Download electroconvection and pattern formation in nematic liquid crystals

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ELECTROCONVECTION AND PATTERN
FORMATION IN NEMATIC LIQUID CRYSTALS
http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40
A dissertation submitted to
Kent State University in partial
fulfillment of the requirements for the
degree of Doctor of Philosophy
by
Gyanu R. Acharya
May, 2009
electronic-Liquid Crystal Dissertations - May 28, 2009
Dissertation written by
Gyanu R. Acharya
M.Sc., Tribhuvan University, Nepal, 1995
Ph.D., Kent State University, 2009
Approved by
, Chair, Doctoral Dissertation Committee
Dr. James T. Gleeson
, Members, Doctoral Dissertation Committee
Dr. David W. Allender
,
Dr. Brett D. Ellman
,
Dr. Oleg D. Lavrentovich
Accepted by
, Chair, Department of Physics
Dr. Bryon D. Anderson
, Dean, College of Arts and Sciences
Dr. Timothy S. Moerland
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TABLE OF CONTENTS
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vii
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xx
ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xxi
1
Introduction
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1
Liquid crystal phase of matter . . . . . . . . . . . . . . . . . .
1
1.2
Thermotropic liquid crystal and its different phases . . . .
2
1.3
Pattern formation . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
1.3.1
Rayleigh-Bénard convection . . . . . . . . . . . . . . . .
8
1.3.2
Taylor-Couette instability . . . . . . . . . . . . . . . . .
11
1.3.3
Nematic electrohydrodynamic instability . . . . . . . .
12
1.4
Spatiotemporal chaos . . . . . . . . . . . . . . . . . . . . . . . .
15
1.5
Dissertation outline . . . . . . . . . . . . . . . . . . . . . . . . . .
17
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
2
Theoretical background
2.1
2.2
. . . . . . . . . . . . . . . . . . . . . . . .
22
Theoretical tools . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
2.1.1
Linear instabilities . . . . . . . . . . . . . . . . . . . . . .
22
2.1.2
Amplitude equations . . . . . . . . . . . . . . . . . . . . .
26
Standard model for nematics . . . . . . . . . . . . . . . . . . .
29
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2.2.1
The molecular field . . . . . . . . . . . . . . . . . . . . . .
31
2.2.2
Dynamic theory of nematics . . . . . . . . . . . . . . . .
35
2.2.3
Nematic viscosities . . . . . . . . . . . . . . . . . . . . . .
38
2.2.4
Carr-Helfrich mechanism and the threshold voltage .
40
Weak electrolyte model . . . . . . . . . . . . . . . . . . . . . . .
46
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
2.3
3
Experimental methods
. . . . . . . . . . . . . . . . . . . . . . . . .
52
Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
3.1.1
Shadowgraphy and optical microscope . . . . . . . . .
52
3.1.2
Temperature control . . . . . . . . . . . . . . . . . . . . .
59
3.1.3
Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
3.2
Liquid Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
3.3
Sample preparation . . . . . . . . . . . . . . . . . . . . . . . . .
64
3.4
Sample cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
3.1
4
Hopf bifurcation and convective patterns
4.1
. . . . . . . . . . .
71
Experiments in I52 . . . . . . . . . . . . . . . . . . . . . . . . . .
73
4.1.1
Threshold voltage and different regimes . . . . . . . .
73
4.1.2
Oblique Hopf instability . . . . . . . . . . . . . . . . . . .
78
4.1.3
Flat fielding . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
4.1.4
Continuous Hopf bifurcation . . . . . . . . . . . . . . . .
83
4.1.5
Defects in NLCs . . . . . . . . . . . . . . . . . . . . . . . .
94
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4.2
Experiments in Phase 5 . . . . . . . . . . . . . . . . . . . . . . .
4.3
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
96
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5
Spatiotemporal chaos in I52
5.1
. . . . . . . . . . . . . . . . . . . . . 115
Four-wave demodulation . . . . . . . . . . . . . . . . . . . . . . 119
5.1.1
Spatial demodulation and critical wave numbers . . 122
5.1.2
Temporal demodulation . . . . . . . . . . . . . . . . . . . 130
5.2
Alternating waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.3
Localized states . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
5.4
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
6
Material parameters and Nusselt numbers
characterization in I52 . . . . . . . . . . . . . . .
6.1
. . . . . . . . . . 150
Freedericksz Transition . . . . . . . . . . . . . . . . . . . . . . . 151
6.1.1
Splay elastic constant . . . . . . . . . . . . . . . . . . . . 153
6.1.2
Bend elastic constant . . . . . . . . . . . . . . . . . . . . 155
6.1.3
Twist elastic constant . . . . . . . . . . . . . . . . . . . . 160
6.1.4
Dielectric anisotropy and conductivity anisotropy . . 162
6.1.5
Bend deformation in electric field . . . . . . . . . . . . 169
6.1.6
Frequency dependence of conductivities and dielectric constants . . . . . . . . . . . . . . . . . . . . . . . . . . 172
6.2
Director relaxation time . . . . . . . . . . . . . . . . . . . . . . . 174
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6.3
Refractive indices . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
6.4
Dynamic light Scattering . . . . . . . . . . . . . . . . . . . . . . 182
6.4.1
Geometry A-splay/twist geometry for measurement
of ηsplay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
6.4.2
Geometry B- bend/twist geometry for measurement
of α0 s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
6.5
6.6
Electric Nusselt number characterization . . . . . . . . . . . . 193
6.5.1
Current flow through the sample cell . . . . . . . . . . 194
6.5.2
Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 195
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
A Matlab codes to calculate Hopf frequency . . . . . . . . . . . . . . 202
B Matlab codes to extract single envelope using two-wave demod-
ulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
C Matlab codes to extract envelope using four-wave demodulation 206
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LIST OF FIGURES
1.1
Schematic of different phases. Top row from left to right; crystalline,
nematic and isotropic and bottom row; schematic A and Schematic C
phases, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2
4
Common calamitic liquid crystals: Top; 4-methoxybenzylidene-4’-butylaniline
(MBBA), the first room temperature liquid crystal having all known
parameters. Bottom; 4-pentyl-4’-cyanobipheny (5CB), the common
LC used in electro-optic display. . . . . . . . . . . . . . . . . . . . . .
1.3
Schematic diagram for Rayleigh-Bénard convection. The fluid is heated
from below by a heat current Q. . . . . . . . . . . . . . . . . . . . . .
1.4
9
(a) Roll pattern for a Boussinesq fluid [22] for circular side walls. (b)
For square side walls [19]. . . . . . . . . . . . . . . . . . . . . . . . .
1.5
5
11
Snapshot of a typical EHC pattern of size 480×640 slightly above onset
in planar cell filled with I52: (a) Oblique modes in a doped planar
sample cell of thickness 10.95 ±0.09 µm (b) Rectangular modes in a
doped planar sample cell of thickness 23.12 ± 15 µm. For both images,
the double arrow shows the direction of unperturbed director. . . . .
2.1
13
Schematic representation of the growth rate as a function of the wave
vector q for various values of rescaled control parameters. . . . . . . .
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2.2
A schematic of the basic deformation in NLCs. The ellipsoids are for
the nematic director orientation after deformation: Examples of (a)
pure splay deformation (b) pure twist deformation and (c) pure bend
deformation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
2.3
Illustration of geometries for Leslie viscosity coefficients. . . . . . . .
37
2.4
Illustration of geometries for Miesowicz viscosity coefficients; (a) ηa :
n ⊥ v, n ⊥ ∇v, (b) ηb : n k v, (c) ηc : n ⊥ v, n k ∇v and (d) η12 = α1 . 39
2.5
Cross section of a roll pattern from different geometry, double arrows denote the director modulations and the symbols + and - denote
the positive and negative induced charges. (a) Planar geometry. (b)
Homeotropic geometry. . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1
41
(a) Principle of shadowgraph method: The incoming light is deflected
according to the refractive index; δ is the thickness of the cell, α the
maximum deflection angle of the light. (b) Experimental setup for
electroconvection; an ac voltage is applied to the conductive coating
glass plates. The convection rolls and the tilt angle of the director are
shown schematically; the dashed points represent the virtual images
and the solid point, the real image; the labels 1 and 2 represent the
3.2
real foci and 3 the virtual focus [1]. . . . . . . . . . . . . . . . . . . .
54
Polarizing microscope with the camera system [4]. . . . . . . . . . . .
56
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3.3
(a) Shadowgram for electroconvection of nematic I52 very near onset
consisting of counter-propagating zig and zag rolls. (b) Spectral density
showing fundamental peaks for the image at (a). (c) Shadowgram for
sample cell I52 at different parameters than that of (a) above onset
having superposition of counter-propagating zig and zag rolls along
with rectangular patterns and active and inactive regions. (d) Its power
spectrum showing higher harmonics dominating the fundamental peaks. 58
3.4
Schematic drawing of the FP82 microscope hot stage. Platinum RTD
measures the temperature of the hot furnace. . . . . . . . . . . . . . .
3.5
59
(a) Chemical formula for I52. (b) Chemical formula for Phase 5 (mixture of 35 wt.- % p-ethyl-p’-methoxy-azoxybenene and 65 wt.-% pbutyl-p’-methoxy-azoxybenzene). . . . . . . . . . . . . . . . . . . . .
3.6
62
Schematic drawing of the top view of commercial cell. The rectangular
shaded region at the center represents the ITO coated ‘active area’.
The dark vertical ellipsoids on either side of the active area represent
the spacer used. These spacers can be of different thickness as desired.
4.1
67
(a) Threshold voltage as a function of applied frequency at different
temperatures in the sample cell I5299. (b) Different regimes in the
sample cell I5234 at 47 ◦ C; the Lifshitz point is 240 Hz and the cut-off
frequency is 420 Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2
75
Typical stationary zigzag pattern slightly above onset during electroconvection showing Williams-Kapustin domains. The double arrow
along the vertical is the direction of unperturbed director. . . . . . .
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4.3
Different patterns obtained during electroconvection in I52: (a) Nearly
normal rolls with dislocation to the upper left corner. (b) Turbulent
structure high above onset. (c) Localized patterns called worms and
(d) Chevron patterns. The length scale represents 100 µm. . . . . . .
77
4.4
Illustration of counter-propagating zig and zag modes. . . . . . . . . .
80
4.5
(a) Background image at zero applied ac voltage. (b) Snapshot of
the image at same illumination as that of the background and ac field
turned-on at ε = 0.01. . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6
Flat fielded image of Fig. 4.5 showing pure zig and zag rolls as indicated
by circles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.7
82
83
(a) Threshold curves showing the variation of onset voltage with driving ac frequency at 25 ◦ C (blue circles), 35 ◦ C (red up triangles) and
50 ◦ C (green diamonds). (b) Fourier transform of the central pixel
values of 2048 images: it is at ε = 0.01, driving frequency of 210 Hz
and corresponds to Hopf frequency of 0.4 Hz. . . . . . . . . . . . . . .
4.8
86
(a) Variation of the charge relaxation time with driving frequency at
25 ◦ C (pink circles), 35 ◦ C (red circles) and 50 ◦ C (blue up triangles).
(b) Variation of Hopf frequency with normalized driving frequency at
three different temperatures. . . . . . . . . . . . . . . . . . . . . . . .
4.9
88
(a) Variation of Hopf frequency with ε > 0 at 50 ◦ C and driving frequency of 51 Hz, Vc =11.73 V. It corresponds to σ⊥ = 64.1×10−9 Ω−1 m−1
and (b) Variation of zig and zag peaks of the Fourier transform with
ε > 0 at 25 ◦ C and 25 Hz, Vc = 12.917 V. It has σ⊥ = 26.49×10−9 Ω−1 m−1 . 90
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4.10 (a) Wave vectors as a function of the normalized driving frequency
at different temperatures. Up triangles, hexagons and square with
cross are for q at 25 ◦ C, 35 ◦ C and 50 ◦ C, respectively. With rise in
temperatures, q slightly decrease at higher frequencies. Down triangles,
diamonds and stars are for p at 25 ◦ C, 35 ◦ C and 50 ◦ C respectively,
both multiplied by d/π to make them dimensionless. (b) Variation of
the angle between the wave vector q and n with normalized driving
frequency. Circles, diamonds and up triangles are for θ at 25 ◦ C, 35
◦
C and 50 ◦ C, respectively. . . . . . . . . . . . . . . . . . . . . . . . .
√
4.11 ωH σ⊥ d3 as a function of ωo τq for sample cell I5246 (up triangles) at
92
57.5 ◦ C and σ⊥ = 8.39 × 10−9 Ω−1 m−1 and I5261 (solid circles)at 43 ◦ C
and σ⊥ = 6.37 × 10−9 Ω−1 m−1 . . . . . . . . . . . . . . . . . . . . . . .
93
4.12 (a) Pure zig mode at t=1280. (b) Zigzag grain boundary at t=11,101.
(c) Pure zag modes at t= 11,752 and (d) Zigzag grain boundary at
t=13,117. The double arrow gives the direction of unperturbed director
and the length scale represents 100 µm. . . . . . . . . . . . . . . . . .
95
4.13 (a) Two grain boundaries with double zag domains and single zig domain at t = 26,440. The double arrow denotes the direction of unperturbed director and the length scales represents 100 µm. (b) Its
envelope extracted from two-wave demodulation. . . . . . . . . . . . .
97
4.14 Threshold curve for cell P59 at 35 ◦ C. It corresponds to the cut-off
frequency of 1050 Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4.15 (a) Oblique stationary rolls at 90 Hz and ε = 0.05. The double arrow
is the direction of unperturbed director. (b) Average FFT of first 120
frames showing the inner oblique modes in the region of interest. . . . 100
4.16 (a) Normal traveling pattern at 95 Hz and ε = 0.016. The double
arrow represents the direction of unperturbed director. (b) Average
Fourier transform of first 120 frames showing normal peaks. . . . . . 101
4.17 Circular Hopf frequency in cell P59 as a function of the normalized
driving frequency. The first vertical short dashed line is for the critical
ωo τq , left of which the pattern is stationary and right of which it is
traveling. The second short dashed vertical line is where the second
discontinuous Hopf bifurcation occurs. . . . . . . . . . . . . . . . . . 102
4.18 (a) Variation of the roll angles with ωo τq . (b) Wave vectors as a function
of the normalized driving frequency in cell P59. Up triangles are for
the wave vector qx̂ and the circles indicate the wave vector pŷ, both
multiplied by d/π to make them dimensionless. The short dashed
vertical line differentiates between OS and NT rolls regime. . . . . . . 104
4.19 ωH as a function of ωo τq in the cell P58. The up triangles are at 35 ◦ C
and the circles are at 40 ◦ C . Short dashed vertical lines are drawn at
critical ωo τq at which the Hopf bifurcation occurs. . . . . . . . . . . . 105
4.20 Variation of ²⊥ (up triangles) and σ⊥ (circles) with ωo τq for the cell
P59. The short dashed vertical line drawn at critical ωo τq separates
the OS and NT modes. . . . . . . . . . . . . . . . . . . . . . . . . . . 106
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4.21 (a) Variation of Ω with ωo τq in cell P59. The left short dashed vertical
line drawn at critical ωo τq separates the OS and NT modes and the
right short dashed vertical line is where the second discontinuity Hopf
frequency occurs. (b) ω 0 in Eq. 4.6 as a function of ωo τq in cell P59,
both at 35 ◦ C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.22 (a) Variation of charge τq with the applied frequency in cell P59 at 35
◦
C. (b) Phase diagram at 25 ◦ C for the planar sample cell of P53. The
Lifshitz point FL =62 Hz and the cut-off frequency Fc = 155 Hz. . . . 108
4.23 snapshot of the EHC pattern at 61 ◦ C for a cell of thickness 23.18±0.24
µm filled with I52+4 wt.% I2 . σ⊥ = 16.8 × 10−9 Ω−1 m−1 and fH is 0.85
Hz at ωo τq = 0.28 and ε = 0.01. The rubbing direction of the cell
plates is in the direction of double arrow. . . . . . . . . . . . . . . . . 110
5.1
Carrier positive signal of high frequency and slowly varying modulating
signal also called the information bearing signal . . . . . . . . . . . . 120
5.2
(a) 2D spatial Fourier transform of the flat fielded image at t = 10,100
showing dominating fundamental modes. Zig and zag fundamental
peaks, normal mode and the higher harmonic peak are shown by solid
circles in the window −50 ≤ m ≤ 50 and −50 ≤ n ≤ 50. (b) 3D view
of the Fourier transform of the same image in the same window. . . . 121
5.3
(a) Average of the individual peaks of a time series of images for
10, 001 ≤ t ≤ 20, 000 in the window −50 < m ≤ 50 and −50 < n ≤ 50
. (b) Individual average modes of the same time series of Fourier
transform 10, 001 ≤ t ≤ 20, 000 in different windows as specified in the
figure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
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5.4
(a) Average of zig and zag Fourier peaks in the window −25 < m ≤ 25
and 0 < n ≤ 50 for the time as in Fig. 5.3. These are the peaks
of interest for spatial demodulation. (b) Time and zig-zag averaged
power spectrum Pav . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.5
2D Gaussian filter in the window 141 ≤ m ≤ 240, 221 ≤ n ≤ 320 used
to filter out the primary oblique spatial Fourier modes. The blue and
red colors correspond to minimum and maximum intensity respectively. 124
5.6
(a) Izag , ( b) Azag , (c) Izig and (d) Azig for the pattern snapshot at t
= 10,100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.7
(a) Time series of zag modes for real, imaginary and absolute values
for 1 ≤ t ≤ 20, 000. (b) Zooms of the real, imaginary and the absolute
parts in the range 10001 ≤ t ≤ 12048. . . . . . . . . . . . . . . . . . . 128
5.8
(a) Time series of zig modes for real, imaginary and absolute values
for 1 ≤ t ≤ 20, 000 (b) Zooms of the real, imaginary and the absolute
parts in the range 10001 ≤ t ≤ 12048. . . . . . . . . . . . . . . . . . . 129
5.9
(a) Inor , (b) Its envelope and (c) Reconstructed image showing zig
and zag envelopes. The blue and the green regions in this image are
regions with high zig and zag contributions to the recorded image. . . 131
5.10 2D plots of | A1 | − | A4 | for t = 10,100.
. . . . . . . . . . . . . . . 133
5.11 a-d; Real, imaginary and the absolute values of the amplitudes at the
center of each envelopes for the time series 10001 ≤ t ≤ 12048 . . . . . 135
5.12 Four wave amplitudes for 10001 ≤ t ≤ 12048. A1 (blue), A2 (red), A3
(green) and A4 (black). . . . . . . . . . . . . . . . . . . . . . . . . . . 136
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5.13 (a) Time average of the patterns from 10, 001 ≤ t ≤ 20, 000 and (b)
Variation of pixels along central row for individual image (upper) and
for the average of images (lower). . . . . . . . . . . . . . . . . . . . . 137
5.14 (a) Oblique modes at t = 1,000 from the sample cell I5234 at ωo τq =
0.38, ε = 0.028 and T = 55 ◦ C. The rubbing direction of the cell plates
is in the vertical direction in the picture; the length scale represents
100 µm. (b) Average of the central 10 × 10 pixel values of the zig (red)
and zag (blue) wave envelopes as a function of time. (c) Zoom in of
(b) showing alternating waves. . . . . . . . . . . . . . . . . . . . . . . 139
5.15 Snapshot of an images at 30 ◦ C. It corresponds to ε = 0.042, ωo τq =
0.74, fH = 0.34 Hz and consists of active and inactive regions. The
rubbing direction of the cell plates is in the vertical direction in the
picture; the length scale represents 100 µm. . . . . . . . . . . . . . . 141
5.16 Time series of worms at interval of 2 s. The hot stage temperature is
30 ◦ C and the frequency of applied ac is 130 Hz. The rubbing direction
of the cell plates is in the vertical direction in the pictures; the length
scale represents 100 µm. . . . . . . . . . . . . . . . . . . . . . . . . . 142
5.17 Intensity variation of the worms along and at right angle to the director.143
5.18 Worms at 30 ◦ C and 150 Hz ac field. The first worm from right covers
almost whole field of view along ±n. The rubbing direction of the
cell plates is in the vertical direction in the picture; the length scale
represents 100 µm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
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6.1
Illustration of the geometries for Freedericksz transition to determine
(a) splay, (b) twist and (c) bend elastic constants. Geometries to the
left are for the magnetic field less than the critical threshold field and
the geometries to the right are for the magnetic field greater than the
critical threshold field. . . . . . . . . . . . . . . . . . . . . . . . . . . 152
6.2
Experimental set up for magnetic field induced Freedericksz transition. 154
6.3
(a) DC photo voltage as a function of the applied ac voltage at H =
0.3209 T > Hc = 0.2476 T at 25 ◦ C. (b) Graph for Hc2 versus Vc2 . The
fit gives Vc2 = −655.9 + 1.07 × 104 Hc2 . . . . . . . . . . . . . . . . . . . 156
6.4
Splay critical magnetic fields as a function of temperature. . . . . . . 157
6.5
(a) The capacitance of sample cell I5233 as a function of the magnetic
field when the sample goes bend deformation. (b) Variation of critical
magnetic field with the temperature for the same deformation. . . . . 159
6.6
(a) Schematic of the cell geometry. The drawing shows the top view
of the cell and n is the direction of undistorted director orientation.(b)
Variation of the capacitance of sample cell I5222 with magnetic field
when the sample goes twist deformation at 25 ◦ C. . . . . . . . . . . . 161
6.7
Variation of Capacitance C (blue) and conductivity σ (red) with the
applied magnetic field. . . . . . . . . . . . . . . . . . . . . . . . . . . 165
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6.8
Variation of inverse H with capacitance and conductivity at 25 ◦ C from
the data of Fig. 6.7 for the cell I5211. The blue up triangles are for
σ verses H −1 . The pink straight line is the fit H −1 = 4.382 × 10−9 −
1.915σ which gives σk = 2.29 × 10−9 Ω−1 m−1 and the red circles are for
capacitance. The green straight line is the fit H −1 = −67.32 × 10−12 +
2.345C which gives Ck = 28.71 pF. . . . . . . . . . . . . . . . . . . . 166
6.9
²⊥ and ²k as a function of temperature for the cell I5211. ∆² = 0 at
60.18 ◦ C. The red circles are for ²⊥ and the blue up triangles are for
²k . The error bars are calculated repeating the experiment on the same
cell after two months. . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
6.10 (a) the dielectric anisotropy and (b) the conductivity anisotropy as a
function of temperature for planar cell I5211 filled with the sample. . 168
6.11 Schematic of the geometry for electric field induced bend Freedericksz
transition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
6.12 Transition curve used to determine the onset of Freedericksz transition
in I52 at 30 ◦ C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
6.13 (a) Variation of critical voltage for bend transition with temperature. (b) Same data of figure 6.13 plotted as (VcF )−2 versus temperature. The fit gives the ∆² = 0 at 62.91 ◦ C. . . . . . . . . . . . . . . 171
6.14 Variation of ²⊥ and σ⊥ with frequency at different temperatures. The
dashed lines are for ²⊥ at temperatures as shown. The solid lines are
for σ⊥ at 25 ◦ C (hexagons connected by cyan solid line), 30 ◦ C (filled
circles connected by dark green solid line) and 45 ◦ C (down triangles
connected by solid black lines). . . . . . . . . . . . . . . . . . . . . . 173
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6.15 (a) Variation of loss with time at 40 ◦ C when the magnetic field is
suddenly ceased. (b) Variation of loge | ∆G | with time at 40 ◦ C
immediately after the magnetic field is off. It is a straight line loge |
∆G |= 30.735 − 0.659t and gives τd = 3.035 s. . . . . . . . . . . . . . 176
6.16 Variation of director relaxation time τd with temperature. . . . . . . . 177
6.17 Refraction of light passing through the liquid crystal in Abbe’s refractometer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
6.18 (a) Variation of refractive indices of pure I52 with temperature. The
blue circles and red up triangles indicate for ne and no respectively
as a function of temperature. (b) Variation of birefringence of pure
I52 with temperature. In both figures (a) and (b), the light source is
He-Ne laser beam.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
6.19 (a) Variation of refractive indices of pure I52 with temperature. The
blue circles and red up triangles are for ne and no respectively. (b) Variation of birefringence of pure I52 with temperature. In both figures (a)
and (b), the source is NaD light. . . . . . . . . . . . . . . . . . . . . . 183
6.20 (a) Geometry A in dynamic light scattering used to measure ηsplay . (b) Vari2
ation of K11 q⊥
with the relaxation frequency Γ1 at 50◦ C. . . . . . . . 185
6.21 (a) Geometry B in dynamic light scattering used to measure α0 s. (b) Variation of correlation function with the delay time at 25 ◦ C. . . . . . . 188
2
/qk2 with the scattering angle in the lab. (b) Varia6.22 (a) Variation of q⊥
2
+ K33 qk2 with relaxation frequency
tion of product of the sum of K22 q⊥
Γ2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
6.23 (a) Linear fit of the equation at 50 ◦ C and (b) At 25 ◦ C. . . . . . . . 192
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6.24 Schematic of lock-in amplifier used to measure in phase and out of
phase currents in the sample cell when ac voltage is applied. . . . . . 194
6.25 (a) In-phase current Ir (blue) and out of phase current Ii (pink) versus
applied voltage V and (b) The real part of reduced Nusselt number Nr
verses the applied voltage. Both graphs are at the same frequency 100
Hz and at 30 ◦ C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
6.26 (a) The variation of the slope dNr /dε and slope dNi /dε with frequency
30 ◦ C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
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LIST OF TABLES
4.1
Summary of sample cells used in pattern characterization in I52. . . .
73
4.2
Summary of sample cells used in EHC in nematic Phase 5. . . . . . .
98
6.1
Summary of sample cells used in parameters characterization. . . . . 151
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ACKNOWLEDGEMENTS
During the course of my dissertation, I have had the pleasure of working under
the supervision of Dr. James T. Gleeson. I would like to thank him for his vision,
guidance, support and patience. This work could have never been accomplished without him. I am confident that the skills I have learnt under his mentorship will serve
me well in my future career. Next, I would like to thank Dr. Gerhard Dangelmayr
and Dr. Iuliana Oprea for their valuable time to analyze the data I had taken and
allowing me to visit the Department of Mathematics, Colorado state university. I
want to thank Dr. David Wiant for the discussions and ideas we shared in Dr. Gleeson’s lab and Tanya Ostapenko for her assistance with refining the finished document.
Also, I sincerely thank all the office staff, who assisted me the entire time I was here.
At last, I would like to thank my family for their unconditional support and
love to bring me up to this day, especially my newborn son whose single smile worked
as the strongest motivation in my work.
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CHAPTER 1
Introduction
1.1 Liquid crystal phase of matter
A liquid crystal (LC) is the state of matter that lies between the solid crystal
state and the isotropic liquid state. It has some properties of a liquid, such as it is
unable to support shear, having high fluidity, formation and combination of droplets,
etc. It is analogous to crystals in that it shows anisotropy (different properties in
different directions) in its optical, electrical and magnetic properties. Since it is the
intermediate state of matter, it is sometimes called a mesophase.
The credit for the discovery of LC goes to an Austrian botanist F. Reinitzer.
He observed two melting points [1–3] on either side of the intermediate phase while
working with cholesterol benzoate in 1888. At a melting point of 144.5 ◦ C, it became
a cloudy liquid and upon further heating, the cloudy liquid turned into clear liquid at
178.5 ◦ C. The cloudy liquid reported by Reinitzer was later found to be a cholesteric
liquid crystal. German physicist O. Lehmann constructed the polarizing microscope
with a heating stage to study the behavior of this intermediate phase under controlled
temperatures. Lehmann and Vorländer qualitatively interpreted the microscopic textural observations. G. Friedel established the nomenclature to describe different types
of LCs using the words nematics, smectics and cholesterics [1, 2] and explained the
close connection between the textures and corresponding structures [4]. He is credited
with detecting the phenomenon of liquid crystal polymorphism (compound showing
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more than one liquid crystalline phase).
Following World War II, the progress on liquid crystal research was slow for several years. However, it gained its momentum in late fifties with work by G. Brown, a
chemist at Kent State University. During the sixties, liquid crystal research flourished
from few centers to many institutions in developed countries. Founding of the Liquid
Crystal Institute and the International Liquid Crystal Conference at Kent State were
two pioneer works among his achievements. In succeeding years, the development of
industrial applications with success in electro-optic information display has made life
easier and reliable. These days, many physicists, chemists, engineers, biologists and
mathematicians are engaged in LC research and applications and hence it has become
an interdisciplinary subject of study.
1.2 Thermotropic liquid crystal and its different phases
Most liquid crystals are organic substances and they reveal liquid crystal phases
either by changing the temperature or by changing the concentration in the solution
or both. Those obtained by changing the temperature are called thermotropic liquid
crystals. They can be pure compounds or mixtures. The basic unit of interaction in
a LC system is called the mesogen, which can be molecule or composite of molecules.
The mesogens of many thermotropic LCs consist of organic molecules composed of a
rigid aromatic core of benzene rings with attached end groups called side-chains and
terminal groups and linkage groups between the rings. If the molecules have flat segments, such as benzene rings, liquid crystallinity is expected to occur more. A rigid
backbone with double bonds describe the long axis. The core may be either straight
or bent. Molecules with straight cores are often referred to as rod-like, or calamitic
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LCs. Their dimensions are typically ∼ 20 Å long and ∼ 5 Å wide. The side-chain
and terminal groups are, for example, alkyl (Cn H2n+1 ), alkoxy (Cn H2n+1 O), acyloxyl,
alkyl carbonate, alkoxy carbonyl, and the nitro and cyano groups. On the other
hand, the linkage groups are simple bonds or groups, such as stilbene, ester, tolane,
azoxy, Schiff base, acetylene and diacetylene. Besides benzene derivatives, other liquid crystals include heterocyclics, organometallics, sterols and some organic salts or
fatty acids. In calamitic LCs, despite the flat character of the benzene rings, the effective molecular shape is not flat and they have rotational freedom around the long
molecular axis. Fig. 1.2 shows chemical structure of most common LCs called MBBA
and 5CB.
Many physical parameters, such as dielectric constants, elastic constants, viscosities, transition temperatures, existence of mesophases and anisotropies are all a
result of how these molecules are arranged. Whether a liquid crystal is chemically
stable or not depends on the central linkage group. Schiff-base LCs are quite unstable. Even though ester, azo and azoxy compounds are quite susceptible to moisture,
temperature change and ultraviolet radiation, they are comparatively stable.
There are three main classes of the calamitic LCs: nematics, cholesterics and
smectics. Smectics are subclassified according to the positional and directional arrangement of the molecules. The molecular structure plays a crucial role in the phase
of LCs. However, some compounds of different shapes may have similar phase structures and same compound might show different phases.
Fig. 1.1 shows the schematic arrangement of crystalline, nematic, isotropic,
smectic A and smectic C phases. The long planar molecules are symbolized by ellipsoids. Nematics are positionally random in that there is no long range order in
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n
n
n
Figure 1.1: Schematics of different phases. Top row: crystalline, nematic and
isotropic and bottom row: Smectic A and smectic C phases.
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O
H3C
N
C4H9
H11C5
CN
Figure 1.2: Common calamitic liquid crystals: Top; 4-methoxybenzylidene-4’butylaniline (MBBA), the first room temperature liquid crystal having all known parameters. Bottom; 4-pentyl-4’-cyanobipheny (5CB), the common LC used in electrooptic display.
the positions of the center of mass of the molecules but they are directionally correlated; the long axis of the molecules tend to align along a preferred direction. This
preferred direction of orientation is denoted by a unit vector n and called the director [5]. The locally preferred direction may vary throughout the medium. Nematics
are centrosymmetric; physical properties along n and -n are equivalent. It is the least
ordered LC phase characterized by only long range orientational order. It is usually
a uniaxial phase in the sense that the macroscopic properties are different along and
perpendicular to that order. In uniaxial nematics, the ordering of the molecules can
be described by the order parameter S given as
1
S = hP2 (cos θ)i = h3 cos2 θ − 1i
2
(1.1)
Here, the bracket denotes an average over many molecules at the same time or the
average over time for the given molecule and θ is the angle between the cylindrical
axis of the molecule and the direction of the director. Eq. 1.1 has the property that if
there is no orientational order, S = 0 and the system is isotropic. For complete order,
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θ = 0 and S = 1. For liquid crystals, S lies between zero and one [6–8]. Cholesteric
or chiral nematics lack inversion symmetry but have physical properties the same as
that of nematics, but the director rotates in a helix about an axis perpendicular to
the director. The distance along the helical axis over which the director rotates by
360◦ is the pitch of the chiral nematic. Since n and -n are equivalent, the structure
repeats every half pitch. A simple nematic is the chiral nematic of infinite pitch.
Smectics differ from nematics in that they stratify. The molecules arrange in
layers and exhibit some correlation in their positioning in addition to orientational
ordering. In each layer of smectic A, the molecules are orientationally ordered with
their long axes perpendicular to the plane of the layer, but are positionally disordered.
They are rotationally symmetric around the director axis as nematics and the layers
can slide freely over one another. In the smectic C phase, the preferred axis is not
perpendicular to the layers, so it is optically biaxial. The director makes an angle
with the layer normal as shown in Fig. 1.1(e). Each layer in both phases acts as a
2D liquid. Smectic B phase is the most ordered phase among A, B and C phases.
Here, the layers emerge to have the periodicity and the rigidity as that of a 2D
solid. Mechanical study confirms it to be a solid having the possibility of shear wave
propagation at low frequency. However, the dielectric measurements do not show its
crystalline behavior [9]. The nematic phase usually appears at higher temperatures
than the smectic phase and the smectic phases occur in the order A →C→B →S
when the temperature decreases. Here, S stands for solid. TBBA ( terephthal-bis(-p
butylaniline)) shows all the discussed phases from solid to isotropic liquid [6, 9, 10]
with rise in temperature.
In thermotropic LCs, transitions occur from the phase of lower symmetry to
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the phase of higher symmetry as the temperature is increased. In lyotropic LCs, it
occurs as the concentration in the liquid solvent changes.
1.3 Pattern formation
Patterns are omnipresent in nature. They are based on replication and periodicity. Animal markings, body segmentation of animals, phyllotaxis, remarkable
shapes of snowflakes in dendritic growth, sunflowers, pinecones, patterns on shells,
ridges on our fingertips, piles of sand, flocking birds, reaction-diffusion systems, undulating ripples of a desert dune, regular or irregular cloud formation are some of the
examples of natural patterns surrounding us. Pattern formation deals with the selforganized, spatially extended visible system which follows common principles behind
similar patterns. It is a branch of nonlinear dynamics which focuses on systems where
nonlinearities work together to form spatial patterns which are stationary, traveling
or disordered in space and time [11–13]. In some cases, the shape of the pattern is
unique among other growth forms. For instance, each snowflake in its six-fold symmetry is different than any other and the arms of a snowflake are of different length.
In previous decades, there has been major progress in the field of pattern formation. Ginzburg-Landau type model systems [14–16] have been used to describe
weakly nonlinear patterns. Near onset of threshold, scientists have advanced their
understanding of time dependent as well as time independent patterns. Due to rapid
progress in collaboration of experiment and theory and advances in computational
power, it has been possible to study complex spatiotemporal patterns [13, 17] in systems of large spatial extent. The increase in computational power and imaging technology has allowed the analysis of many digital images captured during experiments.
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This has been the basis for identifying and studying the nonequilibrium dynamics of
extended systems. The fundamental mechanism and the accompanying mathematics
have brought together scientists and researchers from diverse fields, such as physics,
chemistry, biology, material science, mathematics, medicine, geophysics, surface science, etc. Therefore, it has been an interdisciplinary subject.
Patterns can form via bifurcation: branching off into two parts. It occurs when
a small change made to the control parameters of a spatially uniform state causes a
sudden qualitative or topological change in its behavior when the control parameters
exceed a certain critical value. Often the amplitude of the pattern grows continuously from zero when the control parameter goes beyond its critical value. These
control parameters, such as temperature, pressure, external magnetic or electric field,
etc. determine the characteristic length and the growth speed of the pattern. In
the following sections, I will proceed by introducing pattern forming phenomena in
hydrodynamic systems.
1.3.1 Rayleigh-Bénard convection
Rayleigh-Bénard convection (RBC) is an interesting system among the pattern
forming systems due to its easy access for lab studies, well-known governing hydrodynamic equations, high stability and reproducibility. RBC is the instability of an
isotropic fluid layer (water, methanol, ethanol, CO2 , SF6 are commonly used) confined between two thermally conducting plates and heated from below or cooled from
above so as to produce a temperature gradient with the lower plate at higher temperature than the upper one. A typical schematic of RBC is shown in Fig. 1.3. The fluid
remains at rest for a small temperature gradient. It is referred to as the ‘conducting’
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or the ‘uniform’ fluid. However, the fluid near the lower plate will be less dense due to
thermal expansion. It causes a finite wavelength instability - a fundamental prototype
of pattern formation. If the driving force due to the temperature difference, ∆T , is
enough to overcome the dissipative effects of thermal conduction and viscosity, then
the instability occurs, leading to convective current [18] transporting additional heat.
Bénard did his first intensive experiment in 1900 on a fluid of thin layer
λ
d
Q
Figure 1.3: Schematic diagram for Rayleigh-Bénard convection. The fluid is heated
from below by a heat current Q.
and observed an appearance of hexagonal cells when convective instability occurred.
Rayleigh developed the necessary theory in 1916 and showed that for instability, the
temperature gradient, β must be large enough so that the Rayleigh number (dimensionless ratio of the destabilizing buoyancy force to the stabilizing dissipative force)
given by
Rc =
αβgd4
kν
(1.2)
exceeds a certain critical value. The instability occurs at Rc = 1708, independent of
the considered fluid. Here, g is the acceleration due to gravity, α is the isobaric coefficient of thermal expansion, d is the chamber depth, k is the thermal diffusivity and
ν is the kinematic viscosity (ν = η/ρ) . One can define the reduced Rayleigh number
ε = R/Rc − 1 so that ε = 0 corresponds to the onset of convection. This number can
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be taken as the experimental control parameter. The Prandtl number (σ = ν/k) is
the second control parameter that describes the fluid properties. Rayleigh explained
the phenomena in terms of buoyancy driven instability in which the expected patterns would be stripes of convective rolls rather than the cellular pattern observed by
Bénard. The convection observed by Bénard is driven by temperature dependent surface tension force rather than by buoyancy. Nevertheless, the stripes or roll patterns
formed in buoyancy-driven convection is referred to as Rayleigh-Bénard convection,
whereas the surface tension induced convection is called Marangoni convection. Under different circumstances, the pattern above onset will consist of rolls, hexagons
or squares. For a pure fluid confined between rigid and conducting top and bottom
plates, in absence of non-Boussinesq effects (where the fluid parameters and transport
coefficients are assumed to be temperature dependent), the convective patterns will
be like rolls [19] and for non-Boussinesq effects or for open top surface, hexagonal
patterns are obtained [20]. For poorly conducting boundary plates, the structure will
be squares and in binary fluids, traveling or standing wave patterns may develop [21].
I have observed these waves during electroconvection of nematic liquid crystals via
continuous Hopf bifurcation, which I will explain in Chapter Four. Different groups
working on RBC conclude that the bifurcation is supercritical.
The critical Rayleigh number Rc is the minimum value of R at which the conducting state becomes unstable to disturbance of velocity v given by δv ∼ eiqc x for the
wave vector q in the horizontal plane. The value | q |= qc at which the instability at
Rc occurs is of the order of inverse plate separation. Very near onset, the pattern for
a Boussinesq system consists of straight rolls with perhaps some defects induced by
the side walls. Typical roll patterns for circular side walls and the square side walls
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(a)
(b)
Figure 1.4: (a) Roll pattern for a Boussinesq fluid [22] for circular side walls. (b) For
square side walls [19].
are shown in Fig. 1.4. In a few regions, the rolls split, merge or end. These are called
defects and play a vital role in the dynamics of the pattern. Further above onset,
for ε > 0.5, a qualitatively different state of spatiotemporal chaos called spiral-defect
chaos (SDC) [23,24] occurs in the system with a Prandtl number of order one or less.
The difference between straight rolls and the SDC is that SDC is a bulk property
and no side wall is necessary to produce defects. The system becomes more complex
and interesting even near onset when it is forced to rotate about a vertical axis with
angular velocity ωo . Both experiments and theory have proved that if ωo > ωc , a
critical angular velocity, the primary bifurcation will be supercritical with unstable
parallel rolls [19].
1.3.2 Taylor-Couette instability
Taylor-Couette flow or Taylor Vortex flow (TVF) [18] is another hydrodynamic
example analogous to RBC. The system consists of two concentric cylinders with one
or both of the cylinders rotating along the common axis. The fluid is confined to the
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gap between the cylinders and the resulting fluid flow is called the Taylor-Couette
flow. When the inner cylinder rotates and the outer cylinder is at rest, an azimuthal
flow called ‘Couette flow’ arises. The centrifugal force is larger near the inner rotating cylinder, which leads to an instability above a critical rotation rate resulting
in circulating rolls perpendicular to the axis of the cylinder. As in RBC, the length
scale is d = ro − ri , where ro and ri are outer and inner radii of the cylinders. The
distinction between RBC and TVF is that the buoyancy force is replaced by the centrifugal force due to rotation. For the system with inner cylinder rotating and outer
cylinder fixed, the dimensionless control parameter, also called the Taylor number, is
T = 2ri2 d3 /(ri + ro )(ωo /ν)2 . Here, ωo is the rotation rate of the inner cylinder and ν
is the kinematic viscosity. There are three control parameters which direct the flow:
the radius ratio η = ri /ro , the aspect ratio Γ = L/d where L is the gap length and d
is the gap width and the Reynold number Re = ωo ri /ν. The instability occurs when
the Taylor number exceeds a well defined threshold: T > Tc ' 3416.
1.3.3 Nematic electrohydrodynamic instability
Nematics are very good systems to study pattern forming instabilities. The intrinsic anisotropy is a very important property of such a system. These are extended
systems (λ < L, where λ is the structural wavelength and L is the linear dimension
of the system). The most studied system consists of the horizontal layer of nematics
subjected to an ac electric field of frequency F.
The calamitic liquid crystal molecules are long, anisotropic and described by
their director n. Since n and -n are equivalent, any mathematical term involving
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50
100
150
200
250
300
350
100 µm
400
450
100
200
300
400
500
600
(a)
50
100
150
200
250
300
350
400
50 µm
450
100
200
300
400
500
600
(b)
Figure 1.5: Snapshot of a typical EHC pattern of size 480×640 slightly above onset
in planar cell filled with I52: (a) Oblique modes in a doped planar sample cell of
thickness 10.95 ±0.09 µm (b) Rectangular modes in a doped planar sample cell of
thickness 23.12 ± 15 µm. For both images, the double arrow shows the direction of
unperturbed director.
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physical variables describing the NLCs containing n can only have an even power of
n. The interaction between the director and the applied electric field induces a destabilization mechanism known as the Carr-Helfrich mechanism, which I will discuss in
brief in Chapter Two.
Consider a thin horizontal layer of nematic with the uniform director along
x̂-axis and the electric field is applied along ẑ-axis . For NLCs with positive electrical
conductivity anisotropy (∆σ = σk − σ⊥ ) and negative or slightly positive dielectric
anisotropy (∆² = ²k − ²⊥ ) very near onset, the conductive state becomes unstable
and a periodic structure of oblique or normal rolls appear. Here, σk ( ²k ) and σ⊥
(²⊥ ) are the components of electric conductivities (dielectric constants) parallel and
perpendicular to the director respectively. The bifurcation point depends upon the ac
voltage, frequency and the material parameters. For an NLC sample cell of thickness
∼ 25 µm filled with doped I52 (4-ethyl-2-fluoro-4’-[2-(trans-4-n-pentylcyclohexyl)ethyl]biphenyl) subjected to a frequency of 25 Hz, the instability occurs at Vc = 10
volts, leading to traveling oblique modes. Fig. 1.5 shows typical EHC patterns in I52.
The electrohydrodynamic convection (EHC) of NLCs presents various interesting aspects to study pattern forming phenomena. Firstly, since the nematic layers are
very thin (∼ 10 to 100 µm), the system has a large spatial extension perpendicular
to the direction of the applied field. The large aspect ratio (∼ 1000) is closer to
the theoretical idealization of a system of infinite lateral extent and hence boundary
effects can be ignored. Secondly, the typical time scale in EHC is of the order of 10−3
to 10−1 s. This is significantly faster than the RBC system, which has a time scale
of the order of minutes. As a result, reasonable statistics can be obtained in a short
time. Thirdly, the external parameters, such as applied ac voltage, frequency and
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temperature, are easily controlled, which allow various types of patterns and lastly,
the intrinsic anisotropy of the LCs induce a preferred axis for the orientation of patterns.
The disadvantage of the EHC system lies in the need for uniformity of the cell.
Due to the small thickness of the cell, one micron scale nonuniformity is relatively
large. Also, since the LCs themselves are not good conductors, they must be doped
by some suitable impurities to make it conducting. Due to the conductivity drift with
time, it is hard to reproduce similar patterns under similar physical conditions. Also,
nonuniformity in the alignment might affect the patterns across the larger area of the
cell.
1.4 Spatiotemporal chaos
Spatiotemporal chaos (STC), is a dynamical state that is nonperiodic in both
space and time. The fluctuations arise when a system is driven out of equilibrium,
which play a major role in the dynamics. The STC in spatially extended systems has
its origin in experiments in RBC at low temperatures [23–26]. Sustained STC flow of
electroconvection in NLCs when the magnetic and electric field were simultaneously
applied along ẑ direction in the planar cell is reported [27]. Both theoretically and
practically, the transition from inactive state to convection is found to be subcritical
with the discontinuity between the two states increasing with increase of magnetic
field larger than the critical field. This contradicts the simple EHC geometry where
the transition is supercritical. In homeotropic cells filled with nematics, when the
onset voltage is greater than the critical voltage for bend Freedericksz transition and
the applied frequency is less than the Lifshitz point(FL ), there is superposition of two
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oblique modes, which are chaotic in space and time [28]. For frequencies greater than
FL , the pattern consisted of chaotic traveling normal rolls with defects. In contrast,
the chaotic localized traveling wave states in the EHC of NLC I52 are long lived and
are unpredictable about their spatial and temporal birth and death [22, 29]. These
localized states have unique small widths, vary irregularly in length and always travel
along the direction of director. STC in I52 are observed slightly above onset and
arises as a superposition of four degenerate traveling modes: right and left, traveling
zig and zag. These rolls are extended over the entire convection cell [13,30]. When the
control parameter is increased further in all these systems, all the patterns explained
above will be unstable, leading to turbulent structure.
The solution of coupled Ginzburg-Landau equations can exhibit the dynamics
of slowly varying envelopes of plane wave trains associated with the critical wave
numbers. The spiral defect chaos in RBC is the exception, as it emerges from an
already complicated state. Also, RBC in 2D extended isotropic fluid layers will have
critical wave numbers in every direction, which makes the description through the
finite set of plane wave envelopes more complicated. However, in anisotropic systems, there is only a small number of critical wave numbers which allow a finite set
of plane wave envelopes above the onset where this dynamics can be described by
Ginzburg-Landau type amplitude equations. Hence, EHC in nematic liquid crystals
is a suitable experimental system to study ordered and complex STC patterns by
varying its control parameters, such as the electrical conductivity, driving frequency
and the voltage amplitude. Standard hydrodynamic description which combines the
continuum theory of Ericksen and Leslie with the quasistatic Maxwell equations explain most of the phenomena observed near the onset, except the experimentally
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observed oscillatory instabilities leading to traveling waves. The recently developed
weak electrolyte model by Treiber and Kramer describes this phenomena, which is
explained in Chapter Two.
My main objective of studying the time series of a number of slowly varying
envelopes of these waves through a Fourier demodulation analysis is to perform a
variety of tests using different tools to find different patterns and to check whether
the state observed during the experiment is STC or not. Spatial demodulation that
generates amplitudes of zig and zag waves varying slowly in space, but not in time, is
only partially able to test the validity of STC state. Hence, I carried out a temporal
demodulation separately, which extracts the envelopes of four oblique traveling waves
from the time series of zig and zag amplitudes. It not only gives the idea of temporal
variation of the envelope amplitudes of four waves, but also reduces the computational
effort by dividing full 3D Fourier transform into 2D (spatial) +1D (temporal) Fourier
transform. Characterizing the dynamics of the pattern, location of holes in time and
space, global and local Karhunen-Loeve decomposition in Fourier and physical space
and estimates of Lyapunov exponents, all require time series of data recorded during
electroconvection. Also, the correlation analysis [31] showed that the temporal and
spatial complexities are not independent of each other in systems where the system
size is comparable to the correlation length.
1.5 Dissertation outline
To study EHC and characterize the patterns, I have utilized two samples: I52,
a single component NLC and Phase 5, a mixture of azoxy compounds. This dissertation is divided into six chapters. The first chapter gives a brief introduction about
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liquid crystals, pattern formation and the STC states. In the second chapter, I will explain the theoretical tools to describe nematodynamics, Carr-Helfrich mechanism and
weak electrolyte model to explain the observed phenomena in EHC. Chapter Three
will give details about the electroconvection apparatus and the sample cell preparation. In Chapter Four and Chapter Five, I will explain continuous Hopf bifurcations,
convective patterns and the STC states observed in I52. The last chapter will present
the material parameters and the Nusselt numbers characterization. The concluding
explanations of the dissertation will be on the discontinuous Hopf frequency in Phase
5, and continuous Hopf bifurcation and STC in I52.
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BIBLIOGRAPHY
[1] H. Kelker. History of liquid crystals. Mol. Cryst. Liq. Cryst., 21:1, 1973.
[2] P. J. Collings. Liquid Crystals: Nature’s Delicate Phase of Matter. Princeton
University Press, 2nd edition, 1990.
[3] T. J. Sluckin, D. A. Dunmur, and H. Stegemeyer. Crystals That Flow. Taylor
& Francis, 2004.
[4] D. Demus and L. Richter. Textures of Liquid Crystals. Verlag Chemie, New
York, 1978.
[5] Iam-Choon Khoo. Liquid Crystals: Physical Properties and Nonlinear Optical
Phenomena. John Wiley & Sons, Inc., 1995.
[6] P. G. de Gennes. The Physics of Liquid Crystals. Clarendon Press. Oxford,
1974.
[7] P. J. Collings and M. Hird. Introduction to Liquid Crystals. Taylor & Francis,
1997.
[8] B. J. Frisken. Nematic Liquid Crystals in Electric and Magnetic Fields. PhD
thesis, The University of British Columbia, 1989.
[9] L. Benguigui. Dielectric relaxation in the crystalline smectic-B phase. Phys. Rev.
A, 28(3):1852, 1983.
[10] M. J. Stephen and J. P. Straley. Physics of liquid crystals. Rev. Mod. Phys.,
46(4):618, 1974.
[11] R. Ribotta and A. Joets. Oblique roll instability in an electroconvective
anisotropic fluid. Phys. Rev. Lett., 56(15):1595, 1986.
[12] M. Treiber, N. Éber, Á. Buka, and L. Kramer. Traveling waves in electroconvection of the nematic phase 5: a test of the weak electrolyte model.
J. Phys. II France, 7:649, 1997.
[13] G. Dangelmayr, G. Acharya, J. Gleeson, I. Oprea, and J. Ladd. Diagnosis of spatiotemporal chaos in wave-envelopes of an electroconvection pattern, submitted.
Phys. Rev. E, 2008.
[14] I. S. Aranson and L. Kramer. The world of the complex Ginzburg-Landau equation. Rev. Mod. Phys., 74(1):99, 2002.
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[15] M. Treiber and L. Kramer. Coupled complex Ginzburg-Landau equations for
the weak electrolyte model of electroconvection. Phys. Rev. E., 58(2):99, 1973.
[16] P. E. Cladis and P. Palffy-Muhoray, editors. Spatiotemporal Patterns in Nonequilibrium Complex Systems, chapter one, page 19. Addison-Wesley publishing
company, 1994.
[17] D. Walgraef. Spatiotemporal Pattern Formation. Springer-Verlag New York,
Inc., 1997.
[18] M. C. Cross and P. C. Hohenberg. Pattern formation outside of equilibrium.
Rev. Mod. Phys., 65(3):854, 1993.
[19] G. Ahlers. Experiments with Rayleigh-Bénard convection, 2003.
[20] E. Bodenschatz, J. R. de Bruyn, G. Ahlers, and D. S. Cannell. Transition between patterns in thermal convection. Phys. Rev. Lett., 67(22):3078, 1991.
[21] M. A. Dominguez-Lerma, G. Ahlers, and D. S. Cannell. Rayleigh-Bénard convection in binary mixtures with separation ratios near zero. Phys. Rev. E,
52(6):6159, 1995.
[22] U. Bisang and G. Ahlers.
Bifurcation to worms in electroconvection.
Phys. Rev. E, 60(4):3910, 1999.
[23] S. W. Morris, E. Bodenschatz, D. S. Cannell, and G. Ahlers. Spiral defect chaos
in large aspect ratio Rayleigh-Bénard convection. Phys. Rev. Lett., 71(13):2026,
1993.
[24] S. W. Morris, E. Bodenschatz, D. S. Cannel, and G. Ahlers. The spatiotemporal
structure of spiral-defect chaos. Physica D, 97:164, 1996.
[25] G. Ahlers. Low-temperature studies of the Rayleigh-Bénard instability and turbulence. Phys. Rev. Lett., 33:1185, 1974.
[26] G. Ahlers. Experiments on spatiotemporal chaos and reference there in. Physica
A, 249:18, 1998.
[27] J. T. Gleeson. Sustained spatiotemporal chaotic flow at onset of electroconvection in nematic liquid crystals. Physica A, 239:211, 1997.
[28] S. Zhou and G. Ahlers. Spatiotemporal chaos in electroconvection of homeotropically aligned nematic liquid crystals. Phys. Rev. E, 74:046212, 2006.
[29] M. Dennin, G. Ahlers, and D. S. Cannell. Chaotic localized states near the onset
of electroconvection. Phys. Rev. Lett., 77:2475, 1996.
[30] M. Dennin, G. Ahlers, and D. S. Cannell. Spatiotemporal chaos in electroconvection. Science, 272:388, 1996.
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[31] I. Oprea, I. Triandaf, G. Dangelmayr, and I. I. B. Schwartz. Quantitative and
qualitative characterization of zigzag spatiotemporal chaos in a system of amplitude equations for nematic electroconvection. Chaos, 17:023101, 2007.
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CHAPTER 2
Theoretical background
There are different tools to describe the phenomena observed in nematics under
the influence of an external field. Section 2.1 describes the technique of linear stability analysis and amplitude equations. Section 2.2 presents the relevant equations
that describe the nematics and section 2.3 describes the extension of the standard
model (SM) called the weak electrolyte model (WEM). Many symbols that I use here
have their usual meanings. All vectors are denoted by bold face letters. In cartesian coordinates, I will denote ( x, y, z)=r, the partial derivatives ∂i = ∂/∂xi and
Kronecker delta function by δij . Einstein’s summation conventions are applied for
repeated indices.
2.1 Theoretical tools
The pattern-forming systems as discussed in Chapter One are spatially extended systems and they have some basic features. In this section, I will describe the
linear instabilities and the amplitude equations.
2.1.1 Linear instabilities
Since one can not define a free energy for pattern-forming systems and no thermodynamic extrema are involved, the transition from the homogeneous conductive
state to the convective state is not a phase transition. However, there is a sudden
qualitative change in the characteristics of a solution of a set of equations when the
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control parameter is varied. This is known as bifurcation. It is always within the
same state of matter; in contrast a phase transition is between two different states.
Consider a spatially infinite uniform system near equilibrium. The control parameter, R, takes the system away from equilibrium. At some threshold value R = Rc ,
the system may become unstable with wave vector qc and/or to a fluctuation of particular mode with frequency ωc . When R > Rc , the pattern grows in amplitude,
which depends upon some power of R − Rc . Stationary patterns appear for ωc = 0
and oscillatory patterns for ωc 6= 0. There are four types of local bifurcations of fixedpoint solutions that depend on a single control parameter R [1]. The Hopf bifurcation
is one of them, and it is of the form
∂t u1 = Ru1 − u2 + go u1 (u21 + u22 ),
∂t u2 = u1 + Ru2 +
go u2 (u21
+
(2.1)
u22 )
Here, u1 and u2 are the wave amplitudes. In Eq. 2.1, if go = −1, the normal form has
equilibrium at the origin, which is asymptotically stable for R ≤ 0 (weakly at R = 0)
and unstable for R > 0. There is a unique and circular limit cycle that exists for R > 0
√
and has radius R. This is called the supercritical (forward) Hopf bifurcation. On
the other hand, if go = +1, the origin in the normal form is asymptotically stable for
R < 0 and unstable for R ≥ 0 (weakly at R = 0). There exists a unique and unstable
limit cycle for R < 0 and this is called the subcritical (backward) Hopf bifurcation.
The Hopf bifurcation is a richer phenomenon than the steady state bifurcation in
the sense that it leads to time-dependent nonlinear behavior. A supercritical Hopf
bifurcation in the experiment implies a spontaneous onset of oscillatory behavior. The
difference between the various bifurcations are due to the difference in the symmetry
of equations.
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The basic pattern that emerges in the convecting state of RBC, TVF or EHC
is either stationary or an explicitly time-dependent, traveling mode. For a system
consisting of nonlinear PDEs, the state U, which describes the physical field, takes
the form
∂t U = G[U, ∂x U, ..., R]
(2.2)
Here, R is the control parameter of the system. In the uniform state, U = 0 is the
solution for all values of R. For EHC, the state U includes the velocity field, the
director field and the charge density, and G(U) includes the Navier -Stokes equation,
the electrostatic equations and the constraints, such as conservation of charge and of
angular momentum. Considering the rescaled control parameter, ε = R/Rc − 1, the
instability occurs when ε > 0. In the supercritical case, the homogeneous U = 0 state
becomes linearly unstable at ε = 0. For ε > 0, the system will be pushed from the
conducting state into the convecting U 6= 0 state. For this supercritical bifurcation,
the amplitude of the convecting pattern grows like the square root of ε above onset.
When the control parameter increases and passes through Rc , the amplitude of the
experimentally observed mode begins to grow continuously. Thus, the supercritical
bifurcation is similar to a second order phase transition.
In the case of subcritical bifurcation, when ε increases through zero, the U=
0 state loses stability and the system will end up on some U 6= 0 branch. When ε
decreases, the system does not follow its initial path, but jumps back to U = 0 state.
This hysteresis and discontinuous jump of the amplitude is similar to a first order
phase transition.
When the control parameter is slightly changed, the infinitesimal perturbation
causes the new state U = Uo + δU . For simplicity, I will consider the pattern-forming
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system in 1D and Uo = 0. Assuming the system is spatially infinite, the perturbation
can be expanded in Fourier modes δU =
P
j
uj (x, t) and uj (x, t) = ujo ei(qx−ωt) + cc for
the wave vector q. Here, cc denotes the complex conjugate, uj describes the physical
field and ujo is some basic mode. The growth rate Imω of each mode q behaves as
shown in Fig. 2.1. The homogeneous basic state is stable if all modes are decaying
(Imω < 0) and this is the case for ε < 0. For ε = 0, the instability sets Imω = 0
at bf q=qc . For ε > 0, there is a narrow band of wave vectors bf q − < q < q+ for
which the uniform state is unstable. The instability in Fig. 2.1 can be of two types:
either stationary if Reω = 0 or oscillatory if Reω = ωo 6= 0. So the transition from
stationary to traveling wave patterns is that ωo changes from zero to nonzero value.
Thus, we can distinguish the patterns that grow beyond the threshold into three types
of instabilities. (a) Type Is , stationary periodic ( ωo = 0, qo 6= 0); instabilities are
Imω
q
c
q
ε>0
ε<0
ε=0
Figure 2.1: Schematic representation of the growth rate as a function of the wave
vector q for various values of rescaled control parameters.
spatially periodic and stationary in time. There is the possibility of stationary rolls or
the superposition to form regular patterns such as rectangular, square or hexagonal
in 2D. The region occupied by the (R, q) plane where these stationary patterns exist
is called ‘stationary balloon’.
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(b) Type Io , oscillatory and periodic (ωo 6= 0, qo 6= 0) instabilities are spatially periodic and oscillatory in time. The ideal patten in this case involves oblique traveling
and normal traveling modes, superposition of left and right traveling zig and zag
modes to form standing modes, etc.
(c) Type IIIo (ωo 6= 0, qo = 0) instabilities are spatially uniform and oscillatory in
time. The ideal state does not exhibit any spatial pattern. The case IIIs (ωo = 0,
qo = 0) does not involve pattern formation and will be omitted.
2.1.2 Amplitude equations
In the weakly nonlinear regime, i.e., very near to the threshold of pattern-forming
instabilities, the pattern can be considered as slow modulations in space and time
of a simple basic structure. These slow modulations near the threshold for an instability can be described by the amplitude equations. Consider a system in a basic
homogeneous state (for example, the purely conductive state in RBC). It can reveal
a finite wavelength instability when one of its control parameters is varied. The basic
principle of the derivation of the amplitude equations is the same for many types of
pattern-forming systems. It includes the expansion of the solution U of the full equation of motion, writing the leading term as a product of a slowly varying amplitude
and a primary pattern of faster dependency in space and/or time.
Consider the plane-wave growing solution very near to the threshold. For
steady state instability (qo 6= 0, ωo = 0), the dynamics of the pattern near onset can
be expressed as
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U (x, t) ∝ [A(x, t)eiqc x + cc] + hh
(2.3)
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Here, I have assumed 1D rolls. A(x, t) is the amplitude or the envelope, cc stands
for the complex conjugate and hh stands for higher harmonics proportional to eiqc nx .
To lowest order in ε, the envelope satisfies the equation of the form
τo
∂A
∂2A
= ξo2 2 + εA − go | A |2 A
∂t
∂ x
(2.4)
The coefficients τo , ξo and go can be calculated from the complete equations describing
the physical problem under study. By suitable choice of space, time and amplitude
scales, these coefficients can be scaled out. One should not scale out go , because
for its positive value, the nonlinear term is stabilizing and it results a supercritical
bifurcation, while a negative go gives rise to a destabilizing effect on the amplitude
and the bifurcation is subcritical. The rescaled envelope equation for the supercritical
bifurcation is
∂A
∂ 2A
=
+ εA− | A |2 A
2
∂t
∂x
(2.5)
Eq. 2.5 shows that the amplitude depends explicitly on ε. It can be scaled out by
√
√
rescaling x → x/ ε, t → t/ε, A → εA, which means the amplitude of the pattern
√
grows as ε. For ε > 0, Eq. 2.5 has stationary solutions of the form A = ao eiqx ,
with q 2 = ε − a2o . Since the coefficients in Eq. 2.5 are real, this is called the real
Ginzburg-Landau (RGL) equation. Physical systems which undergo an instability
described by RGL equations are RBC, TVF and flames stabilized in a burner.
In case of oscillatory instability (qo 6= 0, ωo 6= 0), the emerging pattern is
intrinsically time dependent. The linearized equation of motion will be of the form
ei(qc x−ωc t) , in which the system becomes unstable at ε = 0. In this case, Eq. 2.3 takes
the form
U (x, t) ∝ [A(x, t)ei(qc x−ωc t) + B(x, t)e−i(qc x−ωc t) + cc] + hh
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where A and B are right and left traveling wave amplitudes respectively. The amplitude equation obeyed by A is
∂A
∂2A
= (1 + iα) 2 + εA − (1 − iβ) | A |2 A
∂t
∂x
(2.7)
and similar is the case for B. This equation is called the complex Ginzburg-Landau
(CGL) equation and α and β are real coefficients. For left and right traveling zig and
zag waves, there should be four coupled CGL equations, one each for the amplitude
of left traveling zig, left traveling zag, right traveling zig and right traveling zag
waves. Depending upon the nonlinear interaction terms, either the standing waves
are favoured, or if one wave suppresses the other, only a single CGL equation is
enough. In the limit α, β → 0, Eq. 2.7 converts to Eq. 2.5. For traveling waves, there
is a band of traveling wave solutions A = a0 ei(qx−ωt) with Imω = 0. Then,
ω = αq 2 − βa0 2 ,
q 2 = ε − a0 2
(2.8)
In Eq. 2.8, the coefficient α measures the frequency dependence of the wave on the
wave number and β is a measure of the nonlinear dispersion. Physical systems which
undergo an instability described by CGL equations are RBC in binary mixtures, instability of rolls in low Prandtl number RBC and EHC in nematic liquid crystals.
Both RGL and CGL equations are from the assumption that there is a supercritical bifurcation with nonzero wave vector. The amplitude equations are generally
real or complex depending upon whether the bifurcation is stationary or oscillatory.
In the limit, α and β → ∞, CGL equation reduces to the nonlinear Schrödinger
equation which is integrable. The fact that CGL changes to RGL in one limit and to
a integrable equation in another limit makes it more interesting.
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2.2
Standard model for nematics
One of the most striking properties of a liquid crystal is its ability to flow freely
while revealing various crystalline like properties. It is this dual nature which makes
its dynamics not only more complicated but also richer than conventional hydrodynamics. The distinction between the ordinary fluid and the ordered fluid such as
nematics is that in the latter case, the physical properties depend on the orientation
of the director n. When the nematics are taken out of thermodynamic equilibrium
by an external perturbation, the translational motion couples with the orientational
motion of the molecules and the flow disturbs the alignment. On the other hand,
applying an external field will change the alignment and that may induce flow. While
flowing, a nematic continuously changes its appearance due to change in director orientation, establishing a strong coupling between the director and the velocity field.
In an ordinary liquid, the hydrodynamical variables are the density ρ, the velocity v(r,t) or the momentum density ρv and the internal energy density e. Conservation laws and propagating or damped hydrodynamic modes are connected with these
variables [2]. These modes show macroscopic relaxation times τ which depend upon
a certain positive power of wavelength. Thus, in the limit q → 0, τ → ∞. These long
lived modes are called ‘hydrodynamic modes’.
In nematics, an additional variable related to the orientational order of the
molecules comes into existence. It depends upon three independent components: the
amplitude of the order parameter S, and any two of the three components of the
director. The order parameter is not a hydrodynamic variable as any perturbation of
S irrespective of its wavelength relaxes toward the equilibrium value over microscopic
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times. On the other hand, the director is a hydrodynamic variable since the relaxation time of the distortion in orientation depends on the wavelength (τ ∼ λ2 ). This
variable is not connected with any conservation law, but rather with the symmetry
breaking at the N-I phase transition.
The standard model (SM) [3, 4] for nematics consists of the hydrodynamical
equations, which are basically the conservation of mass, linear momentum and angular
momentum and the Maxwell equations, the conservation of charge and the Coulomb’s
law, with the assumption of ohmic conductivity. Here, I will discuss molecular field,
nematodynamics, nematic viscosities and the Carr-Helfrich mechanism. As NLCs are
anisotropic in dielectric permittivity, conductivity, refractive index, magnetic susceptibility, etc., in general, the anisotropy can be written as
bij = b⊥ + ∆bni nj
(2.9)
In Eq. 2.9, ∆b = bk −b⊥ is the anisotropy of the material. The symbols k and ⊥ are for
directions parallel and perpendicular to n. For example, ∆² (²k − ²⊥ ) is the dielectric
anisotropy which is responsible for the alignment of the director by the electric field
whereas the diamagnetic anisotropy ∆χ (χk − χ⊥ ) is responsible for the alignment of
the director by the magnetic field. For ‘positive’ ∆χ materials, χk is less negative than
χ⊥ . ∆σ (σk − σ⊥ ) is the conductivity anisotropy. It is the anisotropy of the refractive
index which is responsible for the image formation of EC pattern by shadowgraph
technique [5], which will be discussed in Chapter Three. If the permanent dipole
moment of each molecule is parallel to its long axis, the dielectric constant ²k will be
greater than ²⊥ and the dipole can be oriented easily by the electric field E along the
molecular axis. The situation will be reversed if ²k is less than ²⊥ . In this case, the
permanent dipole moment of each molecule is perpendicular to the molecular axis.
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Similarly, if χk > χ⊥ , the director aligns parallel to the external magnetic field and
if χk < χ⊥ , the director will be aligned perpendicular to the magnetic field. The
sign of anisotropies can be denoted in compact notation and the liquid crystal which
undergoes EHC can be represented as (+, +), (+, -), (-, +) or (-, -). The first and
second signs in the parentheses are for ∆² and ∆σ respectively.
2.2.1 The molecular field
The electric field E at an arbitrary angle to the director, produces electric
displacement D given by
D = ²o ²⊥ E + ²o ∆²(n · E)n
(2.10)
The total electric energy density arising when a fixed voltage is maintained is
Fele = −
Z E
0
D · dE
(2.11)
Using Eq. 2.10, it simplifies to
1
Fele = − (²o ²⊥ E 2 + ²o ∆²(n · E)2 )
2
(2.12)
The first term is independent of n and is usually omitted. Then,
1
Fele = − ²o ∆²(n · E)2
2
(2.13)
Thus, the electric energy density depends upon the sign of ∆², the angle between
the direction of the director and the direction of the electric field and its magnitude.
For ∆² > 0, the energy is minimized when n is parallel to E and for ∆² < 0, it is
minimized when n is perpendicular to E. Assuming absence of free charge, Maxwell’s
equations for electric displacement D and the electric field E are
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∇ · D = 0,
∇×E=0
(2.14)
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Similarly, in the magnetic field, if H makes an arbitrary angle with the director n,
and the magnetisation M induced by H is
M = χ⊥ H + ∆χ(n · H)n
(2.15)
The magnetic induction B in the presence of magnetic field H, which plays the similar
role as that of electric displacement D in the presence of electric field is given by
B = µo µ⊥ H + µ0 ∆χ(n · H)n
(2.16)
The energy density in the presence of magnetic field is analogous to that in the
presence of electric field and is given by
1
Fmag = − µo ∆χ(n · H)2
2
(2.17)
Thus, the magnetic energy density is a function of ∆χ, the angle between the direction
of the director and the direction of the magnetic field and its magnitude. If ∆χ > 0,
the energy is minimized when n is parallel to H and if ∆χ < 0, it is minimized when
n is perpendicular to H. The magnetic induction B and the magnetic field H must
satisfy the Maxwell’s field equations.
∇ · B = 0,
∇×H=0
(2.18)
Three types of basic deformations that occur in NLCs are splay, twist and
bend deformations, as shown in Fig. 2.2. I will explain these phenomena in details in
Chapter Six. The Frank-Oseen elastic free energy density in the NLCs is then,
1
1
1
Fd = K11 (∇ · n)2 + K22 (n · ∇ × n)2 + K33 (n × ∇ × n)2
2
2
2
(2.19)
where K11 , K22 and K33 are splay, twist and bend elastic constants [6, 7]. They are
of the order of 10−11 N and usually K33 > K11 > K22 . In the presence of external
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(a)
(b)
(c)
Figure 2.2: A schematic of the basic deformation in NLCs. The ellipsoids are for the
nematic director orientation after deformation: Examples of (a) pure splay deformation (b) pure twist deformation and (c) pure bend deformation.
electric and magnetic fields, the total free energy density is
1
1
1
1
1
Ftot = K11 (∇·n)2 + K22 (n·∇×n)2 + K33 (n×∇×n)2 − ²o ∆²(n·E)2 − µ0 ∆χ(n·H)2
2
2
2
2
2
(2.20)
R
In equilibrium, the free energy is F = F d3 r, where the integration is over the sample
volume taking into account the fact that n2 = 1. Then, using the Euler-Lagrange
equations with adequate Lagrange multipliers, one gets
Ã
δF
δn
!
=
i
∂F
∂ ∂F
−
= −λ(r)ni
∂rj ∂ni,j
∂ni
(2.21)
Here ni,j = ∂ni /∂rj , ni = nx , ny , nz , rj = x, y, z. The quantity hi is introduced by
Ã
hi =
δF
δn
!
=
i
∂ ∂F
∂F
−
∂rj ∂ni,j
∂ni
(2.22)
Then, Eq. 2.21 modifies to hi + λ(r)ni = 0. Thus, in equilibrium, the director at
each point is parallel to the molecular field h. Since, the cross product of two parallel
vectors is zero, a necessary but insufficient condition that is independent of λ(r) is
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n×h=0
(2.23)
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From Eq. 2.19 and Eq. 2.22, the total molecular field h due to pure splay, pure twist
and pure bend deformations can be expressed as
hd = hs + ht + hb
(2.24)
Using the notation he and hm for the contribution to the molecular field due to
the external electric and magnetic fields, the total molecular field will be htot =
hd + he + hm . In EHC, when the nematic system is out of equilibrium, the director
distortion exerts bulk torque and the torque per unit volume is
Γ=n×h
(2.25)
For example, the torque due to a magnetic field on NLCs when H and M are at
arbitrary angle is
Γm = M × H
(2.26)
Using Eq. 2.15 and Eq. 2.25, the contribution in the molecular field in the presence
of the magnetic field is
hm = ∆χ(n · H)H
(2.27)
Thus, in the magnetic field, the system will be in equilibrium if
Γd + Γm = 0
(2.28)
From Eq. 2.25 and Eq. 2.28, it can be simplified as
n × hd + n × h m = 0
(2.29)
n × hd + ∆χ(n · H)n × H = 0
(2.30)
or
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or
n × hd = −∆χ(n · H)n × H = −Γm
(2.31)
Thus, for equilibrium in the magnetic field, the torque due to the elastic deformation
should be equal and opposite to the torque due to the magnetic field.
2.2.2 Dynamic theory of nematics
Here, I will discuss in brief about the ‘nematodynamics’ explained by Ericksen
and Leslie [6–8]. With the assumption of incompressible nematics, the constraints on
the velocity and the director fields are
vi,i = 0, ni ni = 1
(2.32)
The momentum balance equation for incompressible fluid is
ρ(
∂v
+ (v · ∇)v) = −∇p + η∇2 v + f
∂t
(2.33)
Here, ρ is the mass density, f is the volume force and it is given by f = ρe E in the
electric field. In Eq. 2.33, the left hand side is the inertia term which is a sum of the
unsteady acceleration term ∂v/∂t and the convective acceleration term (v · ∇)v. The
right hand side consists of the divergence of stress (the sum of the pressure gradient
and the viscosity term) and the other body force. This equation is the Navier-Stokes
equation for an incompressible and isotropic fluid. For nematics, the stress tensor is
of the form [3, 8]
Ã
∂F
Tij = −pδij −
∂nk,j
!
nk,i + tij
(2.34)
In Eq. 2.34, p is the pressure, F is the total free energy and t is the dissipative part
of the viscous stress tensor related to six viscosity coefficients [3, 7, 8].
tij = α1 Akp nk np ni nj + α2 Ni nj + α3 Nj ni + α4 Aij + α5 Aik nk nj + α6 Ajk nk ni (2.35)
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with
1
Aij = (vj,i + vi,j ),
2
1
Ni = ṅi − (vi,k − vk,i )nk
2
(2.36)
where A is the symmetric strain rate tensor and N is the rate of change of the director
relative to the moving fluid. αi ’s are called Leslie viscosity coefficients and are on
the order of 10−3 N sm−2 . For a normal isotropic fluid, only the term α4 Aij remains.
These equations are independent of the replacement of n by -n and reflects the absence
of polarity in NLCs.
The equation of motion for the director is related to the torques Γ and its
moment of inertia (I) [9] as
I
d
(n × dn/dt) = Γtot + Γvisc
dt
(2.37)
where Γtot is the sum of the torque per unit volume on the director due to elastic,
magnetic and electric forces given by
Γtot = −n × htot
(2.38)
The viscous torque in vector form [3, 9] is
Γvis = −n × (γ1 N + γ2 A · n)
(2.39)
γ1 and γ2 in Eq. 2.39 are the shear viscosity coefficients and they are related to the
Leslie viscosity coefficients by
γ1 = α3 − α2
(2.40)
γ2 = α 3 + α 2
γ1 is the viscosity coefficient when the shear is in the xz-plane with the velocity
v lying within the plane of shear and n k v and ∇v along ẑ. It characterizes the torque
associated with the rotation of n. For this reason, γ1 is often called the rotational
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No torque
No contribution to stress
α1
(a)
α5
Torque not equal to zero
(b)
α6
Torque not equal to zero
(c)
( α >0)
3
α2
α3
Torque not equal to zero
(d)
α4
No dependence on n
No torque
Figure 2.3: Illustration of geometries for Leslie viscosity coefficients.
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viscosity or twist viscosity. It generally determines the director relaxation rate. The
coefficient γ2 contributes to the torque exerted on the director due to shear flow and
referred to as the torsion coefficient. These viscosity coefficients do not have their
counterparts in an isotropic fluid. Eq. 2.39 shows that for rotational flow, the viscous
torque is proportional to γ1 and vanishes for N = 0, i.e., the director rotates with
the same angular velocity as the fluid. On the other hand, in irrotational flow, the
viscous torque is proportional to γ2 and vanishes when the director is aligned along
Aij tensor axis.
The total torque per unit volume acting on the director in the absence of external
fields is the sum of the contributions from Eq. 2.25 and Eq. 2.39 as
Γtot = n × hd − n × (γ1 N + γ2 A · n) = n × [hd − (γ1 N + γ2 A · n)]
(2.41)
2.2.3 Nematic viscosities
In Eq. 2.35, actually there are only five independent viscosity coefficients to
describe the dynamics of an incompressible nematic due to the Onsager reciprocity
relation
α2 + α3 = α6 − α5
(2.42)
The fourth term in the right hand side of Eq. 2.35 does not contain information
about the director and is determined only by the fluid velocity field. With the exception of this coefficient α4 , the Leslie viscosities are not identified individually, but
their certain combinations are identified experimentally. Miesowicz [6] introduced
four viscosity coefficients that could be independently measured experimentally by
considering the director orientation with respect to the flow velocity. Assuming the
director orientation with respect to v and ∇v, the Leslie viscosities are the linear
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combination of these three principal Miesowicz viscosities.
α0 s and η 0 s are shown in Fig. 2.3 and in Fig. 2.4. The coefficients α2 and α3 are
(a)
(b)
(c)
(d)
Figure 2.4: Illustration of geometries for Miesowicz viscosity coefficients; (a) ηa : n ⊥
v, n ⊥ ∇v, (b) ηb : n k v, (c) ηc : n ⊥ v, n k ∇v and (d) η12 = α1 .
contained in director angular velocity Ni and do not appear in the terms containing
velocities as well as their gradients. The coefficient α1 corresponds to tensile strain
and it is negative for calamitic LCs. Since the rate of entropy production must be
positive, Leslie coefficients satisfy the following inequalities [10, 11].
α4 ≥ 0
2α1 + 3α4 + 2α5 + 2α6 ≥ 0
(2.43)
2α4 + α5 + α6 ≥ 0
(α3 − α2 )(2α4 + α5 + α6 ) ≥ (α2 + α3 )2 )
γ1 ≡ α3 − α2 ≥ 0
(2.44)
Out of six viscosity coefficients, the sign of α2 and α3 are important to describe
flow behavior of NLCs. With the restriction of Eq. 2.44, if
(a) α2 α3 < 0: It is possible only when α2 < 0 and α3 > 0. Positive α2 and negative
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α3 is prohibited. The behavior raised in some NLCs with this condition is called
‘tumbling’.
(b) α2 α3 > 0: There are two possibilities (b1 ) α2 < α3 < 0, i.e., both are negative.
Then, it must be | α2 |>| α3 |, again due to the restriction of Eq. 2.44. It gives rise to
flow alignment [12] which is the most common phenomenon in NLCs. Positive value
of both of these α0 s would also give rise to flow alignment.
(b2 ) α3 > α2 > 0 which is possible for disc like molecules [10].
Three out of four Miesowicz viscosity coefficients η’s (Fig. 2.4) are related to the
Leslie viscosity coefficients as
1
ηa = α4
2
1
ηb = (α3 + α4 + α6 )
2
ηc = 12 (−α2 + α4 + α5 ) = ηb − γ2
(2.45)
Besides these relations, a stretching type deformation as shown in Fig. 2.4(d)
is also possible and called η12 = α1 . All these viscosities depend on temperature and
pressure [13–15]. I will discuss the measurement of these viscosity coefficients, except
α1 , in Chapter Six.
2.2.4 Carr-Helfrich mechanism and the threshold voltage
When an electric field is applied to a thin layer of (-,+) or slightly (+,+) NLC,
at or near the onset of a certain voltage called the threshold (Vc ), some convective
instabilities with the periodic director distortion can be explained by Carr-Helfrich
mechanism [16, 17].
Consider the simplest case with a thin layer of liquid crystal in the planar
geometry, with (-,+) sample and an electric field applied perpendicular to the director.
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λ
+
−
+
−
+
~ V
d
+
Z
(a)
X
λ
Y
+
−
+
~ V
d
+
−
+
(b)
Figure 2.5: Cross section of a roll pattern from different geometry, double arrows
denote the director modulations and the symbols + and - denote the positive and
negative induced charges. (a) Planar geometry. (b) Homeotropic geometry.
Consider the slight distortion of the director field, so that the angle formed by the
director with the x̂-axis in the xz-plane is θ(x) = θo cos(qx̂ · x) where q is the wave
vector along x̂. The elastic deformation force tries to restore the uniform director
field whereas the force due to the electric field tries to realign the director in the
field’s direction as shown in Fig. 2.5(a). The ions which are current carriers in the
nematic phase have greater mobility in the preferred direction of the molecules than
perpendicular to it. This anisotropy in conductivity causes space charge due to the
ion segregation. The current J is then related to the electric field as
J = σ⊥ E⊥ + σk Ek = σ⊥ E + ∆σ(E · n)n
(2.46)
Here, Ek and E⊥ are the electric fields parallel and perpendicular to the director.
This induces an x-component to J, which leads to charge accumulation ρ(x).
An additional electric field δE is developed along x̂. The charges, moving under the
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influence of the electric field drag the LC molecules with them creating a flow. This
flow helps to maintain the distortion. The space charge distribution which causes convection can be expressed mathematically in terms of the charge conservation equation
and the Gauss’ law simultaneously:
∂Ex
∂Ez
∂ρ
+ σ⊥
=
∂x
∂z
∂t
(2.47)
∂Ex
∂Ez
∇ · D = ²o ²k
+ ²o ²⊥
= ρ(x)
∂x
∂z
Thus, when the uniform solution for the electric field becomes unstable, a spatial
∇ · J = σk
distribution of charge will arise. Consider the dc electric field for simplicity, and the
distortion is only along x̂. Then, Gauss’ law can be expressed as
∂Dx
= ρ(x)
∂x
(2.48)
Assuming the contribution from the viscosity is much greater than that due to
the pressure and the inertial force, the Navier-Stokes equation (Eq. 2.33) for an
anisotropic fluid can be expressed as
η∇2 vz + ρ(x)Ez = 0
(2.49)
where η is defined in Eq. 2.45. The balance of torque is
Γd + Γele| + Γvis = 0
(2.50)
For simplification, say ∆² = 0, which removes the electric torque. Then, the balance
of torque for the applied critical field Ec will be
K33
∂ 2θ
∂vz
− α2
=0
2
∂x
∂x
(2.51)
In the steady-state regime, ρ is independent of time and ∇ · J = 0. To the first order
in θ, Jx = σk Ex + ∆σEc θ. So,
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σk
∂Ex
∂θ
+ ∆σEc
=0
∂x
∂x
(2.52)
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Using Eq. 2.48 and Eq. 2.52,
ρ(x) = ²o ²
∆σ
qEz θo sin(qx)
σk
(2.53)
Thus, the charge density depends on the applied electric field Ec , the director distortion θo , the conductivity anisotropy and the dielectric constant. The torque balance
Eq. 2.51 gives ∂vz /∂x = −K33 qc2 θ/α2 or ∂ 2 vz /∂x2 = K33 q 3 θo sin(qc x) = −ρ(x)Ec /η.
Thus,
Ec2 = −
1 σk ηK33 2
q
²o ² ∆σ α2 c
(2.54)
where qc is the critical wave number. Usually it is expressed as qc ≈ π/d. Expressing
the critical field in terms of critical potential, one gets
Vc2 = −
π 2 σk ηK33
²o ² ∆σ α2
(2.55)
Eq. 2.55 indicates that an increase in the deformation force and the fluid viscosity increase the threshold voltage. On the other hand, an increase in α2 , which
couples the director to the fluid motion, reduces the threshold. The critical field is
proportional to
√
σk , i.e., more conducting liquid crystal makes it easier to carry cur-
rent. Thus, the threshold increases with increasing conductivity.
Since Eq. 2.54 contains only one wave vector qc , this model only explains the
normal rolls and is silent about the oblique wave instability. Also, since the critical
voltage does not depend on the thickness of the sample cell, it is taken as the control
parameter and not the critical field. The threshold voltage I calculated is the most
trivial and considers 1D type distortion with the dc electric field neglecting the flexoelectric effect. This analytical value of the threshold as given by Helfrich, taking into
account of dielectric anisotropy is
Vc2 = −
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π 2 K33
α2 ²o ²k (²⊥ /²k − σ⊥ /σk )/η + σ⊥ ²o ∆²/σk
(2.56)
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Taking ∆² = 0, Eq. 2.56 yields Eq. 2.55. For NLCs with ∆² < 0, Eq. 2.56 gives
higher threshold than that by the simplified expression of Eq. 2.55.
To explain the experimentally observed oblique wave instability, the Bayreuth
group developed a model [3, 18]. This model also uses the hydrodynamic equations
(assuming incompressible fluid), coupled with Gauss equation and charge conservation for the ac electric field, and the balance of torque for the director field. The ac
field removes the ion segregation effect. The nonlinear equations have six unknown
variables: two director components, three velocity components and the electric potential. Linearizing the equations and expanding the variables in Fourier series, the
neutral curve gives the threshold as a function of both wave vectors q= (q, p). The
minima of the curve gives both the threshold value and the oblique angle of the rolls.
Vc2 =
π 2 K eff
²o ∆²eff + | α2 | τq ∆σ eff /η eff
(2.57)
The effective parameters K eff > 0, ∆²eff < 0, ∆σ eff > 0, η eff > 0 are proportional
to corresponding physical quantities: elastic constants, dielectric and conductivity
anisotropies and viscous coefficients, respectively. The linear analysis of SM shows
that the patterns at the threshold can be either stationary normal rolls with the wave
vectors qc = (qc , 0), or a pair of degenerate oblique rolls with the wave vectors of zig
and zag patterns as (qc )zig = (qc , −pc ) and (qc )zag = (qc , pc ).
The SM contains three time scales: the director relaxation time τd is related
to the rotational viscosity γ1 of the LC, the viscous relaxation time τvisc is related
to the kinematic viscosity and the charge relaxation time τq is related to the normal
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components of dielectric constant and the conductivity. They are expressed as
γ1 d2
π 2 K11
ρd2
τvisc =
α4 /2
²o ²⊥
τq =
σ⊥
τd =
(2.58)
Typically τd ∼ 1 s is usually the longest, τvisc ∼ 10−5 s is the shortest [19] and
τq ∼ 10−3 s is the intermediate time scale. Thus, one can claim only two dynamically
active fields; director and charge density in the low frequency ‘conduction regime’.
With nonzero conductivity, when the applied voltage reaches the threshold voltage
and the frequency of electric field is lower than the charge relaxation frequency, a
periodic alignment of the director appears. It is called the ‘Williams domains’. The
spatial periodicity is in the order of sample thickness and the pattern is independent
of the reversal of electric field. In this regime, the charge density will oscillate with
external field, but the director and velocities are stationary. The threshold voltage
is independent of the sample thickness but depends upon the elastic constants, the
dielectric constants, the applied ac frequency and the wave vectors.
When the voltage is increased further above Vc , the distortion amplitude becomes pronounced and the flow velocity increases. The flow is turbulent, long range
orientational order is destroyed and the nematic is capable of light scattering. It is
therefore called dynamic scattering mode (DSM). In the high frequency region, the optical pattern of the perturbed state will have shorter spatial periods than the classical
Williams domains. Above the threshold, these striations look like ‘chevron patterns’.
This regime is called the ‘dielectric regime’. The frequency of applied voltage below
which the regime is conductive and above which it is dielectric is called the cut-off
frequency Fc . It is found to increase with the sample conductivity. Hence, one can
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expect chevron patterns with pure LC sample in an ac field. In the dielectric regime,
ωo τq >> 1 where ωo = 2πF , the threshold is much higher and it is proportional to the
sample thickness. Hence, one can take Ec = Vc /d as the control parameter instead
√
of Vc and Ec ∝ ωo . I will explain the threshold curve and patterns in conduction
and dielectric regimes with necessary shadowgraph images in Chapter Four and the
measurement of time scales τq and τd in Chapter Six.
Many features in the conductive regime at low frequencies like the threshold
voltage as function of external frequency, possibility of oblique and normal rolls, roll
angles are quantitatively described by SM, neglecting the flexoelectric effect [20, 21].
Although the Hopf bifurcation leading to traveling rolls were experimentally observed
in various NLCs like MBBA [22], I52 [23], and Phase 5 [24–26] during electroconvection since last few decades, SM could not describe it either quantitatively or qualitatively. Treiber and Kramer have developed a theory to explain Hopf bifurcation
leading to traveling rolls in the conduction regime which they termed the ‘weak electrolyte model’ (WEM).
2.3 Weak electrolyte model
Traveling rolls during electroconvection in NLCs are found only in thin cells,
only below certain threshold conductivities for the given cell thickness and only within
certain external frequency range. The weak electrolyte model (WEM) [27, 28], which
is an extension of the standard model, explains the Hopf bifurcation leading to traveling rolls during electroconvection in certain nematics very well. WEM also has
same equations for director and velocity fields as that of SM but the idea of ohmic
conductivity is dropped. Instead, it is extended to an electrodiffusion model with
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two active ionic charge carriers e+ and e− with number densities n+ (r, t) and n− (r, t)
respectively. These ionic species have constant mobility tensors µ±
ij :
±
±
µ±
ij = µ⊥ δij + ∆µ ni nj
(2.59)
These mobilities and their anisotropy are independent of number densities and the
electric field E. Also, no transfer of electric charges through the electrodes is assumed,
i.e., Jz+ (z = ±d/2) = Jz− (z = ±d/2) = 0. Blocking electrodes imply that the average
charge density per unit area is constant. In general, the conductivity of a fluid is
related to charge q and mobility µ of the charge carriers as σ = µq 2 . Also, one has
µ = vl/3kT , where v is the mean velocity of the charge carriers, l is their mean free
path, k is the Boltzmann constant and T is the absolute temperature. For anisotropic
fluids, the mean free path along and perpendicular to the director are different and
l⊥ > lk . The ratio of conductivities can be expressed in terms of the mean free path
by
σk
vk l k
=
σ⊥
v⊥ l ⊥
(2.60)
±
±
The ion’s steady state velocity in the presence of electric field is v(⊥,k)
= µ±
(⊥,k) E(⊥,k)
+
−
−
with µ+
k /µ⊥ = µk /µ⊥ and ∆µ/µ⊥ = ∆σ/σ⊥ . The space charge density ρ(r, t) is
given by ρ(r, t) = e[n+ (r, t) − n− (r, t)] with
R
n+ d3 r =
R
n− d3 r and the conductivity
tensor σij (r, t) is given as σij (r, t) = σ⊥ (r, t)[δij + ni nj (µk /µ⊥ − 1)]. Here, ni are the
− −
+
director components and σ⊥ (r, t) = e[µ+
⊥ n (r, t) + µ⊥ n (r, t)]. Neglecting diffusion,
the balance equation for ρ(r, t) and σ⊥ (r, t) are
∂t ρ + ∇ · (vρ + ∆µEσ⊥ ) = 0
(2.61)
eq
−
0
−1
∂t σ⊥ + ∇ · (vσ⊥ + µ+
⊥ µ⊥ ∆µ Eρ) = −τrec (σ⊥ − σ⊥ )
(2.62)
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Besides these two equations, the WEM consists of the standard model equations for the director and the velocity field [29]. The dynamic equations for n(r,t)
and v(r,t) which are common in both SM and WEM are the Erickson-Leslie equations explained earlier. There are six fields of the WEM, namely the potential φ of
the local electric field distortion, local conductivity σ⊥ , two director components (ny ,
nz ) and two velocity fields.
The WEM has two more relevant time scales [23, 27]: (a) the recombination time τrec = 1/2Kr no for the relaxation toward the equilibrium of dissociationrecombination reaction and (b) the carrier transition time τt = d2 /(π 2 Vc0 µ0 ) where
the mobility µ0 =
q
−
µ+
⊥ µ⊥ , the applied ac voltage Vc0 =
q
K11 /²o ∆², which is of the
order of critical voltage and τt ∼ 0.1 s.
There are five dimensionless parameters: the distance from threshold ε =
q
V 2 /Vc2 − 1, external normalized frequency ωo τq , mobility parameters α̃ = π τq τd /τt2 ,
recombination parameter r̃ = τd /τrec and the charge relaxation parameter τq /τd =
²o ²⊥ K11 π 2 /σ⊥ γ1 d2 . Finally, the analytic expression for the circular Hopf frequency
ωH (= 2πfH ) is
q
ωH = Ω 1 − (1/τrec Ω)2
where
(2.63)
v
u
+ −
V 2 ²o ²⊥ u
t µ⊥ µ⊥
Ω = πC 0 3 c
d (1 + ω 02 ) γ1 σ⊥
(2.64)
and
ω 0 = ωo τq
Here, C 0 =
q
1 + (d/π)2 (qc2 + p2c + ∆²qc2 /²⊥ )
1 + (d/π)2 (qc2 + p2c + ∆σqc2 /σ⊥ )
(2.65)
∆σ/σ⊥ C is a dimensionless parameter with C given in [27], Vc is the
threshold voltage, qc and pc are the critical wave numbers along the direction of the
director n and perpendicular to it respectively. Other symbols have already appeared
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in above discussions.
In Eq. 2.63, the condition for Hopf bifurcation to traveling waves is that the ex−1
−1
pression inside the square root must be positive, so τrec
< Ω. The condition τrec
=Ω
gives the codimension-two point separating the traveling wave regime and the stationary wave regime. Also, the traveling mode is favored for thin cells and low con1/2
ductivity of the sample as Ω is proportional to 1/(d3 σ⊥ ).
The standard model is recovered in the limit τt /τq >> 1, τt ωo >> 1 and
q
τd /τq τt /τrec >> 1. WEM explains successfully on the origin of traveling rolls on
I52 [23]. Eq. 2.63 implies that the Hopf frequency depends upon the non-SM mobilq
ity parameters
−
µ+
⊥ µ⊥ . Thus, one can calculate the geometric mean of mobilities by
measuring the Hopf frequency and other parameters of Eq. 2.63. Also, for materials
having ∆² < 0, the Hopf frequency increases with increasing driving frequency.
These equations predict the continuous Hopf bifurcation [23, 24, 28] and continuous variation of wave vectors along the threshold curve. But it is valid only if all the
terms in ω’ are continuous. If any one of the term in Eq. 2.65 is discontinuous, the
Hopf frequency along the threshold curve should be discontinuous. I will explain the
experimentally observed continuous Hopf bifurcation in I52 and Phase 5 resulting in
continuous and discontinuous Hopf frequencies respectively in Chapter Four.
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BIBLIOGRAPHY
[1] M. C. Cross and P. C. Hohenberg. Pattern formation outside of equilibrium.
Rev. Mod. Phys., 65(3):851, July 1993.
[2] P. Oswald and P. Pieranski. Nematic and Cholesteric Liquid Crystals. Taylor &
Francis, 2005.
[3] E. Bodenschatz, W. Zimmermann, and L. Kramer. On electrically driven
pattern-forming instabilities in planar nematics. J. Phys. France, 49:1875, 1988.
[4] L. Kramer, E. Bodenschatz, W. Pesch, W. Thom, and W. Zimmermann. New results on the electrohydrodynamic instability in nematics; invited lecture. Taylor
& Francis, page 699, 1989.
[5] S. Rasenat, G. Hartung, B. L. Winkler, and I. Rehberg. The shadowgraph
method in convection experiments. Experiments in fluids, 7:412, 1989.
[6] I. W. Stewart. The Static and Dynamic Continuum Theory of Liquid Crystals.
Taylor & Francis, 2004.
[7] P. G. de Gennes and J. Prost. The Physics of Liquid Crystals. Clarendon press,
Oxford, 2nd edition, 1993.
[8] F. M. Leslie. Theory of flow phenomena in nematic liquid crystals. Archive for
rational mechanics and analysis, 28:265, 1968.
[9] W. J. A. Goossens. Advances in Liquid Crystals, volume 3. Academic Press,
1978.
[10] T. Carlsson. The possibility of the existence of a positive leslie viscosity α2 . proposed flow behavior of disc like nematic liquid crystals. Mol. Cryst. Liq. Cryst.,
89:57, 1982.
[11] V. V. Belyaev. Physical methods for measuring the viscosity coefficients of nematic liquid crystals. physics-Uspekhi, 44:255, 2001.
[12] T. Carlsson. Theoretical investigation of the shear flow of nematic liquid crystals
with the leslie viscosity α3 > 0: Hydrodynamic analogue of first order phase
transitions. Mol. Cryst. Liq. Cryst., 104:307, 1984.
[13] P. J. Collings and M. Hird. Introduction to Liquid Crystals. Taylor & Francis,
1997.
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[14] M. Cui. Temperature Dependence of Viscoelastic Properties of Nematic Liquid
Crystals. PhD thesis, Liquid Crystal Institute, Kent State University, 2000.
[15] M. Cui and J. R. Kelly. Temperature dependence of viscoelastic properties of
5CB. Mol. Cryst. Liq. Cryst., 331:49, 1999.
[16] W. Helfrich. Conduction-induced alignment of nematic liquid crystals: basic
model and stability considerations. J. Chem. Phys., 51:4092, 1969.
[17] E. F. Carr. Influence of electric fields on the molecular alignment in the liquid
crystal p-(anisalamino)-phenyl acetate. Mol. Cryst. Liq. Cryst., 7:253, 1969.
[18] L. Kramer and A. Buka, editors. Electrohydrodynamic Instabilities in Nematic
Liquid Crystals, chapter six. Springer, 1995.
[19] A. Buka and L. Kramer, editors. Pattern Formation in Liquid Crystals, chapter
four, page 91. Springer Verlag New York, Inc., 1996.
[20] T. Tóth-Katona, N. Éber, and Á. Buka. Flexoelectricity and competition of time
in electroconvection. Phys. Rev. E, 78:036306, 2008.
[21] A. Krekhov and W. Pesch. Nonstandard electroconvection and flexoelectricity
in nematic liquid crystals. Phys. Rev. E, 77:025705, 2008.
[22] I. Rehberg, S. Rasenat, and V. Steinberg. Traveling waves and defect-initiated
turbulence in electroconvecting nematics. Phys. Rev. Lett., 62:756, 1989.
[23] M. Dennin, M. Treiber, G. Ahlers L. Kramer, and D. S. Cannell. Origin of
traveling rolls in electroconvection of nematic liquid crystals. Phys. Rev. Lett.,
76:319, 1996.
[24] M. Treiber, N. Éber, A. Buka, and L. Kramer. Traveling waves in electroconvection of the nematic phase 5: A test of the weak electrolyte model. J. Phys.
II France, 7:649, 1997.
[25] I. Rehberg, B. L. Winkler, Manuel de la Torre Juarez, S. Rasenat, and W. Schöpf.
Pattern formation in liquid crystals. Advances in solid state physics, 29:35, 1989.
[26] Manuel de la Torre Juarez and I. Rehberg. Four-wave resonance in electrohydrodynamic convection. Phys. Rev. A, 42:2096, 1990.
[27] M. Treiber and L. Kramer. Bipolar electrodiffusion model for electroconvection
in nematics. Mol. Cryst., Liq. Cryst., 261:311, 1995.
[28] M. Treiber. On the Theory of the Electrohydrodynamic Instability in Nematic
Liquid Crystals Near Onset. PhD thesis, Universität Bayreuth,Theoretische
Physik II, Universitätstrasse 30, D-95440 Bayreuth, Germany, 1996.
[29] M. Treiber and L. Kramer. Coupled complex Ginzburg-Landau equations for
the weak electrolyte model of electroconvection. Phys. Rev. E, 58:1973, 1998.
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CHAPTER 3
Experimental methods
There are three main elements which play vital role in electroconvection experiments. The first one is the apparatus that includes a polarizing microscope,
temperature-controlled hot stage and electronics to apply an ac voltage. The remaining two elements are the choice of suitable NLCs and the cell assembly. Our lab
has three polarizing microscopes with temperature controlled hot stages. All of them
work under the same principle. In the following sections, I will describe the apparatus, the NLCs used for my experiment under different control parameters, sample
preparation, doping of NLCs and the cell construction.
3.1 Apparatus
The apparatus consists of three parts: the imaging system, temperature controlled hot stage which acts as the sample holder and the electronics to apply an ac
voltage, as well as to measure capacitance and the conductivity of the sample cell at
a desired temperature.
3.1.1 Shadowgraphy and optical microscope
Light propagates uniformly through a homogeneous medium. However, in an
inhomogeneous medium, the optical inhomogeneities may refract light rays. A material that has different properties in different directions is called an anisotropic material
and possesses more than one index of refraction. If a beam of light having both x-
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and y- polarization components are obliquely incident on this material, it travels
along two different directions due to different index of refraction for x-polarized and
y-polarized beam. This is the birefringence phenomenon. Any physical parameter
that is a function of refractive index is different for these two polarizations.
The shadowgraph technique [1, 2] is an optical method that indicates the
nonuniformity in transparent media such as water, glass, liquid crystal layers in thin
cells, etc. It is a very old and well-developed method to visualize the variation in
dielectric constants ²’s and, hence, the refractive index of fluid. Let us consider the
RBC system in which the fluid heated from below and cooled from above gives rise
to convective flow when the temperature exceeds a certain value, or EC in nematics
in which the sample fluid is implanted between two parallel electrodes. The image
of convection mechanism is shown in Fig. 2.5 when the applied ac voltage crosses
a certain threshold. Geometric optics clearly explain the variation in the index of
refraction (n), and hence, the dielectric constant (²) of the sample. As the rays bend
toward the higher value of n, the light rays propagating along ẑ-direction through the
cell focus in a plane over the cell. The brighter and darker regions are, respectively,
for higher and lower values of n.
The principle of shadowgraph is shown in Fig. 3.1. When a parallel beam of
light passes vertically through a convecting layer, the light polarized in the direction of the director deflects and the microscope focused to a plane close above the
sample cell visualizes the 2D shadowgraph image. There is a fundamental difference
between the RBC and EHC in terms of the angle of the outgoing light beam. In the
former case, the angle is directly proportional to the square root of the distance from
the critical point whereas in the latter case, the angle grows linearly with distance.
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(a)
(b)
Figure 3.1: (a) Principle of shadowgraph method: The incoming light is deflected
according to the refractive index; δ is the thickness of the cell, α the maximum
deflection angle of the light. (b) Experimental setup for electroconvection; an ac
voltage is applied to the conductive coating glass plates. The convection rolls and
the tilt angle of the director are shown schematically; the dashed points represent the
virtual images and the solid point, the real image; the labels 1 and 2 represent the
real foci and 3 the virtual focus [1].
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Fig. 3.3 shows the typical shadowgraph images very near onset and above onset and
their Fourier transforms.
The main advantage of shadowgraphy is its simplicity. The optical quality requirements can be of various choices, like single-element lenses, Fresnel lenses, mirrors,
etc. Shadowgraph can be casted on photographic film, on ground-glass or projection
screens, sandy soil or snow [3]. Its disadvantage is the formation of caustics, which
can be confused with other phenomena. For example, a boundary layer can behave as
a cylindrical lens which focuses light into a bright line or a band next to a solid surface
in the shadowgraph. Its position depends upon the distance between the object and
the film.
There are some differences between the schileren and the shadowgraph. Firstly,
the shadowgraph is not a focused optical image; it is a shadow. It responds to the
second spatial derivative of Laplacian (∂ 2 n/∂x2 ) (the schileren image responds to the
first spatial derivative of the refractive index, e.g., ∂n/∂x). Due to its simplicity,
shadowgrams appear in nature frequently without high technology. It allows large
scale visualization without any gross change in the illumination. There are different
types of shadowgraphy: direct shadowgraphy in diverging and parallel light, focused
shadowgraphy, large scale shadowgraphy, microscopic, stereoscopic and holographic
shadowgraphy, computed shadowgraphy and conical shadowgraphy.
I used the polarizing microscope for the shadowgraph. It consists of two
polarizing filters: the polarizer and the analyzer as shown in Fig. 3.2. The polarizer is situated below the specimen stage with its direction of vibration left-to-right
(East-West), although it is usually rotatable. The analyzer, which is usually aligned
North-South, is also rotatable. It is situated over the objective and can be moved in
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Figure 3.2: Polarizing microscope with the camera system [4].
and out of the path as required. When both the polarizer and the analyzer are in
the optical path and are positioned at a right angle to each other, light does not pass
through the system and dark field of view appears in the eyepiece. It is called cross
polarized configuration. Even though the polarizer and the analyzer are essential
components of the polarizing microscope, a wide selection of accessories are available
for the users to configure the microscope to meet its special needs. A circular rotating specimen stage facilitates the orientation of the specimen at a certain angle to
the polarized light. Centering of the objective and the stage ensures that the center
of the stage rotation coincides with the center of the field of view. The achromatic
objectives are available in 4×, 10×, 20× and 40× magnifications. These objectives
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are responsible for the primary image formation and play a vital role in determining
the quality of images a microscope produces. They are responsible for the overall
magnification of a particular specimen and set a resolution limit of the microscope.
An eyepiece with cross hair marks the center of view.
Polarizing microscopy uses either reflected or transmitted light. To study
opaque materials such as mineral oxides and sulfides, reflected light is used. The
microscope can distinguish between the isotropic and anisotropic medium through
which it passes. The technique utilizes optical property of anisotropy to divulge
details about the structure and the composition of materials. Isotropic materials,
including gases, liquids, cubic crystals, etc. reveal same optical properties in all directions. They have a single index of refraction and allow light of any orientation
passing through them. On the other hand, the optical properties of anisotropic materials vary with the vibration of incident light. They act as a beam splitter and divide
light rays into two parts. These two components of light travel at different speeds
through the specimen and have different refractive indices.
The charge-coupled device (CCD) camera connected up at the top of the microscope via camera extension tube is the most common image capture technology
employed in modern optical microscopy. It is most significant for the experimenter
to determine immediately whether the desired image is successfully recorded or not.
It is invaluable due to experimental complexities of many imaging situations. The
frame grabber [5] hooked up to the camera captures maximum individual frames at
28 FPS or a movie at 30 FPS. It is a Video-to-FireWire converter which converts
analog video signals (PAL, NTSC, CCIR, EIA) into uncompressed data streams. Its
maximum video resolution is 640×480 at 60 Hz. The color formats it supports are
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160
50
180
qx
100
200
150
220
200
240
250
300
260
350
280
100 µm
400
450
py
300
320
100
200
300
400
500
600
200
250
(a)
300
350
400
450
(b)
50
210
100
qx
220
150
230
200
240
250
300
250
350
260
400
50 µm
p
y
270
450
100
200
300
(c)
400
500
600
280
250
300
350
400
(d)
Figure 3.3: (a) Shadowgram for electroconvection of nematic I52 very near onset
consisting of counter-propagating zig and zag rolls. (b) Spectral density showing fundamental peaks for the image at (a). (c) Shadowgram for sample cell I52 at different
parameters than that of (a) above onset having superposition of counter-propagating
zig and zag rolls along with rectangular patterns and active and inactive regions. (d)
Its power spectrum showing higher harmonics dominating the fundamental peaks.
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YUV, RGB 32, RGB 24 and RGB 8. The frame grabber digitizes these recorded
images to study different electroconvective patterns found during the experiment.
3.1.2 Temperature control
Most of the physical parameters of LC materials are temperature dependent.
Therefore, it is necessary to stabilize the temperature of the sample cell. The temperature controlled hot stage is shown in Fig. 3.4. The sample under investigation is
placed inside the hot stage. It is subjected to a temperature program and observed
visually. The platinum RTD attached to the flat furnace measures the temperature
which ranges from room temperature to 375 ◦ C. The absolute accuracy is ±0.4◦ C
for the temperature between -20 and 100 ◦ C. The cold air stream of the fan (not
Microscope objective
Sample
Flat furnace with RTD
Inner casing, warm
Heat protection
filter
Outer casing, cold
Protective glass
Light source
Figure 3.4: Schematic drawing of the FP82 microscope hot stage. Platinum RTD
measures the temperature of the hot furnace.
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shown in the diagram) protects the sensitive objective against extreme heat. The
sample is at the center of the furnace. It gets heat from the bottom and the top to
keep the temperature difference within the sample to a minimum. For EC convection
under a microscope, the rotatable stage of the microscope must be at least 12.5 cm in
diameter for the required space of the hot stage. The distance between the front lens
of the objective and the stage must be greater than 2.9 cm when the stage is lowered
completely or the body tube is fully raised. An objective with a working distance of
less than 1.2 cm may have a maximum size of 1.9 cm so as to fit in the opening of
the outer casing.
The protective glass inserted in the hot stage as shown in the Fig. 3.4 stops
premature sample loss through sublimation on heating. Sometimes the sample will
condense on the heat protection filter above it and make the field of view unclear.
During this condition, it is better to take out the heat protection filter and wash it
with ethanol. The hot stage, together with the upper heating plate, can be flipped
up to allow free access to the sample chamber.
3.1.3 Electronics
Potential difference between the plates of the cell causes the convective flow
in the nematic sample cell. Even though dc voltage can also be used, it is not recommended due to its electrolysis effect. The HP 33120A [6] function generator can
produce a variety of signal waveshapes using a signal-generation technique called direct digital synthesis(DDS). The most common ac signal is the sine wave. In fact, all
periodic waveshapes are made up of sine waves of different frequencies, amplitudes
and phases added together. The magnitude of the sine wave is described by the RMS
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or peak-to-peak or the average value. To get an ac signal of the desired magnitude, I
connected it to the step-up transformer and from this transformer to the HP 34401A
multimeter [7]. The step-up transformer helps to get higher output voltage than the
input. From the multimeter, I hooked up to two wires of the cell. Thus, the function
generator gives an ac signal frequency and the multimeter records the amplified ac
voltage.
Even though an ac voltage plays a vital role in EHC, some physical parameters,
such as the dielectric constants and the conductivities can not be ignored. The capacitance bridge works only at 1.0 kHz and measures the capacitance and the conductance
(loss) at that frequency. To measure these parameters at different frequencies, I used
a lock-in amplifier (LI).
3.2
Liquid Crystals
In the past 40 years, electroconvection experiments have utilized a variety of dif-
ferent nematic liquid crystal compounds and mixtures, including Phase 5; a mixture
of azoxy compounds [8–12], Mischung 5; a mixture of phenyl benzoates [13–17], and
MBBA (p-methoxybenzylidene-p-butylaniline); a single component [18–20]. MBBA
is one of the earliest thermotropic, room-temperature nematic liquid crystals having ∆² < 0. Its nematic range is from ∼ 20 ◦ C to 47 ◦ C, its density is ∼ 1.05 gm
cm−3 at 22 ◦ C and its conductivity is ∼ 10−7 Ω−1 m−1 . The advantage of using it is
that practically all of its physical properties have been measured extensively. Even
though it has suitable values of dielectric anisotropy ∆² and conductivity anisotropy
∆σ in order for electroconvection to occur, its chemical stability is problematic. Also,
∆² is roughly -0.5 (depending on temperature), which is low if one desires oblique
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modes [21, 22] with a relatively large angle between the director and the wave vector.
The conductivity of the sample used plays a vital role in EHC of nematics. All
F
(a)
O
ON+
N
O
ON+
N
(b)
Figure 3.5: (a) Chemical formula for I52. (b) Chemical formula for Phase 5 (mixture of 35 wt.- % p-ethyl-p’-methoxy-azoxybenene and 65 wt.-% p-butyl-p’-methoxyazoxybenzene).
NLCs in their pure form are poor conductors and hence, need some kind of dopant to
make it sufficiently conducting. Different dopants are in use, which cause the variation of conductivity with concentration. The most commonly used dopant for MBBA
is tetra-n-butyl-ammonium bromide (TBAB). This dopant, when used in the range
from 0 to 100 parts per million by weight, provides sufficient conductivity. MBBA has
some drawbacks over other samples. Firstly, it has some health risks and hence, one
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should wear gloves and work under a hood while handling it. Also, it is susceptible
to moisture and hence, hard to get in its pure form. It degrades with age and can’t
be used for long times.
I utilized a single component NLC as an alternative to MBBA for pattern formation. It is 4-ethyl-2-fluoro-4’-[2-(trans-4-n-pentylcyclohexyl)-ethyl]biphenyl [21, 23],
usually referred to by its trade name, I52. Its chemical formula is shown in Fig. 3.5(a).
It has a smectic B phase from 13 ◦ C to 24 ◦ C and wide nematic range from 24 ◦ C to
104.8 ◦ C. It is chemically stable and ∆² increases monotonically from -0.054 at 25
◦
C to ∼ 0.034 at 100 ◦ C, passing through zero at 62.96 ◦ C. Thus, ∆² is low enough
to exhibit distinct oblique traveling rolls, with greater oblique angles at low frequencies when adequately doped [21, 24]. Also, I52 reliably exhibits a supercritical Hopf
bifurcation at onset [25], giving rise to counter-propagating zig and zag rolls. Since
I am interested in different types of patterns formation during electroconvection, I
have chosen the temperature range of 25 to 60 ◦ C.
Besides electroconvection in I52, I also worked with a mixture of azoxy compounds called Phase 5 [8]. It is a mixture of 65 wt.-% p-butyl-p’-methoxy-azoxbenzene
and 35 wt.-% p-ethyl-p’-methoxy-azoxybenzene. Its chemical formula is shown in
Fig.3.5(b). Its molar mass is 0.2746 kg and the nematic range is between -5 ◦ C
and 75◦ C. The lower temperature is the transition between the crystal and the nematic phase and the upper temperature is the transition between the nematic and
the isotropic phase. I worked with Phase 5 to see the demodulated images of zig and
zag patterns. However, I ended up with a different kind of Hopf bifurcation, which is
not captured by WEM. I will explain this later in Chapter Four.
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3.3 Sample preparation
The conductivity of pure I52 is 4.2 ×10−10 Ω−1 m−1 at room temperature and
does not vary noticeably at higher temperatures. This conductivity is not sufficient to
carry out EHC experiments on I52. Hence, it should be doped with a suitable impurity. I doped it with ∼ 2 wt.% molecular Iodine (I2 ) as the dopant as described in [21].
It takes two to three weeks to dissolve I2 in I52 solution to get enough conductivity
in the nematic range for which ∆² < 0. Because I2 is highly volatile, the conductivity
of the solution decreases with time. As a result, I could not work with the sample
cell for more than one week and also the conductivity was not high enough. Hence, I
tried doping with ∼ 5 wt.% of I2 in I52 to prepare sample with enough conductivity,
which lasts longer for at least one month. With a higher amount of dopant, the concentration of the solution increases and the solution becomes dark red. It provided
enough conductivity in the order of 4.5 × 10−9 Ω−1 m−1 at room temperature. However, in this case also, I could not use the sample cell for more than one week, due
to its degradation. This is because achieving the desired conductivity requires more
than simply adding iodine to the nematic liquid crystal as the dopant is not highly
soluble. To get a more stable doped solution, I heated the sample for up to 50 ◦ C,
but the I52 evaporated and stuck on the inner side of the lid of the bottle instead of
dissolving. To overcome it, I tried putting the sample solution bottle in the hot oven
at the same temperature as of the heater and the same phenomenon repeated. It was
a great challenge to prepare stable doped solution of higher conductivity which lasts
longer than the cases mentioned above. For an alternate way of heating, I kept the
sample solution bottle in a temperature bath (a brass cylinder of diameter and height
slightly greater than that of the bottle). Even in this case, I could not get a more
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stable sample solution of desired conductivity.
Alternatively, I prepared a sample solution with 0.3 wt.% of HN O3 and increased the concentration up to 1.5 wt.%, but unfortunately, even though the conductivity was ∼ 3 × 10−9 Ω−1 m−1 , it was more unstable than when I2 was used as
dopant. The conductivity decreased to 1/9th of the initial value in just 24 hours.
Thus, I could not use HN O3 as a dopant. Finally, I prepared the I52 solution with 4
wt.% of I2 and kept the bottle in a hot temperature bath at elevated temperature (150
◦
C for 72 hours). The sample became dark red when all Iodine dissolved. It gave
enough conductivity at the desired temperature range and for more than a month
before the sample degraded. As time elapsed, the sample solution turned a light red.
Although temperature this high can subtly affect the liquid crystal’s material properties, particularly the dielectric anisotropy as pointed out in [21], I confirmed that
this change was not only small, but also reproducible. Thus, the sample prepared in
this way was found to give reliable and repeatable results for, not only the electrical
conductivity, but also the electroconvection behavior.
The color of the sample solution gives a rough idea of the conductivity of
the solution. A solution with dark red color has enough conductivity for EHC with
counter-propagating traveling modes at onset. As time elapses, the color changes to
pink red, and after few months, it turns blackish, indicating that it is very weakly
conducting. So, it is better to prepare a smaller quantity as needed. Also, the sample
solution should always be warmed up before filling the cell to get desired conductivity.
Even though a small amount of tetrabutyl ammonium bromide (TBAB) is usually added in Phase 5 sample to get the desired electrical conductivity, I performed
the experiment without dopant, as the conductivity was enough for EHC. To test the
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conductivity of the prepared sample solution, I filled the sample in the planar cell
and measured the conductivity using capacitance bridge.
3.4 Sample cell
For all electroconvection experiments, both for I52 and Phase 5, I used commercial EHC cells [26]. A schematic of the top view of the cell is shown in Fig. 3.6.
The standard EHC cell consists of two transparent glass plates with conductive coating, separated by a spacer and sealed by an epoxy. The inner surface is treated so as
to align the NLC. Proper treatment of the glass plates is necessary to orient the nematics. Without it, the cell will contain many disclinations, which divide the nematic
sample into numerous domains. The director varies in each domain and one has to
work with the single domain. For electroconvection experiments, I used planar cells of
different thickness ranging from 10 µm to 50 µm with an active area ranging from 25
mm2 to 100 mm2 . These cells utilize electro-conductive coating of Indium-Tin-Oxide
(ITO) to make the inner surface conducting. The thickness of these films range from
200-300 Å, with a refractive index of 1.05. The empty cells have resistance in the
order of 10,000 M Ω. The thickness of each cell was measured interferometrically. The
substrate glass is soda-lime and has an index of refraction 1.510 ± 0.015. The inner
glass plates are spin coated with polyimide and rubbed with valvate along the width
(along the horizontal in Fig. 3.6) to get planar alignment. The bottom plate of the
cell extends out on one side and the top plate to the opposite side. This extended
space has conducting stripes for attaching wires. The electrical contact between these
plates and the hookup wires were made by using silver-laden epoxy and cured at room
temperature for up to an hour. Unless otherwise stated, the sample cell stands for
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the cell attached with wires and filled with the sample. For twist and bend Freedericksz transition, I constructed cells in the lab, which I will describe in Chapter Six.
The advantage of commercial cells over the cells made in the lab is that they take
less time to assemble, the cells are easy to fill and have a well-defined active area
whose accurate measurement is very important while calculating the conductivity of
the sample.
Experimentalists have been using two types of cells to study EHC in ne-
1
0
111110
00000
1
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
Figure 3.6: Schematic drawing of the top view of commercial cell. The rectangular
shaded region at the center represents the ITO coated ‘active area’. The dark vertical
ellipsoids on either side of the active area represent the spacer used. These spacers
can be of different thickness as desired.
matics. Depending upon the alignment, they are called the planar (in which the
nematic director is along the plane of the glass plates in the absence of any field)
alignment and the homeotropic (in which the nematic director is at right angle to
the plane) alignment. Accordingly, the former is called the planer cell and the latter,
the homeotropic cell. Generally, researchers use planar cells to study EC in nematics.
Some have utilized homeotropic cells too [27, 28]. For I52, which has a negative ∆²
at lower temperatures in its nematic phase, homeotropic cells are used to study bend
Freedericksz transition. Knowing ∆² and the critical field for the transition, one can
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easily calculate the bend elastic constant. To study the Freedericksz transition and
the EHC, we always need the sample cell having no pretilt angle (small nonzero angle
between the director and the glass).
These cells are filled with sample solution of desired concentration via capillary
action using a digital microdispenser [29]. There is the chance of air bubble in the
cell. If these bubbles are in the active area, they will interfere with EC. Also, the
sample might not fill the cell in its first attempt. Trying to fill the cell from another
extended area of the glass can cause empty region inside the cell. This will be a
trouble during EC.
I will utilize all aforementioned equipment to characterize convective patterns
in I52 and Phase 5 in Chapter Four and Chapter Five. Planar and homeotropic sample cells will be utilized in material parameters characterization (in sample I52) in
Chapter Six.
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BIBLIOGRAPHY
[1] S. Rasnet, G. Hartnug, B. l. Winkler, and I. Rehberg. The shadowgraph method
in convection experiments. Experiments in Fluids, 7:412, 1989.
[2] F. Simoni. Non-linear optics in liquid crystals:basic ideas and perspectives.
Liq. Cryst., 24(1):83, 1998.
[3] G. S. Settles. Schileren and Shadowgraph Techniques. Springer, 2006.
[4] www.microscopyu.com/museum.
[5] The imaging source, LLC USA.
[6] Hewlett-packard company. Printed in USA, August 1997.
[7] Hewlett-packard company. Printed in USA, February 1996.
[8] J. Grebovicz and B. Wunderlich. The glass transition of p-alkyl-p’-alkoxyazoxybenzene mesophases. Mol. Cryst. Liq. Cryst., 76:287, 1981.
[9] I. Rehberg, B. L. Winkler, Manuel de la Torre Juarez, S. Rasenat, and W. Schöpf.
Pattern formation in liquid crystals. Advances in solid state physics, 29:35, 1989.
[10] R. Ribota, A. Joets, and L. Lei. Oblique roll instability in an electroconvective
anisotropic fluid. Phys. Rev. Lett., 56:1595, 1986.
[11] M. Treiber, N. Éber, A. Buka, and L. Kramer. Traveling waves in electroconvection of the nematic phase 5: A test of weak electrolyte model. J. Phys. II
France, 7:649, 1997.
[12] Manuel de la Torre Juareg and I. Rehberg. Four-wave resonance in electrohydrodynamic convection. Phys. Rev. A, 42:2096, 1990.
[13] J. Shi, C. Wang, V. Surendranath, K. Kang, and J. T. Gleeson. Material characterization for electroconvection. Liq. Cryst. , 29:877, 2002.
[14] T. John, U. Behn, and R. Stannarius. Fundamental scaling lows of onoff intermittency in a stochastically driven dissipative pattern-forming system.
Phys. Rev. E, 65:046228, 2002.
[15] T. John and R. Stannarius. Preparation of subharmonic pattern in nematic
electroconvection. Phys. Rev. E, 70:025202, 2004.
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[16] H. Bohatsch and R. Stannarius. Frequency-induced structure transition of nematic electroconvection in twist cell. Phys. Rev. E, 60:5591, 1999.
[17] T. John, R. Stannarius, and U. Behn. On-off intermittency in stochastically
driven electrohydrodynamic convection in nematics. Phys. Rev. Lett., 83:749,
1999.
[18] S. Nasuno, O. Sasaki, and S. Kai. Secondary instabilities in electroconvection in
nematic liquid crystals. Phys. Rev. A, 46:4954, 1992.
[19] I. Rehberg, S. Rasenat, and V. Steinberg. Traveling waves and defect-initiated
turbulence in electroconvecting nematics. Phys. Rev. Lett., 62:756, 1989.
[20] I. Rehberg, F. Horner, and G. Hartung. The measurement of subcritical electroconvection. Journal of Stat. Phys., 64:1017, 1991.
[21] M. Dennin. A Study in Pattern Formation: Electroconvection in Nematic Liquid
Crystals. PhD thesis, University of Santa Barbara, 1995.
[22] A. Buka and L. Kramer, editors. Pattern Formation in Liquid Crystals, chapter
six, page 221. Springer-Verlag New York, Inc., 1995.
[23] U. Finkenzeller, T. Geelhaar, G. Weber, and L. Pohl. Liquid crystalline reference
compounds. Liq. Cryst, 5:313, 1989.
[24] M. Dennin, D. S. Cannell, and G. Ahlers. Patterns of electroconvection in a
nematic liquid crystal. Phys. Rev. E, 57:649, 1998.
[25] M. Dennin, M. Treiber, L. Kramer, G. Ahlers, and D. S. Cannell. Origin of
traveling rolls in electroconvection in nematic liquid crystals. Phys. Rev. Lett,
76:319, 1996.
[26] EHC Co., Japan.
[27] S. Zhou and G. Ahlers. Spatiotemporal chaos in electroconvection of homeotropically aligned nematic liquid crystal. Phys. Rev. E, 74:046212, 2006.
[28] S. Zhou, N. Éber, A. Buka, W. Pesch, and G. Ahlers. Onset of electroconvection
of homeotropically aligned nematic liquid crystals. Phys. Rev. E, 74:046211,
2006.
[29] VWR scientific, West Chester, PA 19380 USA.
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CHAPTER 4
Hopf bifurcation and convective patterns
Electrohydrodynamic convection in planar sample cells of suitable thickness via
the Carr-Helfrich mechanism was presented in Chapter Two. The transition from the
conduction state to the excited state takes place at a well-defined value of the control
parameter where the upper state shows a periodic pattern. The linear theory has
been extended to quasi 3D to describe oblique rolls, which is standard in this system.
Also, the nonlinear theory developed extensively in the conduction regime describes
the stability and the destabilizing mechanism of the roll patterns and more complex
structures.
In conventional electroconvection, initially the director alignment is n=(1, 0, 0),
i.e., the director is along x̂-axis. The applied ac field E is perpendicular to n, so the
dielectric torque stabilizes the ground state. When the applied voltage V exceeds the
threshold Vc , the electroconvection pattern develops. Whether oblique or normal rolls
exist at threshold is a difficult question as it depends upon the material parameters of
given NLC and the frequency of ac field. If oblique rolls present, they always appear
at lower frequencies along the threshold curve. In this case, the symmetry spontaneously breaks with a two-fold degeneracy, resulting zig and zag rolls. The primary
bifurcation is supercritical in the sense that the oblique angle and the amplitude of
√
the patterns grow continuously with distance from the threshold (∝ ε).
The nature of the convection pattern depends on the conductivity of the sample
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used. To get the sufficiently high conductivity, some kind of impurity dopant must
be added on the pure liquid crystals. Appealing new bifurcation phenomena in NLC
systems near onset involves the orientation of the wave vector q with the director n
and the nonlinear interactions between different modes. Many interesting patterns
including stationary and traveling, normal and oblique modes, localized states, stationary rectangles, alternating waves and spatiotemporal chaos states are observed
depending upon the NLC chosen and the control parameters.
In the case of homeotropic sample cells, the first instability is the bend Freedericksz transition that spontaneously breaks the rotational symmetry. The oblique
and normal convection rolls appear as secondary instabilities [1, 2]. The supercritical
bifurcations are either stationary or Hopf bifurcation to traveling waves, depending
upon the sample conductivity. The threshold voltage increases monotonically with
increasing frequency and tends to diverge near the cut-off frequency. The properties
at onset are more complicated than that in planar NLCs because the ground state
formed after the Freedericksz transition is spatially inhomogeneous.
I used many ready-made EHC planar cells during pattern characterization in
I52 and Phase 5 with different thickness ranging from 10 µm to 50 µm. Both these
samples follow standard Carr-Helfrich mechanism, i.e., they are (-,+) at room temperature. The concentration of dopant in I52 ranged from 1.98 wt.% to 11 wt.% and
Phase 5 was used without dopant. Many sample cells could not exhibit any EHC
pattern either due to their thickness, or the conductivity or both. Here, I will only
explain the results obtained from few sample cells with more emphasis in I52.
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4.1 Experiments in I52
Table 4.1 gives the summary of planar sample cells used to study pattern formation. In the following sections, I will explain threshold voltage, oblique Hopf instability, flat fielding, continuous Hopf bifurcation and defects observed during EHC.
Table 4.1: Summary of sample cells used in pattern characterization in I52.
label
I5299
I5234
I5295
I5246
I5261
I5205
4.1.1
thickness, µm I2 , %
purpose
24.99±0.17
4.71 threshold curve
22.34±1.06
5
different regimes
10.95±0.09
10
WEM
22.46±0.22
3.27
WEM
48.61±0.91
3.27
WEM
23.05±0.25
5
grain boundary
Threshold voltage and different regimes
The EHC cells, sample preparation and filling were explained in Chapter
Three. At first, each planar cell filled with doped I52 of different concentration was
inserted in the temperature controlled hot stage. Then, they were put on the rotating table of a microscope. Since, at lower frequencies, ²⊥ and σ⊥ are found to
be frequency dependent, I used a lock-in amplifier to measure them at different frequencies. Then, ac voltage of certain frequency was applied along the ẑ-direction.
The objective lens of magnification 5×, 10×, and 20× were used to observe different
patterns during EHC. In addition to other advantages, the small thickness of the cell
and the lower magnification of the microscope enabled me to capture more rolls in the
individual frame. Before capturing any frame, I studied the variation of the threshold
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voltage (Vc ) as a function of external driving frequency (F), both in the ‘conductive regime’ and the ‘dielectric regime’ at a particular temperature and plotted the
threshold curve. Fig. 4.1(a) shows the variation of Vc with applied ac frequency at
four different temperatures in sample cell I5299. From the graph, it is clear that when
the temperature is increased, the threshold voltage decreases at fixed frequency. This
is due to the fact that | ∆² | and ∆σ both increase with temperature, which reduces
Vc . Also, the cut-off frequency (Fc ) which separates the conductive and the dielectric
regime, increases with temperature. Fc is the basis for choosing the frequency range
in the conductive regime to record the patterns of interest. As explained in Chapter
Two, Vc depends upon the elastic constants, the dielectric anisotropy, the magnitude
of wave vectors in the pattern, the charge relaxation time, the conductivities and the
viscosities of the sample used at given temperature and frequency. However, it is
independent of sample thickness.
At the onset of convection, the charge relaxation time, τq = ²o ²⊥ /σ⊥ is much
smaller than the director relaxation time, τd = γ1 d2 /π 2 K11 . While a generalization to lower frequency is doable in principle, it should be avoided due to possible
electrochemical effect, and one should be careful about frequency dependence of the
dielectric constants and the conductivities.
Many movies and sequences of frames were recorded in the conductive regime
below the Lifshitz point (FL ) to analyze the patterns. FL is the frequency along
the threshold curve in the conduction regime, below which the pattern is oblique
(q = qx̂ + pŷ), and above which it is normal (q = qx̂). Except in a few sample
cells, the initial bifurcation was supercritical Hopf bifurcation, leading to counterpropagating zig and zag rolls at lower frequencies. Along the threshold curve and
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(a)
(b)
Figure 4.1: (a) Threshold voltage as a function of applied frequency at different
temperatures in the sample cell I5299. (b) Different regimes in the sample cell I5234
at 47 ◦ C; the Lifshitz point is 240 Hz and the cut-off frequency is 420 Hz.
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50
100
150
200
250
300
350
50 µm
400
450
100
200
300
400
500
600
Figure 4.2: Typical stationary zigzag pattern slightly above onset during electroconvection showing Williams-Kapustin domains. The double arrow along the vertical is
the direction of unperturbed director.
below Fc , I observed two regimes in sample cell I5234 at 47 ◦ C; the oblique traveling
at lower frequencies and the normal traveling at higher frequencies, with different parameter values as shown in Fig. 4.1(b). However, in the dielectric regime, a decrease
in | ∆² | increases the threshold voltage [3]. In Fig. 4.1(b), I have introduced ωo τq as
the dimensionless normalized frequency, due to the fact that they appear jointly in
WEM equations. Here, ωo = 2πF and F is the frequency of applied ac.
The electrical conductivity plays a decisive role in the mechanism of the onset
of instability. The variation in the conductivities in my experiments was a function of
three factors: concentration of the dopant, temperature and time dependence. At a
constant temperature, the drift in the conductivity with time shifts the onset voltage
for convection. For a given sample cell at a given temperature, the patterns recorded
in the lab are a function of three control parameters: applied ac voltage, the driving
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frequency and the conductivity. As a result, the threshold voltage of convection is
a function of applied frequency and the conductivity, i.e., Vc = Vc (σ⊥ , ωo ). Another
reduced control parameter ε = V 2 /Vc2 − 1 is defined in such a way that at onset, it
is zero; below onset, it is negative and above onset, it is positive. Its value gives the
idea of the distance of the pattern from the threshold.
Fig. 4.2 shows a typical Williams-Kapustin domain for stationary zig and zag
a
b
c
d
Figure 4.3: Different patterns obtained during electroconvection in I52: (a) Nearly
normal rolls with dislocation to the upper left corner. (b) Turbulent structure high
above onset. (c) Localized patterns called worms and (d) Chevron patterns. The
length scale represents 100 µm.
patterns at 45 ◦ C very near onset and Fig. 4.3 shows different patterns formed at
different sets of parameters. The normal rolls, turbulent patterns and the worms
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are in the conductive regime and the chevron patterns are in the dielectric regime.
The normal rolls are formed at higher frequencies and the turbulent structures are
formed high above the onset. The localized structures, called worms, are localized
perpendicular to the direction of the director.
4.1.2 Oblique Hopf instability
A uniform steady state of a physical system in a large spatial domain may lose
stability in two common ways: the stationary or steady state bifurcation, and the
oscillatory instability (or Hopf bifurcation). In this section, I will deal with the Hopf
bifurcation. There are two types of supercritical Hopf bifurcations: normal Hopf bifurcation and oblique Hopf bifurcation. In the first type, the pattern consists of two
counter-propagating traveling waves solutions at onset, expressed by plane waves of
the form ei(±qc x+ωc t) , with two critical wave numbers (±qc , 0), qc > 0. Here, ωc is the
critical Hopf frequency. These traveling waves are called normal traveling waves (NT).
In nematic EHC, it corresponds to roll patterns propagating along the direction of
the director when the driving frequency along the threshold curve in the conduction
regime is ≥ FL .
In the oblique Hopf bifurcation, the pattern consists of two pairs of counterpropagating zig and zag rolls at onset, expressed by plane waves of the form ei(±qc x±pc y+ωc t)
having four critical wave numbers(±qc , ±pc ), qc , pc > 0, located off both reflection
axes. These traveling waves are called oblique traveling waves (OT) and in nematic
EHC, it corresponds to roll patterns propagating in two oblique directions. They
have slowly varying envelopes as stipulated in weakly nonlinear analysis. A scalar
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field, say u, which represents the pattern can be written in the form,
u(x, y, t) ∼ (A1 E1 + A2 E2 + A3 E3 + A4 E4 + cc) + hh
(4.1)
where the Aj = Aj (x, y, t), j=1,..,4, are complex envelopes in four oblique directions
(two counter-propagating pairs). These are varying slowly in comparison to Ej ’s.
Ej ’s are expressed as
E1 = ei(qc x+pc y+ωc t) , E2 = ei(−qc x+pc y+ωc t) , E3 = ei(−qc x−pc y+ωc t) , E4 = ei(qc x−pc y+ωc t)
(4.2)
In Eq. 4.2, ωc is the critical Hopf frequency and the waves Ej arise in the solutions of the linearized system at the onset. Here, x is the coordinate along the
direction of director and y is perpendicular to it. In Eq. 4.1, cc refers to the complex
conjugate expression, and hh to higher harmonics. For a small value of the distance
√
from onset ε, the Aj are of order O( ²). At O(ε), the hh comprise terms of the form
Ai Aj Ei Ej , Ai Āj Ei Ēj (the bar denotes the complex conjugates) and their complex
conjugates, and similarly at higher orders. Thus, the envelopes of the higher harmonics are ‘slaved’ by the basic envelopes Aj . The Aj , in turn satisfy the system of
complex Ginzburg Landau equations [4, 5] and so are the main drivers of the dynamics. These envelopes are invariant under following symmetry operations [4].
x → x + xo : (A1 , A2 , A3 , A4 ) → (eiqc xo A1 , e−iqc xo A2 , e−iqc xo A3 , eiqc xo A4 ),
y → y + yo : (A1 , A2 , A3 , A4 ) → (eipc yo A1 , eipc yo A2 , e−ipc yo A3 , e−ipc yo A4 ),
t → t + to : (A1 , A2 , A3 , A4 ) → eiwc to (A1 , A2 , A3 , A4 ),
x → −x : (A1 , A2 , A3 , A4 ) → (A2 , A1 , A4 , A3 ),
y → −yo : (A1 , A2 , A3 , A4 ) → (A4 , A3 , A2 , A1 )
The waves E2 and E4 are propagating in the directions ±(−qc , pc ) and are referred
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left
right
left
right
Zig
Zag
Figure 4.4: Illustration of counter-propagating zig and zag modes.
to as ‘zig waves’, whereas E1 and E3 propagate in the directions ±(qc , pc ) and are
referred to as ‘zag waves’. Accordingly, A2 , A4 and A1 , A3 are referred to as zig and
zag envelopes, respectively. The zig and zag waves are pairs of counter-propagating
traveling waves moving left (E1 , E4 ) and right (E2 , E3 ) as illustrated in Fig. 4.4. If
the angle between the direction of the director and the wave vectors for the zag modes
is θ, then, for zig modes, it will be π/2 − θ. Also, θ = tan−1 (pc /qc ). With increase of
driving frequency, the angle keeps on decreasing and for normal modes, pc = 0 and
therefore θ = 0◦ .
4.1.3 Flat fielding
Several problems commonly cause uneven illumination (inhomogeneities) in the
final image captured by CCD camera. They are:
• Vignetting in the optics: even in a perfect optical system, some portion of
the focal plane may get more light than others. Usually, the central portion gets a
bit more light than the outer edges.
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• Intrinsic and surface defect of the CCD camera: sometimes, one region of
the silicon is more sensitive to light than others or some chips are prone to showing
artefacts.
• Shadow cast by dust: CCD cameras may be nearly perfect when it is first
made. However, over the time, it can accumulate layers of contaminants. Tiny spots
of dirt and dust can block most of the light reaching the particular pixel.
Flat fielding is the technique used to improve the quality of the captured images by removing artefacts caused by variations in the pixel-to-pixel sensitivity due to
above mentioned reasons. It is a standard calibration procedure in everything, from
pocket digital camera to giant telescopes. To flat field an image from a sequence of
frames, I captured the 8-bit gray scale images of size M × N , with M = 480 pixels in
the vertical direction and N=640 pixels in the horizontal direction. For each setting
under the polarizing microscope, I captured a dark frame (no input light) and the
background. The background is the map of the CCD’s sensitivity to light at zero
ac field, with the same illumination and temperature as that of the raw image. In
order to remove the inhomogeneities in the optical system, the raw images have been
treated according to the equation.
FFI =
RI − DF
× average(F F − DF )
F F − DF
(4.3)
where FFI is the flat fielded image, RI is the raw image, DF is the dark frame and FF
is the background. A background image with inhomogeneities in the optical system,
the raw image and the flat fielded image are shown in Fig. 4.5 along with pure zig
and pure zag modes specified by circles.
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50
100
150
200
250
300
350
400
450
100
200
300
400
500
600
400
500
600
(a)
50
100
150
200
250
300
350
400
450
100
200
300
(b)
Figure 4.5: (a) Background image at zero applied ac voltage. (b) Snapshot of the
image at same illumination as that of the background and ac field turned-on at
ε = 0.01.
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50
100
zag
150
200
zig
250
300
350
100 µm
400
450
100
200
300
400
500
600
Figure 4.6: Flat fielded image of Fig. 4.5 showing pure zig and zag rolls as indicated
by circles.
4.1.4 Continuous Hopf bifurcation
The shortcomings of SM and the origin of WEM were explained in Chapter Two. The beauty of WEM is that it explains the ionic migration, molecular
dissociation-recombination reactions and their consequences on the conductivity. It
provides the basis for explaining the Hopf bifurcation observed at the onset during
EHC in nematics [6, 7]. WEM has modified the concept of static ohmic conductivity
of SM to dynamically active species of positive and negative charge carriers. As a
result, there is distinctive change in the threshold behavior of the EHC instability,
explicitly the traveling modes. Recall the expression for ωH from Eq. 2.62,
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q
ωH = Ω 1 − (1/τrec Ω)2
(4.4)
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where
v
Ω=
u + −
2
u
0 Vc ²o ²⊥ t µ⊥ µ⊥
πC 3
d (1 + ω 02 ) γ1 σ⊥
(4.5)
and
ω 0 = ωo τq
1 + (d/π)2 (qc2 + p2c + ∆²qc2 /²⊥ )
1 + (d/π)2 (qc2 + p2c + ∆σqc2 /σ⊥ )
(4.6)
The WEM expresses Vc , qc and ωH as a function of the applied frequency
ωo τq , normalized to the inverse of charge relaxation time τq . Also, Vc (ωo τq ) and
qc (ωo τq )d/π do not depend on the thickness [7]. This model has clearly predicted
q
−
the Hopf frequency fH to be proportional to (1/d3 ) µ+
⊥ µ⊥ /σ⊥ along with other SM
parameters.
In this section, I will present the results of the measurement of the circular Hopf
frequency ωH , the oblique angle θ, the wave vector q and the thickness dependence
of the Hopf frequency as a function of new variable ωo τq , introduced in WEM. A few
other parameters measured as the function of other variables will also be discussed.
All the results, except the thickness dependence, are from the sample cell I5295.
To measure the different parameters, I used the electroconvection apparatus,
consisting of temperature controlled hot stage (FP82), electronics for applying an ac
voltage and the shadowgraph apparatus for visualization. The cell was illuminated by
polarized light, with the polarization along the director and the resulting shadowgraph
image was monitored by the charge coupled device (CCD) camera mounted on the
microscope at around 30 cm from the sample using a 10× objective.
By increasing the driving frequency and the applied voltage in small steps and
waiting for a few minutes in each step, I measured the onset voltage at different
frequencies to get the threshold curve. Fig. 4.7(a) shows the threshold curve at
three different temperatures. As explained in section 4.1.1, the threshold voltage
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decreases with rise in temperature. These curves give the idea of frequency limit in
the conduction regime to carry out other measurements.
At first, I fixed the temperature of the sample cell at 25 ◦ C and the external
driving frequency to 10 Hz and slowly increased the ac voltage V to the onset voltage
Vc . After Vc was reached, I slightly increased the voltage further (slightly +ε), waited
for few minutes, and then, captured the 8-bit gray scale image of size (480×640) and
calculated its power spectrum in real-time. The size of the image corresponds to an
area of 358.21 µm × 477.61 µm. The power spectrum allowed finer focusing of the
shadowgraph in order to enhance the dominant inner oblique modes (first harmonics)
of the power spectrum right above onset. The relatively strong contributions centered
at the origin in the Fourier transform of the image appears due to improper focusing.
The sample stage, kept on the fully rotatable stage of the microscope, was rotated as
necessary to make sure that the peaks of the power spectrum were symmetric about
the x̂-axis. Then, the stage was fixed for the whole experiment.
To measure the Hopf frequency, I recorded a sequence of 2048 (210 ) images
captured at 28 FPS, very near to the onset of convection. The primary instability for
these images was the supercritical oblique Hopf instability. The central pixel value
of each image was extracted, using a program in Matlab. Its fast Fourier transform
(FFT) was taken and plotted the graph as a function of inverse time taken to capture
the frames. The frequency at which the maximum FFT of the pixel values occurs
is the Hopf frequency (fH ). Fig. 4.7(b) shows the Fourier transform of the central
pixel values of the images at ε = 0.01 and driving frequency of 210 Hz at 25 ◦ C. It
corresponds to the Hopf frequency of 0.4 Hz. I went over the same process at higher
frequencies along the threshold curve at 35 ◦ C and at 50 ◦ C. Finally, I measured
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(a)
2000
1800
1600
f(Frequency)
1400
1200
1000
800
600
400
200
0
0
5
10
15
Frequency (Hz)
(b)
Figure 4.7: (a) Threshold curves showing the variation of onset voltage with driving ac
frequency at 25 ◦ C (blue circles), 35 ◦ C (red up triangles) and 50 ◦ C (green diamonds).
(b) Fourier transform of the central pixel values of 2048 images: it is at ε = 0.01,
driving frequency of 210 Hz and corresponds to Hopf frequency of 0.4 Hz.
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the capacitance and the conductivity of the sample cell using lock-in amplifier at
each frequency at which fH was measured. At each frequency, τq was calculated so
as to find the WEM variable ωo τq . Fig. 4.8(a) gives the variation of τq with the
driving frequency at different temperatures and Fig. 4.8(b) gives the Hopf frequency
graph. From the experiment, it was found that, at a given temperature, fH increased
with the increase of applied frequency in the conduction regime. Also, with rise
in temperature, it increased at a given frequency. It differs from [8] in the sense
that, with rise in temperature, the graphs are more flat and the Hopf frequencies are
decreased at higher temperatures.
To study the variation of Hopf frequency with ε for oblique modes, I measured
fH at Vc =11.73 V, F = 51 Hz at 50 ◦ C. The conductivity recorded by the lock-in
amplifier was σ⊥ = 64.1 × 10−9 Ω−1 m−1 . Then, I ramped up the applied voltage
in small steps, waited for few minutes, and again measured fH as explained above.
The process was repeated until the turbulent regime appeared where the pattern is
aperiodic and fH cannot be measured. From the graph Fig. 4.9(a), it is found that
fH is maximum at the onset and decreases with rise in ε.
The convective roll patterns are more pronounced above onset than that at
onset. To visualize it in the graph, I captured convective oblique roll patterns at
Vc =12.92 V, F= 25 Hz and conductivity of σ⊥ = 26.49 × 10−9 Ω−1 m−1 at 25 ◦ C.
Then, its 2D Fourier transform gives the peaks of zig and the zag rolls, as defined in
section 4.1.2. Keeping the driving frequency and the temperature constant, I ramped
up the voltage in small steps, waited for few minutes in each step and captured the
image. I repeated the same process unless it appeared the turbulent regime. The
variation of these arbitrary zig and zag amplitude is shown in Fig. 4.9(b). From
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(a)
(b)
Figure 4.8: (a) Variation of the charge relaxation time with driving frequency at 25
◦
C (pink circles), 35 ◦ C (red circles) and 50 ◦ C (blue up triangles). (b) Variation of
Hopf frequency with normalized driving frequency at three different temperatures.
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the graph, it is obvious that, at particular ε, when the amplitude of the zag peak
increases, the amplitude of the zig peak decreases and vice versa. However, both of
these amplitudes increase with an increase in ε.
One way to find the temperature dependence of Lifshitz point (FL ), is to
measure the wave vector q or the angle θ between the wave vector and the director
n. For F ≥ FL , q = qx̂ so that θ = 0◦ . To measure these quantities, I set the
temperature of the hot stage at 25 ◦ C, the driving frequency at 10 Hz and increased
the voltage in small steps so that there were oblique traveling rolls slightly above the
onset. Then, I captured a sequence of 100 images. I added the Fourier transform of
these images and computed the first moment of the indices in the region of interest
of one of the peaks of S(q) as
Z
hqi =
qx S(q)dqx
and
Z
hpi =
py S(q)dpy
with
Z
S(q)d2 q = 1
Then, assuming oblique modes, the angles in magnitude were computed for the
zig and zag rolls separately using
θ = tan−1
hpi
hqi
in the first and third quadrant and finally took the average of these two angles. The
same process was repeated at higher driving frequencies along the threshold curve.
The capacitance and the conductivity of the sample cell were measured using the
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(a)
(b)
Figure 4.9: (a) Variation of Hopf frequency with ε > 0 at 50 ◦ C and driving frequency
of 51 Hz, Vc =11.73 V. It corresponds to σ⊥ = 64.1×10−9 Ω−1 m−1 and (b) Variation of
zig and zag peaks of the Fourier transform with ε > 0 at 25 ◦ C and 25 Hz, Vc = 12.917
V. It has σ⊥ = 26.49 × 10−9 Ω−1 m−1 .
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lock-in amplifier, as explained earlier. The whole process was repeated to get different
values of wave vectors and angles at different ωo τq at 35 ◦ C and at 50 ◦ C. Fig. 4.10(a)
shows the variation of the wave vectors with ωo τq at three different temperatures.
From the curve, it is apparent that, pc is a maximum at the lowest value of ωo τq
and with increase of its value, pc keeps on decreasing and becomes a minimum in
the normal mode regime. On the other hand, the variation in the wave vector qc
is very weak. In Fig. 4.10(a), the Lifshitz point decreases with rise in temperature.
Fig. 4.10(b) is the alternative way to express the variation of Lifshitz point with
temperature. Near the Lifshitz point, θ should go to zero as a square root law
√
θ ∝ FL − F . The wave vector information can be used to calculate the spacing of
the rolls and the angle made by the wave vector with the director indicating whether
it is normal or oblique mode.
In addition to the conductivity, the thickness of the sample cell plays a vital
role in the Hopf bifurcation as well as the Hopf frequency. From the WEM, it is
found that ωH is proportional to 1/d3 . Thus, according to the WEM prediction,
√
Hopf bifurcation is preferred by thin cells and 1/ σ⊥ indicates that it prefers the
cell of lower conductivity. Two cells, I5246 and I5261, were filled with a sample of
dopant concentration 3.27 wt.% and the Hopf frequency was measured, as explained
previously. Fig. 4.11 shows the variation of Hopf frequency with different thickness
and the conductivity. These experimental results are the evidence that the WEM has
captured the main driver of EHC, which was missing in the SM.
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(a)
(b)
Figure 4.10: (a) Wave vectors as a function of the normalized driving frequency at
different temperatures. Up triangles, hexagons and square with cross are for q at
25 ◦ C, 35 ◦ C and 50 ◦ C, respectively. With rise in temperatures, q slightly decrease
at higher frequencies. Down triangles, diamonds and stars are for p at 25 ◦ C, 35
◦
C and 50 ◦ C respectively, both multiplied by d/π to make them dimensionless. (b)
Variation of the angle between the wave vector q and n with normalized driving
frequency. Circles, diamonds and up triangles are for θ at 25 ◦ C, 35 ◦ C and 50 ◦ C,
respectively.
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√
Figure 4.11: ωH σ⊥ d3 as a function of ωo τq for sample cell I5246 (up triangles) at
57.5 ◦ C and σ⊥ = 8.39 × 10−9 Ω−1 m−1 and I5261 (solid circles)at 43 ◦ C and σ⊥ =
6.37 × 10−9 Ω−1 m−1 .
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4.1.5 Defects in NLCs
Defects in NLCs refer to regions where a periodic unit (roll pair) ends or begins, and hence, the director is not uniquely defined. There are point or line defects
in nematics called point or line disclinations [9]. A disclination is a discontinuity in
the orientation of the director field n(r) so that n(r) is not a smooth function of
r. These defects appear due to topological, energetic or dynamic reasons. The total
topological charge is a conserved quantity for fixed boundary conditions. All transformations among defects, such as merging and decaying, are allowed only when the
conservation law is obeyed. The number of defects in the equilibrium state depends
upon the energy balance due to elastic, surface and the external field force [10]. In
general, the defects of opposite meandering attract each other and tend to annihilate
into a defectless state. However, with time, the concentration of defects slowly decrease so as to reduce the free energy of the system. One can expect the motion of
defects along the rolls (‘climb’) as the system has translational invariance along this
axis. If the defects climb, the wavelength is changed and if they glide, the orientation
of the roll pattern is changed [11].
On the other hand, striped patterns generally have stability regime in the
ε − q space. Even in the stable regime, patterns have the tendency to reach most
favourable wave vector q’ which coincides with qc very near the onset. During transients, patterns may exist with wave vector mismatch ∆q = q − q0 . Dislocations are
defects or irregularities located at points in the pattern where the additional rolls
end in the bulk. Its formation and dynamics present a possibility to adjust either
magnitude or direction of q so that ∆q = 0. These defects also move more smoothly
parallel to the rolls (climb) than perpendicular (glide) due to same reason as in case
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a
b
c
d
Figure 4.12: (a) Pure zig mode at t=1280. (b) Zigzag grain boundary at t=11,101. (c)
Pure zag modes at t= 11,752 and (d) Zigzag grain boundary at t=13,117. The double
arrow gives the direction of unperturbed director and the length scale represents 100
µm.
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of disclinations [12].
During electroconvection experiment with I5205, I observed line defects called
grain boundaries. The threshold voltage and the Hopf frequency were measured by
the same technique as explained in section 4.1.4. I chose the temperature of 50 ◦ C
and the driving frequency of 25 Hz and the conductivity was 41.5×10−9 Ω−1 m−1 , as
measured by lock-in amplifier. The cut-off frequency was 632 Hz and fH = 0.91 Hz.
I captured a sequence of T=30,000 images at 28 FPS by using a 10× objective near
threshold (ε = 0.18). These were left and right traveling zig and zag modes. Zag
modes were traveling to the left while zig modes were traveling to the right. In the
field of view, there were pure zig rolls, grain boundaries, pure zag rolls and again grain
boundaries as shown in Fig. 4.12. The grain boundary traveled perpendicular to the
rolls. Fig. 4.13(a) shows two grain boundaries with double zag domains and single
zig domain and Fig. 4.13(b) shows its envelope extracted by two-wave demodulation
technique.
4.2 Experiments in Phase 5
The WEM prediction of continuous Hopf bifurcation is in good agreement with
all the results achieved during EHC in I52. Besides I52, Phase 5 also exhibits distinct
oblique patterns during EHC. Hence, I prepared a Phase 5 sample cell to test the
program written in Matlab to demodulate an image having oblique patterns. But, I
found peculiar behavior during its EHC, which is not captured by the WEM.
I used three planar cells labelled P59, P58 and P53, each with lateral dimensional area of 10× 10 mm2 . A summary of sample cells used is given in table 4.2. P58
was used to reproduce the patterns, as observed in P59, and P53 was used to study the
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50
100
150
200
250
300
350
400
450
100
200
300
400
500
600
400
500
600
(a)
50
100
150
200
250
300
350
400
450
100
200
300
(b)
Figure 4.13: (a) Two grain boundaries with double zag domains and single zig domain at t = 26,440. The double arrow denotes the direction of unperturbed director
and the length scales represents 100 µm. (b) Its envelope extracted from two-wave
demodulation.
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Table 4.2: Summary of sample cells used in EHC in nematic Phase 5.
label
P59
P58
P53
thickness, µm alignment
10.63±0.09
planar
10.07±0.08
planar
10.06±0.13
planar
purpose
WEM
WEM
phase diagram
phase diagram. Sample cell preparation and filling were done, as explained in Chapter Three. Even though a small amount of dopant called TBAB (tetrabutylamonium
bromide) is usually added in the sample to get the desired electrical conductivity, I
performed the experiment without any dopant. The conductivity was large enough
to achieve EHC patterns. The conductivity varies between individual cells, with temperature and time elapsed. As a result, respective threshold values vary accordingly.
I inserted the sample cell P59 in the hot stage of the electroconvection apparatus
and kept on the rotatable stage of the microscope. I fixed the stage and measured Vc ,
fH , θ and q for cell P59 with the same technique, as explained in section 4.1.4. By using the frame grabber, I recorded some individual frames which correspond to an area
of 358.21 µm × 477.61 µm. Background frames and the dark frames were recorded
for flat fielding. I recorded the onset voltage at certain steps of driving frequency to
get the threshold curve. Fig. 4.14 is the threshold curve for the sample cell P59. The
cut-off frequency was noted and the experiment was done in the conduction regime.
I checked the power spectrum of the image recorded to confirm that the fundamental
modes were dominating over other modes. Then, I captured a short movie of 2048
frames (M = 480 × N = 640) at 30 FPS very near onset at different frequencies.
Fig. 4.15(a) shows the electroconvection pattern of stationary oblique rolls at 35 ◦ C
with an ac voltage of 10.38 V and driving frequency 90 Hz. The applied voltage
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Figure 4.14: Threshold curve for cell P59 at 35 ◦ C. It corresponds to the cut-off
frequency of 1050 Hz.
was slightly above the threshold, ε = V 2 /Vc2 − 1 = 0.05. Fig. 4.15(b) is the average
Fourier transform of first 120 frames of the movie recorded during electroconvection.
Besides the fundamental peaks, higher harmonic modes are undoubtedly seen. This
figure also indicates dominating oblique modes. The strong peak at the center is an
artefact of the optical system. Using Matlab code, I extracted the central pixel value
of each frame of the individual movie and calculated the Hopf frequency, fH . I found
stationary oblique modes up to the frequency of 90 Hz.
Fig. 4.16(a) gives the electroconvection pattern recorded at 35 ◦ C for the same
sample cell P59 at driving frequency of 95 Hz which corresponds to ε = 0.016 and
Hopf frequency of 2.0 Hz. Fig. 4.16(b) is the average power spectrum of the first 120
images of the movie. Again, the dominating central peak is due to the artefact in the
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100
50
100
150
200
250
300
50 µm
350
400
450
100
200
300
400
500
600
(a)
5
x 10
12
5
x 10
15
10
8
〈P〉
10
6
5
4
0
50
50
n
2
0
m
0
−50
−50
(b)
Figure 4.15: (a) Oblique stationary rolls at 90 Hz and ε = 0.05. The double arrow is
the direction of unperturbed director. (b) Average FFT of first 120 frames showing
the inner oblique modes in the region of interest.
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50
100
150
200
250
300
350
50 µm
400
450
100
200
300
400
500
600
(a)
5
x 10
6
5
x 10
5
5
4
4
3
3
2
1
2
0
50
50
m
1
n
−50
−50
(b)
Figure 4.16: (a) Normal traveling pattern at 95 Hz and ε = 0.016. The double arrow
represents the direction of unperturbed director. (b) Average Fourier transform of
first 120 frames showing normal peaks.
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optical system and nearest dominating modes from the center are the normal modes.
Even without using any filter, there is no contribution from the oblique modes. Thus,
there is a jump in Hopf frequency from fH = 0, corresponding to oblique stationary
modes at F =90 Hz, to fH = 2.0 Hz, corresponding to normal traveling modes at
F=95 Hz. It noticeably shows a discontinuity in the Hopf frequency. Fig. 4.17 shows
Figure 4.17: Circular Hopf frequency in cell P59 as a function of the normalized
driving frequency. The first vertical short dashed line is for the critical ωo τq , left of
which the pattern is stationary and right of which it is traveling. The second short
dashed vertical line is where the second discontinuous Hopf bifurcation occurs.
ωH as a function of dimensionless normalized driving frequency ωo τq . There are two
discontinuities in the graph. To the left of the first vertical short dashed line, the
pattern is oblique stationary and to the right, it is normal traveling. On either side
of the second short dashed line, the pattern is normal traveling.
I measured the wave vectors and the angle between the wave vector q and the
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director n as explained in section 4.1.4. Fig. 4.18(a) shows the variation of roll angles with ωo τq . It shows that as the pattern goes from oblique stationary (OT) to
normal traveling (NT), the angle immediately reaches a minimum value and remains
almost constant throughout the conductive regime. This is different than that observed in [6, 7, 13, 14]. Next, I measured the wave vectors qx̂ and pŷ and plotted the
graph of these quantities against ωo τq . Interestingly, I found that bf q increases faster
in the OS region and slower in the NT region along the threshold curve but p reduces
abruptly in OS/NT transition region and remains almost zero at NT region as shown
in Fig. 4.18(b).
The sample cell P58 of almost equal thickness was used to check whether or
not the discontinuous Hopf bifurcation phenomenon is reproducible. Fig. 4.19 shows
the variation of ωH with ωo τq at different temperatures. At a lower temperature, the
traveling normal modes appear at lower F and at higher temperature, they appear at
higher F. Also, once the NT regime is reached, the curve is almost flat.
Eq. 4.4 predicts the continuous Hopf bifurcation [6, 15, 16] and continuous
variation of wave vectors along the threshold curve. But it is valid only if all the
terms in ω 0 are continuous. If any one of the term in Eq. 4.6 is discontinuous, then
the Hopf frequency along the threshold curve should be discontinuous.
I calculated Ω using Eq. 4.5 and Eq. 4.6. γ1 at 30 ◦ C was taken from [7]. I
found a sharp peak at the critical ωo τq at which the transition between the OS and
NT modes occurred. Fig. 4.22(b) is the phase diagram for different states during
electroconvection in the cell P53 of almost equal thickness as that of cells P59 and
P58. Below FL , there is an OS state in the conduction regime and above FL , there
is an NT state. However, very near to FL in NT state, if the voltage is ramped up
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(a)
(b)
Figure 4.18: (a) Variation of the roll angles with ωo τq . (b) Wave vectors as a function
of the normalized driving frequency in cell P59. Up triangles are for the wave vector
qx̂ and the circles indicate the wave vector pŷ, both multiplied by d/π to make them
dimensionless. The short dashed vertical line differentiates between OS and NT rolls
regime.
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Figure 4.19: ωH as a function of ωo τq in the cell P58. The up triangles are at 35 ◦ C
and the circles are at 40 ◦ C . Short dashed vertical lines are drawn at critical ωo τq at
which the Hopf bifurcation occurs.
above threshold, oblique modes are reproduced. This is very interesting observation
that the OS state is recovered at higher values of ε in the NT state.
4.3 Conclusions
EHC in nematic liquid crystals is an appropriate entrant for experiments exhibiting different patterns and the Hopf bifurcation in nonequilibrium systems. However,
compared to other pattern forming systems, it has some disadvantages, such as many
material parameters involved and mathematical description of the instability is more
complex in comparison to instabilities in simple fluids. Since the instability is driven
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Figure 4.20: Variation of ²⊥ (up triangles) and σ⊥ (circles) with ωo τq for the cell P59.
The short dashed vertical line drawn at critical ωo τq separates the OS and NT modes.
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(a)
(b)
Figure 4.21: (a) Variation of Ω with ωo τq in cell P59. The left short dashed vertical
line drawn at critical ωo τq separates the OS and NT modes and the right short dashed
vertical line is where the second discontinuity Hopf frequency occurs. (b) ω 0 in Eq. 4.6
as a function of ωo τq in cell P59, both at 35 ◦ C.
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(a)
(b)
Figure 4.22: (a) Variation of charge τq with the applied frequency in cell P59 at 35
◦
C. (b) Phase diagram at 25 ◦ C for the planar sample cell of P53. The Lifshitz point
FL =62 Hz and the cut-off frequency Fc = 155 Hz.
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by an ac field, the amplitude and frequency are easily controlled. And hence, by
changing the ac voltage by few volts, the observed pattern can have transition from
ordered pattern to a very disordered turbulent state. Changing the frequency can
cause bifurcation from stationary to traveling states. The applied ac does not break
the up-down symmetry of the system. As a result, two regimes with Williams and
dielectric rolls add to the richness of the system. The interaction between the hydrodynamics and the electric force in the anisotropic fluid is a stunning and enthralling
field of physics.
During EHC in doped I52, the threshold voltage at 40 ◦ C corresponding to
driving frequency of 33.35 Hz is 27.33 V while at 55 ◦ C, it is 14.82V at the same
frequency. At these temperatures, ∆² has increased from -0.0259 to -0.0062 as shown
in Fig. 6.9 while ∆σ has increased from 2.03 × 10−9 Ω−1 m−1 to 4.43 × 10−9 Ω−1 m−1 .
Thus, the threshold voltage decreased with increase in the anisotropies ∆² and ∆σ
which are function of temperature. On the other hand, the frequency of the onset
voltage 30 V at 40 ◦ C is 38.05 Hz. At 55 ◦ C, to get that onset voltage, one should
ramp up the frequency to 95.85 Hz. This frequency is in the dielectric regime for the
threshold curve at 40 ◦ C. The cut-off frequency increases with rise in temperature.
Thus, the increase in conductivity increases the area under conductive regime.
In the threshold curve for I52, there are two regimes in the conduction
regime namely oblique traveling (OT) and normal traveling (NT). The OT consists
of counter-propagating zig and zag waves and NT consists of right and left traveling normal modes. Thus, there is the possibility of six pure traveling modes. But,
there can be up to nine pure modes including three stationary modes (stationary zig,
stationary zag and normal stationary) and their superpositions. For comparatively
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110
50
100
150
200
250
300
350
400
50 µm
450
100
200
300
400
500
600
Figure 4.23: snapshot of the EHC pattern at 61 ◦ C for a cell of thickness 23.18 ± 0.24
µm filled with I52+4 wt.% I2 . σ⊥ = 16.8 × 10−9 Ω−1 m−1 and fH is 0.85 Hz at
ωo τq = 0.28 and ε = 0.01. The rubbing direction of the cell plates is in the direction
of double arrow.
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clean sample cell filled with I52, the primary bifurcation is always supercritical Hopf
bifurcation leading to oblique modes at lower frequencies . Depending upon the conductivity, stationary and traveling oblique and normal modes, rectangular patterns,
alternating waves, STC states including worms are formed. Fig. 4.23 is a typical
EHC pattern traveling along only one direction of the director so that it has only two
modes. the rectangular patterns is apparent here due to superposition of these two
oblique modes.
With increase of temperature, the dielectric constant ²⊥ decreases and the conductivity σ⊥ increases so that the charge relaxation time τq decreases. At lower
frequencies, since both ²⊥ and σ⊥ are frequency dependent, τq obviously depends
upon the frequency as shown in Fig. 4.8. The wave vector q is almost independent of
external driving frequency as well as the temperature. However, p depends on both
control parameters. With increase of ωo τq , it decreases and becomes zero for F ≥ FL .
Also, FL decreases with rise in temperature.
According to WEM predictions, when τrec is not small compared to τq or τd ,
the conductivity itself becomes a new dynamically active variable. This new degree
of freedom leads to a Hopf bifurcation at onset for large value of the ratio of ion
migration, α̃ to the recombination rate, r̃. My results agree with the predictions and
the Hopf frequency for traveling waves increased with ωo τq . One major difference I
observed that does not matches with the results of [8] is the temperature dependence
of the Hopf frequency. In my case, it has increased with rise in temperature. Since the
√
ratio α̃/r̃ is proportional to 1/ σd3 , the Hopf condition is fulfilled for thin cells and
the sample of low dopant concentration. It is in qualitative agreement with Fig. 4.11
and [17, 18].
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During electroconvection in planar sample cells filled with undoped Phase 5,
I observed first instability right at the onset at lower frequencies in the conductive
regime as stationary bifurcation giving rise to oblique rolls. When the driving frequency was slowly increased, there was Hopf bifurcation exhibiting normal traveling
rolls. The experiment was repeated with another sample cell of almost equal thickness
to confirm the observed discontinuity in Hopf frequency. The phenomenon was easily
reproduced. There was continuous variation of ²⊥ and σ⊥ with ωo τq . It is reasonable
to assume that ²k and σk also behave accordingly. The rotational viscosity does not
depend on ωo τq and hence, should not have any role for discontinuity in ωH . But
as there is continuity in q and discontinuity in p in the transition regime between
OS and NT modes along the threshold curve, there must be discontinuity in Hopf
frequency.
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BIBLIOGRAPHY
[1] H. Richter, A. Buka, and I. Rehberg. Electrohydrodynamic convection in a
homeotropically aligned nematic sample. Phys. Rev. E, 51(6):5886, 1995.
[2] S. Zhou, N. Éber, Á. Buka, W. Pesch, and G. Ahlers. Onset of electroconvection
of homeotropically aligned nematic liquid crystals. Phys. Rev. E, 74:046211,
2006.
[3] A. N. Trufanov, L. M. Blinov, and M. I. Barnik. New type of high-frequency
electrohydrodynamic instability in nematic liquid crystals. Sov. Phys. JETP,
51(2):314, Feb. 1980.
[4] G. Dangelmayr and I. Oprea. Modulational stability of traveling waves in 2D
anisotropic systems. J. Nonlin. Sci., 18:1, 2008.
[5] P. E. Cladis and P. Palffy-Muhoray, editors. Spatiotemporal patterns in nonequilibrium complex systems, chapter one, page 19. Addison-Wesley publishing company, 1995.
[6] M. Dennin, M. Treiber, L. Kramer, G. Ahlers, and D. S. Cannell. Origin of
traveling rolls in electroconvection of nematic liquid crystals. Phys.Rev.Lett.,
76:319, 1996.
[7] M. Treiber, N. Éber, Á. Buka, and L. Kramer. Traveling waves in electroconvection of the nematic phase 5: A test of the weak electrolyte model. J.Phys II
France, 7:649, 1997.
[8] M. Dennin. A Study of Pattern Formation: Electroconvection in Nematic Liquid
Crystals. PhD thesis, Department of Physics, University of California, Santa
Barbara, 1995.
[9] P. G. de Gennes. The Physics of Liquid Crystals. Clarendon Press, Oxford,
1974.
[10] R. Repnik, L. Mathelitsch, M. Svetec, and S. Kralj. Physics of defects in nematic
liquid crystals. Eur. J. Phys., 24:481, 2003.
[11] P. Tóth, N. Éber, T. M. Bock, Á. Buka, and L. Kramer. Dynamics of defects in
electroconvection patterns. Europhys. Lett., 57:824, 2002.
[12] Á. Buka, N. Éber, W. Pesch, and L. Kramer. Isotropic and anisotropic electroconvection. Physics Report, 448:115, 1995.
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[13] I. Rehberg, B. L. Winkler, Manuel de la Torre Juarz, S. Rasenat, and W. Schöpf.
Pattern formation in liquid crystals. Ad. solid state Phys., 29:35, 1989.
[14] Manuel de la Torre Juarez and I. Rehberg. Four-wave resonance in electrohydrodynamic convection. Phys. Rev. A, 42:2096, 1990.
[15] M. Treiber. On the Theory of the Electrohydrodynamic Instability in Nematic
Liquid Crystals Near Onset. PhD thesis, Department of Mathematics, University
of Bayreuth, Germany, 1996.
[16] M. Treiber and L. Kramer. Bipolar electrodiffusion model for electroconvection
in nematics. Mol. Cryst. Liq. Cryst., 261:311, 1995.
[17] M. Dennin, G. Ahlers, and D. S. Cannell. Spatiotemporal chaos in electroconvection. Science, 272:388, 1996.
[18] I. Rehberg, S. Rasenat, J. Fineberg, M. de la Torre Juarez, and V. Steinberg.
Temporal modulation of traveling waves. Phys. Rev. Lett., 61(21):2449, 1988.
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CHAPTER 5
Spatiotemporal chaos in I52
Spatiotemporal chaos (STC) is a time dependent dynamical state which is aperiodic in both space and time. It is a purely mathematical term and can be defined as
a deterministic randomness, ‘deterministic’ in the sense that STC arises from intrinsic
causes and not from some unrelated disturbance, and ‘randomness’ in the sense that
it has irregular and unpredictable behavior [1, 2]. It presents a way to understand
complicated behavior as something that is consistent and ordered, instead of extrinsic and inadvertent. The fluctuations in space play a major role in dynamics. Such
fluctuations arise when the system is driven slightly out of equilibrium.
The fluid flow in which it behaves as a system of orderly layers, with no eddies
or irregular fluctuations is called laminar flow or streamline flow. However, when the
fluid flow is disordered in time and space, it is called the turbulent flow where the
fluid acquires fairly different dynamics. It involves momentum diffusion, high momentum convection and rapid variation of pressure and velocity in space and time.
For turbulence to occur [3]:
• The flow must be unpredictable, in the sense that a small uncertainty at a
given initial time will amplify so that it is impossible to provide precise deterministic
prediction of its evolution.
• The flow should be able to mix transported quantities much more rapidly than
if only molecular diffusion process is involved.
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• It must involve a wide range of spatial wavelength.
By nature, a turbulent flow is unstable. At least in certain region of space,
the flow is rotational, i.e., ω = ∇ × v 6= 0. Fully developed turbulence are free
from constraints, such as boundaries, external forces or viscosity. In smaller scale,
the turbulence will be fully developed if the viscosity does not play a direct role in
the dynamics of these scales. The dimensionless Reynolds number Re (ratio of the
inertial force to the viscous force) characterizes whether the flow is laminar or turbulent. Generally, the flow having Re less than 500 is defined as laminar (for NLC cells,
Re ∼ 10−4 ). In turbulent flow, unsteady vortices appear in many scales and interact
with each other. Since laminar-turbulent transition is directed by Reynolds number,
the same transition occurs if the size of the object is increased, or the viscosity of the
fluid is decreased, or the density of fluid is increased.
In nematic EHC, if the applied ac voltage V is far from the onset (ε ≥ 3), in
the conduction regime, the amplitude of the convective patterns as well as the flow
velocities increase. At certain voltage Vt , the Williams domains are distorted and
long-range nematic alignment is disturbed. As a result, the optical axis is rapidly
randomized, leading to strong scattering of light. This is called dynamic scattering
mode (DSM). This mode is considered a transition to turbulence. There are two
DSM regimes; the lower voltage state is called primary or DSM1 while the higher
voltage state is called secondary or DSM2. On the basis of spatial power spectrum of
transmitted light, DSM1 is considered anisotropic turbulence and DSM2 is isotropic
turbulence [4]. The transition is characterized by the sudden increase in the density
of disclination loops where the transition voltage depends upon the ramp rate, sample thickness, anchoring strength and the driving frequency [5]. As in the case of
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conduction regime, there is also turbulence at around the same distance from onset
in the dielectric regime [6]. The first display devices had used the turbidity of DSM
but completely abandoned later because of its low energy performance.
The difference between the regular and the chaotic motion can be expressed in
terms of the power spectrum of the dynamical variables. For regular motion, in the
temporal Fourier transform of a dynamical variable, the power spectrum consists of
a set of sharp δ-functions whereas for chaotic motion, the spectrum will have smooth
components. If ui (t) is the dynamical variable, the correlation function Ci (t) of the
chaotic signal decays at long time [7], usually as
Ci (t) lim ∼ e−t/τcorr
t→∞
(5.1)
Another quantity characterizing the dynamics of the motion is the Lyapunov exponents which describe the separation of orbits of two dynamical variables ui (0) and
ui (0) + δui (0) at t = 0 in phase-space, very close to each other. If the difference δui
grows exponentially in time, such that the characteristic rate of evolution of the exponential, called the Lyapunov exponent, is positive, a spatiotemporal chaotic behavior
is assumed to occur. In general, STC means any type of random behavior resulting
from deterministic equations with regular initial conditions. However, turbulence in
fluid dynamics signifies disordered flow, particularly the flow involving the birth and
transportation of the vorticities [7].
In hydrodynamic systems, STC is found in thermal convection [8] and nematic
EHC [9, 10]. In nematic electroconvection, due to the anisotropy of the system, the
waves travel only in a specific direction. The STC arises in the conduction regime,
where the onset of electroconvection is a supercritical Hopf bifurcation, leading to two
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counter-propagating zig and zag rolls, with broken translational symmetry due to the
presence of physical boundaries [11]. As a result, the accurate amplitude description
of Eq. 4.1 needs a system of four globally coupled complex Ginzburg-Landau equations (GCCGLE) [12]. Recall the equation for the scalar field u, which represents the
pattern,
u(x, y, t) = eiωc t (A1 ei(qc x+pc y) + A2 ei(−qc x+pc y)
+ A3 ei(−qc x−pc y) + A4 ei(qc x−pc y) + cc) + hh
(5.2)
Here, Aj = Aj (x, y, t), j = 1, ..., 4, are complex envelopes varying slowly with respect
to the exponential terms and hh refers to higher order terms. For weakly nonlinear
analysis, the constraints for Aj ’s and the control parameter R are
| ∂Aj /∂t |¿| Aj |¿ 1,
| ∂ 2 Aj /∂x2 |¿| ∂Aj /∂x ¿| Aj |,
| R − Rc |¿ 1
(5.3)
Assume that, a basic, horizontally uniform steady-state solution uo (R) becomes unstable when the control parameter, R passes through a critical value Rc from below.
The neutral stability surface is defined as the set of all points in the (q, p, R)-space
for which the critical eigenvalue σ for Hopf instability is σ(qc2 , p2c , Rc ) = iωc and ωc is
the Hopf frequency. Besides the critical frequency and the wave numbers, the critical
group velocities are also important characteristic quantities of the Hopf instability.
These velocities are defined as
vq =
∂
Imσ(q 2 , p2 , Rc ) |( qc2 , p2c ),
∂q
vp =
∂
Imσ(q 2 , p2 , Rc ) |( qc2 , p2c )
∂p
(5.4)
For normal traveling modes vp = 0. The GCCGLE for A1 [13] is
∂A1
∂A1
∂A1
− vq
− vp
= [ao (R − Rc ) + D̃(∂x , ∂y ) + a1 | A1 |2 + a2 | A2 |2
∂t
∂x
∂y
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+ a3 | A3 |2 + a4 | A4 |2 ]A1 + a5 A2 Ā3 A4 + hh
(5.5)
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Here, an overbar denotes complex conjugate, D̃ is the second order differential operator, D̃ = dqq ∂x2 + 2dqp ∂x∂y + dpp ∂y 2 operating on A1 , and a1 ,....,a5 are five complex
coefficients computable from the original PDE-system. The other three equations
have the same structure.
To study STC states, I used three planar sample cells I5234, I5239 and I5295,
having thickness 22.34 ± 1.06 µm, 10.39 ± 0.08 µm and 10.95 ± 0.09 µm, respectively. The dopant concentration in that order was 5 wt.%, 11 wt.% and 10 wt.%.
Electroconvection apparatus, measuring the capacitance and the conductivity by the
lock-in amplifier and the shadowgraph technique were explained in Chapter Three.
In the following sections, I will explain the different phenomena observed in these
sample cells.
5.1 Four-wave demodulation
Modulation is the process of varying some characteristics of a periodic wave
with an external signal. In radio communication such as AM radio, there is a superposition of the information bearing signal with the carrier signal of high frequency
which can be transmitted in air easily and is capable of traveling long distance. The
characteristics of the carrier signal (amplitude, frequency or phase) vary in accordance with the information bearing signal also called the modulating signal. Fig. 5.1
shows the modulating signal which varies slowly in comparison to the carrier signal.
Demodulation or enveloping is the process of extracting the original modulating signal wave from a modulated carrier wave. There are several demodulation techniques
depending on what parameters of the signal are transformed in the carrier signal.
The envelope is very important as it carries all the information of the wave.
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During EHC in sample cell I5295, the applied voltage was slightly above the
carrier signal
modulating signal
Figure 5.1: Carrier positive signal of high frequency and slowly varying modulating
signal also called the information bearing signal
critical value for the onset of electroconvection. My objective is to extract envelopes
varying slowly in space and in time, as prescribed in a weakly nonlinear analysis
of system of PDEs to draw conclusions about the nature of the dynamical states.
This analysis also yields the value of critical wave number and the critical (Hopf)
frequency.
Unless otherwise stated, the experimental images and the analysis of the data
are from the sample cell I5295. At first, I fixed the temperature of the hot stage at
50 ◦ C and the driving frequency at 51 Hz. ²⊥ and σ⊥ of the sample were 3.19 and
75.11 × 10−9 Ω−1 m−1 respectively. Then, I recorded a sequence of T = 54,000 images
very near the onset (ε = 0.01) at a rate of 27.6 FPS. Each image had size M × N ,
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4
x 10
5
16
x 10
−40
hh
2
−30
14
zig
−20
12
1.5
m
−10
P
zag
zig
0
10
1
zag
10
8
0.5
6
20
0
50
normal
30
4
50
40
2
0
0
n
50
−40
−30
−20
−10
0
n
(a)
10
20
30
40
50
m
−50
−50
(b)
Figure 5.2: (a) 2D spatial Fourier transform of the flat fielded image at t = 10,100
showing dominating fundamental modes. Zig and zag fundamental peaks, normal
mode and the higher harmonic peak are shown by solid circles in the window −50 ≤
m ≤ 50 and −50 ≤ n ≤ 50. (b) 3D view of the Fourier transform of the same image
in the same window.
with M = 480 pixels in the vertical direction and N = 640 pixels in the horizontal
direction. These images were flat fielded by a technique as described in Chapter Four.
I performed a spatial demodulation of each flat fielded image that generates the amplitude of zig and zag waves, varying slowly in space, but not in time by Fourier
decomposition method. At first, I took 2D Fourier transform of each image. Then,
the region around the peak of interest was chosen, setting all other pixel values to
zero. In the second step, I carried out a temporal demodulation, in which I extracted
the envelopes of the four oblique traveling waves from the time series of zig and zag
amplitudes. This separation into spatial and temporal demodulation sidesteps full
3D Fourier transforms, which appreciably reduces the computational effort.
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4
4
4
x 10
8
x 10
x 10
x 10
8
8
4
0
6
5
4
4
n
10
5
4
0
−10
2
20
6
6
>
6
5
−,−
>
8
−,+
7
<P
10
<P
4
x 10
m −10
−20
4
4
−20
−20
m
−10
4
4
x 10
x 10
2
n
x 10
8
x 10
8
2
2
50
n
0
0
−50
1
m
−50
(a)
6
5
4
0
−10
<P+,+>
3
0
50
<P+,−>
〈P〉
4
4
x 10
10
m
4
0
2
n
−20
6
5
2
20
20
n
10
10 m
20
(b)
Figure 5.3: (a) Average of the individual peaks of a time series of images for 10, 001 ≤
t ≤ 20, 000 in the window −50 < m ≤ 50 and −50 < n ≤ 50 . (b) Individual average
modes of the same time series of Fourier transform 10, 001 ≤ t ≤ 20, 000 in different
windows as specified in the figure.
5.1.1 Spatial demodulation and critical wave numbers
I denote the sequence of flat fielded images by I(k, l, t), where k and l are the
vertical and horizontal pixel labels, 0 ≤ k < M and 0 ≤ l < N , and t is time,
0 ≤ t < T . A typical snapshot, recorded at t = 10,100 and its flat fielded image are
shown in Fig. 4.5 (b) and (c). The discrete Fourier transform of an image is denoted
by
F (m, n, t) = (Fs I)(m, n, t) ≡ √
−1 N
−1
X
1 MX
e−2πi(mk/M +nl/N ) I(k, l, t)
M N k=0 l=0
(5.6)
and its spatial power spectrum by P (m, n, t) =| F (m, n, t) |2 , where Fs refers to the
spatial Fourier operator. The wave numbers are identified with the integer labels (m,
n). Given F, the image can be reconstructed via the inverse Fourier transform.
I(k, l, t) =
(Fs−1 F )(k, l, t)
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≡√
−1 N
−1
X
1 MX
e2πi(mk/M +nl/N ) F (m, n, t)
M N m=0 n=0
(5.7)
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For displaying Fourier transforms and power spectra, I chose the window −50 <
m ≤ 50 and −50 < n ≤ 50 . Note that since I is real, F(-m, -n, t)=F̄ (m, n, t).
Fig. 5.2 shows the fundamental dominating peaks. The higher harmonic peaks are
comparatively weaker, indicating that the images in real space are taken very near
onset. Also, the peaks due to normal modes are very weak and hence, the real-time
image is basically a superposition of zig and zag modes.
In Fig. 5.2(b), the power spectrum P of the image at t = 10,100 is displayed
4
4
x 10
x 10
8
4
x 10
10
8
4
x 10
7
7
8
8
6
6
〈P〉
6
5
4
Pav
6
5
4
4
4
2
2
3
0
3
0
2
50
25
n
20
n
1
m
10
−25
0
2
20
1
10
m
(a)
(b)
Figure 5.4: (a) Average of zig and zag Fourier peaks in the window −25 < m ≤ 25
and 0 < n ≤ 50 for the time as in Fig. 5.3. These are the peaks of interest for spatial
demodulation. (b) Time and zig-zag averaged power spectrum Pav .
in the windows −50 < m ≤ 50 and −50 < n ≤ 50. Outside of this window, P
is negligibly small. The small regions with high contribution from the zig and zag
modes can be easily recognized. Fig. 5.3(a) shows the average power spectrum over
the time 10, 100 ≤ t ≤ 20, 000 in the windows −50 < m ≤ 50 and −50 < n ≤ 50. It
is identified very clearly that the zig and zag modes are equally strong. Very weak
second harmonic peaks are also seen. Fig. 5.3(b) gives the zoom into the zig and zag
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regions of the average power spectrum (and their reflections about the origin) over
the same time as in Fig. 5.3(a), but in the window of size 20 × 20. Colorbars are used
to show the relative strength of each peak. To each image, individual zig and zag
components are extracted by filtering out the modes in the corresponding windows
in the spatial Fourier space.
While recording the movie of the EHC patterns or the sequence of frames, it is
0.9
−150
0.8
−100
0.7
−50
0.6
0
0.5
50
0.4
100
0.3
150
0.2
200
0.1
m
−200
−300
−200
−100
0
n
100
200
300
Figure 5.5: 2D Gaussian filter in the window 141 ≤ m ≤ 240, 221 ≤ n ≤ 320 used
to filter out the primary oblique spatial Fourier modes. The blue and red colors
correspond to minimum and maximum intensity respectively.
hard to get pure oblique modes and hence, filter mask is required to minimize the effect
due to normal modes, higher harmonic oblique modes and the optical inhomogeneities.
One way to minimize the effect due to normal modes in EHC of I52 is to record the
sequence of images at lower frequencies. To minimize the effect of higher harmonics
in the spatial Fourier transform, one should record the images very near onset. To
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minimize the effect of optical inhomogeneities, it is recommended to focus the images
in the polarizing microscope, checking the spatial Fourier transform of the recorded
image and refocusing as necessary until the peak at the center in Fourier space is
weaker than the inner oblique modes. In signal processing, I chose 2D Gaussian filter
mask in such a way that its amplitude is maximum at critical pixel values for oblique
modes (mc , nc ) in spatial Fourier space. The width of the filter can be increased
or decreased as required. The filter window of the Gaussian filter is the Gaussian
function of the form
G(x, y) =
1 −[(x−xo )2 +(y−yo )2 ]/2σ2
e
2πσ 2
(5.8)
Here, x and y are the horizontal and vertical distance from the origin respectively.
xo (yo ) is the average of the distribution along the horizontal (vertical) and σ is the
standard deviation of the Gaussian distribution. In 2D, Eq. 5.8 produces a surface
of concentric circles with Gaussian distribution from the center. Since the Fourier
transform of the Gaussian function yields a Gaussian function, it can be multiplied
by the fast Fourier transform of the signal and transform back. The pixel at (mc , nc )
receives the heaviest weight and neighboring pixels get smaller weights as their distance from (mc , nc ) increase. One must be careful in choosing the window: window
too small means throwing away data having valuable information, window too large
means picking modes from other regions which mislead the characteristics of the primary modes. Wave number filtering is done in such a way that only the inner oblique
modes, referred to as primary modes, are kept for further investigation. The remainder analysis is exclusively for these modes.
Fig. 5.4(a) is the zig and zag peaks after filtering out other reflection modes
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for 10, 001 ≤ t ≤ 20000. The zig and zag components are defined by
Izig = Fs−1 (Fzig + F̂zig ),
Izag = Fs−1 (Fzag + F̂zag )
(5.9)
where F̂ (m, n, t) = F̄zig (−m, −n, t), and, analogously, F̂zag . Fig. 5.6(a), Fig. 5.6(c)
and Fig. 5.9(a) show the components Izig , Izag and Inor , respectively, where Inor is
the embedded normal rolls in the real-image.
To complete the spatial demodulation, I determined the critical wave numbers
(mc , nc ). For this, lets define the time-averaged power spectra Pav (m, n) by
Pav (m, n) = hPzig i(−m, n) + hPzag i(m, n)
(5.10)
The critical wave numbers are then defined as the averages with respect to this distribution,
M2
X
(mc , nc ) =
N2
X
(m, n)Pav (m, n)
m=M1 n=N1
M2
N2
X
X
= (19, 18)
(5.11)
Pav (m, n)
m=M1 n=N1
The associated vertical and horizontal wavelengths λv and λh respectively, in physical
units are given by
λv = M/mc × P D and λh = N/nc × P D, where PD = 0.746 µm is the pixel diameter,
giving λv = 18.85 µm and λh = 26.52 µm.
After identifying the critical wave numbers, I can now extract the demodulated
zig and zag envelopes varying slowly only in space and not in time. These envelopes
are defined by
Azig (k, l, t) = (Fs−1 Fzig (k, l, t)e2πi(kmc /M −lnc /N ) + cc
(5.12)
Azag (k, l, t) = (Fs−1 Fzag (k, l, t)e−2πi(kmc /M +lnc /N ) + cc
(5.13)
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50
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450
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(a)
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600
(b)
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400
450
450
100
200
300
(c)
400
500
600
100
200
300
(d)
Figure 5.6: (a) Izag , ( b) Azag , (c) Izig and (d) Azig for the pattern snapshot at t =
10,100.
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Re(Azag)
128
0.5
0
−0.5
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
4
Im(Azag)
x 10
0.5
0
−0.5
 Azag
0.2
0.4
0.6
0.8
1
t
1.2
1
t
1.2
1.1
1.12
1.4
1.6
1.8
2
4
x 10
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0.4
0.2
0
0.2
0.4
0.6
0.8
1.4
1.6
1.8
2
4
x 10
Re(Azag)
(a)
0.5
0
−0.5
1.02
1.04
1.06
1.08
1.14
1.16
1.18
1.2
4
Im(Azag)
x 10
0.5
0
−0.5
1.02
1.04
1.06
1.08
1.1
t
1.12
1.1
t
1.12
1.14
1.16
1.18
1.2
4
x 10
 Azag
1
0.5
0
1.02
1.04
1.06
1.08
1.14
1.16
1.18
1.2
4
x 10
(b)
Figure 5.7: (a) Time series of zag modes for real, imaginary and absolute values for
1 ≤ t ≤ 20, 000. (b) Zooms of the real, imaginary and the absolute parts in the range
10001 ≤ t ≤ 12048.
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Re(Azig)
0.5
0
−0.5
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
4
x 10
Im(Azig)
0.5
0
−0.5
 Azig
0.2
0.4
0.6
0.8
1
t
1.2
1
t
1.2
1.1
1.12
1.4
1.6
1.8
2
4
x 10
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
1.4
1.6
1.8
2
4
x 10
Re(Azig)
(a)
0.5
0
−0.5
1.02
1.04
1.06
1.08
1.14
1.16
1.18
1.2
4
Im(Azig)
x 10
0.5
0
−0.5
 Azig
1.02
1.04
1.06
1.08
1.1
t
1.12
1.1
t
1.12
1.14
1.16
1.18
1.2
4
x 10
0.6
0.4
0.2
0
1.02
1.04
1.06
1.08
1.14
1.16
1.18
1.2
4
x 10
(b)
Figure 5.8: (a) Time series of zig modes for real, imaginary and absolute values for
1 ≤ t ≤ 20, 000 (b) Zooms of the real, imaginary and the absolute parts in the range
10001 ≤ t ≤ 12048.
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Fig. 5.7(a) gives the average of the central 10 × 10 amplitudes for real, imaginary
and the absolute value of zag envelopes for 1 ≤ t ≤ 20, 000 and (b) gives their zoom
for 10001 ≤ t ≤ 12048. The amplitudes are varying in time periodically for short
time interval, but over the time, the absolute value | Azag | is more chaotic in space.
The same is true for the case of zig envelopes, as shown in Fig. 5.8.
The absolute values of Azag and Azig for the snapshot at t = 10,100 are displayed
in Fig. 5.6(b) and (d) respectively. The red (blue) region in these plots is region
with high (low) zig and zag contributions to the recorded image. Fig. 5.9(a) shows
the dominated normal rolls (Inor ) and (b) shows its envelope. Since the normal
rolls are embedded in the real-time image, the reconstructed image is defined as
Izig + Izag + Azig + Azag and is shown in Fig. 5.9(c). Here, zig and zag envelopes are
stacked to the image showing the dominating zig and zag patterns. This superimposed
image is same as the flat fielded image of section 4.1.3. The blue and green regions
in this plot are regions with high zig and zag contributions to the recorded image.
Since the contribution from normal modes is negligible, it is not included in the
reconstructed image. Thus, the pattern is believed to be purely oblique.
5.1.2 Temporal demodulation
Due to the presence of Hopf instability, the time series of the dominant oblique
modes exhibit fast oscillations. The frequency range of these oscillations and Hopf
frequency are determined from the temporal power spectra of the oblique modes.
Given any time series f(t), 0 ≤ t < T , its Fourier transform is denoted by
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1 TX
g(ω) = (Ft f )(ω) ≡ √
f (t)e−2πiωt/T
T t=0
(5.14)
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0.6
0.5
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0.5
100
0.3
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0.2
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0.1
200
0
250
0.4
0.3
250
−0.1
300
300
−0.2
350
−0.3
0.2
350
400
−0.4
400
450
−0.5
450
100
200
300
400
500
600
0.1
100
(a)
200
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400
500
600
(b)
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250
300
350
400
450
100
200
300
400
500
600
(c)
Figure 5.9: (a) Inor , (b) Its envelope and (c) Reconstructed image showing zig and
zag envelopes. The blue and the green regions in this image are regions with high zig
and zag contributions to the recorded image.
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and by s(ω) = | g(ω) |2 , its power spectrum, where Ft refers to the temporal Fourier
operator. In this discrete setting, temporal frequencies are identified with integers ω
in the range T /2 ≤ ω < T /2.
The critical Hopf frequency is determined in a similar manner as the critical
wave numbers. Here, I have used the averages of the temporal power spectra,
hSzig/zag i(ω) =
1
Mω Nω
X
| Gzig/zag (m, n, ω) |2
(5.15)
(m,n)∈Wzig/zag
where Gzig/zag (m, n, ω) = (Ft Fzig/zag )(m, n, ω). Outside the window, hSzig i and hSzag i
are very small, as are the spectra of the individual oblique modes time series. Thus,
I use the zig-zag and left-right average Sav (ω) = hSzig i(ω) + hSzag i(ω) + hSzig i(−ω) +
hSzag i(−ω).
The Hopf frequency ωc in W+ is given as ωc =
X
ω∈W+
X
ωSav (ω)/
Sav (ω). Using ωc ,
ω∈W+
I extracted slow time series from the oblique modes by setting
±
−2πiωc t/T
Fzig/zag
(m, n, t) = Ft−1 G±
zig/zag (m, n, t)e
(5.16)
where
G+
zig/zag (m, n, ω) = M(ω)Gzig/zag (m, n, ω),
(5.17)
G−
zig/zag (m, n, ω) = M(ω)Ḡzig/zag (m, n, ω)
and M(ω) is a frequency filter mask. I chose 1D Gaussian filter for this case. However,
using different mask of the form tanh(x + a) − tanh(x − a) did not make a remarkable
difference. Finally, define envelopes varying slowly in time and space through spatial
Fourier inversion as
+
A1 (k, l, t) = (Fs−1 Fzag
)(k, l, t)e2πi(−mc k/M −nc l/N )
+
A2 (k, l, t) = (Fs−1 Fzig
)(k, l, t)e2πi(mc k/M −nc l/N )
A3 (k, l, t) =
−
(Fs−1 F̃zag
)(k, l, t)e2πi(mc k/M +nc l/N )
+
A4 (k, l, t) = (Fs−1 F̃zig
)(k, l, t)e2πi(−mc k/M +nc l/N )
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−
−
where F̃zig/zag
(m, n, t) = Fzig/zag
(−m, −n, t). The four-wave superposition
Io (k, l, t) = [A1 (k, l, t)e2πi(mc k/M +nc l/N ) + A2 (k, l, t)e2πi(−mc k/M +nc l/N ) +
2πi(−mc k/M −nc l/N )
A3 (k, l, t)e
+ A4 (k, l, t)e
2πi(mc k/M −nc l/N )
2πiωc t/T
]e
(5.19)
+ cc
which is the discrete analogue of Eq. 4.1 and 4.2 without higher order terms, is considered the basic oblique pattern created in the Hopf instability.
Fig. 5.10 shows four envelopes extracted by spatial and temporal demodula A1 
 A2 
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100
 A3 
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300
200
300
400
300
400
 A4 
100
100
200
300
400
100
200
(a)
Figure 5.10: 2D plots of | A1 | − | A4 | for t = 10,100.
tion as discussed above for the snapshot of image at t = 10,100. By observing the
intensity variation in the envelope movies, I concluded that all four envelopes have
significant contribution in the envelope dynamics. In this particular sample for the
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recorded frames, the envelopes of the rolls slowly fluctuate both in space and in time.
At a certain time, one of the modes dominates, and, at another time, the modes have
approximately equal amplitudes. The patterns I observed are the superpositions of
modulated oblique traveling waves. They occur at onset and the GCCGLE [12] provides the correct envelope equations for their description in weakly nonlinear analysis.
Fig. 5.11 gives the average of the central 10 × 10 amplitudes of real, imaginary and
the absolute values for all the envelopes for the time series 10001 ≤ t ≤ 12048. The
absolute value of the average amplitudes shows unpredictable chaotic behavior, both
in space and in time as shown in Fig. 5.12.
The images were taken very carefully at lower frequencies in the conduction
regime so that the inner oblique modes were stronger. Before recording the sequence
of images, I captured an image slightly above onset at particular frequency and calculated its power spectrum in real time. The power spectrum gives the idea whether
the primary modes are stronger or not, whether there is significant contribution from
the higher harmonics, the normal modes and the optical inhomogeneities. One has
to lower the applied voltage to get dominating primary modes, record the images at
lower frequencies to overcome the contribution from normal modes, fine focusing to
reduce artefacts due to optical system and rotate the table to make sure that the
peaks of the power spectrum are symmetric about the x̂-axis. The artefact due to
optical system was not apparent here, unlike in [9]. The fundamental modes of the
average power spectrum in Fig. 5.3(a) are almost as strong, with the zag mode amplitude being 1.03× the zig mode amplitude. But, taking into consideration the power
spectrum for the individual image, as shown in the Fig. 5.2(b), the zig mode is 1.65×
that of the zag mode. Although average power content in the higher harmonics, as
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0.5
Re(A2)
Re(A1)
1
0
−1
1.02
1.04
1.06
1.08
1.1
t
1.12
1.14
1.16
1.18
0
−0.5
1.2
0
1.02
1.04
1.06
1.08
1.1
t
1.12
1.14
1.16
1.18
1.2
1.02
1.04
1.06
1.02
1.04
1.06
1.08
1.1
t
1.12
1.14
1.16
1.18
0
1.2
1.02
1.04
1.06
1.12
1.1
t
1.12
1.16
1.18
1.2
4
x 10
1.14
1.16
1.18
1.2
4
x 10
1.14
1.16
1.18
1.2
4
x 10
1
Re(A4)
Re(A3)
1.02
1.04
1.06
1.08
1.1
t
1.12
1.14
1.16
1.18
0
−1
1.2
1.02
1.04
1.06
1.08
4
x 10
1.1
t
1.12
1.1
t
1.12
1.1
t
1.12
1.14
1.16
1.18
1.2
4
x 10
1
Im(A4)
Im(A3)
1.1
t
1.14
(b)
0
0
1.02
1.04
1.06
1.08
1.1
t
1.12
1.14
1.16
1.18
0
−1
1.2
1.02
1.04
1.06
1.08
4
x 10
1.14
1.16
1.18
1.2
4
x 10
1
 A4 
0.4
3
1.08
4
0.5
A 
1.12
0.2
x 10
0.5
0.2
0
1.1
t
0.4
(a)
−0.5
1.08
4
0.6
−0.5
1.08
x 10
 A2 
1
A 
1.06
0
−0.5
0.8
0.4
1.04
0.5
Im(A2)
Im(A1)
1
−1
1.02
4
x 10
1.02
1.04
1.06
1.08
1.1
t
(c)
1.12
1.14
1.16
1.18
1.2
4
x 10
0.5
0
1.02
1.04
1.06
1.08
1.14
1.16
1.18
1.2
4
x 10
(d)
Figure 5.11: a-d; Real, imaginary and the absolute values of the amplitudes at the
center of each envelopes for the time series 10001 ≤ t ≤ 12048 .
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1.2
 Aj (arbitrary units)
1
0.8
0.6
0.4
0.2
0
1.02
1.04
1.06
1.08
1.1
t
1.12
1.14
1.16
1.18
1.2
4
x 10
Figure 5.12: Four wave amplitudes for 10001 ≤ t ≤ 12048. A1 (blue), A2 (red), A3
(green) and A4 (black).
well as in the normal modes, is very small, as shown in Fig. 5.3(a), they still have
a significant effect on the intensity variation of the pattern. Fig. 5.13(a) shows the
time average of the flat fielded patterns at 10, 001 ≤ t ≤ 20, 000. The time average
exhibits short segments of zig and zag waves distributed irregularly. The maximum of
this time average is very close to the maximum of the individual image. Fig. 5.13(b)
shows the intensity variation along the central row for the individual image (upper)
and for the average pattern (lower). It is obvious from the figure that the intensity
fluctuation from the average pattern is less than that from the individual image.
Besides fundamental peaks, the average power spectrum of Fig.5.3(a) has three
pairs of peaks at ±(2qc , 2pc ), ±(2qc , 0), and ±(2qc , −2pc ). The ratios of these peaks
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relative to the average fundamental peak are 0.05, 0.086 and 0.09. Thus, the dominant structure, in this case, is a nonuniform superposition of counter-propagating
waves.
170
168
pixel values
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n
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300
pixel values
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400
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450
100
200
300
400
500
600
158
n
(a)
(b)
Figure 5.13: (a) Time average of the patterns from 10, 001 ≤ t ≤ 20, 000 and (b)
Variation of pixels along central row for individual image (upper) and for the average
of images (lower).
5.2 Alternating waves
In an alternating wave, the direction reverses periodically. The sinusoidal form
of the wave starts from zero, goes to maximum, decreases to zero, reverses, reaches
a maximum in the opposite direction, returns again to zero and repeats the cycle.
Thus, the wave varies periodically in its magnitude and direction. If the medium
is moving opposite direction of the wave, or if the medium is stationary and there
is superposition of two traveling waves of same frequency propagating in opposite
direction, a standing wave comes into existence with nodes and antinodes. In 1D,
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consider two harmonic waves with same frequency, wavelength and the amplitude
traveling in opposite directions. Then, they can be represented as y1 = yo sin(qx− ωt)
and y2 = yo sin(qx + ωt); where yo is the amplitude of the wave, ω is the angular
frequency and q is the wave vector along x̂-axis. Then, the resultant wave is y =
2yo cos(ωt) sin(qx). It describes a wave that oscillates in time, but has a dependence
on sin(qx). Standing wave patterns are characterized by certain fixed points along
the medium which undergo no displacements. These points are called nodes. these
nodes are the result of destructive interference of two waves. Midway between every
constructive model point are points which undergo maximum displacement. These
points are the antinodes, which oscillate back and forth between maximum positive
displacement and negative displacement. The significance of the alternating wave is
that two AW can superimpose to form standing waves.
The experiment was performed in the sample cell I5234, having active area
A = 25 mm2 , filled with I52 of dopant concentration 5 wt.%. It was inserted in
the hot stage of the EC apparatus and the temperature was maintained at 55 ◦ C.
The sample had ²⊥ = 3.17 and σ⊥ = 11.53 × 10−9 Ω−1 m−1 at 25 Hz, as measured
by the lock-in amplifier. A video of the counter-propagating waves very near onset
(ε = 0.028) was recorded in the .avi format at 30 FPS in the conduction regime,
as explained in Chapter Four. These rolls had a Hopf frequency of 0.83 Hz. A
Matlab program converted the recorded video into a sequence of images (it can read
only up to 3,476 images). Fig. 5.14(a) is a snapshot of the video at t = 1,000.
The image was a superposition of oblique and normal modes. After flat fielding,
each image was transformed to the frequency domain and filtering was done using
Gaussian filter masks. Two-wave spatial demodulation was performed to get the
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50
100
m
150
200
250
300
50
100
150
200
250
300
350
400
n
(a)
7
5
4.5
zigzag amplitude (arbitrary unit)
zigzag amplitudes (arbitrary unit)
6
5
4
3
2
1
0
4
3.5
3
2.5
2
1.5
1
0
0.2
0.4
0.6
0.8
1
1.2
time (min)
(b)
1.4
1.6
1.8
0.5
0
1
2
3
4
5
6
7
time (s)
(c)
Figure 5.14: (a) Oblique modes at t = 1,000 from the sample cell I5234 at ωo τq = 0.38,
ε = 0.028 and T = 55 ◦ C. The rubbing direction of the cell plates is in the vertical
direction in the picture; the length scale represents 100 µm. (b) Average of the central
10 × 10 pixel values of the zig (red) and zag (blue) wave envelopes as a function of
time. (c) Zoom in of (b) showing alternating waves.
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zig and zag envelopes of each image. To understand the spatiotemporal behavior of
the envelopes, I computed local spatial average of the envelopes over a small region
at the center of the domain. The time series of these local averages over the time
1 ≤ t ≤ 3, 476 is shown in the Fig. 5.14(b). From the figure, it is obvious that
the zig mode is dominating over the zag mode. Fig. 5.14(c) is the zoom in the time
interval 901 ≤ t ≤ 1, 110. Here, the amplitudes of zig and zag modes vary periodically
between standing zig and zag modes. The frequency of these waves is comparable to
the Hopf frequency.
5.3 Localized states
Localized states are nonlinear structures, which emerge in nonequilibrium hydrodynamic systems. These are spatially localized traveling waves and are called
worms. These states have been observed in the electroconvection of nematic liquid
crystals [10, 14, 15]. They coexist in a small region, along with the larger uniform
region. These are the superposition of traveling waves localized within an envelope
having small widths normal to n but long and variable wavelength along n. Localized
pulses have been reported [16] for positive ε in convection in water/ethanol mixture
confined in a circular cell. The initial state they observed here was a superposition of
radially inward and outward traveling circular convection rolls. Initially, the pulses
spread in the direction parallel to the rolls. The pulses assumed nearly circular shape
and these localized states can be either by subcritical or by supercritical bifurcation.
Dennin et al have observed worm envelopes almost stationary [14] as well as traveling [10].
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These localized states occur in a system which is intrinsically anisotropic: electroconvection in nematic liquid crystals. For oblique rolls, because of reflection symmetry, there are four possible states that combine to form worms: left and right
traveling, zig and zag rolls. Worms consist of either a combination of left-traveling
zig and zag waves or of the right-traveling zig and zag waves. Even though the width
of the worms does not change much with an increase in applied voltage at a given
frequency, their lengths increase. Also, they will occupy more space at higher voltage.
I observed worms in planar sample cell I5239. At lower frequencies, in the
50
100
150
m
200
250
300
350
400
450
100
200
300
n
400
500
600
Figure 5.15: Snapshot of an images at 30 ◦ C. It corresponds to ε = 0.042, ωo τq = 0.74,
fH = 0.34 Hz and consists of active and inactive regions. The rubbing direction of
the cell plates is in the vertical direction in the picture; the length scale represents
100 µm.
conduction regime, the supercritical Hopf bifurcation resulted active and inactive
regimes as shown in Fig. 5.15 and exhibited STC states [9]. However, at F = 70 Hz
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and onward, I found localized states. They are caused by supercritical bifurcation
until the frequency reached the cut-off frequency of 340 Hz. Fig. 5.16 shows the
worms recorded at 2 s apart. They travel along ±n with frequency of 2.33 Hz.
a
b
c
d
Figure 5.16: Time series of worms at interval of 2 s. The hot stage temperature is 30
◦
C and the frequency of applied ac is 130 Hz. The rubbing direction of the cell plates
is in the vertical direction in the pictures; the length scale represents 100 µm.
Fig. 5.17 shows the profile of the worms of Fig. 5.16 along ±n. It evidently shows
the variation of the intensity of the worm states and also the shifting of the peaks
show that they are traveling. Fig. 5.17 of these four localized states of Fig. 5.16 shows
almost uniform illumination over the size of the system, except at the worm states
where the intensity reduces, illustrating the localization of the worm states. As ε
is increased, the average lifetime of the worms increases until they disappear, either
through interaction with other worms or traveling out of the system. For two worms
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amplitude (arbitrary units)
amplitude (arbitrary units)
a
150
140
130
120
0
50
100
150
200
250
m
b
300
350
400
450
150
140
130
120
0
100
200
300
400
500
600
n
Figure 5.17: Intensity variation of the worms along and at right angle to the director.
at large distance, there does not appear to be any interaction. But, when two worms
are close to each other, they nucleate. If the short worm collides with the long worm,
the short worm sometimes disappears. The length of the shorter worm in Fig. 5.16
has not changed with time, but the length of the longer worm does change with time
and eventually might extend over the whole length of the system, because of periodic
boundary condition. Fig. 5.18 shows worms at 150 Hz, the longest at the right side;
indicating that higher is the frequency, longer is the worm. There is the report of
localized states via subcritical bifurcation [17, 18] too. This means the onset of the
extended plane wave state is subcritical, with discontinuities in amplitudes etc.
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50
100
150
m
200
250
300
350
400
450
100
200
300
n
400
500
600
Figure 5.18: Worms at 30 ◦ C and 150 Hz ac field. The first worm from right covers
almost whole field of view along ±n. The rubbing direction of the cell plates is in the
vertical direction in the picture; the length scale represents 100 µm.
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5.4 Conclusions
I analyzed a complex spatiotemporal pattern that occurred in electroconvection
in nematic liquid crystal I52. The pattern was observed very near onset of convection, which is a supercritical Hopf bifurcation leading to counter-propagating zig and
zag pairs propagating in oblique directions relative to the director. Mathematically,
this kind of instability is described by a system of four coupled Ginzburg-Landau
equations [9, 12, 13], leading the evolution of slowly varying envelopes of four traveling plane waves. I extracted the envelopes from the experimental data using spatial
and temporal demodulation technique. In order to characterize the dynamics of the
pattern, I studied the amplitude variation in the small region at the center of each envelope. The absolute value of the average amplitudes showed chaotic behavior, both
in space and in time as shown in Fig. 5.12. Application of various diagnostic tools
to the envelopes extracted from the images of EHC patterns in sample cell I5239,
including the calculation of average intensities and spatial correlation length, global
and local Karhunen-Loéve decomposition in Fourier space and physical space, the
location of holes in time and space, the identification of coherent vertical structures,
and estimates of Lyapunov exponents are explained elsewhere [9]. The authors have
confirmed the chaotic nature of the EHC patterns, which resulted in some of the
positive Lyapunov exponents, along with some negative values. Lyapunov exponents
provide a qualitative and quantitative characterization of dynamical behavior. These
exponents are related to exponentially fast divergence or convergence of nearby orbits in phase space [19]. Since nearby orbits correspond to nearly indistinguishable
states, exponential orbital divergence indicates that the system behaves quite differently. Any system containing at least one positive Lyapunov exponent is defined to be
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chaotic. 1D maps are characterized by a single Lyapunov exponent which is positive
for chaos, zero for a slightly stable orbit and negative for a periodic orbit.
The patterns in sample cell I5295 had extended actively along the whole space.
The inhomogeneities due to optical system was not apparent as in [9] and primary
modes were dominating even before filtering. As a result, the dynamics carried by
the envelopes were not lost while filtering out unnecessary modes. This pattern is
similar to the pattern reported by Dennin et al [10, 20], which was found in EHC in
nematic I52 slightly above onset, but for different set of material parameters. In the
future, one can determine the SM parameters for these experiments. Together with
estimates of the additional parameters occurring in WEM, this will enable to compute
the parameters in the Ginzburg-Landau system for these kind of experiments, and to
compare the simulated and the recorded envelope dynamics.
In the sample cell I5234, I observed alternating waves which are characterized
by an alternation between the standing waves in the two oblique directions. Theoretical predictions [21] has shown that there is a large region in the parameter space for
alternating waves (AW) with different sets of parameters and the results agree with
them. The frequency of these alternating waves is in the order of Hopf frequency.
Quantitatively different phenomena occurred in the sample cell I5239 above
70 Hz in the conduction regime. These were the localized states called worms. The
rolls and the worms travel in opposite directions. Their growth, decay and length
are apparently unpredictable and random, both in time and space. Thus, they can
be regarded as certain type of STC. The amplitude of the leading and the trailing
edges are different, with the trailing edge decreasing gradually in its amplitude. At
higher frequencies, they are longer than at lower frequencies. When it becomes long
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enough to cover the whole system, it loses its distinct head structure. They always
travel in the direction of their head. They annihilate and nucleate continuously at
short distances. When they have a head-on collision, either the worm with weaker
amplitude disappears or it travels along the direction of the stronger one. When
worms annihilate, the motion at a right angle to n should be driven by the imbalance
between the amplitude of zig and zag components. Since the worms occur very close
to threshold and the bifurcation is supercritical, these states are clearly described
by extension to the CGLE for zig and zag waves that lead to localized, worm like
solution [22].
Out of the many sample cells with different thickness and different impurity
concentration, I observed localized states only in the cell I5239. This concludes that
the appearance of worms in electroconvection must have some sort of relation to σd2 .
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BIBLIOGRAPHY
[1] R. Pool. Chaos theory: how big and advance? science, 245:26, July 1989.
[2] M. C. Cross and P. C. Hohenberg. Spatiotemporal chaos. science, 263:1569,
March 1994.
[3] M. Lesieur. Turbulence in Fluids. Kluwer academic publishers, 2nd edition,
1990.
[4] J. P. McClymer, E. F. Carr and H. Shehadeh. Dynamic scattering modes of
nematic liquid crystals in magnetic field. Addison-Wesley, 1995.
[5] T. T. Katona and J. .T. Gleeson. Conductive versus dielectric defects and
anisotropic vs. isotropic turbulence in liquid crystals-electric power fluctuation
measurements. electronic-Liquid Crystals Communications, May 2003.
[6] S. Chendrasekhar. Liquid Crystals. Cambridge University Press, 2nd edition,
1992.
[7] M. C. Cross and P. C. Hohenberg. Pattern formation outside of equilibrium.
Reviews of Modern Physics, 65(3):854, 1993.
[8] S. W. Morris, E. Bodenschatz, D. S. Cannell, and G. Ahlers. The spatiotemporal
structure of spiral-defect chaos. Physica D, 97:164, 1996.
[9] G. Dangelmayr, G. Acharya, J. Gleeson, I. Oprea, and J. Ladd. Diagnosis of spatiotemporal chaos in wave-envelopes of an electroconvection pattern, submitted.
Phys. Rev. E, 2008.
[10] M. Dennin, G. Ahlers, and D. S. Cannell. Spatiotemporal chaos in electroconvection. Science, 272:388, 1996.
[11] J. Langenberg, G. Pfister, and J. Abshagen. Chaos from hopf bifurcation in a
fluid flow experiment. Phys. Rev. E, 70:046209, 2004.
[12] I. Oprea, I. Triandaf, G. Dangelmayr, and I. I. B. Schwartz. Quantitative and
qualitative characterization of zigzag spatiotemporal chaos in a system of amplitude equations for nematic electroconvection. Chaos, 17:023101, 2007.
[13] G. Dangelmayr and I. Oprea. Modulational stability of traveling waves in 2D
anisotropic systems. J. Nonlin. Sci., 18:1, 2008.
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[14] C. Kamaga and M. Dennin. Modulation of localized states in electroconvection.
Phys. Rev. E, 65:057204, 2002.
[15] U. Bisang and G. Ahlers. Bifurcation to worms in electroconvection. Phys. Rev.
E, 60:3910, 1999.
[16] K. Lerman, E. Bodenschatz, D. S. Cannell, and G. Ahlers. Transient localized
states in 2D binary liquid convection. Phys. Rev. Lett., 70(23):3572, 1993.
[17] Y. Tu. Worm structure in the modified swift-hohenberg equation for electroconvection. Phys. Rev. E, 56(4):R3765, 1997.
[18] U. Bisang and G. Ahlers.
Phys. Rev. E, 60(4):3910, 1999.
Bifurcation to worm in electroconvection.
[19] A. Wolf, J. B. Swift ad H. L. Swinney, and J. A. Vastano. Determining lyapunov
exponents from a time series. Physica D, 16:285, 1985.
[20] M. Dennin. A Study in Pattern Formation: Electroconvection in Nematic Liquid
Crystals. PhD thesis, Department of Physics, University of California, Santa
Barbara, 1995.
[21] G. Dangelmayr and I. Oprea. A bifurcation study of wave patterns for electroconvection in nematic liquid crystals. Mol. Cryst. Liq. Cryst., 413:305, 2004.
[22] H. Riecke and G. D. Granzow. Localization of waves without bistability: worms
in nematic electroconvection. Phys. Rev. E, 81(2):333, 1998.
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CHAPTER 6
Material parameters and Nusselt numbers
characterization in I52
Most of the phenomena in nematics are connected by the fact that external
stimuli easily influence the orientation of the director. These phenomena, such as
onset of EHC, Freedericksz transition [1, 2], N-I phase transition temperature, etc.
depend on the material parameters. Besides, these parameters have their importance
to compare the experimentally observed results with the theory like SM and WEM,
to calculate the coefficients of CGL equations, to estimate the patterns during electroconvection, and so on. Since all the physical properties of a particular LC are
interconnected and they are defined by their molecular structures, it is not possible
to change one physical parameter without affecting the rest.
I studied some of the characteristic properties of I52 as function of temperature namely critical electric and magnetic fields for Freedericksz transition, dielectric
constants (²k and ²⊥ ), conductivities (σk and σ⊥ ), frequency dependence of dielectric
constants and conductivities, director relaxation time (τd ), charge relaxation time (τq )
and refractive indices (nk and n⊥ ). Besides these, I have also measured elastic constants (K11 , K22 and K33 ), Leslie viscosity coefficients (α’s) and Nusselt numbers. A
number of parameters had been measured and reported elsewhere [3,4]. The purpose
of measuring these material parameters is to use them for quantitative comparison
between the WEM and the experimental results and to calculate the coefficients of
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CGL equations in the future. Most of these parameters characterization are based on
Freedericksz transition. Table 6.1 gives the sample cells used, their thickness and the
alignment for various experiments.
Table 6.1: Summary of sample cells used in parameters characterization.
label
I5211
I5222
I5233
I52T
I52A
I52F
I52Z
I52N
thickness, µm
alignment
57.74±0.69
planar
51.51±0.85
planar
54.53 ±0.83 homeotropic
57.15 ±0.86
planar
50.6 ±0.73
planar
47.92 ±1.02
planar
23.8
homeotropic
24.12±0.41
planar
purpose
K11 , σ, ²
K22
K33
τd
α’s
²(F) and σ(F)
T at which ∆² = 0
Electric Nusselt numbers
6.1 Freedericksz Transition
In the presence of external electric or magnetic field, a liquid crystal sample
uniformly aligned between two parallel glass plates undergoes a transition to an elastically deformed state when the strength of the external field exceeds a well defined
threshold. This transition first observed by Freedericksz and Zolina in 1927 is called
the Freedericksz transition . It is a well studied phenomenon and has its applications
in liquid crystal display devices.
In a thin sample cell of NLC having diamagnetic anisotropy, ∆χ (χk − χ⊥ ) > 0,
the magnetic field H causes the liquid crystals in bulk to be parallel to H. This alignment is different from the initial sample alignment if H is equal to or greater than
certain critical field strength. A similar phenomenon occurs in the presence of electric
field.
There are three pure deformations [1, 2, 5] in a uniaxial LC sample when there
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H < Hc
H>H
c
H < Hc
H > Hc
(a)
(b)
z
y
(c)
x
H<Hc
H>H
c
Figure 6.1: Illustration of the geometries for Freedericksz transition to determine (a)
splay, (b) twist and (c) bend elastic constants. Geometries to the left are for the
magnetic field less than the critical threshold field and the geometries to the right are
for the magnetic field greater than the critical threshold field.
is strong anchoring at the boundaries. If initially, the molecular alignment is along
the boundary plane and the field is applied normal to its plane, the LC undergoes
splay deformation (non vanishing divergence of vector field n) for the applied field
greater than the critical threshold field. If the alignment is parallel to the boundary
plane and the field is applied normal to the planar alignment along the plane, the
LC undergoes twist deformation (non vanishing component of curl of n parallel to n)
for the applied field greater than the critical field. Similarly, if the molecular alignment is normal to the boundary plane and the field is applied along the plane, the
LC undergoes bend deformation (non vanishing component of curl of n normal to n)
for the applied field greater than the critical field. These deformations are shown in
Fig. 6.1. The total elastic energy density due to these deformations is
1
F = [K11 (∇ · n)2 + K22 (n · ∇ × n)2 + K33 (n × ∇ × n)2 ]
2
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where K11 , K22 and K33 are the splay, twist and bend elastic constants. In the
presence of a magnetic field, the anisotropy ∆χ produces an excess magnetization M
= ∆χ(n · H) in the direction of n. This magnetization produces a magnetic torque
density Γm = ∆χ(n · H)n × H. Thus, the total free energy density (omitting the
surface energy terms due to elastic anchoring of the liquid crystal at the bounding
surfaces) is given by Eq. 6.1. Even in the presence of H, the unperturbed state
satisfies the conditions for local equilibrium (n · H = 0).
The elastic constants emerge when dealing with almost all important phenomena
related to nematics, such as microscopic textures, flow, light scattering, threshold
voltage in EHC, etc. Scientists have used these ratios for providing characteristics
of the molecular structure [6], liquid crystal display and switching devices. They are
also used to investigate the critical behavior of nematic-smectic A phase transition.
This is because of the fact that the bend and twist elastic constants diverge at this
transition. In a twisted nematic cell, the threshold behavior becomes steep with
increasing ratio of K33 /K11 [7]. ∆χ has its importance in evaluating the orientational
order and must be known for the determination of elastic constants by magnetic-field
induced deformation.
6.1.1
Splay elastic constant
I performed EHC experiments in I52 in the range of temperatures for which ∆² < 0
and I am interested in the physical parameters for the same temperature range. The
NLCs having ∆² < 0 cannot be used for electric splay deformation. The splay, twist
or bend deformation due to external magnetic field gives the critical threshold magnetic field for corresponding deformation. The critical magnetic field for Freedericksz
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Power supply
to detector
Multimeter
Sample
He−Ne laser
Polarizer
Analyzer
Detector
Copper block
Magnet
Magnet
x, E
z, H
Figure 6.2: Experimental set up for magnetic field induced Freedericksz transition.
transition is
π
Hc =
d
s
Kii
µo ∆χ
i = 1, 2, 3
(6.2)
One parameter out of Kii and ∆χ should be known to calculate the other. Since,
both K11 and ∆χ for I52 were not known, I applied the electric and magnetic field,
simultaneously.
Fig. 6.2 shows the schematic of the apparatus used to measure the elastic constants for the NLC I52. The maximum magnetic field from the electromagnet in our
laboratory is 1.3 T. At first, I set the analyzer and the polarizer without the sample
in between the magnetic fields. By passing laser light through the polarizer, I rotated
the analyzer until the reading on the photodetector was at a minimum to confirm
that the polarizer and the analyzer were crossed. To reduce the effect of background
light, I inserted a red filter on the tip of the detector. I measured the thickness of
the ready-made planar cell [8] I5211 interferometrically and attached two wires on
the plates using silver-laden epoxy. After drying, I filled it with a sample of I52 and
inserted it into the copper block fitted with a heater and RTD sensor to heat and
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control the temperature. I positioned the copper block in between the electromagnets and passed laser light through the polarizer, sample cell, analyzer and finally,
through the detector to measure the photo voltage. Fixing the polarizer and analyzer
as crossed polarizers in between the electromagnets, I rotated the sample holder until
I could record the maximum photo voltage. At first, with only a magnetic field applied, I found the critical field for splay deformation. Then, I kept the magnetic field
above the critical threshold field at a certain temperature and recorded the variation
of dc photo voltage with applied ac voltage of driving frequency 1.0 kHz. The graph
of photo voltage versus the applied ac gives the transition field.
When both fields are applied simultaneously, the equation for the transition
field is given by
²o ∆²Vc2 + µo ∆χd2 Hc2 = π 2 K11
(6.3)
Here, Hc is the critical magnetic field for Freedericksz transition and Vc is the critical
ac voltage which causes the Freedericksz transition in the presence of magnetic field
greater than Hc . By varying the magnetic field and repeating the process at certain
temperatures, I was finally able to plot the graph of Vc2 versus Hc2 . Comparing
Eq. 6.3 with the fitting equation from Fig. 6.3, I got π 2 K11 /²o ∆² = −655.9 which
gives K11 = 22.36 × 10−12 N. The coefficient of Hc2 in the equation from Fig. 6.3 can
be used to get ∆χ.
6.1.2 Bend elastic constant
Bend Freedericksz transition takes place in a homeotropic sample cell. To
prepare the homeotropic cell of desired thickness, I followed following steps:
• Take ITO coated glass plates of different sizes.
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(a)
(b)
Figure 6.3: (a) DC photo voltage as a function of the applied ac voltage at H =
0.3209 T > Hc = 0.2476 T at 25 ◦ C. (b) Graph for Hc2 versus Vc2 . The fit gives
Vc2 = −655.9 + 1.07 × 104 Hc2 .
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Figure 6.4: Splay critical magnetic fields as a function of temperature.
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• Clean the glasses by ultrasonic method;
• For this, keep the trough with glasses arranging ITO coating of each glass
towards the same side.
• Heat the glasses in the trough for 30 minutes at 50 ◦ C.
• Remove the trough from the ultrasonic machine and clean the glasses thoroughly with deionized water for 15-20 minutes.
• Put the trough in the hot oven at around 100-120 ◦ C for at least 30 minutes.
• After removing from the hot oven, it is ready for spin coating, making sure that
no dust particles attach to the glass plates. Then, spread polymide solution with the
help of syringe over the ITO glass carefully and uniformly over the area of the plate.
• Cover the lid of the spin coater.
• Rotate the machine in low speed for first few minutes and then in high speed.
• After spin coating, remove the glass and put it in hot plate for about 5-10 minutes.
• Keep the coated glass plates in hot oven for about an hour at 180 ◦ C.
• Finally, use proper mylar space and epoxy to join these plates.
The prepared homeotropic cell was I5233. I measured its thickness and attached
connecting wires as in section 6.1.1. This cell was filled with pure I52 and kept on
the copper oven, along with RTD sensor and heater in between the electromagnets.
A capacitance bridge connected to the hookup wires of the cell gives the capacitance
(C) and conductance (G) as a function of applied field at different temperatures. For
the graphs of H versus C and H versus G, I calculated the bend critical Freedericksz
fields. Fig. 6.5(a) gives the variation in capacitance below and above the Freedericksz
transition and (b) gives the variation of critical fields with temperature. The critical
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(a)
(b)
Figure 6.5: (a) The capacitance of sample cell I5233 as a function of the magnetic
field when the sample goes bend deformation. (b) Variation of critical magnetic field
with the temperature for the same deformation.
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field decreases with increase of temperature. The anisotropy in magnetic susceptibility was calculated when the magnetic and electric fields were applied simultaneously
q
in section 6.1.1. Using Hc = π/d K33 /µo ∆χ, the bend elastic constant at 25 ◦ C is
26.31 × 10−12 N .
6.1.3 Twist elastic constant
The threshold for a twist deformation cannot be detected optically when viewed
along the twist axis. This is because of the large birefringence, ∆n, of the medium
for this direction of propagation. Thus, with experimental geometry for twist cell for
which the director is anchored parallel to the walls at either end and light is incident
normal to the film, the state of polarization of the emergent beam is indistinguishable
from that of the beam emerging from the untwisted nematic [2] and hence, I followed
the dielectric method introduced by Z. Li [9] to measure Hc for twist geometry.
To prepare the cell, I followed following steps:
• Spin coat the clean glass plates with IP255 on the non ITO coated surface.
• Dry them on general hot stage for about 10 minutes and on another hot stage at
225 ◦ C for about an hour.
• Finally, rub the glass at only one direction with valvate for three times and mark
the rubbing direction.
• Fix two gold wires as parallel as possible to the rubbing direction on one of the
plates (of course two glass plates should be antiparallel to overcome the tilt angles).
The diameter of the wire was 51.51± 0.85 µm, as measured by a traveling stage
microscope and the space between the wires was D = (1600± 70) µm. The schematic
of the cell geometry is given in Fig. 6.6. The capacitance of the cell in this geometry
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H
n +
2r
D
(a)
(b)
Figure 6.6: (a) Schematic of the cell geometry. The drawing shows the top view
of the cell and n is the direction of undistorted director orientation.(b) Variation of
the capacitance of sample cell I5222 with magnetic field when the sample goes twist
deformation at 25 ◦ C.
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is [10]
"
C = ²²o ι 2 cosh
Ã
−1
D2 − 2r2
2r2
!#−1
(6.4)
Here, ι is the length of each wire. The prepared cell was I5222. I filled it with
I52 and to measure the transition magnetic field, I joined the gold electrodes to the
capacitance bridge making sure that the wires from capacitance bridge to the sample
cell were nonmagnetic. In case of magnetic wires, there will be fluctuation in C and
G.
Then, I kept the sample cell in between the electromagnets, as explained in
6.1.1. The direction of field was H= H(0, ŷ, 0). I studied the variation of capacitance
and the conductance with the magnetic field and from the graph of H versus C or
G, I calculated Hc for twist deformation. ∆χ was calculated in section 6.1.1. Then,
q
using Hc = π/d K22 /µo ∆χ, the twist elastic constant at 25 ◦ C is 12.84 ×10−12 N.
In all these experiments for Freedericksz transition in I52, the temperature of
the sample was stabilized to ±0.1◦ C using Conductus LTC-10 temperature controller
and the values of Kii ’s decreased by about 25 % while increasing the temperature
from 25 ◦ C to 50 ◦ C.
6.1.4 Dielectric anisotropy and conductivity anisotropy
The dielectric constants along the preferred axis (²k ) and perpendicular to this
axis (²⊥ ) are different due to uniaxial symmetry in NLCs. The difference, ∆² = ²k −²⊥ ,
is called the dielectric anisotropy. Depending upon the chemical structure of the constituent molecules, it can be positive or negative. ²k is greater than ²⊥ when the
permanent dipole moment of each molecule is parallel to the long axis and ²k is less
than ²⊥ when the permanent dipole moment of each molecule is normal to the long
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axis [1] . For positive (negative) ∆², if the nematic axis is parallel (perpendicular)
to the field, it will have the lowest energy state [11]. Due to the fluid nature of the
NLCs, the field required for this kind of realignment is small. The ability to control
the orientation of the nematic axis by a weak field is the basis for various NLCs in
liquid crystal display. The response of NLCs on electric field depends on both the sign
and magnitude of ∆². At room temperature, typical values of ²k and ²⊥ for MBBA
are = 4.7 and = 5.4 respectively.
Many NLCs are poor conductors of electricity. The electrical conductivity is
anisotropic and depends upon the impurity concentration and chemical nature of
the impurities present in the sample. Adequate amount of the conductivity in the
sample is essential for EHC to take place. Patterns obtained during EHC are highly
dependent upon its magnitude; various patterns, such as oblique stationary, counterpropagating zig and zag rolls, localized states obtained during EHC in I52 were explained in Chapter Four. The threshold voltage for EHC in SM, the Hopf frequency
in WEM and the charge relaxation time, all depend upon the conductivity. Generally, nematics are doped with some suitable impurities to increase its conductivity.
An impurity introduced into the sample dissociates into anions A− and cations B +
according to the reaction
AB ⇔ A− + B +
The conductivity depends upon the mobility of positive and negative ions and the
dissociation-recombination rates. In weak electric field and low ionization rate, it is
expressed as [12]
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s
σ = e(µ+ µ− )
KD C
KR
(6.5)
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where µ+ and µ− are the mobilities of positive and negative ions, C is the impurity
concentration measured in cm−3 , and the constants KD and KR are the dissociation
and recombination rates respectively. The temperature dependence of ion mobilities
and ionization-recombination constants control the temperature dependence of conductivity. Even though the conductivity varies by varying the concentration, the ratio
σk /σ⊥ remains constant. One can change this ratio by changing the stereochemical
shape of the dissolved ions [1]. In most cases, σk is greater than σ⊥ and the anisotropy
(∆σ = σk − σ⊥ ) depends on the type of dopant. Einstein’s relation [12] gives the ratio
of conductivities with the mobilities (µ’s) and the diffusion coefficients (D’s) as
σk /σ⊥ = µk /µ⊥ = Dk /D⊥
(6.6)
Depending upon the shape of the impurity molecules, the magnitude σk /σ⊥ varies
from 1.23 to 1.68 for MBBA [12].
For the measurement of ²’s and σ’s, I used the same ready-made EHC cell
I5211. I chose a thicker cell due to the fact that the critical field for Freedericksz
transition depends upon thickness of the sample cell.
To measure ²⊥ , ²k , σ⊥ , σk and their anisotropies at various temperatures, I inserted
the sample cell into a copper oven fitted with heater and platinum RTD and inserted
it in between the electromagnets such that H k ẑ. The temperature of the sample
was stabilized to ± 0.1 ◦ C using Conductus LTC-10 temperature controller. The nonmagnetic hookup wires from the cell were connected to the capacitance bridge. Proper
connection of the wires is necessary to avoid excessive fluctuation of capacitance and
conductance. The magnetic field was ramped up at small steps and the corresponding
capacitances and conductances were recorded at certain temperatures.
Below the Freedericksz transition field, C⊥ and G⊥ are constant. For a sample
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Figure 6.7: Variation of Capacitance C (blue) and conductivity σ (red) with the
applied magnetic field.
of negative ∆² and positive ∆σ, the capacitance decreases, whereas the conductance
increases when the applied field is greater than the critical field strength (Fig. 6.7).
The average of the capacitance and the conductance below critical field gives C⊥ and
G⊥ respectively. To measure Ck and Gk , I plotted the inverse magnetic field versus
the conductance and capacitance, choosing data for which the applied field was high
above critical field. They fit to straight lines and the conductance (capacitance) for
which H −1 is zero (H= ∞) is the required σk (Ck ). However, the slope of these straight
lines for the capacitance versus inverse field is positive and that for the conductance
(and hence, the conductivity) versus inverse field is negative (Fig. 6.8). Then, the
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Figure 6.8: Variation of inverse H with capacitance and conductivity at 25 ◦ C from
the data of Fig. 6.7 for the cell I5211. The blue up triangles are for σ verses H −1 .
The pink straight line is the fit H −1 = 4.382 × 10−9 − 1.915σ which gives σk =
2.29 × 10−9 Ω−1 m−1 and the red circles are for capacitance. The green straight line
is the fit H −1 = −67.32 × 10−12 + 2.345C which gives Ck = 28.71 pF.
dielectric constants and the conductivities were calculated using
dCk,⊥
A²o
Gk,⊥ d
=
A
²k,⊥ =
σk,⊥
(6.7)
I repeated the same process by ramping up the temperature by 5 ◦ C within
the nematic phase range. Fig. 6.9 shows that both ²⊥ and ²k decrease with increasing
temperature. Below 60.18 ◦ C, ²⊥ > ²k , becomes equal at 60.18 ◦ C and on further
increasing the temperature, ²⊥ < ²k . Fig. 6.10 shows the variation of the dielectric
anisotropy and the conductivity anisotropy with temperature for the sample. Both
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Figure 6.9: ²⊥ and ²k as a function of temperature for the cell I5211. ∆² = 0 at
60.18 ◦ C. The red circles are for ²⊥ and the blue up triangles are for ²k . The error
bars are calculated repeating the experiment on the same cell after two months.
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(a)
(b)
Figure 6.10: (a) the dielectric anisotropy and (b) the conductivity anisotropy as a
function of temperature for planar cell I5211 filled with the sample.
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of them increase with increase of temperature.
6.1.5
Bend deformation in electric field
[3] reports that ∆² of I52 depends on temperature, changing its sign from (-,+) to
(+,+) at 75.4 ◦ C as temperature is increased. I followed the concept of electric field
induced bend Freedericksz transition to measure the temperature at which ∆² = 0.
The theoretical value of the critical voltage for bend deformation due to electric field
is
VcF = π(
K33 1/2
)
²◦ ∆²
(6.8)
When the temperature is increased, the onset voltage decreases and it diverges as ∆²
goes to zero.
The schematic of the experiment to measure critical voltage for bend FreederSample
Temperature
Control
Optical
Power
Meter
sample
He−Ne laser
Polarizer
Analyzer
Detector
z
HP 33120A
Function
Generator
HP 34401A
Multimeter
Figure 6.11: Schematic of the geometry for electric field induced bend Freedericksz
transition.
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Figure 6.12: Transition curve used to determine the onset of Freedericksz transition
in I52 at 30 ◦ C.
icksz transition is shown in Fig. 6.11. A homotopic cell I52Z was made as described in
6.1.2 and filled with I52. Proper wire attachments helped to provide ac voltage. The
cell was kept in a closure fitted with a heater and RTD sensor between cross polarizers. A He-Ne laser source was fixed coplanar with the polarizers. The temperature
was controlled by LTC-10, the temperature controller with accuracy of ±0.01◦ C. To
generate an ac field in the sample, a sinusoidal 1 kHz voltage from Hewlett Packard
33120A function generator was amplified in an HP 34401A multimeter and applied
to the sample, as shown in Fig. 6.11. The voltage was ramped up on the cell, which
communicated with the given instruments and IEEE bus. A detector connected to
the optical power meter (model 835) measured the intensity of the transmitted light
from the analyzer. A red filter used in the tip of the detector made sure that only
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(a)
(b)
Figure 6.13: (a) Variation of critical voltage for bend transition with temperature. (b) Same data of figure 6.13 plotted as (VcF )−2 versus temperature. The fit
gives the ∆² = 0 at 62.91 ◦ C.
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light from the laser beam was detected by the detector and not any background light.
One can turn off the room light instead of using the filter.
To determine the critical Freedericksz voltage (VcF ), I measured the intensity of
light transmitted through the sample cell I52Z in the form of power. A transmission
curve at 30 ◦ C is shown in Fig. 6.12. Initially, the intensity is not completely zero due
to slight inhomogeneity of the cell. VCF was calculated by extrapolating the sharp rise
in intensity to a zero value. It is 22.48 V at this temperature. Fig. 6.13 shows the
onset voltage for the Freedericksz transition as a function of temperature. It is clear
that the onset voltage increases with temperature. Fig. 6.13(c) shows the variation
of the reciprocal of (VcF )2 with temperature. The fit obtained is a straight line. Its
extrapolation up to the temperature-axis gives 1/(VcF )2 = 0. Since K33 cannot be
zero, ∆² must be zero. Thus, I got ∆² = 0 at 62.91 ◦ C. The importance of this temperature is that when an ac electric field is applied, EHC takes place in the sample
cell for which ∆² is either negative or slightly positive. I performed all experiments
for EHC below this temperature.
6.1.6 Frequency dependence of conductivities and dielectric constants
This experiment was performed to study the variation of the conductivities and
the dielectric constants and their anisotropies with applied frequency. The lock-in
amplifier, which I will explain in section 6.5, measures the in-phase and out-of-phase
components of the current flowing through the sample cell at different frequencies.
I filled the readymade planar cell I52F with I52 and inserted it into a copper
block, along with the heater and the RTD sensor as in section 6.1.1. The copper
block was kept in between the poles of the electromagnet. A function generator of
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Figure 6.14: Variation of ²⊥ and σ⊥ with frequency at different temperatures. The
dashed lines are for ²⊥ at temperatures as shown. The solid lines are for σ⊥ at 25 ◦ C
(hexagons connected by cyan solid line), 30 ◦ C (filled circles connected by dark green
solid line) and 45 ◦ C (down triangles connected by solid black lines).
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the lock-in amplifier was used to produce a sinusoidal voltage, which is amplified and
applied to the cell. The path-to-ground for the current traversing the cell was through
a current-to-voltage converter. The output signal from this converter was measured
by a lock-in amplifier. The reference signal and the frequency was applied by the
lock-in amplifier itself. For each frequency, the nematic cell was replaced by a pure
resistance and the phase setting of the lock-in was adjusted to zero, the out-of-phase
component. The nematic cell was then reinserted.
At each frequency, I measured the in-phase (X) and out-of-phase (Y) components. The X component gives the conductivity (σ) and the Y component gives the
dielectric constant (²). After calculating the perpendicular components, the magnetic
field was turned on and ramped up in steps to record the X and Y components as
a function of magnetic field. From the graph of capacitance versus field and the
conductivity versus field, the parallel and perpendicular components of σ and ² were
calculated as explained in section 6.1.4.
At lower frequencies, ²⊥ is very dependent on the frequency but independent
at higher frequencies. However, σ⊥ increases with an increase of applied frequency.
This is shown in Fig. 6.14. I have found that ²k and σk also vary with the frequency
as ²⊥ and σ⊥ respectively.
6.2 Director relaxation time
The director relaxation time (τd ) is the time scale introduced in SM [13]. It is
related to the elastic constant, rotational viscosity and the thickness of the sample
cell as
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γ1 d2
τd = 2
π K11
(6.9)
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Since it is connected with the rotational viscosity γ1 , it does not have an isotropic
counterpart.
To measure the director relaxation time, I filled the planar cell I52T with I52
+ 4 wt. % I2 and kept it inside a copper block fitted with the heater and the sensor
as explained in section 6.1.1. The cell was hooked up to the capacitance bridge to
measure its capacitance and conductance. Then, a maximum magnetic field was
applied along the ẑ-direction (initially, the director was along x̂). When the field
exceeded the critical value, splay deformation takes place and at a maximum field,
the director will be along the direction of the field. The program was run to measure
the capacitance and the loss with elapsed time. The sampling rate was chosen as
small as possible. After a while, the whole copper block (along with the sample cell)
was gently removed from the magnetic field. I could switch off the field, but it took
8-10 seconds to drop to a minimum, which was found to be greater than the director
relaxation time. The capacitance and the loss immediately decreased and remained
constant as shown in Fig. 6.15(a).
The loss (G), which is reciprocal of resistance, as a function of time when the
field is switched off can be expressed as [14, 15]
G(t) = G⊥ + (G(H) − G⊥ ) exp(−
2t
)
τd
(6.10)
where the director relaxation time, τd , is given by equation (6.9). Here, d is the
thickness of sample cell, which was measured interferometrically. G(t) is the loss at
any time t, G⊥ is the loss perpendicular to the director and G(H) is the loss as a
function of magnetic field. Denoting ∆G = G(t) − G⊥ , Eq. 6.10 can be written as
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loge (∆G) = loge (G(H) − G⊥ ) −
2t
τd
(6.11)
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(a)
(b)
Figure 6.15: (a) Variation of loss with time at 40 ◦ C when the magnetic field is
suddenly ceased. (b) Variation of loge | ∆G | with time at 40 ◦ C immediately after
the magnetic field is off. It is a straight line loge | ∆G |= 30.735 − 0.659t and gives
τd = 3.035 s.
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Eq. 6.11 is a straight line with the slope 2/τd . Thus, the director relaxation time
Figure 6.16: Variation of director relaxation time τd with temperature.
τd = 2/slope
I calculated τd at a difference of 5 ◦ C in the nematic range from 25 ◦ C to 100
◦
C. Fig. 6.16 gives the variation of τd with temperature. It decreases with rise in
temperature and stays almost constant at higher temperatures in the nematic range.
These values of τd , along with K11 , will be used to calculate γ1 in section 6.4.
6.3 Refractive indices
The dielectric anisotropy of the NLCs causes the difference in velocity of light
polarized along the direction of director and at right angle to it and hence they are
birefringent. If the transmitted beam orients at an angel other than 0o or 90o with
the director when the laser beam passes through the polarizer, then, it will have its
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two components, one along the direction of director and the other at right angle to
it with no phase difference. When passing through the liquid crystal, they will be
out-of-phase and emerge as elliptically polarized. As the electric field of elliptically
polarized light uniformly rotates, it will be parallel to the polarization axis of the
second polarizer two times in each rotation. Consequently, if we introduce a liquid
crystal in between crossed polarizers, the field of view will generally be bright and
without liquid crystals, it will be dark. This phenomenon convinced O. Lehmann
that liquid crystals have anisotropy. Thus, it is the dielectric anisotropy of the liquid
crystal which is responsible for the anisotropy in refractive indices (n2 ∝ ²) provided
the order parameter is non-zero. The average value of the refractive indices in the
nematic phase is given by hn2 i =
1
(n2k
3
+ 2n2⊥ ). Since the dielectric constant is a
function of material density and material density is a function of temperature, this
value,
q
hn2 i, differs from the refractive index of the material in the isotropic phase
(niso ) [12, 16, 17].
The principal refractive indices of liquid crystals, ne and no range from 1.4
to 1.9 and the uniaxial birefringence, ∆n = ne − no can be between 0.02 and 0.4
[18]. Negative birefringence are associated with discotic nematics or columnar phases.
Biaxial liquid crystals have all three refractive indices different with n3 significantly
greater or smaller than the other two.
For molecules having axial symmetry, the Lorentz-Lorentz expression relating
the refractive index to the mean molecular polarizability is given by
n2k − n2⊥
S∆α
=
2
n −1
α
(6.12)
where ∆α = αk - α⊥ and αk and α⊥ are the polarizabilities parallel and perpendicular to the director. Eq. 6.12 relates the refractive indices with the order parameter.
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Since the magnitude of dielectric anisotropy decreases with temperature, so does the
birefringence.
An Abbe’s refractometer [19] is commonly used to measure the refractive indices. A liquid crystal sample is sandwiched into a thin layer between an illuminating
prism and a refracting prism. The refracting prism has the higher refractive index (n)
than the sample whose refractive index has to be measured; in my case the highest
reading in the scale of the refractometer was 1.71 and the highest value of refractive
index of nematic I52 was 1.6465. Light was passed through the illuminating prism.
As shown in Fig. 6.17, the light ray AB has the highest angle of incidence at the LC
and refracting prism interface has the highest value of angle of refraction. All light
rays having an angle smaller than θi will refract left of BC and hence, the region
appears brighter and the region right of BC will appear darker. The refractive index
is calculated from Snell’s law n1 sinθi = n2 sinθr .
To measure the refractive indices of pure I52 in the nematic phase, I cleaned
Illuminating
Prism
A
θi
sample
Refracting
Prism
B
θr
C
Light
Dark
Figure 6.17: Refraction of light passing through the liquid crystal in Abbe’s refractometer.
the area between the illuminating prism and the refracting prism with isopropanol.
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(a)
(b)
Figure 6.18: (a) Variation of refractive indices of pure I52 with temperature. The
blue circles and red up triangles indicate for ne and no respectively as a function of
temperature. (b) Variation of birefringence of pure I52 with temperature. In both
figures (a) and (b), the light source is He-Ne laser beam.
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To verify whether the instrument gives a correct result or not, I measured the refractive index of glycerol and found a value corresponding with the standard value. Next,
I checked the refractive indices of 5CB at 25 ◦ C to verify whether LC can be used
directly on the refracting prism surface or the surface must be treated with lecithin
to get homotopic alignment of LC. Fortunately, I found the refractive indices exactly
the same as found in literature with surface treatment [20, 21].
To control the temperature, I used RTE-111 [22]. Through it, water circulates
to the prism and controls the LC’s temperature. A thermometer is fixed in the refractometer which reads the temperature of LC (or say the temperature of water
circulating in the prism. The temperature recorded by the thermometer attached to
the refractometer shows the same reading as recorded by RTE-111 with a variation
of ±0.01 ◦ C.
At first, I measured refractive indices with NaD (λ = 589 nm) light source. A
rotating polarizer fixed to the eyepiece was used to view the ordinary and extraordinary lines. I could see the ordinary line easily even without a polarizer, but not the
extraordinary line. To view the ordinary line, I focused without a polarizer P. Then, I
used P and rotated it so that I could see the dark and bright regions easily. Adjusting
the crosshair at the region between dark and bright, I noted the readings. To find
the extraordinary line, I rotated P by 90◦ . To measure refractive indices with laser
beam, I used unpolarized light so that it hits almost uniformly on the illuminating
prism. A lens of focal length around an inch very close to the laser source helped to
diffuse light falling on the illuminating prism. The light beam was covered with black
cloth to protect eyes from direct exposure. It is harder to view the crosshair in case
of a laser beam than that of NaD source.
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The birefringence results agree with the results by [3]. With both the laser
beam and the NaD source (Fig. 6.18 and 6.19), the ordinary and the extraordinary
refractive indices decrease with increase of temperature. I was interested in the temperature range in which ∆² < 0 and hence measured only up to 60 ◦ C. But, the nature
of the graph is similar to that of ²⊥ and ²k within the specified range. Fig. 6.18(b)
and Fig. 6.19(b) give the birefringence of I52 for the light source of He-Ne laser and
NaD respectively. Both of them decrease with rise in temperature.
6.4 Dynamic light Scattering
Without considering the state of alignment, NLCs flow easily as conventional
liquids of similar molecules. But if one takes into account the state of alignment, the
flow becomes complicated. This is because the flow depends on the angles, the director
makes with the direction of flow and the velocity gradient. Also, the translational
motion is coupled to orientational motion of the molecules. As a result, the flow
disturbs the alignment and causes director rotation. Due to thermal fluctuation etc.,
the local value of the director n(r,t) is not the same as its equilibrium orientation
no . The local orientational fluctuation δn(r, t) = n(r, t) − no of the director n(r,t )
causes light scattering. This small fluctuation must be perpendicular to n to fulfill
the condition n2 = 1. This fluctuation gives rise to fluctuation in the dielectric
permittivity and hence to light scattering. Defining the wave vector q as q = 2πn/λ,
where n is the average refractive index, the scattering will be strong for small q. It
causes turbid appearance of the nematic. Above TN I , the permittivity fluctuation
will be due to only density fluctuation and a clear liquid results [14, 23].
Taking no along ẑ, the fluctuation at any point r can be described by nx (r) and
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(a)
(b)
Figure 6.19: (a) Variation of refractive indices of pure I52 with temperature. The
blue circles and red up triangles are for ne and no respectively. (b) Variation of
birefringence of pure I52 with temperature. In both figures (a) and (b), the source is
NaD light.
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ny (r). In Fourier components, they can be expressed as
nx,y (r) =
X
nx,y qe(iq·r)
(6.13)
q
For a given component of wave vector q, rotating the coordinate system x, y, z
around the z-axis so that the new x-axis coincides with the unit vector e1 which is
perpendicular to the z-axis in the qz plane. The y-axis then coincides with e2 where
no × q
| no × q |
e2 × no
e1 =
| no × q |
e2 =
(6.14)
In this new system, the components of n(q) along em are nm (q) (m = 1, 2). Then
the distortion free energy density will be
Fd =
1X
| nα (q) |2 (K33 q2k + Kmm q2⊥ )
2 q
(6.15)
Here, qk = qz and q⊥ = q · e1 . The relaxation frequency for the mode m is
Γm (q) =
Kmm q2⊥ + K33 q2k
ηm (q)
(m = 1, 2)
(6.16)
The effective viscosities depend only on the orientation of the wave vector q, Leslie’s
and Miesowicz’s viscosity coefficients as
η1 (q) = γ1 −
η2 (q) = γ1 −
(q2⊥ α3 −q2k α2 )2
q4⊥ ηb +q2⊥ q2k (α1 +α3 +α4 +α5 )+q4k ηc
q2k α22
(6.17)
q2⊥ ηa +q2k ηc
Where ηa , ηb and ηc are Miesowicz viscosities defined in Chapter Two. By choosing
an appropriate combination of incoming and scattered light, scattering angles and
director orientations, I measured the relaxation frequency of pure splay, pure bend
and combination of twist and bend modes. The sample cell I52A filled with pure I52
was used for this experiment. The experiment was done in Dr. S. Sprunt’s lab with
the help of Mr. K. Neupane.
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6.4.1 Geometry A-splay/twist geometry for measurement of ηsplay
The schematic of this geometry is given in Fig. 6.20(a). In this geometry, I
used the sample cell with the director normal to the scattering plane (plane made by
incident wave vector ki and the scattered wave vector kf ). The laser beam incident
normally is polarized parallel to the director whereas the scattered light is polarized
in the scattering plane. The vectors for this geometry are
n
z
i
y
ki
ϕ
x
q
f
kf
(a)
(b)
Figure 6.20: (a) Geometry A in dynamic light scattering used to measure
2
with the relaxation frequency Γ1 at 50◦ C.
ηsplay . (b) Variation of K11 q⊥
i = (0, 0, 1)
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f = (sin φ, cos φ, 0)
n = (0, 0, 1)
ki = ko ne (1, 0, 0)
kf = k0 n0 (cosφ, −sinφ, 0)
q=kf − ki = (n0 cos φ − ne , −n0 sin φ, 0)
qk = q · n = 0
q
q⊥ =| q |= ko (no cos φ − ne )2 + (−no sin φ)2 in terms of scattering angle inside the
r q
liquid crystals and q⊥ = ko ( n2o − sin2 θlab − ne )2 + sin2 θlab in terms of scattering
angle in the lab.
For a given scattering angle, the scattering light will have contribution from both
modes. The geometric factor for these modes are given by
ko ne
sinφ
q⊥
ko
G2 = (i · e2 )(f · n) + (i · n)(f · e2 ) = − (no − ne cosφ)
q⊥
G1 = (i · e1 )(f · n) + (i · n)(f · e1 ) = −
(6.18)
With geometry A, the relaxation frequency for the first mode (m=1) from Eq. 6.16 is
Γ1 =
2
K11 q⊥
,
η1
qk = 0
(6.19)
2
Thus, the slope of the graph of q⊥
versus Γ1 gives K11 /η1 . From the known
value of K11 from section 6.1.1, I calculated η1 = ηsplay and found ηsplay to be 0.018
at 50 ◦ C. But, from Eq. 6.17, ηsplay = γ1 − α32 /ηb . From the measurement of τd and
K11 as discussed in previous sections, the value of γ1 is 0.19569 Pa s. Thus α32 /ηb =
0.18516. This implies
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ηb = 5.40073α32 = Dα32
(6.20)
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where D = 5.40073 (say). From Eq. 6.18, the geometric factor G2 vanishes at particular angle φo at which no − ne cosφo is zero and hence
µ
−1
φo = cos
no
ne
¶
(6.21)
This internal scattering angle is called the magic angle and the scattered intensity at
this angle is only due to mode one with pure splay distortion. In the lab, this angle
will be
v

u
2
u
n
θo = sin−1 no t1 − o2 

ne
(6.22)
Since the refractive indices are function of temperature, this angle also depends upon
the temperature.
6.4.2 Geometry B- bend/twist geometry for measurement of α0 s
This geometry also utilizes a planar cell with the director in the scattering plane.
As shown in Fig. 6.21(a), both the incident and scattered light have the direction of
polarization as in the case of geometry A. In this geometry, both components of q
exist as
qk = ko (−nef f sinφ) = qy
(6.23)
q⊥ = ko (nef f cosφ − no ) = qx
The effective refractive index in this case depends upon the scattering angle φ and is
given by
Ã
nef f =
The vectors for this geometry are
i = (0,0,1)
f = (sin φ, cos φ, 0)
n = (0,1,0)
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sin2 φ cos2 φ
+
n2o
n2e
!−0.5
(6.24)
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z
n
i
y
ki
ϕ
x
q
f
kf
(a)
(b)
Figure 6.21: (a) Geometry B in dynamic light scattering used to measure α0 s.
(b) Variation of correlation function with the delay time at 25 ◦ C.
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nef f =
³
sin2 φ
n2o
+
´
cos2 φ −0.5
2
ne
ki = ko no (1, 0, 0)
kf = ko nef f (cos φ, − sin φ, 0)
q = kf − ki = ko (nef f cos φ − no , −nef f sin φ, 0)
qk = qy = ko nef f sin φ
q⊥ = q − qk = ko (nef f cos φ − no ) = qx in terms of the scattering angle inside the
liquid crystals and q⊥ = ko [ne (1 − sin2 θlab /n2o )0.5 − no ] in terms of scattering angle in
the lab.
In this geometry, the geometric factor G1 vanishes and G2 = −cosφ. The
relaxation frequency due to mode 2 at any scattering angle can be written from
Eq. 6.16 as
Γ2 =
2
K22 q⊥
+ K33 qk2
η2
(6.25)
and η2 is given by Eq. 6.17.
Using the setup in Fig. 6.21(a), I measured the relaxation frequency Γ2 for the
scattering angle ranging from 11◦ to 46◦ at difference of 1◦ (difference of 0.5◦ at around
the magic angle) at the temperature T ranging from 25 ◦ C to 50 ◦ C at difference of
5 ◦ C.
From Eq. 2.44 and Eq. 6.20,
1
2
(α3 + α4 + α6 ) = Dα32 and hence,
α6 = 2Dα32 − α4 − α3
(6.26)
For the geometry shown in Fig. 6.20(a), at around the magic angle (θmagic ± 3o ),
q⊥ < 0.1 % of qk . So, Eq. 6.25 can be modified as
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Γ2 (q) = Γ2 =
K33 qk2
ηbend
(6.27)
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Plotting the graph of Γ2 versus K33 qk2 , I can calculate ηbend .
Also, I had a q scan for relaxation frequency at lower angles from 11o up to the magic
angle to get significant value of q⊥ from which I can get Γ2 as a function of q. Then,
from the relation
α22 qk2
2
q⊥
ηa + qk2 ηc
(6.28)
1
ηa q⊥
ηc
= 2 ( )2 + 2
γ1 − η2
α 2 qk
α2
(6.29)
η2 (q) = γ1 −
This equation can be simplified as
From Eq. 6.29, I can get η2 as a function of q. By linear fitting 1/(η1 − γ2 ) and
2
q⊥
/qk2 , I got slope(m) = ηa /α22 = 56.1176 and intercept(C) = ηc /α22 = 24.311.
ηa = mα22 ⇒ α4 = 2ηa = 2mα22 and ηc = Cα22
Using Eq. 2.43,
1
2
(−α2 + α4 + α5 ) = Cα22 . This gives,
α5 = 2(C − m)α22 + α2
(6.30)
Simplifying Parodi relation (α2 + α3 = α6 − α5 ), Eq. 6.26 and Eq. 6.30, it gives
α2 + α3 = 2Dα32 − 2mα22 − α3 − (2Cα22 − 2mα22 + α2 ). But, γ1 = α3 − α2 . So,
(D − C)α32 + 2(Cγ1 − 1)α3 + (γ1 − Cγ12 ) = 0
(6.31)
Let, D-C = A = -18.91027, 2(Cγ1 −1) = B = 7.52797 and (γ1 −Cγ12 ) = E = −0.73759.
Then,
Aα32 + Bα3 + E = 0
(6.32)
Solving Eq 6.32 gives α3+ = 0.17427 and α3− =0.22382. Correspondingly, α2+ = 0.02169 and α2− = 0.02786.
The role of α2 and α3 to describe flow behaviour in nematics is explained in Chapter
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(a)
(b)
2
/qk2 with the scattering angle in the lab. (b) Variation
Figure 6.22: (a) Variation of q⊥
2
+ K33 qk2 with relaxation frequency Γ2 .
of product of the sum of K22 q⊥
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(a)
(b)
Figure 6.23: (a) Linear fit of the equation at 50 ◦ C and (b) At 25 ◦ C.
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Two. Due to the restriction that γ1 ≥ 0, I chose the value of α3 = 0.17427 and
α2 = −0.02169. With these values,
α4 = 2ηa = 0.0528 Pa s, α5 = −0.05162 Pa s and α6 = 0.10096 Pa s
Requiring the entropy production to be positive, the coefficients calculated satisfy
the inequalities mentioned in Chapter Two (Eq. 2.41 and Eq. 2.42).
From 25 ◦ C to 45 ◦ C at difference of 5 ◦ C, I tried to calculate α0 s in the same
way as it was done for 50 ◦ C, but for these temperatures, I found γ1 < η2 and hence,
the linear fit equation becomes
ηa q⊥
ηc
1
= − 2 ( )2 − 2
η 2 − γ1
α 2 qk
α2
(6.33)
which gives ηc as negative and α0 s as imaginary with the available physical parameters.
For example, ηsplay = 0.0216 Pa s and for the linear fit equation in Fig. 6.23(b), I
found m = 142.614, C = -32.3144, so that α3 = 0.2656+0.0253i.
Since I could measure α’s reliably only at 50 ◦ C within the framework of available
equipment, I analyzed the EHC data at this temperature in Chapter Four and Chapter
Five.
6.5 Electric Nusselt number characterization
In thermal convection, when heat is transferred at a boundary surface within a
fluid, the Nusselt number is the ratio of convective to conductive heat transfer across
the boundary. In electroconvection, there is energy dissipation as electric current
flows through the sample cell when an ac voltage is applied. The electric Nusselt
number is defined as the ratio of electric current with and without convection. Due
to the sinusoidal ac voltage, there are two Nusselt numbers corresponding to the inphase and out-of -phase components of the current [24, 25].
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LI
X
A
M
O
P
B
Y
CH1 Out
CH2 Out
Ref Sine
In Out
1111111111
0
1000000000
Multimeter
1
0
1
0
1
0
1
0
C
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1111
0000
111111111111
000000000000
1
0
1
0
1
0
R
CSP
LI = Lock in Amplifier
CSP = Current Sensitive Preamplifier
SC = Sample Cell
SC
O = Outpot
M = Monitor
P = Power In
Figure 6.24: Schematic of lock-in amplifier used to measure in phase and out of phase
currents in the sample cell when ac voltage is applied.
6.5.1 Current flow through the sample cell
The total current I flowing through the nematic cell is the sum of the conduction and displacement currents:
Z
I=
A
(Jz + ∂t Dz )dxdy
(6.34)
Here, within the SM,
Jconduction = σ⊥ E + ∆σ(n · E)n + ρe v,
(6.35)
ρe is the induced charge density, E is the total electric field intensity given by
√
E = 1/d[ 2V eiωt ẑ − d∇φ] and Jdisp = ∂D/∂t where D = ²o ²⊥ E + ²o ∆²(n · E)n. Also,
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Jz = σ⊥ Ez =
√
2σ⊥ V eiωt /d.
For no convection, ρe = 0, φ = 0, v= 0 and n= x̂ for a planar sample cell. Hence
√
∂Dz
∂Ez
2V iωt
= ²o ²⊥
= i²o ²⊥ ω
e
∂t
∂t
d
(6.36)
Finally, from Eq. 6.34, the total current can be expressed as
√
Io =
2V A
(σ⊥ + iω²o ²⊥ )eiωt .
d
(6.37)
In the convecting state, the current I is modified along with fields n, v and φ. If Ir
and Ii be the in-phase and out-of-phase currents in the convecting state, then, the
real and imaginary reduced electric Nusselt numbers are
Nr =
Ir
Ii
− 1 and Ni = o − 1,
o
Ir
Ii
(6.38)
respectively. Hence, for no convection, Nr = Ni =0.
6.5.2 Experiments
To measure the Nusselt numbers, I used the sample cell I52N. The sample cell
was kept in Instec hot stage in the rotating stage of polarizing microscope. A function
generator of the lock-in amplifier (LI) produces sinusoidal voltage which amplifies and
applies to the cell. The path-to-ground for the current traversing the cell is through
a current to voltage converter. LI ensures the output signal from this converter. LI
itself applies the reference signal and the frequency. For each frequency, I replace the
nematic cell by a pure resistance and adjust the phase setting of the lock in to zero,
the out-of-phase component. Finally, I reinserted the nematic cell.
At certain frequency (driving frequency generally applied during EHC), I increased the applied voltage in steps and waited for several seconds before recording
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(a)
(b)
Figure 6.25: (a) In-phase current Ir (blue) and out of phase current Ii (pink) versus
applied voltage V and (b) The real part of reduced Nusselt number Nr verses the
applied voltage. Both graphs are at the same frequency 100 Hz and at 30 ◦ C.
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the currents Ir and Ii . In the quiescent state, the increase in current for Ir is linear
and when it crosses the threshold, the current deviates from its value. To calculate
Vc from either Ir or Ii data, I fitted a straight line from the data for current much
smaller than that for Vc . The linear fit gives the current below onset of convection.
The reduced Nusselt numbers as function of V were calculated by subtracting unity
from Ir /Iro and Ii /Iio . In this sample cell, I could not observe the cut-off frequency,
even when I reached F= 800 Hz and Vc =50V. However, it can be calculated from the
fitting equation.
It is worthy to note that in Fig. 6.25(a), the blue graph is the curve, the red
Figure 6.26: (a) The variation of the slope dNr /dε and slope dNi /dε with frequency
30 ◦ C.
is the straight line fit with data much less than the threshold voltage. The pink line
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is for the imaginary part of the current, both expressed in µA. Since the value ∆σ
is significant enough, I could see the change in current at threshold voltage. But,
the pink graph is a straight line showing that ∆² is very small in case of I52 and I
could not see a change in out-of-phase current. In the case of poor conductivity, I
could not see the change in Ir too. I tested the first cell at 50 ◦ C and could see the
change in Ir but not at Ii . Again, I tried at 30 ◦ C. In this case, I could not see the
change in Ir and Ii . I worked with this cell for one month and finally used second
cell of almost equal thickness. In this case too, I worked at 30 ◦ C, but could not see
the variational jump in Ii at the threshold voltage. This should be because of low
∆² for I52. The ∆² at 30 ◦ C for I52 is about one tenth that of MBBA. Thus, it is
quite difficult to calculate the imaginary component of Nusselt number in case of I52.
Fig. 6.26 gives the variation of slope (dNi,r /dε) with the frequency. It is very clear
that dNr /dε decreases with frequency, whereas dNi /dε increases.
6.6 Conclusions
I utilized the capacitance bridge to measure ²⊥ and σ⊥ at fixed frequency of
1 kHz and lock-in amplifier at variable frequencies. Magnetic Freedericksz transition
concept was used to measure Kii , ²k and σk . The dielectric anisotropy ∆² increased
from -0.0541 at 25 ◦ C to 0.0335 at 100 ◦ C in the passing through zero at 62.96
◦
C. Similarly, the conductivity anisotropy ∆σ increased from 0.7 × 10−9 Ω−1 m−1 to
11 × 10−9 Ω−1 m−1 in the same temperature range. Since I52 is standard NLC with
(-,+) for wide temperature gap and ∆² is very small, it is the benchmark material for
EHC experiments with dominating oblique rolls pattern. The refractive indices and
the director relaxation time are supporting parameters to calculate α’s in dynamic
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light scattering. These parameters are measured within the framework of available
equipment in the lab to calculate the coefficients of Ginzburg-Landau equations and
the WEM parameters in the future.
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BIBLIOGRAPHY
[1] P. G. de Gennes and J. Prost. The Physics of Liquid Crystals. Oxford University
Press Inc., New York, 2nd edition, 1993.
[2] S. Chandrasekhar. Liquid Crystals. Cambridge University Press, 2nd edition,
1992.
[3] U. Finkenzeller, T. Geelhaar, G. Weber, and L. Pohl. Liquid-crystalline reference
compounds. Liq. Cryst., 5:313, 1989.
[4] M. Dennin. A Study in Pattern Formation: Electroconvection in Nematic Liquid
Crystals. PhD thesis, Department of Physics, University of California, Santa
Barbara, 1995.
[5] I. W. Stewart. The Static and Dynamic Continuum Theory of Liquid Crystals.
Taylor & Francis, 2004.
[6] I. Haller. Elastic constants of the nematic liquid crystal phase of mbba. The
journal of chemical physics, 57(4):1400, 1972.
[7] T. Toyooka, G. Chen, H. Takezoe, and A. Fukuda. Determination of twist elastic
constant k22 in 5CB by four independent light scattering technique. Japanese
journal of applied physics, 26(12):1959, 87.
[8] EHC Co., Japan.
[9] Z. Li. Dielectric method to determine the twist elastic constant in a homogeneous
nematic cell. J. Appl. Phys., 75:1225, 1994.
[10] J. D. Jackson. Classical Electrodynamics. Oxford University Press, 2nd edition,
1993.
[11] E. B. Priestly, P. J. Wojtowicz, and P. Sheng. Introduction to Liquid Crystals.
Plenum press. New York and London, 2nd edition, 1975.
[12] L. M. Blinov and V. G. Chigrinov. Electro-Optic Effects in Liquid Crystal Materials. Springer-Verlag, 1994.
[13] E. Bodenschatz, W. Zimmermann, and L. Kramer. On electrically driven
pattern-forming instabilities in planar nematics. J. Phys. France, 49:1875, 1988.
[14] V. V. Belyaev. Physical methods for measuring the viscosity coefficients of nematic liquid crystals. physics-Uspekhi, 44:255, 2001.
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[15] H. Schad. The rotational viscosity of nematic liquid crystal mixtures at low
temperatures. J. Appl. Phys., 54:4994, 1983.
[16] L. M. Blinov. Electro-Optical and Magneto-Optical Properties of Liquid Crystals.
John Wiley & Sons Ltd., 1983.
[17] P. J. Collings. Liquid Crystals: Nature’s Delicate Phase of Matter. Princeton
University Press, 2002.
[18] S. Elston and R. Sambles. The Optics of Thermotropic Liquid Crystals. Taylor
& Francis, 1998.
[19] Bausch & Lomb Co., USA.
[20] M. Cui. Temperature Dependence of Viscoelastic Properties of Nematic Liquid
Crystals. PhD thesis, Liquid Crystal Institute, Kent State University, 2000.
[21] P. Oswald and P. Pieranski. Nematic and Cholesteric Liquid Crystals. Taylor &
Francis, 2005.
[22] NESLAB instruments Inc., USA.
[23] W. H. De Jeu. Physical Properties of Liquid Crystal Materials. Gordon and
Breach science Publishers Ltd. one park Avenue, NY10016, 1980.
[24] J. T. Gleeson, N. Gheorghiu, and E. Plaut. Electric nusselt number measurement
characterization of electroconvection in nematic liquid crystals. Eur. Phys. J. B,
26:515, 2002.
[25] J. T. Gleeson. Charge transport measurement during turbulent electroconvection. Phys. Rev. E, 63:026306, 2001.
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APPENDIX A
Matlab codes to calculate Hopf frequency
This program calculates the Hopf frequency from given set of data.
function getHopffreq(N)
% load the data file
getdata=load(‘E:\11V51Hz\Random.txt’);
% t is the time taken to record N data at 30 FPS.
t=linspace(0,N/30,N);
getdata1=getdata-mean(getdata);
dd=(max(time)-min(time))/(N-1);
w=linspace(0,1/(2*dd),1+N/2);
% calculate 1D fast Fourier transform
getfft=fft(getdata1);
absval=abs(getfft);
% plot one of the peak of the absolute value of FFT as a function of frequency and label it.
%Display the maximum value (A) of the distribution and its index (I);
[A, I]=max(absval)
%Display the frequency of corresponding maximum value.
w(I)
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APPENDIX B
Matlab codes to extract single envelope using two-wave demodulation
This program extracts zig envelope using two-wave demodulation technique. The
steps are: (a) Take 2D FFT of flat fielded image. (b) Window it to pick the region of
interest for the dominating peak with remaining pixels zero.(c) Transform the pixel
values into four corners in Fourier space and finally take inverse 2D FFT.
function zigenv(number)
m = 480; n=640; %image size
%Flat fielding
Background = imread(‘E:\10.95micronI52\FlatField Frames\Light.bmp’);
BG = double(Background);
DarkFrame = imread(’E:\10.95micronI52\FlatField Frames\Dark.bmp’);
DF = double(DarkFrame);
for k1 = 1:number
stringname =[’E:\10.95micronI52\Image’ int2str(k1) ’ .bmp’];
input1 = imread(stringname);
input = double(input1);
flatfield = (input-DF)*mean2(BG-DF)/(BG-DF);
flatfield1 = flatfield- mean2(flatfield);
% Calculate 2D FFT
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fftimage = fft2(flatfield1);
absval = abs(fftimage);
% Find the index of maximum value in the region of interest in Fourier space.
maxval = 0;
index = 0;
for i1 = m/2+1 : m
for j1 = n/2+1 : n
y = absval(i1, j1);
if y > maxval
index1 = i1;
index2 = j1;
maxval = y;
end;
end;
end;
%Introduce 2D Gaussian filter
filter1 = zeros(m, n);
transzig = zeros(m, n);
% For transformed zig contribution
fw = 10.;%filter width
for i2 = 381 : 480
for j2 = 541:640
filter1(i2,j2) = exp((−((i2 − (index1 − 1))2 + (j2 − (index2 − 1))2 ))/(2 ∗ (f w)2 ));
end;
end;
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%Apply the filter in the spatial transform of the image filteredimg = filter1 .∗ fftimage;
%Transformation of zig states.
for i3 = m/2 + 1 : m
for j3 = n/2 + 1 : n
if i5 < index1 & j5 > = index2
transzig(i3 + m + 1 - index1, j3 + 1 - index2) = filteredimage(i3, j3);
end;
if i3 < index1 & j3 < index2 & j3 ¿= n/2
transzig(i3 + m + 1 - index1,j3 + n +1-index2)=filteredimage(i3, j3);
end;
if i3>= index1 & j3 < index2 & j5 >= n/2
transiag(i3 +1 -index1,j3+ n+1-index2) = filteredimage(i3, j3);
end;
if i3 ≥ index1 & j3 ≥ index2
transzig(i3+1 - index1, j3+ 1 - index2) = filteredimage(i3, j3);
end;
end;
end;
%Calculate the inverse Fourier transform.
absinv = abs(invfft);
imagesc(absinv);
frame(k1) = getframe;
end;
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invfft = ifft2(transzag);
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APPENDIX C
Matlab codes to extract envelope using four-wave demodulation
This program extracts envelope A1 using four-wave demodulation technique. The
steps are: (a) Take 2D FFT of flat fielded image. (b) Window it to pick the region of
interest for the dominating peak with remaining pixels zero.(c)) Take the time series
of data from non-zero pixel values. (d) Calculate its temporal FFT. (f) Window it
and calculate its inverse FFT. (g) Rearrange into original dimensions and finally take
the inverse spatial 2D FFT.
function envelopeA1(T)
%To extract envelope A1
m = 480; n = 640; T=1024;
% m and n are the vertical and horizontal pixel labels and T is the total number of frames to b
wsf = zeros(11, 11, T);
wsft1 = zeros(11, 11, T);
filter1=zeros(1, T);
%Introduce 1D-Gaussian filter
fw = 10.;
fw1 = 2*f w2 ;%filter width
filter1 = zeros(1,T);
for i1 = 6:26
filter1(1, i1)=exp(-((i1 − 16)2 )/fw1);
end;
% Flat fielding
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Background = imread(’E:\10.95micronI52\FlatField Frames\light.bmp’);
BG = double(Background);
Darkframe = imread(’E:\10.95micronI52\FlatField Frames\dark.bmp’);
DF = double(Darkframe);
for k1 = 1:1024
stringname =[’E:\10.95micronI52\Images\Image’ int2str(k1) ’.bmp’];
input1 = imread(stringname);
input = double(input1);
flatfield = (input-DF)*mean2(BG-DF))./(BG-DF);
flatfield1 = flatfield- mean2(flatfield));
% Calculate 2D FFT.
getfft = fft2(flatfield1);
absval = abs(getfft);
% Apply spatial window in the region of interest.
for i2=14:24
for j2=620:630;
wsf(i2-13, j2-619, k1)=getfft(i2,j2);
end;
end;
end;
% Take temporal Fourier transform of each non-zero element
%, window it and take inverse time fourier transform.
for i3 = 1:11
for j3 = 1:11
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wsft=wsf(i3,j3,:);
timefft=fft(wsft);
wtimefft= (timefft(:).’).*filter1;
% windowed time Fourier transform
% temporal inverse FFT
invwtimefft=ifft(wtimefft);
wsft1(i3,j3,:)=invwtimefft;
end;
end;
%Rearrange into original dimensions.
for k=1 : T
for i4=1 : 11
for j4 = 1 : 11
wsf2(i4+13, j4+619)= wsft1(i4, j4, k);
end;
end;
%Take spatial 2D inverse FFT and display it.
invspacefft=ifft2(wsf2);
absA1=abs(invspacefft);
qquad imagesc(absA1);
frame(k)=getframe;
end;
%Record the movie
movie2avi(frame, ’envelopeA1.avi’);
http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40