{"Affiliation":[{"label":"Affiliation","value":"Applied Science, Faculty of","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","classmap":"vivo:EducationalProcess","property":"vivo:departmentOrSchool"},"iri":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","explain":"VIVO-ISF Ontology V1.6 Property; The department or school name within institution; Not intended to be an institution name."},{"label":"Affiliation","value":"Chemical and Biological Engineering, Department of","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","classmap":"vivo:EducationalProcess","property":"vivo:departmentOrSchool"},"iri":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","explain":"VIVO-ISF Ontology V1.6 Property; The department or school name within institution; Not intended to be an institution name."}],"AggregatedSourceRepository":[{"label":"AggregatedSourceRepository","value":"DSpace","attrs":{"lang":"en","ns":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","classmap":"ore:Aggregation","property":"edm:dataProvider"},"iri":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","explain":"A Europeana Data Model Property; The name or identifier of the organization who contributes data indirectly to an aggregation service (e.g. Europeana)"}],"Campus":[{"label":"Campus","value":"UBCV","attrs":{"lang":"en","ns":"https:\/\/open.library.ubc.ca\/terms#degreeCampus","classmap":"oc:ThesisDescription","property":"oc:degreeCampus"},"iri":"https:\/\/open.library.ubc.ca\/terms#degreeCampus","explain":"UBC Open Collections Metadata Components; Local Field; Identifies the name of the campus from which the graduate completed their degree."}],"Creator":[{"label":"Creator","value":"Khullar, Siddharth","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/creator","classmap":"dpla:SourceResource","property":"dcterms:creator"},"iri":"http:\/\/purl.org\/dc\/terms\/creator","explain":"A Dublin Core Terms Property; An entity primarily responsible for making the resource.; Examples of a Contributor include a person, an organization, or a service."}],"DateAvailable":[{"label":"DateAvailable","value":"2011-03-17T21:46:02Z","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/issued","classmap":"edm:WebResource","property":"dcterms:issued"},"iri":"http:\/\/purl.org\/dc\/terms\/issued","explain":"A Dublin Core Terms Property; Date of formal issuance (e.g., publication) of the resource."}],"DateIssued":[{"label":"DateIssued","value":"2007","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/issued","classmap":"oc:SourceResource","property":"dcterms:issued"},"iri":"http:\/\/purl.org\/dc\/terms\/issued","explain":"A Dublin Core Terms Property; Date of formal issuance (e.g., publication) of the resource."}],"Degree":[{"label":"Degree","value":"Master of Applied Science - MASc","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#relatedDegree","classmap":"vivo:ThesisDegree","property":"vivo:relatedDegree"},"iri":"http:\/\/vivoweb.org\/ontology\/core#relatedDegree","explain":"VIVO-ISF Ontology V1.6 Property; The thesis degree; Extended Property specified by UBC, as per https:\/\/wiki.duraspace.org\/display\/VIVO\/Ontology+Editor%27s+Guide"}],"DegreeGrantor":[{"label":"DegreeGrantor","value":"University of British Columbia","attrs":{"lang":"en","ns":"https:\/\/open.library.ubc.ca\/terms#degreeGrantor","classmap":"oc:ThesisDescription","property":"oc:degreeGrantor"},"iri":"https:\/\/open.library.ubc.ca\/terms#degreeGrantor","explain":"UBC Open Collections Metadata Components; Local Field; Indicates the institution where thesis was granted."}],"Description":[{"label":"Description","value":"Liquid crystals are nature's beautiful examples of complex materials which are fundamentally fascinating. Their unusual properties have intrigued researchers from a wide variety fields including biologists, engineers, and even cosmologists.\r\nThis thesis focuses on the dynamics of topological defects occurring near micro-droplets and micro-bubbles as they rise through an aligned nematic liquid crystal. The experiments were conducted in a fabricated flow-cell, and the observations were made using polarized light microscopy with the help of a motion control system.\r\nThe results settle a controversy in the literature regarding the effect of hydrodynamic flow on the motion of defects by providing direct evidence of downstream convection of a Saturn ring defect and its transformation to a hyperbolic point defect. The point defect is convected further in the wake of the drop or bubble as the rising velocity increases. In equilibrium, both defect\r\nconfigurations may persist for long times. But the point defect sometimes spontaneously opens into a Saturn ring, indicating the latter as the globally stable configuration for the conditions used. A quantitative analysis of the rise velocities versus the location of defects yields graphs which are consistent with recent theoretical predictions. Besides these, we also observe interesting multiple drop and bubble interactions leading to the phenomenon of self-assembly and distorted defect structures.","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/description","classmap":"dpla:SourceResource","property":"dcterms:description"},"iri":"http:\/\/purl.org\/dc\/terms\/description","explain":"A Dublin Core Terms Property; An account of the resource.; Description may include but is not limited to: an abstract, a table of contents, a graphical representation, or a free-text account of the resource."}],"DigitalResourceOriginalRecord":[{"label":"DigitalResourceOriginalRecord","value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/32579?expand=metadata","attrs":{"lang":"en","ns":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","classmap":"ore:Aggregation","property":"edm:aggregatedCHO"},"iri":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","explain":"A Europeana Data Model Property; The identifier of the source object, e.g. the Mona Lisa itself. This could be a full linked open date URI or an internal identifier"}],"FullText":[{"label":"FullText","value":"An experimental study of bubbles and droplets rising in a nematic liquid crystal by Siddharth Khullar A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES (Chemical and Biological Engineering) THE UNIVERSITY OF BRITISH COLUMBIA August 2007 \u00a9 Siddharth Khullar, 2007 ABSTRACT Liquid crystals are nature's beautiful examples of complex materials which are fundamentally fascinating. Their unusual properties have intrigued researchers from a wide variety fields including biologists, engineers, and even cosmologists. This thesis focuses on the dynamics of topological defects occurring near micro-droplets and micro-bubbles as they rise through an aligned nematic liquid crystal. The experiments were conducted in a fabricated flow-cell, and the observations were made using polarized light microscopy with the help of a motion control system. The results settle a controversy in the literature regarding the effect of hydrodynamic flow on the motion of defects by providing direct evidence of downstream convection of a Saturn ring defect and its transformation to a hyperbolic point defect. The point defect is convected further in the wake of the drop or bubble as the rising velocity increases. In equilibrium, both defect configurations may persist for long times. But the point defect sometimes spontaneously opens into a Saturn ring, indicating the latter as the globally stable configuration for the conditions used. A quantitative analysis of the rise velocities versus the location of defects yields graphs which are consistent with recent theoretical predictions. Besides these, we also observe interesting multiple drop and bubble interactions leading to the phenomenon of self-assembly and distorted defect structures. ii ACKNOWLEDGEMENTS First, I want to express my heartfelt gratitude to my supervisor Prof. James Feng under whose guidance this work saw its completion. His encouragement and support provided a solid ground on which I could execute my research. He taught me the skill of scientific writing and organised approach to problem solving, which I believe to be the vital ingredients for intellectual research. I would like to extend my thanks to my committee members - Dr. Boris Stoeber and Dr. Royann Petrell who carefully oversaw the completion of this project. Dr. Stoeber's ideas during the setting of my experiments were of valuable assistance and Dr. Petrell's captivating talks about her everyday experiences kept me going on enthusiastically. I appreciate Chunfeng Zhou's help and contribution to this project. I had many interesting chats with him as an officemate. During the initial stages, I enjoyed working with Xiaopeng (Paul) Chen, whose guidance helped me adjust with ease. I am grateful to Dr. Teodor Burghelea for his help in setting up of my experiment. I also appreciate Prof. John Gosline's and Prof. Robert Shadwick's efforts to help us with the imaging system. I owe many thanks to all of my friends, especially Raghavendra and Pramod for their support and suggestions. I want to thank my family members for keeping up my morale, throughout my masters' programme. My mom deserves a special mention for taking care of me through all these years, and her unfailing belief in my abilities. It is because of her scientific aptitude and diligence that kindled my liking towards science. I appreciate our department faculty and staff for their cooperation and assistance. iii T A B L E O F C O N T E N T S ABSTRACT ii ACKNOWLEDGEMENTS iii TABLE OF CONTENTS iv LIST OF FIGURES vi LIST OF TABLES ix 1 INTRODUCTION 1 1.1 Discovery of Liquid Crystals (LCs) 1 1.2 Description of LCs 1 1.3 LC Nomenclature 3 1.3.1 The director .'. 3 1.3.2 The order parameter 3 1.3.3 Anchoring (surface alignment) 4 1.3.4 Elasticity and Free energy 6 1.4 Defects in Liquid crystals 6 1.5 Hyperbolic point and ring defects 8 1.6 Stability of Defects 9 1.6.1 Effect of particle size 9 1.6.2 Effect of External fields 11 1.6.3 Effect of flow 13 1.7 Objective and Scope of this thesis 14 2 EXPERIMENTAL MATERIALS AND METHODS 15 2.1 Observation of Liquid Crystals 15 2.2 Polarized light 15 2.2.1 Light Propagation through a liquid crystalline medium 16 2.3 Quantitative Imaging using Abrio 17 2.3.1 Restriction imposed by Abrio 18 2.4 Schlieren Texture 19 2.5 Cell fabrication and preparation 22 2.6 Uniform Planar Wall Anchoring 23 2.7 Anchoring at 5CB-air interface 25 2.8 Motion Control System and microscope 26 iv 2.9 Micro-bubble and micro-drop production 28 2.10 Rising bubble in homeotropically wall anchored cell 29 2.11 Materials used and their properties 33 3 RESULTS 35 3.1 Defects on static bubbles and droplets 35 3.1.1 Comparison with computed profde 36 3.1.2 Comparison with quantitative imaging using Abrio 37 3.1.3 Spontaneous and induced point defect to Saturn-ring transition 39 3.2 Oil drops in motion 41 3.3 Air bubbles in motion 45 3.3.1 Elastic relaxation 47 3.3.2 Loop breaking 48 3.4 Multi particle interactions 49 3.4.1 Self assembly among micro-droplets and micro-bubbles 49 3.4.2 Deformed defect structures 51 3.5 Conclusions 53 3.6 Recommendations for future work 54 4 REFERENCES 56 5 APPENDIX 60 5.1 Structure and properties of 5CB 60 5.2 Abrio's Working Layout 61 5.3 Abrio's Specifications 62 v LIST OF F IGURES Figure 1: A pictorial representation of molecular organization in different phases 2 Figure 2: An illustration depicting the director (shown by the arrow) and the meaning of order parameter in liquid crystals 4 Figure 3: Anchoring phenomenon in liquid crystals. The arrow shows the anchoring direction.. 5 Figure 4: Geometric arrangement of the director leading to hyperbolic and radial types of point defects 7 Figure 5: Schematic showing the director profile of a hyperbolic satellite point defect (a) and a Saturn-ring defect (b), surrounding a particle with homeotropic anchoring at its surface. The orientation is vertical in the far field 8 Figure 6: Schlieren pattern in a thin sample of liquid crystal (no wall anchoring) as seen through crossed polarizers (a) , and the corresponding Abrio snapshot (b) with the superimposed orientation vectors 21 Figure 7: Schematic of a fabricated cell 23 Figure 8: Schematic showing the direction of light path through the plane of observation (X-Y) in a wedged cell 24 Figure 9: Abrio snapshot of a wedged cell leading to the appearance of retardance bands 24 Figure 10: Crossed polarizers image (a) and the corresponding Abrio snapshot (b) of a thin drop of LC placed on a glass slide 25 Figure 11: Abrio snapshot of a 5CB film sitting on a single glass slide exposed to air. The intensity indicates the measured retardance and the superimposed vectors represent the LC-director. Note that the abrupt and repetitive 90 degree turns in the vectors is due to a restriction in Abrio's working and is to be correctly interpreted by rotating through 90 degrees, as mentioned before. The anchoring near air-LC interface is homeotropic 26 Figure 12: Schematic of the experimental setup 27 Figure 13: A photograph showing the experimental setup 28 Figure 14: Schematic of a bubble rising in a cell with homeotropic wall anchoring. The anchoring at the bubble surface being homeotropic, will lead to a relaxed director (shown in red) profile 31 Figure 15: A bubble moving in a cell with homeotropic wall anchoring (a) Retardance intensity Abrio snapshot; (b) False colour azimuth profile with superimposed orientation vectors; (c) Retardance vs distance plot 32 vi Figure 16: Schematic of a drop\/bubble with homeotropic anchoring in a cell with planar wall anchoring. One of the polarizer's optic axis (P) is parallel to the alignment direction and the other polarizer (A) is crossed at 90 degrees to the first one. Light path is indicated by the arrows. 35 Figure 17: Static defects surrounding oil drops as viewed through crossed polarizers, (a) Saturn-ring defect (drop radius a = 30.2 um); (b) point defect (a = 21.8 urn); (c) computed director field and light intensity graph for a point defect 36 Figure 18: Abrio snapshot (b) of an oil droplet possessing a dipolar point defect with the corresponding crossed polarizers' image (a). The white vectors represent the measured orientation profile. The red lines are the superimposed edited director orientations, some of which are turned by 90\u00b0 whenever the retardance falls outside the measurable range (as discussed in section 2.3.1). The LC - drop interface is demarcated by the black circular ring 38 Figure 19: Spontaneous point-to-ring transformation around a stationary bubble of radius a = 100 um. The images were taken through crossed polarizers, and (a) shows the characteristic birefringence pattern for the point defect (cf. Figure 17(c)) 39 Figure 20: Time series of snapshots showing convection of a Saturn-ring defect (present around a rising oil droplet) and its ultimate convergence to a steady state point defect. The images were taken with single polarizer for clarity 43 Figure 21: (a) Position of the ring or point defect as a function of Ericksen number (Er = rjva\/K) oil droplet run of Figure 20. For the Saturn ring (squares), rj is the vertical distance between the drop center and the edge of the ring. For the satellite point defect (circles), rj is the distance from the drop center. A = 15.1. The inset shows the velocity overshoot that coincides with the ring-to-point transition. Arrows A and B correspond to the first and last data points in the inset; (b) Numerical prediction of steady-state defect positions for A = 30, after Fig. 7 of [Zhou et al. 2007], \u00a9Cambridge University Press 44 Figure 22: Position of the point defect as a function of the instantaneous Ericksen number (Er) for accelerating and decelerating bubbles. Insets Bl and B2 show the bright birefringent loops with the point defect at the bottom, and C shows a computed birefringent pattern roughly corresponding to B2. The solid arrow indicates the final stabilization of the point defect to its equilibrium position (inset A) 46 Figure 23: Time series (a) - (e) of the elastic retraction process on a static bubble. The instantaneous loop retraction velocities (vioop) are given except for (b) which is the start of a new reference frame 47 vii Figure 24: Annihilation of a pair of opposite point defects with opposite charges. The length of the loop shrinks at a rate of 27.4 \/mVs, so that each point defect moves at a speed of 13.7 \/mVs. The scale bar in (a) corresponds to 100 \/mi 48 Figure 25: Self-assembly of oil droplets in LC medium. The droplets are in the size range of ~ 5 to 10 microns 50 Figure 26: Self assembly at a large scale. Air bubbles, -100 \/\/m in diameter, self assembling in vertically aligned liquid crystal 51 Figure 27: Deformed defect structure can be observed between a pair of static air bubbles in (a) and (b) above. Another pair of static bubbles (c) at separation distance d~ \\0a form a stretched-gum type of defect 52 Figure 28: Molecular representation of liquid crystal 5CB 60 viii LIST OF TABLES Table 1: Material properties 33 Table 2: Additional properties of 5CB 60 ix 1 INTRODUCTION* 1.1 Discovery of Liquid Crystals (LCs) The discovery of liquid crystals dates back to more than 150 years ago when Virchow, Mettenheimer and Valentin found that a compound from the nerve core exhibited strange behaviour when viewed with polarizing light. Not realizing the uniqueness of this new phase, they labelled the compound as a 'living crystal'. However, the official discovery of liquid crystals is credited to Otto Lehmann, a German physicist and Friedrich Reinitzer, an Austrian botanist. Reinitzer, while conducting experiments on an organic compound named cholesteryl benzoate, was surprised by the fact that it had two melting points. At 145.5 \u00b0C the solid crystal melted into a cloudy liquid which existed until 178.5 \u00b0C where the cloudiness suddenly disappeared, giving way to a clear transparent liquid. Puzzled by this mysterious behaviour, he turned to Lehmann who confirmed his findings and later coined the term 'liquid crystal', indicating a state with properties between a liquid and a solid. This idea of a new fourth state of matter was disputed by the scientific community, but was accepted a few decades later with numerous reports and conclusive experiments of liquid crystalline behaviour becoming prevalent. Liquid crystals' use for display applications was developed in 1960's thus fuelling an increased research focus in every aspect of liquid crystal science involving chemists, physicists, applied scientists and engineers. Today, these materials are at the core of many technological innovations among which liquid crystal displays and liquid crystal thermometers are the best known. Liquid crystal displays make use of a thin sample of twisted liquid crystal sandwiched between crossed-polarizers. By applying an electric field, the pixels comprising of individual liquid crystal cells can be made to act as a light valve, thus controlling the on-off state of individual pixel. 1.2 Description of LCs Liquid crystals' fundamental unit is called a 'mesogen'. Mesogen, which is derived from Greek, meaning 'species in between', illustrates the idea that the liquid crystalline state behaves in a * A version of this thesis has been submitted for publication [Khullar et al. 2007]. 1 manner which is intermediate between crystalline solids and amorphous liquids. Owing to the molecules' anisometric shape (rod-like and disc-like being the most common) and their non-covalent interactions, a gamut of interesting properties differentiate these from conventional solids or liquids. These include the ability of molecules to orient collectively, spontaneously or in the presence of electric or magnetic field and to anchor preferentially to a binding surface, and the optical property of birefringence among others. Figure 1: A pictorial representation of molecular organization in different phases Figure 1 shows a comparison of molecular order among different phases. Crystalline solids have orientational as well as positional order. Liquids have neither orientational nor positional order. In between these two, are liquid crystals, which have orientational order but partial or no positional order. Liquid crystalline molecules fall into two groups: thermotropes and lyotropes. Thermotropic liquid crystals or thermotropes, are characterized by the presence of phase transitions which depend on temperature. In general, a temperature change can induce a transition from solid crystalline phase to liquid crystal phase and further to an isotropic phase. Inside the liquid crystal phase, the strength of intermolecular interactions is a direct function of the temperature. Lyotropic compounds, on the other hand, exhibit liquid crystalline behaviour when dissolved in proper solvents and the state is concentration dependent. For example, amphiphilic molecules consisting of a short polar head with a long polar tail (as in surfactants), when dissolved in water, may exhibit liquid crystalline behaviour. At critical concentrations these aggregate into specialized structures giving rise to vesicles and micelles. Solutions of polypeptide enantiomers Liquid Phase Gas Phase like polybenzyl-L-glutamate (PBLG) and polybenzyl-D-glutamate (PBDG) also add to the class of lyotropic liquid crystals. Liquid crystals can also be classified according to molecular size, into small molecule and polymeric liquid crystals. Either can be thermotropic or lyotropic [Collings and Hird 1997]. For polymeric liquid crystals, the mesogenic unit is incorporated into the main chain or side chain of a polymer, thus combining the properties of polymers and liquid crystals. Another special class of polymer liquid crystal consists of molecules (typically biological) suspended in high concentrations, an example being the tobacco mosaic virus. Colloidal suspensions of this rod shaped virus exhibit liquid crystalline phases, such as the nematic and smectic phases. 1.3 LC Nomenclature Liquid crystals come in different flavours, which can be distinguished by the dimensionality of translational order. Nematic phases have no translational order while smectic, columnar and other phases can have single or multi-dimensional translational orders. Here, we will be concerned with the nematic phase only. 1.3.1 The director The director, represented by the unit vector n, denotes the average of the directions carried by individual molecules over a volume V, which is large compared to the molecular dimensions but small with respect to typical distortion lengths in the nematic. Figure 2 illustrates an arrangement of rod-like molecules for which the director n can be drawn (shown by the arrow). The molecular arrangement remains symmetric even when the molecules are flipped over, thus keeping the director the same. For this reason n and -n are equivalent i.e. n = -n. The director can be defined at all points in the sample except at singularities (or regions of defects). 1.3.2 The order parameter The order parameter helps describe the long-range order of properties (thermodynamic or structural) which repeat uniformly in a system. For liquid crystals, it is easy to convince oneself that a nematic state will be more ordered than the isotropic phase. The degree of orientational order can be measured in terms of the order parameter s, first introduced by Tsvetkov [1942]: 3 s = ^ <3cos29-1 > (1) where, 9 is the angle between the mesogen's long molecular axis and the preferred orientation direction n, and the angular brackets denote an average over all molecular orientations (see Figure 2). In a perfect alignment case (most ordered state) s = 1, while for most randomly ordered state (isotropic phase) s = 0. In liquid crystalline state, s lies between 0 and 1. For thermotropes, s has a strong temperature dependency in the nematic phase. e Figure 2: An illustration depicting the director (shown by the arrow) and the meaning of order parameter in l iquid crystals. 1.3.3 Anchoring (surface alignment) When a nematic phase comes in contact with another phase (solid, liquid or gas), an interface is created. Close to the interface, the nematic molecules take a fixed mean orientation which is transmitted into the bulk phase via elastic forces. This ability of the liquid crystal molecules to bind and align preferentially to a surface is known as anchoring (Figure 3). It is analogous to the phenomenon of 'epitaxy' of solids on substrates. The anchoring direction in liquid crystals is determined by the choice of surface in contact and can be controlled by mechanical and chemical treatments of the substrate. Perpendicular anchoring occurs when the molecules align perpendicular to the surface. It is also known as homeotropic anchoring. When the molecules lie 'flat' in the plane of the surface in contact, it is called planar anchoring. The molecules can also occupy positions oblique to the surface in which case it is known as tilted anchoring. In reality 4 there is no rigid homeotropic or planar anchoring, as there will always be some small deviations in the angle which the molecules make with the surface. S u b s t r a t e ( i sot rop ic phase) Figure 3: Anchoring phenomenon in liquid crystals. The arrow shows the anchoring direction. An anchoring strength can be described in terms of polar (0) and azimuthal (cp) deviations of the molecular orientation from the anchoring direction (also called easy axis). Often the anchoring energy function, w, is written as [Kleman and Lavrentovich 2003]: where, (90, cpo) define the equilibrium director orientation. The coefficients of proportionality w9 and w
T; kb being the Boltzmann constant (1.381 x 1023 J\/K) and T is the temperature. For numerically calculated values of Wa\/K ~ > 10, both ring and satellite point defects are stable solutions with the satellite enjoying somewhat lower energy than the ring, although the estimate is not concrete. This energy difference lessens as the value of Wa\/K decreases, indicating that Saturn-ring may be unfavourable for larger particles. The Frank theory cannot resolve the defect cores. Hence for using Frank theory throughout the sample region, the defect cores are typically assigned an energy value, which is then added to the free energy F. This yields results which are highly dependent on defect size and the core energy, which may be one of the reasons of mismatch of stable Saturn rings predicted numerically [Stark 2001] (on particle sizes < 1 pm) and those observed experimentally [Gu and Abbott 2000; Mondain-Monval et al. 1999] (on large particles on the order of tens of microns). Such a discontinuity in the calculation of molecular orientation at singularities (defect cores) was avoided in simulations by Feng and Zhou [2004], which uses a mean-field theory to calculate the stability of defect types. The results are in general agreement with previous suggestions; that the Saturn-ring's stability decreases with increase of particle size although the threshold for absolute stability (global minimum) of Saturn-ring defects differs (being on micrometer scale) from that previously calculated. Experimental observations are needed to establish a consistent picture on the range of stability of these two defect types in terms of particle sizes and the strength of surface anchoring. One such study involving in-situ variation of bubble sizes was conducted by Voltz et al. [2006], as described below. 10 1.6.1.2 Experimental The experimental verification of the theoretically predicted particle-size-induced transition between defects has been hindered by the lack of dynamic size variation of the particles. The experiments by Voltz et al. [2006] aimed to demonstrate this transition in-situ. They used a high pressure technique to vary the size of air bubbles confined in a planarly anchored LC cell. They observed that, as the bubble size is reduced there is a transition from hyperbolic hedgehog defect configuration to Saturn-ring configuration. This transition was predicted to occur between bubble radii of 1 pm to 5 pm. Experiments by Monval et al. [1999] focussed on the weak surface anchoring on water droplets suspended in host LC medium. They observed primarily, the occurrence of quadrupolar equatorial ring defects for particles on the scale of few tens of microns with occasional occurrences of hedgehog defects also. This suggests that the two defect types can be meta-stable in conditions of weak anchoring. However, they did not mention the range of anchoring energy ratio, Wa\/K, applicable to their experiments. Other experiments [Loudet and Poulin 2001] on colloidal particles (evolving from phase separation mechanism ) in liquid crystals, indicate the formation of quadrupolar distortions on droplets with sizes as large as few microns in diameter. These quadrupolar droplets grow in size by coalescence due to elastic quadrupolar attraction, ultimately reaching a critical radius where the quadrupolar ring loses its energetic favourability and transforms into a dipolar hedgehog defect. Although the trend of Saturn-ring structure being preferred for lower W values, is in agreement with previous theoretical calculations, the upper limit on the radius (~ 3.75 pm) of their drops possessing Saturn-rings, is higher than that theoretically predicted [Lubensky et al. 1998; Ruhwandl and Terentjev 1997; Stark 1999]. 1.6.2 Effect of External fields The magnetic susceptibility of liquid crystal molecules is sufficient to induce distortions in the director in the presence of field. When a magnetic field is applied, the molecules experience a net torque, which tends to rotate them parallel or perpendicular to the direction of the applied field depending on whether the diamagnetic susceptibility of the liquid crystal is positive or negative. 11 The effect of a strong magnetic field is thus, to compress the field lines to create high densities of elastic and magnetic free energies which has an effect on the stability of the defect type. 1.6.2.1 Computational Stark [1999] and Grollau et al. [2003] have calculated that a magnetic field of sufficient strength will result in Saturn-ring achieving lower energy than a dipole even if the dipole was the preferred configuration in the absence of field. A transition from dipole to ring will occur by the application of magnetic field, and for smaller particles, the Saturn-ring remains meta-stable while the field is turned off. 1.6.2.2 Experimental Gu and Abbott [2000] investigated the presence of defects on solid particles, treated to impose homeotropic surface anchoring, in an LC cell with planar anchoring. They observed the microspheres (40 pm to 100 um in diameter in a 120 pm thick cell) to be occupied by stable (some > 1 month) Saturn-ring defects, contrary to previous predictions [Lubensky et al. 1998; Stark 2001]. However, this was later predicted computationally [Grollau et al. 2003; Stark 2002], to be a confinement induced effect, akin to the application of magnetic field. Besides these, they also observed some spheres to be surrounded by defects of a dipolar nature, and suggested that they may have been formed by spontaneous transformation from Saturn-ring defects. However, such transitions were not captured, so the details of their occurrences were not clear. Also, it was not verified whether an isotropic droplet suspended in LC medium and possessing a dipolar state would be converted to a quadrupolar state by the application of field, although their experiments showed the expansion of stable Saturn-rings upon application of electric field. More convincing experimental evidence was provided later by Loudet & Poulin [2001]. They used phase separated silicone oil droplets suspended in bulk LC phase. In equilibrium and absence of field, these droplets possessed a dipolar point defect. Upon application of electric field it opened up into a quadrupolar equatorial ring defect. Further, the threshold field for this conversion depended on the strength surface anchoring (Wa). As the droplet size increased, the tendency to achieve dipolar state also increased, and thus due to a higher activation energy, the droplets required a higher threshold field to induce this transition. The transition was found to occur in a span of less than 1 second and was believed to be because of instability of hyperbolic 12 hedgehog rather than thermodynamically driven transition between metastable states. Upon cessation of field, the dipolar state always returned, with relaxation time of the order of few tens of seconds. So under their experimental conditions the metastability of the two configurations was either weak or absent. 1.6.3 Effect of flow The effect of a flow field on the defects has received far less attention. Flow of nematics is more interesting and complex than classical isotropic fluids due to the coupling of director field and velocity. Motion and flow are involved in various dynamic processes in nematic dispersions, e.g. the self-assembly of micro-droplets [Poulin and Weitz 1998], and a thorough understanding of the hydrodynamic effects on defect evolution will be valuable. 1.6.3.1 Contradictory Computational predictions So far, there have been several computational studies. Stark and Ventzki [2002] predicted that, flow around a particle will move a satellite point defect against the direction of flow for finite Ericksen numbers Er, defined as the ratio of viscous to elastic forces: Er = rjav\/K, where rj is the viscosity of the medium, a and v are the particle radius and velocity, and K is the bulk elastic constant. This counter-intuitive motion of defect was attributed to a minimized resistance to flow. It was also postulated that, for large Er, the satellite point defect would open up to form a more stable Saturn-ring defect, which can then move upstream. This prediction was later contradicted by Yoneya et al. [2005] whose calculations predicted that an initial satellite point defect near a particle is stable under flow, and that its motion would be along the direction of flow. For flow direction from particle to defect, the defect would get convected far in the wake and would not transform to a Saturn-ring defect. An initial Saturn-ring was predicted to be continuously swept in the flow direction, and ultimately converge to a stable hedgehog point defect in the wake. More recently, simulations by Zhou et al. [2007] suggested that a defect ring around a rising drop convects downstream as the rise velocity increases, and may be transformed into a point defect in the wake. Since the defect is not a physical entity, its apparent 'convection' reflects rearrangement of the orientational field rather than flow of materials. Thus, it is not intuitively obvious which 13 prediction is correct. Furthermore, the intricate details of nematodynamics, involving the coupling of molecular anisotropy with hydrodynamics of flow, make it difficult to unravel the contradictory predictions. 1.7 Objective and Scope of this thesis This work involves the experimental study of defect dynamics around drops and bubbles rising in a nematic liquid crystal. It aims to resolve the contradictions in literature regarding the behaviour of a hyperbolic point defect and a Saturn-ring defect under hydrodynamic flow. In particular, we seek a clear picture of the stability and the direction of motion of a hyperbolic point defect with respect to the flow direction and whether flow can induce a structural transition of either defect type. We present direct experimental evidence for the downstream convection of a point defect, and its subsequent retraction in the wake of a rising bubble. Flow induced defect transformation, from Saturn-ring to a point defect, was observed. A surprising spontaneous transition, from satellite point defect to a Saturn-ring defect, on static drops and bubbles was also captured. Quantitative analysis involving measurements of flow velocities and defect locations yields data that is consistent with previous theoretical calculations. 14 2 EXPERIMENTAL MATERIALS AND METHODS This chapter describes the working principles that are involved in the experimental observation and which are necessary for the understanding of the results. It also provides the details on the experimental setup and the methods used. 2.1 Observation of Liquid Crystals Owing to their optical anisotropy, liquid crystals have traditionally been observed with the help of polarized light microscopy. Samples of liquid crystals sandwiched between glass slides, yield regions of a good contrast (due to the varying orientation of optic axis). In addition, because of their turbidity due to light scattering by director fluctuations [Durand et al. 1969], care has to be taken to make the nematic sample thickness to 'sub-millimetre' scale, lest the optical clarity and uniformity of anchoring be sacrificed. Due to the dynamic nature of our experiments, we have customized an experimental system for the purpose of flow observation of moving bubbles and droplets. This involves the use of a polarizing microscope and an accurately (with speed precision of few pm\/s) controlled motion-control system coupled to the microscope. The experimental runs are conducted in a fabricated micro-flow-cell. An experimental protocol for this fabrication process has been established and benchmarked, following which we conduct the desired runs. We also incorporate the relatively new technique of measuring retardances and molecular orientation using a quantitative imaging system, first developed by Oldenbourg [Guang and Oldenbourg 1994; Oldenbourg and Mei 1995] and later commercialized by CRI Inc as Abrio\u2122. We have used it to determine the director orientation in a nematic liquid crystal and around bubbles and drops submerged in the nematic medium. 2.2 Polarized light Light, because of its electromagnetic nature, is subject to polarization. Polarization of light describes the direction of oscillation of electric or magnetic field in a plane perpendicular to the direction of travel. Mathematically, the amplitude of electric field for a light wave polarized in x direction and travelling in the z direction can be described by the wave equation: 15 E(z, t ) = E\u00b0 X s in[27Wt - 27rzA, + % ] (5) Here, E \u00b0 X is the maximum amplitude of the light wave, v is the frequency, X is the wavelength and cp0 is a constant specifying its absolute phase. Thus, in the above instance, light is depicted as being linearly polarized. However, depending on the amplitude (E) and phase difference (Acp), it can also be circularly or elliptically polarized; circular and linear polarization being special cases of elliptical polarization. Typically, to convert unpolarized light to polarized light, optical materials known as polarizers are used. 2.2.1 Light Propagation through a liquid crystalline medium Birefringence or double refraction, is the decomposition of a light ray into two rays - the ordinary ray (o-ray) and the extraordinary ray (e-ray), when it passes through an anisotropic medium. It is manifested as a result of orientation-dependent differences in the refractive index of a material. The o-wave behaves as in travelling in an isotropic medium, but for the e-wave, the refractive index depends on the direction of propagation. Due to the anisotropic nature of liquid crystal molecules, this property of birefringence can easily be tapped to study their orientation under polarized light. Mathematically, Here, ne and n0 are the refractive indices experienced by the extraordinary and ordinary rays, respectively. A related term, the retardance (optical path difference) is defined by the relative phase shift between the ordinary and extraordinary rays, as they emerge from the anisotropic material. In general, for a material having uniform refractive indices ni and n2 and thickness t: In other words, Retardance = Birefringence times the Thickness of the birefringent medium Birefringence = |ne - n0| (6) Retardance, AO = (ni - n2) \u2022 t (7) 16 For calamitic liquid crystals (e.g. 5CB), the plane of polarization of the e-wave always contains the director n, whereas the o-wave is always polarized normally to n. Consider a nematic medium confined between two glass plates and placed between crossed polarizers. Let the director n be in the plane of the plates. Unpolarized light upon passing through the first polarizer becomes linearly polarized with intensity I0 = E2, where E is the amplitude of the wave. When this wave passes through the nematic sample, it splits into two mutually perpendicular ordinary and extraordinary waves. As discussed before due to the different refractive indices experienced by these waves as they exit they emerge with a net phase difference , A O = 2nd (ne - n0) \/ Xo, where A<, is the wavelength of light in vacuum. The second polarizer (also called the analyzer) with optic axis perpendicular to the first one (crossed state), transforms this phase difference into the pattern of transmitted light intensity, thus eventually yielding [Kleman and Lavrentovich 2003]: with the condition that n is perpendicular to z-axis, and (3 is the angle between local n and polarization direction of incident light. However, if n makes a constant angle 9 with z-axis, the transmitted intensity would be: Thus, wherever n is parallel to either polarizer the propagating wave is either purely ordinary or purely extraordinary, and the corresponding region in the sample would appear dark. Similarly, n making an angle of 45\u00b0 with the polarizer axis would result in regions of maximum intensity. 2.3 Quantitative Imaging using Abrio The use of a quantitative imaging system is made for the measurement of liquid crystal director in the region of interest. The imaging system is akin to using a rotating polarizer, with the advantage that it eliminates the 90\u00b0 orientation ambiguity present when viewing through crossed 7 = \/0sin2 2(3sin2 ^\u2014 (ne \u2014 n0) , (8) (9) 17 polarizers. It uses a liquid crystal compensator device and special image processing algorithms to measure optical polarization parameters at many points simultaneously, in fast time intervals, thus generating a birefringence map from measured intensities. This map depicts the retardance and slow axis orientation at the pixel level. Abrio has been used previously to successfully measure details of living cells, viz. the spindles in dividing cells, monitor vesicle trafficking, observe cell-to-cell interactions, and monitor cytoskeletal reconstruction [Katoh et al. 1997; Oldenbourg 1999; Oldenbourg et al. 1998]. An application of Abrio to observe liquid crystalline state appeared recently [Wu and Mather 2005] where Abrio was used successfully to map the orientation of liquid crystalline polymer near the nematic isotropic interface. The elementary components of the system include: a CCD digital camera, a liquid-crystal (LC) compensator optic, a circular polarizer\/interference filter optic (CP\/IF) and an image processing software (see appendix for layout). The CCD camera is connected to the LC compensator and is interfaced with the computer via USB. The compensator fits into the analyzer position in the microscope. The circular polarizer creates monochromatic circularly polarized light, while the LC compensator changes the polarization states according to commands from the software. Note that for the operation of Abrio's quantitative imaging module, other optical elements like the linear polarizers, have to be disengaged completely from the light path and the LC compensator should be inserted in place of the analyzer. The liquid crystal we use (see section 2.11) is positively uniaxial [Cognard 1982], hence the actual director orientation is parallel to the slow axis direction calculated by Abrio. With the highest magnification objective of 40x, retardances at a pixel level of 0.25 pm can be measured, and the orientation vector's length is averaged over 8 such pixels (2 pm). Abrio's other specifications can be found at the appendix. As a reference standard for Abrio's prediction capability on the setup we conducted a series of benchmarks, some of which are mentioned later in this chapter. 2.3.1 Restriction imposed by Abrio A limitation of Abrio for samples which produce retardation greater than half the wavelength of light used (X12 = 273 nm) is that it depicts the calculated slow-axis director in a manner which is turned by 90 degrees with respect to the true orientation, for sample retardance values lying 18 between every mXI2 and (m+l)X\/2 , where m =1,3,5... According to Oldenbourg [Goldman and Spector 2005], the expressions Abrio uses for calculating retardance and azimuth are: Retardance, ^A2 + B2 R - arctan R= 180\u00b0 -arctan JA2 + B2 ifi2 + I3 -21,>0and ifl2 + l 3 -21, <0 (10) Azimuth, 1 IA O^arctanl-) (11) Here, I,, I2, I3 and I4 are the measured instantaneous image intensity values for different compensator settings and A and B are the intermediate results based on image intensities. However, if the actual retardance is in the range X12 < Ro < X, the measured retardance is: R = X - Ro and azimuth is turned by 90\u00b0, and if X