harmonics form a complete set of orthonormal functions and thus form an
orthonormal basis of the field of quadratically integrable functions. Therefore an arbitrary
B.2. Clebsch–Gordan coefficients 117
function f(ˆ
u)on the unit sphere can be expanded in terms of a linear combination of
spherical harmonics. Thus,
f(ˆ
u)=
∞
μ=0
μ
ν=−μ
fμν Yμν (ˆ
u),(B.8)
where the fμν coefficients are given by
f(ˆ
u)=
S2
dˆ
ufμν Y∗
μν (ˆ
u).(B.9)
The series in equation (B.8) is convergent if the μcoefficients decay sufficiently fast [122,
130].
B.2 Clebsch–Gordan coefficients
The spherical harmonics Yμν (ˆ
u)transform under rotations
Yμν(ˆ
u)=
ν
Dμ
νν(Ω)Yμν (ˆ
u),(B.10)
and the inverse transformation
Yμν (ˆ
u)=
ν
Dμ
νν(Ω−1)Yμν(ˆ
u),(B.11)
where ˆ
uand ˆ
uare the orientation of rwith respect to XY Z and XYZ, respectively,
and Ωdenotes the rotation carrying XY Z into coincidence with XYZ(see Ref. [99]).
Equation (B.8) defines the rotation matrix Dμ
νν(Ω−1)where μ=1,2,··· ;ν, ν=
−μ, ··· ,μ. The rotation matrices (also called the representation coefficients of the ro-
tation group SO(3)) are functions of the rotation Ω={φ, θ, ψ}(Euler angles) which
carries the initial frame XY Z into coincidence with the final frame XYZ.
Equation (B.8) allows to write the product Yμ1ν
1(ˆ
u)Yμ2ν
2(ˆ
u)[see (B.5)]:
Yμ1ν
1(ˆ
u
1)Yμ2ν
2(ˆ
u
2)=
ν1ν2
Dμ1
ν1ν
1(Ω)Dμ2
ν2ν
2(Ω) Yμ1ν1(ˆ
u1)Yμ2ν2(ˆ
u2).(B.12)
For fixed μ1,μ
2,the interest is to find a linear combination Fμν (ˆ
u1,ˆ
u2)of products of
spherical harmonics that transforms according to a single rotation matrix Dμ:
Fμν(ˆ
u
1,ˆ
u
2)=
ν
Dμ
νν(Ω)Fμν (ˆ
u1,ˆ
u2).(B.13)
118 Chapter B. Spherical harmonics and their tensor representation
In terms of the product of spherical harmonics Fμν (ˆ
u1,ˆ
u2)is defined as:
Fμν(ˆ
u
1,ˆ
u
2)=
ν1ν2
C(μ1μ2μ;ν1ν2ν)Yμ1ν1Yμ2ν2,(B.14)
where C(μ1μ2μ;ν1ν2ν)are the Clebsch–Gordan coefficients (see e.g. Ref. [99]). The
linear combinations Fμν(ˆ
u1,ˆ
u2)are normalized to unity to ensure that C(μ1μ2μ;ν1ν2ν)
in Eq. (B.12) is a real valued unitary transformation. The above argument can also be
extended to construct rotational invariants form triple products of spherical harmonics
Yμ1ν1Yμ2ν2Yμ3ν3.
In the following we list a few properties of the Clebsch–Gordan coefficients:
Symmetry properties
C(μ1μ2μ3;ν1ν2ν3)=(−1)μ1+μ2+μ3C(μ1μ2μ3;−ν1−ν2−ν3)(B.15)
=(−1)μ1+μ2+μ3C(μ2μ1μ3;ν2ν1ν3)(B.16)
=(−1)μ1+ν12μ3+1
2μ2+11
2
C(μ1μ3μ2;ν1−ν3−ν2)(B.17)
=(−1)μ2+ν22μ3+1
2μ1+11
2
C(μ3μ2μ1;−ν3ν2−ν1)(B.18)
=(−1)μ1+ν12μ3+1
2μ2+11
2
C(μ2μ3μ1;ν3−ν1ν2)(B.19)
=(−1)μ2+ν22μ3+1
2μ1+11
2
C(μ2μ3μ1;ν2ν3−ν1)(B.20)
Orthogonality relations
μ1μ2
C(μ1μ2μ;ν1ν2ν)C(μ1μ2μ;ν1ν2ν)=δμμδνν,(B.21)
μ1μ2
C(μ1μ2μ;ν1ν2ν)C(μ1μ2μ;ν
1ν
2ν)=δμ1μ
1δμ2μ
2.(B.22)
Series expansion (Rotational invariants)
Referred to an arbitrary space–fixed reference frame a two–molecule quantity f(r−
r,ˆ
u,ˆ
u)(e.g. pair potential, pair correlation function, etc.) may be represented as an
infinite sum of spherical invariants.
f(r−r;ˆu,ˆu)=
l1l2l
f(l1l2l;r−r)Φ(l1l2l;ˆu,ˆu,ˆux),(B.23)
B.3. Spherical components of symmetric traceless tensors 119
where f(lil2l;r−r)are the harmonic expansion coefficients given by
f(l1l2l;x)=(2l1+ 1)(2l2+1)
4π(2l+1)
S2
dˆu
S2
dˆuf(r−r;ˆu,ˆu)Φ(l1l2l;ˆu,ˆu,ˆux).(B.24)
Here, the rotational invariants Φ(l1l2l;ˆu,ˆu,ˆux)are given by
Φ(l1l2l;ˆu,ˆu,ˆux)=
m1m2m
C(l1l2l;m1m2m)Yl1m1(ˆu)Yl2m2(ˆu)Y∗
lm(ˆux),(B.25)
where Ylm are the spherical harmonics and C(l1l2l;m1m2m)are the Clebsch–Gordan
coefficients. The notation in Eqs. (B.23) and (B.25) represents explicitly the following:
l1l2l→
∞
l1=0
∞
l2=0
|l1+l2|
l=|l1−l2|
and
m1m2m→
l1
m1=−l1
l2
m2=−l2
l
m=−l
δm,m1+m2.(B.26)
B.3 Spherical components of symmetric traceless tensors
An arbitrary Cartesian vector r=(rx,r
y,r
z)is written in terms of its spherical compo-
nents rμwith μ=0,±1in terms of the orthogonal basis
e0=(0,0,1) ,e±1=∓1
√2(1,±i, 0) ,(B.27)
where
e∗
μeν=δμν and e∗
μ=(−1)μe−μ.(B.28)
Hence r=rμeμwhere the rμcoefficients are given by the relation rμ=r·eμ.
B.3.1 Arbitrary rank tensors
Akrank tensor can be expressed in terms of the basis [113]:
e(k)
μ=k!(2μ)!(2k−1)!!
(k−μ)!μ!(k+μ)!(2μ−1)!!1/2
ek−μ
0eμ
1,μ≥0.(B.29)
with e(k)
−μ=(−1)μe(k)∗
μ.(B.30)
This tensor basis is orthogonal and complete, therefore:
e(k)∗
μe(k)
ν=δμν and e(k)
μe(k)∗
μ=Δ(k),(B.31)
120 Chapter B. Spherical harmonics and their tensor representation
where Δ(k)is the 2krank isotropic tensor addressed in App. A1
It follows directly from Eqs. (B.10), (B.12) and the orthogonality of the spherical
harmonics that, in terms of the unit vector ˆ
u= (sin θcos φ, sin θsin φ, cos θ):
Ykμ(ˆ
u)=(2k+ 1)!!
4πk!1/2
(ˆ
uk)μ,(B.32)
ˆ
ukˆ
uk=k!
(2k−1)!! ,(B.33)
S2
ˆ
ukˆ
ukdˆ
u=4πk!
(2k+ 1)!!δkkΔ(k).(B.34)
Further identities and properties of the spherical harmonics and their tensor repre-
sentation can be found in Refs. [99,113].
1In the same manner as in the previous Appendix (App. A) Here the notation (k)refers to a 2krank
tensor.
Bibliography
[1] A. Mertelj, D. Lisjak, M. Drofenik, and M. ˇ
Copiˇc, “Ferromagnetism in suspensions
of magnetic platelets in liquid crystal,” Nature, vol. 504, pp. 237–241, 2013.
[2] A. R. Morgan, A. B. Dawson, H. S. Mckenzie, T. S. Skelhon, R. Beanland, H. P.
Franks, and S. A. Bon, “Chemotaxis of catalytic silica–manganese oxide match-
stick particles,” Mater. Horiz., vol. 1, pp. 65–68, 2014.
[3] C. Su, J. Lien, and H. Lin, “Liquid crystal mixture and liquid crystal display panel,”
2013. US Patent 8,558,973.
[4] C. Wang, W. Chu, R. Chiou, S. You, H. Zhang, and Y. Tan, “Liquid crystal
compound and liquid crystal mixture,” 2013. US Patent 8,388,861.
[5] R. Baetens, B. P. Jelle, and A. Gustavsen, “Properties, requirements and possibil-
ities of smart windows for dynamic daylight and solar energy control in buildings:
A state-of-the-art review,” Sol. Energ. Mat. Sol. Cells, vol. 94, pp. 87–105, 2010.
[6] J. P. Lagerwall and G. Scalia, “A new era for liquid crystal research: Applications
of liquid crystals in soft matter nano-, bio-and microtechnology,” Current Applied
Physics, vol. 12, pp. 1387–1412, 2012.
[7] S.-H. Sun, M.-J. Lee, Y.-H. Lee, W. Lee, X. Song, and C.-Y. Chen, “Immunoassays
for the cancer biomarker ca125 based on a large-birefringence nematic liquid-
crystal mixture,” Biomed. Opt. Express, vol. 6, pp. 245–256, 2015.
[8] S. C. Glotzer and M. J. Solomon, “Anisotropy of building blocks and their assembly
into complex structures,” Nat. Mater., vol. 6, pp. 557–562, 2007.
[9] J. Pelaez and M. Wilson, “Molecular orientational and dipolar correlation in the
liquid crystal mixture e7: a molecular dynamics simulation study at a fully atomistic
level,” Phys. Chem. Chem. Phys., vol. 9, pp. 2968–2975, 2007.
[10] C. L. Risi, A. F. Neto, P. R. Fernandes, A. R. Sampaio, E. Akpinar, and M. B.
Santos, “Shear viscosity and rheology of ternary and quaternary lyotropic liquid
crystals in discotic and calamitic nematic phases,” Rheol. Acta, vol. 54, pp. 529–
543, 2015.
121
122 BIBLIOGRAPHY
[11] M. Shuai, A. Klittnick, Y. Shen, G. P. Smith, M. R. Tuchband, C. Zhu, R. G.
Petschek, A. Mertelj, D. Lisjak, M. ˇ
Copiˇc, et al., “Spontaneous liquid crystal and
ferromagnetic ordering of colloidal magnetic nanoplates,” Nat. Commun., vol. 7,
p. 10394, 2016.
[12] P. Woolston and J. S. van Duijneveldt, “Isotropic-nematic phase transition of
polydisperse clay rods,,” J. Chem. Phys, vol. 142, 2015.
[13] R. M. van der Kooij and H. N. W. Lekkerkerker, “Liquid-crystalline phase behavior
of a colloidal rod-plate mixture,” Phys. Rev. Lett., vol. 84, pp. 781–784, 2000.
[14] X. Zhou, H. Chen, and M. Iwamoto, “Orientational order in binary mixtures of
hard gaussian overlap molecules,” J. Chem. Phys., vol. 120, p. 1832, 2004.
[15] S. Varga, K. Purdy, A. Galindo, S. Fraden, and G. Jackson, “Nematic-nematic
phase separation in binary mixtures of thick and thin hard rods: Results from
onsager-like theories,” Phys. Rev. E, vol. 72, p. 051704, 2005.
[16] K. R. Purdy, S. Varga, A. Galindo, G. Jackson, and S. Fraden, “Nematic phase
transitions in mixtures of thin and thick colloidal rods,” Phys. Rev. Lett., vol. 94,
2005.
[17] M. Dijkstra, “Entropy-driven demixing in a binary mixture of thick and thin sphe-
rocylinders.” 2006.
[18] T. van Westen, T. J. H. Vlugt, and J. Gross, “The isotropic-nematic and nematic-
nematic phase transition of binary mixtures of tangent hard-sphere chain fluids:
An analytical equation of state,” J. Chem. Phys., vol. 140, p. 034504, 2014.
[19] P. G. de Gennes, The physics of liquid crystals. Clarendon Press, Oxford, 1993.
[20] C. Oseen, “The theory of liquid crystals,” J. Chem. Soc. Faraday Trans., vol. 29,
pp. 883–899, 1933.
[21] F. C. Frank, “I. liquid crystals. on the theory of liquid crystals,” Discuss. Faraday
Soc., vol. 25, pp. 19–28, 1958.
[22] S. Hess, Tensors for Physics. Springer Intl., Switzerland, 2015.
[23] I. Pardowitz and S. Hess, “Of the theory of irreversible processes in molecular
liquids and liquid crystals, non-equilibrium phenomena associated with the second
and forth rank alignment tensors,” Physica A, vol. 100, 1980.
[24] L. Onsager, “The effects of shape on the interaction of colloidal particles,” Ann.
N. Y. Acad. Sci., vol. 51, 1949.
BIBLIOGRAPHY 123
[25] R. Evans, “The nature of the liquid-vapour interface and other topics in the statis-
tical mechanics of non-uniform, classical fluids,” Adv. Phys., vol. 28, pp. 143–200,
1979.
[26] R. Evans, Density functionals in the theory of nonuniform fluids. Marcel Dekker,
New York, 1992.
[27] P. Tarazona, “A density functional theory of melting,” Mol. Phys., vol. 52, pp. 81–
96, 1984.
[28] P. Tarazona, J. A. Cuesta, and Y. Mart´ınez-Rat´on, Density Functional of Hard
Particle Systems, vol. 753. Springer Verlag, 2008.
[29] A. Poniewierski and R. Holyst, “Density-functional theory for systems of hard
rods,” Phys. Rev. A, vol. 41, pp. 6871–6880, Jun 1990.
[30] Y. Singh and K. Singh, “Density-functional theory of curvature elasticity in ne-
matic liquids. iii. numerical results for the berne-pechukas pair-potential model,”
Phys. Rev. A, vol. 33, p. 3481, 1985.
[31] K. Singh and Y. Singh, “Density-functional theory of curvature elasticity in ne-
matic liquids. ii. effect of long-range dispersion interactions,” Phys. Rev. A, vol. 34,
p. 548, 1986.
[32] K. Singh and Y. Singh, “Density-functional theory of curvature elasticity in ne-
matic liquids. i,” Phys. Rev. A, vol. 35, p. 3535, 1987.
[33] M. Dennison, M. Dijkstra, and R. van Roij, “Phase diagram and effective shape
of semiflexible colloidal rods and biopolymers,” Phys. Rev. Lett., vol. 106, 2011.
[34] M. Dennison, M. Dijkstra, and R. van Roij, “The effects of shape and flexibility on
bio-engineered fd-virus suspensions,” The Journal of Chemical Physics, vol. 135,
no. 14, 2011.
[35] A. Malijevsky, G. Jackson, and S. Varga, “Many-fluid onsager density functional
theories for orientational ordering in mixtures of anisotropic hard-body fluids,” J.
Chem. Phys, vol. 129, 2008.
[36] R. Wittkowski, H. L¨owen, and H. R. Brand, “Derivation of a three-dimensional
phase-field-crystal model for liquid crystals from density functional theory,” Phys.
Rev. E, vol. 82, p. 031708, Sep 2010.
[37] J. D. Parsons, “Nematic ordering in a system of rods,” Phys. Rev. A, vol. 19,
pp. 1225–1230, 1979.
[38] S.-D. Lee, “A numerical investigation of nematic ordering based on a simple hard-
rod model,” J. Chem. Phys, vol. 87, pp. 4972–4974, 1987.
124 BIBLIOGRAPHY
[39] T. V. Ramakrishnan and M. Yussouff, “First-principles order-parameter theory of
freezing,” Phys. Rev. B, vol. 19, pp. 2775–2794, 1979.
[40] A. Poniewierski and J. Stecki, “Statistical theory of the elastic constants of ne-
matic liquid crystals,” Mol. Phys., vol. 38, pp. 1931–1940, 1979.
[41] A. Poniewierski and J. Stecki, “Statistical theory of the frank elastic constants,”
Phys. Rev. A, vol. 25, pp. 2368–2370, Apr 1982.
[42] H. L¨owen J. Phys. Condens. Matter, vol. 20, p. 404201, 2008.
[43] H. L¨owen, “Colloidal dispersions in external fields,” J. Phys.: Condens. Matter,
vol. 24, p. 460201, 2012.
[44] D. Guu, J. Dhont, and M. Lettinga, “Dispersions and mixtures of particles with
complex architectures in shear flow,” Eur. Phys. J. ST, vol. 222, pp. 2739–2755,
2013.
[45] A. M. Menzel Phys. Rep., vol. 554, pp. 1 – 45, 2015.
[46] P. D. Olmsted, The effect of shear flow on the isotropic-nematic transition in
liquid crystals. PhD thesis, University of Illinois, Urbana-Champaign, 1991.
[47] Z. Dogic and S. Fraden, Phase behavior of rod–like viruses and virus–sphere mix-
tures. Wiley Verlag, 2007.
[48] G. Rien¨acker and S. Hess, “Orientational dynamics of nematic liquid crystals under
shear flow,” Physica A, vol. 267, 1999.
[49] G. Rien¨acker, Orientational dynamics of nematic liquid crystals in a shear flow.
Shaker, Verlag, 2000.
[50] D. A. Strehober, H. Engel, and S. H. L. Klapp, “Oscillatory motion of sheared
nanorods beyond the nematic phase,” Phys. Rev. E, vol. 88, 2013.
[51] G. Rien¨acker, M. Kr¨oger, and S. Hess, “Chaotic orientational behavior of a nematic
liquid crystal sybjected to a steady shear flow,” Phys. Rev. E, vol. 66, 2002.
[52] B. Chakrabarti, M. Das, C. Dasgupta, S. Ramaswamy, and A. Sood, “Spatiotem-
poral rheochaos in nematic hydrodynamics,” Phys. Rev. Lett., vol. 92, p. 055501,
2004.
[53] S. M. Fielding and P. D. Olmsted, “Kinetics of the shear banding instability in
startup flows,” Phys. Rev. E, vol. 68, p. 036313, 2003.
[54] P. D. Olmsted Rheol. Acta, vol. 47, pp. 283–300, 2008.
BIBLIOGRAPHY 125
[55] Y.-G. Tao, W. den Otter, and W. Briels, “Kayaking and wagging of rods in shear
flow,” Phys. Rev. Lett., vol. 95, p. 237802, 2005.
[56] M. Ripoll, P. Holmqvist, R. G. Winkler, G. Gompper, J. K. G. Dhont, and M. P.
Lettinga, “Attractive colloidal rods in shear flow,” Phys. Rev. Lett., vol. 101,
p. 168302, 2008.
[57] S. Hess, “Pre– and post–transitional behavior of the flow alignment and flow–
induced phase transition in liquid crystals,” Z. Naturforsch., vol. 30a, 1975.
[58] C. Pereira Borgmeyer and S. Hess, “Unified description of the flow alignment and
viscosity in the isotropic and nematic phases of liquid crystals,” J. Non-Equilib.
Thermodyn., vol. 20, pp. 359–384, 1995.
[59] P. D. Olmsted and C.-Y. D. Lu, “Phase separation of rigid-rod suspensions in
shear flow,” Phys. Rev. E, vol. 60, pp. 4397–4415, Oct 1999.
[60] M. Lettinga, Z. Dogic, H. Wang, and J. Vermant, “Flow behavior of colloidal
rodlike viruses in the nematic phase,” Langmuir, vol. 21, pp. 8048–8057, 2005.
[61] S. Heidenreich, P. Ilg, and S. Hess, “Robustness of the periodic and chaotic ori-
entational behavior of tumbling nematic liquid crystals,” Phys. Rev. E, vol. 73,
p. 061710, Jun 2006.
[62] S. Heidenreich, S. Hess, S. Klapp, M. G. Forest, R. Zhou, and X. Yang, “Oscil-
lating hydrodynamical jets in steady shear of nano-rod dispersions,” in THE XV
INTERNATIONAL CONGRESS ON RHEOLOGY: The Society of Rheology 80th
Annual Meeting, vol. 1027, pp. 168–170, AIP Publishing, 2008.
[63] S. Heidenreich, Orientational Dynamics and Flow Properties of Polar and Non-
Polar Hard-Rod Fluids. PhD thesis, 2009.
[64] H. Rehage and H. Hoffmann, “Rheological properties of viscoelastic surfactant
systems,” The Journal of Physical Chemistry, vol. 92, no. 16, pp. 4712–4719,
1988.
[65] H. Rehage and H. Hoffmann, “Viscoelastic surfactant solutions: model systems
for rheological research,” Molecular Physics, vol. 74, no. 5, pp. 933–973, 1991.
[66] S. M. Fielding and P. Olmsted, “Nonlinear dynamics of an interface between shear
bands,” Phys. Rev. Lett., vol. 96, p. 104502, 2006.
[67] J. K. G. Dhont and W. J. Briels, Rod-Like Brownian Particles in Shear Flow:
Sections 3.1 3.9, pp. 147–216. Wiley-VCH Verlag GmbH & Co. KGaA, 2007.
[68] J. K. Dhont and W. J. Briels, “Gradient and vorticity banding,” Rheol. Acta,
vol. 47, pp. 257–281, 2008.
126 BIBLIOGRAPHY
[69] L. Chen, C. Zukoski, B. Ackerson, H. Hanley, G. Straty, J. Barker, and C. Glinka,
“Structural changes and orientational order in a sheared colloidal suspension,”
Phys. Rev. Lett., vol. 69, p. 688, 1992.
[70] J. Decruppe, R. Cressely, R. Makhloufi, and E. Cappelaere, “Flow birefringence
experiments showing a shear-banding structure in a ctab solution,” Colloid Polym.
Sci., vol. 273, pp. 346–351, 1995.
[71] H. Reinken, “Shear banding and constitutive instabilities in nematogenic fluids,”
2016.
[72] J. Goveas and P. Olmsted, “A minimal model for vorticity and gradient banding
in complex fluids,” Eur. Phys. J. E, vol. 6, pp. 79–89, 2001.
[73] N. Spenley, X. Yuan, and M. Cates, “Nonmonotonic constitutive laws and the
formation of shear-banded flows,” J. Phys. II, vol. 6, pp. 551–571, 1996.
[74] P. Olmsted, O. Radulescu, and C.-Y. Lu, “Johnson–segalman model with a diffu-
sion term in cylindrical couette flow,” J. Rheol., vol. 44, pp. 257–275, 2000.
[75] M. Das, B. Chakrabarti, C. Dasgupta, S. Ramaswamy, and A. K. Sood, “Routes
to spatiotemporal chaos in the rheology of nematogenic fluids,” Phys. Rev. E,
vol. 71, p. 021707, 2005.
[76] E. F. Gramsbergen, L. Longa, and W. H. de Jeu, “Landau theory of the nematic-
isotropic phase transition,” Phys. Rep., vol. 135, pp. 195–257, 1986.
[77] N. J. Mottram and C. J. Newton, “Introduction to q-tensor theory,” arXiv preprint
arXiv:1409.3542, 2014.
[78] W. Maier and A. Saupe, “Eine einfache molekular-statistische theorie der nema-
tischen kristallinfl¨ussigen phase. teil l1.,” Zeitschrift f¨ur Naturforschung A, vol. 14,
pp. 882–889, 1959.
[79] P. Kaiser, W. Wiese, and S. Hess, “Stability and instability of an uniaxial alignment
against biaxial distortions in the isotropic and nematic phases of liquid crystals,”
J. Non-Equilib. Thermodyn., vol. 17, pp. 153–170, 1992.
[80] S. Hess, “Fokker–planck equation approach to flow–alignment in liquid crystals,”
Z. Naturforsch., vol. 31a, 1976.
[81] H. Mori, E. C. Gartland Jr, J. R. Kelly, and P. J. Bos, “Multidimensional director
modeling using the q tensor representation in a liquid crystal cell and its application
to the πcell with patterned electrodes,” Jpn. J. Appl. Phys., vol. 38, p. 135, 1999.
[82] D. Frenkel, Statistical mechanics of liquid crystals. Les Houches session LI, 1989.
BIBLIOGRAPHY 127
[83] P. Sheng, “Boundary-layer phase transition in nematic liquid crystals,” Phys. Rev.
A, vol. 26, pp. 1610–1617, 1982.
[84] B. Jerome, “Surface effects and anchoring in liquid crystals,” Rep. Prog. Phys.,
vol. 54, p. 391, 1991.
[85] M. Ruths, S. Steinberg, and J. N. Israelachvili, “Effects of confinement and shear
on the properties of thin films of thermotropic liquid crystal,” Langmuir, vol. 12,
pp. 6637–6650, 1996.
[86] O. Manyuhina, A.-M. Cazabat, and M. B. Amar, “Instability patterns in ultra-
thin nematic films: Comparison between theory and experiment,” Europhys. Lett.,
vol. 92, p. 16005, 2010.
[87] R. Pathria and P. D. Beale, Statistical Mechanics. Academic Press, third edi-
tion ed., 2011.
[88] J.-P. Hansen and I. R. McDonald, Theory of simple liquids. Elsevier, 1990.
[89] P. Egelstaff, An introduction to the liquid state. Elsevier, 2012.
[90] J. E. Mayer, “Contribution to statistical mechanics,” J. Chem. Phys., vol. 10,
pp. 629–643, 1942.
[91] J. R. Solana, Perturbation Theories for the Thermodynamic Properties of Fluids
and Solids. CRC Press, 2013.
[92] D. J. Cleaver, C. M. Care, M. P. Allen, and M. P. Neal, “Extension and gen-
eralization of the gay-berne potential,” Phys. Rev. E, vol. 54, pp. 559–567, Jul
1996.
[93] B. M. Mulder, “The excluded volume of hard sphero-zonotopes,” Mol. Phys.,
vol. 103, 2005.
[94] A. Isihara, “Theory of anisotropic colloidal solutions,” J. Chem. Phys., vol. 19,
pp. 1142–1147, 1951.
[95] P. Hohenberg and W. Kohn, “Inhomogeneous electron gas,” Phys. Rev., vol. 136,
pp. B864–B871, Nov 1964.
[96] N. D. Mermin, “Thermal properties of the inhomogeneous electron gas,” Phys.
Rev., vol. 137, pp. A1441–A1443, Mar 1965.
[97] H. L¨owen, “Density functional theory of inhomogeneous classical fluids: recent
developments and new perspectives,” J. Phys.: Condens. Matter, vol. 14, p. 11897,
2002.
128 BIBLIOGRAPHY
[98] E. Keshavarzi and G. Parsafar, “The direct correlation function and its interpreta-
tion via the linear isotherm regularity,” J. Phys. Soc. Jpn., vol. 70, pp. 1979–1985,
2001.
[99] C. G. Gray and K. E. Gubbins, Theory of molecular liquids. Clarendon Press,
Oxford, 1984.
[100] M. Rex, H. H. Wensink, and H. L¨owen, “Dynamical density functional theory for
anisotropic colloidal particles,” Phys. Rev. E, vol. 76, p. 021403, Aug 2007.
[101] Y. Rosenfeld, “Free-energy model for the inhomogeneous hard-sphere fluid mixture
and density-functional theory of freezing,” Phys. Rev. Lett., vol. 63, pp. 980–983,
Aug 1989.
[102] H. Hansen-Goos and K. Mecke, “Fundamental measure theory for inhomogeneous
fluids of nonspherical hard particles,” Phys. Rev. Lett., vol. 102, p. 018302, 2009.
[103] R. Wittmann, M. Marechal, and K. Meck, “Fundamental mixed measure theory
for non-spherical colloids,” Europhys. Lett., vol. 109, p. 26003, 2015.
[104] W. Curtin, “Density-functional theory of the solid-liquid interface,” Phys. Rev.
Lett., vol. 59, p. 1228, 1987.
[105] H. B. Callen, Thermodynamics & an Intro. to Thermostatistics. John Wiley &
Sons, 2006.
[106] E. Velasco and L. Mederos, “A theory for the liquid-crystalline phase behavior of
the gay–berne model,” J. Chem. Phys., vol. 109, pp. 2361–2370, 1998.
[107] S.-D. Lee, “The onsager-type theory for nematic ordering of finite-length hard
ellipsoids,” J. Chem. Phys, vol. 89, pp. 7036–7037, 1988.
[108] P. Bolhuis and D. Frenkel, “Tracing the phase boundaries of hard spherocylinders,”
J. Chem. Phys., vol. 106, pp. 666–687, 1997.
[109] A. Cuetos, B. Martinez-Haya, S. Lago, and L. Rull, “Use of parsons-lee and onsager
theories to predict nematic and demixing behavior in binary mixtures of hard rods
and hard spheres,” Physical Review E, vol. 75, 2007.
[110] H. L¨owen, “A phase-field-crystal model for liquid crystals,” J. Phys.: Condens.
Matter, vol. 22, p. 364105, 2010.
[111] R. Lugo-Fr´ıas and S. H. L. Klapp, “Binary mixtures of rod-like colloids under
shear: microscopically-based equilibrium theory and order-parameter dynamics,”
J. Phys.: Condens. Matter, vol. 28, p. 244022, 2016.
BIBLIOGRAPHY 129
[112] S. Grandner, S. Heidenreich, S. Hess, and S. H. L. Klapp, “Polar nano-rods under
shear: From equilibrium to chaos,” Eur. Phys. J. E, vol. 24, 2007.
[113] F. McCourt, J. Beenakkeran, W. Koehler, and I. Kuscer, Nonequilibrium phenom-
ena in polyatomic gases. Clarendon Press, Oxford, 1991.
[114] G. J. Vroege and H. N. W. Lekkerkerker, “Phase transitions in lyotropic colloidal
and polymer liquid crystals,” Rep. Prog. Phys., vol. 55, p. 1241, 1992.
[115] M. Kr¨oger, A. Ammar, and F. Chinesta, “Consistent closure schemes for statistical
models of anisotropic fluids,” J. Non-Newtonian Fluid Mech., vol. 149, 2008.
[116] M. Kr¨oger, “Simple models for complex nonquilibrium fluids,” Phys. Rep., vol. 390,
2003.
[117] J. Feng, C. V. Chaubal, and L. G. Leal, “Closure approximations for the doi
theory: Which to use in simulating complex flows of liquid-crystalline polymers?,”
J. Rheol., vol. 42, p. 1095, 1998.
[118] J. Han, Y. Luo, W. Wang, P. Zhang, and Z. Zhang, “From microscopic theory
to macroscopic theory: a systematic study on modeling for liquid crystals,” Arch.
Rational Mech. Anal., pp. 1–69, 2014.
[119] H. H. Wensink, “Spontaneous sense inversion in helical mesophases,” Europhys.
Lett., vol. 3, 2014.
[120] L. Feigin and D. Svergun, Structure Analysis by Small-Angle X-Ray and Neutron
Scattering. Plenum Press, New York, 1987.
[121] K. R. Elder, N. Provatas, J. Berry, P. Stefanovic, and M. Grant, “Phase–field
crystal modeling and classical density functional theory of freezing,” Phys. Rev.
B, vol. 75, p. 064107, 2007.
[122] R. Courant and D. Hilbert, Methods of Mathematical Physics. Interscience, New
York, 1989.
[123] C. V. Achim, R. Wittkowski, and H. L¨owen, “Stability of liquid crystalline phases
in the phase-field-crystal model,” Phys. Rev. E, vol. 83, p. 061712, 2011.
[124] H. Emmerich, H. L¨owen, R. Wittkowski, T. Gruhn, G. I. T´oth, G. Tegze, and
L. Gr´an´asy, “Phase-field-crystal models for condensed matter dynamics on atomic
length and diffusive time scales: an overview,” Adv. Phys., vol. 61, pp. 665–743,
2012.
[125] N. F. Carnahan and K. E. Starling, “Equation of state for nonattracting rigid
spheres,” J. Chem. Phys., vol. 51, pp. 635–636, 1969.
130 BIBLIOGRAPHY
[126] P. J. Camp and M. P. Allen, “Hard ellipsoid rod-plate mixtures: Onsager theory
and computer simulations,” Physica A, vol. 229, pp. 410–427, 1996.
[127] T. Dingemans, L. Madsen, N. Zafiropoulos, L. Wenbin, and E. Samulski, “Uniaxial
and biaxial nematic liquid crystals,” Phil. Trans. R. Soc. A, vol. 364, pp. 2681–
2696, 2006.
[128] M. J. Solomon and P. T. Spicer, “Microstructural regimes of colloidal rod suspen-
sions, gels, and glasses,” Soft Matter, vol. 6, 2010.
[129] J. Mewis and N. Wagner, Colloidal Suspension Rheology. Cambridge, 2012.
[130] G. B. Arfken, H. J. Weber, and F. E. Harris, Mathematical methods for physicists:
A comprehensive guide. Academic press, 2011.
[131] S. Dussi and M. Dijkstra, “Entropy-driven formation of chiral nematic phases by
computer simulations,” Nat. Commun., vol. 7, 2016.
[132] G. Taylor, “The motion of ellipsoidal particles in a viscous fluid,” Proc. R. Soc.
A, vol. 103, pp. 58–61, 1923.
[133] M. Fialkowski, “Viscous properties of binary mixtures of nematic liquid crystals,”
Phys. Rev. E, vol. 53, pp. 721–726, 1996.
[134] A. M. Menzel, A. Saha, C. Hoell, and H. L¨owen, “Dynamical density functional
theory for microswimmers,” The Journal of Chemical Physics, vol. 144, no. 2,
p. 024115, 2016.
[135] J. Ericksen, “Hydrostatic theory of liquid crystals,” Arch. Rational Mech. Anal.,
vol. 9, pp. 371–378, 1962.
[136] F. Leslie, “Some constitutive equations for liquid crystals,” Arch. Rational Mech.
Anal., vol. 28, pp. 265–283, 1968.
[137] M. Doi and S. Edwards, “Dynamics of rod–like macromolecules in concentrated
solution. pts. 1 and 2,” J. Chem. Soc., vol. 74, 1978.
[138] C. Eckart, “The thermodynamics of irreversible processes. i. the simple fluid,”
Phys. Rev., vol. 58, p. 267, 1940.
[139] S. R. De Groot and P. Mazur, Non-equilibrium thermodynamics. Dover, 1983.
[140] R. Dong, “On the anisotropy of nuclear spin relaxation in a nematic liquid crystal,”
Chem. Phys. Lett., vol. 9, pp. 600–602, 1971.
[141] L. Onsager, “Reciprocal relations in irreversible processes. i.,” Phys. Rev., vol. 37,
pp. 405–426, Feb 1931.
BIBLIOGRAPHY 131
[142] Z. Dogic, Liquid crystalline phase transitions in virus and virus/polymer suspen-
sions. PhD thesis, Citeseer, 2000.
[143] K. R. Purdy, Liquid Crystal Phase Transitions of Monodisperse and Bidisperse
Suspensions of Rodlike Colloidal Virus. PhD thesis, Brandeis University, 2004.
[144] D. Chakraborty, C. Dasgupta, and A. K. Sood, “Banded spatiotemporal chaos in
sheared nematogenic fluids,” Phys. Rev. E, vol. 82, p. 065301, 2010.
[145] R. Ganapathy, S. Majumdar, and A. K. Sood, “Spatiotemporal nematodynamics
in wormlike micelles en route to rheochaos,” Phys. Rev. E, vol. 78, p. 021504,
Aug 2008.
[146] S. Hess and M. Kr¨oger, “Regular and chaotic orientational and rheological behavior
of liquid crystals,” J. Phys.: Condens. Matter, vol. 16, 2004.
[147] S. Hess and M. Kr¨oger, in Computer Simulations of Liquid Crystals and Polymers,
pp. 295–333. Springer Verlag, 2005.
[148] R. Lugo-Fr´ıas, H. Reinken, and S. H. L. Klapp, “Shear banding in nematogenic
fluids with oscillating orientational dynamics,” Eur. Phys. J. E Soft Matter, p. sub-
mitted, 2016.
[149] O. Hess and S. Hess, “Nonlinear fluid behavior: from shear thinning to shear
thickening,” Physica A, vol. 207, 1994.
[150] M. E. Cates, D. A. Head, and A. Ajdari, “Rheological chaos in a scalar shear-
thickening model,” Phys. Rev. E, vol. 66, p. 025202, 2002.
[151] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical
recipes in C. Cambridge university press Cambridge, 1996.
[152] O. Radulescu and P. Olmsted, “Matched asymptotic solutions for the steady
banded flow of the diffusive johnson–segalman model in various geometries,” J.
Non-Newton. Fluid, vol. 91, pp. 143–164, 2000.
[153] J. Adams, S. Fielding, and P. Olmsted, “The interplay between boundary condi-
tions and flow geometries in shear banding: Hysteresis, band configurations, and
surface transitions,” J. Non-Newtonian Fluid Mech., vol. 151, pp. 101–118, 2008.
[154] T. Xu and V. A. Davis, “Rheology and shear-induced textures of silver nanowire
lyotropic liquid crystals,” J. Nanomaterials, vol. 2015, 2015.
[155] B. Saint-Michel, T. Gibaud, M. Leocmach, and S. Manneville, “Local oscillatory
rheology from echography,” Phys. Rev. Applied, vol. 5, p. 034014, Mar 2016.
132 BIBLIOGRAPHY
[156] D. A. Strehober, E. Sch¨oll, and S. H. L. Klapp, “Feedback control of flow alignment
in sheared liquid crystals,” Phys. Rev. E, vol. 88, 2013.
[157] J. L. Synge and A. Schild, Tensor calculus, vol. 5. Courier Corporation, 1969.
[158] H. Nguyen-Sch¨afer and J.-P. Schmidt, Tensor Analysis. Springer, 2014.
[159] K. B. Petersen, M. S. Pedersen, et al.,The matrix cookbook. PhD thesis, 2008.
[160] N. Herdegen and S. Hess, “Nonlinear flow behavior of the boltzmann gas,” Physica
A, vol. 115, pp. 281–299, 1982.
[161] W. Loose and S. Hess, “Nonequilibrium velocity distribution function of gases:
Kinetic theory and molecular dynamics,” Phys. Rev. A, vol. 37, p. 2099, 1988.
[162] R. Hornreich and S. Shtrikman, “Landau theory of blue phases,” Mol. Cryst. Liq.
Cryst., vol. 165, pp. 183–211, 1988.
[163] R. Hornreich and S. Shtrikman, “Field-induced hexagonal blue phases in posi-
tive and negative dielectric anisotropy systems: Phase diagrams and topological
properties,” Phys. Rev. A, vol. 41, p. 1978, 1990.
[164] G. Burns, Introduction to group theory with applications: materials science and
technology. Academic Press, 2014.
Acknowledgements
First of all I would like to thank my supervisor Prof. Dr. Sabine Klapp for giving me the
chance to be park of her group. Thanks for all your help, advice and constant support. It
was really challenging, I learned a lot and I had a lot of fun. Also, I would like to thank
Dr. Andreas Menzel and Prof. Dr. Michael Lehmann for being part of my dissertation
committee and taking the time to read my thesis.
Thanks a lot to all the past, present and future members of the AG Klapp: From day
one you made me feel welcome and helped me a lot along the way. Specially I would
like to thank Ken, Florian, Henning, Nicola, Robert, Arzu, David, Stavros, Alex, Sascha
and Sarah with whom I probably spent more time discussing things that had nothing to
do with physics.
I will also like to acknowledge financial support from the Research Training Group 1558.
To my friends in Mexico, New Zealand, Germany, England, Spain and the US: I salute
you, G.G.N.M.U!
Finally I would like to thank Kathrin; I find it difficult to express my appreciation because
it is boundless. Thank you for your love, patience and support.
Este trabajo se lo dedico a mis padres.
De gente bien nacida es agradecer los beneficios que reciben
y uno de los pecados que m´as a Dios ofende es la ingratitud.
–El Quijote de la Mancha
Miguel de Cervantes Saavedra
