Nematic textures constructed by conformal mapping on corrugated substrates

概要

博士論文

Nematic textures constructed by conformal
mapping on corrugated substrates
（等角写像による基板上のネマティック液晶の
配向パターンの構築）

馬原

天志朗

令和４年

Contents
1.

2.

3.

4.

Introduction of liquid crystals ..........................................................................................1
1.1.

Overview of researches on liquid crystals in contact with substrates ........................1

1.2.

Nematic liquid crystal phase .....................................................................................2

1.3.

Topological defects in nematic liquid crystals ..........................................................3

1.4.

General models for nematic liquid crystal.................................................................5

1.5.

Purpose of the present thesis ....................................................................................8

1.6.

The organization of this thesis ................................................................................10

Complex analysis for liquid crystals................................................................................11
2.1.

The method of conformal mapping ........................................................................11

2.2.

Nematic texture with complex potential .................................................................13

2.3.

Construction of complex potential for disclinations ...............................................15

2.4.

Free energy model by analogy with vortices in superfluid on a flat space ...............18

2.5.

Free energy model of nematic liquid crystal on a curved metric .............................21

2.6.

Geometric potential by conformal anomaly ............................................................25

2.7.

Topological charge of vortex and disclination.........................................................28

Nematic texture on a sinusoidal substrate ......................................................................31
3.1.

Infinite cell without upper substrate .......................................................................31

3.2.

Stability of disclinations on sinusoidal substrates ...................................................32

3.3.

Finite cell with upper substrate ...............................................................................36

3.4.

Stabilization process of disclinations in a finite cell ................................................40

Disclination driven by the flexoelectric effect ................................................................43
4.1.

Flexoelectric free energy derived with complex analysis .........................................43

4.2.

Interaction term with electric field..........................................................................45

4.3.

The case with a uniform electric field .....................................................................46

4.4.

The case with a deformed electric field due to a sinusoidal substrate.....................47

4.5.

Disclination migration by an electric field ..............................................................48

4.6.

Threshold voltage for switching states ....................................................................50

Summary ................................................................................................................................54
Appendices ............................................................................................................................56
A.

Schwarz-Christoffel transformation ...............................................................................56

B.

Geometric interpretation of the flexoelectric effect .......................................................61

References .............................................................................................................................63
Acknowledgments ..................................................................................................................65

1. Introduction of liquid crystals

1.1. Overview of researches on liquid crystals in contact with substrates
Liquid crystals are known to have intermediate properties between solid and isotropic
liquid. This is because liquid crystal molecules are rod or disc like shape and have
properties such that the positions of their center of mass are random but their
orientational order is nonzero. Liquid crystals show several phases; nematic phase where
the orientations of the rod like molecules align in a specific direction to some extent,
smectic phase where a modulation of the distribution of the center of mass of molecules
arises in addition to the molecular alignment, cholesteric phase where nematic layers
stack like a spiral, discotic phase where disc like molecules are stacked in cylinders.
Figure 1.1 shows typical examples of these phases. In this thesis, we focus on the nematic
liquid crystal phase and discuss the orientation of the nematic liquid crystal contacting
to a substrate that has a spatial modulation pattern.
In recent years, a number of researches on the structures of nematic liquid crystals
(LCs) in contact with micro-structured substrates have been reported. The purpose of
these researches is not only a fundamental description of liquid crystals, but also
practical applications such as the design of bistable LC display like a zenithally bistable
device (ZBD) [1-13], and trapping a colloid particle [14-16].
The ZBD is a kind of LC displays which utilizes the bistability of orientations of LC
molecules at the surface of a substrate. Similar to conventional twisted-nematic displays,
ZBD’s have two substrates separated by about 5 µm each of which is overcoated with a
polymer layer to make the liquid crystal molecules align in a specified direction [4,7-13].
The difference between ZBD’s and ordinary LC displays is that one of the substrates in
a ZBD has a corrugated pattern structure that induces two or more metastable

nematic

cholesteric

smectic

discotic

Fig. 1.1 Typical liquid crystal phases that are formed by rod-like or disc-like molecules.

1

(a)

(b)

Fig. 1.2 Nematic textures induced by a sinusoidal substrate. Boundary condition is assumed
as a strong homeotropic anchoring on the sinusoidal substrate. There are two metastable
states; (a) defect-free state and (b) defect state. The orange dot and blue dots in (b) represent
the positions of + 1⁄2 and − 1⁄2 disclinations, respectively.

orientations (see Fig.1.2). These different spatial patterns of the director fields result in
different optical properties such as dark and light states [7-9]. While a standard LC cell
requires a steady electric power to display images, bistable LC requires a power
consumption only when the states are switched. Thus, the bistable LC can act as a lowpowered device which is suitable for displays that do not require frequent image
updating such as electronic books, e-paper, signage, smart cards, watches, etc.

1.2. Nematic liquid crystal phase
The nematic liquid crystal phase is the closest to the isotropic liquid phase among
various LC phases due to its low viscosity. As the nonzero orientational order gives
optical anisotropy like solid, the nematic liquid crystal phase has both of solid properties
and liquid properties, i.e., an optical anisotropy like a solid and a high fluidity like the
isotropic liquid. According to these features, nematic liquid crystals have been developed
as an optical medium used in a display.
Nematic states are described by the nematic director 𝒏 and the scalar order
parameter 𝑆. The nematic director 𝒏 is a unit vector and is defined by the mean field of
local directors of longitudinal axis of molecules. Since the head and the tail of rod like
molecules cannot be distinguished on the macroscopic level, the nematic director has the
inversion symmetry with respect to 𝒏 ↔ −𝒏.
2

The scalar order parameter 𝑆 measures the degree of local alignment of molecules.
𝑆 = 0 represents the state that orientations of molecules are perfectly random and looks
like an isotropic liquid. On the other hand, 𝑆 = 1 is the state that orientations of
molecules are perfectly aligned to one direction. For 0 < 𝑆 < 1 , molecules have the
orientational order to some extent. In 3D spaces, 𝑆 is measured by a statistical mean as
1
1
𝑆 = 〈3 cos 2 𝜃 − 1〉 = ∫ (3 cos 2 𝜃 − 1)𝑓(𝜃)𝑑𝑉 ,
2
2 𝑉

(1. 1)

where 𝜃 is the orientation angles of a molecule between its longitudinal axes and the
nematic director, and 𝑓(𝜃) is a statistical distribution function of the molecular
orientation angle. 𝑓(𝜃) is determined by the physical condition such as the temperature
and material parameters and gets sharp around 𝜃 = 0 for a strong nematic phase.

1.3. Topological defects in nematic liquid crystals
Topological defects in nematic liquid crystals are singular points of nematic director
field, for which the director is not well-defined at the center of their cores [17-19].
Topological defects in nematic liquid crystals are often called “disclinations”, which we
also use in this thesis.
Disclinations are known to arise when a transition from isotropic phase to liquid
crystal phase occurs after a temperature quench. This is because the orientation of the
nematic director is random in the initial stage of the ordering process, which leads to the
emergence of singular points of the director field. In this process, disclinations with
opposite signs attract each other and annihilate immediately.
While the disclinations generated by the temperature quench are usually unstable,
stable disclinations are also observed in a system where a nematic liquid crystal is in
contact with a curved surface or a curved boundary [1,4,11-13,20-24]. Even if the surface
has the same topology as flat one and no topological constraints require disclinations to
exist, a curved surface can cause the unbinding of pairs of disclinations with topological
charges +1/2 and −1/2 (defined in Eq. ...

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