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electronic-Liquid Crystal Dissertations - May 28, 2009 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 ii electronic-Liquid Crystal Dissertations - May 28, 2009 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 iii electronic-Liquid Crystal Dissertations - May 28, 2009 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 iv electronic-Liquid Crystal Dissertations - May 28, 2009 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 v electronic-Liquid Crystal Dissertations - May 28, 2009 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 vi electronic-Liquid Crystal Dissertations - May 28, 2009 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. . . . . . . . http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 vii 25 electronic-Liquid Crystal Dissertations - May 28, 2009 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 viii electronic-Liquid Crystal Dissertations - May 28, 2009 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. . . . . . . http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 ix 76 electronic-Liquid Crystal Dissertations - May 28, 2009 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 x electronic-Liquid Crystal Dissertations - May 28, 2009 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. . . . . . . . . . . . . . . . . . . . . . . . . . . . http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 xi 99 electronic-Liquid Crystal Dissertations - May 28, 2009 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 xii electronic-Liquid Crystal Dissertations - May 28, 2009 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 xiii electronic-Liquid Crystal Dissertations - May 28, 2009 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 xiv electronic-Liquid Crystal Dissertations - May 28, 2009 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 xv electronic-Liquid Crystal Dissertations - May 28, 2009 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 xvi electronic-Liquid Crystal Dissertations - May 28, 2009 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 xvii electronic-Liquid Crystal Dissertations - May 28, 2009 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 xviii electronic-Liquid Crystal Dissertations - May 28, 2009 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 xix electronic-Liquid Crystal Dissertations - May 28, 2009 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 xx electronic-Liquid Crystal Dissertations - May 28, 2009 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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 xxi electronic-Liquid Crystal Dissertations - May 28, 2009 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 1 electronic-Liquid Crystal Dissertations - May 28, 2009 2 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 3 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 4 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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 5 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, http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 6 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 7 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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 8 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’ http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 9 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 10 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 11 (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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 12 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 13 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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 14 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 15 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 16 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 17 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 18 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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 19 electronic-Liquid Crystal Dissertations - May 28, 2009 20 [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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 21 [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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 22 electronic-Liquid Crystal Dissertations - May 28, 2009 23 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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 24 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 25 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’. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 26 (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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 U (x, t) ∝ [A(x, t)eiqc x + cc] + hh (2.3) electronic-Liquid Crystal Dissertations - May 28, 2009 27 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 (2.6) electronic-Liquid Crystal Dissertations - May 28, 2009 28 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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 29 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 30 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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 31 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 ∇ · D = 0, ∇×E=0 (2.14) electronic-Liquid Crystal Dissertations - May 28, 2009 32 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 33 (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 à 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 à 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 n×h=0 (2.23) electronic-Liquid Crystal Dissertations - May 28, 2009 34 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 35 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] à ∂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) http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 36 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 37 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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 38 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 39 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 40 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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 41 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 42 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, http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 σk ∂Ex ∂θ + ∆σEc =0 ∂x ∂x (2.52) electronic-Liquid Crystal Dissertations - May 28, 2009 43 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 = − http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 π 2 K33 α2 ²o ²k (²⊥ /²k − σ⊥ /σk )/η + σ⊥ ²o ∆²/σk (2.56) electronic-Liquid Crystal Dissertations - May 28, 2009 44 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 45 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 46 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 47 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) http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 48 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 49 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. 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Cryst., 104:307, 1984. [13] P. J. Collings and M. Hird. Introduction to Liquid Crystals. Taylor & Francis, 1997. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 50 electronic-Liquid Crystal Dissertations - May 28, 2009 51 [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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 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- http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 52 electronic-Liquid Crystal Dissertations - May 28, 2009 53 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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 54 (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]. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 55 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 56 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 57 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 58 160 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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 59 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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 60 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 61 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 62 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 63 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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 64 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 65 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 66 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 67 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 68 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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 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. 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E, 70:025202, 2004. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 69 electronic-Liquid Crystal Dissertations - May 28, 2009 70 [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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 71 electronic-Liquid Crystal Dissertations - May 28, 2009 72 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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 73 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 74 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 75 (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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 76 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 77 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 78 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 79 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 80 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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 81 • 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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 82 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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 83 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, http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 q ωH = Ω 1 − (1/τrec Ω)2 (4.4) electronic-Liquid Crystal Dissertations - May 28, 2009 84 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 85 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 86 (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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 87 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 88 (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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 89 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 90 (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 . http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 91 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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 92 (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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 93 √ 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 . http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 94 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 95 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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 96 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 97 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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 98 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 99 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 101 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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 102 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 103 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 104 (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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 105 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 106 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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 107 (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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 108 (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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 109 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 111 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]. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 112 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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 113 electronic-Liquid Crystal Dissertations - May 28, 2009 114 [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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 115 electronic-Liquid Crystal Dissertations - May 28, 2009 116 • 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 117 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 118 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 + a3 | A3 |2 + a4 | A4 |2 ]A1 + a5 A2 Ā3 A4 + hh (5.5) electronic-Liquid Crystal Dissertations - May 28, 2009 119 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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 120 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 , http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 121 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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 122 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) http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 ≡√ −1 N −1 X 1 MX e2πi(mk/M +nl/N ) F (m, n, t) M N m=0 n=0 (5.7) electronic-Liquid Crystal Dissertations - May 28, 2009 123 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 124 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 125 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 126 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) http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 127 50 50 100 100 150 150 200 200 250 250 300 300 350 350 400 400 450 450 100 200 300 400 500 600 100 200 (a) 300 400 500 600 400 500 600 (b) 50 50 100 100 150 150 200 200 250 250 300 300 350 350 400 400 450 450 100 200 300 (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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 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 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(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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 129 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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 130 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 −1 1 TX g(ω) = (Ft f )(ω) ≡ √ f (t)e−2πiωt/T T t=0 (5.14) electronic-Liquid Crystal Dissertations - May 28, 2009 131 0.6 0.5 50 50 0.4 100 0.5 100 0.3 150 0.2 150 200 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) 200 300 400 500 600 (b) 50 100 150 200 250 300 350 400 450 100 200 300 400 500 600 (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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 132 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 ) http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 (5.18) electronic-Liquid Crystal Dissertations - May 28, 2009 133 − − 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 50 50 100 100 150 150 200 200 250 250 300 300 100 200 300 400 100 A3 100 200 200 300 300 200 300 400 300 400 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 134 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 135 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 . http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 136 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 137 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 50 100 150 166 164 162 160 200 158 0 100 200 300 0 100 200 300 250 n 400 500 600 400 500 600 170 300 pixel values 168 350 400 166 164 162 160 450 100 200 300 400 500 600 158 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, http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 138 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 139 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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 140 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]. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 141 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 142 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 143 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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 144 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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 145 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 146 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 147 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 . http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 148 electronic-Liquid Crystal Dissertations - May 28, 2009 149 [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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 150 electronic-Liquid Crystal Dissertations - May 28, 2009 151 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 152 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 (6.1) electronic-Liquid Crystal Dissertations - May 28, 2009 153 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 154 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 155 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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 156 (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 . http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 157 Figure 6.4: Splay critical magnetic fields as a function of temperature. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 158 • 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 159 (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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 160 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 161 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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 162 is [10] " C = ²²o ι 2 cosh à −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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 163 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] http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 s σ = e(µ+ µ− ) KD C KR (6.5) electronic-Liquid Crystal Dissertations - May 28, 2009 164 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 165 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 166 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 167 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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 168 (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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 169 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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 170 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 171 (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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 172 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 173 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). http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 174 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 γ1 d2 τd = 2 π K11 (6.9) electronic-Liquid Crystal Dissertations - May 28, 2009 175 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 loge (∆G) = loge (G(H) − G⊥ ) − 2t τd (6.11) electronic-Liquid Crystal Dissertations - May 28, 2009 176 (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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 177 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 178 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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 179 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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 180 (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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 181 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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 182 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 183 (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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 184 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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 185 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) http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 186 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 ηb = 5.40073α32 = Dα32 (6.20) electronic-Liquid Crystal Dissertations - May 28, 2009 187 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 à nef f = The vectors for this geometry are i = (0,0,1) f = (sin φ, cos φ, 0) n = (0,1,0) http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 sin2 φ cos2 φ + n2o n2e !−0.5 (6.24) electronic-Liquid Crystal Dissertations - May 28, 2009 188 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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 189 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 Γ2 (q) = Γ2 = K33 qk2 ηbend (6.27) electronic-Liquid Crystal Dissertations - May 28, 2009 190 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 191 (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⊥ http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 192 (a) (b) Figure 6.23: (a) Linear fit of the equation at 50 ◦ C and (b) At 25 ◦ C. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 193 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]. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 194 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, http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 195 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 196 (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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 197 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 198 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 199 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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 200 electronic-Liquid Crystal Dissertations - May 28, 2009 201 [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. http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 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) http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 202 electronic-Liquid Crystal Dissertations - May 28, 2009 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 203 electronic-Liquid Crystal Dissertations - May 28, 2009 204 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; http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 205 %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; http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 invfft = ifft2(transzag); electronic-Liquid Crystal Dissertations - May 28, 2009 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 206 electronic-Liquid Crystal Dissertations - May 28, 2009 207 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 http://www.e-lc.org/dissertations/docs/2009_05_27_22_49_40 electronic-Liquid Crystal Dissertations - May 28, 2009 208 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