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TRANSCRIPT
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Three Dimensional Finite Element
Modelling of Liquid CrystalElectro-Hydrodynamics
by
Eero Johannes Willman
A thesis submitted for the degree of Doctor of Philosophy of
University College London
Faculty of Engineering
Department of Electronic & Electrical Engineering
University College London
The United Kingdom
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I, Eero Johannes Willman, confirm that the work presented in this thesis is my own.
Where information has been derived from other sources, I confirm that this has been
indicated in the thesis.
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Contents
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Outline of the Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Development of a 3-D Finite Element Computer Model . . . 3
1.2.2 Modelling of Weak Anchoring in the Landau-de Gennes Theory 4
1.2.3 Modelling a Post Aligned Bistable Nematic LC Device . . . . 5
1.3 Achievements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Liquid Crystals 8
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Liquid Crystal Phases . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Liquid Crystal Order . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3.1 Uniaxial Order . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3.2 Biaxial Order and the Q-Tensor . . . . . . . . . . . . . . . . . 12
2.4 Defects and Disclinations . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.5 Dielectric Properties and Flexoelectric
Polarisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.6 Optical Properties of LCs . . . . . . . . . . . . . . . . . . . . . . . . 18
2.6.1 Jones Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3 Theoretical Framework 21
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3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2 Mean Field Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.3 Molecular Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.4 Continuum Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.4.1 Liquid Crystal Elasticity . . . . . . . . . . . . . . . . . . . . . 25
3.4.2 Thermotropic Energy . . . . . . . . . . . . . . . . . . . . . . . 31
3.4.3 External Interactions . . . . . . . . . . . . . . . . . . . . . . . 32
3.5 Static Equilibrium Q-Tensor Fields . . . . . . . . . . . . . . . . . . . 33
3.6 Q-Tensor Hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . 34
3.6.1 Conservation of Energy . . . . . . . . . . . . . . . . . . . . . . 35
3.6.2 Frictional Forces . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.6.3 The Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.6.4 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . 39
3.6.5 Choice of the Dissipation Function . . . . . . . . . . . . . . . 40
3.6.6 Explicit Expressions for the LC-Hydrodynamics . . . . . . . . 41
3.7 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . 42
4 Modelling of the Liquid CrystalSolid Surface Interface 444.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.2.1 Classification of Different Anchoring Types . . . . . . . . . . . 46
4.2.2 Anchoring Mechanisms . . . . . . . . . . . . . . . . . . . . . . 47
4.2.3 Experimental Measurement of Anchoring Strengths . . . . . . 49
4.3 Review of Currently Used Weak Anchoring Expressions . . . . . . . . 51
4.3.1 Weak Anchoring in Oseen-Frank Theory . . . . . . . . . . . . 51
4.3.2 Weak Anchoring in the Landau-de Gennes
Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
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4.4 The Anchoring Energy Density of an Anisotropic Surface in the Landau-
de Gennes Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.4.1 Determining Values for the Anchoring Energy Coefficients . . 58
4.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.5.1 Comparison between the Landau-de Gennes and Oseen-Frank
Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.5.2 Effect of Order Variations on the Effective Anchoring Strength 63
4.6 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . 66
5 Finite Elements Implementation 69
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.2 The Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . 71
5.2.1 Weighted Residuals Method . . . . . . . . . . . . . . . . . . . 73
5.2.2 Variational Method . . . . . . . . . . . . . . . . . . . . . . . . 74
5.2.3 Enforcing Constraints and Boundary Conditions . . . . . . . . 765.2.4 Solution Process . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.3 Shape Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.3.1 Analytic and Numerical Integration of Shape Functions . . . . 82
5.4 General Overview of the Program . . . . . . . . . . . . . . . . . . . . 83
5.5 Electrostatic Potential . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.6 Q-Tensor Implementation . . . . . . . . . . . . . . . . . . . . . . . . 87
5.6.1 Newtons Method . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.6.2 Time Integration . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.7 Implementation of the Hydrodynamics . . . . . . . . . . . . . . . . . 92
5.7.1 Enforcement of Incompressibility . . . . . . . . . . . . . . . . 93
6 Mesh Adaptation 98
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
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6.2 Mesh Adaptation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.2.1 Assessment of the Error . . . . . . . . . . . . . . . . . . . . . 100
6.2.2 Adapting the Spatial Discretisation . . . . . . . . . . . . . . . 101
6.3 Overview of the Mesh Adaption Algorithm . . . . . . . . . . . . . . . 104
6.4 Example Defect Movement in a Confined Nematic Liquid CrystalDroplet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.5 Hierarchicalp-Refinement . . . . . . . . . . . . . . . . . . . . . . . . 108
6.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
7 Validation and Examples 115
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
7.2 Three Elastic Constant Formulation . . . . . . . . . . . . . . . . . . . 116
7.3 Switching Dynamics of a TN-Cell, with Back flow . . . . . . . . . . . 117
7.4 Defect Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
7.5 Defect Loops in the Zenithally Bistable Device . . . . . . . . . . . . . 1217.6 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . 126
8 Modelling of the Post Aligned Bistable Nematic Liquid Crystal
Structure 129
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
8.2 Overwiew of the The PABN Device . . . . . . . . . . . . . . . . . . . 130
8.3 Modelling the PABN Device . . . . . . . . . . . . . . . . . . . . . . . 131
8.3.1 The Geometry of the Modelling Window . . . . . . . . . . . . 132
8.4 Modelling Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
8.4.1 A Topological Study of a Single Corner . . . . . . . . . . . . . 133
8.4.2 Modelling the Full StructureThe Two Stable States . . . . 1378.4.3 Modelling the Full StructureThe Switching
Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
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8.5 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . 142
9 Summary and Future Work 145
9.1 Summary or Achievements . . . . . . . . . . . . . . . . . . . . . . . . 146
9.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
A Values of Material Parameters Used in this Work 149
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List of Figures
2.1 The molecular configurations of the isotropic, nematic and smectic A
and C phases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 The nematic directornand the scalar order parameter S. . . . . . . 12
2.3 (a) IsotropicQ-tensor, (b) uniaxial Q-tensor, (c) biaxialQ-tensor, (d)
uniaxial Q-tensor, but with negative scalar order parameter. . . . . . 14
2.4 Director profiles for defects of whole m= 1 and m= 12
strengths. . 16
3.1 Splay, bend and twist deformations. . . . . . . . . . . . . . . . . . . . 26
3.2 Bulk energy as a function of order parameter for various temperatures 32
4.1 Twist angle in a cell of thickness d. Dashed line, strong anchoring.
Solid line, weak anchoring. . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2 Normalised anisotropic parts of the anchoring energy density for a
surface with e = [1, 0, 0], v1 = [0, 1, 0] and v2 = [0, 0, 1]. (a) R = 1.
(b) R= 3. (c) R= 0. (d) R=
. (R= W2/W1) . . . . . . . . . . . 58
4.3 Eigenvalues of aQ-tensor that minimises the surface energy density as
a function ofR, when Se is unity. . . . . . . . . . . . . . . . . . . . 60
4.4 (a) Scalar order parameter Sand (b) biaxiality parameter P as func-
tions of the distance from the surface (in m) and the ratioR between
W2 and W1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
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4.5 Normalised eigenvalues of Q at the surface as a function of W2 for
R= 1, 3 and
, whenais set according to expression 4.23 (no markers)
and for the linear case as= 0 and R= 1 (circles). . . . . . . . . . . 62
4.6 (a)(c) Tilt and twist angles as a function ofV, with a constantR= 13
.
(d)(f) Tilt and twist angles as a function ofR, with a constant appliedvoltageV = 1.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.7 Ratio of the effective azimuthal anchoring strength coefficient andW1
as a function ofR . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.1 Local coordinates of a tetrahedron. . . . . . . . . . . . . . . . . . . . 82
5.2 Flowchart of the program execution. . . . . . . . . . . . . . . . . . . 85
5.3 Container with 90 bend for testing the stabilised Stokes flow. . . . . 96
5.4 Flow magnitude (top row) and pressure (bottom row) solutions ob-
tained using three different values the stabilisation parameter =
104, 106 and 109). . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.1 Element refinement by the red-green method. Bisected edges are drawn
in bold. Original nodes are labelled with capital letters whereas new
nodes resulting from edge bisection are labelled using lower case letters. 103
6.2 Example of error introduced by linear interpolation of the components
of a Q-tensor field representing a rotation of the director field of a
constant order. Black dots represent the original nodes and gray dots
the new added node. . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.3 Partial 3-Dimensional view of initial unrefined mesh for LC droplet
inside a cube of fixed isotropic dielectric material. Approximately a
quarter of the dielectric region (coloured white) and half of the liquid
crystal (coloured grey) are shown. . . . . . . . . . . . . . . . . . . . 108
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6.4 (a), (c), (d) 2-Dimensional slices through the centre of a nematic
droplet during switching by an external electric field. Director colour
indicates scalar order parameter and background electric potential.
(b), (d), (e) 3-Dimensional views of corresponding meshes. . . . . . . 109
6.5 (a) Second, third and fourth order hierarchical shape functions for a
one dimensional finite elements implementation. (b) Example of super-
position of first and second order hierarchical element shape functions.
Linear element (dashed line) is p-refined by the addition of a second
order (solid line) shape function. . . . . . . . . . . . . . . . . . . . . . 110
6.6 (a) 12
defect in two dimensions (left) and the one dimensional director
profile through the centre (right). (b) Eigenvalues of the Q-tensor in
the one dimensional case plotted against the z dimension. . . . . . . . 112
6.7 Comparison between results obtained using hierarchical elements of
different order. (a) Total free energy as a function of element size. (b)
The effective number of degrees of freedom as a function of element size114
6.8 Magnitudes of higher order hierarchical degrees of freedom as a func-
tion of the z-dimension. The number of 1-D elements is 50, resulting
in an element size of 2 nm. . . . . . . . . . . . . . . . . . . . . . . . . 114
7.1 Comparison of tilt angles at z = 0.5m as a function of time using
the Oseen-Frank (dashed line) and the Landau-de Gennes (solid line)
theories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
7.2 Switching dynamics of a twisted nematic cell, with and without flow
of the LC material. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
7.3 The two initial director configurations for the defect annihilation cases
(a) and (b), and the corresponding flow solutions (c) and (d) at time
= 20 s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
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7.4 Defect positions with respect to time for the two initial configurations,
with and without flow. In both cases when flow is ignored, identical re-
sults are obtained. The solid line represents the position of the positive
defect and the dashed line the position of the negative defect. . . . . 120
7.5 The continuous (a) and discontinuous (b) states found in the two di-
mensional representation of the ZBD grating structure. . . . . . . . . 123
7.6 Three different surface profiles for the ZBD structure, with the height
of the slip region set to 0, 0.5 and 1 times the ridge height in (a), (b)
and (c) respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
7.7 Iso-surfaces of reduced order parameter showing the locations of the
defect lines. Circles are drawn to indicate the regions of the 12
defect
transitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
8.1 The geometries of the 3-D modelling windows (a) for the full device,
and (b) for the isolated corner. . . . . . . . . . . . . . . . . . . . . . . 134
8.2 Director profiles for the horizontal (a) and continuous vertical (b) states
on a regular grid along the (x, y) plane through the centre of the iso-
lated corner structure at z= 0.3m. The discontinuous vertical state
is not shown, as it appears nearly identical to the continuous vertical
state from this point of view. . . . . . . . . . . . . . . . . . . . . . . 135
8.3 The director field on a regular grid along the diagonal (x =
y, z)
plane through the separated corner structure. (a) Stable continuous
vertical configuration, (b) stable discontinuous vertical configuration . 135
8.4 Defect line along a post edge during switching. (a) a magnified view of
(x, y) plane atz= 0.3 m cutting through the post. Darker background
colour indicates a reduction in the order parameter near the defect core.
(b) 3-D view of same post edge with a dark iso-surface for the order
parameter showing the extent of the line defect. . . . . . . . . . . . . 136
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8.5 Sums of elastic, thermotropic and surface energies for the four director
configurations using the modified thermotropic coefficients (black) and
for the 5CB material (white). The energies are normalised with respect
to the respective horizontal states. . . . . . . . . . . . . . . . . . . . . 137
8.6 The director field (x, y) plane at z= 0.3m for the (a) planar and (b)
tilted states. The planar (c) and tilted (d) states in the (x= y, z) plane
running diagonally through the modelling window. In (a) and (b),
the background color corresponds to the z-component of the director,
where positive z direction is out of the page. . . . . . . . . . . . . . . 139
8.7 The tilt angles of the stable planar and tilted states along a corner of
the modelling window as a function ofz. . . . . . . . . . . . . . . . . 140
8.8 Simulation results of planar to tilted to planar switching. . . . . . . . 142
8.9 The sum of the total thermotropic, elastic and surface anchoring ener-
gies during the planar-tilted-planar switching sequence. . . . . . . . . 143
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List of Symbols and Abbreviations
ij the Kroenecker Delta
ijk the Levi-Civita anti-symmetric tensor
n liquid crystal director
S scalar order parameter
S0 equilibrium order parameter
P biaxiality parameter
Q Q-tensor representing the nematic distribution of order
i eigenvalue of theQ-tensor, corresponding to the eigenvector i
E electric field
D electric displacement field
liquid crystal relative permittivity tensor
relative permittivity parallel to n
relative permittivity perpendicular to n
dielectric anisotropy, = P flexoelectric polarisation vector
e11 flexoelectric splay coefficient
e33 flexoelectric bend coefficient
n refractive index parallel ton
n refractive index perpendicular to n
n birefringence, n= n
n
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fd elastic distortion energy density
fth thermotropic energy density
ff electric field induced energy density
fs surface energy density
K11 splay elastic coefficient in the Oseen-Frank theory
K22 twist elastic coefficient in the Oseen-Frank theory
K33 bend elastic coefficient in the Oseen-Frank theory
L1
L6 elastic coefficients in theQ-tensor theory
T, Tc, T temperature, clearing temperature and nematic-isotropic
transition temperatures respectively
A= a(T T) temperature dependent thermotropic energy coefficient in theLandau-de gennes theory
B, C thermotropic energy coefficients in the Landau-de gennes theory
v flow field
p hydrostatic pressure
Dij symmetric flow gradient tensor
Wij antisymmetric flow gradient tensor
1, 2, 1 6 Ericksen-Leslie viscosities1, 2, 1 6 Qian-Sheng viscositiese easy axis of anchoring
v1,v2 principal axes of weak anisotropic anchoring
tilt angle
twist angle
as isotropic anchoring strength coefficient
Wi anchoring strength corresponding tovi
R anchoring anisotropy ratioR= W2/W1
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Ni finite element shape function at node i
r,s,t local tetrahedral coordinates
surface normal unit vector
q1 q5 five independent components of theQ-tensorLC Liquid Crystal
ZBD Zenithally bistable Device
PABN Post Aligned Bistable Nematic
TN Twisted Nematic
FE Finite Element
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Acknowledgements
I would like to thank the following people who have contributed to this Ph.D. and
made the past few years both enjoyable and unforgettable.
I am grateful to my supervisors Dr. Anbal Fernandez and Dr. Sally Day who
have provided me with guidance and have patiently helped me with all aspects related
to this work.
I would also like to thank Dr. Richard James, Dr. Mark Gardner and Dr. Jeroen
Beeckman with whom I had the pleasure of sharing the office with. The long hours
spent in the office never felt like a chore, and I cant imagine the outcome of this work
without their expertise and advice.
Other people who have been helpful include Mr. David Selviah, who was my
M.Phil./Ph.D. transfer thesis examiner and has been a useful resource of construc-
tive critique and many what if questions. Also, a considerable portion of this Ph.D.
deals with the modelling of bistable liquid crystal devices. Many informative conver-sations on this topic have been held with Dr. Christopher Newton from HP Labs and
Dr. Cliff Jones from ZBD Displays.
The whole Ph.D. experience would not be complete without both past and
present friends and flatmates who have contributed indirectly to this work by livening
up the non-academic moments.
Last but not least, Id like to thank my parents who have always been supportive
and made this work financially possible.
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Abstract
Liquid crystals (LC) are used in new applications of increasing complexity and smaller
dimensions. This includes complicated electrode patterns and devices incorporating
three dimensional geometric shapes, e.g. grating surfaces and colloidal dispersions.
In these cases, defects in the liquid crystal director field often play an important part
in the operation of the device. Modelling of these devices not only allows for a faster
and cheaper means of optimising the design, but sometimes also provides information
that would be difficult to obtain experimentally.
As device dimensions shrink and complex geometries are introduced, one and two
dimensional approximations become increasingly inaccurate. For this reason, a three
dimensional finite element computer model for calculating the liquid crystal electro-
hydrodynamics is programmed. The program uses the Q-tensor description allowing
for variations in the liquid crystal order and is capable of accurately modelling defects
in the director field.The aligning effect solid surfaces has on liquid crystals, known as anchoring, is
essential to the operation of nearly all LC devices. A simplifying assumption often
made in LC modelling is that of strong anchoring (the LC orientation is fixed at the
LC- solid surface interface). However, in small scale structures with high electric fields
and curved surfaces this assumption is often not accurate. A general expression that
can be used to represent various weak anchoring types in the Landau-de Gennes theory
is introduced. It is shown how experimentally measurable values can be assigned to
the coefficients of the expression.
Using theQ-tensor model incorporating the weak anchoring expression, the oper-
ation of the Post Aligned Bistable Nematic (PABN) device is modelled. Two stable
states, one of higher and the other of lower director tilt angle, are identified. Then,
the switching dynamics between these two states is simulated.
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Chapter 1
Introduction
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1.1 Motivation
Nematic liquid crystals (LC) possess anisotropic properties making them useful in a
wide range of electro-optical applications. Traditionally these include for example LC
displays and beam steering devices for optical communication. However, nematic LCs
also find new applications as solvents for micro emulsions and particle dispersions, in
e.g. bio-molecular sensors[1] or in the self-assembly of crystal structures[2].
Traditional applications can be relatively simple; some LC material sandwiched
between two glass plates with electrodes. In these cases the orientation of the liquid
crystal director varies in a continuous fashion throughout the device. However, the
drive for devices with higher resolution and faster switching implies smaller dimen-
sions and more complicated electrode shapes. In addition, applications increasingly
incorporate complex three dimensional geometries, as is the case e.g. with some
bistable devices and colloids. Frequently this results in discontinuities in the director
field orientation, known as defects or disclinations.Computer modelling often allows for faster and cheaper design and optimisation
of novel LC devices than manufacturing actual prototype devices. Furthermore, ad-
ditional information that may be difficult or impossible to gather experimentally can
be obtained.
In general, modelling of a device involves two steps: First, the orientation of the
liquid crystal is found. Then, based on the previously obtained director field the cor-
responding optical performance of the device can be calculated. Different methods
for finding the alignment of the liquid crystal exist. It is possible to consider the
interactions between each LC molecule one by one on a molecular or even atomistic
scale. However, currently this process is computationally too expensive and time
consuming for practical device modelling due to the large number of molecules in-
volved. Instead, continuum elastic theories that describe the LC material in terms of
local averages of the molecules can be used. Two continuum theories that have been
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extensively used are the so-called Oseen-Frank theory [3, 4, 5] and the Landau-de
Gennes theory [6]. The Oseen-Frank theory represents the local average orientation
of the LC molecules with the unit vector n, known as the director. The molecular
order is assumed constant and uniaxial, limiting the validity of the theory to rela-
tively large, defect free structures. When defects are present, the Landau-de Gennes
theory which allows for biaxiality and variations in the order parameter gives a better
description. In this theory, the liquid crystal is represented using the rank two, trace-
less, symmetric tensor order parameter, the Q-tensor. The Ericksen-Leslie [7, 8] and
Qian-Sheng [9] formalisms are extensions to the Oseen-Frank and Landau-de Gennes
theories respectively that include the effect of flow of the LC material.
1.2 Outline of the Work
The work described in this thesis concentrates on the static and dynamic three dimen-
sional computer modelling of the Q-tensor field in small scale LC devices containing
topological defects. Three main topics can be identified:
1.2.1 Development of a 3-D Finite Element Computer Model
The finite element method has been used to discretise the equations of the Landau-de
Gennes theory [6] and its extension, the Qian-Sheng formalism [9], in three dimen-
sions. Previously, the Qian-Sheng formalism has been used in one and two dimen-
sional modelling of LCs (e.g. [10, 11, 12]), but to my knowledge, this is the first three
dimensional finite element implementation of the theory. A Brezzi-Pitkaranta stabil-
isation scheme [13] has been used in the flow solver making it possible to use linear
elements for both the flow and pressure solutions without the commonly encountered
instability of the pressure solution [14].
A three dimensional mesh adaptation algorithm which performs local h-refinement
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in regions selected using an empirical error indicator has been implemented. This, in
conjunction with a stable non-linear Crank-Nicholson time integrator with variable
time step makes modelling of three dimensional defect dynamics feasible on a standard
PC workstation. The finite element program can be used for the modelling of both the
switching dynamics and the static equilibrium states of arbitrarily shaped domains
including multiple electrodes and non-liquid crystal regions.
1.2.2 Modelling of Weak Anchoring in the Landau-de Gennes
Theory
The operation of LC devices relies on the aligning effect of anchoring the LC to the
solid surfaces of the cells. This effect can be achieved by treating the surfaces by a
number of means. The physical/chemical processes behind the anchoring are com-
plex and not always well known. Instead, a phenomenological approach describing
the observed effect the surfaces have on the LC as an energy density is more usefulin device modelling. The assumption of a surface energy density that varies in a
Wsin2 fashion as the director at the surface deviates from the preferred easy direc-
tion by an angle has become common (known as the Rapini-Papoular assumption
[15]). However, usually the anchoring is anisotropic, the polar and azimuthal anchor-
ing strengths being unequal. For this reason, various generalisations that take into
account the difference between the two directions have been proposed in the Oseen-
Frank theory (e.g. [16, 17, 18]). The Landau-de Gennes theory has been used in the
past to explain various aspects of the fundamental physics of the solid surface-LC
interface. However, the inclusion of anisotropic weak anchoring characterised by ex-
perimentally measurable parameters into a numerical model has so far not received
much attention within this framework.
Here, a power expansion on the Q-tensor and two mutually orthogonal unit vec-
tors is used as a surface energy density. The expression is shown to simplify in the
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limit of uniaxial constant order parameter to a well known anisotropic anchoring ex-
pression in the Oseen-Frank theory. This makes it possible to assign experimentally
measurable values with a physical meaning to the coefficients of the tensor order pa-
rameter expansion. The two expressions in the Oseen-Frank and Landau-de Gennes
are compared using numerical simulations and shown to agree well. The validity of
the assumption of constant uniaxial order used in the determination of the coeffi-
cients of the expansion is examined by measuring the effective polar and azimuthal
anchoring strengths by simulating the torque balance method.
1.2.3 Modelling a Post Aligned Bistable Nematic LC Device
Bistable LC devices have two distinct stable configurations to which the director field
may relax, and in which they remain without applied holding voltages. Advantages of
bistability include lower power consumption and the possibility of passive addressing
of high resolution LC devices.
The switching dynamics and the two stable states of the Post Aligned Bistable
Nematic (PABN) LC device [19, 20] are modelled using the finite element implemen-
tation of the Landau-de Gennes theory. In the past, the Oseen-Frank theory has
been used to find the two stable director configurations [21], but the dynamics of the
switching has not been reported.
The two stable states are found to be separated by a pair of line defects extending
along the edges of the post. These defect lines act as energy barriers separating the
two stable states. In order to switch between the two topologically distinct states,
energy must be provided by externally applied electric fields.
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1.3 Achievements
The work described in this thesis has resulted in the following publications, confer-
ences and prizes:
Publications
R. James, E. Willman, F. A. Fernandez and S. E. Day, Finite-Element Mod-elling of Liquid Crystal Hydrodynamics with a Variable Degree of Order, IEEE
Transactions on Electron Devices, 53, no. 7, (2006).
E. Willman, F. A. Fernandez, R. James and S. E. Day, Computer Modellingof Weak Anisotropic Anchoring of Nematic Liquid Crystals in the Landau-de
Gennes theory, IEEE Transactions on Electron Devices, 54, pp. 2630-2637,
(2007).
E. Willman, F. A. Fernandez, R. James and S. E. Day, Switching Dynamics of
a Post Aligned Bistable Nematic Liquid Crystal Device, IEEE J. Disp. Tech.,
4, pp. 276-281 (2008).
R. James, E. Willman, F. A. Fernandez and S. E Day, Computer Modelingof Liquid Crystal Hydrodynamics, IEEE Transactions on Magnetics, 44, pp.
814-817, (2008).
J. Beekman, F. A. Fernandez, R. James, E. Willman and K. Neyts, FiniteElement Analysis of Liquid Crystal Optical Waveguides, 12th International
Topical Meeting on Optics of Liquid Crystals, Puebla, Mexico. (2007)
S. E. Day, E. Willman, R. James and F. A. Fernandez, P-67.4: Defect Loops inthe Zenithally Bistable Device, Society for Information Display International
Symposium Digest of Technical Papers, 39, pp. 1034-1039, (2008)
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Conferences
2D and 3D Modelling of Liquid Crystal Hydrodynamics Including Order Pa-rameter Changes, International Workshop on Liquid Crystals for Photonics,
April 2628 2006, Ghent (Belgium), Oral Presentation.
Three Dimensional Modelling of Nematic Liquid Crystal Devices,Flexoelectricityin Liquid Crystals, September 19 2006, Oxford, Poster Presentation.
Prizes
Winner of SHARP-SID Best Student award 2008.
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Chapter 2
Liquid Crystals
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2.1 Introduction
Liquid Crystal (LC) is a general term used for a type of mesophase of matter that
exists between the solid and liquid phases. LC materials consist typically of organic
molecules that are free to move about and flow like a liquid, while retaining a degree
of orientational and sometimes positional order [6, 22, 23].
Different LC phases can be classified according to the distribution of molecular
order. LC materials exist in different phases depending on the temperature or concen-
tration of a solvent. When the phase depends on the temperature, the LC material
is said to be thermotropic, and when it depends on the the concentration of a solvent
the LC is said to be lyotropic.
Lyotropic LC materials consist of amphiphilic molecules with a hydrophobic tail
and a hydrophilic head [22]. When mixed with a polar solvent (e.g. water), the
molecules tend to arrange themselves so that the tails group together, while the
hydrophilic heads are attracted to the solvent. Soaps are an example of lyotropicliquid crystals.
Thermotropic LC materials consist usually of rigid, anisotropically shaped molecules.
The molecules are generally shaped either like rods (calamitic) or disks (discotic).
Variations in these are possible, e.gwedge shaped or bent-core mesogens have been
observed [24].
Currently, most electro optic LC devices make use of calamitic thermotropic ma-
terials in the nematic phase. For this reason, throughout the rest of this thesis,
it is understood that referring to liquid crystals means thermotropic calamitic LC
materials, unless otherwise stated.
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Figure 2.1: The molecular configurations of the isotropic, nematic and smectic A andC phases.
2.2 Liquid Crystal Phases
Thermotropic LC materials undergo phase transitions as the temperature is varied.
At high temperatures the LC material is in the isotropic phase, where the molecules
are randomly distributed. No long range positional or orientational order exists. As
the temperature is lowered, at some critical temperature a phase transition occurs.
Depending on the exact compound, the LC material becomes either nematic or smec-
tic.
In the nematic phase the LC molecules are free to move (no positional order), but
an average direction along which the molecules tend to orient their long axes can be
observed (long range orientational order exists). This is known as the director and
represented by the unit vector n.
In the smectic phase both positional and orientational order can be identified: TheLC molecules tend to arrange themselves in layers of identical orientation. Depending
on the orientation of the molecules within the layers, the smectic phases can further
be classified into sub categories A, B, C, . . .
Additionally, cholesteric or chiral variants of the nematic and smectic phases exist.
The chiral nematic phase exhibits a continuous twisting of the molecules perpendicu-
lar to the long axis of the molecules. In the chiral smectic phases, a finite twist angle
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from one layer to another can be observed. The distance over which the director
undergoes a full 360 rotation is known as the chiral pitch length.
2.3 Liquid Crystal Order
Typically when below the nematic-isotropic transition temperature, nematic LC ma-
terials exist in a uniaxial configuration in the bulk. That is, a single axis of symmetry
exists. However, biaxial order, when more than one axis of symmetry exist, may
occur e.g. near confining surfaces or in the vicinity of defects.
2.3.1 Uniaxial Order
The uniaxial nematic phase can be characterised by the degree of orientational order,
S, and the macroscopic average direction of the constituent molecules, n. The scalar
order parameter Scan be defined as a measure of the degree of orientational order.
In a small volume containing Nmolecules, with the orientations of their long axes
denoted by the unit vectorsu, the scalar order parameter can be defined as the second
order Legendre polynomial:
S = 1
2 3cos2 1
= 1
2N
Ni=1
3 (n ui)2 1
, (2.1)
where is the angle between each molecule and the nematic director n (see Fig.
2.2). In the isotropic phase, where no order exists, S = 0. In the nematic phase
S is typically within the range from 0.4 to 0.7, depending on the temperature. A
negative scalar order parameter is also possible. This corresponds to the molecules
lying randomly oriented in a plane perpendicular to n.
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Figure 2.2: The nematic director nand the scalar order parameter S.
Many experimentally measurable parameters of a LC material are related to the
value of the order parameter, and it can be determined e.g. by means of NMR
spectroscopy, Raman scattering, X-ray scattering or birefringence studies [23, 22, 6].
2.3.2 Biaxial Order and theQ-Tensor
In the case of a biaxial distribution of the LC molecules, more than one order param-
eter is needed. It is then more convenient to characterise the LC material in terms of
a tensor order parameter called the Q-tensor.
The Q-tensor is a symmetric traceless rank 2 tensor (a three by three matrix). Q
has 9 components, but only five of them are independent. This gives three spatial
degrees of freedom and two orientational degrees of freedom. The three eigenvalues
1, 2and3ofQare a measure of the nematic order in the three orthogonal directions
defined by the corresponding eigenvectors n, k and l.
The Q-tensor can be written in terms of the eigenvalues and eigenvectors as:
Q= 1(n n) +2(k k) +3(l l). (2.2)
However, since only two of the eigenvalues are independent, the definition S=1
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and P = 12
(2 3) can be made. Then the Q-tensor can be written in terms of thescalar order parameter S, the biaxiality parameter Pand the three eigenvectorsn, k
and las:
Qij =S
2(3ninj ij) +P(kikj lilj). (2.3)
When the eigenvectors coincide with the x,y andzaxes of the frame of reference,
the eigenvalues appear along the diagonal of the Q-tensor:
Q=
1 0 0
0 2 0
0 0 3
=
S 0 0
0 S2
+P 0
0 0 S2 P
(2.4)
A Visual Representation of the Q-Tensor
Figure 2.3 is a visual representation of the different distributions of nematic order
that can be described using theQ-tensor description. The pictured cuboids or boxes
can be imagined to contain rigid rods representing LC molecules, and to be shaken
in order to simulate the effect of thermal vibrations. The relative lengths of the
sides of the boxes then affect the average orientations of the contained rods and are
proportional to the eigenvalues of the Q-tensor describing the corresponding order
distribution within the box (with an additional positive factor to avoid negative side
lengths):
a) 1 = 2 = 3 = 0. The three eigenvalues are equal (and zero due to the
tracelessness of the Q-tensor) in the disordered isotropic phase. In this case,
the sides of the box are of equal lengths so that the container does not impose
a preferred direction on the rods.
b) 2 = 3 =1
21. The dominant eigenvalue is positive while the other two
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(a) (b)
(c) (d)
Figure 2.3: (a) Isotropic Q-tensor, (b) uniaxial Q-tensor, (c) biaxial Q-tensor, (d)uniaxial Q-tensor, but with negative scalar order parameter.
are equal and negative, resulting in the uniaxial configuration S = 1 and
P =2 3= 0. The rods are most likely to be oriented in the direction alongthe longest side1, with smaller but equal probabilities of being oriented in the
directions corresponding to 2 and 3.
c) 1= 2= 3. In the biaxial configuration the three eigenvalues are different,so that in this case S= 1 and P = 3 2 > 0. The lengths of the sides ofthe box are then related by 1> 3> 2.
d) 1 < 0, 2 = 3 =121. The dominant eigenvalue is negative while thetwo others are positive and equal, resulting in the uniaxial configuration with
S=1 1.
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Figure 2.4: Director profiles for defects of whole m= 1 and m= 12
strengths.
2.5 Dielectric Properties and Flexoelectric
Polarisation
The anisotropy in shape of the LC molecules affects its dielectric permittivity and
magnetic susceptibilities. The dielectric permittivity when measured parallel to the
long axis of the molecules, , is different from that measured perpendicular to the
same axis, . The dielectric anisotropy, defined as = , may be eitherpositive or negative depending on the specific LC compound. The permittivity may
then be expressed as a tensor in terms of the director:
ij =ij+ ninj . (2.6)
An approximation to (2.6) written in terms of the Q-tensor is:
ij =ij+
2
3S0Qij+
1
3ij
, (2.7)
where S0 is the equilibrium order parameter of the LC material. The magnetic sus-
ceptibility tensor may be defined in a similar fashion.
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2.6 Optical Properties of LCs
The dielectric anisotropy of LCs discussed previously extends to the optical frequen-
cies, resulting in an anisotropic refractive index. Two indexes of refraction and their
differences are defined,n, n and n= n n (Often these are referred to as theordinary and extraordinary indexes of refraction respectively). The subscripts have
the same meaning as described in the case for the dielectric anisotropy.
The speed of an electromagnetic wave propagating in an isotropic medium is
v =c/n, where c is the speed of light in vacuum and n is the refractive index of the
material. The electric field of light propagating through a sample of LC material can
be decomposed into two orthogonal components. If the orientation of the director
is such that the two components of the electric field experience different values of
the refractive index, the two components will propagate at different velocities. This
results in a change of the polarisation state of the propagating light.
Most LC display devices consist of a layer of LC material whose orientation maybe controlled by some configuration of transparent electrodes sandwiched between a
pair of polarisers. Incoming unpolarised light, typically from a back light, is polarised
by the first polariser. The linearly polarised light then passes through the LC layer
which may change the orientation of the polarisation of the light, depending on the
orientation of the director. Finally the light is either transmitted or blocked by the
second polariser, so that the device may appear bright or dark. The electrodes are
used for creating electric fields which align the LC director in such a way that the
polarisation of the light is parallel to the last polariser for the bright and perpendicular
to it for the dark state.
Different approaches for calculating the optical output of an LC device exist.
The Jones [32] and Berreman [33] methods are two commonly used methods. These
methods are valid when the lateral variation in the director field is small over distances
comparable to the wavelength of the propagating light. When lateral variations in
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the director field are rapid, diffractive effects not taken into account by the Jones or
the Berreman approaches become significant. In that case a grating method [34, 35]
which takes into account lateral variations in the refractive index is more accurate.
2.6.1 Jones Calculus
Jones calculus (or Jones method) [32] is probably the simplest method used in cal-
culating LC optics. Only changes in the light polarisation are considered, but not
reflections or diffractive effects.The linearly polarised light (propagating in the z-direction) is described by a Jones
vector J = [Ex, Ey]T, which represents the polarisation state of the wavefront. The
medium through which the light propagates, is considered to consist ofk layers. Each
layer is described by a 2 2 Jones matrix, M. The combined effect of the layers onthe polarisation state of the propagating wavefront is then described by a series of
multiplications:
Jk=MkMk1 . . . M1J0, (2.10)
where J0 and Jk are the incoming and outgoing Jones vectors.
Each Jones matrix may represent either an optical element (e.g. a polariser or a
retarder) or a slice of the LC material. In general, M may be written as:
M= S()NS(), (2.11)
where S() is a rotation matrix, with defined in the (x, y) plane:
S=
cos() sin()sin() cos()
, (2.12)
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and N describes the retardation independent of its orientation:
N=
exp(ix) 00 exp(iy)
. (2.13)
xy = is the relative phase difference introduced to the polarisation of theelectromagnetic field propagating in thez-direction as it passes through the medium.
In the case of a slice of LC material with director tilt and twist angles and
respectively, the two phase angles are calculated from the refractive indexes in the xand y directions and the thickness d of the layer as:
x = nx2
d,
y = n2
d, (2.14)
where is the wavelength of the propagating light. The refractive index in the x-
direction depends on , and is obtained using [36]:
1
n2x=
sin2()
n2+
cos2()
n2(2.15)
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Chapter 3
Theoretical Framework
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3.1 Introduction
Liquid crystal device modelling is typically a two step process: First, the LC director
field orientation within the device is estimated. Then, the corresponding optical
performance can be calculated. In this chapter, a theoretical background for the
method used throughout the rest of this thesis for calculating LC director fields is
introduced.
For completeness, this chapter starts with a brief review of some well known
theories that can be used to describe LC physics, but are in general not suitable for
practical device modelling. In section 3.2, statistical mean field theories explaining
LC phase changes are introduced. Then, in section 3.3 methods and applications of
molecular simulations are outlined.
A good estimate of the orientation of the LC director field over length scales
comparable to LC device dimensions can be obtained using arguments based on con-
tinuum elasticity. This is the approach taken here, and the majority of this chapter,starting from section 3.4, is devoted to explaining the underlying theory.
3.2 Mean Field Theories
Mean field theories attempt to explain what happens to a large number of molecules
by making the assumption that on average all the molecular interactions are equal.
This means that the macroscopic properties of many molecules can be deduced from
the microscopic properties of only a few. Two such theories are the Onsager hard-rod
theory [37] and the Maier-Saupe theory [38, 39, 40]. Both of these theories describe
the nematic-isotropic phase transition.
In the Onsager theory, the constituent molecules are considered to be hard rods,
whose lengths are much greater than their widths. The basic assumption is that of a
balance of positional and orientational entropy of the rods that cannot interpenetrate
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each other. An interaction potential for a pair of rods is written in terms of the
relative positions and orientations of them and the concentration of rods. The solution
of the Onsager theory is independent of the temperature and predicts a first order
isotropic to nematic phase transition occurring when the concentration of molecules
is sufficiently high, it is an early proof that shape anisotropy alone is sufficient to
induce nematic order.
In the Maier-Saupe theory, an intermolecular attractive contribution due to van
der Waals force is additionally taken into account. Furthermore, the probability of
finding a molecule being oriented at a given angle from the director can be written
as a function involving the temperature of the system, making it possible to predict
a first order thermotropic nematic to isotropic transition.
3.3 Molecular Simulations
In contrast to mean field theories, molecular theories consider a large number of
individual molecules or particles (usually some simplified representation is used).
Reviews of the method and its many variations can be found e.g. in [41, 42, 43].
Due to the involved computational cost the number of simulated particles is neces-
sarily limited to far less than what is required in full-scale device modelling. However,
molecular simulations have been used to explain links between molecular and observed
bulk macroscopic properties of LC materials. For example, the values of elastic con-stants, viscous parameters and flexoelectric coefficients can be estimated in this way
[44, 45, 46, 30].
The core of a molecular simulation is an interaction potential which represents
the pairwise potential energies between each of the the constituent molecules. Dif-
ferent assumption on the form of the interaction potential have been made in the
past. For example, both hard and soft particle interaction potentials are possible.
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sample of LC material can be written as a function of the director or the Q-tensor.
The LC material then prefers to exist in a state (director orientation, order parameter
distribution) that minimises the total energy within that region. The energy density
consist of a number of terms, each accounting for some physical property of the
material and its interaction with external effects. The total free energy within a
sample with boundaries is given by:
F= fd+fth ff d + fs d, (3.1)where fd is the elastic distortion energy density, fth is the thermotropic (or Landau)
energy density,ff is an external field induced energy density andfsis a surface energy
density appearing at interfaces between the LC material and its surroundings. Each
of these terms will be described in more detail in the following sections.
3.4.1 Liquid Crystal Elasticity
A distortion energy density, fd is written as a function of the director and its spa-
tial derivatives. For nematics, this term is minimised when the director field is in
an undistorted configuration and for chiral LCs the minimum occurs when a twist
deformation with a pitch length pis present.
The distortion energy density introduced by Oseen [3], Frank [4] and Zocher [5]
identifies three possible distortion types of the bulk nematic director field. These are
the so-called splay, twistand benddistortions, depicted in figure (3.1).
From Figs. (3.1) a, b and c, showing the vector presentation of the possible
distortions in a director field with n = [0, 0, 1], it is easy to verify that the vector
expressions satisfying the distortions are given by:
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Splay = nx,x+ny,y = n,Twist = nx,y ny,x = n n,Bend = nx,z+ny,z = n n,
(3.2)
where a comma in the subscript indicates differentiation with respect to the direction
following it. The bulk distortion energy density can then be written as the weighted
sum of the terms in (3.2) squared:
fd =1
2K11( n)2 +1
2K22(n n)2 +1
2K33(n n)2, (3.3)
where K11,K22 and K33 are elastic constants assigned to the three distortion types.
(a) (b) (c)
(d) (e) (f)
Figure 3.1: Splay, bend and twist deformations.
A more rigorous way of obtaining the general distortion energy of a nematic LC
material is to write the elastic energy density as a power expansion in all the possible
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Typically values for the elastic constants lie in the range 5-15 pN and usually the
relation K33 > K11
K22 holds (see e.g. [6] p. 103-105). It is common to make a
single elastic coefficient assumption K= K11 = K22 = K33 to simplify calculations.
In this case, after some manipulations of (3.13) (see e.g. p. 23 in [51] for details), the
elastic energy density reduces to:
fd 12
K|n|2 =12
Kni,jni,j. (3.14)
The elastic energy density can also be expressed in terms of the Q-tensor and its
spatial derivatives as:
fd =1
2L1Qij,kQij,k+
1
2L2Qij,jQik,k
+1
2L3Qik,jQij,k+
1
2L4likQljQij,k
+1
2L6QlkQij,lQij,k, (3.15)
where Li are elastic coefficients. The relation between the elastic coefficients in equa-
tions (3.13) and (3.15) can be found by replacing the Q-tensor in (3.15) by its uniax-
ial definition S02
(3ninj ij) and comparing the expressions. This has been done in[52, 53], resulting in:
L1= 1
27S20
(K33
K11+ 3K22)
L2= 2
9S20(K11 K22 K24)
L3= 2
9S20K24 (3.16)
L4= 8
p09S20K22
L6= 2
27S30(K33 K11)
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The single elastic coefficients simplification in terms ofQ is
fd 12
L1Qij,kQij,k.
3.4.2 Thermotropic Energy
A thermotropic energy density, fth, is used to describe the LC order variations. The
bulk, or thermotropic, energy density fth is a power expansion on the tensor order
parameter:
fb=1
2A(T)Tr(Q2) +
1
3B(T)Tr(Q3) +
1
4C(T)Tr(Q2)2 +O(Q5), (3.17)
whereA,B and Care temperature dependent material parameters. Expression (3.17)
describes the first order nematic-isotropic phase transition with respect to tempera-
ture T. In practice, B and Care assumed independent of temperature and only the
lowest order material parameter A is taken as A(T) = a(T T) [6]. The values ofa, B and Ccan be determined e.g. by fitting expression (3.17) with experimentally
obtained data of order parameter variation with respect to temperature [22] (p. 250).
Substituting Q in terms of the scalar order parameter Sand the biaxiality param-
eter Pas defined in (2.3) into (3.17) gives:
fth =
3
4AS
2
+
1
4BS
3
+
9
16 CS
4
+ (A BS+3
2CS
2
)P
2
+CP
4
. (3.18)
The value ofSthat minimises (3.18) at any given temperature is known as the
equilibrium order parameterS0, and is given by S0= B +
B2 24AC/(6C). Thebulk energy, as written here, always favours uniaxiality, i.e. P0 = 0. Higher order
terms would be needed in order to describe a nematic LC with bulk biaxiality P0= 0[54] or [6] p. 82-84. Such materials have recently been observed experimentally in
[24].
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0.4 0.2 0 0.2 0.4 0.6 0.8
0
S
f b
T>Tc
T=Tc
T=T*
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vector as defined in (2.8) and (2.9). The permittivity tensor can be defined in terms
of the director or the Q-tensor as in (2.6) and (2.7) respectively. A similar expression
can be written for magnetic fields, where the magnetic susceptibility tensor replaces
the dielectric tensor .
In addition to interactions between external electric or magnetic fields, solid sur-
faces in contact with the LC material have an aligning effect on the director field.
This effect, known as anchoring, can be either strongor weak. In the case of strong
anchoring the surface energy density, fs, is assumed infinite and the director or Q-
tensor is fixed at the surface. When the anchoring is weak, the surface energy density
is some finite function involving the director or the Q-tensor. The effect of solid sur-
faces on the LC can be complex and the surface energy density is described in more
detail in chapter 4.
3.5 Static Equilibrium Q-Tensor Fields
In the continuum elastic theory explained in section 3.4, a free energy density f is
written in terms of the Q-tensor and its spatial derivatives f = f(Qij, Qij,k). LC
configurations resulting in minima in the total energy for the complete region of
interest are stable. These are the states to which the LC director field and order
parameter distribution relaxes to in the limit of time (here, tens of milliseconds
is enough in most cases of interest).The process of finding these states is a task of variational calculus, seee.g. [55, 31].
Stable LC configurations correspond to nulls of the first variation of the total energy
with respect to the Q-tensor:
F=
f
QijQij+
f
Qij,kkQij
d = 0. (3.20)
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Integrating the second term by parts:
F=
( f
Qij k f
Qij,k)Qij d +
kf
Qij,kQij d = 0, (3.21)
where is a unit vector normal to the bounding surface . Since Qij is an arbitrary
variation, in order for (3.21) to be true, the following must be satisfied:
f
Qij k f
Qij,k= 0 in . (3.22)
fQij,k
k = 0 on . (3.23)
These are the Euler-Lagrange equations for the problem. Analytic solutions that
satisfy the Euler-Lagrange equations are usually only possible in simplified cases,
whereas in most cases numerical methods must be used.
3.6 Q-Tensor Hydrodynamics
In the previous sections, only the orientation and order distribution of the LC material
has been considered. However, since LCs are fluids, they flow and this needs to
be taken into account for a more comprehensive description of the material. It is
known that director re-orientation induces flow and similarly flow causes director re-
orientation. An example of this is the observed optical bounce [56] due to backflow
in a twisted nematic cell after a holding voltage is removed.
Probably the most successful theory describing the liquid crystal hydrodynamics is
that by Ericksen and Leslie [7, 8]. This theory, commonly known as the Ericksen-Leslie
(EL) theory describes the viscous behaviour of liquid crystals with six phenomeno-
logical coefficients (known as Leslie viscosities) 1 6, but taking into account theParodi relation 2+ 3 = 6 5, only five of these are independent [57]. In gen-
eral, the Leslie coefficients are not directly experimentally measurable, but can be
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obtained from the four shear viscosities 1, 2, 3 and 12 and rotational viscosity 1
[58, 59]. Alternatively these can be estimated by means of molecular simulations or
by interpolating from known viscous coefficients for other materials using knowledge
of other material properties [60, 61].
The EL theory uses the vector description for the LC orientation, and does not
take into account order parameter variations making it unsuitable for describing cases
where topological defects are present. Other dynamic descriptions that do take into
account variations in the LC order have been proposed in the past in e.g. the Beris-
Edwards [62] and the Qian-Sheng formulations [9]. Both of these approaches yield
qualitatively similar results [12, 63], but the Qian-Sheng equations reduce in the limit
of constant uniaxial order to the EL theory allowing for direct mapping of viscous
coefficients between the two theories. Because of this, the Qian-Sheng equations are
chosen for this work.
The hydrodynamic equations of LC materials can be derived by starting from the
conservation of linear and angular momentum as is done with the EL theory and the
original derivation of the Qian-Sheng formalism. However, more recently in [64, 65]
it is argued that these assumptions are not strictly valid when the LC is described
using the Q-tensor with variable order. Instead, a more general approach starting
from principles of conservation of energy (but reducing to the same final equations)
is proposed. Following the approach presented in [64, 65], the theoretical background
of the Qian-Sheng equations is outlined in the following sections 3.6.13.6.6 .
3.6.1 Conservation of Energy
The basic idea is to balance the rate of change of energy against frictional losses in
the form of a Rayleigh dissipation function [66]:
W+R = 0, (3.24)
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where W is the time rate of change of energy (power) and R is the dissipationaccounting for frictional losses. In (3.24), variations with respect to the rate of change
of the Q-tensor, Q, and the flow field, v, are taken ensuring minimum restrained
dissipation. The total power of the system is the sum of the rate of change of the
kinetic,T, and potential,F, energy of the system:
W= T + F, (3.25)
The equations of motion need to be frame invariant. This can be achieved by
writing the dissipation in terms of the tensors Q, D and N. D and N are the
symmetric velocity gradient tensor and the co-rotational time derivative respectively,
and are related to the total flow gradient tensor vi,j as follows:
vi,j =Dij+ Wij, (3.26)
whereDij = 1
2(vi,j+vj,i) is the symmetric andWij =
1
2(vi,jvj,i) is the anti-symmetric
(also known as the vorticity tensor) part of flow gradient tensor. N is a measure of
the rotational rate of change of the Q-tensor with respect to the background flow
field:
Nij = Qij+ QikWkj WikQkj , (3.27)
where Qis the total or material time derivative measuring the rate of change ofQ in
the flow field v, and is defined in the usual manner as:
Qij =
tQij+ vkQij,k. (3.28)
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3.6.2 Frictional Forces
The dissipation functionRrepresents the effect of friction and can in the most general
form be written as a power expansion of the tensors Q, D and N. Then, the total
dissipation within a region is given by:
R =
R(Q,N,D) d. (3.29)
The variation of the dissipation with respect to Q and v then takes the form:
R =
R
QijQij+
R
vi,jjvi
d. (3.30)
Integrating the second term by parts gives:
R =
R
QijQij j( R
vi,j)vi
d
+
Rvi,jjvi d. (3.31)
The derivatives in the volume integral are then evaluated using the chain rule of
differentiation:
R
Qij=
R
Nij, (3.32)
and
R
vi,j=
R
Nkl
NklWab
Wabvi,j
+ R
Dkl
Dklvi,j
. (3.33)
Taking into account symmetries of the involved tensors, equation (3.31) simplifies to:
R=
R
Nij
Qij+ j Qik RNkj
R
Nik
Qkj vi d. (3.34)
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The surface integral in (3.31) reduce to zero, since v= 0 along when boundary
conditions for v are enforced.
3.6.3 The Power
The total power of a sample of LC material equals the sum of the rate of change of
the kinetic and potential energies:
W=
T +
F, (3.35)
where the kinetic energy isT =
1
2vivid and the potential energyF is the free
energy of the LC material as defined in equation (3.1). InT, is the the density ofthe LC material.
The rate of change of the kinetic energy, after introducing the hydrostatic pressure
p as a Lagrange multiplier to enforce incompressibility, vi,i = 0, of the LC material
and integrating by parts is:
T =
(vivi+j(pij)vi) d
vipij j d. (3.36)
The time rate of change of potential energy is given by:
F=
f
Qij
Qij+ f
Qij,k
dQij,k
dt d. (3.37)
Using the identity ddt
Qij,k = Qij,k Qij,lvl,k, and integrating by parts in (3.37) gives:
F =
f
Qij k f
Qij,k
Qij+k
Qij,l
f
Qij,k
vl
d
+
k
f
Qij,kQij,k k f
Qij,kQij,lvl
d. (3.38)
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relation between the EL-viscosities and the coefficients can be determined by replac-
ing Q by its uniaxial definition Qij
= 12
S0(3ninj
ij
) and comparing the resulting
terms with the dissipation function in the EL theory [65].
3.6.6 Explicit Expressions for the LC-Hydrodynamics
After performing the steps outlined above and rearranging terms in the viscous ten-
sor, the equations for the hydrodynamics can be written explicitly. The Qian-Sheng
formalism governing theQ-tensor evolution is given by:
1Nij = 12
2Dij fQij
+kf
(Qij,k), (3.47)
and the flow of the LC material is governed by:
vi=jji, (3.48)
where is as defined in (3.45), with the viscous stress tensor written as:
vij = 1QijQklDkl+4Dij+ 5QikDkj+ 6QjkDki
+1
22Nij 1QikNkj +1QjkNki. (3.49)
Additionally the incompressibility of the LC material should satisfy:
vi,i= 0. (3.50)
In expressions (3.47) and (3.49), 1, 4, 5, 6, 1 and 2 are viscous coefficients
consisting of linear combinations of1 5. The values of the coefficientsand are
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related to the six viscous coefficients 1 to 6 in the EL theory by [9]:
1 = 29S20
(3 2)
2 = 2
3S0(6 5)
1 = 4
9S201 (3.51)
4 = 1
2S0(5+6) +4
5 = 2
3S05
6 = 23S0
6
In cases when the effect of flow is not considered, (3.48) can be ignored and the
Q-tensor evolution (3.47) simplifies to:
1
tQij = f
Qij+k
f
Qij,k(3.52)
3.7 Discussion and Conclusions
The theoretical background for the equations used in this work for describing the
physics and modelling the operation of LC devices has been presented.
A Landau-de Gennes free energy density taking into account elastic deformations
and allowing for order parameter variations and biaxiality induced by externally ap-
plied electric fields and/or aligning solid surfaces is used. The elastic energy contri-
bution reduces in the limit of constant uniaxial order to the well known Oseen-Frank
elastic description of nematics with three independent elastic coefficients, allowing
for realistic treatment of the LC elasticity. Similarly, the thermotropic energy con-
tribution, the essence of the Landau-de Gennes approach, allows for localised order
variations making a continuum description of defects possible.
In regions of the LC material where order variations are allowed but are not
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Chapter 4
Modelling of the Liquid
CrystalSolid Surface Interface
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4.1 Introduction
Solid surfaces in contact with a LC break the symmetry of the nematic phase, resulting
in a non-arbitrary orientation of the director field. This effect of interfaces imposing
an orientation on the director is commonly known as anchoring.
The operation of virtually all LC devices relies in some way on anchoring. In
traditional display devices the solid surfaces are typically the glass plates between
which the LC material is sandwiched. Other possible solid surface-LC interfaces
include for example spacers used to keep the cell thickness constant throughout the
device or colloidal particles immersed in the LC material for various applications,e.g.
[1, 2].
The simplest way of including the effect of anchoring into a continuum model
is by fixing the director or the Q-tensor at the interface. This is known as strong
anchoring. Alternatively, the aligning effect can be included by introducing a surface
anchoring energy density which is minimised when the director n is parallel to theanchoring directione(also known as the easy direction). In this case, known as weak
anchoring, nor Q may vary at the surface with an associated change in energy.
Sometimes weak anchoring gives a more realistic description of the aligning effect
than strong anchoring and is an important feature to be included in an LC device
model. This is especially true in the case of very small structures where torques on
the director due to high electric fields or elastic forces may become comparable to
even the high (but in reality finite) anchoring energies.
In the Oseen-Frank theory, it has become standard practise to include the effect
of weak anchoring by making the well known Rapini-Papoular (RP) assumption [15]
or some generalisation of it (see section 4.3.1). In the Landau-de Gennes theory,
however, although the fundamental physics of the surface interface has been examined,
the anchoring phenomenon has received less attention from the LC device modelling
point of view.
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The purpose of this chapter is to study the modelling of weak anchoring in LC de-
vices using the Landau-de Gennes theory. First, in section 4.2 various anchoring types
are introduced, physical reasons for the aligning effect of various solid surfaces are
given and methods for measuring the anchoring strength are presented. In sections
4.3.1 and 4.3.2, surface energy densities in the Oseen-Frank and Landau-de Gennes
theories respectively are reviewed. Then, starting from section 4.4 new work is pre-
sented. A general power expansion on the Q-tensor and two unit vectors describing
the local geometry of the surface in contact with the LC material is proposed to rep-
resent the surface energy density. It is shown that in the limit of constant uniaxial
order, the proposed expression reduces to a well known anisotropic generalisation of
the RP expression by Zhao, Wu and Iwamoto [17, 18], developed in the Oseen-Frank
framework. In this limit, experimentally measurable values with a physical meaning
in the Oseen-Frank theory can be scaled and assigned to the scalar coefficients of
the Q-tensor expansion. The validity of this assumption is examined by comparing
results of numerical experiments using both theories.
4.2 Background
4.2.1 Classification of Different Anchoring Types
Different anchoring types can be classified depending on the orientation of the easy
direction with respect to the aligning surface. When the easy direction is in the
plane of the surface the anchoring is said to be planar. Planar anchoring can be
homogeneous or degenerate. In the case of homogeneous planar anchoring only a
single easy direction exists. In the case of degenerate planar anchoring all directions
in the plane are equal and the director field may rotate in the plane. It is also
possible that the easy direction is not in the plane of the surface. That is, a pre-tilt
exists. In the degenerate case, the result is conical degenerate anchoring. When the
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along the grooves in order to minimise the elastic distortion energy, resulting in pla-
nar homogeneous anchoring. Based on this argument, Berreman [78] has proposed
an expression relating the width and separation of the grooves and the bulk elastic
constants to the anchoring strength . For example, rubbing of polyimide or oblique
evaporation of inorganic compounds produce grooved or rough surfaces favouring pla-
nar alignment [6, 79]. It is also argued that the rubbing process orients the polyimide
chains in one direction, along which the LC molecules then align.
If the surface is covered with a film of a surfactant consisting of aliphatic chains
oriented perpendicular to the surface, the LC molecules at the interface may partially
penetrate the chains and adopt their orientation. This method can be used to produce
surfaces with homeotropic anchoring [80, 81].
It is also suggestede.gin [82, 83, 31] that surface electric fields due to the presence
of ions or the so-called ordoelectric polarisation can have an effect on the strength of
the anchoring and the orientation of the easy direction e.
Also, non-structured interfaces (not necessarily with a solid surface) have an align-
ing effect on the LC (see e.g. [74] and references therein). In this case, changes in
the properties of the LC material in a thin region (in the order of nanometres) near
the surface are responsible for the alignment. This includes changes in the density of
the LC, gradients in the order parameter and monolayers of smectic phases.
By geometric arguments, a non-structured or isotropic solid surface should pro-
duce planar degenerate anchoring with zero azimuthal anchoring strength. It has been
shown long ago that this is not necessarily the case [84]. Two different phenomena
have been reported to be responsible for a finite azimuthal anchoring strength in LC
cells with untreated surfaces: These are the flow [77, 81] and the memory [85, 86, 87]
alignment.
The flow alignment occurs when a cell is filled with an LC material in the nematic
phase. In this case the alignment tends to be in the filling direction. The flow
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nitude. From this information it is then possible to estimate the anchoring strength
by fitting parameters to a model. The torque can be generated either by applying
external electric/magnetic fields (field on techniques) [88, 89] or by a distortion in
the director field due to the chosen geometry of the test cell used in the measurement
(field off techniques).
Perhaps the simplest (field off) technique is the torque balance method [90, 91]
which relies on calculating the elastic torque energy in a twisted nematic cell of
thickness d and with a known total twist angle t between the anchoring directions
on both surfaces. The distortion in the bulk produces an elastic torque that causes
the director at the surfaces to deviate from the easy directions on both surfaces by
angles which can be found experimentally e.g. by measuring the retardation of
polarised light transmitted through the cell. In the case of zero tilt, the twist angle
varies linearly through the cell, see figure 4.1. The total energy is then a sum of the
bulk distortion energy and the anchoring energies:
Ftot = Fd+ 2Fs. (4.1)
The total bulk twist distortion energyFd in a cell with = t 2radians of twistis:
Fd=K22
2d
2. (4.2)
The surface energy Fs at each interface is taken as the RP anchoring energy:
Fs=Wsin2 . (4.3)
In (4.2) and (4.3) W and K22 are the azimuthal anchoring energy strength and the
twist elastic constant respectively. By minimising equation (4.1), with respect to
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(and using 2 cos sin= sin 2), a balance between the two opposing torques
and the resulting value ofW
can be found:
W = K22
d sin2, (4.4)
Figure 4.1: Twist angle in a cell of thickness d. Dashed line, strong anchoring. Solidline, weak anchoring.
4.3 Review of Currently Used Weak Anchoring
Expressions
4.3.1 Weak Anchoring in Oseen-Frank Theory
Probably the first and best known expression describing the weak anchoring effect in
the Oseen-Frank theory is the Rapini-Papoular (RP) expression [15]. This assumes
that the anchoring energy density increases in a sin2 fashion as the director deviates
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from the easy direction:
FRP =Wsin2(), (4.5)
where W is a scalar value known as the anchoring strength, and is the angle of
departure of the director n from the easy direction e. Alternatively, this can be
written as:
FRP = W(n e)2
. (4.6)
One weakness of (4.5), is its inability to distinguish between different directions
of angular departures from e. This means that the difference between polar and
azimuthal anchoring strengths cannot be taken into account. Furthermore, it has been
suggested that higher order terms (e.g. terms in sin4 ) should be taken into account
when is large [92]. Despite this, the RP anchoring is a widely used approximation
and often used as a reference to which other anchoring representations are compared.
Various generalisations to 4.5 exist. One that differentiates between polar and
azimuthal anchoring strengths is (e.g. [16]):
FRPgen=A1sin2( e) +A2sin2( e), (4.7)
where A1 and A2 refer to polar and azimuthal anchoring strengths and , , e and
e to the tilt and azimuthal angles of the director and easy direction, respectively.
However, this approach completely decouples the two angles in an unrealistic way
giving rise to complications: Firstly, the decoupling of the two angles makes the
anchoring energy density discontinuous with respect to and [18]. Secondly, the
azimuthal anchoring energy density should also depend on the tilt angle of the director
and this effect is not included. Furthermore, expression (4.7) is periodic with a period
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of radians, resulting in a bistable anchoring when the tilt angle of the easy direction
lies in the range 0< e< /2.
It has later been shown by Zhao, Wu and Iwamoto [17, 18], that a representation
of the anisotropic surface energy density without the complications outlined above is:
FZWI = B1sin2() cos2( 0)
+B2sin2() sin2( 0), (4.8)
where (, ) are angular deviations of the director fromein a local coordinate system
defined by the orthonormal vector triplet (v1,v2, e) describing the principal axes of
anchoring. Equation (4.8) can also be expressed more compactly as [17]:
FZWI=B1(v1 n)2 +B2(v2 n)2 (4.9)
where B1 and B2 are anchoring strength coefficients corresponding to deformations
in the (v1, e) and (v2, e) planes respectively.
4.3.2 Weak Anchoring in the Landau-de Gennes
Theory
In the Landau-de Gennes theory the anchoring energy density is written as a function
of the Q-tensor. This means that order variations also affect the surface energy.
Perhaps the simplest way of approximating the anchoring effect of an aligning
surface is by means of a penalty type expression [70, 93]:
Fpen=WTr
(Q Q0)2
, (4.10)
where Q0 is the preferred easy Q-tensor. Clearly the energy density is minimised
when Q = Q0. Expression (4.10) shows a sin2 variation with respect to angular
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departures from the easy direction [93]. However, similarly to the original Rapini-
Papoular expression (4.5), the penalty anchoring does not distinguish between polar
and azimuthal anchoring strengths.
Another expression for the surface energy density in the Landau de-Gennes theory
describes the effect of an isotropic surface on a LC material, i.e. a surface giving
degenerate alignment, where only the director tilt is constrained. This is a Landau
power series expansion on the surface normal unit vector v and Q [94]:
Fexp= c1(v Q v) +c2Tr(Q2) +c3(v Q v)2 +c4(v Q2 v). (4.11)
Here,ciare scalar coefficients that determine the preferred tilt angle and surface order.
Expression 4.11 has been used e.g. in [95, 96] to study anchoring transitions. Slow
convergence (order of 1000 Newton iterations) of numerical schemes with (4.11) as a
surface energy term has been reported in [95], making the expression computationally
too expensive for the modelling of device dynamics.
An expression for anisotropic anchoring, linear in Q, has been studied in [97]:
fs= Tr(H Q), (4.12)
where H is a symmetric traceless tensor describing the symmetry of the surface.
However, since this expression is linear there is no control over the surface order
parameter which tends to either positive or negative infinity depending on the exact
form ofHand the anchoring strength.
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4.4 The Anchoring Energy Density of an Anisotropic
Surface in the Landau-de Gennes Theory
A generalisation with a reduction in symmetry as compared to (4.11), that can be
written as a power expansion truncated to 2nd order on the Q-tensor and two or-
thogonal unit vectors whose directions are determined by the surface treatment is
presented here:
Fs = asTr(Q2) +
+ W1(v1 Q v1) +W2(v2 Q v2) +W3(v1 Q v2) +X1(v1 Q v1)2
+ X2(v2 Q v2)2 +X3(v1 Q v2)2 +X4(v1 Q2 v1) +X5(v2 Q2 v2)
+ X6(v1 Q2 v2) +X7(v1 Q v1)(v2 Q v2)
+ X8(v1 Q v2)(v1 Q v2) +X9(v1 Q v2)(v1 Q v1)
+ X10(v1
Q
v2)(v2
Q
v2), (4.13)
where Wi and Xi are anchoring strength coefficients. The simplest case that still
allows for anisotropic anchoring with a preferred order parameter is when the scalar
coefficients W3and Xi are zero. In this case the surface anchoring energy reduces to:
Fs=asTr(Q2) +W1(v1 Q v1) +W2(v2 Q v2). (4.14)
The principal axes of anchoring (e,v1,v2) are