TOPOLOGICAL DEFECTS IN
LYOTROPIC AND THERMOTROPIC NEMATICS
A dissertation submitted
to Kent State University in partial
fulfillment of the requirements for the
degree of Doctor of Philosophy
by
Young-Ki Kim
August 2015
© Copyright
All rights reserved
Except for previously published materials
Dissertation written by Young-Ki Kim
B. S., Korea University, Republic of Korea, 2007
M. Eng., Hanyang University, Republic of Korea, 2009
Ph.D., Kent State University, USA. 2015
Approved by
, Chair, Doctoral Dissertation Committee Dr. Oleg D. Lavrentovich
, Member, Doctoral Dissertation Committee Dr. Hiroshi Yokoyama
, Member, Doctoral Dissertation Committee Dr. Liang-Chy Chien
, Member, Outside Discipline Dr. Samuel Sprunt
, Member, Graduate Faculty Representative Dr. Elizabeth Mann
Accepted by
, Director, Chemical Physics Interdisciplinary Program Dr. Hiroshi Yokoyama
, Dean, College of Arts and Sciences Dr. James L. Blank
ii
TABLE OF CONTENTS
LIST OF FIGURES AND TABLES ……………………………………………………vii
ACKNOWLEDGEMENTS…………………………………………………………..xxiii
CHAPTER 1. INTRODUCTION…………………………………………………………1
1.1. Liquid Crystal Phases..………………………………………………………..1
1.2. Topological Defects in Nematics..…………………………………………….5
1.3. Biaixial Nematic Liquid Crystal….………………………………………….12
1.4. Scope and Objectives of the Dissertation…………………………………….16
CHAPTER 2. MORPHOGENESIS OF DEFECTS AND TACTOIDS DURING
ISOTROPIC-NEMATIC PHASE TRANSITION IN SELF-ASSEMBLED LYOTROPIC
CHROMONIC LIQUID CRYSTALS…………………………………………………...19
2.1. Introduction………………………………………………………………….19
2.2. General Properties of LCLCs and Experimental Techniques……………….22
2.3. Topological Characteristics of Point Defects in 2D N: Disclinations and
Boojums………………………………...………………………………………...27
iii 2.4. Early Stages of I-to-N Transition: Tactoids, Boojums and Disclinations…....31
2.5. Shape of Positive N Tactoids with Two Cusps……………………………….35
2.6. Late Stages of I-to-N Transition: I Tactoids as Disclination Cores…………37
2.7. Homogeneous N; Cores of Semi-integer Disclinations………………………42
2.8. N-to-I Transition: I Tactoids and Multiply Connected Tactoids……………..44
2.9. Shape of N and I Tactoids in Frozen Director Field…………………………50
2.10. Conclusions………………………………………………………………...56
CHAPTER 3. DOMAIN WALLS AND ANCHORING TRANSITIONS MIMICKING
NEMATIC BIAXIALITY IN THE THERMOTROPIC OXADIAZOLE BENT-CORE
LIQUID CRYSTAL C7………………………………………………………………….58
3.1. Introduction………………………………………………………………….58
3.2. Materials and Techniques……………………………………………………60
3.3. Alignment, Anchoring Transition and Domain Walls………………………..62
3.3.1. Polarizing Optical Microscopy………………………………………..62
3.3.2. Maps of Retardance and Director Field by LC-Polscope
Observation…………………………………………………………………..64
3.3.3. Behavior of Textures in the Electric Field……………………………..67
iv 3.3.4. Fluorescence Confocal Polarizing Microscopy of the Surface Anchoring
Transition…………………………………………………………………….70
3.3.5. Thickness-dependent Anchoring Transition and Electric Double Layers
of Ions…...…………………………………………………………………...72
3.3.6. Influence of Thermal Degradation of C7 on the Alignment…………...77
3.4.Conclusions…………………………………………………………………...79
CHAPTER 4. SURFACE ALIGNMENT, ANCHORING TRANSITIONS, OPTICAL
PROPERTIES, AND TOPOLOGICAL DEFECTS IN THE THERMOTROPIC
NEMATIC PHASE OF AN ORGANO-SILOXANE TETRAPODES…………………..81
4.1. Introduction………………………………………………………………….81
4.2. Material and Techniques……………………………………………………..82
4.3. Search of Biaxiality in the Cell with a Homeotropic Alignment.…………….85
4.4. Topological Defects………………………………………………………….98
4.4.1. Escape of Director in a Round Capillary………………………………98
4.4.2. Point Defects in Droplets Suspended in Isotropic Fluid……………...101
4.4.3. Point Defects at Colloidal Spheres in the Tetrapode Material………..105
4.5.Conclusions………………………………………………………………….108
v CHAPTER 5. DIRECTOR REORIENTATION OF A NEMATIC LIQUID CRYSTAL
BY THERMAL EXPANSION…………………………………………………………111
5.1. Introduction………………………………………………………………...111
5.2. Methods………………………………………………………………...…..113
5.2.1. NLC Materials, Capillaries, Alignment, Temperature Control………113
5.2.2. Fluorescent Tracers of Flow…………………………………………113
5.2.3. Fluorescent Anisometric Dye………………………………………..114
5.3. Contraction / Expansion, Flow, and Realignment in Flat Capillary………...115
5.4. Director Profile in Shear ( xz ) Plane………………………………………..118
5.5. Thermal Expansion Effects in Other Types of LCs…………………………122
5.5.1. Thermotropic and Lyotropic Chromonic LC………………………...122
5.5.2. Complex-Shaped LC, DT6Py6E6…………………………….……..123
5.6. Thermal Expansion Effects in Other Types of Geometries…………………136
5.7. Applications………………………………………………………………...140
5.8. Discussion and Conclusion…………………………………………………141
CHAPTER 6. SUMMARY…………………………………………………………….148
REFERENCES…………………………………………………………………………152
vi
LIST OF FIGURES AND TABLES
Figure 1.1. Schematic illustration of various nematic phase: (a) uniaxial nematic, (b) chiral nematic, and (c) twist-bend nematic...………………………………….……………………3
Figure 1.2. (a) Anisotropy of physical properties in the (a) uniaxial nematics ( Nu ) along and perpendicular to the main director nˆ and (b) the physical properties different along
three mutually perpendicular directors nˆ , mˆ , and ˆl in biaxial nematics ( Nb )………..…3
Figure 1.3. Examples of three types of topological defects in Nu : (a) linear disclination of strength m 1/2 ; (b) a point defect in the bulk of LC droplet (hedgehog) caused by a homeotropic anchoring of nˆ at the LC droplet surface; (c) surface point defects
(boojums) at the LC droplet surface with a tangential anchoring of .…..……………...6
Figure 1.4. (a) Nuclei of the thermotropic N mixture E7 emerging from the I phase as viewed in a polarizing microscope with crossed polarizers; the I phase appear black; (b) spherical droplets of a thermotropic nematic n-butoxyphenyl ester of nonyloxybenzoic acid dispersed in a glycerin-lecithin matrix, viewed in a polarizing microscope with a single polarizer (no analyzer). In both systems, each N droplet shows two surface point defects- boojums (some are marked by white arrows) and disclination loops (black arrows). The topological defects occur as a result of balance of surface anchoring and elastic distortion energy; (c) principal scheme of director distortions.……………………………………...10
vii Figure 1.5. Candidate materials for biaxial nematic phases with nontrivial molecular shapes: Oxadiazole bent-core LCs, (a) C7 and (b) C12; (c) Azo-substituted bent-core LC,
A131; Bent-core LCs with four lateral flexible chains and shape-persistent derivatives of
(d) thiadiazole, (e) benzodithiophene and (f) fluorenone; (g) Tetrapode shaped LC; (h)
Cyclic (ring-like) LC...... …………………………………….…………………………..13
Figure. 2.1. (a) Molecular structure of DSCG. (b) Schematic structure of chromonic aggregates in I and N phases. (c) Phase diagram of DSCG dispersed in water as the function of temperature T and DSCG concentration. The numbers correspond to the sections that describe the scenarios of phase transition; the dots indicate the approximate location of the system on the phase diagram and the arrows indicate whether the system is cooled down or heated up; the dot with the number 2.7 and no arrow corresponds to the discussion of disclination cores in section 2.7. (d) T dependence of the area occupied by the N phase in biphasic region, for heating and cooling; the rate of temperature change is 1C/mino . Note the difference in the two curves, associated with the different type of director distortions in the system.………………...………………………………………22
Figure 2.2. FCPM images of the I-N biphasic region in DSCG water solution doped with fluorescent dye: (a) in-plane and (b) vertical cross-section of the cell. The tilt angle of I-N interface with respect to the bounding plate is about (777) o .…………………..…...... 25
viii Figure 2.3. Director configurations for (a-d) disclinations m 1/ 2 , 1 , (e) positive
1 1 boojum m inside a positive cusp, (f) negative boojum m associated 22π 22π with a negative cusp, (g) schematic definition of angular parameters.…………………....28
Figure 2.4. LC-PolScope textures of nucleating N tactoids during the I-to-N phase transition in DSCG. (a) The N tactoids feature two cusps with point defects-boojums and tangential orientation of the director at the I-N interface. (b-e) The tactoids grow and merge as temperature is reduced. Note that a merger of two differently oriented 2c tactoids in (c) produces a 3c tactoid with a disclination m 1/ 2 in the center (d,e).………..…..31
Figure 2.5. Anisotropic shape of a small 2c tactoid: (a) LC-PolScope texture and (b) reconstructed shape; (c) phase retardance measured along the long l and short u axes of tactoid…….…………………………………………………………………………....32
Figure 2.6. LC-PolScope textures of N tactoids with (a) 2 cusps, (b) 4 cusps, (c) 3 cusps and one disclination m 1/2 , (d-f) coalescence of two large tactoids into a single tactoid with a pair of oppositely charged disclinations m 1/2 in the bulk, 4 positive and 2 negative cusps at the I-N interface………………………………………………………..34
Figure 2.7. (a) LC-PolScope textures of a round I domain associated with a m 1 director field around it; (b, c) the I tactoid shrinks as T is reduced and then (d) splits into two disclinations of m 1/ 2. (e) T dependence of characteristic size A / π and area A of the I tactoid. (f, g) Retardance profile of the I tactoid shown in parts (b, c), respectively.
(h) Retardance profile of the two disclinations in part (d) separating from each other…....37
ix Figure 2.8. (a) LC-PolScope textures of a 4c negative tactoid associated with a m 1 director field around it; (b, c) the I tactoid shrinks as the temperature is reduced and then
(d) splits into two disclinations of m 1/ 2 . (e) T dependence of characteristic size
A / π and area A of the 4c negative tactoid. (f, g) Retardance profile of the 4c negative tactoid shown in parts (b, c), respectively. (h) Retardance profile of the two disclinations in part (d) separating from each other…………………………...………………………..38
Figure 2.9. Temperature dependence of the free energy difference between the single-core m 1 disclination and a pair of m 1/ 2 disclinations..………...……………………….40
Figure 2.10. (a) LC-PolScope textures of m 1/ 2 disclination and a 1c negative tactoid at the core of the disclination, as a function of increasing temperature T . (b) Optical retardance across the core of defect shown in part (a) for 25.0o C and 30.4o C . (c) across the expanded I core of the defect in the biphasic region, corresponding to
T 30.5Co . (d) T dependence of characteristic size and area of the 1c negative tactoid…………………………………………………………………………………….42
Figure 2.11. (a) LC-PolScope textures of m 1/ 2 disclination and a 3c negative tactoid at the core of the disclination, as a function of increasing temperature T . (b) Optical retardance across the core of defect shown in part (a) for 25.0Co and 30.6Co . (c) across the expanded I core of the defect in the biphasic region, corresponding to
T 30.7o C . (d) T dependence of characteristic size and area of the 3c negative tactoid…………………………………………………………………………………….43
x Figure 2.12. (a) LC-PolScope textures of a 2c negative tactoid nucleated in the uniform
N part of texture. (b) Temperature dependence of characteristic size and area of the 2c negative tactoid. (c) Optical retardance before and after the appearance of the 2c tactoid…………………………………………………………………………………….44
Figure 2.13. LC-PolScope textures of I-N-I tactoid-in-tactoid multiply connected forms with different number of the isotropic inclusions: (a) I 1 , cc2 , (b) I 1 , cc3 (c) I 2 , c 2 , c 0 and (d) I 6, c 12, c 2 .……………….....48
Figure 2.14. Scheme of Wulff construction of an equilibrium shape of a tactoid of a constant area from the polar plot of the surface tension.………………………………....51
Figure 2.15. (a) Polar plots of anisotropic surface tension, equation (2.14), corresponding to the frozen director fields with different topological charges m shown in Figs. 2.3a-d.
(b) The corresponding equilibrium shapes of the negative and positive tactoids, equations
(2.15). The physically relevant parts of the shape do not include the triangular “ears” attached to the cusp points.………………………….…………………………………....53
Figure 2.16. Illustration of how the surface anchoring shapes up the (a) positive N and (b) negative I tactoids with a frozen director of strength m 1/ 2 ...……………..………....54
Figure 3.1. (a) Chemical structure and (b) space filling model of the oxadiazole bent-core liquid crystal C7. (c) Birefringence n of C7 as a function of the relative temperature
t T TNI …………………………………...…………………………………………....60
xi Figure 3.2. Schlieren textures of C7 in the SE1211 cell ( d 4 . 5 μm ) at (a) t 1.0, (b)
2 0 . 0 , and (c) 4 5 . 0 Co observed under a POM. A and P are crossed analyzer and polarizer directions, respectively.……………………………………………...………....62
Figure 3.3. Temperature dependence of POM textures in the SE5661 cells of different thickness: (a, b) d 13 μm ( t* 5 . 5 C o ) and (d, f) 4 . 5 μm ( t* 0 . 9 C o ). The left column shows standard Schlieren textures at elevated temperatures tt *, and the right column shows well-defined SDs at tt *. Red arrows point towards the SDs..………....63
Figure 3.4. Retardation maps of SD textures in the SE5661 cell ( d 1.1μm ) at (a) tt 0.4 (*0C) o , (b) 2 4 . 8 , (c) 3 6 . 8 , and (d) 46.8Co ; ticks show the projection of nˆ ; the central parts of the SDs feature horizontally aligned nˆ and thus yield a higher value of . The SDs separate regions with tilted nˆ and thus reduced . (e) Variation of across the SD along the pathway shown by dotted lines in (a-d)…..……………….65
Figure 3.5. Retardation map with the director field projected on the plane of the SE5661 cell ( d 1.1μm ) at tt 46.8Co (*) . Director configurations of DWs associated with
(b) bend-splay and (c) twist deformations…………………………...…………………...66
Figure 3.6. Retardation maps of the SE5661 cell ( d 1.1μm ) at tt 36.8Co (*)
under the action of the vertical electric field; (a) E 0 , (b) 3 . 6, and (c) 5.5 Vrms /μm . (d)
Change of across a DW (along dotted lines) as a function of E ……………………...68
xii Figure 3.7. (a) FCPM intensity IF C P M in the rubbed planar cell (PI2555) of C7 as a function of the angle and the temperature t . (b) POM texture of the SE5661 cell ( d 13 μm ) between crossed polarizers at tt37.0C(*)o ; arrows indicate DWs.
FCPM textures scanned in (c) the horizontal ( ,xy ) plan at the squared region in b) and (d- f) the vertical ( ,xz ) plan along dotted line in c) under the action of the vertical electric
field; (d) E 0 , (e) 1. 5, and (f) 4.6 Vrms /μm ; arrows indicate DWs...………………...70
Figure 3.8. Retardation maps of the rubbed SE5661 cell ( d 1 . 3μm ) at tt 46.5Co (*) under the action of the in-plane electric field; (a) E 0 , (b) 30 min after E 29.8 V/mm, and (c) 30 minutes after the field is off. (d) profile across an electrode (along red solid lines) in Figs. 3.8a-c…...……………………………………....76
Figure 3.9. (a) Alignment dependence on thermal degradation of C7 in the SE5661 cells of d 4.5 μm . Temperature dependence of POM textures (b) 0, (c) 24, (d) 72 hours after the cell was kept at T 220Co . The first column shows the POM textures of tangential alignment at tt *, the second column shows the POM textures of tilted alignment with
DWs at t* t t H , the third column shows emerging homeotropic domains at ttH , and the forth column shows the cell entirely covered by homeotropic domains at tt42.4o C( Figure 4.1. Molecular structure of an organo-siloxane tetrapode………………………..82 xiii Figure 4.2. (a) Dielectric anisotropy ε and (b) crossover frequency fc as a function of T ..…………………………………………………………….………………………….84 Figure 4.3. Plot of the anchoring transition temperature T * as a function of cell thickness d ………...…………………………………………………………………...…………..86 Figure 4.4. Transition from (a) dark uniaxial texture to (b-d) birefringent textures in the homeotropic cell ( d 4 . 5 μm ) at (a) T 45o C , (b) 40 C(o * ) T , (c) 3 5 Co , and (d) 2 0 Co ; inset in (a) is a conoscopic image of the cell…..…………………………………..87 Figure 4.5. (a-d) Retardation maps and (f) local retardation as a function of temperature T in a homeotropic cell ( d 4.5 μm ) at (a) T 45Co , (b) 40C(*)o T , (c) 3 5 Co , (d) 3 0 Co , and (e) 2 0 Co ...…………………………………………….……………………..87 Figure 4.6. Change of POM texture and conoscopic pattern (a, b, d, f, h) without and (c, e, g, i) with a vertical field in a homeotropic cell ( d 6.9μm ) at (a) T 45Co , (b, c) 40C(*)o T , (d, e) 2 0 Co , (f, g) 0Co , and (h, i) 5Co ; scale bar is 100μm . (j) Transmittance vs. voltage curves for (b-i)..…………………………………..…………..89 Figure 4.7. (a) Change of conoscopic texture as a function of temperature T in a homeotropic cell; no electric field; (b-d) restoration of the Malthese cross of the uniaxial homeotropic state at TT * by a vertical electric field ( d 6.9μm ) at (b) T 35Co , (b, c) 0Co , (d) 5Co . Red arrow in (a) indicates the shift direction of the Malthese cross...... 92 xiv Figure 4.8. (a) Experimental setup to record the light transmittance as a function of cell rotation by around b axis and by around c axis, in the presence of in-plane field; the directions of polarizers for and rotation are indicated by solid and dotted lines, respectively. Intensity profile as a function of the angle (b) and (c) for different temperatures with an in-plane field in a planar cell ( d 20μm ); cartoons illustrate the verifiable geometries of monoclinic symmetry in which the low-frequency dielectric tensor and optical tensor have different directions.………………………………..…………….94 Figure 4.9. (a) FCPM intensity in a homogeneous planar cell of tetrapode material as a function of the angle . Vertical optical slices ( xz- scan) of FCPM textures in the homeotropic cell ( d 20μm ) at (b) TT45C(*)o , (c) 25C(*)o T , and (d) 2 5 Co …....……………………………………………………………………………………....97 Figure 4.10. Escaped director configuration in a round capillary for (a) Nu and (b) a hypothetical Nb ; black solid, blue dashed, and red bold line represent the main director nˆ , secondary director mˆ , and disclination core, respectively. (c-g) POM textures of tetrapode in the round capillaries; (c) capillary of D 50μm shows an escaped configuration at TT42o C( *) and a distorted texture due to an anchoring transition at (d) TT36o C( *) ; large capillary ( D 150μm ) shows smooth texture of an escaped configuration with no biaxial features in the entire temperature range, (d) T 45 , (e) 25 , and (f) 25o C . White dashed lines in (c-g) indicate the inner walls of capillary…….…..99 xv Figure 4.11. Director profile in the tetrapode droplet with tangential surface anchoring: (a) Nu bipolar droplet with two point defects-boojums of mm121 at poles; (b) hypothetical Nb bipolar droplet with a singular disclination m 1 (red bold line) formed by secondary director; (c) hypothetical Nb droplet with a single boojum of m 2 . Textures of Nu bipolar droplets of tetrapode in a glycerol at (d) T 45 , (e) 25 , and (f) 2 5 Co . White dashed line indicates the droplet symmetry axis. Scale bar is 10μm ...... 101 Figure 4.12. Director profile in tetrapode droplets with perpendicular surface anchoring: (a) Nu radial droplet with a point defect hedgehog of m 1 at the center of droplet; Hypothetical Nb droplets with normal anchoring for the principal director, (b) with singular disclination m 1 (red bold line) formed by secondary director, (c) with four disclinations of m 1/ 2 each, and (d) a single surface point defect-boojum with m 2 . The textures of tetrapode droplets in lecithin-glycerol mixture at (e) T 45 , (f) 25 , and (g) 0Co ……………..…………………...... ………………………103 Figure 4.13. Point defects boojums mm121 at the poles of colloidal spheres (diameter D 10μm ) in the planar cell ( d 20μm ) at (a) T 45 , (b) 25 , and (c) 25o C : (d) chain of colloidal particles with point boojums at 25o C ; scheme of director configuration (e) with point boojums in Nu and (f) with new disclination defects in Nb that are singular in the secondary director mˆ . Black solid, blue dashed, and red bold line in (e) and (f) indicate main director nˆ , secondary director mˆ , and disclination core.…………………………105 xvi Figure 4.14. Isolated point defects “hedgehogs” formed next to each colloidal sphere ( D 10μm ) in the planar cell ( d 20μm ) at (a) T 45 , (b) 25 , and (c) 2 5 Co : (d) chain of two colloidal particles with point defects at 2 5 Co ; (e) scheme of director configuration with a point defect “hedgehog” around the particles in Nu and (f, g) hypothetical schemes of director field around the particle with line defects in Nb . Black solid, blue dashed, and red bold line indicate main director nˆ , secondary director mˆ , and disclination core.……………………………………………………………………………………..107 Figure 5.1 (a) Capillary of d 50μm filled with E7 and fluorescent particles. (b) Time sequence of POM textures at x 22mm , when the sample is cooled from 25 to 2 0 Co ( 30o C/min ). The texture changes from dark at constant T 25Co to bright because of the NLC flow (as seen by following the particle marked by a circle) that tilts the director. As T stabilizes at 2 0 Co , the birefringent texture relaxes back to the dark texture of a uniform homeotropic NLC. The insets show the time evolution of the corresponding conoscopic textures. (c) Conoscopic textures of different locations along x . At fixed T , the textures represent a Maltese cross characteristic of a homeotropic uniaxial NLC. As T varies from 25 to 20Co ( 30C/mino ), the isogyres split; the split distance 2a increases with the distance ||x from the capillary center……………………………………………………..…………………….……….116 xvii Figure 5.2. (a) Maximum flow velocity as a function of coordinate x along the capillary axis in the regime of cooling ( T 5Co and 3 0 Co / m i n ). (b) The same for heating ( T 5Co and 3 0 Co / m i n ); capillary of thickness d 50μm filled with E7. (c) Relaxation time tr as the function of cell thickness d and the initial temperature T0 of the homeotropic E7 cell; T 5Co and 30o C/min …………………………………………………………...... 117 Figure 5.3. (a) FCPM texture of the vertical cross-section (xy , ) of a homeotropic cell of d 20μm filled with MCL-6815; the probing beam is polarized along P ; scale bar is 20μm . The inset on the left-hand side shows a conoscopic pattern. T is fixed at 1 5 Co o o . (b) The same during cooling from 15 to 2 5 C ( =15C/min ). (c) IFCPM as a function of z for the initial state. (d) The same for the thermally contracted state; the z - dependence of IFCPM show two pronounced maxima and three minima, near the bounding plates and in the middle plane. (e) The reconstructed director profile in the homeotropic state at fixed T 15Co . (f) The reconstructed bow-shaped director profile in the cooling- induced tilted state at t 8s ...... 119 Figure 5.4. . Horizontal flow velocity and director configuration. Vertical cross-section of a capillary filled with a thermally expanding NLC, as a function of horizontal coordinate x and vertical coordinate z .……………………………………………………………120 xviii Figure 5.5. Thermally induced splitting of crossed conoscopic pattern in the homeotropic cells filled with various thermotropic and lyotropic uniaxial NLCs. The cells are filled o o o with (a) 5CB (T0 2 5 C ), (b) MLC-6815 (T0 5C), (c) ZLI-2806 (T0 1 5 C ), and (d) o lyotropic solution of Sunset Yellow in water, 3 3 %wt (T0 1 5 C ); in all cases, the cell thickness is d 20μm , temperature change T 5Co and rate of change 30o C/min ...... 122 Figure 5.6. Chemical structure and space filling model of the bent-core liquid crystal DT6Py6E6………………..………………………..……...…………………………….123 Figure 5.7. Top: changes in the conoscopic pattern with time while (a) cooling and (b) heating a 13μm thick homeotropic sample of DT6Py6E6. Bottom: data for the intensity of 6 3 3n m light transmitted through the homeotropic sample placed between crossed polarizers, as a function of time, following the temperature changes indicated in (c) cooling and (d) heating....……..………………………..……...………………………………...125 Figure 5.8. Schematic of the technique used to produce a thermal gradient in plane of a homeotropically aligned sample of DT6Py6E6 in the nematic phase. The inset shows the corresponding stationary conoscopic pattern, indicating a steady uniaxial state when the gap between hot and cold plates was varied as shown……………………………………………….………………………..……...….126 xix Figure 5.9. Thermal shrinkage and expansion of a 13μm thick homeotropic sample of DT6Py6E6 during (a) cooling and (b) heating, as visualized by the displacement of the meniscus near the edge of cell observed under the polarizing microscope with parallel polarizers. The dotted black line indicates the edge of the top plate covering the LC layer.…………………………………………………………………………………….127 Figure 5.10. Movement of fluorescent particles in a homeotropic sample of DT6Py6E6 observed by FCPM on (a) cooling and (b) heating the sample; 3 0 Co / m i n . Black area is the LC (no fluorescent intensity) and green dots are the fluorescent particles suspended in the LC. Arrows indicate the direction of particle motion. The particles in the solid circles are fixed by adhesion to the cell surfaces..………………………………………………………………………………..128 Figure 5.11. Splitting of conoscopic isogyres caused by material flow after pushing on the surfaces of a cell containing a 20μm layer of homeotropically aligned DT6Py6E6………………………………………………………………………………129 Figure 5.12. FCPM textures of a 13μm thick homeotropic sample of DT6Py6E6 doped with 0.01wt % of the fluorescent dye BTBP. (a) and (b): In-plane ( xy scan) textures. (c): Vertical optical slices ( xz scan). P indicates the polarizer axis.………………….………………………………………………………………….130 xx Figure 5.13. (a) Raman spectrum of DT6Py6E6. (b) CARS signal intensity in a homogeneous planar cell of the standard calamitic nematic 5CB as a function of the angle C A R S measured between the rubbing direction and polarization of probing beams. (c) CARS signal intensity as a function of C A R S in a homogeneous planar cell of DT6Py6E6. (d–f) Epi-detected CARS texture in a 20μm thick homeotropic DT6Py6E6 sample, stabilized at (d) 5 5 Co , (f) at 5 0 Co , and (e) during temperature change (transient state) o o between 5 5 C and 5 0 C . Note darker field in (d) and (f), and a brighter field in (e). PC A R S indicates the polarization direction of the probing laser beams.………………...……..132 Figure 5.14. Stretching of disclinations by thermally driven flow in the E7 cell with tangential anchoring ( d 20μm ). (a) Stable nearly vertical disclination at a fixed o o temperature, T0 25C . (b) Stretching of disclinations along the flow; T0 25C , T 5Co , and 30C/mino . When the temperature stabilizes, the disclinations relax to the original state (a).………………………...……...………………………………...136 Figure 5.15. Molecular reorientation by thermally driven flow in a uniform planar cell filled with E7 ( d 20μm ). (a,b.c) Sequence of POM textures between the crossed polarizers, for the initial, flow-induced and final equilibrium states. R indicates the rubbing direction; the initial director orientation is along R . (d,e,f) FCPM ( xz, scan) textures o o corresponding to a,b,c, respectively. T0 40C (at a and d), T 5C, and 30o C/min ………………………………………………………………….….……137 xxi Figure 5.16. Thermally activated flows and molecular reorientation in a round capillary. (a) cooling; (b) heating. A point defect-hedgehog at the capillary axis is marked by a red circle. Inset in a shows the director profile around the defect core. T 5Co , and 3 0 Co / m i n (‘ ’ sign for a and ‘ ’ sign for b).……………………………………139 Table 3.1. Dependence of an anchoring transition temperature t * on the concentration c of TBAB in the SE5661 cells of d 1 . 1 and 4 . 5 μm ……………….………………..….74 xxii ACKNOWLEDGEMENTS I am sincerely grateful to my advisor, Dr. Oleg D. Lavrentovich, who has guided me through each and every aspect of this work. I am deeply indebted to him for his constant support and encouragement. It was a wonderful experience to study in the Chemical Physics Interdisciplinary Program. I thank to the Faculty members of Liquid Crystal Institute for sharing their knowledge and experience with me. I also thank to all students and the Administrative and Technical Staff of Liquid crystal for their kind help and great memories. I would like to especially thank to lab member, S. Shiyanovskii, S. Zhou, J. Xiang, C. Peng, G. Cukrov, T. Turiv, I. Lazo, V. Borshch, B. Li, R. Ligas, A. Tishue, B. Senyuk, H. –S. Park and L. Tortora for the fruitful discussions, their valuable advice, and great memories. I acknowledge support from the Kent State University Fellowship, and Graduate Student Senate travel grant. I am profoundly grateful to my wife, Suran, and my family always standing by my side. xxiii CHAPTER 1 INTRODUCTION 1.1. Liquid Crystal Phases Liquid crystals (LCs), also known as mesophase, are the thermodynamically stable state of matter possessing the intermediate properties between the isotropic liquid and the crystalline solid. The LCs exhibit anisotropy in their optical, mechanical, and electromagnetic properties.[1-3] Because of the intriguing ability of LCs to change orientation of molecules in response to very weak physical and chemical cues, such as electromagnetic field, surface modifications, and pressure gradients, LCs are finding numerous applications and are currently one of the most widely used chemicals in our life. In general, LCs are classified into two large categories: lyotropic and thermotropic systems. The thermotropic LCs exhibit mesophases depending on the temperature. Their building units are usually composed of individual organic molecules that have a feature of pronounced shape anisotropy, such as rod and disk. The thermotropic LCs have been successfully used in display applications.[3, 4] In contrast, a lyotropic LC is formed by the dissolution of lyotropic LC molecules in a solvent (usually water). The phases of lyotropic LCs are controlled by the concentration of lyotropic LC molecules in solvent, and, in some cases, such as the lyotropic chromonic liquid crystals (LCLCs), also by temperature. 1 In this dissertation, we explore the topological defects in both lyotropic and thermotropic LCs, and study interesting optical features mimicking a behavior of biaxial nematic phases. In the Introduction, we present a brief introduction of various types of nematic phases, and a short review of topological defects. The nematics are of the following types. Uniaxial and biaxial nematic phase. A nematic (N) is the mesophase exhibiting a long-range orientational order of molecules or their aggregates but showing no long- range positional order. The so-called uniaxial nematic ( N u ) has a single direction of preferred orientation, specified by a director nnˆ ˆ , Fig. 1.1a. The Nu phase shows anisotropy of physical properties with two principal directions, along and perpendicular to the director nˆ , Fig. 1.2a.[1] The best example of Nu is a LC formed by rod like molecules (Fig. 1.2a) that are used in current LC applications such as LCD TVs, and monitors.[3, 4] 2 FIG. 1.1. Schematic illustration of various nematic phase: (a) uniaxial nematic, (b) chiral nematic, and (c) twist-bend nematic. FIG. 1.2. (a) Anisotropy of physical properties in the (a) uniaxial nematics ( Nu ) along and perpendicular to the main director nˆ and (b) the physical properties different along three mutually perpendicular directors nˆ , mˆ , and ˆl in biaxial nematics ( Nb ). 3 The biaxial nematic ( Nb ) has a orthorhombic symmetry ( D2h ) with physical properties different when measured along three mutually perpendicular directors nnˆ ˆ , mmˆ ˆ , and ˆllˆ , Fig. 1.2b.[1] Since the theoretical description in 1970,[5] the Nb phase has been claimed in various LC materials but its existence is still debated. Cholesteric phase. When chiral molecules are added into a nematic, the director nˆ becomes twisted. The resulting structure is termed a cholesteric or chiral nematic, Fig. 1.1b.[1, 2] The axis around which the director twists is called the helical axis hˆ . The distance over which the director makes a full rotation of 2π is called the pitch p . Due to the periodic helical structure of cholesterics, the optical properties of the phase are significantly different from the properties of normal nematics. In particular, the cholesterics show a selective reflection of circularly polarized light in a certain spectral range. This optical phenomenon is of a great interest for reflective display applications.[6] Twist-bend nematic phase. The twist-bent nematic is the recently discovered phase, in which the local director follows an oblique helicoid, maintaining a constant oblique angle with the helix axis and experiencing twist and bend, Fig. 1.1c.[7, 8] The oblique helicoids have a nanoscale pitch as shown by transmission electron microscopy.[7, 8] The new twist-bend nematic represents a structural link between the uniaxial nematic (no tilt) and a chiral nematic (helicoids with right-angle tilt). 4 1.2. Topological Defects in Nematics Whenever a nonhomogeneous state cannot be eliminated by continuous variations of the order parameter, it is called topological stable, or simply, a topological defects.[2] Topological defects play an important role in morphogenesis (from the Greek morphê shape and genesis creation) of phase transitions in cosmological models and in condensed matter.[9-14] In both cases, one deals with a high-temperature symmetric phase and a low- temperature phase in which the symmetry is broken. In the model of early Universe proposed by Kibble,[15] topological defects such as cosmic strings form during the phase transition, when domains of the new state grow and merge. The order parameter is assumed to be uniform within each domain. When the domains with different “orientation” of space-time merge, their junctions have a certain probability of producing defects. Similar effects are expected in condensed matter systems, ranging from superfluids [14, 16] to solids [17]. One of the simplest experimental systems to explore the interplay of phase transitions and topological defects is a uniaxial nematic LC. In the Nu phase, the molecules (or their aggregates) are aligned along the director nˆ (nˆ 2 1) with the property nnˆ ˆ that stems from a non-polar character of ordering. In three dimensional (3D) space, the N phase allows three types of topologically stable defects: linear defects (disclinations, Fig. 1.3a), point defects in the bulk (hedgehogs, Fig. 1.3b) and point defect at the surfaces (boojums, Fig.1.3c).[2] 5 FIG. 1.3. Examples of three types of topological defects in Nu : (a) linear disclination of strength m 1/2 ; (b) a point defect in the bulk of LC droplet (hedgehog) caused by a homeotropic anchoring of nˆ at the LC droplet surface; (c) surface point defects (boojums) at the LC droplet surface with a tangential anchoring of . The isotropic-nematic (I-N) transition is of the first order. Besides the I-N transition temperature TNI , there are two other important temperatures T * and T ** , characterizing the spinodal points: TT* NI is the limit of metastability of the I phase upon cooling and TT** NI is the limit of overheating the N phase. The critical size R* of N nuclei, determined by the gain in the bulk condensation energy and the loss in the surface energy of the I-N interface, is about 10nm near T * .[18] As these nuclei grow and coalesce, they can produce topological defects at the points of junction. Chuang et al [19] and Bowick et al [20] performed the Kibble-mechanism-inspired experiments on the I-N transition and described the dynamics of ensuing defect networks. Mostly disclinations 6 were observed, with hedgehogs appearing seldom; boojums were not recorded. A probability of forming a disclination is significant ( ~1/ π when there are three merging domains [21]). In the analysis of defects emerging during the I-to-N phase transition, it is usually assumed that the director is roughly uniform, nrˆ c onst , within each N nucleus.[20, 22] This assumption mirrors the cosmological model, in which each expanding bubble preserves spatial uniformity of the scalar field in its interior and at its surface.[15, 23] In other words, the I-N interfacial tension is considered as “isotropic”, i.e. independent of director orientation at the surface. As they grow, however, one needs to account for the anisotropy of the I-N interface, as discussed below. Surface properties of LCs are anisotropic because the molecular interactions set up a preferred orientation of nˆ at an interface, called an “easy direction”. For the I-N interface, it is convenient to introduce an “easy angle” between the normal υˆ to interface and the easy direction, Fig. 1.4c. Depending on the details of anisotropic molecular interactions, the easy direction might be perpendicular ( 0 ) to the interface (Fig. 1.3b), conically tilted ( 0 π /2) or tangential, Fig. 1.3c ( π /2, so that nˆ can adopt any orientation in the plane of interface). The anisotropic potential of I-N interface can be described in the so-called Rapini-Papoular form [24] useful for analytical analysis: 2 0 1 w sin , (1.1) 7 where 0 is the orientation-independent part of the surface tension and w is the surface anchoring coefficient. For small angles, the Rapini-Papoular potential describes the work 2 0w needed to realign the director. The surface anchoring by itself is capable of setting up stable topological defects in the interior of each and every N domain,[25, 26] in order to satisfy the theorems of Poincaré and Gauss that demand a certain number of singularities in the vector field (such as nˆ ) defined on surfaces with a non-zero Euler characteristic Eu ( Eu 2 for a sphere). Whether or not the director in an N nucleus of the size R will follow the “easy axis”, 2 depends on the balance of the surface anchoring energy ~ 0wR and the elastic cost of bulk deformations ~ KR , where K is the Frank elastic constant. The last estimate is constructed by multiplying the elastic energy density ~/KR2 by the volume ~ R3 of the distorted domain. The ratio of bulk elastic constant K and the surface anchoring strength w defines the so-called de Gennes-Kleman extrapolation length, Kw/ 0 . When the N nucleus is small, R , the director within it can be assumed to be uniform, since 2 2 KRwR 0 . When R and thus 0wR KR , the surface anchoring conditions need to be satisfied, which necessitates the existence of topological defects in each and every N drop. For example, a large spherical drop with 0 must contain at least one point defect-hedgehog; a radial director field is one of the possible configurations satisfying the perpendicular surface anchoring. For typical thermotropic N materials, near 7 6 2 the I-N transition, K 2 pN and 0w ~ 10 10 J/m ,[27] thus ~ 1 10 μm . The assumption of a uniform director within the nuclei is thus valid when the nuclei are of 8 a submicron size. However, each N nucleus that is larger than must carry topological defects as an intrinsic feature caused by surface anisotropy. Point defects and disclination loops in 3D nematic nuclei of thermotropic liquid crystals. Experimental exploration of morphogenesis of nuclei in the thermotropic LCs is hindered by the fact that the temperatures TNI , T *, and T **, are all very close (one degree or so) to each other. Nevertheless, by carefully stabilizing the temperature, one can obtain and observe large N droplets that coexist with the I background, Fig. 1.4a. For a commercially available nematic mixture E7, Faetti and Palleschi [27] demonstrated experimentally that the easy angle between nˆ and the normal υˆ to the I-N interface is in the range 4 8oo - 6 5 . The tilted easy axis implies that there is a non-zero vector projection onto the interface, nˆ υˆ υˆ nˆ . According to the Poincaré theorem, such a field must contain singularities of a total strength mi equal the Euler characteristic of a surface, i mEiu ; for a sphere, Eu 2 . The strength mi of each defect is determined as a i number of rotations of vector nˆ υˆ υˆ nˆ as one circumnavigates the defect core once. The two defects on the poles of each droplet in Fig. 1.4a, represent these singularities, called boojums, each of strength m 1. 9 FIG. 1.4. (a) Nuclei of the thermotropic N mixture E7 emerging from the I phase as viewed in a polarizing microscope with crossed polarizers; the I phase appear black; (b) spherical droplets of a thermotropic nematic n-butoxyphenyl ester of nonyloxybenzoic acid dispersed in a glycerin-lecithin matrix, viewed in a polarizing microscope with a single polarizer (no analyzer). In both systems, each N droplet shows two surface point defects- boojums (some are marked by white arrows) and disclination loops (black arrows). The topological defects occur as a result of balance of surface anchoring and elastic distortion energy; (c) principal scheme of director distortions.[25, 26] Besides the 2D topological charges m , one can also introduce a 3D characteristic for each boojum, defined as 1 nnˆ ˆ m A nnˆ d12 d ˆ υˆ 1 N (1.2) 42 12 where 12, is the pair of coordinates on a semispherical surface surrounding the boojum from the N side and N is an integer.[25] In the last expression, the director field is treated as a vector and υˆ is assumed to be directed outward the N domain. Note that the 10 definition of A would be ambiguous if nˆ were treated as a director with the states nˆ and nˆ being equivalent: Replacing nˆ with nˆ in the definition of A reverses the sign of A . If there are no disclination lines in the interior of the N phase, the ambiguity is easily removed by regarding nˆ as a vector rather than a director.[25] In Fig. 1.4a-c, the disclination loop near the equatorial plane of the droplets divides the surface into two parts with opposite signs of the scalar product nˆ υˆ , S with nˆ υˆ 0 and S with nˆ υˆ 0 . The conservation laws connecting the 2D and 3D characteristics of the boojums in the presence of equatorial disclination has been derived in Ref. [25] as 1 NmEiiu / 21 that stem from the Gauss theorem. In the case shown in Fig. ii2 2 2 1.4a-c, ANm111sin/ 2,1,1 and ANm222sin/ 2 ,0,1 , which obeys the conservation laws above. The surface-anchoring and topology dictated scenario of defect formation is applicable to any N nucleus that is larger than the de Gennes-Kleman length, so that it is energetically preferable to satisfy the surface anchoring conditions at the expense of the elastic deformations associated with defects. The phenomenon is not restricted to the I-N phase transition. Similar defect-rich textures are observed in equilibrium nematic droplets with a fixed size, dispersed in an immiscible isotropic fluid, such as glycerin [25], Fig. 1.4b. The concrete set of defects depends on the easy angle that can be controlled in experiments.[25] If varies from some nonzero value to 0 , then the boojums shown in Fig. 1.4 should disappear. The disclination loop shrinks into a point defect-hedgehog at the surface (which reduces the elastic energy). The ensuing point defect can leave the 11 surface when 0 and go into the center of drop, thus establishing either a 3D radial structure of nˆ or a more complex structure, depending on the elastic anisotropy of the material.[28] If the easy angle changes towards its maximum value /2, then the disclination loop seen in Fig. 1.4b gradually disappears, and the N drop features only two boojums that are sufficient to satisfy the tangential boundary conditions.[25] 1.3. Biaxial Nematic Liquid Crystal A biaxial nematic phase is of orthorhombic symmetry with physical properties different along three mutually perpendicular directors nnˆ ˆ , mmˆ ˆ , and ˆllˆ , Fig. 1.2b. Since the Nb phase was theoretically described by Freiser in 1970,[5] there has been a growing interest in the search for Nb phases. The Nb phase was first observed experimentally by Yu and Saupe [29] in a lyotropic mixture potassium laurate / 1-decanol / water system. Although the biaxial order has been observed by other researchers in this and other lyotropic nematics,[30-39] the issue remains controversial, as the studies by Berejnov et al. [40, 41] suggest that the biaxial order is only transient and when the samples are left intact, they eventually relax into the uniaxial state. The low-molecular weight thermotropic version of Nb is of a special interest, as it would allow one in principle to construct fast-switching displays and other electro-optic devices.[42-44] However, the existence of a thermotropic Nb is even less clear than its lyotropic counterpart. Simple rod-like mesogens explored so far do not yield a spontaneous biaxial order, although some of them do show a field-induced biaxiality with very fast 12 (nanoseconds) electro-optic response.[45] Significant synthetic efforts thus extended to novel and more complex molecular architectures, Fig. 1.5. Among these biaxial nematic candidates, the most studies materials are the so-called bent-core molecules, Fig. 1.5a- f.[46-54] FIG. 1.5. Candidate materials for biaxial nematic phases with nontrivial molecular shapes: Oxadiazole bent-core LCs, (a) C7 and (b) C12; (c) Azo-substituted bent-core LC, A131; Bent-core LCs with four lateral flexible chains and shape-persistent derivatives of (d) thiadiazole, (e) benzodithiophene and (f) fluorenone; (g) Tetrapode shaped LC; (h) Cyclic (ring-like) LC. 13 In the bent-core materials with oxadiazole units (Fig. 1.5a,b), Nb had been identified in the studies of X-ray diffraction (XRD),[47, 55] NMR & conoscopy,[46] and electro-optical switching.[43, 56-58] Subsequently, reexamination of how materials behave in the external fields and in confined geometries with topological defects, led to a conclusion that two oxadiazole compounds, abbreviated C7 (Fig. 1.5a) and C12 (Fig. 1.5b), are uniaxial nematics that mimic the behavior of Nb because of effects such as surface anchoring transitions.[59, 60] However, as the so-called “secondary disclinations” was recently described as a new evidence of Nb in C7, [61] the existence of Nb phase in this material becomes once again a controversial issue. The Nb phase was also claimed in the azo-substituted bent-core nematic A131 (Fig. 1.5c), following XRD and DSC,[48] NMR,[62] conoscopy,[53, 63] and Raman scattering experiments.[52]. More recent NMR studies resulted in a conclusion that it is necessary to test the material for possible conformational changes [51] and that a transition within the nematic range of A131, identified as the one from the uniaxial to the biaxial phase, can be associated with a slowing down of the molecular rotations around the long molecular axis.[64] Two other similar azo compounds have also been claimed to feature the Nb phase on the basis of XRD [48, 65] and conoscopy [63] studies. However, explorations of electro- and magneto-optics, surface alignment and defects showed that the nematic order in A131 is uniaxial.[60, 66, 67] One of the reasons for the controversy is that the uniaxial nematic phase can mimic the features of the Nb phase in a variety of forms. Very often this mimicking behavior is 14 rooted in the complexity of surface alignment of molecules with nontrivial shapes such as bent-core [59, 67] and tetrapodes [68] mesogens. For instance, the tetrapode material (Fig. 1.5g) had been claimed to exhibit a Nb phase based on the studies of IR spectroscopy,[69] NMR,[70] light scattering,[71] and electro-optics,[72] although 129 X e N M R [73] and high- resolution adiabatic scanning calorimetry[74] could not detect a discernible N -ub N transition. At the time of the beginning of this dissertation project, the claimed existence of the Nb phase has not been challenged for bent-core mesogens with oxadiazole [47, 49, 75] and azo-groups,[48, 63, 65] for strongly asymmetric units,[76, 77] and for the shape- persistent derivatives of benzodithiophene (Fig. 1.5.e) [78, 79] and fluorenone (Fig. 1.5.f).[63, 80-82] Other molecular geometries, such as mesogens with four flexible chains,[83, 84] tetrapode derivatives [69, 71, 85-88] different from the one explored in Ref. [68] also show Nb features that have not been disputed. Finally, there is an interesting but underexplored class of compounds with cyclic (ring-like) mesogens (Fig. 1.5h) synthesized by Percec [89-91] with the Nb features in phase behavior [89, 90] and in structure of wall defects.[91] 15 1.4. Scope and Objectives of the Dissertation The scope of this dissertation is the study of topological defects in both lyotropic and thermotropic LCs as well as the material characterization with nontrivial molecular shape of LCs. The study involves various experimental techniques, such as polarizing optical microscopy, fluorescence polarizing optical microscopy, coherent anti-Stokes Raman scattering microscopy, LC-PolScope, conoscopy, electro-optics, and so on. The objectives of the dissertation are to study: (i) the structure of nuclei and topological defects in the first-order phase transition between the nematic and isotropic phases in lyotropic chromonic LCs; (ii) the features of topological defects, surface alignment, anchoring transitions, and optical properties in thermotropic nematics with nontrivial molecular shapes that have been claimed to exhibit the biaxial nematic phase; (iii) the role of thermal expansion (on heating) and contraction (on cooling) of liquid crystals in director structures and optical properties. The dissertation is organized as follows. Chapter 2 describes the intriguing features of topological defects and tactoids in lyotropic chromonic liquid crystal. In this chapter, we illustrate how the balance of anisotropic surface energy and internal elasticity shapes the tactoids and influences the topological defects. Chapter 3, 4 describes the studies of topological defects, surface alignment, anchoring transitions, and optical properties of materials with nontrivial molecular shapes that have been previously claimed to exhibit the biaxial nematic phase. In particular, 16 Chapter 3 deals with an oxadiazole bent-core thermotropic LC, C7 and Chapter 4 deals with a thermotropic nematic phase of organo-siloxane tetrapodes. Chapter 5 describes new thermo-mechanical and thermo-optical phenomena that the temperature-induced density variation of nematic LCs can trigger through molecular reorientation. In Chapter 6, the results of this dissertation are summarized. The following publications cover the topics discussed in the dissertation: [1] B. Senyuk, Y. –K. Kim, L. Tortora, S. –T. Shin, S. V. Shiyanovskii, and O. D. Lavrentovich, “Surface alignment, anchoring transitions, optical properties and topological defects in nematic bent-core materials C7 and C12”, Molecular crystals and liquid crystals, 540, 20 (2011). [2] Y. –K. Kim, M. Majumdar, B. I. Senyuk, L. Tortora, J. Seltmann, M. Lehmann, A. Jákli, J. T. Gleeson, O. D. Lavrentovich, and S. Sprunt, “Search for biaxiality in a shape- persistent bent-core nematic liquid crystal”, Soft Matter, 8, 880 (2012). [3] Y. –K. Kim, B. Senyuk, and O. D. Lavrentovich, “Molecular reorientation of a nematic liquid crystal by thermal expansion”, Nature Communications, 3, 1133 (2012). [4] O. D. Lavrentovich, Y. –K. Kim, and B. I. Senyuk, “Optical manifestation of thermal expansion of a nematic liquid crystal”, Proc. SPIE, OP211, 84750G (2012) 17 [5] Y. –K. Kim, S. V. Shiyanovskii, and O. D. Lavrentovich, “Morphogenesis of defects and tactoids during isotropic-nematic phase transition in self-assembled lyotropic chromonic liquid crystals”, J. Phys.: Condens. Matter, 25, 404202 (2013). [6] Y. –K. Kim, B. Senyuk, S. –T. Shin, A. Kohlmeier, G. H. Mehl, and O. D. Lavrentovich, “Surface alignment, anchoring transitions, optical properties, and topological defects in the thermotropic nematic phase of organo-siloxane tetrapodes”, Soft Matter, 10, 500 (2014). [7] Y. –K. Kim, R. Breckon, S. Chakraborty, M. Gao, S. N. Sprunt, J. T. Gleeson, R. J. Twieg, A. Jákli, and O. D. Lavrentovich, “Properties of the broad-range nematic phase of a laterally linked H-shaped liquid crystal dimer”, Liq. Cryst., 41, 1345 (2014). [8] Y. –K. Kim, G. Cukrov, J. Xiang, S. –T. Shin, and O. D. Lavrentovich, “Domain walls and anchoring transitions mimicking nematic biaxiality in the oxadiazole bent-core liquid crystal C7”, Soft Matter, 11, 3963 (2015). 18 CHAPTER 2 MORPHOGENESIS OF DEFECTS AND TACTOIDS DURING ISOTROPIC- NEMATIC PHASE TRANSITION IN SELF-ASSEMBLED LYOTROPIC CHROMONIC LIQUID CRYSTALS 2.1. Introduction In the models of early universe proposed by Kibble, topological defects form during the phase transition, when domains of the new state grow and merge. When the domains with different orientation merge, their junctions have a certain probability to produce defects. The similar effect are expected in condensed matter systems. An interesting experimental system to explore the interplay of phase transitions and topological defects is the so-called lyotropic chromonic liquid crystal (LCLC). The main distinctive feature of LCLCs is that their “building units” are not of a fixed shape, representing self-assembled molecular aggregates.[92-95] The range of materials which form LCLCs includes dyes, drugs [92-95], nucleotides [96] and DNA oligomers [97, 98]. Typically, the LCLC molecule is plank-like or disc-like with an aromatic flat core and peripheral polar groups. In water, the cores stack face-to-face. The aggregates, bound by weak non-covalent interactions, are polydisperse, with the length distribution that depends on concentration, temperature, ionic strength, etc.[93, 99-102] The LCLCs show a 19 temperature- and concentration-triggered first order I-N transition with a broad ( 5 -15 Co ) coexistence region. Therefore, the LCLCs allow one to observe the formation of numerous types of domains and topological defect by Kibble mechanism because the nucleation, growing, and coalescence of domains can be controllable by varying the temperature T . Another intriguing feature of LCLCs is that the equilibrium shape of (nematic and isotropic) nuclei is very different with the one in thermotropic LCs. The thermotropic N droplets dispersed either in their own melt (Fig. 1.4a) or in a foreign isotropic fluid (Fig. 1.4b) is nearly spherical. The reason is the high surface tension 0 of the I-N interface 52 and a relatively weak surface anchoring 0w : 0 ~ 10J/m , while 7 6 2 0w ~ (10 10 ) J/m [103, 104], so that w ~ 0.1 0.01 . The surface energy 217 FRs 0 ~10J of a thermotropic N droplet of a radius R ~1μm or larger overweighs 2 318 the elastic energy of internal distortions FKRKRe nˆ ~~ 210J . When the materials are of lyotropic type, however, the interplay of surface and bulk effects during phase transitions in LCs is enriched. If the lyotropic N phase is formed by building units of a size in the range 10 100 nm , the interfacial surface energy is expected 752 to be weaker than in the thermotropic case; experimentally, 0 ~ (1010) J/m .[105, 106] Furthermore, the surface anchoring anisotropy in lyotropic LCs might be more pronounced than in their thermotropic counterparts. The experimental estimates range from w 4 [107] for water dispersions of carbon nanotubes to w ~10100 ( 52 72 0w ~ (0.5 5) 10 J/m and 0 ~ 5 10 J/m ) for vanadium pentoxide 20 dispersions.[108, 109] On the other hand, the Frank elastic constants in the lyotropic LCs are nearly of the same order as those in thermotropic LCs.[102] It is thus expected that the structure of nuclei in the I-N phase transitions would be highly nontrivial, both in terms of 2 2 their shape and the interior director structure, as the representative energies 0R , 0wR , and KR might vary in a much wider range than in the thermotropic LCs. The main goal of this chapter is to describe experimentally the surface anchoring- controlled morphogenesis of tactoids and accompanying topological defects during the I- to-N and N-to-I transitions. This chapter is organized as follows. In section 2.2, we present a general properties of LCLCs and experimental techniques. Section 2.3 introduces the elements of topological description of point defects in 2D. Section 2.4 describes the early stages of the I-N phase transition, in which the nucleating, growing and coalescing of N domains occur. The simplest N tactoid that has two cusps and two boojums at the poles is discussed in Section 2.5. The scenarios of the later stages of the I-N transitions are presented in Section 2.6. Section 2.7 describes the cores of m 1/2 disclinations in the homogeneous N phase showing non-circular shape. The reverse transition, from the homogeneous N phase with disclinations, into the I phase, is described in Section 2.8. In Section 2.8, we also present tactoid-in-tactoid scenarios with multiply connectivity. Finally, in Section 2.9, we use the Wulff construction to describe the shape of topologically nontrivial N and I tactoids. 21 2.2. General Properties of LCLCs and Experimental Techniques FIG. 2.1. (a) Molecular structure of DSCG. (b) Schematic structure of chromonic aggregates in I and N phases. (c) Phase diagram of DSCG dispersed in water as the function of temperature T and DSCG concentration. The numbers correspond to the sections that describe the scenarios of phase transition; the dots indicate the approximate location of the system on the phase diagram and the arrows indicate whether the system is cooled down or heated up; the dot with the number 2.7 and no arrow corresponds to the discussion of disclination cores in section 2.7. (d) T dependence of the area occupied by the N phase in biphasic region, for heating and cooling; the rate of temperature change is 1Co / min . Note the difference in the two curves, associated with the different type of director distortions in the system. 22 We explore water solutions of disodium cromoglycate (DSCG, Sigma-Aldrich), one of the first studied LCLCs,[92, 94] Fig. 2.1a. In water, the DSCG molecules stack on top of each other face-to-face (the so-called H-aggregation) to minimize the areas of unfavorable contact with water.[101, 110, 111] The stacking distance is ( 0 . 3 3 - 0 . 3 4 ) n m .[101, 110] The important difference is that in LCLCs, there are no chemical bonds to fix the length of aggregates. We used a DSCG concentration c w t1 6 % ( 0 . 3 7 m o l / k g ). The relevant portion of phase diagram is shown in Fig. 2.1c; note a broad biphasic region in which the N and I phases co-exist. Fig. 2.1d shows how the area of the N phase changes with T in the biphasic region. Although the trend is close to the linear one, the curves for heating and cooling are somewhat different, since the director pattern in the sample is different upon cooling and heating. From the experimental point of view, it is important to explore a pseudo-2D geometry, to mitigate complications associated with the effects such as depth-dependent nˆ . 2D geometry also allows one to apply a quantitative methods of optical microscopy such as mapping of the optical retardance and orientation of nˆ .[112] If the sample is thin, the undesirable director distortions along the light propagation direction are suppressed and the image represents a 2D pattern of the director field that can be reconstructed by using a microscope with LC-PolScope universal compensator.[113] The samples were prepared as thin slabs of thickness d (1-5) μm between two glass plates, spin-coated with un-rubbed polyimide SE-7511(Nissan, Inc.) for tangential 23 anchoring. The cell thickness d was set by glass spheres mixed with UV epoxy (NOA 65, Norland Products, Inc.) applied at the periphery of cells to seal them and to prevent evaporation of water. The temperature T was controlled with the Linkam controller TMS94 and hot stage LTS350 (Linkam Scientific Instruments) with precision of 0.01 Co . In both cooling and heating, the temperature rate was typically 0.1o C / min ; we waited until the expansion of N or I tactoids would stop before changing T again. The textures were examined by a polarizing microscope equipped with LC- PolScope (Abrio Imaging System). The LC-PolScope uses a monochromatic illumination at 5 4 6n m and maps optical retardance ( ,xy ) and orientation of the slow axis in the sample.[112] The maximum measured ( ,xy ) in the LC-PolScope is listed as 2 7 3 n m , but we found by testing wedge samples with a variable retardance that the device does not produce accurate measurement for any value of ( ,xy ) larger than about 2 4 0 n m . We thus restricted the thickness d of our cells to set ( ,xy ) between 0 and 2 4 0 n m . For a tangentially anchored N, nndeo, where ne and no are the extraordinary and ordinary refractive indices. For DSCG, optical birefringence is negative, nneo 0.02 .[111, 114] The slow axis is thus perpendicular to the optic axis nˆ . The LC-PolScope was set up to map the local orientation nˆ xy, (rather than the slow axis), by headed nails. Note that the nail’s head is an artificial feature of LC-PolScope software that is not related to any particular property of the LCLC, as nnˆ ˆ and there is no director tilt at the N- substrate interface. The accuracy with which the LC-PolScope determines the orientation 24 of the optic axis is better than 0 .1o . The map of retardance (,)xy allows us to trace the changes in the degree of the scalar order parameter S , as in the first approximation,[1] neo n S . FIG. 2.2. FCPM images of the I-N biphasic region in DSCG water solution doped with fluorescent dye: (a) in-plane and (b) vertical cross-section of the cell. The tilt angle of I-N interface with respect to the bounding plate is about (77 7)o . 25 The surface anchoring of DSCG at bounding plates is tangential. The director configuration can be treated as 2D, as the bounding plates suppress the out-of-plane distortions. These distortions are better suppressed in thinner samples. However, different contact angles between the N and I phases and the bounding plates might still distort nˆ in the vertical cross-section of the cell. To explore the menisci, we used fluorescent confocal polarizing microscopy (FCPM) that images vertical cross-sections of the samples.[115] The DSCG solution was doped with a water-soluble fluorescent dye Acridine Orange (Sigma-Aldrich); it concentrates predominantly in the I phase. In this particular experiment, we used thick cells of d 20μm to enlarge the menisci. Fig. 2.2 shows that the I-N interface is nearly perpendicular to the bounding plates, with the N side forming a contact angle in the range around 777 o with the substrate. Since is not very different from π/2 , the meniscus profile should not significantly alter the shape of N and I regions that is determined from the LC-PolScope images of thin samples. In a sample of d 2 μm , the meniscus effect would lead to a 0.2 μm inaccuracy in measured lateral distances, which is less than the optical resolution (about 0 . 5 μm ). In order to estimate the surface tension 0 of the I-N interface, we used the pendant drop technique.[105, 116] The 16%wt solution of DSCG was centrifuged at 4400 rpm at N-I biphasic temperature, T = 38o C , to achieve macroscopic phase separation. The I and N phases were carefully filled into separate syringes. The corresponding densities were measured with DE45 Density Meter (Mettler Toledo) at o 3 3 T 38 C : N 1.08 g / cm and I 1.06 g / cm . The centrifuged I solution was 26 transferred into the rectangular bath and the pipette containing the N solution was inserted into the I bath kept at 3 8 Co . The N solution was pushed out from the pipette to form a pendant drop. The interfacial tension 0 was determined by fitting the drop profile with the theoretical plots.[105, 116] Since the droplets were stable only within minutes, we 42 could only estimate the order of magnitude, 0 1 0 J / m . 2.3. Topological Characteristics of Point Defects in 2D N: Disclinations and Boojums In what follows, we operate with the 2D description of topological defects in the N domains, Fig. 2.3. The 2D director field is parameterized as nnxy,cos, sin , where is the angle between the director and a fixed axis x in the ( ,xy ) plane, and is the polar angle of the polar coordinate system ( ,r ) , 1 Fig. 2.3g. The order parameter space of a 2D N represents a circle SZ/ 2 with two 1 opposite points being identical to each other. The first homotopy group of SZ/ 2 is 1 nontrivial, π12SZ/ {0, 1/ 2, 1,...} , which implies that in the interior of the 2D domains, there might be topologically stable point disclinations of various “strength” or “topological charge”.[2] The disclination strength is introduced as an integer or semi- integer number, 11nnyx m nxy n d 2π 0 0, 1/ 2, 1,... (2.1) 2π 2π 27 For example, a simple radial configuration nnxy,cos,sin yields m 1. Four main disclination types with m 1/ 2, 1 are shown in Figs. 2.3a-d; as an example of m 1 disclinations, in Fig. 2.3c, we show a circular director pattern, nnxy,cos π / 2, sin π /2 . FIG. 2.3. Director configurations for (a-d) disclinations m 1/ 2, 1 , (e) positive 1 1 boojum m inside a positive cusp, (f) negative boojum m associated 22π 22π with a negative cusp, (g) schematic definition of angular parameters. 28 The second type of point defects, boojums, are observed in the experiments with DSCG as defects located at the cusps of I-N interface. The “strength” of a boojum is determined not only by the behavior of but also by the angle N between the two shoulders on the opposite sides of the cusp (measured in the N phase anti-clockwise; we direct the x axis along one of the shoulders): 11 N nnyx m nxy n d N 0 . (2.2) 2π 0 2π For a flat I-N interface, N π and the radial director field nnxy,cos, sin yields m 1/ 2 , an intuitively clear result, as such a boojum represents 1/ 2 of the radial bulk disclination m 1. When the interface is not flat, N π, equation (2.2) results in k m , (2.3) 22π where k is an integer and π N is the angle measured between the right shoulder and the continuation of the left shoulder of the cusp. The convenience of last notation is that the sign of discriminates between the positive cusp (N phase protruding into the I phase, 0 N π,0 π ), and the negative cusp (I phase protruding into the N phase, π N 2π, π 0 ). The sign of m and k is determined uniquely by the comparison of the traveling direction around the defect core and the sense of director rotation.[2] Fig. 2.3e shows a positive boojum with k 1, while Fig. 2.3f shows a negative boojum with k 1. This 29 boojums have the lowest energies and therefore are observed in our experiment. A ‘zero’ boojum with k 0 is unstable as it exists only with a cusp and can be smoothly eliminated simultaneously with the cusp, 0. It is important to stress that for the tangential surface anchoring at the I-N interface (easy angle π /2 ), the distinction between the disclinations and boojums is of an energetic rather than a topological origin. The presence of cusps filled with boojums is dictated by the balance of surface and elastic energies; when such a balance yields 0, the boojums acquire semi-integer or integer strength and can leave the interface and move into the interior of the N domains. If the 2D picture is extended to 3D (along the normal to the LCLC cell), then the point disclinations would correspond to linear disclinations and the point boojums to the linear surface disclinations parallel to the third dimension axis. 30 2.4. Early Stages of I-to-N Transition: Tactoids, Boojums and Disclinations FIG. 2.4. LC-PolScope textures of nucleating N tactoids during the I-to-N phase transition in DSCG. (a) The N tactoids feature two cusps with point defects-boojums and tangential orientation of the director at the I-N interface. (b-e) The tactoids grow and merge as temperature is reduced. Note that a merger of two differently oriented 2c tactoids in (c) produces a 3c tactoid with a disclination m 1/2 in the center (d,e). Fig. 2.4 illustrates appearance of the N phase through elongated tactoids with a two- cusped shape, or “2c tactoids”, when T is lowered from the I phase into the biphasic phase, see label “2.4” in Fig. 2.1c. There are two surface defects-boojums at the cusps. The I-N interface near the cusp forms an angle 0 π , Fig. 2.5b, so that the corresponding boojum is of a strength m 1/ 2 / 2π , Fig. 2.3e. The director is tangential to the I-N interface, as clear from the image of a large tactoid in Fig. 2.4a. In the cusp regions, the 31 retardance is reduced as compared to the interior of tactoid. The reasons are (i) non-flat meniscus, (ii) finite width of the interfacial region and decrease of the scalar order parameter in order to reduce the energy of strong director distortions, (iii) realignment of nˆ along the vertical axis, an effect similar to the “escape into the third dimension”.[1] FIG. 2.5. Anisotropic shape of a small 2c tactoid: (a) LC-PolScope texture and (b) reconstructed shape; (c) phase retardance measured along the long l and short u axes of tactoid. Coalescence also produces “negative” cusps of protruding I phase, the N part of which is filled with a boojum of a negative strength 1/ 20 m , Fig. 2.3f. Three negative cusps are clearly visible in Figs. 2.4e and 2.6f. Since each N domain is topologically equivalent to a disk, for the fixed in plane tangential surface anchoring, the total director reorientation measured by circumnavigating the I-N boundary, should be equal 2π minus the angle that describes missing orientations of the I-N interface. If there are ic1,2,... boojums at the interface of strengths mi associated with the cusps of angles 32 i and jn1,2 ,. . . disclinations of charges mj in the interior, then the conservation law for the topological charges involved is written as cn cn i 1 mmij 1 or kmij1. (2.4) ij2 2 ij cn For a smooth round disk, i 0 , the last relationship reduces to mmij1. ij In our experiments, only low energy boojums are observed: a positive boojum, k 1, inside the positive cusp, Fig. 2.3e and a negative boojums k 1 in the negative cusp, Fig. 2.3f. Then Eq. (2.4) can be rewritten in terms of the excess number of positive cusps c over the negative ones c that is determined by the total charge of disclinations: n ccm 21 j . (2.5) j As an illustration, the bottom-right part of Fig. 2.6b shows a tactoid with 3 positive cusps and one negative cusp, and no bulk disclinations, thus 3 1 2 . When such a tactoid coalesces with a regular 2c tactoid the negative surface boojum evolves into a full bulk disclination m 1/ 2 , Fig. 2.6c. The next merger between a disclination-free 4c tactoid and 3c tactoid with a disclination, Fig. 2.6d-e, produces a large 6c tactoid with 4 positive cusps, 2 negative cusp, one disclination m 1/ 2 and one disclination m 1/ 2, so that the total number of cusps and the topological charges are again satisfying equation (2.5), as 4 2 2(1 1/2 1/2) , Fig. 2.6f. Mergers of defects in the bulk and at the I-N interface is possible, but it does not change the sum total. 33 FIG. 2.6. LC-PolScope textures of N tactoids with (a) 2 cusps, (b) 4 cusps, (c) 3 cusps and one disclination m 1/2 , (d-f) coalescence of two large tactoids into a single tactoid with a pair of oppositely charged disclinations m 1/2 in the bulk, 4 positive and 2 negative cusps at the I-N interface. The color in LC-PolScope textures of the tactoids in Fig. 2.6 is gradually changing from the reddish to yellow as the temperature is lowered and the area occupied by the N phase increases. It shows that the N domains at higher temperatures have a higher concentration of DSCG (and thus higher birefringence) than their counterparts at the lower temperatures.[111] 34 2.5. Shape of Positive N Tactoids with Two Cusps The shape of negative and positive tactoids is determined by balance of the elastic energy of the director in the N phase and by the I-N interfacial tension energy. The problem is mathematically challenged and can be solved analytically only for certain simplified situations, see, for example, Refs. [117-120] and references therein. One of those is a special case of crystals, in which the elastic bulk forces are infinitely large as compared to the anisotropic surface forces.[121] In such a case, the crystal shape is described by the classic Wulff construction based on the angular dependence of . The Wulff construction has been applied to the elastically “rigid” and uniform N drops with nˆ const . As discussed by Herring,[122, 123] if the angular dependence on the angle between nˆ and υˆ is pronounced (but remains a smooth function of with continuous derivatives), the equilibrium shape should be bounded by a number of smoothly curved (not flat) surfaces which intersect in sharp edges. Herring [122] cited the observations by Zocher of nematic tactoids with two sharp cusps [124] as an experimental confirmation of Wulff construction. The only experimental case with unperturbed director, nˆ const , in the interior of tactoids was reported recently by Puech et al [107] for a water dispersion of carbon nanotubes. The inverse Wulff construction led to the estimate w 4 . Kaznacheev et al [108, 109] found that for vanadium pentoxide dispersion in water, surface anisotropy is even stronger, w ~10100 . In our case, the director field is clearly distorted within the 2c tactoids, being tangential to the I-N interface. We are not aware of any analytical results for such 35 geometry. Recently, van der Schoot et al [125] performed numerical simulations of the shape of positive 2c tactoids taking into account the anisotropic surface tension in the form aa02c o s2 and internal elasticity in one-constant approximation with the Frank 2 modulus K . In our notations, 0 1cosw , 002aa and 02wa 2 . The shapes and director textures within the tactoids were simulated in Ref. [125] for various 2aA 2aA values of 2 and 0 , where A is the surface of the 2D tactoid. K π K π Unfortunately, the upper limit of was only 100, which limited the comparison of our experimental shapes with those simulated in Ref. [125], to very small tactoids. For a relatively small tactoid in Fig. 2.5, of a surface area A 200 μm2 , we measure the ratio lu/ of the long axis (length l ) to the short axis (width u ) to be around 1 . 3 0 . 1 and the cusp angle to be N 1.050.05 . The radius of curvature of the I-N interface at the cusps is less than the resolving power of our microscope, thus smaller than 0 . 5 μm . Comparison to the simulation results in figure 9 of Ref. [125] shows that the experimentally observed N tactoids correspond to the so-called “III regime” of Ref. [125]. In this regime, nˆ is always parallel to the I-N interface, while the shape is modestly anisotropic, 2 , and has two pronounced cusps with boojums. Comparison with figure 8 in Ref. [125] leads to a rough estimate / ~ 0.85 0.1, which suggests a rather strong anisotropy of the interfacial tension, on the order of w ~10 . 36 2.6. Late Stages of I-to-N Transition: I Tactoids as Disclination Cores FIG. 2.7. (a) LC-PolScope textures of a round I domain associated with a m 1 director field around it; (b, c) the I tactoid shrinks as T is reduced and then (d) splits into two disclinations of m 1/ 2 . (e) T dependence of characteristic size A / π and area A of the I tactoid. (f, g) Retardance profile of the I tactoid shown in parts (b, c), respectively. (h) Retardance profile of the two disclinations in part (d) separating from each other. At the late stages of the I-to-N transition, when many large domains coalesce, they occasionally trap I islands around which the director rotates by 2π or 2π , which corresponds to integer strength m 1 (Fig. 2.7) and m 1 (Fig. 2.8) disclinations. Topologically, these are allowed in the 2D case. When T is reduced and the I islands at the core shrinks, the integer disclinations always split into pairs of semi-integer disclinations, either m 1/ 2 (Fig. 2.7), or m 1/ 2 (Fig. 2.8). In 3D LCs, the integer 37 disclinations do not show such a split, as nˆ simply realigns parallel to the defect’s axis (“escape into the third dimension”). It is only for samples thinner than 0 . 5 μm that one can suppress the escape by a very strong tangential anchoring of nˆ .[126] In our case of LCLC textures, the escape is apparently suppressed at higher thicknesses as the integer cores split into pairs of semi-integer disclinations. The effect indicates that the director remains mostly parallel to the ( ,xy ) plane. FIG. 2.8. (a) LC-PolScope textures of a 4c negative tactoid associated with a m 1 director field around it; (b, c) the I tactoid shrinks as the temperature is reduced and then (d) splits into two disclinations of m 1/ 2 . (e) T dependence of characteristic size A / π and area A of the 4c negative tactoid. (f, g) Retardance profile of the 4c negative tactoid shown in parts (b, c), respectively. (h) Retardance profile of the two disclinations in part (d) separating from each other. 38 The elastic energy of the disclination with the director perpendicular to its axis 2 2 grows as m : fmKRrfeccπ ln/ , where K is the average value of the Frank splay and bend elastic constants, R is the size of the system or the distance to the closest disclination, rc and fc are the radius and the energy of the core, respectively. The elastic energy ~ m2 thus favors splitting of the integer disclinations into pairs of semi-integer defects. On the other hand, the interfacial energy of the I-N interface should favor the m 1 structure with a single isotropic core. Consider the effect in a greater detail. The energy of m 1 disclination (per unit length, assuming nˆ is perpendicular to the axis) is comprised of the elastic energy fKRre,11 π ln/ of the director, the excess 2 energy of the I core, fffrcond,1IN1 π , where fI and fN are the free energy densities of the I and N phases, respectively, and the energy of the I-N interface frIN,110 2π . Minimization of ffff1e,1cond,1IN,1 with respect to r1 yields 1 2 rKff10IN0 2 . For the split pair of disclinations m 1/ 2 , 2 ffIN π separated by a distance L , the elastic energy is fKRrKLre,pair1/21/2π ln/ln/ 2 , 2 2 while fffrcond,pairIN1/22 π and fIN,pair2π r 1/2 0 2 w . The latter estimate accounts for a change of director orientation from tangential to normal when one circumnavigates the defect core shown in Fig. 2.3a. To estimate at which temperature the defect m 1 can split into the pair of m 1/ 2 defects, we assume that at the moment of 39 splitting, the areas of the cores are equal, i.e. rr1/21 /2. Then the difference in the free energies of the two configurations can be written as π KL ffrw1pair10 ln2 π 22 . (2.6) 2 2r1 The first term favors splitting while the second term stabilizes the single-core disclination. Assuming that the defect core follows the experimentally observed behavior A , we approximate its temperature dependence as ATTT2 IN r1 tanh1 , (2.7) 2π 22TTIN where TI and TN are the upper and lower boundaries of the biphasic region, respectively. FIG. 2.9. Temperature dependence of the free energy difference between the single-core m 1 disclination and a pair of m 1/ 2 disclinations. 40 Then equation (2.6) predicts that the quantity ff1 p a i r can change the sign from negative to positive as T is lowered. An example of ff1 pair vs. T behavior is shown in 52 Fig. 2.9 for the following parameters: K 6 p N ,[102] 2π 0 2210J/m w , A / π 5 μm , L 100 μm ; according to this dependency, the single core will be expected to split into two cores at about 3 0 . 3 Co . Similar consideration can be applied to m 1 disclination splitting into m 1/ 2 defects, Fig. 2.8. The qualitative features of the model above, such as relative stability of integer defects with a large isotropic core at high temperatures, their splitting into pairs at low temperatures and separation of the semi-integer defects of the same sign are clearly observed in the experiments, Figs. 2.7 and 2.8. For a quantitative comparison, much more information on the materials parameters needs to be gathered. Note that in the homogeneous N phase, we observe only semi-integer disclinations that we discuss in a greater detail below. 41 2.7. Homogeneous N; Cores of Semi-integer Disclinations FIG. 2.10. (a) LC-PolScope textures of m 1/ 2 disclination and a 1c negative tactoid at the core of the disclination, as a function of increasing temperature T . (b) Optical retardance across the core of defect shown in part (a) for 25.0 Co and 30.4 Co . (c) across the expanded I core of the defect in the biphasic region, corresponding to T 30.5Co . (d) T dependence of characteristic size and area of the 1c negative tactoid. Since the director anchoring is degenerate in the plane of glass plates, the N textures typically contain a certain number of semi-integer disclinations, Figs. 2.10a and 2.11a. Even in the deep N phase, well below the T range of the biphasic region, the cores of disclinations in LCLC show unusual and interesting features. 42 FIG. 2.11. (a) LC-PolScope textures of m 1/ 2 disclination and a 3c negative tactoid at the core of the disclination, as a function of increasing temperature T . (b) Optical retardance across the core of defect shown in part (a) for 25.0 Co and 30.6 Co . (c) across the expanded I core of the defect in the biphasic region, corresponding to T 30.7Co . (d) T dependence of characteristic size and area of the 3c negative tactoid. First, the cores are not circular and feature one (in case of m 1/ 2 ) or three (in case of m 1/ 2 ) cusp-like irregularities, located in the region where the director is forced to be parallel to the radius-vector emanating from the geometrical center of the defect. Second, the distance over which the structure shows a substantial change in the degree of orientational order, is macroscopic, on the order of 10 μm . In Figs. 2.10b,c, we show the optical retardance nndeo profile measured across the disclination m 1/ 2 core. Far away from the core, the retardance is practically constant. If the temperature is raised, the far-field retardance slightly decreases, as the LCLC aggregates 43 become shorter so that birefringence nneo and scalar order parameter S are reduced. Within the core region of the linear size of about 10 μm , drops sharply to zero, being practically a linear function of the distance to the center at which r 00 . 2.8. N-to-I Transition: I Tactoids and Multiply Connected Tactoids FIG. 2.12. (a) LC-PolScope textures of a 2c negative tactoid nucleated in the uniform N part of texture. (b) Temperature dependence of characteristic size and area of the 2c negative tactoid. (c) Optical retardance before and after the appearance of the 2c tactoid. 44 When T is increased to some critical value, the cores of disclinations start to expand dramatically, see Fig. 2.10 for m 1/ 2 and Fig. 2.11 for m 1/ 2 . To characterize the temperature behavior of the cores, we use the normalized square root A / π of the isotropic (black in LC-PolScope textures) area A as a measure of core size. As seen from data in Figs. 2.10d and 2.11d, the T dependence of the core size for m 1/ 2 defects is close to ATTconst c , where Tc is some critical point at which the core starts its expansion; it slightly varies from defect to defect (even if they are of the same charge), indicating an influence of local effects such as surface irregularities. Defect-free uniform portions of the N phase also nucleate I islands, in the shape of negative tactoids with two cusps, Fig. 2.12. The T dependence of the negative 2c tactoid size, also measured as A / π , Fig. 2.12b, is very similar to the behavior observed for the I disclination cores, Figs. 2.10d and 2.11d. The T dependence of the core size has been analyzed by Mottram and Sluckin [127] for m 1/ 2 , under an assumption that the core represents a round island of an I phase. The disclination energy is then comprised of the elastic energy of the director distortions outside the core, surface energy of the I-N interface 2π 01/2r , and the 22 condensation energy fI f N π r 1/2 S I S N T NI T r 1/2 , where SI and SN are the entropy densities of the I and N phase, respectively. The core size was shown to grow with T , following the parametric dependence 0 K T r1/2 T NI 2 (2.8) SI S N r 1/28 S I S N r 1/2 45 22 with rT 1/2 /0 and rT 1/2 /0. In the case of the biphasic region of LCLC, the core size of disclinations does not follow this model, since experimentally, r1/2 const T Tc , so that rT 1/2 /0 but 22 rT1/2 /0 . The distinctive behavior of the core defects in LCLCs is brought about by the broad T range of I-N coexistence in which the total area (volume) of each of the two phases shows an approximately linear dependence on T , Fig. 2.1d. The total energy of a singular m 1/ 2 disclination can be approximated as 2 ffffKR1/2, 1/2cond,er rrffr 1/2IN, 1/21/21/2 IN1/2 ln / π 2π 1/2 , (2.9) 4 1 2 where the equilibrium core size rKff1/21/2IN1/2 /2 is ffIN determined by the anisotropic surface energy 1/20 ,,wm that is the function of the I surface tension, surface anchoring strength w and the strength of disclination. The negative tactoids contain negative cusps, i.e. regions of the I phase protruding into the N phase at the points where the director pattern of the disclination forces the director to become perpendicular to the I-N interface. In principle, the LC can simply reduce its scalar order parameter in these regions to reduce the energy of director distortions or realign the director along the normal to the bounding plates, Figs. 2.10 and 2.11. As already indicated in the discussion of the splitting of integer disclination, the second mechanism is less likely. In the N regions around the negative cusps, one finds negative boojums with m 1/ 2 / 2π . Similarly to the conservation law (2.4), one can write 46 down a conservation law for the negative tactoid surrounded by the director field that corresponds to a disclination with the strength mout : c i mmi out 1, (2.10) i 2π which leads to the relationship between the number of positive and negative cusps and mout ccm 21 out . (2.11) For a uniform far field, mout 0 , there are two more negative cusps than the positive ones, for mout 1/2 , there should be at least three negative cusps, etc. Comparison of Figs. 2.10, 2.11 and 2.12 demonstrates that the 1c, 3c and 2c isotropic tactoids occur in a close proximity to each other on the temperature scale, but still at somewhat different values of Tc . The cores of the disclinations are not necessarily the ones that start to expand first; sometimes the 2c isotropic tactoids nucleate at temperatures lower than the temperature at which the disclination core already present in the system starts to expand. The overall expansion of the isotropic tactoids with temperature follows the temperature behavior of the isotropic phase fraction shown in Fig. 2.1d. The dynamics of the phase transition can be used to create I-N-I tactoids that are non-simply connected. These represent N tactoids that contain large I tactoids inside them, Fig. 2.13; “large” here means larger than the disclination core in a homogeneous N phase. 47 A practical approach to create I-N-I tactoids is to nucleate first N tactoids by cooling the system from the I phase and then rapidly heat it up, remaining within the biphasic region. FIG. 2.13. LC-PolScope textures of I-N-I tactoid-in-tactoid multiply connected forms with different number of the isotropic inclusions: (a) I 1, cc2 , (b) I 1, cc3 (c) I 2 , c 2 , c 0 and (d) I 6, c 12, c 2 . The conservation law for these more complicated geometries in Fig. 2.13 is written as a generalization of equations (2.4) and (2.10): 48 cn cn i 1 mmIij 1 or kmIij 1 , (2.12) ij2π 2 ij where c is the number of boojums at all the I-N interfaces, mi is their individual strength, i are the cusp angles corresponding to the i -th boojum, n is the number of disclinations in the N interior of the I-N-I tactoid, with topological strengths mj , and I is the number of I tactoids inside the N tactoid. Similarly to the previously described cases of simply- connected N and I tactoids, the relationship (2.12) can be rewritten in terms of the excess number of positive cusps c over the negative ones c residing at all the I-N interfaces of the multiply connected I-N-I tactoid: n ccIm 21 j . (2.13) j As an illustration, Fig. 2.13a shows two I - N - I tactoids with I 1 and two different realizations of the condition cc0 . Fig. 2.13b illustrates the case of cc3. Fig. 2.13c shows an I - N - I tactoid with two I regions. In this particular case, there are no cusps on two round I-N interfaces; the only cusps are two negative cusps at the interior 2c I tactoid. Finally, Fig. 2.13d shows six 2c I tactoids inside one N tactoid, cc 10 . 49 2.9. Shape of N and I Tactoids in Frozen Director Field As already indicated, the shape of the positive and negative tactoids is determined by the balance of the anisotropic surface tension and nematic elasticity of the region either inside the positive tactoid or outside the negative tactoid. There is no analytical solution of this problem. On the other hand, the LCLC tactoids observed in this work show some general features such as multiple cusps, the number of which is determined by the topological structure of associated disclinations. Therefore, it is of interest to describe these forms using the simplest possible approximation, namely, by assuming that the director field is frozen in its distorted state, as determined by the disclination structures written in polar coordinates (r , ) as nxycos m const , n sin m const ; examples of different m ’s are shown in Figs. 2.3a-d. This approximation is equivalent to an assumption of an infinitely large elastic constants of the N phase. With the director field frozen, the shape of the I-N boundary of the tactoids is determined by the angular 2 dependence of the surface tension that we take in the form 0 1cosw . We place the geometrical center of the tactoid (positive or negative) at the center of disclination. The angular dependence of the surface energy at the I-N interface drawn in the frozen director field then becomes dependent on the disclination configuration ()m and leads 2 0 1cos1constwm , (2.14) 50 where the constant in the argument specifies the overall orientation of the director pattern; it can be put equal 0. Applying the principle of Wulff construction of equilibrium shapes of perfect crystals [121, 128-132] to the disclination-specific expression (2.14) for the anisotropic interfacial tension, we construct the equilibrium shapes of negative and positive 2D tactoids under the condition of a constant surface area and a frozen director field. The principle is illustrated in Fig. 2.14. FIG. 2.14. Scheme of Wulff construction of an equilibrium shape of a tactoid of a constant area from the polar plot of the surface tension. 51 From some origin O , we draw a line segment OA along the direction , of a length . Repeating for all angles , we obtain the polar plot of anisotropic tension . In each point A of , raise the normal to the segment OA . The equilibrium shape is the pedal of the polar plot , i.e., the convex envelope of all these normal. The coordinates xy, of the shape curve are expressed in terms of the polar angle between the normal to the I-N interface and the x -axis and the radius-vector r O A . As easy to see from Fig. 2.14, r cos . By taking the derivative with respect to , r sin ' , one obtains two equations that provide a parametric description of the tactoid’s shape: r 22' and arctan'/ . In the Cartesian coordinates, the shape is parametrized as x cos ' sin ; (2.15) y sin ' cos . Fig. 2.15a shows different polar plots of surface potential with w 2 for different director configurations, shown in Figs. 2.3a-d, as well as for the uniform director field, m 0. The corresponding shapes of tactoids are shown in Fig. 2.15b. These shapes should be understood as the shape separating either the I interior from the N exterior (negative tactoids) or the N interior from the I exterior (positive tactoids). 52 FIG. 2.15. (a) Polar plots of anisotropic surface tension, equation (2.14), corresponding to the frozen director fields with different topological charges m shown in Figs. 2.3a-d. (b) The corresponding equilibrium shapes of the negative and positive tactoids, equations (2.15). The physically relevant parts of the shape do not include the triangular “ears” attached to the cusp points. Three configurations, namely, m 1;1/ 2; 0 for a chosen value of surface anisotropy w 2 show shape contour that crosses itself, Fig. 2.15b. These crossings are associated with zeroes of the radius of curvature of the envelope, which is Rrr''''222 .[121, 130-132] Whenever '' 0 , the resulting shape is rounded, with the I-N interface showing all possible orientations in the plane. If '' 0 , the pedal develops cusps of angular discontinuity: some of the orientations of the I-N interface are missing from the equilibrium shape. The criterion "0 for missing 53 orientations for a tactoid associated with the director field of strength m and the surface tension dependency given by equation (2.15), reads 22 1141cos1210wmmmw2 . (2.16) For m 1/ 2,1, this condition is never fulfilled and the shape of the tactoids is circular. Interestingly, as the surface anchoring increases, the core of the m 1/ 2 disclination shifts towards the axis at which the director is normal to the I-N interface, see Fig. 2.15b; in this particular model, however, the core remains circular. For all other director configurations, the center of the core coincides with the center of the director field. FIG. 2.16. Illustration of how the surface anchoring shapes up the (a) positive N and (b) negative I tactoids with a frozen director of strength m 1/ 2 . 54 For m 1/ 2 configurations, inequality (2.16) predicts '' 0 when w 2 / 7 . Fig. 2.16 shows a progressive change in the shape of 3c tactoid with m 1/ 2 as the surface anisotropy increases from the values below the critical value w1/2 2 / 7 ( w 0 .1 ; 0 . 2 5 ) to the values above w1/2 ( w 0 . 4 ; 1). When ww 1 /2 , the I-N interface is round with all possible orientations. When ww 1 /2 , some orientations are missing. The geometrical indicator of missing orientations are triangular “ears” attached to the cusps in which nˆ is perpendicular to the interface. These ears do not correspond to physical portions of the shape. Our analysis is equally applicable to the positive 3c tactoids shown in Fig. 2.16a and to the negative 3c tactoids shown in Fig. 2.16b. For m 1, the condition of missing orientations of the I-N interface reads ww1 1/ 7 and for a classic 2c tactoid with m 0, ww0 1. 55 2.10. Conclusions In this chapter, we illustrate how the balance of anisotropic surface energy and internal elasticity determine complex developments of tactoidal forms and topological defects emerging in the dynamic phase transitions between the isotropic (I) and nematic (N) phases. In the case of a thermotropic liquid crystal, the equilibrium shape of new phase is close to spherical (circle in 2D) because the surface energy is more dominant than the elastic energy of internal distortions. The situation is very different in the lyotropic chromonic liquid crystal DSCG that was studied in details for a 2D geometry of confinement. The nuclei of new phase appear as strongly anisometric islands with pronounced cusps at which the I-N interface shows a singular behavior and the director forms surface defects-boojums. The reason for the existence of defects is the anisotropy of interfacial tension, i.e., surface anchoring of the director. The growing nuclei of the N phase merge with each other and often create bulk defects-disclinations of semi-integer or even integer strength in a process similar to the Kibble mechanism. The presence of interior disclinations influences the number of cusps featured by the closed I-N interface. We derive conservation laws that relate the number of positive and negative cusps c to the topological strength m of the defects inside and outside the simply-connected I and N tactoids. Similar relationships are established for the experimentally observed multiply connected tactoids. 56 The integer-strength disclinations that might exist in the biphasic region, split into pairs of semi-integer disclinations upon approaching the homogeneous N phase. In the homogeneous N phase, well below the temperatures of biphasic region, only semi-integer disclinations survive. The cores of these disclinations are of an extraordinary large radius (tens of microns) and of a non-circular shape. The N-to-I transition features isotropic 2c tactoids nucleating in the uniform parts of the texture and tactoids with one or three cusps nucleating at the cores of m 1/ 2 and m 1/ 2 disclinations, respectively. The size of these isotropic cores r1 /2 increases with temperature T , approximately as rTT1/2 const c , which is consistent with the overall temperature dependence of the volume fractions of coexisting I and N phases. Finally, we illustrate how the orientational anisotropy of interfacial tension leads to the formation of tactoidal shapes with cusps, by applying a generalized Wulff construction to the topologically nontrivial frozen director fields. 57 CHAPTER 3 DOMAIN WALLS AND ANCHORING TRANSITIONS MIMICKING NEMATIC BIAXIALITY IN THE THERMOTROPIC OXADIAZOLE BENT-CORE LIQUID CRYSTAL C7 3.1. Introduction The search for the biaxial nematic ( Nb ) liquid crystals (LCs) represents an active and fascinating area of studies with arguments presented both in favor [29, 36, 43, 46-48, 52-58, 62, 63, 69, 71, 85-88, 90, 91, 133, 134] and against [40, 41, 59, 60, 66-68, 135-138] their existence. Among the thermotropic biaxial candidates, the most studied material is the oxadiazole bent-core material 4,4’(1,3,4-oxadiazole-2,5-diyl) di-p-heptylbenzoate derived from 2,5-bis-(p-hydroxyphenyl)-1,3,4-oxadiazole, abbreviated ODBP-Ph-C7, or simply C7 (Figs. 3.1a,b). The Nb phase of C7 has been suggested by X-ray diffraction (XRD),[47, 55] NMR,[46] and electro-optical studies.[43, 56-58] These suggestions were challenged [59, 60, 135-137] on the grounds that the observed features represent a mimicking behavior of the uniaxial nematic ( Nu ) phase rather than a true biaxial order in the bulk of the specimen. For example, the Nu phase of C7 might mimic a biaxial behavior through the appearance of smectic cybotactic groups,[136, 137] and surface anchoring 58 transitions in which the uniaxial director nˆ (depicting the average direction of long molecular axes) tilts as the sample’s temperature T is varied.[59, 60] Recently, a new evidence has been presented in favor of the Nb phase of C7, based on the polarizing optical microscopy (POM) textures exhibiting the so-called “secondary disclinations” (SDs).[61] The POM showed a secondary Schlieren texture growing into a Nu texture on lowering the temperature and disappearing on heating.[61] The appearance of SDs was attributed to the N -ub N phase transition. The SDs were associated with disclinations in the orientational order of the short molecular axes, i.e., with the secondary director (different from nˆ ) of the Nb phase.[61] In this chapter, we apply a set of imaging techniques, namely, POM, LC-PolScope, and fluorescence polarizing optical microscopy (FCPM), to explore SDs in C7 and their temperature and electric field-induced behavior. Our studies demonstrate that the SDs are caused by a surface anchoring transition, i.e., by realignment of the uniaxial director nˆ from being parallel to the bounding plates to being tilted. The SDs thus represent domain walls (DWs) that separates regions of tilted director with different azimuthal direction. The anchoring transition is associated with the balance of two different mechanisms responsible for the alignment of C7: alignment by the polyimide substrate and by the electric field within the surface electric double layer formed by ionic impurities. The findings demonstrate a uniaxial nature of the nematic order in C7 and are thus consistent with the previous claims of uniaxial character of C7.[59, 60] The DWs observed in C7 are similar to the DWs observed in another uniaxial nematic LC, with the H-shaped molecules.[139] 59 3.2. Materials and Techniques FIG. 3.1. (a) Chemical structure and (b) space filling model of the oxadiazole bent-core liquid crystal C7. (c) Birefringence n of C7 as a function of the relative temperature tTTNI . Figs. 3.1a,b show the chemical structure of the oxadiazole bent-core LC, C7. The phase diagram confirmed by a POM upon cooling is Cr(148o C)Sm(173 o C) N(222 o C)I where Cr, Sm, N, and I are crystal, smectic, nematic, and isotropic phase, respectively; the phase diagram is consistent with the previous studies,[46, 47, 59] if one does not discriminate between the two nematic ( Nu and Nb ) phases. The cells were assembled from flat glass substrates with alignment layers. The cell thickness d was set by glass spacers mixed with UV glue NOA 65 (Norland Products, 60 INC.) that was also used to seal the cells. Because C7 shows signs of aging at elevated temperatures in presence of oxygen,[59] we performed the experiments with freshly prepared cells within 5 hours or less. In order to prevent possible surface memory effect,[59] we injected C7 into the cells at the temperatures above the I-N transition temperature TNI and carried out the experiments within the temperature range of the N phase. The temperature was controlled by a LTS420 hot stage with a T95-HS controller (both Linkam Instruments), with 0.01 Co accuracy. The rate of temperature change 0.4 C/mo in was deliberately slow to minimize the flows caused by thermal expansions that might represent yet another facet of a uniaxial behavior that mimics the behavior of Nb .[138, 140, 141] In the analysis of optical textures, it is important to know the temperature dependence of birefringence n . The latter is presented in Fig. 3.1c as a function of the relative temperature tTTNI ; the dependence was measured in planar cells of thickness d 1.1μm with polyimide PI2555 (HD Microsystems) serving as planar alignment layers. 61 3.3. Alignment, Anchoring Transition and Domain Walls To re-enact the appearance of SDs, we fabricated cells with two polyimide aligning agents, SE1211 and SE5661 (both supplied by Nissan Chemical Industries) deposited onto the indium-tin-oxide (ITO) electrodes at glass substrates. The polyimide layers were spin- coated from solutions and then baked but not rubbed. 3.3.1. Polarizing Optical Microscopy In the SE1211 cells, one observes the classic Schlieren textures characteristic of the uniaxial order in the entire temperature range of the N phase, Fig. 3.2.[142] The Schlieren textures feature point defect-boojums (with four brushes of extinction emerging from their centers) and vertical disclinations of semi-integer strength (with two dark brushes emanating from their cores), Fig. 3.2.[143] FIG. 3.2. Schlieren textures of C7 in the SE1211 cell ( d 4.5 μm ) at (a) t 1.0, (b) 20.0 , and (c) 45.0o C observed under a POM. A and P are crossed analyzer and polarizer directions, respectively. 62 FIG. 3.3. Temperature dependence of POM textures in the SE5661 cells of different thickness: (a, b) d 13 μm ( t* 5.5o C ) and (d, f) 4.5 μm ( t* 0.9o C ). The left column shows standard Schlieren textures at elevated temperatures tt *, and the right column shows well-defined SDs at tt *. Red arrows point towards the SDs. In the SE5661 cells, one observes two different textures as the temperature is varied. Above some critical temperature t *, the textures are of the classic Nu Schlieren type, Figs. 3.3a,c. Below t *, secondary textures with SDs form, Figs. 3.3b,d. The SDs 63 show a reversible behavior, disappearing once the temperature is raised above t * and re- appearing again when the sample is cooled below t * . The transition temperature t * increases as the cell thickness d decreases, Fig. 3.3; t* 5.5o C when d 13μm , and t* 0.9 C o when d 4 . 5μm . In very thin cell, d 1 . 1μm , SDs appear simultaneously with the formation of the N phase; t* 0 C o . These thin cells are convenient for the quantitative analysis of the textures through LC-PolScope observations that map the pattern of optical retardance of the cell, Fig. 3.4, as described in the next section. 3.3.2. Maps of Retardance and Director Field by LC-Polscope Observation The LC-PolScope (Abrio Imaging System) allows one to map the optical retardance ndeff of the studied cells as the function of in-plane coordinates xy, , where nnoe nneffo (3.1) 2222 nneosincos is the effective birefringence determined by the ordinary no and extraordinary ne refractive indices; is the angle between nˆ and the bounding plate. When 0 , the optical retardance reaches its maximum, max nd , since in this case neff n n e n o . In addition to mapping xy, , LC-PolScope also shows the local projection of the optic axis nˆ onto the cell’s plane, Fig. 3.4.[113, 144, 145] 64 FIG. 3.4. Retardation maps of SD textures in the SE5661 cell ( d 1 . 1μm ) at (a) tt 0.4 (*0C) o , (b) 2 4 . 8 , (c) 3 6 . 8 , and (d) 46.8Co ; ticks show the projection of nˆ ; the central parts of the SDs feature horizontally aligned nˆ and thus yield a higher value of . The SDs separate regions with tilted nˆ and thus reduced . (e) Variation of across the SD along the pathway shown by dotted lines in (a-d). Figs. 3.4a-d show the temperature dependence of LC-PolScope textures in the SE5661 cell of d 1.1μm . The SDs exist in the entire temperature range of the N phase o (t*0C ), manifesting themselves as bands of high retardance Γmax , separating regions of lower < Γmax . As t decreases, measured in the surrounding domains decreases, Fig. 3.4e. Such a behavior is consistent with the idea that the SDs are domain walls (DWs) separating regions with the titled nˆ , 0. At the center of the DW, nˆ has to be parallel to the plates, 0, in order to connect the two regions with different azimuthal directions 65 of the tilt 0, Fig. 3.5. Fig. 3.5 shows the details of director field in the DWs that can be associated with splay-bend (Fig. 3.5b) and twist (Fig. 3.5c) deformations, depending on how the DW is oriented with respect to the projection of nˆ onto the cell’s substrates. FIG. 3.5. Retardation map with the director field projected on the plane of the SE5661 cell ( d 1.1μm ) at tt 46.8o C ( *) . Director configurations of DWs associated with (b) bend-splay and (c) twist deformations. Outside the DW, the director tilt increases as the temperature is lowered below t *, thus both neff and decrease, according to Eq. (3.1), see Fig. 3.4e. At the center of DW, nˆ remains parallel to the bounding plates before and after the anchoring transition. 66 One would expect the measured retardance to reach its maximum value max nd there. Although the peaks at the DW, Fig. 3.4e, limited lateral resolution of the LC-PolScope (being of the same order as the width of the DWs, about 1μm ) does not allow one to map the within the DWs accurately; the value of at the center of the DWs is somewhat decreased as compared to the expected value max nd because of the surrounding regions with the tilted director. However, this problem of limited in-plane resolution is eliminated in the experiments with the electric field, as described below. 3.3.3. Behavior of Textures in the Electric Field In order to further confirm the model of DWs, we performed an experiment in which an alternating current (AC) field E (sinusoidal wave of frequency f 1kHz ) was applied across the cell, between two ITO electrodes on the glass plates, Fig. 3.6. C7 has a negative anisotropy of dielectric permittivity: (2C)17.1 t o at f 1kHz ,[59] which means that the field realigns nˆ perpendicularly to itself. In absence of E , the retardation outside the DWs is very small, Fig. 3.6a. An applied E increases in these homogeneous regions, Fig. 3.6b, up to the point (achieved at E 5.5 V/rms μm) when of the entire sample is uniformly high and the DWs are no longer distinguishable (Figs. 3.6c,d). This behavior is consistent with the model of DWs separating differently tilted domains. Namely, the electric field forces the tilted nˆ to realign everywhere parallel to the bounding plates, thus eliminating the DWs. 67 FIG. 3.6. Retardation maps of the SE5661 cell ( d 1.1μm ) at tt 36.8Co (*) under the action of the vertical electric field; (a) E 0 , (b) 3.6, and (c) 5.5 V/rms μm . (d) Change of across a DW (along dotted lines) as a function of E . The tilt angle achieved as the result of surface anchoring transition below t * can be estimated from the data on t and Eq. (3.1), by using the values ne n 2 n / 3 and no n n /3 determined by the measured n (Fig. 3.1c) and the approximate value of the averaged refractive index n 1.60 .[59, 146] At t 36.8o C , with the estimated 68 o o ne 1 . 7 5 and no 1 . 5 3, one finds that ( 0E ) 5 0 (Fig. 3.6a), (3.6V/E μm)28 (Fig. 3.6b), and (E 5.5 V/μm) 1o (Fig. 3.6c) in the SE5661 cell of d 1.1μm . Figs. 3.6c,d show that of the sample in the electric field of amplitude 5 . 5 V /rms μm is about 2 4 3 n m . Since d 1 . 1μm , this result implies that the quantity / 0d . 2 2 is in an excellent agreement with the value of n measured independently in the planar well aligned field-free cells at t 3 6 . 8 Co , Fig. 3.1c. On the basis of the POM and LC-PolScope studies, we conclude that the SDs are not associated with the appearance of the secondary director and represent DWs that emerge during a temperature-induced surface anchoring transition in the uniaxial nematic cells with certain types of surface aligning layers (such as SE5661). Below t *, the director nˆ deviates from the tangential alignment; the regions with different azimuthal directions of the tilt are bridged by the DWs. The tilted orientation of nˆ below t * is also confirmed by the fact that in POM and LC-PolScope textures, the DWs either join two half integer disclinations or form closed loops (Fig. 3.3). Below t *, the projection of the tiled nˆ onto the plane of the cell is a vector (as opposed to the director), thus the half-integer disclinations no longer exist as isolated defects and must be associated with the DWs.[2] The fact that t * depends on d (Figs. 3.3 and 3.4) provides another evidence that the DWs are associated with the effects of confinement rather than with the Nub -N phase transition in the bulk; the thermodynamic stability of the biaxial nematic Nb should not depend on the type of surface alignment and on variations of d in the range of micrometers. 69 3.3.4. Fluorescence Confocal Polarizing Microscopy of the Surface Anchoring Transition FIG. 3.7. (a) FCPM intensity IFCPM in the rubbed planar cell (PI2555) of C7 as a function of the angle and the temperature t . (b) POM texture of the SE5661 cell ( d 13 μm ) between crossed polarizers at tt 37.0Co (*) ; arrows indicate DWs. FCPM textures scanned in (c) the horizontal (,)xy plan at the squared region in b) and (d-f) the vertical (,)xz plan along dotted line in c) under the action of the vertical electric field; (d) E 0 , (e) 1.5, and (f) 4.6 Vrms /μm ; arrows indicate DWs. 70 For the direct demonstration of the anchoring transition and director tilt at tt *, the SE5661 cell was investigated using the 3D microscopy technique, so-called fluorescence confocal polarizing microscopy (FCPM).[147] C7 was doped with a small amount ( 0 . 0 1 %wt ) of the fluorescence dye N,N’-Bis(2,5-di-tert-butylphenyl)-3,4,9,10- perylenedicarboximide (BTBP, Sigma Aldrich). Planar (PI2555) cells were used to establish that the transition dipole dˆ of BTBP molecules is parallel to nˆ of C7 by measuring the azimuthal angular dependence of the fluorescence intensity IF C P M (Fig. 3.7a).[147, 148] IF C P M reaches its maximum when the angle between nˆ and the 4 polarization of probing light is zero; IFCPM cos .[147, 148] FCPM textures of PI2555 cells show that IFCPM does not change when the electric field is applied across the cell, confirming strict tangential alignment of nˆ . FCPM shows an evidence of an anchoring transition in the SE5661 cells, Figs. 3.7c- f. At t 37.0Co , the SE5661 cell of thickness d 13 μm viewed by a POM with two crossed polarizers shows the DWs (indicated by arrows in Fig. 3.7b). The same region of the sample was scanned in the FCPM mode, in the plane parallel to the bounding plates, Fig. 3.7c, and in the plane perpendicular to the bounding plates, Figs. 3.7d-f. The probing beam was circularly polarized in order to detect only the polar angle of the director tilt. In both the horizontal (,)xy scans (Fig. 3.7c) and the vertical (,)xz scans (Fig. 3.7d), the DWs feature a higher intensity of fluorescence as compared to that of the surrounding regions. This feature is consistent with the idea that nˆ in the center of DWs is parallel to the bounding plates and is tilted in the regions outside the DWs. Furthermore, the vertical 71 scans obtained in the presence of E (Figs. 3.7e,f) show that IF C P M increases as E is increased. This clearly demonstrates that the electric field realigns nˆ towards the horizontal planes (parallel to the bounding glass plates). 3.3.5. Thickness-dependent Anchoring Transition and Electric Double Layers of Ions The appearance of anchoring transition at t * for some aligning substrates and the fact that t * depends on the cell thickness can be explained by the aligning action of electric double layers formed by ionic impurities near the substrates. The electric double layers create local electric fields acting on the director near the surfaces.[149, 150] The surface anchoring potential for a tangentially anchored substrate is, in Rapini-Papoular approximation, WW sin2 , (3.2) where the positive definite anchoring coefficient WWW0 i 0 can be represented as a sum of an “intrinsic” coefficient W0 0 and an “electrostatic” Wi 0 contributions, caused by the dielectric torque of the local electric field on the dielectrically anisotropic LC. The latter was estimated by Barbero and Durand [149, 151] to be 2 D Wi 2 , (3.3) 20 where is surface density of charges, is the average dielectric permittivity of the LC, 0 is the electric constant, and D is the thickness of the double layers, also known as the 72 Debye screening length (see also Ref. [152] for the range of validity of Eq. (3.3)). The local electric field is perpendicular to the bounding surfaces. Therefore, since C7 is of a negative dielectric anisotropy, 0 , the local field tends to align the director parallel to the surface, so that the electrostatic coefficient Wi 0 facilitates tangential anchoring. When the intrinsic coefficient W0 ( 0 for SE5661) and the electrostatic coefficient Wi ( 0 ) are compensate each other, the contribution of quadratic term of surface potential (Eq. 3.2) becomes weaker while the effect of 4th order term becomes dominant.[104] As a result, the system can be of the local minima for the tilted state between tangential and homeotropic state. The intrinsic anchoring at polyimides such as SE5661 is typically homeotropic, W0 0 . This anchoring is a local property of the interface, and is thus thickness – independent. The electrostatic part Wi 0, however, is thickness dependent because the surface density of absorbed ions at the interface depends on the total volume of the LC, so that 0dd/2 D ; the value of 0 depends on availability of surface absorption sites and adsorption energy.[151] In LCs, D is relatively large, on the order of 0.1-1μm ,[153-155] and thus comparable to the range of cell thicknesses explored in this work. The electrostatic contribution to the surface anchoring is expected to reach its maximum 22 positive value WiD /2 0 in thick samples, d D . As the cells become 222 thinner, the anchoring coefficient decreases Wi00( ) D / 2 d / ( d 2 D ) , which implies that the anchoring strength responsible for tangential alignment is weaker in thinner cells. This behavior is in a qualitative agreement with our experiments, in which 73 the thicker cells show the widest temperature range of the stable tangential alignment, and the thin cells show a very narrow or non-existent region of tangential alignment. Table 3.1. Dependence of an anchoring transition temperature t * on the concentration c of TBAB in the SE5661 cells of d 1 . 1 and 4 . 5 μm . c w t0% c1.0 wt % c 2.9 wt % td*(4.5 μm) 0.9o C 31.1Co No Transition td*(1.1 μm) 0Co 11.0Co 24.0Co To further demonstrate the role of ions on the thickness-dependent anchoring transition, we performed the following two experiments. First, the transition temperature t * was measured as a function of concentration c of the salt tetrabutylammonium bromide (TBAB, Sigma Aldrich), added to C7. As shown in Table 3.1, t * drops significantly with the increase of c , as expected. At higher c , the electric field of the double layers becomes stronger and imposes a stronger tangential alignment force. For high salt concentration c 2.9 wt % and in thick cells, d 4.5 μm , there is no anchoring transition (and thus no DWs), as the tangential alignment persists in the entire range of the nematic phase. 74 The second experiment was designed to create an in-plane gradients of the ionic additives by applying the electric field in the plane of the cell. The SE5661 layer was spin- coated on a top glass substrate (no electrodes) and a bottom glass substrate with interdigitated ITO electrodes; the gap between electrodes was 1mm . Two substrates were rubbed in the anti-parallel fashion and subsequently were assembled with d 1 . 3μm ; the rubbing direction is perpendicular to an in-plane field. Figs. 3.8a-c show the retardance maps of the cell at tt46.5C(*)o for different values of the in-plane electric field. In absence of the field, there is no significant variation of , Figs. 3.8a, d. Once the in-plane direct current (DC) field is applied, measured in the cathode region increases, as compared to the rest of the cell, (Figs. 3.8b,d), and the higher value of on the cathode is maintained even after the field is removed (Figs. 3.8c,d). The in-plane DC field carries the negative ions to the cathode thus enhancing the tangential alignment, as evidenced by a higher value of . Interestingly, a clear trend of thickness–dependent anchoring with the tangential contribution becoming weaker in thin cells has been recently observed by Ataalla, Barbero, and Komitov for the LC, MLC-6608, with 0 in cells treated with SE1211.[156] The authors observed that the anchoring strength of homeotropic alignment increases as the cells become thinner, which is equivalent to the weakening of the electrostatically induced tangential anchoring Wi at small d . The effect is similar to the phenomenon of thickness dependent t * for SE5661 cells observed in our work 75 FIG. 3.8. Retardation maps of the rubbed SE5661 cell ( d 1.3μm ) at tt 46.5Co (*) under the action of the in-plane electric field; (a) E 0 , (b) 30 min after E 29.8 V/mm, and (c) 30 minutes after the field is off. (d) profile across an electrode (along red solid lines) in Figs. 3.8a-c. 76 3.3.6. Influence of Thermal Degradation of C7 on the Alignment It was reported that the C7 material is chemically unstable and experiences chemical degradation / decomposition when it is kept at the temperature of N phase ( 222C173CooT ) in an oxygen environment.[59] This is the reason why all our experiments were performed with fresh cells within 5 hours or less. To verify how the chemical degradation of C7 influences the alignment features, the director alignment in a SE5661 cell of d 4 . 5 μm was observed over time while keeping the temperature fixed at T 220Co . The fresh cells exhibit the phase diagram as in the previous studies [46, 47, 59], and show an anchoring transition from tangential (1st column in Fig. 3.9b) to tilted alignment with the emergence of DWs at t * (2nd column in Fig. 3.9b). In the cells kept at elevated temperature for 24 hours, both TNI and the temperature range of tangential alignment decrease. Moreover, one observes another textural transition, from a tilted to homeotropic alignment state, at a certain temperature t H , Figs. 3.9c. After a more prolonged exposure to the high temperature, 42-72 hours, the temperature range of tangential alignment expands, while TNI and the range of tilted alignment decrease, Fig. 3.9a. We also observed formation of air bubbles and crystal aggregates (red dotted circle in Fig. 3.9d) that did not melt even at T 270Co . Therefore, the surface alignment of C7 is strongly affected by degradation at elevated temperatures. 77 FIG. 3.9. (a) Alignment dependence on thermal degradation of C7 in the SE5661 cells of d 4.5 μm . Temperature dependence of POM textures (b) 0, (c) 24, (d) 72 hours after the cell was kept at T 220Co . The first column shows the POM textures of tangential alignment at tt *, the second column shows the POM textures of tilted alignment with DWs at ttt* H , the third column shows emerging homeotropic domains at ttH , and the forth column shows the cell entirely covered by homeotropic domains at tt42.4o C(<) H . Red dotted circle in (d) indicates the crystal aggregates. 78 3.4. Conclusions In this chapter, we demonstrate that the so-called secondary disclinations (SDs) observed in the nematic C7 represent domain walls (DWs) that occur in a uniaxial nematic phase as a result of the surface anchoring transition triggered by temperature changes. This transition is observed for a certain aligning material (SE5661) but not for other aligning materials (such as PI2555 and SE1211). In the cells with the SE5661 aligning layer, above some temperature t *, the director nˆ is parallel to the bounding plates. Below t *, the director tilts away from the substrates. Directors in different regions of the cell tilt into different azimuthal directions. Topologically, these regions have to be bridged by the DWs in which the director remains tangential, Fig. 3.5. The observed DWs are not associated with the biaxial nematic phase, as follows from the facts that (a) the DWs are observed only with some alignment layers and not with the others; (b) the temperature range of stability of DWs (its upper limit t *) depends on the thickness of cells; (c) the DWs can be removed by modest electric fields that realign the director from tilted orientation towards an orientation parallel to the bounding plates. The issue of temperature-induced anchoring transitions in bent core mesogens is by itself an interesting problem. These transitions have been observed in C7 and C12, [59, 60] as well in some other materials.[60, 66, 67] Very interesting is the fact that the transition temperature t * depends on the cell thickness. The natural reason for thickness- dependent anchoring transitions is the presence of ionic impurities in the samples that form electric double layers and thus create local electric fields acting on the director.[149, 150] 79 Since the dielectric anisotropy of C7 is negative, the vertical electric field of the double layers tends to align the director tangentially; this tendency competes with the perpendicular alignment caused by the polyimide layers. Additional experiments with variable concentration of added salts and in-plane gradients of the ions support the model of the anchoring transition as a balance of polyimide and electric double layers alignment tendencies. We also demonstrated that degradation of C7 at the temperatures corresponding to its nematic phase dramatically influence the surface anchoring of the material. We conclude that C7 represents a uniaxial nematic phase in the entire temperature region between the isotropic melt and the smectic phase. The conclusion confirms an earlier statement on the uniaxial nature of C7 based on the studies of topological defects [59] and a magneto-optical response.[60] The study demonstrates yet another facet by which a regular uniaxial nematic mimics the properties of a biaxial nematic phase, this time through thickness dependent anchoring transition from a tangential to tilted alignment of the director. 80 CHAPTER 4 SURFACE ALIGNMENT, ANCHORING TRANSITIONS, OPTICAL PROPERTIES, AND TOPOLOGICAL DEFECTS IN THE THERMOTROPIC NEMATIC PHASE OF AN ORGANO-SILOXANE TETRAPODES 4.1. Introduction In this chapter, we explore an organo-siloxane material (Fig. 4.1) with a molecular structure that resembles a tetrapode.[69-74, 86, 88] Originally,[69] the phase sequence for oo the tetrapode material was determined as N(37C)bu N(47C)I , where Nb , Nu , and I stands for biaxial nematic, uniaxial nematic and isotropic phase, respectively. Later on, Figueirinhas et al.[70] presented the phase sequence of a mixture with a nematic deuterated o o o probe as Tg(-30 C) N b (0 C) N u (47 C)I , here Tg is the glass transition temperature, based on extensive solid state NMR investigations, with the results being in line with those reported in Ref.[157]. The difference in the transition temperatures, compared to the pure sample[69] was attributed to the admixing of the probe, indicating that the stability of the Nb phase is strongly affected by external stimuli. A combination of XRD studies and fast field cycling experiments suggest that a local C2h symmetry of the material in the nematic state, supporting the view that local clustering may be crucial for the explanation of the formation of a biaxial nematic phase.[86] Polineni et al.[72] present an additional evidence 81 o of a N -ub N transition of the tetrapode material shown in Fig. 4.1 at 3 7 C , as originally reported.[69] In this chapter, we use electro-optical and optical microscopy (polarizing microscopy, conoscopy, FCPM) techniques to explore the nature of the nematic phase of the tetrapode material shown in Fig. 4.1. The samples represent (i) flat layers of thickness between 4 μm to 50 μm confined between two solid plates (ii) round capillaries of diameter 50 μm and 150 μm ; (iii) spherical or nearly spherical freely suspended droplets of diameter between 5 μm and 20 μm . The observed electro-optical, surface and topological features of these samples show that the material has a uniaxial order in the whole nematic temperature range studied, 25C46CooT . 4.2. Material and Techniques Fig. 4.1. Molecular structure of an organo-siloxane tetrapode. 82 Fig. 4.1 shows the chemical structure of the tetrapodic material. Four mesogens are connected to the siloxane core through four siloxane spacers. We confirmed the phase oo o transition as Tg (-27C)N(46C)I in the regime of cooling with the rate 0 . 1 C / m i n ; the transition temperatures agree well, within 1- 3 Co , with the previous studies,[70, 86] if one does not discriminate between the two versions of the N phase. The flat cells were assembled from parallel glass plates with transparent indium tin oxide (ITO) electrodes. For planar alignment, the ITO glass substrates were spin coated with polyimide PI2555 (HD Microsystems); the polymer was rubbed unidirectionally. For homeotropic alignment (the director nˆ is perpendicular to bounding plates), the glass plates were treated with a weak solution of lecithin in hexane. The cell thickness d was set by spacers mixed with UV-curable glue NOA 65 (Norland Products, INC.) that was also used for sealing the cells. The material was filled into the cells in the I phase. To prevent a possible memory effect, we performed most of the experiments for the temperatures above 25o C , which is slightly higher than the glass o transition temperature Tg 27C . The temperature T was controlled by a hot stage LTS350 with a controller TMS94 (both Linkam Instruments) with 0.01o C accuracy. A typical rate of temperature change was 0.1o C/min to minimize the effects caused by thermal expansion.[138, 140] Cooling was assisted by a circulation of liquid nitrogen. 83 Fig. 4.2. (a) Dielectric anisotropy ε and (b) crossover frequency fc as a function of T . In order to use the electric field as the means of director reorientation, we measured the dielectric properties of tetrapode with a precision LCR meter 4284A (Hewlett Packard) in the cell of thickness d 10 μm . We measured the dielectric permittivity for the directions parallel ( ) and perpendicular ( ) to the director nˆ in homeotropic and planar cells, respectively, and thus determined the dielectric anisotropy = . In the homeotropic cells, the director experiences a surface reorientation (an anchoring transition) when the temperature is lowered, as we discuss in a greater detail later. To reinforce the homeotropic alignment during the dielectric measurement of ε , the cells were kept in the magnetic field of 1.4 T , directed normally to the cell. It allowed us to keep nˆ oo perpendicular to the glass plates in the range of temperatures 20 CTTNI (46 C), where 84 TNI is the N-I transition temperature; the diamagnetic anisotropy of tetrapode material is positive. The dielectric anisotropy ε changes its sign depending on temperature and frequency, Fig. 4.2a. Fig. 4.2b shows the crossover frequency fc that separates the regions of different sign of ε , as the function of T being in line with those reported earlier.[72] The solid line is a tentative extrapolation of the dependency to lower temperatures. The condition ε >0 can be used to distinguish the surface anchoring transition of a uniaxial Nu phase from a hypothetical appearance of two optic axes if the material was a Nb phase. 4.3. Search of Biaxiality in the Cell with a Homeotropic Alignment Optical tests of homeotropic cells offer a straightforward approach to determine whether the phase is Nu or Nb of orthorhombic symmetry provided there are no surface induced transitional effects. The homeotropic Nu shows no birefringence for the orthoscopic transmission of light, regardless of its polarization. The cell is dark when viewed between two crossed polarizers. The homeotropic cell filled with the Nb phase of orthorhombic symmetry, however, should show in-plane birefringence, nxy=0 n y n x ; its polarizing optical microscopy (POM) textures can be bright, depending on the orientation of the secondary directors mˆ and ˆl with respect to the crossed polarizers. At relatively high temperatures, above a critical temperature T * , the tetrapode nematic in the homeotropic cells always show the standard homeotropic dark texture when 85 viewed between two crossed polarizers, since the optic axis nˆ is parallel to the direction of polarized light propagation. The value of T *varies in a wide range31oo CT* 44 C , depending on the cell thickness d ; T * decreases as d increases (Fig. 4.3). Below T *, the dark homeotropic texture (Fig. 4.4a) becomes birefringent (Figs. 4.4b-d) with brightness increasing when the temperature decreases. Fig. 4.3. Plot of the anchoring transition temperature T * as a function of cell thickness d . 86 Fig. 4.4. Transition from (a) dark uniaxial texture to (b-d) birefringent textures in the homeotropic cell ( d 4.5 μm ) at (a) T 45o C , (b) 40o C( T *) , (c) 35o C , and (d) 2 0 Co ; inset in (a) is a conoscopic image of the cell. Fig. 4.5. (a-d) Retardation maps and (f) local retardation as a function of temperature T in a homeotropic cell ( d 4.5 μm ) at (a) T 45o C , (b) 40o C( T *) , (c) 35Co , (d) 30Co , and (e) 20Co . 87 Using LC-Polscope (Abrio Imaging System), we mapped the optical retardance across the cells (Figs. 4.5a-e) and also measured the change of retardation (Fig. 4.5f) in a preselected location as the function of temperature for the homeotropic cell of thickness d 4 . 5 μm . The transition between the textures is reversible; the birefringent texture becomes dark when the temperature increases above T * . The very fact that the temperature T * depends on the cell thickness (Fig. 4.3), suggests that the transition at T * is not a bulk phase transition such as the Nu -to- Nb transition, but could be a feature associated with surface effects. The data below support this conjecture. There are two possible mechanisms for the transformation from the dark to the bright birefringent texture as the temperature changes: (i) the phase transition from Nu to Nb and (ii) an anchoring transition, i.e. surface-mediated realignment of nˆ in the Nu phase [59, 67] (a combinations of both is also possible, of course). Anchoring transitions caused by temperature changes are well documented in various nematic LCs [150, 158, 159] and occur whenever there are two competing tendencies in setting the direction of surface orientation of nˆ with different temperature dependencies. In the case of tetrapodes, these tendencies can be associated with the temperature evolution of molecular conformations, surface interactions and packing. As discussed in Ref. [160], strong electric and magnetic fields could affect the molecular packing and the symmetry of the mesophase made of complex molecules such as tetrapodes. What is important is that if nˆ aligns parallel to the applied electric field, there is a simple way to discriminate between the two mechanisms by applying the electric field normal to the substrates without inducing any 88 phase change.[66, 67] In the Nu case, the main director nˆ would be aligned along the field and the homeotropic cell would restore the dark POM texture, while Nb would remain birefringent because the secondary directors mˆ and ˆl lead to the in-plane anisotropy and birefringence. FIG. 4.6. Change of POM texture and conoscopic pattern (a, b, d, f, h) without and (c, e, g, i) with a vertical field in a homeotropic cell ( d 6.9μm ) at (a) T 45o C , (b, c) 40o C( T *) , (d, e) 20o C , (f, g) 0Co , and (h, i) 5Co ; scale bar is 100μm . (j) Transmittance vs. voltage curves for (b-i). 89 Fig. 4.6 shows the change of POM texture (Figs. 6a-i), conoscopic texture (inset in Figs. 4.6a-i), and transmittance (Fig. 4.6j) by applying the vertical electric field (sinusoidal wave) with a frequency below fc in a homeotropic cell ( d 6 . 9 μm ). To align the director nˆ parallel to the field, the frequency f should be below ~10 Hz at T 5Co , Fig. 4.2b. Note that for low frequencies, the measured dielectric anisotropy ε is influenced by finite electric conductivity; the condition ε >0 means that the combined action of the dielectric and electric current torques leads to the alignment of nˆ along the field. Another complication is that a high electric field might cause a dielectric breakdown.[161, 162] We indeed observed such a breakdown in our system for voltages higher than 200250V , which is close to or smaller than the voltage needed to realign the director into a homeotropic state when the temperature is very low, T 5Co . To avoid dielectric breakdown, we performed the electric field experiments only in the range 5oo C When TT * , the homeotropic POM texture (Fig. 4.6a) corresponds to a symmetric Malthese cross (inset in Fig. 4.6a) in the conoscopic view, indicating that nxy 0 and nˆ is aligned normal to the bounding plates. As the temperature decreases below T * , the POM texture becomes birefringent and the conoscopic cross becomes blurry (Figs. 4.6b,d,f,h and inset), indicating some misalignment. By applying a vertical field, however, we could always restore the dark POM texture and the symmetric Malthese cross (Figs. 4.6c,e,g,i and inset). Fig. 4.6j shows that the transmittance of a He-Ne laser beam (wave length 633nm ) passing through the nematic cell and two crossed 90 polarizers always reduces to zero when the field is increased. This result demonstrates that the transformation of textures below T * in the homeotropic cell is an anchoring transition rather than an N -ub N phase transition. An alternative explanation would be that the electric field used in the experiments suppresses the biaxial order. Such a possibility seems unusual, as the electric field is directed along the principal director. In contrast to the o results of Polineli et al.,[72] we do not observe an Nub -N transition neither at 3 7 C nor at any other temperature in the homeotropic cells within the range 5 Coo < < 4T 6 C . We note that the surface tilt of the director becomes rather strong as the temperature decreases below the anchoring transition point T * and thus at lower temperatures one needs higher electric fields to restore the uniaxial homeotropic configuration, Fig. 4.6. Fig. 4.7 presents a more detailed examination of the conoscopic patterns as the function of temperature (Fig. 4.7a) and field (Figs. 4.7b-d). The temperature induced modification of the conoscopic pattern is mostly in the shift of the center of the Malthese cross, which is consistent with the tilt in a Nu phase rather than with the appearance of Nb phase. The symmetric Malthese cross and the homeotropic uniaxial state of the cells are always restored at all the temperatures studied, if the applied electric field is of the proper frequency (to align nˆ parallel to the field) and amplitude (to overcome the surface anchoring forces that do not favour a homeotropic state below T *). In order to avoid a false biaxiality of the samples caused by thermal expansion,[138, 140] the rate of temperature change was kept low, 0.1o C/min . 91 FIG. 4.7. (a) Change of conoscopic texture as a function of temperature T in a homeotropic cell; no electric field; (b-d) restoration of the Malthese cross of the uniaxial homeotropic state at TT * by a vertical electric field ( d 6.9μm ) at (b) T 35Co , (b, c) 0Co , (d) 5Co . Red arrow in (a) indicates the shift direction of the Malthese cross. 92 The optical studies presented above present an argument against the existence of an Nb phase with orthorhombic symmetry. As discussed by Tschierske and Photinos,[163] in the biaxial nematic phase Nb of a lower monoclinic symmetry, the principal axes of the low-frequency dielectric tensor ( ) and the refractive index tensor ( n) might not be parallel to each other. In that is the case, “a biaxial nematic phase with monoclinic symmetry can under special conditions appear optically isotropic due to the deviation of direction of the optical axis from the direction of the tilt axis”.[163] In terms of the experimental situation described in Figs. 4.3-4.7, it means that the field-oriented state that appear dark (“isotropic”) when viewed between crossed polarizers “from above” (with light propagating parallel to the applied electric field) might still be an Nb phase of monoclinic symmetry. If this is the case, then optical observations with light propagating perpendicularly to the direction of the applied electric field would help to discriminate between such a monoclinic Nb state and a normal uniaxial Nu state. If the nematic is uniaxial, the direction of the applied electric field will be the axis of symmetry. If the nematic is biaxial of monoclinic symmetry, then the field direction would not be the symmetry direction of the optical response. The results of additional tests of a low- symmetry state are presented in Fig. 4.8 and discussed below. 93 FIG. 4.8. (a) Experimental setup to record the light transmittance as a function of cell rotation by around b axis and by around c axis, in the presence of in-plane field; The directions of polarizers for and rotation are indicated by solid and dotted lines, respectively. Intensity profile as a function of the angle (b) and (c) for different temperatures with an in-plane field in a planar cell ( d 20μm ); cartoons illustrate the verifiable geometries of monoclinic symmetry in which the low-frequency dielectric tensor and optical tensor have different directions. 94 To distinguish between a monoclinic biaxial and uniaxial states, we prepared planar cells ( d 20μm ) in which the electric field was applied in the plane parallel to the bounding glass plates. Both substrates are rubbed in an antiparallel fashion and the rubbing direction is tilted by 3o from the direction of field. We recorded the intensity of 6 3 3n m light transmitted through the cell and a pair of crossed polarizers as the function of temperature and applied sinusoidal AC electric field: 2.92 ( f 1 0H z , T 40 Co ), 4.12 o o o (1 0H z , 2 0 C ), 7 . 2 8 (1Hz, 0C), 1 5 . 1 2Vrms / μm (1Hz, 5C). The transmission was recorded as the function of two types of cell rotation. First, the cell was rotated by an angle around the axis “b” that is perpendicular to the plane of the cell (Figs. 4.8a,b, T 40,20,0,-5Co ). This experiment tests the potential tilt of the optic tensor with respect to the low-frequency dielectric tensor in the plane of cell, see the scheme in Fig. 4.8b. Second, the cell was rotated by an angle around the axis “ c ” that is perpendicular to the electric field but is parallel to the plane of cell (Figs. 4.8a,c, T 40,20Co ). This experiment tests whether the optical ellipsoid is tilted away from the dielectric ellipsoid in the cross-section of the cell. In the second case, the experiments could be conducted only for temperatures above 20o C. In both experiments, Figs. 4.8b,c, the intensity of transmitted light changes symmetrically around the values 0 , thus indicating that the dielectric and refractive index principal axes are parallel to each other and to the direction of the applied electric field in the plane of the cell. Similarly to the results obtained for the homeotropic cells, Figs. 4.3-4.7, the data in Fig. 4.8 for the planar cells support the conclusion that the nematic 95 phase in the range of temperatures studied is a uniaxial rather than biaxial (orthorhombic or monoclinic) nematic. Another clear evidence of the fact that the textural changes correspond to an anchoring transition in homeotropic cells of the Nb state rather than to the appearance of Nb state was obtained with a fluorescence confocal polarizing microscopy (FCPM). The tetrapode material was doped with a small amount ( 0.01wt % ) of a fluorescent dye N,N’- bis(2,5-di-tert-butylphenyl)-3,4,9,10-perylenedicarboximide (BTBP, Sigma-Aldrich). First, rubbed planar cells were prepared to establish the fluorescent dipole dˆ of BTBP dye. The fluorescent intensity IFCPM depends on the angle between the polarization of ˆ 4 probing beam Pˆ and d as IFCPM cos .[147, 148] Fig. 4.9a shows IFCPM of the tetrapode nematic as a function of at TT45C(*)o , 20Co (*) T , and 2 5 Co . In the whole N temperature range, dˆ of BTBP is parallel to the main director nˆ of tetrapode nematic; IFCPM is max when is along the rubbing direction. Figs. 4.9b-d show the vertical slices of FCPM texture in a homeotropic cell of thickness d 20 μm; the cell was gently rubbed in anti-parallel fashion to reinforce unidirectional director tilting. In all cases, Pˆ is parallel to the bounding plates. At TT45o C (*) , the FCPM texture (Fig. 4.9b) has a low IFCPM which is corresponding to the dark POM texture indicating nˆ is aligned normal to the substrates. If the transformation of POM and conoscopic texture is not caused by the anchoring transition but by Nub -N phase transition at TT *, IFCPM o should be still low because of 90 . However, at TT * , IFCPM increases which 96 represents the tilting of nˆ , Figs. 4.9c,d. Increase of IF C P M as T decreases apparently demonstrates the tilting of director nˆ . It is possible that N -ub N phase transition and an anchoring transition occurs simultaneously but we demonstrated that there is no N -ub N phase transition with the electro-optic measurement (Figs. 4.6 and 4.7) and the observation of topological defects which is discussed below. FIG. 4.9. (a) FCPM intensity in a homogeneous planar cell of tetrapode material as a function of the angle . Vertical optical slices ( xz- scan) of FCPM textures in the homeotropic cell ( d 20μm ) at (b) TT45o C(*) , (c) 25C(*)o T , and (d) 25Co . 97 4.4. Topological Defects 4.4.1. Escape of Director in a Round Capillary Nu and Nb have different sets of topological defects associated with a single ˆ director nˆ in Nu and three directors nˆ , mˆ , l in Nb .[2] Their features provide a useful phase/symmetry identification tool provided the investigation is carried not too close to the N -ub N phase boundary (to avoid pre-transitional textural effects). In the bulk of Nu , the topologically stable defects are point defects such as a “hedgehog” with a radial nˆ()r and linear defects-disclinations of strength m 1/2 . Disclinations with m 1 are not topologically stable. Even if the configuration with m 1 is enforced by boundary conditions, for example, by confining Nu into a round capillary, nˆ would realign along the axis of capillary,[164, 165] a process called an “escape into the third dimension”, Fig. 4.10a. In the Nb , isolated point defects cannot be stable.[2] A radial configuration of one director in Nb indicates that the two other directors are defined at a spherical surface and thus should form additional singularities emanating from the center of defect (bold red line in Fig. 4.10b). The strength of disclination can be either 1/ 2 or 1. The disclination with m 1 are topologically stable with a singular core that cannot escape. Note that all these features of topological defects in Nb should be relevant regardless of whether the symmetry of Nb is orthorhombic or monoclinic. 98 FIG. 4.10. Escaped director configuration in a round capillary for (a) Nu and (b) a hypothetical Nb ; black solid, blue dashed, and red bold line represent the main director nˆ , secondary director mˆ , and disclination core, respectively. (c-g) POM textures of tetrapode in the round capillaries; (c) capillary of D 50μm shows an escaped configuration at TT42o C( *) and a distorted texture due to an anchoring transition at (d) TT36o C(*) ; large capillary ( D 150μm ) shows smooth texture of an escaped configuration with no biaxial features in the entire temperature range, (d) T 45 , (e) 25 , and (f) 25o C . White dashed lines in (c-g) indicate the inner walls of capillary. 99 Prior to filling LC, the round quartz capillary was treated with lecithin for homeotropic alignment. Figs. 4.10c-g show the textural change as a function of temperature in the round capillary with 50μm(Figs. 4.10c,d) and 150μm(Figs. 4.10e-g) inner diameter ( D ). In the round capillaries with a homeotropic anchoring at bounding surface, the tetrapode samples show different textures, depending on D and T . The narrow capillary, D 50 μm, at TT *, shows the typical POM texture of the “escaped” director. When the capillary axis is parallel to one of the two crossed polarizers, one observes dark extinction bands in the center and near the capillary walls, Fig. 4.10c. The three dark bands are separated by two bright bands that correspond to the director orientation tilted with respect to the two polarizers. As the temperature decreases below T *, the anchoring transition results in appearance of numerous defects and strongly deformed director, Fig. 4.10d. In the wider capillary, D 150 μm, the anchoring transition manifests itself in the tilt of the director at the inner wall of the capillary, thus the bright bands become wider, in the entire temperature range explored, 25oo C T 46 C , Figs. 4.10e-g. The textures remain smooth and show no singularities, which is consistent with the escaped configuration characteristic for Nu ; if the material were a Nb , this nonsingular configuration would be replaced with singularities which we do not observe. 100 4.4.2. Point Defects in Droplets Suspended in Isotropic Fluid FIG. 4.11. Director profile in the tetrapode droplet with tangential surface anchoring: (a) Nu bipolar droplet with two point defects-boojums of mm121 at poles; (b) hypothetical Nb bipolar droplet with a singular disclination m 1 (red bold line) formed by secondary director; (c) hypothetical Nb droplet with a single boojum of m 2 . Textures of Nu bipolar droplets of tetrapode in a glycerol at (d) T 45 , (e) 25 , and (f) 25o C . White dashed line indicates the droplet symmetry axis. Scale bar is 10μm. 101 The surface anchoring determines the equilibrium state of director structure that in the case of spherical droplets must possess some number of topological defects, because of the Poincaré and Gauss theorems.[158, 166] For the tangential anchoring, according to the Poincaré theorem, the spherical surface must contain point defects-boojums with the total strength mi measured as the sum of two-dimensional topological charges mi equal 2. i A typical realization in Nu is the bipolar structure with two boojums at the poles, mm121, Fig. 4.11a. In Nb , these boojums of m 1 cannot exist as isolated objects and they should represent the ends of singular disclinations of m 1 terminating at the surface; these disclinations do not vanish in the bulk (there is no “escape” mechanism for them), but they can split into disclinations of m 1/ 2 ; in that case, one would observe four exit points at the surface of the droplet, each of the strength m 1/ 2 .[167] One possible realization is shown in Fig. 4.11b. Another possible Nb structure is a single boojum of m 2 that results when the disclinations shrink into a point at the surface, Fig. 4.11c. The value m 2 is the minimum value of the topological charge that an isolated surface point defect-boojum might have in Nb . We produced the droplets of tetrapode material with tangential anchoring by dispersing it in glycerol (Sigma-Aldrich), Figs. 4.11d-f. An important question is whether the limited solubility of glycerol in the liquid crystal (that is certainly possible, especially at elevated temperatures [168]), can dramatically change the phase diagram. We found that the temperature of melting of the droplets in glycerol is the same as for a bulk material (within a 0.2o C range, which is a typical variation from sample to sample), thus it is 102 unlikely that glycerol alters the phase state and symmetry of the nematic material. The droplets with tangential anchoring show two isolated boojums in the entire studied temperature range 25C46CooT . No additional defects such as disclinations expected for the Nb phase (bold line in Fig. 4.11b) are observed in these droplets. FIG. 4.12. Director profile in tetrapode droplets with perpendicular surface anchoring: (a) Nu radial droplet with a point defect hedgehog of m 1 at the center of droplet; Hypothetical Nb droplets with normal anchoring for the principal director, (b) with singular disclination m 1 (red bold line) formed by secondary director, (c) with four disclinations of m 1/ 2 each, and (d) a single surface point defect-boojum with m 2 . The textures of tetrapode droplets in lecithin-glycerol mixture at (e) T 45 , (f) 25 , and (g) 0Co . Under the perpendicular anchoring condition, the equilibrium state of an Nu droplet is expected to contain a point defect of strength N 1 with a radial director field at 103 the periphery, Fig. 4.12a. Its detailed core structure might be complex and contain a loop configuration, but the main point is that the defect is isolated, i.e., free of other defects attached to it. In Nb , however, the isolated point defects do not exist, as the appearance of the secondary directors implies that the point defect in the main director nˆ is connected to the surface by singular disclinations formed by the secondary directors, Figs. 4.12b,c. The disclination textures would most likely relax by minimizing the length of defects and forming a structure with one point defect-boojum at the surface, Fig. 4.12d; the latter is conceptually the same texture as in Fig. 4.11c, with the interchange of the main and secondary directors. To obtain the homeotropic anchoring, we added a small amount of a surfactant (lecithin) to glycerol. At high temperatures TT *, the droplets show crossed extinction brushes when observed between two crossed polarizers and an isolated point defect in the center, indicating the radial structure, Fig. 4.12a. As the temperature decreases below some critical value T *, the crossed extinction lines start to deform. These changes of texture can be associated either with the surface anchoring transition or with the hedgehog losing its approximate radial configuration and acquiring deformations other than splay. The transition is reversible. Whatever the reason, the most important feature is that the hedgehog remains an isolated singularity that keeps its location at the center of droplet in the entire range 25oo C46T C . We find no disclinations associated with the homeotropic droplets below T *. 104 4.4.3. Point Defects at Colloidal Spheres in the Tetrapode Material FIG. 4.13. Point defects boojums mm121 at the poles of colloidal spheres (diameter D 10μm ) in the planar cell ( d 20μm ) at (a) T 45 , (b) 25 , and (c) 25Co : (d) chain of colloidal particles with point boojums at 25Co ; scheme of director configuration (e) with point boojums in Nu and (f) with new disclination defects in Nb that are singular in the secondary director mˆ . Black solid, blue dashed, and red bold line in (e) and (f) indicate main director nˆ , secondary director mˆ , and disclination core. 105 We also explored the behavior of surface point defects - boojums and isolated point defects - hedgehog produced by spherical colloids dispersed in the tetrapode. The spherical particles ( D 10μm ) of borosilicate glass were added into the material and studied in homogeneous planar cells of d 20μm . The glass yields a tangential anchoring and the texture shows a quadrupolar distortion of nˆ around the spheres with two point defects, boojums, at the poles (Figs. 4.13a-c and 14.3e). The axis connecting the two boojums of the same sphere is parallel to the overall director direction nˆ 0 set by rubbing. Their existence follows from the same topological requirement mi 2 as the existence of two boojums inside a Nu drop. If the material surrounding i the sphere is Nb , the isolated boojums cannot exist and should be accompanied by the singular disclinations, as shown in Fig. 4.13f. In our experiments, we find that the two boojums at the poles of the colloidal spheres have no disclinations attached, in the entire range 25C46CooT . It might be possible that the disclinations are simply invisible under POM. However, their presence should be manifested in the anisotropic forces of interaction of neighbouring spheres. In Nu , two tangentially anchored spheres attract each other in such a way that the line that connects their centres is tilted with respect to the overall director nˆ 0 .[169] In Nb case, the disclinations should arrange the neighbouring spheres into a straight chain parallel to nˆ 0 , in order to minimize the length of disclinations. We do not observe this expected rearrangement, as the clusters of spheres show tilted arrangements expected for the Nu environment, Fig. 4.13d. 106 FIG. 4.14. Isolated point defects “hedgehogs” formed next to each colloidal sphere ( D 10μm ) in the planar cell ( d 20μm ) at (a) T 45 , (b) 25 , and (c) 2 5 Co : (d) chain of two colloidal particles with point defects at 25Co ; (e) scheme of director configuration with a point defect “hedgehog” around the particles in Nu and (f, g) hypothetical schemes of director field around the particle with line defects in Nb . Black solid, blue dashed, and red bold line indicate main director nˆ , secondary director mˆ , and disclination core. 107 To explore the isolated point defects, so called “hyperbolic hedgehog” associated with homeotropically anchored director at the colloidal spheres,[169] the particles were treated with a surfactant octadecyltrichlorosilane (OTS). The particles were studied in a homogeneous planar cell ( 2d 0 μm) . As shown in Fig. 4.14e, the spheres induce an isolated point defect that serves to compensate the topological charge of the sphere. In the entire temperature range 25C46CooT , the hedgehog retains its configuration, showing no appearance of the attached disclinations, Fig. 4.14a-c. The chain of particles was also stable, aligned along the direction of the elastic dipole, Fig. 4.14d, showing no rearrangements expected when the material acquires secondary directors. 4.5. Conclusions We studied the optical, surface, and topological properties of nematic organo- siloxane tetrapodes to verify the existence of a biaxial nematic phase. In homeotropic cells, we observe a surface reorientation of the director below a certain temperature, TT *; T * varies from 3 1 Co to 4 4 Co depending on the cell thickness. We found that this transition is not necessarily associated with the appearance of the biaxial nematic phase, as one can restore the uniaxial optical character of the texture by applying a sufficiently strong electric field and aligning the liquid crystal director parallel to the field; the optical properties reveal a uniaxial character of the state. The possibility of a monoclinic biaxial order has been also verified in electro-optical response of the planar cells. The data, 108 presented in Fig. 4.8, are consistent with the uniaxial order as opposed to a monoclinic biaxial order. The external electric or magnetic field might in principle modify the packing of molecules in the nematic phase as evidenced by the field-induced shifts of melting points.[170-176] The field can also alter the symmetry of phase.[45, 146, 177-181] For example, an electric field applied normally to the director nˆ of the uniaxial nematic with a negative dielectric anisotropy can produce a biaxial modification of the order parameter.[45, 146, 177-180] However, the field typically reduces the symmetry; we are not aware of the opposite effect of the field-induced transformation of the biaxial phase into a uniaxial phase. To avoid potential complications with the field-induced changes, we also tested the tetrapode nematic from the point of view of topological defects. The studies of topological defects in various confinement geometries also support the conclusion of a uniaxial nematic order in the tetrapode material. In these studies, neither external electric nor magnetic fields are involved, and the liquid crystalline structures are determined by the surface anchoring conditions at the boundaries and by the intrinsic elasticity and symmetry of the nematic order. In round capillaries of a sufficiently large diameter, the director shows the “escape into the third dimension” texture that is stable in the entire nematic range of temperatures. In the cases of spherical confinement, such as nematic droplets dispersed in glycerine or spherical colloids dispersed in the tetrapode nematic, we observe isolated point defects boojums and hedgehogs, which are consistent with the uniaxial type of ordering but not with the biaxial order. We thus conclude that in our experiments on optical, surface and topological features, the organo- 109 siloxane tetrapode liquid crystal shows a uniaxial nematic phase behavior in the range of temperatures 25C46CooT . This is at variance with the results based on other experimental techniques; a comprehensive picture of the molecular packing in the material is still outstanding. 110 CHAPTER 5 DIRECTOR REORIENTATION OF A NEMATIC LIQUID CRYSTAL BY THERMAL EXPANSION 5.1. Introduction Materials expand when heated and contract when cooled because of temperature- induced changes in distances between molecules and atoms. Thermal expansion does not change overall structure of the material. Expanding mercury in a thermometer remains an isotropic fluid and an expanding crystal preserves its crystallographic symmetry. In this chapter we explore whether thermal expansion and contraction of a nematic liquid crystal (NLC) can change the structure by realigning the constituent molecules. Reorientation of the optic axis nˆ is used in information displays [3] driven by the electric field, biomedical and chemical sensing based on surface interactions,[182-184] and in microfluidic and optofluidic devices controlled by pressure gradients.[185-187] In the description of dynamic response of NLC to the external fields, it is assumed that the NLC density and volume are always constant, independent of molecular reorientation and materials flow.[1] This “incompressible flow” assumption is a natural extension of a similar simplification in the hydrodynamics of isotropic fluids.[188] The simplification is 111 well justified when the velocities of interest are slower than the speed of sound. However, when the fluid is heated or cooled, it expands or shrinks. The change of volume must be accompanied by flow, because of the mass conservation principle. Surprisingly, even for the isotropic fluids, this thermo-mechanical effect has not been pursued in the literature until 2004, when Yariv and Brenner explained how an unsteady temperature T field causes a flow of an isotropic fluid.[189] In this chapter, we demonstrate that volume changes during heating and cooling of NLC generate flows that cause pronounced reorientation of molecules and of the optic axis nˆ . For the NLC in a long rectangular capillary with open ends, we establish the flow velocity dependence on the distance from the center of the sample and determine the flow- induced distortions of in the vertical cross section of the capillary, using 3D optical microscopy. Under the action of the flow-induced shear, the director tilts in two opposite directions in the top and the bottom parts of the capillary. The director tilts cause dramatic changes of the optical appearance of the uniaxial NLC sample that mimic optical features expected of the biaxial NLC. We show that the expansion/contraction flows can be used for transport of colloidal particles in the NLC. We propose an analytical model that explains the flow, director tilt and optical features as a result of thermal expansion/contraction. The effect is general for many different NLC materials, both thermotropic and lyotropic, and represents a new powerful tool to control optical properties of NLCs and transport of particles in them. The ability to realign the optic axis by the changing T offers numerous applications in sensing, microfluidic, optofluidic, lab-on-a- 112 chip devices. It allows one to build devices with a very simple design as no pressure pumps are needed to generate flow and realign molecules. 5.2. Methods 5.2.1. NLC Materials, Capillaries, Alignment, and Temperature Control We used rectangular capillaries of thickness d 50μm , width w 1m m and length 5c m made of fused quartz (Fiber Optics Center). The capillaries were flashed with a weak solution of lecithin in hexane, to set the homeotropic alignment of nˆ , and filled with E7 which has the wide nematic range 62.5C60CooT . Thermally induced expansions/contractions were directed along the capillary axis x since the coefficient of volume expansion of fused quartz ( 1.710/C6o ) is much lower than that of E7 ( 7.8 104o / C at 2 5 Co , as measured in our laboratory). The temperature was controlled by a hot stage LTS350 (Linkam Instruments) with a Linkam controller TMS94 with 0. 01 Co precision and a typical rate of temperature change 0.5C/so . Cooling was assisted by a circulation of liquid nitrogen. 5.2.2. Fluorescent Tracers of Flow To trace the flow patterns, we added a small amount ( 1%wt ) of fluorescently labeled polydivinylbenzene latex spheres of diameter 2μm (Seragen Diagnostics, Inc.) to 113 33 the NLC. The density of spheres p 1.0610kg/m matched closely the density of E7 33 E7 1.03 10 kg/m [190] to prevent sedimentation. Sedimentation is also prevented by elastic levitation in the NLC bulk.[191] The three coordinates (,,)x y z of the particles were determined by FCPM.[147] The vertical optical slicing of the samples by FCPM showed that the latex beads levitated practically in the middle plane of the capillary, at z ( 2 3 3 ) μm in capillaries of thickness d 50μm . The trajectories of fluorescent tracers were analyzed with Image Pro 6.2 (Media Cybernetics) in order to extract the maximum velocity vx of particles along the x axis during the temperature change. 5.2.3. Fluorescent Anisometric Dye In FCPM observations, the nematic mixture MLC-6815 (birefringence n 0.0517 at 2 5 Co and wavelength 5 5 0n m) was doped with a tiny amount ( 0.01%wt ) of a fluorescent dye N,N’-Bis(2,6-dimethylphenyl)-3,4,9,10-perylenetetracarboxylic diimide (Sigma-Aldrich). The dye molecules are strongly elongated and align parallel to nˆ . 114 5.3. Contraction / Expansion, Flow, and Realignment in Flat Capillary The simplest geometry to observe and characterize the new thermo-optical effect is to place a typical NLC, such as E7, in a rectangular long quartz capillary that orients nˆ perpendicularly to the top and bottom flat boundaries, along the vertical z axis, Fig. 5.1. The ends of capillary are open. The capillary of d 50μm is much thinner than its width 1mm (along the y axis) and length 5c m (along the x axis), Fig. 5.1a. At the lateral edges of the capillary and in the meniscus regions near the open ends, nˆ deviates from the z axis; however, extension of these distortions is only a few tens of micrometers and it does not affect much the perfect homeotropic area of size 1mm 5cm . In what follows, we describe the properties of the initial homeotropic area, away from the distorted edges. The optical appearance of the homeotropic NLC viewed between two crossed polarizers of POM is very different when T is fixed and when it is changing, Fig. 5.1. When T const , the texture is dark, Fig. 5.1b, as the optic axis is along z axis. Once T changes, the texture brightens (Fig. 5.1b). The bright appearance is caused by tilt away from the z axis and toward the expansion direction (the x axis), as established by conoscopic observations, with the sample illuminated by a converging cone of light.[192] The conoscopic texture of the initial homeotropic state represents a symmetric Maltese cross (the two end insets in Fig. 5.1b and the central image at x 0 in Fig. 5.1c). When T changes, the cross splits into two isogyres, indicating that nˆ tilts to the left and to the right from z axis, remaining within the xz, plane.[59] The splitting of isogyres (and thus the tilt of ) increases with the distance x from the capillary center, Fig. 5.1c. 115 FIG. 5.1. (a) Capillary of d 50μm filled with E7 and fluorescent particles. (b) Time sequence of POM textures at x 22mm , when the sample is cooled from 25 to 2 0 Co ( 30o C/min ). The texture changes from dark at constant T 25o C to bright because of the NLC flow (as seen by following the particle marked by a circle) that tilts the director. As T stabilizes at 2 0 Co , the birefringent texture relaxes back to the dark texture of a uniform homeotropic NLC. The insets show the time evolution of the corresponding conoscopic textures. (c) Conoscopic textures of different locations along x . At fixed T , the textures represent a Maltese cross characteristic of a homeotropic uniaxial NLC. As T varies from 25 to 20o C ( 30o C/min ), the isogyres split; the split distance 2a increases with the distance ||x from the capillary center. 116 FIG. 5.2. (a) Maximum flow velocity as a function of coordinate x along the capillary axis in the regime of cooling ( T 5Co and 3 0 Co / m i n ). (b) The same for heating ( T 5Co and 30C/mino ); capillary of thickness d 50μm filled with E7. (c) Relaxation time tr as the function of cell thickness d and the initial temperature T0 of the homeotropic E7 cell; T 5Co and 30C/mino . The director tilt is associated with the material flow, as visualized by fluorescent latex spheres of diameter 2μm added to the NLC, Fig. 5.1b. Following trajectories of the particles, we determined the maximum flow velocity vx during the temperature change, as a function of horizontal coordinate x , for different initial temperatures T( t 0) T0 of the o sample, but fixed increment TTT 0 , and change rate Tt/30 C/min ; vx is proportional to x and increases as T0 decreases (Figs. 5.2a,b). The transient dynamics of o o NLC depends on T0 , T , , and d . In particular, for E7 with T0 25C , T 5C, 30o C/min , d 50μm , the bright texture persists for 25s . To characterize the flow relaxation, we measured the time interval tr between the start of temperature change, t 0 117 , and the time tr when the dispersed particles stopped to move. This relaxation time increases with ||T , || , and d , and decreases at higher T0 ; a typical dependence tr (,) d T0 is shown in Fig. 5.2c. At low T and in thick samples, tr is no longer in the range of seconds and can be in the range of many minutes and even hours. 5.4. Director Profile in Shear ( xz ) Plane To elucidate the link between the director tilts and the material flows, we used fluorescent confocal polarizing microscopy (FCPM) to map the director profile as a function of time and all three spatial coordinates.[147] Of especial interest is the director distortion in the shear (xy , ) plane in Fig. 5.1a. The resolution of 3D images of the director profile is greatly improved when the NLC birefringence is low.[147] Because of this, we performed experiments not only with highly birefringent E7 but also with the nematic MLC-6815 (EM Industries) of low birefringence, doped with a fluorescent dye. The sample is probed by a linearly polarized laser beam that causes the dye to fluoresce. The intensity of fluorescence IFCPM depends on the angle between the polarization P of the 4 beam and nˆ : IFCPM cos .[147]. The optical scan of the sample in the vertical plane (,)xz, Figs. 5.3 a,b, allows us to extract the data on IxzFCPM (,) , Figs. 5.3c,d and thus nˆ(,)xz, Figs. 5.3e,f. 118 FIG. 5.3. (a) FCPM texture of the vertical cross-section (xy , ) of a homeotropic cell of d 20μm filled with MCL-6815; the probing beam is polarized along P ; scale bar is 20μm . The inset on the left-hand side shows a conoscopic pattern. T is fixed at 1 5 Co o o . (b) The same during cooling from 15 to 25C ( =15C/min ). (c) IFCPM as a function of z for the initial state. (d) The same for the thermally contracted state; the z - dependence of IFCPM show two pronounced maxima and three minima, near the bounding plates and in the middle plane. (e) The reconstructed director profile in the homeotropic state at fixed T 15o C . (f) The reconstructed bow-shaped director profile in the cooling- induced tilted state at t 8s. 119 At fixed temperature, IFCPM (,) x z is close to zero, as nˆ is perpendicular to P , Figs. 5.3a, c. A small background signal is due to director fluctuations. When the temperature changes, the fluorescent intensity IzF C P M () profile shows two well resolved maxima (Figs. 5.3b,d). It implies that remains vertical at the cell’s boundaries zd 0, , and in the middle plane, zd /2, but tilts into two opposite directions in the top and bottom parts of the cell, forming a bow-type structure, Fig. 5.3f. The maximum director tilt measured with respect to the z axis is about 20o . The non-uniform director profile along the z axis explains the bright birefringent textures and splitting of isogyres in Figs. 5.1b,c. As the temperature stabilizes, the NLC relaxes back to the initial homeotropic state. FIG. 5.4. Horizontal flow velocity and director configuration. Vertical cross-section of a capillary filled with a thermally expanding NLC, as a function of horizontal coordinate x and vertical coordinate z . 120 Fig. 5.4 summarizes schematically the experimental data on the horizontal velocity and flow-induced director distortions in the thermally expanding NLC. The velocity and director field are shown for the vertical cross-section (,)xz of the capillary, which is also the shear plane. The homeotropic-to-bow state transition just established is similar to the one observed in Refs. [193-197], with that difference that in the prior works, the textural transformations were created by the pressure-driven Poiseuille flows, while in this work the effect results from thermal expansion; there are no pumps. Note that possible temperature gradients are not the cause of the observed effect, although they can be certainly added to the list of control parameters if desired. To elucidate the effect of temperature gradients, we conducted a separate experiment (Fig. 5.8) to demonstrate that this effect is insignificant compared to the effect of temperature variation with time. The homeotropic sample of E7 ( d 50μm ) was placed as a bridge between a cold plate kept at 2 0 Co and a hot plate kept at 5 5 Co . The Malthese cross of the conoscopic texture maintained its fourfold symmetry characteristic of the unperturbed homeotropic state for all locations along the LC bridge and did not change when the distance between the two plates was very slowly varied between 1 and 8mm. 121 5.5. Thermal Expansion Effects in Other Types of LCs 5.5.1. Thermotropic and Lyotropic LCs FIG. 5.5. Thermally induced splitting of crossed conoscopic pattern in the homeotropic cells filled with various thermotropic and lyotropic uniaxial NLCs. The cells are filled o o o with (a) 5CB (T0 25C ), (b) MLC-6815 (T0 5C), (c) ZLI-2806 (T0 15 C ), and (d) o lyotropic solution of Sunset Yellow in water, 33wt % (T0 15 C ); in all cases, the cell thickness is d 20μm , temperature change T 5Co and rate of change 30o C/min . 122 The thermally induced director realignment described above and summarized in Fig. 5.4 refers to an initially uniform thermotropic NLC. In Fig. 5.5, we demonstrate that the same phenomenon can be staged in many other NLCs, such as the thermotropic liquid crystal pentylcyanobiphenyl (5CB), mixtures MLC-6815 and ZLI-2806 and the lyotropic NLC Sunset Yellow. At a fixed temperature, conoscopy textures of homeotropic samples of all these materials show a stable dark Maltese cross (first column in Fig. 5.5). As the temperature varies, the cross splits into two symmetric isogyres indicating the tilt of nˆ (middle column in Fig. 5.5). After a certain relaxation time, the split isogyres relax back to the initial Maltese cross (right column in Fig. 5.5). 5.5.2. Complex-Shaped LC, DT6Py6E6 FIG. 5.6. Chemical structure and space filling model of the bent-core liquid crystal DT6Py6E6. 123 The splitting of isogyres should be even more pronounced in LCs formed by molecules of a complex shape, for which thermal expansion is expected to be stronger than in E7. One example is a non-symmetrically substituted thiadiazole (abbreviated DT6Py6E6) with molecules of a pronounced bent-core shape (Fig. 5.6). At lower temperature of N phase, the material exhibits temperature-expansion-triggered splitting of isogyres for very long tr that may suggest a biaxial phase. However, the characteristic ‘‘biaxial’’ splitting of isogyres is in fact transient and results from a tilt of the uniaxial director triggered by a change in T and a significant thermal contraction/expansion of the material. Optical Conoscopy. Optical conoscopy was performed on homeotropic samples of DT6Py6E6 in the entire temperature range of N phase. In equilibrium at a fixed T , the conoscopic patterns consisted of two straight, crossed isogyres, a signature of optical uniaxiality. However, as Figs. 5.7a,b show for a 13μm thick sample, a transient change in conoscopic pattern occurs while the T is changing (in either heating or cooling). The data o o are recorded during and after T0 5C at 1C/min . Figs. 5.7c,d display the corresponding change in transmission of 633nm laser light through the sample placed between crossed polarizers. During the T change, the sample shows both an increase in light transmission and a separation of the initially crossed isogyres; after tr , both effects relax back to the initial, uniaxial condition. The transient splitting of isogyres was observed o o for temperatures TTNI from 35 C to 120 C ; over this range, the separation of 124 isogyres increases somewhat from higher to lower T , while tr varies enormously, from ~1 s to ~1h r . FIG. 5.7. Top: changes in the conoscopic pattern with time while (a) cooling and (b) heating a 13μm thick homeotropic sample of DT6Py6E6. Bottom: data for the intensity of 6 3 3n m light transmitted through the homeotropic sample placed between crossed polarizers, as a function of time, following the temperature changes indicated in (c) cooling and (d) heating. As we discussed in Section 5.4, the effects demonstrated in Fig. 5.7 could depend on a spatial gradient in T across the field of observation, as well as on its temporal rate of change. Since the thermal conductivity of a nematic is anisotropic, thermal gradients may cause a realignment of the optical axis.[198-200] For a homeotropic sample, this would correspond to a tilting of the director away from the cell normal. In a typical hot stage, a 125 temperature variation across the sample plane is unavoidable because of the presence of the transparent window necessary for optical observations. We conducted a separate experiment to demonstrate that. In order to establish a fixed thermal gradient, a homeotropic sample was placed as a bridge between a cold plate kept at 52o C and a separate hot plate at 9 1 Co (Fig. 5.8). The temperature of each plate was precisely regulated by a combination of a ceramic heating element and a refrigerated circulating water bath. As the gap between two plates was varied between 1 and 8mm , we observed that the conoscopic texture in the middle of the cell always maintained the features (fourfold symmetric cross pattern) of a uniaxial, homeotropically aligned nematic. FIG. 5.8. Schematic of the technique used to produce a thermal gradient in plane of a homeotropically aligned sample of DT6Py6E6 in the nematic phase. The inset shows the corresponding stationary conoscopic pattern, indicating a steady uniaxial state when the gap between hot and cold plates was varied as shown. 126 Optical Microscopy. Fig. 5.9 shows the thermal expansion and shrinkage in a homeotropic sample produced by changing the temperature from 120 Co to 4 0 Co and back to 120 Co with 1 Co / m i n . The expansion and shrinkage are seen as a motion of the meniscus line at the LC-air interface in the cell. We independently determined the coefficient of thermal expansion by injecting the liquid crystal into a cylindrical quartz capillary ( D 320μm ), and recording the displacement of the meniscus as a function of temperature. We obtained the value 1 9 1 0 / C4o . For comparison, the value of in the standard calamitic liquid crystal 5CB is 610/C4o ; thus, is significantly larger in DT6Py6E6 than in a conventional calamitic nematic. FIG. 5.9. Thermal shrinkage and expansion of a 13μm thick homeotropic sample of DT6Py6E6 during (a) cooling and (b) heating, as visualized by the displacement of the meniscus near the edge of cell observed under the polarizing microscope with parallel polarizers. The dotted black line indicates the edge of the top plate covering the LC layer. 127 Fluorescence Confocal Polarizing Microscopy. The thermal expansion and shrinkage induce flows in the sample, which we verified by observing the motion of small fluorescent spheres of diameter D 0.19μm added at a concentration of 1%wt (by volume) to the liquid crystal in 20μm thick homeotropic cells. In particular, we used FCPM to show that a change of temperature is accompanied by a displacement of particles in the cell. FIG. 5.10. Movement of fluorescent particles in a homeotropic sample of DT6Py6E6 observed by FCPM on (a) cooling and (b) heating the sample; 30C/mino . Black area is the LC (no fluorescent intensity) and green dots are the fluorescent particles suspended in the LC. Arrows indicate the direction of particle motion. The particles in the solid circles are fixed by adhesion to the cell surfaces. 128 Fig. 5.10 displays typical results. Once T is stabilized, the particles stop moving. The direction of particle displacement correlates with the direction of movement of the air– liquid crystal interface in Fig. 5.9. The particle motion thus reveals that there is a material flow in the bulk of the cell along the direction of sample contraction/expansion. At lower T , the displacements of the particles become larger (for the same ). In LCs, material flows are able to realign the director.[1] By simply pushing against the cell substrates, we demonstrated (Fig. 5.11) that an in-plane flow in a homeotropic cell of DT6Py6E6 can generate a splitting of isogyres similar to that observed in Fig. 5.7. FIG. 5.11. Splitting of conoscopic isogyres caused by material flow after pushing on the surfaces of a cell containing a 20μm layer of homeotropically aligned DT6Py6E6. 129 We now turn to additional results, which confirm that during a T change, the uniaxial director tilts away from the equilibrium direction perpendicular to the glass plates in the homeotropic cell. In the first experiment, the director tilt was visualized by FCPM with a polarized probe beam. Pure DT6Py6E6 was doped with a tiny amount ( 0 . 0 1 %wt ) of BTBP. In the material, the transition dipole dˆ of BTBP is parallel to the uniaxial director nˆ of DT6Py6E6. FIG. 5.12. FCPM textures of a 13μm thick homeotropic sample of DT6Py6E6 doped with 0.01wt % of the fluorescent dye BTBP. (a) and (b): In-plane ( xy scan) textures. (c): Vertical optical slices ( xz scan). P indicates the polarizer axis. 130 Fig. 5.12 shows a sequence of FCPM images for a 13μm thick homeotropic cell. In all cases, the probing light impinges normally onto the horizontal cell. For homeotropic alignment, the polarization of probing light is perpendicular to the director, π /2 and the FCPM texture is dark, the FCPM intensity IFCPM is close to zero (orientational fluctuations always yield some non-zero contribution to IF C P M even in the perfect homeotropic state), as seen for the equilibrium states at fixed temperatures 50o C and 5 5 Co . The textural appearance is fully compatible with the uniaxial homeotropic structure in which dˆ and nˆ are both perpendicular to the plane of the cell. However, in the transient regime (during and immediately after a 5Co temperature steps), the FCPM textures become bright as seen in the top view of the cell (Figs. 5.12a, b) and in the vertical optical cross-section of the cell (Fig. 5.12c). A strong increase in the intensity IFCPM of fluorescent light indicates that is temporarily tilted away from the normal to the cell, so that is different from π /2. Coherent Anti-Stokes Raman Scattering. As an additional confirmation that temperature variation induces a tilting of , we employed the label-free technique of coherent anti-Stokes Raman scattering (CARS) microcopy[201, 202] to probe directly the orientation of pre-selected chemical bonds in the LC phases.[203, 204] Our set-up is fully described in Ref. [203]. 131 FIG. 5.13. (a) Raman spectrum of DT6Py6E6. (b) CARS signal intensity in a homogeneous planar cell of the standard calamitic nematic 5CB as a function of the angle CARS measured between the rubbing direction and polarization of probing beams. (c) CARS signal intensity as a function of CARS in a homogeneous planar cell of DT6Py6E6. (d–f) Epi-detected CARS texture in a 20μm thick homeotropic DT6Py6E6 sample, stabilized at (d) 55o C , (f) at 50o C , and (e) during temperature change (transient state) o o between 55 C and 50 C . Note darker field in (d) and (f), and a brighter field in (e). PCARS indicates the polarization direction of the probing laser beams. 132 Fig. 5.13a shows a portion of the Raman spectrum of DT6Py6E6, from which we identify the peak corresponding to the carbon–carbon triple bond at 220 7c m -1 .[205] This bond was selected to trace the orientation of DT6Py6E6 molecules in the sample during the temperature changes. Because it is rigid, there is no bending of the bond that could degrade the fidelity of CARS images. We used low power ( 0 . 1W ) irradiation to prevent damage to the sample. In order to establish how the CARS signal intensity depends on the orientation of preselected chemical bonds with respect to the direction of linear polarization PC A R S of the probing (pump and Stokes) laser beams, we first examined homogeneous planar cells of thickness 20μm , filled with either a standard calamitic nematic, pentylcyanobiphenyl (5CB), or with DT6Py6E6. For 5CB, we collected the CARS signal at the frequency 2223cm-1 , corresponding to vibrations of the CN triple bond. The CARS intensity from the planar sample of 5CB is presented in Fig. 5.13b as a function of the angle CARS between the rubbing direction (equivalently, the director nˆ ) and the polarization direction PCARS . The CN bond is parallel to the long axis of the 5CB molecule and therefore, on average, to . As a result, cosCARS P CARS nˆ and the CARS signal reaches a pronounced maximum when PCARS is parallel to the rubbing o o direction, CARS 0 , and decreases sharply when ||CARS decreases to about 60 . In the oo range 60 |CARS | 90 , the CARS signal intensity shows a very weak angular dependence. This behavior is expected for the illumination geometry used, as the electric fields Ep of the pump and Es of the Stokes beams are parallel to each other, so that one 133 2 3 2 3 expects IEECARS p s , where is the third-order susceptibility.[203] The calculated angular dependence of ICARS provides a satisfactory fit of the experimental data in Fig. 5.13b. In DT6Py6E6, the carbon–carbon triple bonds are tilted with respect to the long axis of the molecule, by about 10o (Fig. 5.6), and thus tilted with respect to the director. This feature is clearly reflected in the angular dependency of the CARS signal measured for the planar cell, Fig. 5.13c. The IC A R S shows two local peaks, shifted to the right and to o the left by a few degrees from the rubbing axis, where CARS 0 . As in the case of 5CB, o the signal rapidly decreases as ||CARS increases to approximately 60 , and then depends o o only weakly on ||CARS between 60 and 90 . We now consider the CARS data for a homeotropically aligned 20μm sample of DT6Py6E6. Fig. 5.13d–f show the sequence of CARS images obtained during a temperature change from 55 to 6 0 Co . In the initial state, thermally stabilized at 5 5 Co , the texture is dark, as the angle between the polarization of the probing beams and the CC bonds is around 80o for the homeotropic sample (as noted previously, the bonds are tilted by ~10o with respect to the director, which is now perpendicular to the plane of incidence). When the temperature is quickly reduced to 50Co , the signal increases, and the texture becomes bright, Fig. 5.13e, indicating tilt of the CC bonds away from the cell normal and toward the direction of PCARS . Using the data in Fig. 5.13c, the average angle between the bond direction and the horizontal plane in Fig. 5.13e is estimated to be 134 oo in the range 45||60CARS . This tilt of the molecules is transient. Once the temperature is stabilized at 50o C , the texture relaxes back to the dark state in ~ 30 min, indicating that the angle between the CC bonds and PC A R S returns to the initial value for the homeotropic state (compare Figs. 5.13d,f). The CARS observations thus demonstrate a transient tilt of the DT6Py6E6 molecules, in agreement with the data obtained by FCPM (Fig. 5.12). The transient director tilt detected in DT6Py6E6 should not be confused with the surface anchoring transition observed as a function of temperature in other homeotropically aligned bent-core materials.[59, 67] In the latter cases, the director establishes an equilibrium tilted configuration that does not change with time, as long as the temperature remains fixed. By contrast, in the case of DT6Py6E6, the tilt is an out-of-equilibrium feature that is observed only during a finite time interval that depends on the rate of temperature change. Although both effects mimic rather closely optical features expected for a biaxial nematic sample, their natures are very different from each other and from a true thermodynamic biaxial state. It is also worth noting a very recent report on the appearance of a secondary Schlieren texture upon temperature change in an unaligned BCN,[61] which was considered as an indication of biaxiality. It would be of interest to explore whether this effect is related to the finite rate of temperature change, i.e., represents an effect similar to the one described above for homeotropically aligned DT6Py6E6. 135 5.6. Thermal Expansion Effects in Other Types of Geometries FIG. 5.14. Stretching of disclinations by thermally driven flow in the E7 cell with tangential anchoring ( d 20μm ). (a) Stable nearly vertical disclination at a fixed o o temperature, T0 25 C . (b) Stretching of disclinations along the flow; T0 25 C , T 5Co , and 30C/mino . When the temperature stabilizes, the disclinations relax to the original state (a). Fig. 5.14 shows the thermally induced reorientation of molecules in a tangentially anchored cells ( nˆ is in the plane of cells, but there is no preferred orientation in that plane). The sample contains linear defects disclinations [2] around which nˆ reorients by 180o . The disclinations are topologically stable as there is no continuous deformation of the director field that can transform them into a uniform director state. The ends of each disclination are located at the opposite plates. If the temperature is fixed, the disclinations are more or less parallel to the vertical axis z to minimize the energy of elastic distortions (Fig. 5.14a). As the temperature is changed to generate NLC flow, the disclinations are 136 stretched in the direction of flow, but their ends remain pinned at the surface irregularities (Fig. 5.14b). The stretching of disclinations in the thermally activated NLC can be used for controlled in-plane switching of the optical axis, as nˆ rotates by 90o in going from one side of the disclination to the other (Fig. 5.14a (inset)). FIG. 5.15. Molecular reorientation by thermally driven flow in a uniform planar cell filled with E7 ( d 20μm ). (a,b.c) Sequence of POM textures between the crossed polarizers, for the initial, flow-induced and final equilibrium states. R indicates the rubbing direction; the initial director orientation is along R . (d,e,f) FCPM ( xz, scan) textures corresponding o o o to a,b,c, respectively. T0 40C (at a and d), T 5C, and 30C/min . Fig. 5.15 shows the molecular reorientation by thermally induced flow in a homogeneous planar cell, with nˆ parallel to a single direction in the plane of the cell. To achieve the planar anchoring, we spin-coated polyimide PI-2555 (Dupont) onto the glass substrates and then rubbed it unidirectionally. The cell was filled with E7 doped with the 137 anisometric fluorescent dye, BTBP. At the fixed temperature, the POM texture is dark (Fig. 5.15a) as nˆ is perpendicular to the polarizer. Thermally induced flow in the direction perpendicular to the rubbing direction causes deviations of the director from the original state, seen as slight brightening of the POM texture (Fig. 5.15b); the texture relaxes back to the original state when the temperature is stabilized (Fig. 5.15c). The effect is more pronounced in the FCPM mode of observations that visualizes the vertical cross-section of the cell along the plane perpendicular to the rubbing direction (Figs. 5.15d,e,f). As the polarization of the probing beam in FCPM is perpendicular to the rubbing direction, the states with a stable temperature are dark (Figs. 5.15d,f), whereas the contraction-induced state is bright (Fig. 5.15e). Fig. 5.16 shows the thermo-mechanical effect in a round quartz capillary (inner diameter 100μm). The capillary was flashed with a weak solution of lecithin in hexane, to set the homeotropic alignment of nˆ (perpendicular to the capillary walls), and filled with E7. To minimize the elastic energy, nˆ realigns from the radial directions near the walls to an axial orientation in the center of capillary. This ‘escaped’ configuration often produces point defects at the axis, the so-called hedgehogs [2] (inset in Fig. 5.16a). As the temperature varies, NLC experiences transient flow, director reorientation, and displacement of hedgehogs. 138 FIG. 5.16. Thermally activated flows and molecular reorientation in a round capillary. (a) cooling; (b) heating. A point defect-hedgehog at the capillary axis is marked by a red circle. Inset in a shows the director profile around the defect core. T 5Co , and 30o C/min (‘ ’ sign for a and ‘ ’ sign for b). 139 The experiments above clearly demonstrate the suitability of the effect for optofluidic applications, in which relatively small temperature variations cause dramatic optical changes. There are no fundamental obstacles for similar effects in other types of LCs, such as chiral (cholesteric) phases and smectic phases with partial positional order. 5.7. Applications High Rates of Temperature Changes. In the experiments above, the temperature was controlled by the hot stage with a relatively modest maximum rate 0.5o C/s . Much higher values of can easily be achieved using electrodes, for example, transparent and conductive ITO. We verified that by applying the electric field ( 2 0V , f 1Khz ) to the ITO electrodes on the typical LC cells, one can increase to ( 3 - 5 ) Co and flow velocities to about 100μm/s. Transport of Levitating Colloidal Particles. Besides optofluidics, the described thermo-optical effect in LCs can also be used for a controlled transport of particles. In the nematic host, colloidal particles impose director distortions.[206] Elastic repulsion of these distortions from the bounding plates allows the particles to levitate in the NLC bulk.[191] This levitation effect is an important advantage of the LCs as compared to isotropic fluids in microfluidic applications, as it prevents trapping of the particles at boundaries. The fluorescent latex spheres of diameter 2μm used trace the flows, Fig. 5.1b, in the 140 expanding/shrinking slabs of NLC survive multiple thermal cycles, without being immobilized, remaining in the bulk of the material and resisting sedimentation. 5.8. Discussion and Conclusion The most striking feature of the observed molecular realignment is that the effect is caused simply by thermally induced changes in density. The thermo-optical effect is uniquely attached to the orientational order and absence of the long-range positional order in the NLC. In a regular isotropic fluid, such an effect is impossible as there is no long- range orientational order and flow cannot align the molecules. In a solid crystal, the effect is not possible because the changes in positional order are not sufficiently large to trigger molecular realignments. In a bounded NLC, the flow induced by thermal expansion and contraction is maximum in the central part of the sample but should vanish at the walls because of the no-slip conditions. The ensuing velocity gradients impose a reorienting torque on the director. Below we propose a simple analytical model of the observed thermo-mechanic and thermo-optical effects in the NLC. The mechanism of thermally induced flow is clear from the mass conservation equation,[189] / t v , which connects the time derivative of the fluid density to the spatial gradients of its velocity. The density of a NLC slab thermally expanding or shrinking along its axis x can be presented as (tt )(1) 0 , where 00=() TT is the initial density of NLC. The mass conservation equation then immediately yields vtx , i.e., a non-zero velocity along the axis x that depends linearly on the distance 141 from the geometrical center x 0 , rate , and the coefficient of thermal expansion. A more detailed consideration should take into account the boundary conditions and other (slower) components of velocity. The experimental conditions imply a low Reynolds 6 11 number Re/||~10 dvx 2 , where ||0.28Kgms2 (Ref. [190]) is the anisotropic viscosity of E7 in the geometry under consideration.[206] Therefore, the velocity should satisfy the Stokes equation 2v (here is the pressure gradient and is the dynamic viscosity) (9), as well as a no-slip condition vx 0 and no- penetration condition at the bounding walls. The solution for vx then follows as: zz vxx 61 . (5.1) dd The z component of flow velocity is much weaker at distances of interest, xd 2zd , 0 zd, since vvzx . Here and in what follows, we neglect the small 6x corrections T 102 as compared to 1 and assume that the velocity is not affected by the director orientation. The flow along the x axis realigns nˆ towards the x axis, Fig. 5.3f. The viscous vx vx reorienting torque [1, 206] 2 is proportional to the shear rate that is vanishing at z z the walls and at the middle plane zd /2, according to Eq. (5.1). The viscous torque is 2 opposed by the elastic torque K3 that tends to keep nˆ along the z axis; here K3 is the z2 142 Frank elastic constant of bend.[1] The balance of the two torques for small angles ()z between nˆ and the vertical z axis determines the flow-induced director profile: 2 zz 2 (z ) x z 1 1 . (5.2) K3 d d The model is in good agreement with the experiments. First, vx in Eq. (5.1) increases with the rate and with the coordinate x , being very close to the experimental data in Figs. 5.2a,b. For example, Eq. (5.1) predicts vx 13μm /s at zd /2 and x 22mm if one uses the experimental data 7.810/C4o , 0.5C/so , T 5Co and d 50μm . The measured velocities are 10μm/s, Figs. 5.2a,b. The small difference is understandable because the particles-tracers have a finite size and because they are located somewhat below the middle plane, as determined by FCPM. Second, the director profile, Eq. (5.2), is of the bow type, with nˆ remaining homeotropic at z 0; d /2; d , and tilted in the regions 0/2zd and dzd/2, precisely as in Fig. 5.3f. Note that at the middle plane, zd /2, the velocity is maximum, Eq. (5.1), but the velocity gradient and thus tilt, according to Eq. (5.2), are zero, as in the experiment, Fig. 5.3b,d,f. Using 11 11 2 0.28Kg m s , K3 1.95 10 N ,[2] and d 50μm in Eq. (5.2), one finds a significant tilt of nˆ , about 34o at z 0.2 d ,0.8 d and x 22mm , sufficient to cause the bright birefringent states, Fig. 5.1b. The thermally-induced flow produces a non-uniform director profile, Fig. 5.3b,d,f. The tilt angle ()z is negative in the top half of sample and positive in the bottom part, 143 Fig. 5.3f. When one tests the specimen with a technique such as polarizing microscopy, conoscopy, X-ray, etc. that probes a response integrated over the sample thickness, the thermo-optical effect can be confused with the formation of a biaxial NLC. The response integrated over the uniaxial NLC with the single local optical axis varying from point to point and the response from a genuine biaxial NLC with two optic axes defined in each point of the specimen and titled with respect to each other would be very similar to each other. Consider, for example, the split isogyres in the conoscopic patterns, Figs. 5.1 b,c, Figs. 5.3 b, that are often considered as a signature of the biaxial NLC with two local optic axes with a non-zero angle between them. The separation 2a of the two isogyres is a 2an quantitative measure[192] of : sin , where 2R is the diameter of the field 2NA2R of view, Fig. 5.1 c, NA is the numerical aperture of the microscope’s objective, n is the average refractive index of the NLC. A zero split (a Maltese cross pattern, see inset in Fig. 5.3 a) implies 0 and a uniaxial character of the homeotropic NLC. However, if the director of the uniaxial NLC is not uniform, ()z , as in Eq. (5.2), then the isogyres would be also split, by a distance [59, 67] 21an d 2 ()zdz , (5.3) 2NARd0 1 d where ()zdz . Inserting into Eq. (5.3) the values 0 and (z ) const from d 0 Eq. (5.2), one finds that in a thermally expanding/shrinking uniaxial NLC the isogyres are split, with a separation 144 2an2 xd (5.4) 2RK3 210NA that is proportional to the distance x from the center of capillary (as in the experiment, Fig. 5.1c), cell thickness d , temperature change rate , and thermal expansion coefficient . Therefore, the split isogyres are characteristic of not only a biaxial NLC but also a uniaxial NLC that is thermally expanding or shrinking. Note that the phase diagrams of LCs are typically established by changing the temperature of the specimen; in these studies, one should be aware that thermal expansion/contraction of a uniaxial LC can mimic the appearance of biaxial states. The statement remains valid not only for conoscopic patterns but also for other techniques of characterization with an integrated response, such as polarizing microscopy, X-ray diffraction, and so on. Another important feature to bear in mind is that the temperature-triggered structural changes can be long lived, as suggested by Fig. 5.2c. Moreover, the director tilt might develop into the “peak” structure [193, 197] that is topologically distinct from the homeotropic state and can remain in a metastable state for a long time, requiring nucleation of disclination defects to relax back. The corresponding splitting of isogyres will be trapped as a long-lived metastable state. In conclusion, we demonstrated experimentally that temperature-induced density variations of NLC trigger flows that cause reorientation of molecules and optic axis. The physical mechanisms of coupled expansion, flow, flow-induced shear, resulting director reorientation and its optical features such as splitting of isogyres were explained by a 145 simple model that agrees well with the experiments. In order to demonstrate the essence of phenomena, we focused on relatively well studied materials such as E7. Even in this case, a modest rate of temperature changes 0 . 5 Co / s causes a director tilt by tens of degrees and flow velocities on the order of 10μm / s; velocities on the order of 100μm / s are achieved in experiments with a higher created by electric heating. In the future, it will be of interest to expand the studies of the thermal expansion-induced phenomena in LCs to the case when the source of heating is a focused laser beam. It would allow one to enrich the set of control parameters by adding the anisotropic thermophoretic effects (motion caused by a static thermal gradient, extensively explored for isotropic fluids [207]) and reorienting torque resulting from coupling between the electric field of light and the director. The described thermo-optical and thermo-mechanical effects should be even more pronounced in LCs formed by molecules of a complex shape, for which thermal expansion is expected to be stronger than in E7. One example is a non-symmetrically substituted triazole abbreviated DT6Py6E6, with molecules of a pronounced bent-core shape, in which one can observe temperature-expansion-triggered splitting of isogyres that mimics appearance of a biaxial nematic phase.[138] The thermo-optical phenomenon should be carefully accounted for in establishing the phase diagrams of LCs with phase transitions brought about by temperature changes, as the expansion/contraction of uniaxial states results in features mimicking those of biaxial phases. The NLC cells activated by unsteady temperature can be used for simultaneous thermo-mechanical and thermo-optical effects, such as transport of particles levitating in the NLC bulk with concomitant reorientation of optic axis around them. The simplicity of the observed phenomena that do 146 not require pumps nor even electrodes to produce dramatic optical and mechanical changes suggests that they will find applications in a variety of fields, including sensors, photonics, lab-on-a-chip, micro- and optofluidics. 147 CHAPTER 6 SUMMARY Topological defects play an important role in many physical processes ranging from morphogenesis of phase transitions in condensed matter system to the response to surface confinement and application of external fields. In this dissertation, we investigate the topological defects both in lyotropic and thermotropic nematics in order to characterize the studied materials. The summary of this dissertation is as follows: 1. We explore the structure of nuclei and topological defects that appear during the first-order phase transition between the nematic and isotropic phases in lyotropic chromonic liquid crystals. The defects emerge as a result of two mechanisms: (1) surface- anisotropy that endows each nematic nucleus (‘tactoid’) with topological defects thanks to preferential (tangential) orientation of the director at the closed isotropic–nematic interface, and (2) Kibble mechanism with defects forming when differently oriented nematic tactoids merge with each other. Different scenarios of phase transition involve positive (nematic- in-isotropic) and negative (isotropic-in-nematic) tactoids with nontrivial topology of the director field and also multiply connected tactoid-in-tactoid configurations. The closed isotropic–nematic interface limiting a tactoid shows a certain number of cusps; the lips of the interface on the opposite sides of the cusp make an angle different from π . The nematic 148 side of each cusp contains a point defect-boojum. The number of cusps shows how many times the director becomes perpendicular to the isotropic–nematic interface when one circumnavigates the closed boundary of the tactoid. We derive conservation laws that connect the number of cusps c to the topological strength m of defects in the nematic part of the simply connected and multiply connected tactoids. We demonstrate how the elastic anisotropy of the nematic phase results in a non-circular shape of the disclination cores. A generalized Wulff construction is used to derive the shape of isotropic and nematic tactoids as a function of isotropic–nematic interfacial tension anisotropy in the approximation of frozen director field of various topological charges m . The complex shapes and structures of tactoids and topological defects demonstrate an important role of surface anisotropy in morphogenesis of phase transitions in liquid crystals. 2. We investigate intriguing defects, the so-called “secondary disclinations” that were recently described as a new evidence of a biaxial nematic phase in an oxadiazole bent- core thermotropic liquid crystal C7. With an assortment of optical techniques such as polarizing optical microscopy, LC-PolScope, and fluorescence confocal polarizing microscopy, we demonstrate that the secondary disclinations represent non-singular domain walls formed in an uniaxial nematic during the surface anchoring transition, in which surface orientation of the director changes from tangential (parallel to the bounding plates) to tilted. Each domain wall separates two regions with the director tilted in opposite azimuthal directions. At the centre of the wall, the director remains parallel to the bonding plates. The domain walls can be easily removed by applying a modest electric field. The anchoring transition is explained by the balance of (a) the intrinsic perpendicular surface 149 anchoring produced by the polyimide aligning layer and (b) tangential alignment caused by ionic impurities forming electric double layers. The model is supported by the fact that the temperature of the tangential-tilted anchoring transition decreases as the cell thickness increases and as the concentration of ionic species (added salt) increases. We also demonstrate that the surface alignment is strongly affected by thermal degradation of the samples. The study shows that C7 exhibits only a uniaxial nematic phase and demonstrate yet another mechanism (formation of “secondary disclinations”) by which a uniaxial nematic can mimic a biaxial nematic behavior. 3. We explore topological defects, optical textures, and surface anchoring properties of another biaxial candidate material, organo-siloxane “tetrapode”, in order to determine the existence of the biaxial nematic phase. We demonstrate that the optical, structural, and topological features are compatible with the uniaxial nematic order rather than with the biaxial nematic order, in the entire nematic temperature range 25C46CooT studied. For homeotropic alignment, the material experiences surface anchoring transition, but the director can be realigned into an optically uniaxial texture by applying a sufficiently strong electric field. The topological features of textures in cylindrical capillaries, in spherical droplets and around colloidal inclusions are consistent with the uniaxial character of the long-range nematic order. In particular, we observe isolated surface point defect-boojums and bulk point defects-hedgehog that can exist only in the uniaxial nematic. 150 4. We describe a new effect in which the orientation of nematic liquid crystal molecules is altered by thermal expansion. Thermal expansion (or contraction) causes the nematic liquid crystal to flow; the flow imposes a realigning torque on the nematic liquid crystal molecules and the optic axis. The optical and mechanical responses activated by a simple temperature change can be used in sensing, photonics, microfluidic, optofluidic and lab-on-a-chip applications as they do not require externally imposed gradients of temperature, pressure, surface realignment, nor electromagnetic fields. The effect has important ramifications for the current search of the biaxial nematic phase as the optical features of thermally induced structural changes in the uniaxial nematic liquid crystal mimic the features expected of the biaxial nematic liquid crystal. 151 REFERENCES [1] P. G. de Gennes and J. Prost, The physics of liquid crystals (Clarendon Press, Oxford, 1993). [2] M. Kleman and O. D. Lavrentovich, Soft matter physics: An introduction (Springer, New York, 2003). [3] D.-K. Yang and S.-T. 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