Liquid Crystals: New Perspectives [1 ed.] 1789450403, 9781789450408

Table of contents :
Cover
Half-Title Page
Title Page
Copyright Page
Contents
Preface
1. Singular Optics of Liquid Crystal Defects
1.1. Prelude from carrots
1.2. Liquid crystals, optics and defects: a long-standing trilogy
1.3. Polarization optics of liquid crystals: basic ingredients
1.3.1. The few liquid crystal phases at play in this chapter
1.3.2. Liquid crystals anisotropy and its main optical consequence
1.3.3. Polarization state representation in the paraxial regime
1.3.4. Polarization state evolution through uniform director fields
1.3.5. Effective birefringence
1.4. Liquid crystal reorientation under external fields
1.5. Customary optics from liquid crystal defects
1.5.1. Localized defects structures in frustrated cholesteric films
1.5.2. Elongated defects structures in frustrated cholesteric films
1.5.3. Regular optics from other topological structures
1.5.4. Assembling photonic building blocks with liquid crystal defects
1.6. From regular to singular optics
1.6.1. What is singular optics?
1.6.2. A nod to liquid crystal defects
1.6.3. Singular paraxial light beams
1.6.4. Generic singular beam shaping strategies
1.7. Advent of self-engineered singular optical elements enabled by liquid crystals defects
1.7.1. Optical vortices from a cholesteric slab: dynamic phase option
1.7.2. Optical vortices from a nematic droplet: geometric phase option
1.8. Singular optical functions based on defects: a decade of advances
1.8.1. Custom-made singular dynamic phase diffractive
1.8.2. Spontaneous singular geometric phase optics
1.8.3. Directed self-engineered geometric phase optics
1.8.4. From single to arrays of optical vortices
1.9. Emerging optical functionalities enabled by liquid crystal defects
1.9.1. Spectrally and spatially adaptive optical vortex coronagraphy
1.9.2. Multispectral management of optical orbital angular momentum
1.10. Conclusion
1.11. References
2. Control of Micro-Particles with Liquid Crystals
2.1. Introduction
2.2. Control of micro-particles by liquid crystal-enabled electrokinetics
2.2.1. Liquid-crystal enabled electrophoresis
2.2.2. Liquid crystal-enabled electro-osmosis
2.3. Controlled dynamics of microswimmers in nematic liquid crystals
2.4. Conclusion
2.5. Acknowledgments
2.6. References
3. Thermomechanical Effects in Liquid Crystals
3.1. Introduction
3.2. The Ericksen–Leslie equations
3.2.1. Conservation equations
3.2.2. Molecular field
3.2.3. Constitutive equations
3.3. Molecular dynamics simulations of the thermomechanical effect
3.3.1. Molecular models
3.3.2. Constrained ensembles
3.3.3. Computation of the transport coefficients
3.3.4. Analysis of the results
3.4. Experimental evidence of the thermomechanical effect
3.4.1. The static Éber and Jánossy experiment
3.4.2. Another static experiment proposed in the literature
3.4.3. Continuous rotation of translationally invariant configurations
3.4.4. Drift of cholesteric fingers under homeotropic anchoring
3.5. The thermohydrodynamical effect
3.5.1. A proposal for measuring the TH Leslie coefficient μ: theoretical
prediction
3.5.2. About the measurement of the TH Akopyan and Zel’dovich
coefficients
3.6. Conclusions and perspectives
3.7. References
4. Physics of the Dowser Texture
4.1. Introduction
4.1.1. Disclinations and monopoles
4.1.2. Road to the dowser texture
4.1.3. The dowser texture
4.2. Generation of the dowser texture
4.2.1. Setups called “Dowsons Colliders”
4.2.2. “Classical” generation of the dowser texture
4.2.3. Accelerated generation of the dowser texture using the DDC2
setup
4.3. Flow-assisted homeotropic ⇒ dowser transition
4.3.1. Experiment using the DDC2 setup
4.3.2. Flow-assisted bowser-dowser transformation in capillaries
4.3.3. Flow-assisted homeotropic-dowser transition in the CDC2 setup
4.3.4. Theory of the flow-assisted homeotropic-dowser transition
4.3.5. Summary and discussion of experimental results
4.4. Rheotropism
4.4.1. The first evidence of the rheotropism
4.4.2. Synchronous winding of the dowser field
4.4.3. Asynchronous winding of the dowser field
4.4.4. Hybrid winding of the dowser field with CDC2
4.4.5. Rheotropic behavior of π- and 2π-walls
4.4.6. Action of an alternating Poiseuille flow on wound up dowser fields
4.5. Cuneitropism, solitary 2π-walls
4.5.1. Generation of π-walls by a magnetic field
4.5.2. Generation and relaxation of circular 2π-walls
4.5.3. Cuneitropic origin of the circular 2π-wall
4.6. Electrotropism
4.6.1. Definition of the electrotropism
4.6.2. Flexo-electric polarization
4.6.3. Setup
4.6.4. The first evidence of the flexo-electric polarization
4.6.5. Measurements of the flexo-electric polarization
4.7. Electro-osmosis
4.7.1. One-gap system of electrodes
4.7.2. Two-gap system of electrodes
4.7.3. Convection of the dowser field
4.8. Dowser texture as a natural universe of nematic monopoles
4.8.1. Structures and topological charges of nematic monopoles
4.8.2. Pair of dowsons d+ and d- seen as a pair of monopoles
4.8.3. Generation of monopole–antimonopole pairs by breaking 2π-walls
4.9. Motions of dowsons in a wound up dowser field
4.9.1. Single dowson in a wound up dowser field
4.9.2. The Lorentz-like force
4.9.3. Velocity of dowsons in wound up dowser fields
4.9.4. The race of dowsons
4.9.5. Trajectories of dowsons observed in natural light
4.9.6. Trajectories of dowsons observed in polarized light
4.10. Collisions of dowsons
4.10.1. Pair of dowsons (d+,d-) inserted in a wound up dowser field
4.10.2. Cross-section for annihilation of dowsons’ pairs
4.10.3. Rheotropic control of the collisions outcome
4.11. Motions of dowsons in homogeneous fields
4.12. Stabilization of dowsons systems by inhomogeneous fields with
defects
4.12.1. Gedanken experiment
4.12.2. Triplet of dowsons stabilized in MBBA by a quadrupolar electric
field
4.12.3. Septet of dowsons in MBBA stabilized by a quadrupolar electric
field
4.12.4. Dowsons d+ stabilized by corner singularities of the electric field
4.13. Dowser field submitted to boundary conditions with more complex
geometries and topologies
4.13.1. Ground state of the dowser field in an annular droplet
4.13.2. Wound up metastable states of the dowser field in the annular
droplet
4.13.3. Dowser field in a square network of channels, four-arm junctions
4.13.4. Triangular network, six-arm junctions
4.13.5. Three-arm junctions
4.13.6. General discussion of n-arm junctions
4.14. Flow-induced bowson-dowson transformation
4.15. Instability of the dowson’s d- position in the stagnation point
4.16. Appendix 1: equation of motion of the dowser field
4.16.1. Elastic torque
4.16.2. Viscous torques
4.16.3. Magnetic torque
4.16.4. Electric torque
4.17. References
5. Spontaneous Emergence of Chirality
5.1. Introduction
5.2. Chirality: a historical tour
5.2.1. Chirality and optics
5.2.2. Chiral symmetry breaking and its misuse
5.2.3. Spontaneous emergence of chirality or chiral structures in liquid
crystals
5.2.4. Spontaneous emergence of chirality due to confinement
5.2.5. Spontaneous emergence of chirality due to cylindrical confinement
5.2.6. Some misconceptions about optical rotation
5.3. Concluding remarks
5.4. Acknowledgments
5.5. References
List of Authors
Index

Citation preview

Liquid Crystals

SCIENCES Physics of Soft Matter, Field Director – Françoise Brochard-Wyart Lyotropic and Thermotropic Liquid Crystals, Subject Heads – Pawel Pieranski and Maria Helena Godinho

Liquid Crystals New Perspectives

Coordinated by

Pawel Pieranski Maria Helena Godinho

First published 2021 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd 27-37 St George’s Road London SW19 4EU UK

John Wiley & Sons, Inc. 111 River Street Hoboken, NJ 07030 USA

www.iste.co.uk

www.wiley.com

© ISTE Ltd 2021 The rights of Pawel Pieranski and Maria Helena Godinho to be identified as the author of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress Control Number: 2021934282 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-78945-040-8 ERC code: PE3 Condensed Matter Physics PE3_13 Structure and dynamics of disordered systems: soft matter (gels, colloids, liquid crystals, etc.), liquids, glasses, defects, etc.

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

Chapter 1. Singular Optics of Liquid Crystal Defects . . . . . . . . .

1

Etienne BRASSELET 1.1. Prelude from carrots . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Liquid crystals, optics and defects: a long-standing trilogy. . . . . 1.3. Polarization optics of liquid crystals: basic ingredients . . . . . . . 1.3.1. The few liquid crystal phases at play in this chapter . . . . . 1.3.2. Liquid crystals anisotropy and its main optical consequence 1.3.3. Polarization state representation in the paraxial regime . . . 1.3.4. Polarization state evolution through uniform director fields . 1.3.5. Effective birefringence . . . . . . . . . . . . . . . . . . . . . . . 1.3.6. Polarization state evolution through twisted director fields . 1.4. Liquid crystal reorientation under external fields . . . . . . . . . . 1.5. Customary optics from liquid crystal defects . . . . . . . . . . . . . 1.5.1. Localized defects structures in frustrated cholesteric films . 1.5.2. Elongated defects structures in frustrated cholesteric films . 1.5.3. Regular optics from other topological structures . . . . . . . 1.5.4. Assembling photonic building blocks with liquid crystal defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6. From regular to singular optics . . . . . . . . . . . . . . . . . . . . . 1.6.1. What is singular optics? . . . . . . . . . . . . . . . . . . . . . . 1.6.2. A nod to liquid crystal defects . . . . . . . . . . . . . . . . . . 1.6.3. Singular paraxial light beams . . . . . . . . . . . . . . . . . . .

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1.6.4. Generic singular beam shaping strategies . . . . . . . . . . . . . 1.7. Advent of self-engineered singular optical elements enabled by liquid crystals defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.1. Optical vortices from a cholesteric slab: dynamic phase option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.2. Optical vortices from a nematic droplet: geometric phase option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8. Singular optical functions based on defects: a decade of advances . 1.8.1. Custom-made singular dynamic phase diffractive optics . . . . 1.8.2. Spontaneous singular geometric phase optics . . . . . . . . . . 1.8.3. Directed self-engineered geometric phase optics . . . . . . . . 1.8.4. From single to arrays of optical vortices . . . . . . . . . . . . . 1.9. Emerging optical functionalities enabled by liquid crystal defects . 1.9.1. Spectrally and spatially adaptive optical vortex coronagraphy 1.9.2. Multispectral management of optical orbital angular momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.11. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 2. Control of Micro-Particles with Liquid Crystals . . . . . .

81

Chenhui PENG and Oleg D. LAVRENTOVICH 2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Control of micro-particles by liquid crystal-enabled electrokinetics 2.2.1. Liquid-crystal enabled electrophoresis . . . . . . . . . . . . . . 2.2.2. Liquid crystal-enabled electro-osmosis . . . . . . . . . . . . . . 2.3. Controlled dynamics of microswimmers in nematic liquid crystals . 2.4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 3. Thermomechanical Effects in Liquid Crystals. . . . . . .

81 82 85 91 96 104 107 107 117

Patrick OSWALD, Alain DEQUIDT and Guilhem POY 3.1. Introduction . . . . . . . . . . . . 3.2. The Ericksen–Leslie equations 3.2.1. Conservation equations . 3.2.2. Molecular field . . . . . . 3.2.3. Constitutive equations . .

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Contents

3.3. Molecular dynamics simulations of the thermomechanical effect . 3.3.1. Molecular models . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2. Constrained ensembles . . . . . . . . . . . . . . . . . . . . . . . 3.3.3. Computation of the transport coefficients . . . . . . . . . . . . 3.3.4. Analysis of the results . . . . . . . . . . . . . . . . . . . . . . . 3.4. Experimental evidence of the thermomechanical effect . . . . . . . 3.4.1. The static Éber and Jánossy experiment . . . . . . . . . . . . . 3.4.2. Another static experiment proposed in the literature . . . . . 3.4.3. Continuous rotation of translationally invariant configurations . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.4. Drift of cholesteric fingers under homeotropic anchoring . . 3.5. The thermohydrodynamical effect . . . . . . . . . . . . . . . . . . . 3.5.1. A proposal for measuring the TH Leslie coefficient μ: theoretical prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2. About the measurement of the TH Akopyan and Zel’dovich coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6. Conclusions and perspectives . . . . . . . . . . . . . . . . . . . . . . 3.7. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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130 130 131 133 134 135 136 140

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178 184 185

Chapter 4. Physics of the Dowser Texture . . . . . . . . . . . . . . . . .

193

Pawel PIERANSKI and Maria Helena GODINHO 4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1. Disclinations and monopoles . . . . . . . . . . . . . . 4.1.2. Road to the dowser texture . . . . . . . . . . . . . . . 4.1.3. The dowser texture . . . . . . . . . . . . . . . . . . . . 4.2. Generation of the dowser texture . . . . . . . . . . . . . . . 4.2.1. Setups called “Dowsons Colliders” . . . . . . . . . . 4.2.2. “Classical” generation of the dowser texture . . . . . 4.2.3. Accelerated generation of the dowser texture using the DDC2 setup . . . . . . . . . . . . . . . . . . . . . . . . . .

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4.3. Flow-assisted homeotropic ⇒ dowser transition . . . . . . . . . . . 4.3.1. Experiment using the DDC2 setup . . . . . . . . . . . . . . . . 4.3.2. Flow-assisted bowser-dowser transformation in capillaries . 4.3.3. Flow-assisted homeotropic-dowser transition in the CDC2 setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4. Theory of the flow-assisted homeotropic-dowser transition . 4.3.5. Summary and discussion of experimental results . . . . . . .

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4.4. Rheotropism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1. The first evidence of the rheotropism . . . . . . . . . . . . . . . 4.4.2. Synchronous winding of the dowser field . . . . . . . . . . . . . 4.4.3. Asynchronous winding of the dowser field . . . . . . . . . . . . 4.4.4. Hybrid winding of the dowser field with CDC2 . . . . . . . . . 4.4.5. Rheotropic behavior of π- and 2π-walls . . . . . . . . . . . . 4.4.6. Action of an alternating Poiseuille flow on wound up dowser fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5. Cuneitropism, solitary 2π-walls . . . . . . . . . . . . . . . . . . . . . 4.5.1. Generation of π-walls by a magnetic field . . . . . . . . . . . . 4.5.2. Generation and relaxation of circular 2π-walls . . . . . . . . . 4.5.3. Cuneitropic origin of the circular 2π-wall . . . . . . . . . . . . 4.6. Electrotropism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1. Definition of the electrotropism . . . . . . . . . . . . . . . . . . 4.6.2. Flexo-electric polarization . . . . . . . . . . . . . . . . . . . . . . 4.6.3. Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.4. The first evidence of the flexo-electric polarization . . . . . . . 4.6.5. Measurements of the flexo-electric polarization . . . . . . . . . 4.7. Electro-osmosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.1. One-gap system of electrodes . . . . . . . . . . . . . . . . . . . . 4.7.2. Two-gap system of electrodes . . . . . . . . . . . . . . . . . . . 4.7.3. Convection of the dowser field . . . . . . . . . . . . . . . . . . . 4.8. Dowser texture as a natural universe of nematic monopoles . . . . . 4.8.1. Structures and topological charges of nematic monopoles . . . 4.8.2. Pair of dowsons d+ and d- seen as a pair of monopoles . . . . 4.8.3. Generation of monopole–antimonopole pairs by breaking 2π-walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9. Motions of dowsons in a wound up dowser field . . . . . . . . . . . . 4.9.1. Single dowson in a wound up dowser field . . . . . . . . . . . . 4.9.2. The Lorentz-like force . . . . . . . . . . . . . . . . . . . . . . . . 4.9.3. Velocity of dowsons in wound up dowser fields. . . . . . . . . 4.9.4. The race of dowsons . . . . . . . . . . . . . . . . . . . . . . . . . 4.9.5. Trajectories of dowsons observed in natural light . . . . . . . . 4.9.6. Trajectories of dowsons observed in polarized light . . . . . . 4.10. Collisions of dowsons . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.10.1. Pair of dowsons (d+,d-) inserted in a wound up dowser field 4.10.2. Cross-section for annihilation of dowsons’ pairs . . . . . . . . 4.10.3. Rheotropic control of the collisions outcome . . . . . . . . . . 4.11. Motions of dowsons in homogeneous fields . . . . . . . . . . . . . .

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Contents

4.12. Stabilization of dowsons systems by inhomogeneous fields with defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.12.1. Gedanken experiment . . . . . . . . . . . . . . . . . . . . . . 4.12.2. Triplet of dowsons stabilized in MBBA by a quadrupolar electric field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.12.3. Septet of dowsons in MBBA stabilized by a quadrupolar electric field` . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.12.4. Dowsons d+ stabilized by corner singularities of the electric field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.13. Dowser field submitted to boundary conditions with more complex geometries and topologies . . . . . . . . . . . . . . . . . . . . 4.13.1. Ground state of the dowser field in an annular droplet . . 4.13.2. Wound up metastable states of the dowser field in the annular droplet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.13.3. Dowser field in a square network of channels, four-arm junctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.13.4. Triangular network, six-arm junctions . . . . . . . . . . . . 4.13.5. Three-arm junctions . . . . . . . . . . . . . . . . . . . . . . . 4.13.6. General discussion of n-arm junctions . . . . . . . . . . . . 4.14. Flow-induced bowson-dowson transformation . . . . . . . . . . 4.15. Instability of the dowson’s d- position in the stagnation point . 4.16. Appendix 1: equation of motion of the dowser field . . . . . . . 4.16.1. Elastic torque . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.16.2. Viscous torques . . . . . . . . . . . . . . . . . . . . . . . . . 4.16.3. Magnetic torque . . . . . . . . . . . . . . . . . . . . . . . . . 4.16.4. Electric torque . . . . . . . . . . . . . . . . . . . . . . . . . . 4.17. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 5. Spontaneous Emergence of Chirality . . . . . . . . . . . .

311

Mohan SRINIVASARAO 5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Chirality: a historical tour . . . . . . . . . . . . . . . . . . . . . . 5.2.1. Chirality and optics . . . . . . . . . . . . . . . . . . . . . . . 5.2.2. Chiral symmetry breaking and its misuse . . . . . . . . . . 5.2.3. Spontaneous emergence of chirality or chiral structures in liquid crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.4. Spontaneous emergence of chirality due to confinement . 5.2.5. Spontaneous emergence of chirality due to cylindrical confinement . . . . . . . . . . . . . . . . . . . . . . . . 5.2.6. Some misconceptions about optical rotation . . . . . . . .

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5.3. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

341 342 342

List of Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

347

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

349

Preface

The story of liquid crystals is that of a transgression of frontiers and a breaking of common beliefs. Indeed, liquid crystals were discovered by a biochemist, Friedrich Reinitzer, who shared his perplexity with a physicist, Otto Lehmann. Subsequently, crystallographers, mineralogists, experts in optics, magnetism, thermodynamics, chemists, etc., contributed to the recognition of liquid crystals as new states of matter, called mesophases by George Friedel, endowed with structures intermediate to those of isotropic liquids and crystals. Mesophases are so fragile that they can be easily perturbed by moderate mechanical, electric, magnetic or thermal stresses that generate various defects, such as disclinations, dislocations, walls, monopoles or multipoles. Remarkably, these defects were also observed to appear spontaneously during phase transitions. Inspired by experimental discoveries, theorists and mathematicians became fascinated by the symmetries of mesophases and recognized these beautiful and mysterious defects as topological singularities of vectorial, tensorial or complex fields. This led to the introduction of the concept of order parameters, resulting from symmetry breaking. In this frame, the generation of topological defects during certain phase transitions appeared as a natural, expected consequence. It was thus appreciated that the physics of liquid crystals and their defects had a permeable border with the Kibble–Zurek mechanism in cosmology. The discovery of liquid crystals was made in substances extracted from carrots. Prior to that, Mettenheimer and Virchov observed birefringent lamellar mesophases in aqueous solutions of myelin, a substance extracted from nerve tissues. Mesophases

Liquid Crystals, coordinated by Pawel P IERANSKI, Maria Helena G ODINHO. © ISTE Ltd 2021.

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also appear in solutions of cellulosic compounds or in suspensions of elongated particles, such as cellulose nanocrystals, tobacco mosaic viruses or bacteria. Therefore, from their origin, liquid crystals share a border with living matter. In this book, the borders with cosmology and living matter, as well as other borders will be crossed several times. The title Liquid Crystals – New Perspectives announces that when borders are crossed, new perspectives are unlocked. The first chapter, Singular Optics of Liquid Crystal Defects, explores a terra incognita: the light–matter interaction that occurs when light and/or matter are endowed with topological defects. It tells how the physics of topological defects in liquid crystals has progressively enriched liquid crystal optics as singular optics have developed. This is rooted in a few generic features: the self-organization capabilities of soft matter systems and their sensitivity to external stimuli. Remarkably, all of the assets of liquid crystals that make them ubiquitous in our daily life as liquid crystal displays, now open up several options to explore fundamental optical phenomena, based on the coupling between the polarization and spatial degrees of freedom. Also, this allows us to envision the development of future applications, for instance in optical imaging, beam shaping and the manipulation of matter by light. The second chapter, Control of Micro-Particles with Liquid Crystals, transgresses two frontiers. First, that between liquid crystals and electro-kinetic phenomena. The authors show how electrophoresis and electro-osmosis, well known from studies with isotropic liquids, acquire new unexpected features when the anisotropy and nonlinear hydrodynamics of liquid crystals are switched on. The second transgression is that of the frontier with living matter. The quasi-random swimming motion of flagellated bacteria in isotropic liquids becomes ordered in lyotropic liquid crystals and even more fascinating in the presence of topological defects, disclinations. The third chapter, Thermomechanical Effects in Liquid Crystals, crosses the border between liquid crystals and out-of-equilibrium thermodynamics. Anisotropy and nonlinear hydrodynamics of liquid crystals radically alter the phenomena driven by thermal gradients. For example, the threshold of the Rayleigh–Bénard instability in nematics is known to be lowered by a factor of 1000 with respect to isotropic liquids. The thermomechanical and thermohydrodynamical effects discussed in this chapter are even more original because they do not exist in isotropic fluids. Once again, they become fascinating in the presence of topological defects. This chapter breaks the common belief about a unique explanation, in terms of the Leslie theory, of the Lehmann effect, i.e. the rotation of cholesteric droplets driven by a thermal gradient. The fourth chapter, Physics of the Dowser Texture, transgresses the common belief about the ephemeral character of a distorted nematic texture, bearing varying names in the past: splay-bend state, H state, inversion wall, quasi-planar or

Preface

xiii

flow-aligned. Dubbed as the dowser texture because of its resemblance with the wooden tool of dowsers, this texture is in fact not unstable but only metastable and in certain conditions can be preserved indefinitely. As its complex order parameter eiϕ is degenerated with respect to the phase ϕ, the dowser texture is sensitive to vector fields and as a consequence it is endowed with unprecedented properties called cuneitropism, rheotropism and electrotropism. The dowser texture also appears as a natural universe of nematic monopoles and antimonopoles that can be set in motion and brought to collisions, resulting in the annihilation of monopole–antimonopole pairs, analogous to the annihilation of electron-positron pairs in hadron colliders. The fifth chapter, Spontaneous Emergence of Chirality, deals with nematics, which are made up of achiral molecules. It begins with a very detailed historical tour leading to the birth of the concept of chirality. The tour starts with the discovery of double refraction in Iceland spar crystals by Erasmus Bartolinus in 1669, followed by the crucial contributions of Huygens, Malus, Arago, Brewster, Biot, Fresnel and Faraday, leading to the epoch marking the discovery of molecular chirality by Louis Pasteur. The work of Pasteur gave birth to a common belief that the optical activity of materials results from the existence of molecular chirality. The observations of chiral textures in lyotropic nematics discussed in this fifth chapter break this common belief and unlock new perspectives. We would like to thank Mme Françoise Brochard-Wyart for the invitation to write this book. Its contours were fixed thanks to Tigran Galstian, who invited six of the authors to the 18th Conference on Optics of Liquid Crystals, in Quebec. The printing in color of this book was sponsored by : 1) Marie Curie Grant N¡838199, 2) Department of Physics and Materials Science at The University of Memphis, Memphis, Tennessee (USA), 3) National Science Foundation (USA) grant DMR-1905053 entitled “Active colloids with tunable interactions in liquid crystals”, 4) Laboratoire de Physique at Ecole Normale Supérieure de Lyon, Lyon (France), 5) Laboratoire de Physique des Solides at Université Paris-Saclay, Orsay (France), 6) i3N/CENIMAT, NOVA School of Science and Technology, NOVA University of Lisbon (Portugal), 7) School of Materials Science and Engineering at Georgia Institute of Technology, Atlanta, Georgia (USA). The writing of this book lasted for about one year. Once the work was finished, we were tempted to say, following the Eulogy of Leonhard Euler by Marquis de Condorcet, that: ...the pleasure to work is a sweeter reward than glory... On behalf of all the authors March 2021

1

Singular Optics of Liquid Crystal Defects Etienne B RASSELET Laboratoire Ondes et Matière d’Aquitaine (LOMA), Talence, France

1.1. Prelude from carrots In the late 19th century, a carrot entered into the history of material sciences: as Reinitzer scrutinized the temperature behavior of cholesterol derivatives extracted from a carrot, he noticed “two melting points” (Reinitzer 1989). He thought that his discovery was worthy for consideration by physicists and contacted Lehmann in 1888 for further investigation. These were the early days of thermotropic liquid crystals, for which the temperature is the state parameter. Singularly, defects and light, with all of its colors, are ubiquitous to the observation of liquid crystals (Dierking 2003). This chapter focuses on the singular optical aspects of liquid crystal defects, which are discussed in section 1.6 after having reviewed the regular counterparts in section 1.5. For the sake of consistency, the chapter starts by providing the reader with the necessary, but non-exhaustive, information about liquid crystal structures, optics and behavior under external fields. A jump start is given by Figure 1.1, where the carrot’s orientation rotates by 2π per full-turn around the normal direction to the table, providing an artistic wink at liquid crystals, light and defects. 1.2. Liquid crystals, optics and defects: a long-standing trilogy The pioneering experimental investigations by Lehmann were made possible by an instrument he developed, a temperature-controlled polarizing optical microscope

Liquid Crystals, coordinated by Pawel P IERANSKI, Maria Helena G ODINHO. © ISTE Ltd 2021. Liquid Crystals: New Perspectives, First Edition. Pawel Pieranski and Maria Helena Godinho. © ISTE Ltd 2021. Published by ISTE Ltd and John Wiley & Sons, Inc.

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Liquid Crystals

(see Figure 1.2(a), in which Lehmann is standing nearby his apparatus). Two of his drawings representing the optical observations of the materials he received from Reinitzer, are presented in Figures 1.2(b) and (c). These sketches simply and beautifully celebrate the links between liquid crystals, optics and defects, which still stand vividly today, as this chapter will illustrate. In fact, these pictures demonstrate how irregular structures spontaneously occur in liquid crystals, which, once tamed, pave the way for several modern optics applications worth exploring.

Figure 1.1. Colorful arrangement of carrots recalling an orientational defect with unit charge. Picture taken from https://app.emaze.com/user/lisaadjeroud

On one hand, the black crosses in Figure 1.2(b) are related to local axisymmetric supramolecular structuring of the anisotropic optical properties of liquid crystals. As we shall see, this is one of the situations enabling the interplay between the topology of liquid crystals and that of light. On the other hand, the threads running randomly through a uniform background in Figure 1.2(c) correspond to complex clusters of defects of various kinds, called “oily streaks” by Reitnizer in his first letter to Lehmann in 1888 (Kelker 1973). This illustrates how liquid crystal structures can either act as optical scatterers or be considered as tangible information recorded in the fluid, which will be also discussed. It took a few decades of abundant debates, experiments and conceptual developments (Kelker 1973, 1988; Lagerwall 2013) before Friedel articulated that liquid crystals represent mesomorphic states of matter, with the potential to express a rich polymorphism (Friedel 1922). In the latter work, Friedel also introduced the terms smectic, nematic and cholesteric. Such terminology refers to the different kinds of spatial organization of the constituting anisotropic building blocks of liquid crystals, and is still in use today. Nowadays, the classification is much richer and we refer to Oswald and Pieranski (2005) for a comprehensive overview.

Singular Optics of Liquid Crystal Defects






3



Figure 1.2. (a) Lehmann posing close to his crystallization microscope. (b and c) Sketches by Lehmann of some of his observations, dated from April 4, 1888 (Kelker and Knoll 1989) and August 30, 1889 (Kelker 1988), respectively, which emphasize the emergence of spontaneous irregular structures in liquid crystals

1.3. Polarization optics of liquid crystals: basic ingredients 1.3.1. The few liquid crystal phases at play in this chapter In this chapter, three types of thermotropic liquid crystals will enter the discussion: smectic A, nematic and cholesteric. All three types are made up of elongated molecules represented as cylinders in Figure 1.3, where n is a unit vector called director that refers to the local average orientation of the molecules and satisfies the equivalence n ≡ −n. From a structural point of view, the smectic A phase combines a positional order along one spatial dimension, illustrated as layers in Figure 1.3(a), with an orientational order associated with a director pointing along the normal to the layers. The nematic phase, illustrated in Figure 1.3(b), is only characterized by an orientational order. Finally, the cholesteric phase merely corresponds to a nematic phase endowed with supramolecular chirality, where the director field adopts a one dimensional helical order, in the absence of external constraints (see Figure 1.3(c)). A cholesteric liquid crystal is thus characterized by (i) its helix pitch and (ii) its right or left handedness. The helix pitch p is the distance over which the director rotates by 2π, although the space inversion invariance of the director implies that the physical period is p/2. 1.3.2. Liquid crystals anisotropy and its main optical consequence As suggested by the axisymmetric representation of the liquid crystal building blocks in Figure 1.3, here, we will deal with uniaxial smectic A and nematic phases. In other words, any of their physical property P (electrical, magnetic, optical, thermal, etc.) is characterized by two scalar quantities: one associated with the direction along n (P ) and the other associated with a plane perpendicular to it (P⊥ ). Regarding the cholesteric phase, it is usually prepared by adding a small fraction of a chiral dopant

4

Liquid Crystals

into a nematic phase. This has the advantage of preserving the physical properties of the host phase, while the helix pitch can be tuned at will, typically from infinity to a few hundred nanometers, by adjusting the nature and concentration of the dopant. 



 








  





  

  

    

  

Figure 1.3. Illustration of the three liquid crystal phases appearing in this chapter. Note that in panels (a) and (b) the orientation of the molecules fluctuates with time. In panel (c), the depicted regular helix made up of one molecule is only for the sake of illustration, recalling that there is no positional order and that there are microscopic orientational fluctuations

Discussing the anisotropic nature of liquid crystals leads to the introduction of one of its most salient features when dealing with direct observations by optical means: their birefringence. This refers to the fact that the electronic response of the material irradiated by light depends on its polarization state, which informs on how the real electric field vector E oscillates. Considering a fully polarized monochromatic paraxial light beam1, the trajectory of the tip of the electric field vector E at a given position lies in a plane transverse to the beam propagation direction and is elliptical. The latter ellipse is called the polarization ellipse. Disregarding the field intensity, the latter ellipse is specified by two parameters: its orientation angle 0 ≤ ψ ≤ π and its ellipticity angle −π/4 ≤ χ ≤ π/4 (see Figure 1.4). Note that the angles ψ and χ allow an arbitrary polarization state to be represented unequivocally by a point on the unit sphere, called the Poincaré sphere of polarization (Born and Wolf 2013), whose longitude and latitude are respectively given by the angles 2ψ and 2χ. The birefringence is defined as the difference between the refractive indices experienced by linearly polarized light oriented along a direction parallel and perpendicular to n, namely, dn = n − n⊥ . The birefringence explains the extent to which liquid crystals can modify the polarization state of light.

1 That is, neglecting the electric field component along the beam propagation direction and assuming that the two transverse components are mutually coherent. A well-known example is the uniformly polarized Gaussian beam.

Singular Optics of Liquid Crystal Defects

5

In order to quantitatively grasp how the polarization state of light can be modified as light propagates through liquid crystals, as a consequence of their optical anisotropy, we introduce a few simple, yet relevant, polarization optics facts that enable us to handle the concepts presented in this chapter. Of course, the optics of liquid crystals are by no means restricted to such a simple view. We can refer to Yeh and Gu (2009); Khoo and Wu (1993) for further reading about applied and fundamental aspects. 





























 


 
 

 


  


 


Figure 1.4. Polarization ellipses and their characteristic angles (ψ, χ). Linear, elliptical and circular polarization states refer to χ = 0 (a), 0 < |χ| < π/4 (b) and |χ| = π/4 (c), respectively. We define the polarization handedness as that of the helix formed by the electric field vector E at a given time. Namely, for a wave propagating toward z > 0, sketches with χ < 0 (blue color) referring to right handedness, while those with χ > 0 (red color) refer to left handedness

1.3.3. Polarization state representation in the paraxial regime Let us consider, from now on, a fully polarized monochromatic paraxial light beam and treat it as a plane wave, hence neglecting its longitudinal field component inherent to any real-world beam (Lax et al. 1975). We choose a complex field representation and consider light propagation in a vacuum toward z > 0, with a wave vector k0 = k0 z = 2π/λ0 z, λ0 being the wavelength in the vacuum. The electric field is thus represented, up to an unimportant phase factor, as E = E0 exp(−iωt + ik0 z) e

[1.1]

where E0 is the amplitude, ω is the angular frequency, t is the time and e is a complex unit vector that fully describes the polarization state. The connection to the real fields is simply made, noting that E is the real part of E, while the magnetic field is retrieved from Maxwell’s equations. Using the previously introduced angles ψ and χ that define the polarization ellipse (see Figure 1.4(b)), the general expression of the polarization vector e is, up to an unimportant phase factor, e(ψ, χ) = (cos ψ cos χ − i sin ψ sin χ) x + (sin ψ cos χ + i cos ψ sin χ) y

[1.2]

6

Liquid Crystals

where (x, y, z) refers to the unit vectors in a Cartesian coordinate system. This vector is at the basis of a matrix formalism introduced by Jones (1941) and named after him, which allows us to handle the polarization evolution through optical systems. The Jones vector is customarily represented as a two-dimensional column vector, whose components are ex = e · x and ey = e · y up to an unimportant phase factor. Table 1.1 summarizes the situations depicted in Figure 1.4, for which it is useful to introduce the rotation matrix that operates the change of coordinates of a vector upon a rotation by an oriented angle α around z (right-hand rule: α > 0 for a rotation from x to y): R(α) =

  cos α − sin α sin α cos α

[1.3]

Polarization ellipse Polarization parameters (ψ, χ) Figure 1.4(a)

(ψ, 0)

Figure 1.4(b)

(ψ, χ)

Figure 1.4(c)

(undetermined, ±π/4)

Jones vector   cos ψ sin ψ   cos χ R(ψ) i sin χ   1 1 √ 2 ±i

Table 1.1. Connection between various polarization quantities

In addition, in the framework of the adopted complex representation, it is worth recalling the decomposition of an arbitrary polarization state e0 on any of the infinite number of basis consisting of two orthogonal polarization states (e, e⊥ ), for which the orthogonality condition e · e∗⊥ = 0 implies e⊥ = e(ψ + π/2, −χ). Namely, e0 = (e0 · e∗ ) e + (e0 · e∗⊥ ) e⊥

[1.4]

and we note that the linear basis (x, y) and the circular basis (c+ , c− ), where c± = √12 (x ± iy), respectively refer to left-handed and right-handed circular polarization states, and are the most often used ones. 1.3.4. Polarization state evolution through uniform director fields Using the Jones formulation introduced above, the description of how the polarization state of light is modified as light propagates through liquid crystals can be readily understood on simple grounds. It all starts with the fact that, when light propagates through an isotropic dielectric medium having a refractive index n, the electric field experiences an extra propagating phase exp[ik0 (n − 1)z] with respect

Singular Optics of Liquid Crystal Defects

7

to what happens in the vacuum, while the polarization state is maintained during the propagation. This is a priori no longer the case when the medium is uniaxial. Here, we address the simplest case of a liquid crystal, characterized by a uniform director n, whose orientation defines the optical axis direction, defining e0 as the polarization vector at z = 0 in the (x, y) frame. Two situations are discussed as follows: (i) n parallel to k0 and (ii) n perpendicular to k0 . – Case n  k0 : the electric field propagates while experiencing the refractive index n⊥ , whatever the polarization state. The polarization vector thus remains constant, whatever the propagation distance. Consequently, the electric field at z is expressed as E = E0 exp(−iωt + ik0 n⊥ z) e0

[1.5]

– Case n ⊥ k0 : the electric field propagates while experiencing refractive properties that now depend on the polarization state. Let us define an arbitrary orientation of the director in the (x, y) plane n = cos φ x + sin φ y and introduce the director frame (x , y  ) that is rotated by an angle φ around z, see Figure 1.5(a) in the case of θ = π/2. By definition, the refractive index experienced by the electric field component along the x axis is n , while along the y  axis it is n⊥ . The electric field at z in the (x, y) frame is therefore given in a matrix form as2   ik n z 0 e 0  R(−φ) e0 E = E0 exp(−iωt) R(φ) 0 eik0 n⊥ z

[1.6]

The above equation can be recast as E = E0 exp(−iωt + iΦdyn ) e

[1.7]

¯ z, with n ¯ = (n + n⊥ )/2, refers to a phase of a dynamic where Φdyn = k0 n (i.e. propagation) origin. In addition, the expression of the polarization vector is e=

  Δ i sin 2φ sin Δ cos Δ 2 + i sin 2 cos 2φ 2 e0 Δ i sin 2φ sin Δ cos Δ 2 2 − i sin 2 cos 2φ

[1.8]

where Δ = k0 dn z refers to a phase of an anisotropic origin, associated with the material birefringence dn = n − n⊥ . Discarding the Fresnel reflections at the interfaces, equation [1.8] thus provides a general expression describing the effect of a homogeneous anisotropic optical slab of thickness z and birefringence dn having an 2 Here, we use the tensorial calculus result stating that the expression in the (x, y) frame of a second-order tensor T , rotated by an angle φ around z, is given by [R(φ)] T [RT (φ)], where (.)T refers to the transpose operation. Also, applying a matrix to a vector implies column representation of the vector.

8

Liquid Crystals

in-plane optical axis upon normal incidence, which is usually the case for birefringent optical retarders. The generalization to an arbitrary orientation of the optical axis is addressed in the following section. 1.3.5. Effective birefringence Still considering a liquid crystal slab in the (x, y) plane, illuminated at normal incidence by a field given by equation [1.1], the director now has an arbitrary orientation, as depicted in Figure 1.5(a). Introducing the usual spherical angles (θ, φ), the director is thus defined as n = sin θ cos φ x + sin θ sin φ y + cos θ z. We learn from the optics of uniaxial media (Born and Wolf 2013) that there are two solutions for the forward propagating waves, dubbed as ordinary and extraordinary, which both have their wave vector aligned along the z axis. The electric field of the ordinary (o) wave oscillates perpendicularly to the plane defined by the vectors (n, z) and its propagation is associated with the refractive index no = n⊥ , whereas the extraordinary wave (e) lies in the plane (n, z) and is associated with the refractive index  1/2 ne (θ) = n n⊥ / n2 cos2 θ + n2⊥ sin2 θ

[1.9]

This leads to the introduction of the effective birefringence dneff = ne (θ) − n⊥ . This terminology follows the fact that the polarization evolution during propagation will be that of an effective slab with its optical axis lying in the (x, y) plane and a birefringence dneff . In other words, one can use equation [1.8], but replace dn by dneff , whose variation with respect to the tilt angle is illustrated in Figure 1.5(b). Such an effective polarization behavior should not make us forget that the e-wave is endowed with non-intuitive behavior (as “extraordinary” terminology suggests), even within a plane-wave framework. Indeed, the tensorial relationship between the electric and displacement fields implies that the electric field is generally not perpendicular to the wave vector (except for θ = 0 or π/2). Consequently, the direction of the energy flow given by the Poynting vector Pe , is tilted by an angle δ with respect to the wave vector ke (see Figure 1.5(c)). Using the Maxwell equation ∇ · De = 0, one can show that δ is given by tan δ = [(n2 − n2⊥ )/(2n2 n2⊥ )]n2e (θ) sin 2θ. In contrast, the Poynting vector and the wave vector are collinear for the ordinary wave. The naked eye real-world manifestation of it is the well-known double refraction phenomenon, when an incident beam entering a uniaxial slab may split into two orthogonally linearly polarized beams having distinct propagation directions, depending on the incident polarization state.

Singular Optics of Liquid Crystal Defects

9

1.3.6. Polarization state evolution through twisted director fields In the previous sections, we have considered situations where the director field is a constant, matching the unperturbed smectic or nematic phases, as illustrated in Figures 1.3(a) and (b). The case of cholesteric phases, even undistorted as shown in Figure 1.3(c), is associated with a richer optical behavior, as the medium is now both anisotropic and space variant, as reviewed in Belyakov et al. (1979). In a similar approach to that in the case of a uniform director field illuminated by a plane-wave propagating toward z > 0, see equation [1.1], we will discard the Fresnel reflections at interfaces when dealing with slabs. Also, we assume a cholesteric liquid crystal characterized by a uniform helix, whose revolution axis is referred to as the unit vector h (h and −h being equivalent) and discuss two situations: (i) h is parallel to k0 and (ii) h is perpendicular to k0 . Namely, we will consider a cholesteric slab with an input facet at z = 0 defined according to h  k0 : n(z) = cos(qz) x + χ sin(qz) y

[1.10]

h ⊥ k0 : n(x) = cos(qx) y + χ sin(qx) z

[1.11]

where q = 2π/p is the helical wave vector magnitude and χ = ±1 refers to right or left helix handedness, respectively. The helix pitch is thus an extra length scale that adds to the wavelength and it is not surprising that the ratio λ0 /p, or equivalently q/k0 , plays a role in the definition of distinct regimes in both situations (i) and (ii). With no aim of being exhaustive, we present a selected number of them that will be useful later. 



 










0.2



     

 

     

0.1





 

 



0

0

30

60

90



 






Figure 1.5. (a) Definition of the director in the spherical coordinate system. (b) Effective birefringence dneff versus θ at fixed average refractive index n ¯ = 1.6. (c) Extraordinary electric field (Ee ), displacement field (De ), Poynting vector (Pe ) and wave vector (ke ) in the plane (x , z). In contrast to the o-wave, Pe × ke = 0 when θ = (0, π/2), which signs the double refraction phenomenon

10

Liquid Crystals

1.3.6.1. Propagation parallel to the cholesteric helix: Mauguin regime The Mauguin regime refers to the so-called adiabatic propagation of light, when a linearly polarized light with azimuth angle ψ, either parallel or perpendicular to n, propagates without any alteration of its polarization state in the rotating frame associated with the director (see Figure 1.6). As stated by Mauguin at the end of his work (Mauguin 1911), this happens “when p is large with respect to λ”. Eo (z) = E0 exp(−iωt + ik0 n⊥ z) n(z) × z

[1.12]

Ee (z) = E0 exp(−iωt + ik0 n z) n(z)

[1.13]

The birefringent phase retardation between the e-wave and the o-wave thus accumulates during the propagation of light, as if the director field is untwisted, Δ = k0 (n − n⊥ )z and there is no e/o energy exchange. We usually refer to such a propagation regime as adiabatic, which has a practical importance as it is at the basis of the so-called twisted nematics technology in liquid crystal displays (Yeh and Gu 2009). 









 

Figure 1.6. Illustration of the Mauguin regime for a “long-pitch” left-handed cholesteric, whose simplified picturing is shown in purple color, over a material twist angle of π/2. For a sufficiently small wavelength with respect to the cholesteric pitch, the linearly polarized o-wave and e waves are optical propagation modes: their polarization vectors follow the twisted director field

How large the pitch should be, in order to satisfy the Mauguin regime deserves a comment. Indeed, quoting Mauguin, the present case has been introduced above within the limit λ0  p, though the Mauguin criterion usually reads λ0  p dn in textbooks. However, a careful look at the electromagnetic treatment by Mauguin himself, and others after him, indicates that the criterion emerging from the search√of the optical modes is λ2  14 (¯ n2 /¯ ) p2 dn2 with ¯ = (n2 + n2⊥ )/2, noting that n ¯ / ¯ deviates from unity by less than 1% for a birefringence up to dn = 0.4. This emphasizes the robustness of the Mauguin regime with respect to the “small” parameter λ/(p dn).

Singular Optics of Liquid Crystal Defects

11

1.3.6.2. Propagation parallel to the cholesteric helix: circular Bragg reflection As the dielectric tensor of a cholesteric varies along the helix axis, the existence of a backward wave is expected. This can be qualitatively understood by viewing the cholesteric as a stack of infinitely thin uniaxial slabs, whose in-plane optical axis orientation varies along z. The key point is that the anisotropic Fresnel reflection at each interface couples with the orthogonal polarization vectors. Considering the forward circular polarization vector c± , the process imparts an extra dephasing of a structural origin to the backward component c∓ , whose sign depends on the handedness of both light and matter. When the latter dephasing compensates the dynamic phase arising from backward propagation, all the of elementary contributions to the backward field interfere constructively: this is the circular Bragg reflection phenomenon. This only occurs when the helical structure of the electric field of a circular polarization state and the cholesteric helix are co-handed. In contrast, light is transmitted when the optical and material helices are contra-handed (see Figure 1.7 for a right-handed cholesteric helix). Note that the handedness of the circular polarization state is preserved at reflection as both the propagation direction and the polarization vector are flipped, which differs from the reflection off of a usual mirror. 


   

 
 
   

 



 

Figure 1.7. Circular Bragg reflection for a right-handed cholesteric. Typically, for wavelengths satisfying n⊥ p < λ0 < n p, one of the two circularly polarization states – the one with the same handedness as the cholesteric helix – is not allowed to propagate (a), while the other does (b). The black arrows refer to the electric field vectors at a given time, while the red helices are the corresponding helices of the vectors’ tip and the indices (i,r,t) refer to incident, reflected and transmitted optical waves, respectively

This situation refers to the existence of a polarization-dependent photonic bandgap, which corresponds to a wavelength range for which the propagation of one

12

Liquid Crystals

of the two circularly polarization states is forbidden, and is associated to an evanescent field for a sample with finite thickness. The selective circularly polarized reflections of cholesterics have been known, experimentally, for a long time (Friedel 1922). In fact, even those who are not familiar with the science of liquid crystals may have already had experience with the circular Bragg reflection outdoor, by simply observing the “chiral colors” of some beetles (Feller et al. 2017). Still, the detailed theoretical understanding of this resonant phenomenon took a long time, and we refer to Oseen (1933); De Vries (1951); Conners (1968); Kats (1971) as prominent contributions in the 20th century. Note that liquid crystals are not the only kind of materials exhibiting circular Bragg reflections, which may be encountered in any structurally chiral optical media (Faryad and Lakhtakia 2014). In addition, the resonant nature of the phenomenon associated with structural periodicity implies that substantial circular Bragg reflectance requires a sufficiently thick cholesteric slab. Indeed, the evanescent nature of the co-handed circularly polarized forward wave is expected to tunnel a slab with finite thickness, leading to non-ideal reflectance. How thick should a cholesteric slab be to behave as a mirror? This is quantitatively ascertained from the expression of the reflectance R for a slab with thickness L, by neglecting the reflection from the boundaries (see equation [3.10] in Belyakov et al. 1979). By treating the optical anisotropy as a perturbation we get the following simplified expression (Yeh and Gu 2009), R=

κ2 Q2 coth (QL) + (k − q)2 2

[1.14]

√ where Q2 = κ2 − (k − q)2 , κ = kε/2, ε = (n2 − n2⊥ )/(n2 + n2⊥ ) and k = 2π ¯ λ0  the wave vector inside the medium, noting that ε plays the role of the small parameter of the simplified problem. From Figure 1.8, we typically see that L > 10p gives a virtually perfect discriminatory circular polarization reflector over the photonic bandgap, whose edges correspond to λ∓ 0 = pn⊥, in the limit of small ε. Concretely, taking the usual material parameters n ¯ = 1.6 and dn = 0.2, we get a good circular Bragg reflector at λ0 ∼ 500 nm, with a ∼ 50 nm width bandgap for p ∼ 300 nm and L > 10p. We notice that the expression for the resonance wavelength defined as √ k = q often reads as λ0 = p¯ n in literature, while its rigorous expression is λ = p ¯ (Belyakov 0 √ et al. 1979). However, as mentioned previously, n ¯ ¯, hence the distinction is not significant. A similar comment holds √ for the band edges. Indeed, their rigorous expression corresponds to k± = q/ 1 ∓ ε (Belyakov et al. 1979), which simplifies to λ∓ 0 = pn⊥, in the limit of small ε. Also, we note the spectral oscillations of the reflectance outside the photonic bandgap in Figure 1.8, whose early quantitative experimental study can be found in Dreher et al. (1971). These oscillations are not associated with the reflections at the boundaries of the slab, which have been

Singular Optics of Liquid Crystal Defects

13

neglected. Instead, they result from the back and forth transfer of energy between the forward and the backward waves, as a consequence of the dephasing between the helical polarization source and the radiated backward wave, which presents a fruitful analogy with the nonlinear wave mixing processes discussed in Fu et al. (1987). 

 1 L=2p

 L=5p L=10p  k  /q

 

0.8 0.6



k+ /q

0.4 0.2 0 0.5

0.6

0.7

0.8

0.9

1

1.1








1.2

1.3

1.4

1.5

Figure 1.8. Reflectance spectrum of a normally incident co-handed circularly polarized entering a cholesteric slab with thickness L and pitch p, versus the reduced spatial frequency k/q. The values k± /q refer to the edges of the photonic bandgap associated with the cholesteric medium. The plot is made using n⊥ = 1.5 and n = 1.7

1.3.6.3. Propagation perpendicular to the cholesteric helix Here, we consider two optical regimes depending on the ratio λ0 /p, say, large and small with respect to unity. In order to qualitatively introduce this choice, we quote Friedel’s argument about the apparent negative optical anisotropy of cholesterics: “The negative optical axis of the cholesteric phase has, with respect to the molecule, an orientation normal to that of the positive optical axis of the nematic phase. This strongly supports the idea that this negative optical axis of cholesteric substances is only an apparent optical axis, due to a very strong twist around a direction normal to the actually positive optical axis of the elementary volume”. The mentioned apparent optical axis in the case of a strong twist can be reformulated as follows: when the pitch is small enough with respect to the wavelength, see Figure 1.9(a). The effective relative dielectric tensor at play, eff , is the x-average of the spatially modulated one. Namely, ⎞ ⎛ ⊥ 0 0 [1.15] eff = Rx (qx) local Rx (−qx)x = ⎝ 0 ¯ 0⎠ 0 0 ¯ in the (x, y, z) reference frame, where Rx refers to the 3D rotation matrix around x and local = diag(⊥ , ⊥ ,  ) is the local relative dielectric permittivity diagonal tensor in the director frame, with ⊥ = n2⊥ and  = n2 . The o-polarization vector √ is thus perpendicular to h and is associated with the refractive index ¯, whereas

14

Liquid Crystals

the e-polarization vector is parallel to h and associated with n⊥ . One thus retrieves a behavior similar to that discussed in section 1.3.4 when n ⊥ k0 . In contrast, when the pitch is large enough with respect to the wavelength (see Figure 1.9(b)), the light actually probes the local values of the spatially modulated dielectric tensor. In turn, the o-polarization vector is parallel to h and is associated with the refractive index n⊥ , whereas the e-polarization vector is perpendicular to h and associated with a spatially modulated refractive, as previously discussed in section 1.3.5, namely, ne (π/2 − qx), which is independent of the cholesteric handedness3. Consequently, the e-wave is diffracted by the medium, while the o-wave remains unaltered. A closely related experimental attempt has been reported in Subacius et al. (1997), which only partially mimics the presented ideal situation (see the inset of Figure 1.9(b)). In fact, any attempt to prepare a homogeneous in-plane cholesteric axis in the liquid crystal layer sandwiched between two substrates is doomed to fail under homogeneous orientational boundary conditions that are incompatible with a homogeneous helical director field. This leads to complex 3D orientational perturbations (Shiyanovskii et al. 2001), whose detailed optical diffraction consequences are non-trivial. 






   





  

 



 


























Figure 1.9. Definition and behavior of o- and e-waves incident on a cholesteric perpendicular to the cholesteric helix. (a) Regime p  λ0 : the medium behaves as negative uniaxial media with its optical axis directed along the helix axis. (b) Regime p  λ0 : the medium behaves as an anisotropic diffraction phase grating that maximizes the depth of the spatially modulated refractive index for the e-wave. The (i) wave vectors ke , i integer, refer to the diffraction orders. Inset: related experimental observations from (Subacius et al. 1997); top image: transmission image of the stripe pattern illuminated by e-polarized light; bottom image: far-field diffraction pattern

3 Note that ne (π/2 − qx)  n + dn cos(2qx) to a good approximation, with a relative error 2 less than 1% for the typical parameters n⊥ = 1.5 and n = 1.7. We thus deal with an almost sinusoidal phase grating.

Singular Optics of Liquid Crystal Defects

15

1.4. Liquid crystal reorientation under external fields Besides the polarization optics enabled by liquid crystals, another key ingredient deals with the manipulation of the director field by external fields, which is a necessary step in reconfiguring and adjusting any optical functionality. This is of course an extremely wide topic and we will purposely restrict the presentation below to the situations of interest here, which mainly deal with the director reorientation under electric fields. Considering liquid crystals as non-polar dielectrics4, the presence of an electric field polarizes the neutral medium. The induced polarization density is expressed as P = 0 χ(e) E, where 0 is the vacuum permittivity, χ(e) =  − 1 is the dielectric susceptibility tensor and the curly bold symbols refer to real vectors. The electric torque density acting on the polarized medium is therefore Γelectric = P × E = E × E. By expressing the dielectric permittivity tensor and the electric field into the director basis, we get ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⊥ 0 0 E⊥ E⊥ Γelectric = 0 ⎝ 0 ⊥ 0 ⎠⎝E⊥ ⎠ × ⎝E⊥ ⎠ = 0 a (n · E)(n × E) [1.16] 0 0  E E where a =  − ⊥ is the anisotropy of the dielectric permittivity, whose magnitude and sign depend on the frequency of the field. Therefore, the electric field director tends to reorient the director either along or perpendicularly to it, depending on the sign of the anisotropy, as illustrated in Figure 1.10. Also, the equivalence n ≡ −n transpires from the quadratic dependence of the torque on the director. Introducing the angle β between the director and the electric field vector, we deduce from equation [1.16] that β = 0 corresponds to a stable (respectively, unstable) orientation for a > 0 (respectively, a < 0), whereas β = π/2 corresponds to an unstable (respectively, stable) orientation for a > 0 (respectively, a < 0). Note that while equation [1.16] applies whatever the frequency of the electric field, it is relevant to consider the time-averaged torque when the frequency is large enough, with respect to the inverse of all of the other involved characteristic time scales in the problem. In particular, this is usually the case when dealing with the effect of optical fields. In such a situation, the complex representation for the field, introduced in section 1.3.3, is well suited and the effective electric torque is expressed as Γelectric =

1 0 a (n · E)(n × E∗ ) . 2

[1.17]

4 Liquid crystals usually contain ionic impurities that may lead to director reorientation, induced by charge displacements for DC electric fields or AC fields with low enough frequency. At a large enough frequency, typically above 1 kHz, ionic impurities can be considered as immobile.

16

Liquid Crystals




E













E

 


 






Figure 1.10. Illustration of the director reorientation in the presence of an electric field for positive (a) and negative (b) dielectric anisotropy

The extension to an external magnetic field B is readily made for liquid crystals without spontaneous magnetization, by noting that the smallness of the components of the magnetic susceptibility tensor χ(m) implies that the induced magnetization density (m) is expressed as M = μ−1 B, where μ0 is the vacuum permeability. The magnetic 0 χ torque density acting on the magnetized medium is Γmagnetic = M×B, and can thus be rewritten similarly as equation [1.16] according to Γmagnetic = μ−1 0 χa (n · B)(n × B) (m)

[1.18]

(m)

where χa = χ − χ⊥ . The same applies for the electric torque hold regarding the use of complex fields for oscillating magnetic fields. Note that for optical fields, both electric and magnetic fields are simultaneously at play. Still, since E ∼ Bc and 0 μ0 c2 = 1, where c is the speed of light in vacuum, and |χa /a |  1 implies that the magnetic contribution to the optical torque can be neglected. In general, determining the director field in the presence of external fields implies solving a non-trivial mechanical problem that involves elastic contributions arising from the gradients of the director field, orientational viscous dissipation, as well as the possible coupling between the positional (i.e. the existence of flow field) and the orientational degrees of freedom. This goes far beyond the scope of the present chapter and we refer to Oswald and Pieranski (2005) for a comprehensive overview of the problem. 1.5. Customary optics from liquid crystal defects So far we have introduced a few simple optical behaviors involving homogeneous liquid crystal director fields (gradient-free or with helical order), hence defect-free systems. Defects have long been considered as a scientific curiosity rather than an asset, though defects are ubiquitous to liquid crystals, as recalled in section 1.2. Nevertheless, the non-trivial topological structures of liquid crystals have been

Singular Optics of Liquid Crystal Defects

17

studied for their “regular” optical consequences – a possibly intriguing statement at first sight. Here, the adjective “regular” refers to an optical behavior that we would customarily encounter with defect-free spatial modulation of the dielectric tensor. Accordingly, we present a selected number of liquid crystal structures hosting either singular defects (i.e. the director field is undefined at some places) or non-singular defects (i.e. the director field is continuous everywhere) and whose usual optical functionalities have been explored (such as diffracting, scattering, lensing, steering, waveguiding, lasing and the self-assembly of optical materials mediated by liquid crystal defects). 1.5.1. Localized defects structures in frustrated cholesteric films 1.5.1.1. Brief historical survey This section introduces localized optically inhomogeneous domains that may appear in the frustrated cholesteric films unveiled in 1974 (Kawachi et al. 1974; Haas and Adams 1974b). This refers to cholesteric slabs sandwiched between two parallel substrates, ensuring perpendicular orientational boundary conditions that are incompatible with a uniform helical order, hence leading to inhomogeneous distortion of the director field. If the level of spatial confinement assessed by the ratio p/L, where L is the thickness of the film is large enough, the cholesteric helix is totally unwound and a stable uniform director field n0 oriented along the normal to the plane of the film takes place (see Figure 1.11(a)). The frustrated helical structuring can nevertheless be released under the application of any stimulus, inducing sufficiently large distortions of the uniform state. This can be done in several ways, for instance, by using temperature, electric or optical fields to name a few. In particular, when p/L ∼ 1, localized chiral supramolecular structures surrounded by uniform director fields may survive after the source of perturbation is switched off (see Figure 1.11(b)). It took more than three decades to decipher the details of such elastic quasi-objects, which can be found in literature under the generic names spherulite or bubble. The early double-twist torus guess by Kawachi et al., see Figure 1.11(c)5, was only completed two decades later by Pirkl et al. in 1993 (see Figure 1.11(d)). The proposed triple-twist structure has a double topological character6, where

5 This figure introduces the so-called “nail convention”: molecular orientations are represented by segments, nails and points when molecules are, respectively, parallel, oblique or normal to the plane or to the surface on which they are drawn. The acute ends of nails correspond to molecular extremities pointing toward the observer. 6 The “double topological character” terminology is purposely used to echo the title of a paper by Bouligand et al. (1978), which explores the generic issues associated with such topological coexistence, going beyond the particular case of liquid crystals.

18

Liquid Crystals

singular distortions of the director field (point defects) coexist with non-singular ones (disclination loop). The latter prediction of a triple-twist director field has been eventually ascertained experimentally by Smalyukh et al. (2010) (see Figure 1.11(e)), as well as the structural variants shown in Figures 1.11(f) and (g). Nowadays, it is understood that the anchoring strength of the perpendicular boundary conditions play a role in the detailed topological structure and that a wealth of localized topological director fields can be hosted in frustrated cholesteric films (Ackerman et al. 2014; Ackerman and Smalyukh 2017). More generally, cholesteric films offer a rich experimental playground for exploring the topology of ordered condensed matter that would hardly be assessed otherwise (Smalyukh 2020). 

 












 







 

 

 









Figure 1.11. (a) Frustrated cholesteric film with unwound helix when p/L is large enough. (b) Early observation between crossed linear polarizers (XPOL) of the metastable local release of frustration and (c) its conjectured double-twist torus director field (Kawachi et al. 1974). (d) Conjectured triple-twist director field involving capping point defects capping (Pirkl et al. 1993). (e) Experimentally reconstructed 3D director field (black streamlines) with their related XPOL images (Smalyukh et al. 2010). f) and g) Alternative configurations for the capping defects (Smalyukh et al. 2010). Color code: Red lines refer to non-singular disclination loops, blue circles refer to singular point defects and blue lines refer to singular disclination loops

1.5.1.2. Tunable and reconfigurable diffraction gratings In practice, localized chiral elastic quasi-objects may appear in a collective manner, as is the case when applying a uniform voltage difference between the two substrates sandwiching the cholesteric film, or when applying a temperature quench. The ensuing capability for mass production – filling centimeter-sized areas – opened up exploration into the production of spatially modulated optical phase elements with macroscopic aperture sizes since the earliest works (Haas and Adams 1974b; Bhide et al. 1976). An example dealing with a self-assembled 2D hexagonal lattice is shown in Figure 1.12(a). The extension to large-scale lattices with arbitrary symmetry has been demonstrated via point-by-point laser writing (Ackerman et al. 2012b) (see Figure 1.12(b) for the example of a square lattice). Moreover, the diffraction features of such phase gratings can be finely tuned, electrically, independent of the way it has been created, and erase/rewrite cycles are possible.

Singular Optics of Liquid Crystal Defects













19


     

Figure 1.12. Diffractive optics from metastable localized defect structures in frustrated cholesteric films. (a) Diffraction grating made up of a 2D hexagonal lattice. Top: XPOL images of the structures. Bottom: far-field diffraction pattern. Scale bar: 30 μm. Adapted from (Varanytsia and Chien 2014). (b) Same as panel (a) but for a 2D square lattice. The voltage is increased from left to right, illustrating that the diffracting capabilities of the phase grating are electrically tunable. Scale bar: 100 μm. Adapted from (Ackerman et al. 2012b)

1.5.1.3. Lensing and beam steering Since localized cholesteric structures are inherently endowed with slowly varying dielectric tensor contributions when p λ0 (section 1.3.6.3), they have also been envisioned as individual electrically tunable micro-lenses (Hamdi et al. 2011). Still, the ensuing strong refractive index gradients invite questioning into its relevance in terms of optical imaging, which has been eluded so far. Noting that more complex architectures made up of self-assembled bound localized elastic excitations have been shown to behave as multiple-lens (Loussert and Brasselet 2014) (see Figure 1.13(a)), previously discussed methods to generate on-demand lattices of localized birefringent structures might open up for the development of adaptive lenslet arrays. Beam steering is another optical function that has recently been explored, this time considering beam propagation in the plane of the sample (Hess et al. 2020). In that case, a linearly polarized light beam with its polarization vector oriented along the director is considered and the light confinement inside of the cholesteric slab is ensured using substrates whose refractive index is smaller than both n⊥ and n . Under such conditions, the optical scattering characteristics depend on the “impact parameter”, that is the ratio a/R (see Figure 1.13(b)). Here, the next step toward agile beam steering implies taking control of the location and/or size of the structure, which can be done, for instance, by motion control via optical trapping strategy (Smalyukh et al. 2007) and electrically tunable shrinkage (Pirkl et al. 1993).

20

Liquid Crystals






 



 



 



Figure 1.13. (a) Refractive optics from localized defect structures in frustrated cholesteric films. Experimental lensing of normally incident light to the film. Top-left: XPOL monochromatic image of the structure. Top-right: image of the focal plane for incident natural white light. Bottom row: idem for dimer- and trimer-like bound structures. Scale bar: 10 μm. Adapted from (Loussert and Brasselet 2014). (b) Steering of a e-polarized light beam propagating in the plane of the frustrated cholesteric film, see incident wave vector kin and electric field vector ein . The deflection of the beam that results from its interaction with the liquid crystal structure strongly depends on the dimensionless impact parameter a/R, as illustrated here for three cases that correspond to a/R  1, a/R ∼ 1/2 and a/R = 0 from top to bottom. Scale bar: 100 μm. Adapted from (Hess et al. 2020)

1.5.2. Elongated defects structures in frustrated cholesteric films 1.5.2.1. Brief historical survey Frustrated cholesterics films can also host elongated topological structures in addition to localized ones, as evidenced in Kawachi et al. (1974); Haas and Adams (1974a). Such structures are generically dubbed as cholesteric fingers. Four of them [CFn, n = (1, 2, 3, 4)] have been identified owing to a directional phase transition experiment and statistical analysis of the profile of their intensity cross-section (see Figures 1.14(a) and (b)) (Baudry et al. 1998). This classification of the fingers with their respective correct topology has been reviewed in Oswald et al. (2000). We notice that the structure of the CF1 was first proposed by Press and Arrot (1976), that of the CF2 by Gil and Gilli (1998) and Baudry et al. (1999) (who recognized their strong connection with the localized structures presented in section. 1.5.1), that of the CF3 by Cladis and Kléman (1972) and that of the CF4 by Baudry et al. (1998). All these proposed structures are summarized in Figure 1.14(c), which shows the transverse cross-section of the director field. All of them were experimentally confirmed later (Smalyukh et al. 2005), using the fluorescence confocal polarizing microscopy technique (Smalyukh et al. 2001), see Figure 1.14(d). This figure allows the appreciation of the topological diversity of the transverse cross-sections of

Singular Optics of Liquid Crystal Defects

21

   





























































   
 




  


fingers, as they can host only non-singular defects (CF1 and CF2), only singular defects (CF3) or both singular and non-singular defects (CF4).

 




































Figure 1.14. (a) Natural white light imaging of the four types of cholesteric fingers in a directional growth experiment, where the structures nucleate here from the phase transition front between the cholesteric mesophase and the isotropic liquid that are shadowed with red and blue colors, respectively. (b) Intensity profile along a line a—b of the left panel. (a and b) Adapted from (Baudry et al. 1998). (c) Transverse director field cross-section of the originally proposed topological structures of the four kinds of fingers. Adapted from (Baudry et al. 1998; Baudry et al. 1999). (d) Experimentally confirmed reconstruction of the transverse director field cross-section of the four cholesteric fingers in usual frustrated film at uniform temperature. Open circles, closed circles and closed square markers, respectively, refer to non-singular disclination lines with charge − 12 , + 12 and +1, whereas up/down triangles refer to singular π-twist disclination lines of the opposite sign. Adapted from (Smalyukh et al. 2005)

1.5.2.2. Tunable and reconfigurable diffraction gratings Similar to the case of localized topological structures, elongated structures have also been considered as diffracting optical elements, whose characteristics can be controlled by external fields. The example of a spontaneously formed straight array of fingers is shown in Figure 1.15(a), where the spontaneous packing

22

Liquid Crystals

inhomogeneities of the assembly of fingers results in diffraction blurring, hence questioning a viable applicative use. This can be solved by using artificial optical writing, with a continuous optical release of frustration, while displacing the sample in its plane. The ensuing quality improvement of the regularity of the obtained patterns is illustrated in Figure 1.15(b). Curved patterns can also be recorded (see Figure 1.15(c)), offering an option for creating holographic phase plates from dots and curved segments or lines. As previously anticipated (Haas and Adams 1974a), this also supports the fact that frustrated cholesteric films can be used as rewritable optical data storage material systems, not only at the local level, owing to the topological diversity of the localized structures surveyed in section 1.5.1.1. 














Figure 1.15. (a) Left: XPOL image of a spontaneous 1D phase grating made of CF1s. Right: corresponding far-field diffraction pattern. Adapted from (Varanytsia and Chien 2014). (b) Top: Artificially created version of panel (a) using optical writing strategy. Bottom: corresponding far-field diffraction pattern. (c) Recording arbitrary curved two-dimensional pattern. (b and c) Adapted from (Ackerman et al. 2012a)

1.5.2.3. Waveguiding and beam steering In analogy with optical fibers enabling the transverse confinement of light due to the radial gradient of the refractive index, the tube-like refractive nature of cholesteric fingers makes them potential candidates to guide light. So far, this has been addressed theoretically in the case of a straight CF2, for which optical propagating eigenmodes have been calculated (Poy and Žumer 2020). The fundamental eigenmode is found to be essentially confined in the central part of the cross-section of the finger, as illustrated in Figure 1.16(a). Its inhomogeneous polarization state tends to be locally linearly polarized along the projection of the director field in the transverse cross-section of the finger. This can be qualitatively understood from the fact that light confinement takes place in the regions of the largest refractive index and that dn > 0. In addition, the moderate amount of the estimated scattering losses due to the structure itself under realistic geometrical and material parameters, as well as the bending losses shown in Figure 1.16(b), make an experimental implementation plausible.

Singular Optics of Liquid Crystal Defects

23

 









 



 







 





Figure 1.16. (a) Simulated director field of a CF2 structure. Inset: transverse intensity profile of the fundamental mode. (b) Left panels: simulated XPOL images of undulating CF2s with a characteristic radius of curvature of 200 λ0 (top) and 80 λ0 (bottom). Right panels: corresponding z-averaged beam intensity when the fundamental mode is launched at x = xin along the finger. Adapted from (Poy and Žumer 2020)

Beam steering based on CF3 is another option that has been recently explored (Hess et al. 2020); see Figures 1.17(a) and (b), which show the simulated structure and, for the sake of illustration, its ability to steer an incident e-polarized light beam confined in 2D, inside the frustrated cholesteric film. Depending on the incidence angle on the structure, the latter behaves as either a mirror or a kind of birefringent beam displacer. The latter behavior can be qualitatively understood by considering the structure as a material slab with thickness Leff and refractive index neff lower than that associated with the surrounding unperturbed director field n . This situation is known to lead to a total internal reflection phenomenon above a critical incidence c angle θin = arcsin(neff /n ), while an incident beam will be partially transmitted c otherwise. Taking neff = n⊥ for a crude estimate we get θin 60◦ with the material parameters used in Hess et al. (2020), which allows us to grasp the reflecting versus c transmitting behavior shown in Figure 1.17(b), when θin = 70◦ > θin and c θin = 55◦ < θin , respectively. Within this simplistic approach, the lateral shift d r −θin ) c (see Figure 1.17(b)), is estimated as d = Leff sin(θ , where when θin < θin cos θr n θr = arcsin( n⊥ sin θin ) is the refracted angle inside the effective slab. The simulation result d 10 μm for θin = 55◦ gives Leff 15 μm, which approximately corresponds to the half-width D/2 of the structure, as observed between the crossed linear polarizers (see Figure 1.17(b)). Such an estimate echoes the simulated energy flow, as its redirection mainly takes place in the inner half of the structure. The latter qualitative reasoning should, however, not elude the fact that the interaction of a light beam with the twisted anisotropic structure does not follow the Mauguin regime and e/o energy coupling takes place, as illustrated in Figure 1.17(b), c when θin < θin . Moreover, the experimental counterpart shown in Figure 1.17(c), exhibiting partial transmission above the total internal reflection angle derived from the effective slab picture, pinpoints another subtlety associated with the singular

24

Liquid Crystals

π-twist disclinations that show up in experiments. We refer to Poy and Žumer (2020) for further details. 



 








  





















  










  
  


 





Figure 1.17. (a) Simulated director field of a structure that corresponds to a CF3 free from its pair of singular π-twist disclinations of the opposite sign (see Figure 1.14(c)), owing to weak perpendicular anchoring. (b) Corresponding simulated behavior of an input e-polarized light beam for two different incidence angles θin . The intensity of the o-polarized contribution is enhanced by a factor of 10 to make it visible. (c) Experimental counterpart of panel (b). Adapted from (Hess et al. 2020)

1.5.3. Regular optics from other topological structures 1.5.3.1. Waveguiding along non-singular and singular disclination lines Non-singular disclination lines enabled optical guiding: Studies on liquid crystal based optical fibers, whose core corresponds to a non-singular disclination line, theoretically started three decades ago in the framework of radial escaped axisymmetric director fields in nematics (Lin et al. 1991; Lin and Palffy-Muhoray 1992), whose structure is sketched in the top part in Figure 1.18(a). In that case, both TE (i.e. the electric field lies in the transverse plane) and TM (i.e. the magnetic field lies in the transverse plane) optical modes have been evaluated in the linear regime, defined as a situation when the optical field does not modify the director field. By definition, TE modes correspond to o-polarization, for which the optical torque density vanishes since n · Eo = 0, see equation [1.17]. This is no longer true for TM modes, which can therefore lead to having the nonlinear propagation regime as the director – hence the refractive index profile – depending on the optical intensity. This case has been addressed numerically in Lin and Palffy-Muhoray (1994) by imposing ˇ cula et al. (2016). a constant order parameter, and more recently in Canˇ Note that when considering optical modes fully confined in the cylinder of radius R defining the radial escaped director profile, we deal with an inhomogeneous and anisotropic variant of graded-index optical fibers. In that case, the TE mode is ordinary polarized and experiences a constant refractive index n⊥ , whereas the TM mode is

Singular Optics of Liquid Crystal Defects

25

extraordinary polarized and experiences the radial gradients of the refractive index, with ne running from n⊥ on-axis to n at the periphery. Therefore, a slab of material acts as a convergent lens for dn < 0 and although negative birefringence is rather unusual, the theoretical investigation of its optical consequences is formally worth considering, as illustrated in Figure 1.18(a). In contrast, the effect is that of a divergent lens when dn > 0, which eventually makes the light interact with the boundary of the capillary tube that encloses the liquid crystal. The latter situation has been considered both in theory and experimentally by using an incident off-centered collimated light beam propagating along the direction of the cylinder (see section I.5 in Oswald and Pieranski 2005), whose diameter is small enough to show up a ray-optics behavior. 




 











 



Figure 1.18. Simulated lensing effect experienced by light propagating on-axis along a non-singular nematic radial escaped disclination in nematics (a) and non-singular cholesteric double-twist cylinder (b). (a) Focusing of a radially polarized beam incident on a radial escaped slab. Top: light-matter interaction geometry. Bottom left: intensity (grayscale) and polarization (red arrows) distribution of the incident electric field at a given time. Bottom right: meridional cross-section of the director field (red segments) ˇ cula et al. 2016). (b) Same and of the optical intensity distribution. Adapted from (Canˇ as in (a) for an azimuthally polarized beam incident on a double-twist cylinder slab. Adapted from (Pišljar and Ravnik 2018). Parameters: (a) R  6 μm and L  5 μm; (b) R  2 μm and L  8 μm; (a and b) w/R ∼ 1/2, dn = −0.05 and λ0 =390 nm

A similar lensing effect occurs when light propagates on-axis along a double-twist cylinder cholesterics structure (Pišljar and Ravnik 2018) (see Figure 1.18(b)). This time the lensing only affects the TM mode, which corresponds to the extraordinary polarization state and focusing (defocusing) also takes place for dn < 0 (dn > 0). This result could appear as contradictory with the lensing effect shown in Figure 1.13(a), as the corresponding structures also host a double-twist cylinder. However, the latter refers to a 2π director twist along the diameter of the structure, while Figure 1.18(b) shows a π director twist. Consequently, the outer (inner) parts of the structures shown in Figure 1.13(a) operate converging (diverging)

26

Liquid Crystals

lensing for the positive birefringence used in these experiments. This offers a qualitative explanation of the experimental observation of the virtual focal plane at the rear of the illuminated structure and the real one at the front in Hamdi et al. (2011). The waveguiding properties of both the radial escaped nematic profile and the double-twist cholesteric profile follow naturally from the above-mentioned convergent lensing effect for negative birefringence. Indeed, a guided mode can be supported when the focusing effect of a structured material counterbalances the natural defocusing effect arising from the diffraction of light as it propagates. This is illustrated in Figure 1.19(a) for the straight double-twist cylinder geometry (Pišljar ˇ cula et al. (2016) for the radial escape and Ravnik 2018) and we refer to Canˇ geometry. Within the explored parameters, the propagation scattering losses inherent to the director gradients are minimized when the characteristic radius w of the incident beam is roughly half the radius R of the double-twist cylinder, reaching ∼ 50% power loss after a propagation distance of ∼ 300λ0 . In addition, the bending losses have also been estimated by imposing a constant curvature to the double-twist cylinder, as illustrated in Figure 1.19(b). Comparing the waveguiding performances of a CF2 with that of a double-twist cylinder, and noting that a positive birefringence is the norm for liquid crystals, it appears that the former situation is more promising for further experimental development of a reconfigurable integrated photonics platform based on liquid crystal topological solitons. 






 











Figure 1.19. (a) Simulated waveguiding of an azimuthally polarized beam with a diameter that minimizes the propagation scattering losses and launched at z = zin along the axis of a double-twist cylinder with radius R = 3λ0 and dn < 0; see Figure 1.18(b) for light-matter interaction geometry. (b) Same as in (a) for bent structures with a radius of curvature of 900 λ0 (top) and 150 λ0 (bottom). Adapted from (Pišljar and Ravnik 2018)

Singular disclination lines enabled optical guiding: Perhaps non-intuitively since material singularity promotes light scattering, optical waveguiding can also occur along singular disclination lines. An experimental demonstration of it has been reported using liquid crystal filaments of smectic A (see Figure 1.3(a)), grown in water with surfactant added above its critical micellar concentration (Peddireddy

Singular Optics of Liquid Crystal Defects

27

et al. 2013) as a result of interfacial instability. The myelin-like growth of these filaments recalls the prehistory of liquid crystals, when lyotropic liquid crystals7 were discovered in the middle of the 19th century. Similar to the concentric cylinder of the phospholipid bilayer structure of myelin, the structure of smectic filaments corresponds to a concentric organization of the smectic layers around a singular disclination line, where the director orientation is not defined, as depicted in Figure 1.20(a). This figure also shows a polarizing microscope image of a segment capped with onion-shaped hemispheres, from which a focused laser beam can be precisely injected and further guided along the filament, which acts as an optical fiber (see Figure 1.20(b)). 























Figure 1.20. (a) Top: concentric lamellar structure of a smectic A filament ended by the hemisphere where the red line refers to the singular disclination core, decorated by a purely radial distribution of the director. Bottom: XPOL natural white light imaging, where the white crosses indicate the orientation of the polarizers. (b) Optical waveguiding inside a smectic fiber, fed with a focused visible laser beam at one end, as indicated by the arrow. The optical energy flow is visualized by looking at the fluorescence of dye molecules that dope the liquid crystal. Inset: ray-optics cross-section sketch of the whispering gallery propagation of light due to successive total internal reflections. (c) Same as in (b) for a smectic fiber segment having a roughly 10 times smaller diameter. Inset: enlargement of the fiber output. Adapted from (Peddireddy et al. 2013)

The observed spiraling circulation of light inside such fibers with a large enough diameter suggests a whispering gallery type waveguiding process, named after the “Whispering Gallery” in St. Paul’s Cathedral in London. There, an acoustic wave propagation phenomena, analyzed by Rayleigh at the beginning of the 20th century (Rayleigh 1910), takes place. Quoting Rayleigh, “The phenomena of the whispering gallery, of which there is a good and accessible example in St. Paul’s cathedral, indicate that sonorous vibrations have a tendency to cling to a concave surface”. The latter is illustrated in the inset of Figure 1.20(b) within a ray-optics picture, where successive total internal reflections occur at the water/smectic interface, recalling that the liquid crystal refractive indices are substantially larger than that of the

7 Lyotropic liquid crystals are mixtures of amphiphilic anisotropic building blocks with isotropic solvents, such as water with an excess of surfactants.

28

Liquid Crystals

surrounding aqueous solution. For fibers with a smaller diameter (see Figure 1.20(c)), the propagation should instead be analog of earlier explored situations of optical fibers with liquid crystal cores with purely radial director profiles (Lin et al. 1991). These results unveil a practical path to produce self-engineered soft optical fibers, whose structural perfection (smooth wall at molecular scale, fixed radius), enabled by topological constraints and large refractive index contrast between the core and the cladding, could motivate the development of a directed growth strategy, which has been implemented in the last decade for elongated topological structures in frustrated cholesteric films (section 1.5.2). 1.5.3.2. Lasing from structures hosting point, line or loop defects So far we have discussed the optical properties of liquid crystal topological structures when light probes the material properties. Here, we address the situations where liquid crystals act as source of light, once they have been doped with laser dyes and pumped by pulsed light. Two kinds of optically pumped liquid crystal microlasers will be discussed, depending on the confinement method used to build up an optical resonance. Namely, liquid crystal optical cavities relying on either whispering gallery resonances or resonances associated with the existence of a photonic bandgap. Whispering gallery lasing: Recalling the ray-optics picture shown in the inset of Figure 1.20(b), one understands that optical resonances can take place when light circulates on itself once the constructive interference condition is fulfilled. By adding gain to the medium and an external source of energy (optical pumping in our case) on top of the latter optical feedback provided by a whispering gallery mode, laser emission can take place. Such a lasing scheme was experimentally demonstrated in the early 1960s using millimeter-sized solid-state spherical resonators (Garrett et al. 1961), and the first microlaser enabled by whispering gallery resonant modes was reported in the 1980s, using spherical droplets made of isotropic liquid in suspension in the air (Tzeng et al. 1984). The self-structuring of liquid crystals around topological defects and their voltage-tunable capabilities came to play in 2009 (Humar et al. 2009). In the latter work, nematic spherical droplets immersed in an isotropic surrounding medium were used, ensuring perpendicular boundary conditions. If not too small, such droplets adopt a purely radial director configuration centered on a hedgehog singular point defect and support TM lasing modes in a privileged manner (see Figure 1.21(a)).8 A typical experimental emission spectrum is displayed in Figure 1.21(b), which

8 This holds for liquid crystals with dn > 0, since TM and TE modes are associated with n and n⊥ , respectively. Ensuing refractive index contrast, with the surrounding medium being larger for TM than for TE modes, the former are better confined and reach the lasing threshold first.

Singular Optics of Liquid Crystal Defects

29

exhibits a comb of resonances with free spectral range Δλ and linewidth δλ associated with quality factors as high as Q = λ/δλ ∼ 104 (Humar et al. 2009). Although a wave-optics approach allows the derivation of the equations describing the spatial profiles and wavelengths of the lasing modes, one can understand the main lasing characteristics and behavior from a simple approximate expression for the resonant wavelengths. This is done by considering the circulation of light along a circular perimeter of radius R, satisfying constructive interference after one full turn. Namely, 2πRneff = lλ, where neff is the effective refractive index experienced by the mode and l integer is its azimuthal order. Recalling that the polarization vector of the TM mode is locally oriented parallel to the director field, for lasing in the visible domain we typically get l = 2πRn /λ ∼ 100 and Δλ = λ/l  10 nm for R ∼ 10 μm. This also allows us to understand that the frequency comb blueshift is applied along the optical pumping direction for positive dielectric anisotropy as an external electric field (see Figure 1.21(c)). Indeed, since the director tends to realign along the electric field, there is an overall orientation mismatch between the director and the polarization vector of the TM mode, which leads to local refractive index decrease (see section 1.3.5), so do neff and the lasing wavelength.

 

 
























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Figure 1.21. (a) TM lasing illustration in an optically pumped radial nematic droplet with typical diameter ∼ 10 μm, immersed in a medium whose refractive index is lower that those of the liquid crystal, and sandwiched between two flat substrates between which an external electric field Eext can be applied. White arrows: TM electric field vector. Dark lines: radial streamlines of the director field at Eext = 0. Blue point: hedgehog point defect. (b) Typical emission spectrum. (c) Typical dependence of the emission spectrum versus external electric field. (d) Illustration of the director field as Eext increases from the bottom to the top: the central point defect (blue point) progressively evolves toward an equatorial disclination loop lying in the (x, y) plane (red points). Adapted from (Humar et al. 2009)

The latter behavior has been discussed in the framework on a structural transition from a singular point defect to a singular disclination loop of charge +1/2 (Humar et al. 2009), as depicted in Figure 1.21(d). Still, if the applied electric field is not enough to trigger the hysteretic point-to-loop transformation, the spectral tuning is

30

Liquid Crystals

fully reversible. The effects of topological transitions on lasing have also been experimentally addressed in other situations. For instance, when using nematics with negative dielectric anisotropy, for which another structural scenario evolving toward a radial escaped disclination line of charge +1 has been argued (Xu et al. 1992). In that case, it has been observed that the quality factor is less deteriorated than for positive dielectric anisotropy (Sofi et al. 2017; Sofi and Dhara 2019b). Another example concerns the topological transitions driven by changes in the orientational boundary conditions at the interface between the nematic droplet and the surrounding medium that can involve the following sequence: surface point defects → disclination loop → bulk point defect, see (Humar and Muševiˇc 2011) for an early demonstration. Finally, we note that lasing supported by whispering gallery modes have also been observed in other liquid crystal droplet systems than nematic ones. One can mention smectic A (Kumar et al. 2015), smectic C (Sofi et al. 2019) and smectic C∗ (Sofi and Dhara 2019a), as well as cholesteric (Humar et al. 2016; Wang et al. 2016) droplets. Elongated systems have also been explored, such as the smectic A microfibers shown in Figure 1.20 (Peddireddy et al. 2013), though their lack of spatial control makes them trickier to scrutinize. In all cases, it is the topology-controlled director field of a confined volume of liquid crystals that drives the lasing properties and ensuing application potential (Humar 2016). Photonic bandgap lasing: optical cavities can also be obtained without a tangible reflecting interface as such, but by instead exploiting periodic modulations of the refractive index and/or the laser gain providing spatially distributed feedback via backward Bragg scattering (Kogelnik and Shank 1971). Owing to the circular Bragg reflection phenomenon discussed in section 1.3.6.2, dye-doped cholesterics appeared as natural candidates for distributed feedback lasing, as pointed out in the early 1970s (Goldberg and Schnur 1973). The peculiarity of the photonic bandgap makes low-threshold lasing observable at the edge of the bandgap where the density of photon states drastically increases, which was only reported experimentally in the late 1990s using cholesteric films (Kopp et al. 1998). It took another two decades for cholesteric topological structures to emerge in this area (Humar and Muševiˇc 2010). The idea is to use spherical cholesteric droplets immersed in a medium ensuring a tangential director boundary condition. When the droplet radius is sufficiently large with respect to the cholesteric pitch, experiments reveal that such droplets preferentially adopt an onion-like structure, associated with a radially distributed cholesteric helix that is present almost everywhere in the bulk of the droplet (see Figure 1.22(a)). “Almost” because the topological constraints imposed by the boundary conditions imply the existence of a defect structure, which was first described by Frank and Pryce, as stated in Robinson et al. (1958). The so-called Frank-Pryce model involves a non-singular radial disclination line with winding number +2 ending with a singular point defect with charge +2, which has been

Singular Optics of Liquid Crystal Defects

31

revisited recently by numerical simulations. The latter predict a non-singular disclination line of charge +1 forming a double-helix, ending with two singular point defects with charge +1 at the surface (Seˇc et al. 2012) and fusing at the center of the droplet (see Figure 1.22(a)). The band edge lasing of these topological spherical photonic bandgap resonators (see Figure 1.22(b)) is therefore quasi-omnidirectional, as illustrated in Figure 1.22(c). “Quasi” because the radial defect structure is expected to induce optical scattering losses, preventing lasing over a solid angle centered on its axis. Moreover, recalling that the bandgap is centered at wavelength n ¯ p (section 1.3.6.2), lasing wavelength control can be obtained via the dependence of the helix pitch on temperature (see Figure 1.22(d)). 






































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Figure 1.22. (a) Left: illustration of the 3D director field associated with the radial distribution of the cholesteric helix inside a droplet with tangential boundary conditions. Top right: illustration of the numerically predicted double-helix non-singular disclination lines forming the radial defect structure (Seˇc et al. 2012). Bottom right: direct natural white light imaging of a droplet with p ∼ 350 nm exhibiting a faintly visible radial defect from (Tkachenko and Brasselet 2014). (b) Experimental demonstration of photonic band edge lasing. Inset: direct lasing observation in the presence of a background illumination. (c) Lasing intensity angular spectrum of a 50 μm diameter droplet. (d) Temperature controlled lasing spectrum. Adapted from (Humar and Muševiˇc 2011)

After one decade’s worth of advances, we can now mention the realization of solid-state versions made up of polymerizable materials (Humar et al. 2016), as well as shape-reconfigurable versions made up of cholesterics encapsulated in elastomeric shells (Lee et al. 2018). These two examples emphasize interesting perspectives regarding advanced lasing control and implantability in a third-party environment, including biological ones, for instance toward sensing or lighting applications. 1.5.4. Assembling photonic building blocks with liquid crystal defects Now that liquid crystal based integrated optical waveguides, optical resonators and light sources are available, the development of liquid crystal integrated micro-photonics platforms also appears to be worth considering (Muševiˇc 2016). The development of soft-matter, rather than hard-matter photonic systems implies

32

Liquid Crystals

self-assembling processes enabling the large scale structuring of soft photonic building blocks that would be difficult to produce otherwise. In other words, one needs mutual interaction between individual elements. This is exactly what happens in liquid crystal colloids, namely suspensions of microscopic objects inside liquid crystals, whose experimental first steps can be traced back to the beginning of the 1970s, when solid (Rault 1971) or fluid (Cladis et al. 1971) microparticles dispersed in liquid crystals were found to decorate the director field streamlines and exhibit 1D chaining, hence suggesting elastic interaction mediated by the liquid crystal bath. The topological ingredient only came about in the late 1990s when the role of topological defects was pinpointed (Poulin et al. 1997), which kickstarted a lot of developments; see (Blanc et al. 2013; Muševiˇc 2018; Smalyukh 2018) for a few surveys. This is illustrated in Figure 1.23(a) for a few isotropic spherical microparticles immersed in a uniform nematic background n0 , which have been identified to mimic various kinds of dominant elastic multipoles9. The multipolar nature of a particle depends on several factors that are mainly a means of engineering, such as the nature and strength of the boundary conditions at the particle/nematic interface, the confinement and the possible presence of external fields. In the presence of two or more colloidal particles, mutual interaction takes place from the overlap between the distorted regions of individual particles. The repulsive/attractive nature of the interaction depends on the increase/decrease in the total free energy of the system. The ensuing “colloidal crystallization” in one, two and three dimensions has been observed (Muševiˇc et al. 2006; Nych et al. 2013), as illustrated in Figure 1.23(b). Despite the fact that binding energy between colloidal particles can be orders of magnitude larger than kB T , the power law decay of the elastic interaction potential requires the development of tedious packing protocols. Usually, colloidal crystals are grown from a given initial distribution of colloidal particles, by using optical tweezers to individually direct every particle towards their crystallization site. External fields can also be used to enhance the interaction range (Simoni et al. 2014). Colloidal crystallization is also possible by exploiting the generation of a random network of disclination loops resulting from the thermal quenching from the isotropic to nematic phase. Indeed, the relaxation process can lead to defect-matter entangled states, where a single or many singular disclination loops tie knots and link around many colloidal particles (see Figure 1.23(c)). Oneand two-dimensional colloidal crystals can be obtained using this approach (Ravnik et al. 2007; Tkalec et al. 2011), noting that 2D self-assembly has only been observed in the case of a uniform cholesteric director background. An all-liquid-crystal variant of such a strategy has also been demonstrated in frustrated cholesteric films after orientational quenching from a spatially extended release of frustration to the relaxed

9 This terminology echoes the solutions to the Laplace equation for the director field in terms of a linear superposition of multipoles at a sufficiently large distance from the colloidal particle, when the elastic anisotropy of liquid crystals is neglected.

Singular Optics of Liquid Crystal Defects

33

state. This can lead to the 2D packing of localized defect structures (section 1.5.1) that are bound together by an encircling looped defect structure (Loussert and Brasselet 2014) (see Figure 1.24).  

  

   









 










 

 









Figure 1.23. (a) Elastic multipoles from spherical particles. From left to right: perpendicular, perpendicular, bipolar tangential and tilted conically degenerate anchoring boundary conditions. Top: XPOL images. Bottom: sketch of the axisymmetric director streamlines; dot: singular hyperbolic hedgehog point defect; line: singular disclination loop; half-dots: singular surface point defects (also called boojums). Adapted from (Senyuk et al. 2016). (b) Colloidal crystals. From left to right: 1D dipolar crystal (bright field), 2D binary quadrupolar crystal (bright field) where the inset emphasizes the two sub-lattices, 3D crystal made of 6 × 6 × 3 dipolar colloids (XPOL). Adapted from (Muševiˇc et al. 2006; Ognysta et al. 2011; Nych et al. 2013). (c) Entangled structure made up of a 2D assembly of colloidal particles and a disclination loop with charge −1/2 forming a trefoil knot (top inset), obtained after quenching across the isotropic-nematic phase transition. Here, the director field background corresponds to a cholesteric helix perpendicular to the (x, y) plane of the sample with thickness p/4 (bottom inset). Adapted from (Tkalec et al. 2011)

Note that studies of liquid crystal colloids go far beyond the use of building blocks with spherical geometry (Muševiˇc 2018; Smalyukh 2018), which among others gives access to high-order multipoles with no atomic orbitals analog (Senyuk et al. 2019). This invites further development of colloidal liquid crystal engineering that could prove useful for future photonic technologies.

34

Liquid Crystals







Figure 1.24. XPOL images of self-assembled packing of localized defect structures in a frustrated cholesteric film obtained after releasing frustration over areas much larger than p2 . Adapted from (Loussert and Brasselet 2014)

1.6. From regular to singular optics 1.6.1. What is singular optics? So far, we have discussed situations where liquid crystal defects play a role in the control of light, however, without an optical topological signature resulting from the light-matter interaction in the presence of non-trivial topological material structure. From now on, we will address such an issue, starting by the introduction of what is today called “singular optics”, as coined by Soskin. Quoting Soskin’s work (Soskin and Vasnetsov 2001): “Optical singularities investigations are treated in the Singular Optics, a branch of modern optics studying new important features of light, which are absent in the traditional optics of waves with smooth wave fronts. More precisely, there are three levels of optical singularities: (i) ray singularities (caustics) considered by catastrophe optics, (ii) singularities of plane polarized waves (scalar fields) and (iii) polarization singularities of vector light fields” and it is remarkable that all these levels are rooted in the 1830s (Berry 2000). It happens that we are surrounded by all of these kinds of singularities in our daily life, mentioning, for instance, rainbows for (i), light scattering from irregular surfaces for (ii) and skylight for (iii). In what follows, we skip the most popular case of caustics that one can easily observe without chasing a rainbow – for instance, by merely looking at the effect of a drinking glass or rippling water on light – and we focus on a brief presentation of scalar and vectorial singularities in the context of the two examples given above. Scalar optical singularities refer to locations where the optical phase is undetermined, hence being associated with zeros of amplitude. They have been identified to result from interfering waves (at least three plane waves are necessary) in an acoustic context (Nye and Berry 1974) and are recognized as a generic manifestation of scalar waves, whatever their origin. As such, the random scattering of a coherent light source is a simple way to generate them (Baranova et al. 1982), as illustrated in Figure 1.25(a). In this case, the phase singularities only have two

Singular Optics of Liquid Crystal Defects

35

possible topological charges, namely l = ±1, which stands for a phase winding around a closed-path circuit around the singularity by ±2π, respectively. This reflects the non-generic character of singularities with higher order charges |l| > 1, which are unstable and split into unit-charge singularities under perturbation. Experimentally, interferometry is a simple and efficient way to ascertain both the magnitude and sign of l, since phase singularities are associated with dislocations of the fringe pattern, as illustrated in Figure 1.25(b).

Figure 1.25. (a) Calculated intensity and phase of the random superposition of 40 coherent monochromatic plane waves. Up and down triangles denote elementary singularities with opposite charge. (b) Observing an optical random field by looking at the scattered light from tracing paper illuminated by a laser beam. Left: plain intensity distribution. Right: intensity pattern when a plane wave like collimated reference beam is added. Forked patterns associated with the emergence of one fringe reveal unit charge phase singularities and their up/down character discriminates the sign of the topological charge (photos by M. Ghadimi Nassiri)

Vectorial optical singularities refer to locations where one of the characteristics of the polarization state is not defined. In particular, skylight polarization illustrates the situation when the polarization itself can be undefined or, in other words, when the light is fully depolarized. This occurs at four viewing directions usually called the neutral points: two of them are located near to the Sun, while the other two stand near to the point whose direction is diametrically opposite to that of the Sun (in short, the anti-Sun). Two of them are shown in Figure 1.26(a), which displays one of the maps produced by Brewster, who identified the lines of equal degree of polarization in the atmosphere (Brewster 1847). Surprisingly, it seems that the polarization structure around the neutral points has only recently been experimentally measured (Horváth et al. 1998) (see Figure 1.26(b)). The distribution of orientation of the partially polarized light in the neighborhood of each of the four neutral points exhibits a singular pattern associated with a topological charge of 1/2. Remarkably, both the number of neutral points and the generic nature of their polarization

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Liquid Crystals

structure can be deduced from simple topological arguments and the non-generic nature high-order polarization singularities, as articulated in Berry et al. (2004).  

   
    





  
   

     













Figure 1.26. (a) Map of lines of equal degree of polarization in the atmosphere in a plane (projected on a disk using seeing angles as coordinates) passing through the zenith of the observer, and perpendicular to the line joining the observer and the Sun. The enlargement shows the two points (P) surrounding the Sun (S) where light is fully depolarized. Adapted from (Brewster 1863). (b) Left: angular map of the degree of polarization (full polarization refers to 100%) around one of the four skylight polarization singularities (here the Arago point). Right: corresponding map of the polarization azimuth, where each segment refers to the major axis of the local polarization ellipse. Adapted from (Horváth et al. 1998)

Another kind of polarization singularity is worth mentioning here, namely, places where the polarization ellipse is a circle in the plane of observation, to which we refer to the (x, y) plane, considering in the context of this chapter, monochromatic and paraxial light fields propagating along the z axis. Here, the undetermined quantity is the polarization azimuth angle ψ and this usually refers to points called C points. Generically, there are three kinds of configuration for the polarization ellipses around a C point. This is illustrated in Figure 1.27(a), which highlights two of the three characteristics pertaining to the classification articulated in Nye and Berry (1974). First, the C points are associated with an index ±1/2 that refers to the signed number of full turns made by the polarization orientation around an anti-clockwise closed-path circuit encircling the singularity. Second, the number of straight polarization azimuth streamlines emerging from the C point, 1 or 3 (see the thick solid lines in Figure 1.27(a)). There is an additional property related to the elliptic or hyperbolic shape of the contour lines of constant eigenvalues of the real part of the 2 × 2 polarization matrix, whose elements are given by Ei Ej∗ with i, j = x, y. Finally, recalling that a C point can be right-handed or left-handed, there are 12 types of C points in total that can all be constructed (Dennis 2002). Experimentally, the rich topology of polarization structures can be explored by simple means. An illustrative example relevant to the optics of liquid crystals corresponds to polarization resolved conoscopic patterns from uniformly oriented slabs of nematic liquid crystals (Kiselev et al. 2008; Buinyi et al. 2009) (see Figure 1.27(b)).

Singular Optics of Liquid Crystal Defects



  

  



  

37











Figure 1.27. (a) Polarization ellipses around generic C points, whose naming was introduced in Berry and Hannay (1977). The thin lines refer to the streamlines of the major axes of the polarization ellipses. The thick solid lines refer to the lines satisfying ψ(r, φ) = φ, where φ is the polar angle in the (x, y) plane. Adapted from (Nye and Berry 1974). (b) Polarization-resolved conoscopic patterns from a nematic liquid crystal slab with perpendicular boundary conditions lying in the (x, y) plane and a x-polarized incident light. The labeling of C points refers to that of panel (a) and the solid lines refer to the locations of linear polarization state. Left-handed and right-handed polarizations are indicated by blue and red colored regions, respectively. Left: simulations. Right: experiments. Adapted from (Kiselev et al. 2008)

To end this brief (and incomplete) survey of optical singularities, let us mention that polarization singularities have also been scrutinized in the non-paraxial light regime. In that case, the longitudinal component of the light field can no longer be neglected and the polarization ellipses can no longer be considered to lie in parallel planes. In other words, one should not mix the points of “true” circular polarization, for which the polarization ellipse described by the tip of the real electric field vector E is a circle, with that of “apparent” circular polarization, for which the projection of the polarization ellipse on a given plane of interest is a circle. In addition, there is an intriguing connection between scalar and vectorial singularities: the phase singularities of the scalar quantity E · E occurs at the locations of the true C points in the general case (Berry and Dennis 2001), which is a powerful way to retrieve them. 1.6.2. A nod to liquid crystal defects The universality of topology invites analogies between the vectorial optical singularities and elementary disclination lines with a topological charge of ±1/2 in nematics: – First, between generic singular points of no-polarization and singular elementary disclination lines in uniaxial nematics. This is done by considering that the order parameter plays the role of the degree of polarization and that the director n (recalling the equivalence n ≡ −n) plays the role of the electric field E (recalling that the equivalence ψ ≡ −ψ for the polarization azimuth can be rewritten E ≡ −E).

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Liquid Crystals

In fact, the singularity issue for the orientation of either n or E is solved by dissolving the notion of orientation in the core of the defect. This is illustrated in the left parts of Figures 1.28(a) and (b), where the segments referring to the orientation of either n or E merge at the singularity denoted by the dot. Note that the orientation vanishes continuously in both cases; see Figure B.IV.13 in Oswald and Pieranski (2005) for the order parameter nearby a disclination line, and Figure 1.26(b) for the degree of polarization around skylight neutral points. – Second, between C points and non-singular elementary disclination lines in nematics, when the tensorial order parameter includes both uniaxial and biaxial features. In that case, the core region of a purely uniaxial singular disclination can relax into a non-singular core by escaping into biaxiality while remaining in the nematic mesophase, as demonstrated by Lyuksyutov (1978). This is illustrated in Figure 1.28 where the main axes of the ellipses lying in the plane refer to the eigenvalues of the tensorial order parameter. Note that the circle located at the place of the disclination implies purely uniaxial character, as is the case at sufficiently large distances from the core. In other words, there is a doughnut of biaxiality centered on the disclination. Here, the analogy stands by comparing the patterns of the ellipses describing the tensorial order parameter of the right parts of Figure 1.28, with that defining the polarization state in Figure 1.27(a), though noting that there is no material analog for the handedness pertaining to the light field (at least for chirality-free structures). Such qualitative analogy leaves the question of escape into biaxiality of the lemon versus monstar types for disclinations with charge 1/2 open. 




Figure 1.28. Illustration of singularity removal of nematic disclination lines with topological charge 1/2 (a) and −1/2 (b) by an escape into biaxiality. The biaxial contribution to the tensorial order parameter is neglected in the left parts and is taken into account in the right parts. The dots refer to a singular core and the main axes of the ellipses lying in the plane perpendicular to the disclination line refer to the eigenvalues of the tensorial order parameter, whose traceless nature implies the use of an offset for the sake of illustration

1.6.3. Singular paraxial light beams A practical aspect of singular optics concerns the generation of light beams endowed with phase singularities, from which one can build up other types of

Singular Optics of Liquid Crystal Defects

39

singularities, as we shall see. Among the existing families of singular paraxial beams, the Laguerre–Gauss holds a special place. Indeed, they represent an exact orthogonal infinite-dimensional basis for the scalar paraxial Helmholtz equation, with each element of the basis (or mode) being characterized by a pair of indices, l ∈ Z and p ∈ N, that are associated with the azimuthal and radial transverse degrees of freedom, respectively. When omitting the propagation factor exp(−iωt + ik0 z) associated with beams propagating toward z > 0, the electric field amplitude El,p of a Laguerre–Gauss mode in vacuum, with an optical power P is expressed in the cylindrical coordinate system as

 √ |l|  

 r 2 2r2 r2 p! 2P 1 |l| El,p (r, φ, z) = exp − Lp (p + |l|)! π0 c w(z) w(z) w(z)2 w(z)2    k0 r 2 z z × exp i [1.19] + lφ − (2p + |l| + 1)arctan 2 2 2(z + z0 ) z0 p |l| (|l|+p)! m refers to the associated Laguerre where Lp (x) = m=0 (|l|+m)!(p−m)!m! (−x) polynomials, w0 is the beam waist radius, z0 = k0 w02 /2 and 2 w(z) = w0 1 + (z/z0 ) . As one can see from equation [1.19], the azimuthal and radial dependencies are separable and El,p ∝ exp(ilφ), which implies on-axis phase singularity (or optical vortex) with topological charge l. In addition, the radial index p corresponds to the number of zeros of amplitude along the radial coordinate. As an example, the intensity and phase profiles for l = (1, 2, 3) and p = 0 are shown in Figure 1.29(a) for z = 0, where the spatial extension of the beam is the smallest and noting that the beam waist radius w0 acts as a ruler. Namely, one can construct an infinite number of Laguerre–Gauss basis, by simply varying w0 . In turn, a relevant choice of w0 is usually desirable in order to minimize the number of main contributing modes when describing paraxial fields. The interest in the Laguerre–Gauss modes has developed strongly following the identification that beams with an amplitude proportional to exp(ilφ), carry l orbital angular momentum per photon along the propagation direction (Allen et al. 1992). From that moment, the orbital angular momentum of light was understood as easily accessible experimentally as a complementary source of angular momentum, recalling that fully and uniformly polarized light beams carry sin(2χ) spin angular momentum per photon along the propagation direction. While many studies were indeed initially triggered by the above connection between phase singularities and orbital angular momentum, it should nevertheless be noted that the presence of phase singularity as such and the resulting annular intensity distribution, as well as the modal character associated with the optical vortex, have all contributed to the emergence of many fundamental and technology-oriented advances over the last three decades (Shen et al. 2019).

40



Liquid Crystals












 

 


 



E

 

E

 






  





 


 




  




 



Figure 1.29. (a) Maps of intensity (top) and phase (bottom) of Laguerre–Gauss modes with indices l = (1, 2, 3) and p = 0, at z = 0. (b) Maps of intensity (top) and polarization ellipticity angle χ (bottom) of the superposition E = E0,0 c+ + E1,0 c− at z = 0. Yellow marker: on-axis C point (χ = π/4). Dashed circle: location of linear polarization states (χ = 0). Arrows: E vector pattern at a given time along the dashed circle. (c) Maps of intensity (top) and polarization azimuth angle ψ (bottom) of the superposition E = E−1,0 c+ + E1,0 c− at z = 0. Arrows: E vector pattern at a given time. All plots display a 4w0 × 4w0 region centered on the origin in the (x, y) plane

Recalling that the electric field of a scalar beam is associated with a uniform polarization state, El,p = El,p e, we exemplify how simple the generation of beams endowed with vectorial singularities can be, while emphasizing the connection between scalar and vectorial optical singularities: – The first example consists of the coherent superposition of two Laguerre–Gauss beams with azimuthal indices 0 and l and opposite circular polarization states e = (c+ , c− ). The resulting field, E = E0,0 c+ + El,0 c− , is endowed with an on-axis C point with index l/2, see Figure 1.29(b) for l = 1, which leads to a generic C point of the lemon type (ψ(r, φ) = φ/2). In addition, showing the distribution of E at a given time along the circle at which the polarization state is linear allows us to clarify possible confusion when looking at the polarization pattern, such as that given in Figure 1.26(b). Indeed, there is an azimuthally varying phase delay of π over a full turn for the electric field in the local frame cos φ x + sin φ y of its polarization ellipse, which ensures continuous spatial variation of E at all times. – The second example consists of the coherent superposition of two Laguerre–Gauss beams with opposite azimuthal indices and circular polarization states, E = E−l,0 c+ + El,0 c− . This leads to a so-called vector beam of order l, where the polarization state is linear everywhere, but whose azimuth varies as ψ = lφ; see Figure 1.29(c) for l = 1. Note that here the light is fully polarized everywhere and the singularity in the polarization ellipse azimuth leads to a null amplitude along the propagation axis.

Singular Optics of Liquid Crystal Defects

41

1.6.4. Generic singular beam shaping strategies Recalling that Laguerre–Gauss beams are modal solutions of spherical laser cavities, usual optical cavities designed to operate in the fundamental Gaussian mode associated with (l, p) = (0, 0) can be upgraded in order to emit light in a predetermined high-order mode, owing to the insertion of an intracavity element inducing mode-selective losses. However, this is neither very practical nor versatile. Alternative approaches have been developed to shape phase singularities, either at the source (see (Omatsu et al. 2017) for a recent review on vortex lasers), or in free space, by placing an adequate optical element in the course of a fundamental Gaussian beam. During the last three decades, various technologies have been developed in order to produce user-friendly extracavity optical vortex generators. Nowadays, there are many commercially available solutions that allow the production of optical vortex beams with arbitrary topological charge l on demand. Hereafter, we present the working principle of three main singular beam shaping strategies used in free space in the framework of paraxial light fields. All of them rely on the creation of space-variant transmittance, or reflectance masks imparting a pure phase factor exp(ilφ) to an incident beam as it interacts with the element: – Spiral phase plates: the use of an isotropic material sculpted into a helical shape to produce a helical wavefront is a rather intuitive option. In transmission, this consists of a phase plate made up of a non-absorbing material whose thickness h varies azimuthally according to h(φ) = lλφ/(n − 1), where n is the refractive index of the material. Indeed, as it passes through the element, a normally incident and centered Gaussian beam acquires the soughtafter extra phase factor (2π/λ)(n − 1)h(φ) = lφ. A noteworthy realization can be found in Bryngdahl (1973) in the context of interferometric metrology; however, manufacturing developments were only revived 20 years later. Modern lithographic and 3D printing technologies now allow the fabrication of high-quality transmissive spiral phase plates of various sizes, topological charges and operating wavelengths, an example being shown in Figure 1.30(a). Reflective analogs – helical mirrors – requiring a step height of lλ/2 have also been considered, with experimental demonstrations of optical vortex beams with l up to 104 (Fickler et al. 2016). – Diffracting optics: the 3D manufacturing constraints of spiral phase plates can be released by making use of the optical holographic techniques that bloomed in the 1960s, owing to the advent of lasers. Indeed, there is no need for an object to exist to be reconstructed optically, one only needs to synthesize it from its mathematical description. In our situation of interest, the simplest approach consists of a 2D structured optical phase mask having a “forked” profile defined by the phase profile Φ(x, y) = l arctan(y/x) + 2πx/Λ (see Figure 1.30(b)). The latter mask diffracts a normally incident and centered Gaussian beam into a charge l vortex beam at an angle λ/Λ from the z axis in the (x, z) plane, noting that usually one has λ  Λ  w. The development of liquid crystal spatial phase modulators – 2D

42

Liquid Crystals

pixelated (usually reflective) devices where every liquid crystal pixel acts as an electrically controlled uniform phase retarder – has made this holographic approach widely used, since arbitrary phase profiles can be encoded and refreshed at will. – Geometric phase optics: besides previous scalar approaches, it is also possible to produce flat-optics vortex beam shapers of a vectorial type. This refers to the space-variant anisotropic optical elements proposed by Bhandari (1997), for which the incident polarization state dictates how the output light field is structured. The basic idea can be grasped by looking at the effect of a uniform half-wave retarder having its optical axis oriented at an angle Ψ from the x axis on a incident circularly polarized light field. This is done by using equation [1.8], which gives cσ −−−−−→ iei2σΨ c−σ , Ψ,Δ=π

[1.20]

where σ = ±1. As expected from a half-wave retarder, the handedness is reversed, but an extra polarization-dependent phase 2σΨ of structural origin also appears – a geometric phase. Azimuthally varying half-wave retarders characterized by Ψ = qφ, where q is a half-integer or an integer, thus act as a polarization-dependent vortex phase mask with l = 2σq (see Figure 1.30(c) for q = 1). The first demonstration was made in 2002 (Biener et al. 2002) using space-variant subwavalength gratings operating in the mid-infrared. Nowadays, geometric phase vortex generators are commercially available from a wide range of structured materials, such as nanostructured glasses, metallic or dielectric metasurfaces, photo-patterned liquid crystal polymers, and liquid crystals with space-variant tangential boundary conditions. Importantly, in contrast to the first two approaches relying on the space-variant dynamic phase, which intrinsically depends on wavelength, geometric phase optical elements only suffer from conversion efficiency losses when detuned from the half-wave condition, which is ascertained from equation [1.8] for arbitrary Δ: cσ −−−→ cos(Δ/2) cσ + i sin(Δ/2) ei2σΨ c−σ Ψ,Δ

[1.21]

Moreover, recalling that beams endowed with vectorial optical singularities can be obtained from the superposition of contra-circularly polarized vortex beams (section 1.6.3), the vectorial nature of geometric phase optical vortex generators appears vividly. Indeed, when taking an incident Gaussian beam, beams with on-axis non-generic C point with index q are obtained from elements with Ψ = qφ and Δ = π/2 for the incident circular polarization state, whereas vector beams with index 2q are obtained from elements with Ψ = qφ and Δ = π for incident linear polarization state.

Singular Optics of Liquid Crystal Defects

43

Figure 1.30. Main kinds of vortex beam shaping strategies. (a) Optical profilometry image of laser 3D-printed spiral phase plate with 60 μm diameter designed for l = 5 at 633 nm wavelength. Color coding of the plate thickness from blue to red emphasizes the azimuthally varying height of the element. Adapted from (Brasselet et al. 2010). (b) Phase map Φ(x, y) of a forked phase grating mask with nominal pitch Λ for l = 1. (c) Optical axis orientation angle map Ψ(x, y) of a geometric phase mask for l = ±2. (d) Simulated far field intensity profiles of optical vortex beams generated by a phase mask with transmittance exp(ilφ) (black curves), compared to the Laguerre–Gauss profiles |El,0 |2 (red curves) for l = 1 and l = 10, choosing ad hoc normalization with respect to the spatial frequency κmax , at which the intensity is maximal

Since Laguerre–Gauss modes have been introduced as prototypical singular paraxial beams, a general remark applies to all of the three types of elements presented above. In fact, none of them generate Laguerre–Gauss beams. This is obvious when looking at the field emerging from an optical mask with transmittance exp(ilφ), namely, a Gaussian amplitude multiplied by an azimuthally varying singular phase profile. The phase singularity implies strong diffraction as light propagates, which leads to z-dependent ringing intensity profile that eventually shapes into a Laguerre–Gauss-like doughnut intensity profile in the far field. This is illustrated in Figure 1.30(d), where the comparison between the simulated far field radial intensity profile10 and the Laguerre–Gauss mode with p = 0 is shown for l = 1 and l = 10. This emphasizes a departure from a (l, 0) mode that increases with |l| to the benefit of high-order radial modes. To overcome such a limitation, we need to

10 This is evaluated from the Fourier transform of a Gaussian field, with the planar wavefront multiplied by the singular phase mask transmittance exp(ilφ), which gives I(κ) ∝ 2  ∞  Jl (κr)E0,0 (r, 0, 0)rdr  , where the reciprocal coordinate κ refers to spatial frequencies. 0

44

Liquid Crystals

shape both the magnitude and phase of the electric field. Such complex amplitude modulation can be done in various ways, including using a phase-only spatial light modulator, where the basic trick to modulate the amplitude is to redirect light to either zeroth or higher diffraction orders, see, for instance, for a recent survey of modal issues (Forbes et al. 2016). Also, it is worth mentioning that the presence of a material singularity by design is not a necessary condition to seed light beams with singular features, as the few examples described above could have suggested. In fact, neither a material singularity nor a (non-singular) material structuring is needed to engineer singular light beams in a controlled manner. Indeed, uniform uniaxial and biaxial crystals offer a wealth of options for doing the job (Fadeyeva et al. 2010; Turpin et al. 2016). What needs to be structured is the light-matter interaction. Artificial structuring of matter, however, remains a powerful technological approach and liquid crystals do well in terms of reconfigurable and adjustable singular optical devices. In this context, the self-organization of liquid crystals into a variety of topological structures offers a route to shape light in a singular fashion based on Nature-assisted approaches, which are introduced in section 1.7 and whose recent advances in the last decade are detailed in section 1.8. Remarkably, all of the assets of liquid crystals that have made them enter into our daily life as liquid crystal displays can now be deployed toward the emergence of novel adaptive and reconfigurable optical elements, with performances that would hardly be reached otherwise, which is the topic of section 1.9. 1.7. Advent of self-engineered singular optical elements enabled by liquid crystals defects 1.7.1. Optical vortices from a cholesteric slab: dynamic phase option In section 1.5, which was dedicated to the regular optics of liquid crystal defect structures, we saw that various kinds of 1D diffraction gratings can spontaneously occur in cholesteric liquid crystal slabs with either tangential or perpendicular orientational boundary conditions. Still, the spontaneous emergence of a one-dimensional periodic director pattern is prone to experience local symmetry breaking, for instance due to orientational fluctuations or experimental imperfections of a sample, resulting in the appearance of edge dislocations. Two examples are shown in Figures 1.31(a) and (b), which are analog of Figures 1.15(a) and 1.9(b), but with the presence of a defect, respectively. This was recognized in 2000 as a pixel-free self-engineered version of a singular diffraction grating (Voloschenko and Lavrentovich 2000). The experimental demonstration of the production of an optical vortex beam from a Gaussian beam is shown in Figure 1.31(c), where the nth diffraction order exhibits

Singular Optics of Liquid Crystal Defects

45

|n| points of null intensity. This was interpreted as the generation of optical vortex beams with topological charge l = n and splitting into |l| vortices with elementary charge vortices for |l| > 1. This supports the fact that the refractive index periodicity corresponds to p for stripe patterns, and not p/2 as a glance at Figure 1.31(b) would suggest, because the boundary conditions break the symmetry between two adjacent stripes separated by a distance p/2, so that the actual periodicity is p (Shiyanovskii et al. 2001). Also, the observation of a substantial amount of energy diffracted in various diffraction orders other than the first one, points out serious deviation from the perfectly blazed singular phase grating discussed in the preceding section and shown in Figure 1.30(b), which diffracts light into a single direction. 




  

  


  

 






 



Figure 1.31. Two distinct experimental situations leading to the spontaneous generation of a forked distorted director pattern in the plane of a cholesteric slab, whose thickness is approximately equal to the pitch p. (a) XPOL observation of a 12.5 μm thick cholesteric slab with perpendicular boundary conditions, where fingers have been grown under the application of a magnetic field. Adapted from (Ishikawa and Lavrentovich 1999). (b) Fluorescent confocal microscopy observation of a 14 μm thick cholesteric slab with tangential boundary conditions, where a striped pattern has been grown under the application of an electric field. (c) Different sets of diffraction orders n observed on a screen in the far field zone. (b and c) Adapted from (Voloschenko and Lavrentovich 2000), where s/2 should read s in equation (2) (Lavrentovich, private communication)

The above limitations, in terms of high-charge splitting and diffraction efficiency, probably explain that such a refractive route to self-engineered vortex beam generators has not been further pursued. In addition, recent alternative attempts based on the use of electro-convective dislocation patterns in nematic slabs (Yunda et al. 2018) also suffer from crippling practical limitations. 1.7.2. Optical vortices from a nematic droplet: geometric phase option In 2009, a self-engineered geometric phase vortex generator was demonstrated by using radial nematic droplets trapped in laser tweezers (Brasselet et al. 2009). The experiment was carried out by using an aqueous solution, in which nematic spherical droplets spontaneously formed by mechanical stirring are dispersed, and to which a surfactant is added in order to promote a radial configuration for the director in the

46

Liquid Crystals

bulk of every droplet. Then, a droplet is optically 3D-trapped and self-centered in the focal region of a tightly focused light beam owing to intensity-gradient optical forces. The axisymmetric director field makes the droplet behave like a geometric phase optical vortex mask with q = 1, thus imparting an optical phase singularity with topological charge l = 2σ to σ-polarized incident light, as illustrated in Figure 1.32(a). However, in contrast to the usual flat-optics geometric phase vortex masks, ideally characterized by a constant birefringence phase retardation satisfying Δ = π for the operating wavelength, the 3D nature of the droplet is associated with an effective retardance that depends on the radial coordinate. In turn, whatever the wavelength, the incident beam in only partially converted into an optical vortex beam, as illustrated in Figure 1.32(b), where the intensity distribution of the co- and contra-circularly polarized components of the output light beam are shown. Experimentally, the fraction of the output optical power associated with vortex shaping reaches η ∼ 0.4 for ∼ 3 μm-diameter droplets, for a liquid crystal with birefringence dn ∼ 0.2. Note that the latter limitation can be used to experimentally ascertain the helical phase profile of the contra-circularly polarized output light. This is done by determining the phase difference between the ±σ-polarized components (δΦ) from the polarization azimuth angle (ψ) of the elliptically polarized output light around a closed-path circuit encircling the propagation axis, indeed, up to an unimportant constant, δΦ = 2ψ. A typical result is displayed in Figure 1.32(c), which exhibits a linear azimuthal phase ramp, as expected from the emergence of an optical phase singularity with a topological charge 2. 



  















 
















  

Figure 1.32. (a) Optical vortex generation from a radial nematic droplet trapped in a tightly focused σ-polarized laser beam. Magenta surfaces: spherical incident wavefronts. Blue and red surfaces: intertwined helical output wavefronts of the output vortex field with l = 2σ. Inset: XPOL imaging of an optically trapped droplet, λ0 = 1.06 μm. (b) Typical intensity profiles of the two output circular components. (c) Phase difference between the −σ and σ output light field components around a circle whose center matches that of the droplet and measured in the equatorial plane of the droplet (here σ = +1). Markers: experimental data. Red line: linear phase ramp. Inset bright field image of the droplet. (d) White light vortex generation from a halogen source extracted by circular polarization filtering and its spectral components at 488, 532 and 633 nm wavelengths. Adapted from (Brasselet et al. 2009)

Singular Optics of Liquid Crystal Defects

47

In addition to the specific merits related to the spherical symmetry of radial nematic droplets, such as the self-centering of the optical element with the beam to be processed as well as an omnidirectional behavior, other assets pertaining to the geometric phase approach deserve to be highlighted. First, when dealing with liquid crystal topological defects, one has access to well-defined spatial distribution of the optical axis at the micron scale without resorting to nano-/microfabrication technologies. Second, as discussed in 1.6.4, polychromatic operation can be considered, which is illustrated in the case of radial nematic droplets in Figure 1.32(d). Third, one can readily imagine benefiting from the sensitivity of liquid crystals to external fields in order to adjust the optical performances or control the self-structuring process itself, which will be discussed in the next section, which is devoted to the survey of a decade of advances since 2009. 1.8. Singular optical functions based on defects: a decade of advances 1.8.1. Custom-made singular dynamic phase diffractive optics The production of artificial diffraction gratings made of cholesteric fingers, written by either point-by-point or continuous scanning of a focused laser beam irradiating a frustrated cholesteric slab (see sections 1.5.1.2 and 1.5.2.2), can also be used to create singular diffracting optics. This is illustrated in Figure 1.33, where the generation of optical vortices from 1D and 2D forked gratings are shown. Despite the versatility offered by the direct laser writing of arbitrary patterns of electrically tunable/erasable metastable structures, so far, unavoidable symmetry breaking prevents the obtainment of axisymmetric optical vortex beams. This is emphasized in the two examples given in Figure 1.33. Indeed, in the panel (a), even the first diffraction order exhibit substantial vortex splitting that gets worst for higher order diffraction orders, whereas in panel (b), all of the (n, m) diffraction orders associated with an optical singularity having a topological charge l = −(n + m) exhibit broken axisymmetry. In turn, the practical interest of this approach toward the development of high-quality optical vortex masks is yet to be demonstrated. Nevertheless, further investigations seem useful, recalling that the characteristic size of the recorded structures is of the order of the cholesteric pitch, which can be as small as a few hundred of nanometers. This could overcome the spatial resolution limitations of conventional liquid crystal spatial light modulators whose pixel-size is larger than several micrometers. 1.8.2. Spontaneous singular geometric phase optics 1.8.2.1. Tunable vortex generators from random umbilics in nematics In 1973, Rapini unveiled a kind of non-singular nematic liquid crystal defect that he called umbilics (Rapini 1973). Umbilics spontaneously appear in nematic films

48

Liquid Crystals

with perpendicular boundary conditions under the application of an electric field oriented along the normal to the plane of the film, provided that the dielectric anisotropy is negative and the electric field exceeds a threshold value. These defects are associated with a continuous 3D director field whose approximate universal analytical expression has been derived for an isolated umbilic and under the small reorientation approximation (Rapini 1973). Namely, in the Cartesian reference frame, n = (sin ϑ cos ϕ, sin ϑ sin ϕ, cos ϑ), where the expressions of the angles (ϑ, ϕ) in cylindrical coordinates (r, φ, z) are ϑ(r, z) = ϑ∞ a(r) sin(πz/L) ,

[1.22]

ϕ(φ) = qφ + φ0 ,

[1.23]

where q = ±1 is the topological charge of the defect, φ0 is a constant, L is the film thickness, 0 ≤ a ≤ 1 is the universal normalized radial profile of the tilt amplitude defined by a second-order nonlinear differential equation, and ϑ∞ is the asymptotic value of ϑ at large r that increases with the applied voltage difference U across the film. Typical calculated cross-sections of the 3D director field are illustrated in Figure 1.34(a), where rc is the core radius, which decreases with the applied voltage, as shown in Figure 1.34(b), where the universal profile a(r/rc ) is also displayed.









 





 






Figure 1.33. Experimental demonstrations of singular diffraction gratings produced by laser-written metastable defect structures in frustrated cholesteric films. (a) Left: XPOL image of a 1D double-fork grating made of cholesteric fingers. Right: central part of the observed far-field diffraction pattern whose enlargement points out vortex splitting both in intensity (top) and phase (bottom). Adapted from (Ackerman et al. 2012a). (b) Left: XPOL image of a 2D grating endowed with two elementary dislocations along the x and y axes. The thin white and yellow lines are guides for the eyes in order to visualize the two intertwined orthogonal 1D fork gratings. Right: central part of the experimental (top) and theoretical (bottom) far-field diffraction pattern, where (κx , κy ) refers to the reciprocal space (spatial frequencies). Adapted from (Ackerman et al. 2012b)

Umbilics are usually studied in films with thickness L = 10 − 100 μm (see Figure 1.34(c)). Such thicknesses allows the satisfaction of the half-wave retardation

Singular Optics of Liquid Crystal Defects

49

condition in the visible domain, using typical nematics subjected to few volts of applied voltage, thus leading to voltage-tunable flat-optics geometric phase optical vortex masks with q = ±1, as demonstrated in 2011 (Brasselet and Loussert 2011). Almost (because of the core) pure vortex can be obtained from a Gaussian beam, whatever the wavelength, provided that the asymptotic birefringent phase retardation Δ∞ can reach Δ∞ = N π, with N odd integer, and the incident beam waist is sufficiently large with respect to the defect core radius. This imposes an upper bound for the operating wavelength, typically λmax = 2dnL, which corresponds to Δ∞ = π when the director reorientation amplitude saturates (ϑ → π/2), noting that equation [1.22] is no longer acceptable in that case. For high birefringence nematics and thickness of a few hundreds of microns, THz vortex generators are therefore technically accessible, while liquid crystal absorption in the UV imposes the lower bound for the operating wavelength. 







 








Figure 1.34. (a) Calculated meridional (left panel) and transverse (at z = L/2, right panel) cross-sections of the director field of an umbilic with q = 1 and φ0 = 0. The color scale refers to the magnitude of the projection of the director in the (x, y) plane, here taking ϑ∞ = 30◦ . (b) Typical calculated dependence of the core radius rc versus the ˜ = U/UF , where UF is the so-called Fréedericksz threshold voltage reduced voltage U (usually of the order of a few volts), above which the umbilics appear. Adapted from (Brasselet 2012). (c) Left: typical XPOL imaging of a random assembly of umbilics. Right: enlargment of defect with q = ±1 (top/bottom), where the white lines refer to the in-plane projection of director streamlines near to a defect. Solid and dashed lines for q = 1 refer to umbilics of the splay and twist types, respectively, as defined by Rapini (1973). Adapted from (Brasselet and Loussert 2011)

Voltage-controlled vortex purity in the visible range is illustrated in Figure 1.35(a), where η(U ) is shown for blue, green and red laser radiation, which gives up to η > 0.99 in all cases. This overcomes the limitation of the radial nematic droplet requiring circular polarization filtering to extract the vortex beam, as shown in the top part of Figure 1.35(b), where the total intensity profile of the far-field output beam exhibits a doughnut shape, whatever the incident polarization state, here for N = 7. Moreover, the vortex generation can be switched on and off on-demand

50

Liquid Crystals

by adjusting the voltage to values that give either odd or even N , respectively, as illustrated in Figure 1.35(b) for periodic switching between N = 6 and N = 7. 



 



 



 

 



 




 




 



 

 











Figure 1.35. (a) Experimental voltage-tunable vortex purity of an isolated umbilic for an incident Gaussian beam at 488 nm (blue triangles), 532 nm (green squares) and 633 nm (red circles) wavelength. Parameters: beam waist radius at sample plane sample is 20 μm, L = 15 μm. Solid lines: guide for the eyes. (b) Far-field output radial intensity profiles for time-dependent applied voltage giving either Δ∞ = 6π (U = U6π ) or Δ∞ = 7π (U = U7π ). The reported data corresponds to azimuthal averaging of the recorded intensity distribution (see above panels). (a and b) Adapted from (Brasselet and Loussert 2011). (c) Calculated phase (top) and measured intensity profile (bottom) at the output of the sample placed between parallel linear polarizers (PPOL), whose passing direction is set along the x axis for U = U2π (left) and U = U4π (right). Parameters: monochromatic natural light, L = 30 μm. Adapted from (Brasselet 2012). All data refers to the nematic mixture MLC-2079

A single umbilic can also be used to generate a number of optical vortices by exploiting what could seem to be a drawback at first sight: the existence of a core that prevents the obtainment of an ideally flat birefringent phase retardation. Indeed, space-variant retardance allows the creation of sets of elementary optical vortices with topological charge ±1 that are arranged in a quadrupolar manner (Brasselet 2012). The idea is to use a normally incident linearly polarized incident field, whose arbitrary orientation is given by the unit vector u, and to select the linearly polarized component of the output field oriented along u. We refer to such configurations as parallel linear polarizers (PPOL). To illustrate this, we choose a linearly polarized plane wave with u = x for the incident field, for which the output PPOL field is given from equations [1.7] and [1.8]: EPPOL (r, φ) out

∝ E0 e

i

Δ(r) 2

 Δ(r) Δ(r) cos + i sin cos 2ϕ(φ) x 2 2

[1.24]

L where Δ(r) = 2π λ 0 [ne (ϑ(r, z)) − n⊥ ]dz. The field amplitude is null at locations (r∗ , φ∗ ) satisfying {Δ(r∗ ), ϕ(φ∗ )} = {N π, N π/4} with N odd integer and these

Singular Optics of Liquid Crystal Defects

51

points correspond to elementary phase singularities, whose nature and spatial arrangement can be identified from the phase distribution

 Δ(r) ΦPPOL (r, φ) = arctan tan cos 2ϕ(φ) , out 2

[1.25]

see the top part of Figure 1.35(c), where the calculations are made for Δ∞ = (2π, 4π), q = 1 and φ0 = 0. The corresponding experimental intensity profiles exhibiting 4 and 8 points of zeros of intensities are also shown in the bottom part of Figure 1.35(c). The latter figure emphasizes that the number of quadrupoles, as well as their radial location, can be changed by electrical means. The azimuthal position of the vertices can also be controlled by rotating the incident polarization vector u in the (x, y) plane. All of these features work in a wide range of optical wavelengths (Brasselet 2012). 1.8.2.2. Tunable vortex generators from random disclinations in nematics The promising demonstrations that umbilics can behave as self-engineered geometric phase optical vortex masks, with good optical performances and capabilities for on-demand management of scalar and vectorial optical singularities at arbitrary wavelength (Brasselet and Loussert 2011; Brasselet 2012), triggered further investigations. In particular, the use of umbilics leaves the question of topological diversity open, namely, can it be extended to arbitrary integer or half-integer values of q ? This question was addressed in 2013 by considering disclinations in nematic films with degenerate tangential boundary conditions (Loussert et al. 2013). In fact, random networks of disclinations with topological charge q = ±1/2 and q = ±1 can be observed in such samples, noting that higher order disclinations are structurally unstable (see Figure 1.36(a)). However, by doping usual nematics with non-mesomorphic molecules, it is possible to stabilize high-strength defects (Madhusudana and Pratibha 1983). Applying this recipe, the conversion of Gaussian light beams into scalar and vectorial singular ones, with a broad topological diversity and electrically tunable operating wavelength, has been reported for six different kinds of disclinations with q = (−3/2, −1, −1/2, 1/2, 1, 3/2). This is summarized in Figures 1.36(b)–(d) where streamlines of the director field around the singular defect core of a disclination with charge q are shown in the panel (b), and the corresponding XPOL observations are reported in the panel (c). As is the case for umbilics, the vortex purity can be optimized by applying an electric field along the normal to the plane of the film, however, this time provided that the dielectric anisotropy is positive, in order to reduce the birefringent phase retardation to an odd multiple of π. Vortex purity up to η 0.99 can be reached for |q| ≤ 1 and up to η 0.95 for |q| = 3/2, noting that half-integer disclinations cannot lead to axisymmetric director fields when the director is electrically reoriented out-of-plane. Indeed, axisymmetric 3D director fields are not compatible with the equivalence n ≡ −n (which is the reason why umbilics correspond to q = ±1). The singular

52

Liquid Crystals

beam shaping enabled by each of these structures is experimentally illustrated in Figure 1.36(d), which shows the interferograms resulting from the coaxial and coherent superposition of the output beam, generated from a circularly polarized incident Gaussian beam with a reference Gaussian beam. The characterization of optical phase singularities with topological charge l is made by noting that it leads to a |l|-arm spiraling pattern, whose handedness is associated with the sign of l. 



 


 

 

 

 

 








 






Figure 1.36. (a) Typical XPOL observation of a random distribution of disclinations in a pure nematic film with degenerate tangential boundary conditions. (b) Illustration of in-plane director streamlines around a disclination of charge q. Adapted from (Madhusudana and Pratibha 1983). (c) Experimental XPOL natural white light imaging of six different disclinations owing to the appropriate doping of usual nematics. Scale bar: 10 μm. (d) Interference patterns that result from the coaxial superposition of the vortex beam with a coherent reference Gaussian beam. All data refers to film thickness L = 6 μm. (c and d) Adapted from (Loussert et al. 2013)

While these results brought proof of principle that topological diversity limitations can be formally waived by using appropriate liquid crystal defects, the random distribution of spontaneously generated topological structures (that possibly mutually interact) is a drawback that needs to be solved in the development of self-engineered geometric phase optical elements. Several approaches have been proposed to address such an issue, which are surveyed in the following section. 1.8.3. Directed self-engineered geometric phase optics 1.8.3.1. Field-induced localization of umbilics in nematics To get rid of the random spontaneous emergence of the standard umbilics presented in section 1.8.2.1, implies breaking the translational invariance in the plane of a nematic film with perpendicular boundary conditions. Controlling the latter symmetry breaking is a familiar issue for the liquid crystal displays industry, since

Singular Optics of Liquid Crystal Defects

53

the advent of so-called VA (vertical alignment) technology in 1996 by Fujitsu, and several variants have been developed since then in order to achieve better viewing angles, response times and contrast ratios. In all cases, the idea is to direct the reorientation plane of the director in a patterned manner, however, without the primary requirement of having a continuous azimuthal distribution given by equation [1.23].  #    


 






  

   

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+


Figure 1.37. Illustration of several approaches to generate localized umbilics in a nematic film with perpendicular boundary conditions, by using a localized external stimuli seeding an axisymmetric director field, hence leading to a localized umbilic with topological charge q = 1. (a) Peripheral electric seeding from pixelated electrodes. Adapted from (Loussert and Brasselet 2014). (b) Peripheral magnetic seeding from an annular permanent magnet. Adapted from (Brasselet 2018). (c) Point-like seeding from mechanical stress. Adapted from (Migara et al. 2018). (d) Optical seeding from a moderately focused circularly polarized laser beam. Adapted from (Brasselet 2009, 2010b). See the text for details

Hereafter, we review four distinct options that have been investigated in recent years, which were all designed for the production of localized umbilics with q = 1. We refer to Figure 1.37 for a graphical summary. – Electric field seeding: a straightforward option relies on finite size, ideally disk-shaped, electrode on one of the two side of the film. By doing so, the electric

54

Liquid Crystals

field lines are no longer perpendicular to the plane of the film at the rim of the finite-size electrode, leading to a non-zero electric torque below the customary threshold voltage UF . This torque seeds the appearance of an umbilic, which is further grown by increasing the electric field amplitude. Meridional and in-plane cross-section sketches of an initial attempt based on a 1D array of square-shaped electrodes (Loussert and Brasselet 2014) are shown in the top part of Figure 1.37. The latter design also includes a possible non-zero applied voltage (labeled “2”) that is independent of that (labeled “1”) inducing the umbilic. This allows tighter lateral confinement by virtually imposing ϑ = 0 outside the pixels, providing the use of so-called dual frequency nematics, for which the sign of a depends on the used frequency, here with a (f1 ) < 0 and a (f2 ) > 0. Although this approach does not suit the creation of large aperture devices, it might be useful for the development of integrated singular beam shaping needs, for which the independent and parallel processing of several input optical signals would be of interest, and would require the mitigation of cross-talks between nearby pixels. We refer to section 1.9.2 for an example of application. Also, the pixel size implies a trade-off for the incident beam waist: if too small, a drop in the vortex purity occurs due to the low retardance in the defect cores, whereas, if too large, only a fraction on the incident field is processed, while cross-talk occurs for a dense array of pixels. Vortex purity up to η 0.95 has been reported in the visible domain for pixels with 60 μm×60 μm area, which requires post-polarization filtering to ensure a good quality vortex beam (see the bottom part of Figure 1.37). Consequently, this prevents the generation of vector beams whose quality competes with commercial conventional solutions. – Magnetic field seeding: this approach has been recently developed by analogy with the above electric field strategy, by exploiting the curved magnetic field lines of the axisymmetric magnetic field structure generated by an annular permanent magnet, as depicted in the top part of Figure 1.37 (Brasselet 2018). Under the action of the static magnetic field alone, the director is only slightly reoriented in a radial manner nearby the periphery of the clear aperture of the magnet. A single self-centered umbilic is then further grown by the application of a customary electric field, as is the case for random umbilics. This leads to a director structure whose swirling pattern (see the middle part of Figure 1.37), refers to an umbilic of a splay-twist nature. This results from the competition between the radial magnetic boundary conditions at the periphery, which promote a splay-type umbilic (i.e. φ0 = 0), and energetically favorable twist elastic distortions in usual nematics, which promote a twist-type umbilic (i.e. φ0 = π/2) near the core of the defect, where the elastic gradients are maximal. This has no consequence regarding the genesis of an optical phase singularity; however, recalling equation [1.21], this adds an extra radial dependence 2σφ0 (r) to the phase profile of the processed light field, which impacts the output propagation features (Kravets and Brasselet 2020).

Singular Optics of Liquid Crystal Defects

55

Remarkably, this approach allows the easy and reproducible creation of voltage-tunable vortex phase masks which have large clear aperture up to 1 cm2 and vortex purity up to η 0.99 over the whole visible domain, which makes post-polarization filtering unnecessary for most practical uses. This also means that good quality vector beams can be obtained from usual millimeter-sized linearly polarized incident beams. This is illustrated at the bottom of Figure 1.37: at 750 nm wavelength, the intensity profile of a vector beam obtained from a x-polarized Gaussian beam with 2 mm diameter is shown on the left, and the vectorial polarization structure is illustrated on the right by placing output polarizers oriented along the x axis. To date, there are no equivalent high-tech solutions that combine broadly tunable operating wavelength, centimeter-size clear aperture and high-quality topological ordering, which makes this approach worth developing further. – Mechanical stress seeding: this approach consists of using external point pressure that defines the location of the core of the umbilic (Migara et al. 2018), which thus differs from the two previous “peripheral seeding” strategies. Another distinctive feature of this method is that it relies on the interplay between the positional and orientational degrees of freedom of the liquid crystal (see Figure 1.37). Indeed, applying point-like mechanical stress induces an outward flow field that generates a transient localized umbilic of the splay type. Once this flow field is coupled with the presence of an electric field whose magnitude exceeds the Féedericksz threshold, this leads to an umbilic of the twist type, provided that one of the two possible twisted director distortion profiles (left-handed or right-handed) uniformly develops around the core. The latter is ensured by slightly doping the nematic with a chiral dopant that triggers the right/left symmetry breaking. Although optical vortex generation has been demonstrated, a quantitative analysis of the optical performances has not been experimentally reported. – Optical field seeding: in 2009, light itself was unveiled as a way to generate localized umbilics via the subtle role of the longitudinal electric field component Ez that is inherent to real-world beams, as Maxwell’s equation ∇ · D = 0 suggests (Brasselet 2009). To present the effect, we look at the optical torque density exerted by a normally incident circularly polarized Gaussian beam on the unperturbed director field n = z, however, without neglecting Ez as usually done in such a light-matter interaction geometry. Equation [1.17] thus gives Γ = 12 0 a Re(−Ez∗ Eφ er + Ez∗ Er eφ ) .

[1.26]

Therefore, non-zero radial and azimuthal components of the optical torque, respectively, excite azimuthal and radial director distortions leading to the generation of a localized umbilic, as illustrated in Figure 1.37. In this figure, a typical example

56

Liquid Crystals

of an optical vortex beam obtained from an incident circularly polarized coaxial probe beam (as well as its interferential characterization) are also shown (Brasselet 2010b). However, the obtained all-optically rewritable optical vortex masks that can be activated on demand both in time and space suffer from serious practical limitations: the vortex purity is inherently limited by a strongly non-uniform retardance profile and axisymmetry spontaneously breaks at large pump beam power (Brasselet 2010a). This thresholdless phenomenon is nevertheless intriguing in the context of the optical Fréedericksz transition (Zolot’ko et al. 1980) that usually deals with light-induced defect-free bell-shaped director profiles above a threshold (at normal incidence) and which is the optical analog of the more familiar electric Fréedericksz transition (see El Ketara et al. (2021) for recent developments). This opens up the possibility of self-induced singular beam shaping, which is addressed in the following section. 1.8.3.2. Self-induced singular beam shaping It is well-known that the regular optical reorientation of nematics associated with the customary optical Fréedericksz transition leads to strong self-focusing effects (Zolot’ko et al. 1980) as a result of light-induced Gaussian-like director reorientation amplitude. In short, a self-centered nonlinear refractive lens is created, which represents an emblematic nonlinear optical phenomenon in liquid crystals. Therefore, the topological optical reorientation from a circularly polarized light beam can be viewed as a first step toward nonlinear singular optics of liquid crystals. Here, the vortex purity increases with the incident beam power P , η = ηL + ηNL (P ) (El Ketara and Brasselet 2012), as illustrated in Figure 1.38. A technical note should be stressed: the vortex purity in the linear regime (when P → 0) is not zero even in the absence of a light-induced umbilic. This results from the non-zero longitudinal field component for a light beam, as discussed earlier when light propagates along the optical axis of uniaxial crystals (Volyar et al. 2002). It has been reported that the vortex purity is limited to modest values up to η ∼ 0.5 even in the presence of an applied voltage enhancing the optically induced reorientation. Moreover, ringing radial modulation of the generated vortex beam appears as the incident power increases (see Figure 1.38). An alternative all-optical option based on light-induced thermal effects using absorbing liquid crystals has also been discussed in Budagovsky et al. (2015a). Indeed, for sufficiently large absorbed power, a local nematic-to-isotropic phase transition leads to a self-centered axisymmetric director field outside the isotropic core, as the director tends to align along the normal to the nematic/isotropic interface. This can be viewed as the inverse version of the radial droplet vortex generator presented in section 1.7.2; however, only poor vortex purity of a few hundredths is obtained with this method. To date, such limitations confine the all-optical approach of self-induced singular beam shaping to fundamental exploration rather than practical developments. The above optical performance limitations are elegantly overcome when using a photo-conductor as one of the two substrates defining the nematic slab (Barboza

Singular Optics of Liquid Crystal Defects

57

et al. 2012), as depicted in Figure 1.38. The idea relies on a photo-controlled electric field experienced by the nematic owing to light-induced charge redistribution in the photo-sensitive substrate. This leads to an increase in the electric field’s amplitude that follows the transverse intensity distribution of the incident beam. Therefore, the electric Fréedericksz threshold can be triggered by light in the presence of an applied bias voltage. Since the optical intensity varies radially for an input Gaussian beam, the electric field lines are tilted with respect to the normal orientation of the cell, which seeds the growth of an umbilic with q = 1. The use of a nematic with a < 0 enables a director tilt from 0 to π/2, providing flexible access to a half-wave plate condition, taking the beam power P and bias voltage U as control parameters. Still, reaching optimal performances necessitates the consideration of the incident beam waist, cell thickness and the nature of the materials. This is illustrated in Figure 1.38, which shows a vortex purity reaching η 0.97 at large enough U in the first attempt dating from 2012 (Barboza et al. 2012). Similar purity performance using values of applied voltage typically 10 times lower have been reported since then (see, for instance, Kravets and Brasselet (2018)).      
 

  

     





      











 



 

 







   

 
   





   



 






























































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Figure 1.38. Main self-induced optical vortex generation processes based on light-induced umbilics with q = 1 in nematic films. Adapted from (El Ketara and Brasselet 2012; Barboza et al. 2012, 2013; Budagovsky et al. 2016; Kravets et al. 2019). See the text for details

Interestingly, the photo-electrical approach is not restricted to negative dielectric anisotropies for the applied electric field, to the use of AC field, nor to the use of a photo-conductive substrate. This has been reported in Budagovsky et al. (2015b), where a nematic film with perpendicular boundary conditions prepared in the usual

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Liquid Crystals

way from two glass substrates is submitted to a DC field, for which a > 0. In this case, a localized umbilic is created as a result of light-induced removal of DC field screening by surface electric charges. Further experimental investigations with nematics having either a > 0 or a < 0 have led to maximal vortex purity of η = 0.17 and η = 0.32, respectively (see Figure 1.38) (Budagovsky et al. 2016). Optimizing the parameters of the system, better performances can be reached as demonstrated in Kravets et al. (2019), where η > 0.9 has been reported in the case where a < 0, (see the de-screening dynamics in Figure 1.38). The DC option thus also offers decent optical performances when compared to the AC approach, though longer characteristic times appear less attractive from an applied point of view. 1.8.4. From single to arrays of optical vortices Anticipating the possible development of integrated liquid crystal singular photonic technologies, the generation of localized defect structures received some attention, focusing on the parallel singular optical processing of optical signals. Several approaches have been explored, as overviewed in Figure 1.39. Some of them correspond to the straightforward extension of strategies presented above, such as the use of multiple virtual or real localized electrodes instead of a single electrode (Barboza et al. 2013; Loussert and Brasselet 2014; Nassiri and Brasselet 2018) (see Figures 1.39(a) and 1.39(d), respectively). Other methods also emerged, such as self-assembled umbilical structures with alternating topological charges q = ±1 that cover electrodes with macroscopic dimensions under an AC electric field with appropriate time-varying amplitude envelope, as illustrated in Figure 1.39(b). Self-assembled arrays of umbilics have also been reported in ion-doped nematics in a specific range of parameters for the amplitude and frequency of the applied voltage (Sasaki et al. 2016; Salamon et al. 2018) (see Figure 1.39(c)). Interestingly these efforts were not restricted to defect structures in nematics and one can mention singular optics studies involving the spontaneous generation of ordered 2D arrays of focal conic domains in smectic A films (Son et al. 2014) or topological solitons in frustrated cholesteric films (Yang and Brasselet 2013), as shown in Figures 1.39(e) and (f), respectively. The latter attempts, however, suffer from limitations regarding vortex purity that does not exceed η ∼ 0.5. A next step worth considering would be the induction of arrays endowed with a broader “topological bandwidth” than those restricted to intertwined lattices of defects with topological charges q ± 1. 1.9. Emerging optical functionalities enabled by liquid crystal defects Over the last decade, liquid crystal topological defects have emerged as a special class of optical elements – geometric phase optical elements – that couple the

Singular Optics of Liquid Crystal Defects

59

polarization state of light with its spatial degrees of freedom.11 Naturally, the wealth of readily accessible topological director fields associated with liquid crystals’ defects make the latter well-suited to singular beam shaping purposes, as illustrated in the previous section. Of course, this should not ignore the impressive technological developments of artificially structured space-variant optically anisotropic materials since the demonstration by Biener et al. (2002). Indeed, progress in engineering eventually led to user-friendly, commercially available geometric phase optical elements – some of them made up of patterned liquid crystalline materials, as mentioned in section 1.6.4 – that are routinely used in research laboratories. It is therefore legitimate to ask if the singular optics of the defects of liquid crystals are doomed to remain an entertaining scientific curiosity. Without predicting the future, hereafter we illustrate how the self-organization capabilities of liquid crystals allow for the exploration of novel adaptive and reconfigurable singular optical devices with performances that would be barely reachable otherwise. 1.9.1. Spectrally and spatially adaptive optical vortex coronagraphy In this section, we address how the use of liquid crystal defects has recently been studied in the framework of high dynamic range astronomical imaging. Specifically, such imaging techniques are required for the spectral study of the atmospheres of planets similar to those of the Solar system, which is a difficult task due to the very high optical contrast and proximity between planets and their hosting stars. These techniques are based on instruments – coronagraphs – allowing for the suppression of the light of a star without modifying the light of the nearby orbiting planets. Coronagraphs date back to the 1930s when Lyot tackled the problem of observing the Sun’s corona without requiring a total solar eclipse (Lyot 1931). Lyot’s invention resulted in the creation of an artificial eclipse by placing an obstructing mask at the focal plane of a telescope and another one at the exit pupil plane, thereby selectively suppressing sunlight in favor of corona light, as illustrated in Lyot’s sketch of his instrument in Figure 1.40(a). Since then, high-contrast coronagraphic imaging technology is ever-improving and is nowadays implemented not only in ground-based telescopes but also in space telescopes (including the Hubble Space Telescope, as well as its planned successors). With the aim of reaching high-contrast coronagraphic imaging capability at angular distances from the bright source as close as possible to the theoretical diffraction limit of the telescope, several concepts have emerged (Guyon et al. 2006) and a promising one – optical vortex coronagraphy – is the topic of this section.

11 This refers to the spin–orbit interactions of light, which involve geometric phases originating from different processes (Bliokh et al. 2015) and this chapter restricts to the geometric (Pancharatnam–Berry) phase resulting from space-variant anisotropic media.

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Liquid Crystals









 

















Figure 1.39. (a–d) Field-induced arrays of umbilics in nematic films with perpendicular boundary conditions. (a) Photo-electric generation of hexagonal vortex lattices. Top: multiple-site optical writing light field where the dashed line hexagonal pattern is a guide for the eyes. Bottom: XPOL imaging showing the generated array of defects. Adapted from (Barboza et al. 2013). (b) XPOL imaging of a spatially extended square-shaped electrode under a flat (top) or pulsed (bottom) amplitude envelope for the applied AC electric field. Adapted from (Migara and Song 2018). (c) XPOL imaging of a self-organized array of defects in ion-doped nematic under AC electric field in a 0.5 mm × 0.5 mm square-shaped pixel defined from the overlap of perpendicular striped electrodes. Adapted from (Salamon et al. 2018). (d) XPOL imaging of an array of defects generated by a pixel matrix made of 50 μm × 50 μm square-shaped electrodes. Adapted from (Nassiri and Brasselet 2018). (e) Hexagonal array of selfassembled focal conic domain in open-air smectic A film deposited on glass. Adapted from (Son et al. 2014). (f) Effective polarimetric reconstruction of a hexagonal array of self-assembled localized defect structures in frustrated cholesteric film. Luminance refers to the birefringent phase retardation and the “hsv” colormap refers to the in-plane orientation angle of the optical axis (photo by M. Rafayelyan)

1.9.1.1. Optical vortex coronagraphy: basic principles Optical vortex coronagraphy relies on imparting a helical phase profile associated with an even topological charge l to the light field in the focal plane of the telescope, which makes it belong to the coronagraph of the phase type as opposed to the amplitude type approach initiated by Lyot. This technique emerged in 2005 (Mawet et al. 2005; Foo et al. 2005) and the first image of exoplanets from a vortex coronagraph was reported in 2010 using the Hale Telescope (Serabyn et al. 2010). Optical vortex coronagraphs now equip all main ground-based telescopes (Absil et al. 2016) and technological developments are ongoing. The principle of operation is illustrated in Figure 1.40(b) for l = 2. The optical vortex phase mask is placed at the focal plane of a telescope with a disk-shaped entrance pupil, and is centered on

Singular Optics of Liquid Crystal Defects

61

the starlight point spread function. In turn, the light is ideally rejected outside the geometric image of the input pupil plane. Placing an iris in the latter plane can thus prevent starlight from reaching the detector. In contrast, off-axis light entering the telescope is almost unaltered by the singular phase mask for angular separation larger than the diffraction limit αdiff = 0.61λ/R, where R is the radius of the input circular aperture, and can thus be imaged. 













 



















Figure 1.40. (a) Lyot’s Drawing of his coronagraph. Copyright: Observatoire de Paris, Patrimoine Scientifique de l’Observatoire de Meudon. P1 : input pupil plane; P2 : focal plane of the telescope; P3 : exit pupil plane; P4 : imaging plane. (b) Principle of the optical vortex coronagraph for on-axis observation of a point source placed at the infinity, placing an optical vortex mask with l = 2 in the plane P2 , see phase profile of the mask in the inset on the left. Thick vertical lines refer to opaque apertures and Li refers to lenses with focal length fi . Inset on the right: intensity distribution (luminance) with color coded phase in the plane P3

The creation of a nodal area in the exit pupil plane is a remarkable and non-trivial wave-optics manifestation worth grasping, at least from a mathematical point of view. In fact, this can be done with the simple knowledge of the effect of a thin spherical lens in the paraxial regime. Namely, using polar coordinates (r, φ), the relationship between the field E− located at a distance d− before a lens with focal length f , and the field E+ located at a distance d+ after it is, up to a propagation phase associated with the axial optical path between the two planes (Collins 1970), E+ (r, φ) ∝ −

ik 2πA



2π ∞ 0

ik

E− (ρ, θ)e 2A [−2ρr cos(θ−φ)+Br

2

+Cρ2 ]

ρdρdθ ,

[1.27]

0

where k = 2π/λ, A = [f 2 − (f − d− )(f − d+ )]/f , B = (f − d− )/f and C = (−) (f − d+ )/f . Accordingly, the field E2 right before the focal plane P2 is given by inserting {f, d− , d+ } = {f1 , 0, f1 } into equation [1.27], taking E1 (r < R) = E0 and  2π E1 (r > R) = 0 at P1 . Accounting for the identities 0 exp[iξ cos(θ − ϕ) + ilθ]dθ = 2πil exp(ilϕ)Jl (ξ) and Jl (−ξ) = (−1) Jl (ξ) where Jl is the Bessel function of the

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Liquid Crystals

first kind of order l, we get (−)

E2 (r, φ) ∝ E0

J1 (kRr/f1 ) exp[ikr2 /(2f1 )] . kRr/f1 (+)

Straightforwardly, the field E2 (+)

[1.28]

right after P2 is

(−)

E2 (r, φ) = E2 (r, φ) exp(ilφ)

[1.29]

and the expression of the field E3 in the plane P3 in the absence of a circular aperture in the exit pupil plane is obtained by inserting {f, d− , d+ } = {f2 , f2 , f2 (1 + f2 /f1 )} into equation [1.27], which cancels the quadratic phase and gives 

∞

E3 (r, φ) ∝ E0 exp(ilφ)

J1 (kρR1 /f1 )Jl (kρr/f2 )dρ.

[1.30]

0

In the absence of a vortex mask, i.e. l = 0, the above equation gives  E3 (r, φ) ∝ E0 l=0

1 if r < R = Rf2 /f1 0 if r > R

[1.31]

which allows for the identification of the plane P3 as the exit pupil plane. In contrast, for non-zero even , the exit pupil field can be rewritten as  E3 (r, φ) ∝ E0 l even

0 if r < R (R /r) exp(ilφ) R1|l|−1 (R /r) if r > R

,

[1.32]

with R1|l|−1 the radial Zernike polynomial. Therefore, for non-zero even l, it is ideally possible to reject on-axis illumination completely by placing a circular aperture with a radius smaller than R . Note that the above analytical developments emphasize that the quadratic phase vanishes at exit pupil plane P3 ; therefore, we can rely on Fourier transforms to simulate the intensity distribution I4 of the observed image in the plane P4 numerically, which lies in the focal plane of the lens L3 . Namely, considering an arbitrary incident paraxial field Ein and complex amplitude transmission masks τi placed at Pi , we obtain   2 I4 ∝ F τ3 F −1 [τ2 F [τ1 Ein ]]  where F (F −1 ) denotes the (inverse) Fourier transform.

[1.33]

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63

1.9.1.2. A soft route to optical vortex coronagraphy

 

 



    0

10



1

Normalized intensity




Normalized transmission

In 2016, optical vortex coronagraphy using umbilical defects as geometric phase optical vortex mask was explored in laboratory both in the visible domain and in the monochromatic regime (Aleksanyan and Brasselet 2016). The experiment first consists of selecting one of the few static defects after the annihilation dynamics between nearby topological defects with opposite topological charges has taken place. Typically, experiments are performed two hours or more after the voltage has been switched on and whose steady value is set at a voltage Δ∞ = π (see section 1.8.2.1). Then, a defect is placed on-axis in the focal plane of a telescope according to the sketch shown in Figure 1.40(b), and starlight is mimicked by an on-axis incident quasi-plane wave. Importantly, the core region of the umbilic prevents from the ideal realization of a nodal area at the exit pupil plane. Still, placing the nematic sample between crossed circular polarizers can achieve decent coronagraphic performances, as qualitatively illustrated in Figure 1.41(a), where the pseudo-nodal areas are shown at P3 for q = ±1. Note that the observed intensity pattern with fourfold rotational symmetry in the exit pupil plane for q = −1 results from the elastic anisotropy of liquid crystals. Indeed, umbilics with q = −1 involves both splay and twist elastic constants in the plane of the nematic film (Rapini 1973), which induces fourfold rotationally symmetric modulations of ϑ and ϕ.

0

1

2

3




4

5

0

 10

-5



10

-10

0

2.5

5




Figure 1.41. Laboratory demonstration of optical vortex coronagraphy using liquid crystal umbilical defects in a nematic film. (a) Typical intensity distribution at the plane P3 using umbilics with charge q = ±1. (b) Total power collected through aperture with radius 0.75R at the exit pupil plane P3 as a function of the reduced incidence angle for α/αdiff with respect to the telescope axis, for q = ±1. (c) Azimuth-averaged angular intensity profile at P4 for on-axis incident light when the coronagraph is off (no aperture at P3 , black curve) and on (Lyot stop with radius 0.75R at P3 , color curves for q = ±1). Solid curves: experiments. Dashed curves: simulations. All data refer to R = 1 mm, f1 = f2 = 45 mm, f3 = 300 mm and λ = 633 nm. Adapted from (Aleksanyan and Brasselet 2016)

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Liquid Crystals

Quantitatively, the diffraction limited rejection capabilities are shown in Figure 1.41(b) where the measured total optical power coming out of the circular aperture at P3 is shown as a function of the incidence angle. Finally, the high dynamic range imaging performances are evaluated by recording the image at P4 of the emulated on-axis distant point source with and without the presence of a circular aperture at P3 (aka the Lyot stop); see Figure 1.41(c) that displays azimuth-averaged angular intensity profiles. A peak-to-peak azimuth-averaged intensity rejection ratio of ∼ 1000 associated with a typical power rejection rate ∼ 50, defined as the ratio between the power outside and inside the Lyot stop, are obtained at λ = 633 nm. Simulations’ attempts to support the experimental observations are shown as the solid curve in Figure 1.41(b) and the dashed curves in Figure 1.41(c). The calculations are made according to equation [1.33] using τ1 = circ(r/R) and τ3 = circ(r/[0.75R]) for the circular apertures at P1 and P3 , respectively, with circ(u) = 1 for u < 1 and circ(u) = 0 for u > 1. In addition, the complex transmittance mask associated with the umbilic is obtained from equations [1.7] and [1.8] with e0 = c± and post-selection of the c∓ polarization state component, which gives τ2 (r, φ) = sin[Δ(r)/2] exp[iΔ(r)/2 ± 2iφ] .

[1.34]

2 Moreover, Δ(r) = πa2 (r), which assumes θ∞  1 in the series expansion of the birefringent phase retardation (see section 1.8.2.1), and rc is evaluated at the voltage value providing Δ∞ = π for the used materials.

These results call for several comments. In fact, the expected optical rejection performances, although encouraging for a preliminary study, are typically two decades lower than the expected performances (see Figure 1.41(c)). This encourages further improvements both theoretical and experimental. Firstly, the Rapini model consider an isolated umbilic with r-independent φ0 (see equation [1.23]), and is valid in the limit of small reorientation amplitude. Since none of these assumptions is safely satisfied, a substantial overestimation of the coronagraphic performance compared to the experiment can be easily understood. Secondly, the recent magneto-electric generation of umbilic has solved the single-defect issue as well as the need for large clear apertures (see section 1.8.3.1). Noteworthy is that the core size exhibits a drastic downsizing under high voltage, which we attribute to the saturation of the director tilt reorientation (see Figure 1.42(a)), while the swirl of the transverse director pattern progressively vanishes (see Figure 1.42(b)). The latter trends are both in favor of better performances that deserve further investigation in order to eventually get rid of circular polarization filtering, which is desirable to reduce acquisition time when imaging faint celestial objects. Also, we note recent attempts to tame the swirl of umbilics by on-axis cascading of oppositely swirled defects, which leads to an effective vortex mask with topological charge four (see Figure 1.42(c)). Today, the difficulty of achieving a uniform optical retardance profile outside of the core appears to be the main limitation preventing reaching higher

Singular Optics of Liquid Crystal Defects

65

rejection capabilities and there is room for improvement, and it is worth recalling the uniqueness of the spectral agility of such optical vortex masks. 


  

  

  

  

 


 


  


  
















 

Figure 1.42. (a,b) Evolution of the core and transverse swirl of localized magnetoelectrically generated umbilics. (a) Observation of the umbilic core by placing the sample between crossed circular polarizers under white light incoherent illumination spectrally filtered at 532 nm wavelength. The data refers to applied voltage values for which Δ∞ = N π, N beign an odd integer. (b) XPOL observation of the swirl in the neighborhood of the defect using white light incoherent illumination. Adapted from (Brasselet 2018). (c) Illustration of the cascading strategy enabling to mitigate the swirl by mutual compensation of defects with opposite swirl at the expense of doubling the topological charge of the effective optical vortex mask. Adapted from (Kravets and Brasselet 2020)

1.9.1.3. Multiple-star optical vortex coronagraphy Now that more than 4000 exoplanets detected by various means have been confirmed,12 the development of coronagraphic technologies will undoubtedly continue for many years to come as will optical vortex coronagraphs. However, the latter can only selectively reject the light from a single star at a time while the amount of multiple-star systems is expected to be non-negligible. This points to the need for novel instruments to image planets hosted in those systems, whose number is about 200 to date.13 Worlds with two suns, which have entered the collective imagination since the release of Star Wars, (see Figure 1.43(a)), are no longer restricted to the realm of science fiction. Indeed, the detection of single (Doyle and et al. 2011) and multiple (Orosz and et al. 2012) planetary systems orbiting a pair of stars started almost a decade ago while a few attempts to null binary stars (Crepp et al. 2010; Cady et al. 2011; Kühn and Patapis 2016) have also since emerged.

12 Source: http://exoplanet.eu/catalog, accessed December 2020. 13 Source: https://www.univie.ac.at/adg/schwarz/multiple.html, accessed December 2020.

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In this context, a multiple-vortex coronagraph enabling the simultaneous and selective nulling of multiple-star systems has been proposed (Aleksanyan et al. 2017), soon after preliminary results based on randomly distributed umbilics (see Figure 1.43(b)). The idea is to superimpose the centroids of the stellar images with the location of defects in the focal plane P2 . However, without precise control of the defect positions, this method is not a viable option since one has to adapt to the position of the stars, and not the contrary, as is the case in the illustration in Figure 1.43(b). This issue is solved by exploiting the photo-electrical generation of localized umbilics relying on a photoconductive substrate (see section 1.8.3.2). Namely, N independently steerable writing light beams create N vortex masks Mi adapted to a system of N stars Si , as illustrated in Figure 1.43(c) for a triple-star system. Coronagraphic laboratory demonstration is implemented by adding a faint off-axis illumination – aka the planet P, see in Figure 1.43(d), nearby the star S1 . 





 



 
















































Figure 1.43. (a) Binary star sunset image from the movie Star Wars: A New Hope. (b) Binary star vortex coronagraphy laboratory demonstration in the plane P4 from a pair of randomly generated umbilics, to which the angular position of the stars is adapted. I0 is the maximal intensity when the coronagraph is off. Inset: XPOL image of the pair of defects in the focal plane P2 , scale bar: 200 μm. Unpublished data from (Aleksanyan and Brasselet 2016). (c) Airy spot of three stars (Si ) in P2 without (top, direct image) and with (bottom, XPOL image) the optical activation of adaptive localized vortex masks (Mi ) generated by a photo-electrical approach based on a photoconductive substrate. (d) Triple-star/planet coronagraphic demonstration in P4 , both images being normalized to their own maximal intensity Imax . Parameters: R = 2 mm, f1 = f2 = 200 mm, f3 = 500 mm, a Lyot stop with radius 0.75R and λ = 633 nm. (c and d) Adapted from (Aleksanyan et al. 2017)

Above developments carried out in the framework of high contrast astronomical imaging should not ignore other opportunities for optical imaging techniques relying on optical vortices, such as stimulated emission depletion microscopy (STED) (Hell and Wichmann 1994) and spiral phase contrast microscopy (Furhapter et al. 2005). In addition, we note that the discussed limitations associated with space-variant optical retardance could be turned into benefits whenever singular complex amplitude transfer functions are desirable. All this offers many possibilities to explore.

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1.9.2. Multispectral management of optical orbital angular momentum Liquid crystals spatial light modulators nowadays remain a prime choice to shape an arbitrary structured light field. Recalling that each of their pixels operate as a uniform optical retarder with an electrically controlled dynamic phase, what we were to use structured pixels with an electrically controllable geometric phase instead ? This would give access to spatial light modulation at the level of both one pixel and their assembly, hence opening novel beam shaping possibilities. This was suggested and implemented in 2018 by using arrays of individually tunable pixelated umbilics using electrode patterning (Nassiri and Brasselet 2018), whose individual operating mode has been presented in section 1.8.3.1. The generic idea of the proposed approach is to perform the multispectral management of the photon orbital angular momentum. This is done by generalizing the monochromatic behavior of a geometric phase optical vortex mask with structural topological charge q to a polychromatic field consisting of a discrete set of wavelength {λn } which is individually processed by a set of wavelength tunable singular beam shapers associated with a discrete set of charge {qn }. To illustrate this, equations [1.7] and [1.8] are rewritten  in the circular polarization basis (c+ , c− ). Considering an input light field Ein = λn En (an c+ +bn c− ), where an and bn are complex constants, the output field is expressed as Eout =

 λn

 En eiΦn

cos Δ2n i sin Δ2n e−2iqn φ i sin Δ2n e+2iqn φ cos Δ2n

  an bn

[1.35]

where Φn and Δn are the dynamic phase and the birefringence phase retardation at wavelength λn , respectively. The orbital angular momentum state of the c± -polarized channel n is therefore shifted by δln = ±2qn when Δn = Nn π with Nn an odd integer while it remains unaffected for an even integer. The corresponding implementation scheme is shown in Figure 1.44(a). Experimentally, 1D arrays with q = 1 have been implemented, which correspond to lines of the 2D array shown in Figure 1.39(d), and the identification of the vortex response of a single pixel by interferential means is shown in Figure 1.44(b). Despite dealing here with a simple and specific situation (1D array and q = 1 for all n), this allows for the unveiling of a few novel beam shaping opportunities based on spectrally controlled optical orbital angular momentum. Namely: – Ultra-broadband scalar and vectorial vortex beam shaping: when choosing odd Nn for all n, we ideally obtain a superposition of pure optical vortices independently of the incident polarization state according to equation [1.35]. The experimental implementation is depicted in Figure 1.44(c) for incident circular polarization, namely {an = 1, bn = 0} or {an = 0, bn = 1}, which leads to polychromatic scalar vortex generation. The vectorial analog straightforwardly arises from the use

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of an incident linear polarization state, namely {|an | = |bn |} and {arg(an /bn ) = constant}, which gives polychromatic vector beams. The extension to arbitrary incident polarization states independently controlled for each topological pixel makes it possible to consider even more possibilities for structured polychromatic beams.

  












    



  
  

  
  






   


 









   



Figure 1.44. (a) Principle of multispectral modulation of the orbital angular momentum content of a set of discrete spectral channels λn by an array of q-plates of order qn . (b) Interferometric demonstration of δl = 2 performed by an individual umbilical pixel adjusted to odd N at 550 nm wavelength. (c) Multispectral optical vortex generation with incident circularly polarized incident spectral channel and output contra-circular polarization filtering. The leftmost data corresponds to the images of the seven spectral channels λn = 473, 496, 518, 539, 559, 582, and 611 nm with ∼10 nm full-width at half-maximum in the plane of the array and the rightmost image refers to the ensuing far-field polychromatic vortex. (d) Demonstration of on-demand polychromatic superposition of orbital angular momentum states. Left: Nn odd except for the blue channel. Right: Nn odd except for the red channel. Brightness and contrast are adjusted to make the on-axis spectral channel spot clearly visible. (e) Calculated spatiotemporal beam shaping from spectral vortex modulation of an incident cσ polarized chirped Gaussian optical pulse with ∼40 fs duration, in the hyperspectral limit (i.e. continuous spectral processing). Left: total far field intensity profile for q = +1 and a bell-shaped optical vortex purity that is maximal at the central pulse frequency. κ refers to spatial frequencies in the transverse plane (arbitrary units). Right: intensity profiles of the c±σ -polarized field components carrying 0 (red) and 2σ (green) orbital angular momentum per photon, respectively. Adapted from (Nassiri and Brasselet 2018) where the misprinted set {λn } for present panel (c) is corrected here

– Spectral vortex modulation: the proposed liquid crystal spatial modulator with topological pixels comes with a unique added-value. Indeed, tailor-made wavelength-dependent orbital angular momentum control can be achieved. This is illustrated in Figure 1.44(d), which demonstrates that on-axis intensity of any spectral component λn of a polychromatic field can be activated on demand by switching Nn

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from an odd to an even value. In the context of STED microscopy (Hell and Wichmann 1994), where a vortex beam ensures selective annular deactivation of fluorophores while a non-vortex beam activates them on-axis, this means that the imaging wavelength(s) can be adapted to the fluorescence properties of the sample in a flexible and selective manner. Other STED-inspired technologies, for instance in the field of optical nanolithography, could also benefit from such features. – Spatiotemporal shaping of ultrashort optical pulses: optical pulse shaping is a mature technology having a huge range of applications, for instance in spectroscopy and lightwave communications (Weiner 2011). Spectral vortex modulation has also been proposed to bring enhanced flexibility to this technology (Nassiri and Brasselet 2018). The basic scheme relies on Fourier-transform pulse shaping. It consists of decomposing an incident pulse into its constituent spectral components that are individually modulated. Once recombined, the spectral components reconstruct an output waveform given by the Fourier transform of the reshaped spectrum according to the scheme presented in Figure 1.44(a). Here, the idea is to modulate the orbital angular momentum along the temporal coordinate of an incident pulse whose frequency changes along its temporal waveform. This is illustrated by simulations in Figure 1.44(e) which show the generation of an “optical bottle pulse” for a birefringent phase retardation dispersion Δ(ω) = π + β(ω − ω0 ) where the constant β weights the amount of dispersion. In this case, the central angular frequency component ω0 is fully converted into a vortex state while others experience partial vortex transformation according to the frequency-dependent vortex purity parameter η(ω) = sin2 [Δ(ω)/2]. The resulting non-stationary spin and orbital angular momentum points out a possible interest for spatio-temporal shaping of light-matter angular momentum transfer in the fields of optical spectroscopy, optical manipulation, or high-energy physics (laser-induced particle acceleration). These preliminary attempts obviously leave room for improvement, especially regarding the purity that ranges so far from 0.8 up to 0.95 depending on the operating conditions. Still, the programmable manipulation of light both in space and time via the spectral management of the photon orbital angular momentum provides us with a promising toolbox for many situations, from continuous lightwaves to ultrashort optical pulses. Moreover, since its general principle relies on phase singularities, it can be formally adapted to any wavelength range as well as other kinds of scalar waves. 1.10. Conclusion This chapter has attempted to highlight how the physics of topological defects in liquid crystals has progressively enriched liquid crystal optics as singular optics has developed. This is rooted in a few generic features: the self-organization capabilities of soft matter systems, their sensitivity to external stimuli and the light-matter

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interaction when light and/or matter is endowed with topological properties. While it is too early to assert that the topological character of liquid crystal supramolecular structures will give rise to topological photonic technologies coming out of research laboratories, further developments of beam shaping and imaging applications are nevertheless underway. Moreover, it already appears that fundamental research in this field has an exciting future ahead of it. In particular, we can mention a few barely explored, or as yet unexplored, options, such as the possibility of achieving modal optical vortex generators for both for the azimuthal and radial degrees of freedom (see section 1.6.4) out of self-engineered structures. An interesting step in that direction has been reported very recently in the framework of microlasers made of liquid crystals defect structures placed in a Fabry-Pérot microcavity (Papiˇc et al. 2020). Another topic concerns a recently introduced route to broadband geometric phase optical elements made up of cholesteric liquid crystals that are based on the interplay between the circular Bragg reflection (see section 1.3.6.2) and the geometric phase, which occurred in 2016 (Kobashi et al. 2016a; Rafayelyan et al. 2016; Rafayelyan and Brasselet 2016; Kobashi et al. 2016b; Barboza et al. 2016). The underlying physics of the latter “Bragg-Berry” geometric phase optical elements can be explored in Figure 1.7(a). Indeed, the circular Bragg reflection corresponds to a reflective half-wave plate whose effective optical axis corresponds to the in-plane orientation of the director at the input facet of the cholesteric Bragg reflector, hence satisfying the sought-after condition for pure geometric phase shaping. Until now, Bragg–Berry type optical elements have only been prepared from artificially structured cholesteric films. We may therefore wonder whether cholesteric defects could provide enhanced structural and optical performances as nematic defects have allowed. Speaking of cholesterics, the recent developments of waveguiding optics mediated by topological solitons in frustrated cholesteric films (Hess et al. 2020) also invite for exploring their singular optics features. The topological legacy of carrots is not over yet. 1.11. References Absil, O., Mawet, D., Karlsson, M., Carlomagno, B., Christiaens, V., Defrère, D., Delacroix, C., Femenía Castellá, B., Forsberg, P., Girard, J., et. al. (2016). Three years of harvest with the vector vortex coronagraph in the thermal infrared. Proc. SPIE, 9908, 99080. Ackerman, P.J. and Smalyukh, I.I. (2017). Diversity of knot solitons in liquid crystals manifested by linking of preimages in torons and hopfions. Phys. Rev. X, 7, 011006. Ackerman, P.J., Qi, Z., Lin, Y., Twombly, C.W., Laviada, M.J., Lansac, Y., Smalyukh, I.I. (2012a). Laser-directed hierarchical assembly of liquid crystal defects and control of optical phase singularities. Sci. Rep., 2, 414. Ackerman, P.J., Qi, Z., Smalyukh, I.I. (2012b). Optical generation of crystalline, quasicrystalline, and arbitrary arrays of torons in confined cholesteric liquid crystals for patterning of optical vortices in laser beams. Phys. Rev. E, 86, 021703.

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2

Control of Micro-Particles with Liquid Crystals Chenhui PENG1 and Oleg D. LAVRENTOVICH2 1

Department of Physics and Materials Science, University of Memphis, Tennessee, USA 2 Advanced Materials and Liquid Crystal Institute, Department of Physics, Kent State University, Ohio, USA

2.1. Introduction This chapter deals with the dynamics of living and inanimate micro-particles controlled by a liquid crystal (LC) environment. The first theme is nonlinear liquid crystal-enabled electrokinetics (LCEK) in an LC electrolyte. Director gradients created either by colloidal particles or by patterning of surface alignment in the presence of an externally applied electric field produce space charges that trigger flows with velocities proportional to the square of the electric field. As a result, an LC electrolyte enables transport of colloids of any shape and composition (including fluid droplets and gas bubbles), even when the particles show neither bulk nor surface charges. Various potential applications of LCEK, such as microscale sorting and guided transport, are discussed. The second theme focuses on non-toxic lyotropic chromonic liquid crystals (LCLCs) to control the individual and collective dynamics of microswimmers such as flagellated bacteria. The LC environment controls the spatial distribution of microswimmers, a transition from bidirectional apolar swimming to unidirectional polar swimming, and the geometry of swimming trajectories. The system represents an experimental

Liquid Crystals, coordinated by Pawel PIERANSKI, Maria Helena GODINHO. © ISTE Ltd 2021. Liquid Crystals: New Perspectives, First Edition. Pawel Pieranski and Maria Helena Godinho. © ISTE Ltd 2021. Published by ISTE Ltd and John Wiley & Sons, Inc.

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realization of active matter in which activity and orientational order can be tuned independently from each other. 2.2. Control of micro-particles by liquid crystal-enabled electrokinetics Microscale manipulation of fluids and colloidal particles by an electrical field, called electrokinetics, finds a broad range of applications in microfluidics, information displays, medical diagnostics, biosensing and sorting (Bazant et al. 2009; Bazant & Squires 2010; Ramos 2011; Aranson 2013; Dobnikar et al. 2013; Zöttl & Stark 2016; Peng et al. 2019a, 2019b). Electrokinetic phenomena are typically explored when the carrier medium represents an isotropic aqueous electrolyte. The mechanism by which an electric field causes electrokinetics in an electrolyte depends primarily on how the electric charges are separated in space. In classic linear electrokinetics, the charges are separated through dissociation of surface groups at the water-solid interface. For example, glass placed in water at pH>3 becomes negatively charged by releasing positively charged protons and forming negatively charged SiO− groups. These negative charges immobilized at the surface of the particle attract a cloud of positive charges; together they form an electric double layer, typically of a subnanometer or a nanometer thickness. If a uniform electric field E is applied along a flat glass-water interface, it will move charges of opposite polarities into opposite directions. The two opposing Coulomb forces of an equal amplitude create a relative shift of the solid and the electrolyte. If the solid is immobilized, the electric field would cause a flow of the electrolyte. The effect is called a linear electro-osmosis. If the solid is a small particle freely suspended in the electrolyte, it will move by pushing the electrolyte backward; the ensuing effect is called electrophoresis. The mechanism is easy to understand by considering a sphere. The external uniform field creates a torque at the poles on the sphere where the electric double layers are parallel to the field; these torques tend to realign the local electric field of the electric double layers, thus pushing the electrolyte and the particle in opposite directions. Since far away from the particle the electrolyte does not move, the relative motion of the two systems of charges near the interface propels the particle with a speed v = εε 0ς E / η that is either parallel or antiparallel to the direction of the field, depending on the polarity of immobilized charges. Here, ε and η are the dielectric permittivity and viscosity of electrolyte, respectively, ε 0 is the electric constant and ς is the zeta potential that does not depend on the applied electric field (Morgan & Green 2003). Note that the electrophoretic motion is force-free, similarly to the microscale swimming at low Reynolds number: since the system is electrically neutral, the forces on positive and

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negative charges are equal in amplitude and it is the spatial separation of these charges that produces propulsion. The linear electrophoresis could be driven only by a direct-current (DC) field, which creates problems in applications, such as the screening of the field by free ions and chemical reactions at electrodes. It has always been desirable to find nonlinear mechanisms of electrokinetics that could be powered by the alternating-current (AC) field. One of these is the so-called induced-charge electrokinetics (ICEK), in which the velocities of the flows or the particles grow as the square of the field, as has been established experimentally (Levitan et al. 2005; Gangwal et al. 2008; Wu & Li 2009; Daghighi et al. 2013; Zhang and Li 2013; Peng et al. 2014; Yan et al. 2016; Zhang et al. 2017) and theoretically (Murtsovkin 1996; Bazant & Squires 2004, 2010; Squires & Bazant 2004; Daghighi et al. 2011; Daghighi & Li 2013). The square dependence allows one to use the AC driving, which avoids detrimental effects such as electrode blocking and electrochemical reactions. In contrast to the linear mechanism, the space charges are formed by the applied electric field, so that the zeta potential is induced by the field and grows with it. Such an effect of a field-induced electric double layer was first discussed by Dukhin et al. (1987a, 1987b) who, in 1987, discovered that conductive particles of ion-exchangers move with the velocity proportional to the square of the field and explained that the effective zeta potential should be proportional to the applied field and the particle radius a , ς ∝ aE. Similar nonlinear effects were observed not only for particles of ionic exchangers but also for strongly polarizable metallic particles (Simonov & Shilov 1977; Gamayunov & Murtsovkin 1987; Murtsovkin & Mantrov 1990; Peng et al. 2014). Bazant and Squires (2010) generalized a broad range of effects in which the field induces spatial separation of charges and triggers electro-osmotic flows or electrophoretic transport. They coined the term ICEK (see Bazant and Squires 2010 and references therein). In the Bazant-Squires model, the field acting on a polarizable (metal) sphere of a radius a placed in an isotropic electrolyte causes the electric carriers within the metal to shift toward the surface of the particle. The field also drives the ions present in the aqueous electrolyte toward the surface. Together, the charges within the metal and in the electrolyte form an electric double layer, which expels the field lines, creating a component of the field directed tangentially to the surface. This tangential field component drags the ions around the particle and thus triggers the electrokinetic flows. For a spherical particle, the flows around its two hemispheres are mirror images of each other and the effect does not create any net flow nor does it make the sphere electrophoretically mobile. The local flow velocity should be proportional to ς E ∝ E 2 , since ς ∝ E (Squires & Bazant 2004). This dependency as well as the quadrupolar symmetry of the flow pattern around a

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gold sphere are confirmed in direct experiments with gold spheres (Peng et al. 2014). A metallic sphere cannot be electrophoretically mobile in a uniform field, since the electro-osmotic flows on its two sides cancel each other out. However, if the particle is asymmetric, for example, if it represents a metal-dielectric Janus sphere, the induced-charge electro-osmotic flows adopt a dipolar symmetry, with one hemisphere showing a stronger flow than the other (Peng et al. 2014). If such a sphere is freely suspended in the electrolyte, it will be transported with the velocity v ∝ E 2 by an applied DC or AC field; this induced-charge electrophoresis (ICEP) of Janus spheres has been demonstrated by Gangwal et al. (2008). When an isotropic electrolyte is replaced with a nematic LC, a new nonlinear mechanism emerges, the so-called liquid crystal-enabled electrokinetics (LCEK), which is remarkably different from the isotropic scenario (Lavrentovich et al. 2010; Hernàndez-Navarro et al. 2013, 2014, 2015a, 2015b; Lazo & Lavrentovich 2013; Lazo et al. 2014; Peng et al. 2015; Calderer et al. 2016; Tovkach et al. 2016; Conklin & Viñals 2017; Paladugu et al. 2017; Tovkach et al. 2017; Conklin et al. 2018a, 2018b; Peng et al. 2018; Straube et al. 2018a; Li et al. 2020; Sahu et al. 2020), as reviewed by Lavrentovich (2014, 2016, 2017) and Peng & Lavrentovich (2019). LCEK is caused by field-triggered space charge separation that occurs when two conditions are fulfilled: (1) the LC shows a non-vanishing anisotropy of the dielectric permittivity or electric conductivity; and (2) the direction of local molecular orientation of the LC, the so-called director nˆ , varies in space. In this section, we will first discuss the LC-enabled electrophoresis (LCEP) of particles and how an LC medium used as an anisotropic electrolyte could control their dynamics. The spatially varying director field in LCEP is created by the particle itself, through the surface alignment of the director at the particle’s surface. Next, we discuss the LC-enabled electro-osmosis (LCEO), in which the space charges are separated by a spatially varying director created by a pattern of surface alignment at the bounding plates of the cell. The LCEO flows can be triggered without involvement of any colloidal particles. These flows can carry particles that themselves are not electrophoretically active, a property of importance for applications such as micromixing (Peng et al. 2015), separation and sorting (Peng et al. 2018). For example, a tangentially anchored sphere with a quadrupolar director field is not electrophoretically active but can be moved by the LCEO flows in a cell with an appropriately patterned director. The separation of LCEP and LCEO is purely semantic, as in both cases the underlying mechanism is the same: the separation of charges by the electric field acting on a spatially varying director field. The only difference is what creates these distortions: the particle itself (the LCEP case; Lavrentovich et al. 2010; Lazo & Lavrentovich 2013; Lazo et al. 2014) or other agents, such as surface patterning of the director field or application of external electromagnetic fields (LCEO; Peng et al. 2015).

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2.2.1. Liquid-crystal enabled electrophoresis A uniaxial nematic LC exhibits a long-range orientational order (de Gennes & Prost 1993), with the average direction of orientation specified by a non-polar unit director nˆ . A sufficiently large colloidal sphere placed in a uniformly aligned nematic creates director distortions around itself that depend on the type of surface anchoring. When the surface anchoring sets the director perpendicularly to the surface, a distorted director field is of a dipolar symmetry with a point defect, called a hyperbolic hedgehog (Poulin et al. 1997; Figure 2.1(a)). The hedgehog’s core forms at some distance from the sphere and in a planar cell that has an equal probability to be on the left (as in Figure 2.1(a)) or on the right side of the sphere. The structure can be described by an elastic dipole vector p directed from the hedgehog toward the sphere. The hyperbolic hedgehog can be transformed into a disclination ring around the sphere if the sample thickness is comparable to the diameter of sphere, or when an electric or magnetic field is applied to align the director (Stark 1999; Gu & Abbott 2000; Figure 2.1(b)). If the sphere sets a tangential anchoring, the distortions around the sphere adopt quadrupolar symmetry with two surface point defects-boojums at the poles (Figure 2.1(c)).

Figure 2.1. Director configurations around a colloidal sphere in a uniform nematic LC. (a) A sphere with normal surface anchoring adopts a dipolar structure with a hyperbolic hedgehog, in this case on the left-hand side; p represents the elastic dipole; (b) a sphere with normal anchoring in a shallow cell adopts a quadrupolar structure with an equatorial disclination ring; and (c) a sphere with tangential anchoring induces quadrupolar deformations in the surrounding uniformly aligned nematic with two point defects at the poles

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A colloidal particle with a dipolar configuration in Figure 2.1(a) becomes LCEP-active in an external electric field (Figure 2.2(a)); it moves with the velocity that is proportional to E 2 (Figure 2.2(b)). The two other configurations in Figure 2.1(b,c) with quadrupolar symmetry of the director field cannot be moved by LCEP.

Figure 2.2. Nonlinear LCEP of a glass sphere powered by an AC electric field (Lavrentovich et al. 2010). (a) Time sequence of two colloidal spheres moving in opposite directions defined by the orientation of the structural dipole p; R represents the uniform rubbing direction; and (b) quadratic dependence of the LCEP velocity on the applied AC electric field

To understand how the electric field separates charges in the LC medium and what the role of the symmetry in the LCEP transport shown in Figure 2.2 is, we consider the behavior of ions around colloidal particles in the presence of an electric field (Figure 2.3). For simplicity, we limit ourselves with a two-dimensional (2D) version of Figure 2.1 and assume that the bounding plates set a uniform planar orientation of the nematic director along the x-axis, nˆ 0 = nx , ny = (1,0) (Lazo et al.

(

)

2014). The 2D analogs of spheres, the disk-like particles, are glued to the substrate. Consider first an electrically non-conductive disk with a perpendicular anchoring at the surface with a quadrupolar symmetry of the director distortions around it (Figure 2.3(a)); this is a 2D analog of a sphere in Figure 2.1(b). An electric field E applied along the x -axis forces positive charges to move to the right and negative charges to the left. However, the ions also tend to shift along the y -axis. The reason is the anisotropy of electric conductivity σ = σ || / σ ⊥ − 1 that is typically positive (anisotropy of dielectric permittivity plays a similar role but we limit ourselves with an intuitively simpler case of conductivity mechanism (Lazo et al. 2014; Paladugu et al. 2017; Tovkach et al. 2017)). Suppose for a moment that the anisotropy is infinitely strong, σ → ∞ . Then the ions can only move along the director lines. The director lines are curved because of the presence of the disk in Figure 2.3(a) which

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sets a perpendicular surface alignment of the director. As a result of the electric field action, the positive ions will move along the director lines and accumulate on the left-hand side of the disk in Figure 2.3(a), since they cannot penetrate the insulating particle. Similarly, the negative ions will accumulate on the right-hand side. The field-induced charge density should be proportional to the applied field, ρ ∝ E . Once the ion clouds form in the distorted region, they experience a Coulomb force of a density f ∝ ρ E , which drags the ions around the disk. These moving ions produce electro-osmotic flows around the sphere, shown by thin curved arrows in Figure 2.3(a). When the field polarity is reversed (E is antiparallel to the x -axis), the charges of the ion clouds are also reversed, but the driving force f ∝ ρ E ∝ E 2 preserves its sign, which means that the flows polarity is not sensitive to the field polarity. It is thus expected that the velocity of the electro-osmotic flow around the disk grows as E 2 , thus allowing one to create steady flows with an AC driving. The occurrence of electro-osmotic flows around particles with perpendicular anchoring and quadrupolar director distortions with velocities that scale as E 2 has been confirmed experimentally (Lazo et al. 2014; Figure 2.3(b)); the flow pattern is obtained by tracking trajectories of tiny tracers added to the nematic. Because of the quadrupolar symmetry of the director distortions and electro-osmotic flows around the disk, the nematic fluid only circulates around the particle but there is no net flow along the x - or y-axis. Such a particle shows no LCEP mobility in a uniform electric field. Consider now a sphere with a tangential anchoring that also produces quadrupolar director distortions (Figure 2.3(c,d)). The electric field E that acts along the x -axis would build a negatively charged cloud on the left side and a positively charged cloud on the right side of the sphere (Figure 2.3(c)). The polarity of space charges is thus opposite to the case of a sphere with perpendicular anchoring in Figure 2.3(a). In other words, the polarity of field-induced charges depends on the sign of the director gradients ∂ϕ ∂y , where ϕ is the angle between the local director nˆ and the overall director that is set to be along the x -axis by the substrates, nˆ = (1,0) . Hence, the spheres with perpendicular and tangential anchoring produce opposite polarities of electro-osmotic flows. The perpendicularly anchored sphere is of a “puller” type (Figure 2.3(b)), and the tangentially anchored sphere is of a “pusher” type (Figure 2.3(d)). Mirror symmetry of the flow patterns in Figure 2.3(a–d) with respect to the x - and y-axes shows that neither of these spheres could produce a net flow and neither can move if set free. Consider finally a particle with a dipolar symmetry of director deformations rather than the quadrupolar one (Figure 2.3(e,f)). The dipolar director pattern translates into the dipolar symmetry of electric field-induced space charges and a broken fore-aft

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symmetry of the electro-osmotic flows (Figure 2.3(e,f)). The particle in Figure 2.3(e,f) pumps the nematic fluid from the right side to the left if it is glued to the substrate of the cell. If such a particle is free, it would move electrophoretically, from left to right, with a velocity ∝ E 2 , which explains the mechanism of LCEP that can be driven either by a DC or an AC field (Lavrentovich et al. 2010).

Figure 2.3. Mechanism of charge separation and electro-osmotic flows of nematic around particles with different surface anchoring (Lazo et al. 2014). (a) Charge separation and induced flow around a disk/sphere with perpendicular anchoring; the flow pattern is of a “puller” type; (b) experimental map of electro-osmotic flows around a sphere with perpendicular anchoring and a Saturn ring configuration in a thin nematic layer; (c) charged clouds around a disk/sphere with tangential anchoring; the flow pattern of quadrupolar symmetry is of a “pusher” type; (d) experimentally mapped flow patterns around a tangentially anchored sphere; (e) separation of charges and electro-osmotic flows created by an electric field around a disk/sphere with perpendicular anchoring and director distortions of dipolar symmetry; and (f) experimental map of the flow pattern around a sphere with dipolar director configuration and a hedgehog on the right-hand side

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As follows from the discussion above, it is the anisotropic nature of the nematic rather than the particular bulk properties of the colloidal particles that enables an LCEP transport of particles in it; this feature makes the LCEP mechanism very different from the ICEP in an isotropic electrolyte. The first work on LCEP (Lavrentovich et al. 2010) demonstrated that both dielectric (such as glass) and metallic (such as gold) spheres move in an AC field, as long as the accompanying director distortions are of a dipolar symmetry, as in Figures 2.1(a), 2.2 and 2.3(e,f). Moreover, the particle does not even need to be solid. Water droplets, dispersed in a thermotropic hydrophobic nematic and stabilized by surfactant sodium dodecyl sulphate (SDS), which induces a perpendicular surface alignment and dipolar structure, have also been demonstrated to be mobile thanks to the LCEP mechanism by Hernàndez-Navarro et al. (2013, 2015(a)). The LCEP transport is parallel to the overall director nˆ 0 set by the surface anchoring at the plates of the cell (Figure 2.2(a), Lavrentovich et al. 2010; Lazo & Lavrentovich 2013). Although the LCEP velocity is proportional to the square of the electric field v ∝ E 2 (Figure 2.2(b)), the direction of motion is polar, being defined by the polarity of the vector p. In Figure 2.2(a), spheres with opposite directions of p move in opposite directions. Moreover, since the director nˆ 0 at the bounding plates can be pre-designed into a spatially varying pattern, the nematic medium allows us to control not only the polarity of the motion but also the geometry of trajectory, since it follows the local nˆ 0 (Lavrentovich et al. 2010; Lazo & Lavrentovich 2013; Peng et al. 2015). The motion of particles along the local pre-designed director nˆ 0 is another advantage of LCEP over ICEP, since in ICEP, a spherical particle moves in any direction perpendicular to the electric field (Gangwal et al. 2008). It is important to realize that since the velocity and the field in LCEP are in a nonlinear quadratic relationship, the direction of propulsion of a particle can not only be collinear with the electric field (as in the linear electrophoresis) but also confined to the plane that is perpendicular to the field (Lazo & Lavrentovich 2013). Theoretically, the dependence of LCEP velocity on the applied electric field is predicted to be (Lazo et al. 2014; Paladugu et al. 2017; Tovkach et al. 2017; Conklin et al. 2018a) v = αεε 0η −1 ( ε − σ ) aE 2 , where ε 0 is the permittivity in vacuum, η is the effective nematic viscosity, ε = ε || ε ⊥ − 1 is the anisotropic permittivity, σ = σ || σ ⊥ − 1 is the anisotropic conductivity, || and ⊥ refer to the orientation parallel and perpendicular to nˆ 0 , respectively, and α is a numerical parameter at the order of 1. The signs of ε and σ define whether the LCEP velocity is parallel or antiparallel to p; the latter case is shown in Figure 2.2(a). By

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using a mixture of two nematics, one with ε >0 and another with ε 0 and pure HNG shows ε < 0 ; and (b) temperature dependence of the LCEP velocity, anisotropic permittivity ε and conductivity σ in the mixture of 5CB and HNG. The data are taken from Paladugu et al. (2017)

The LCEP mechanism can be combined with other effects to impact dynamics of colloidal particles. Combining LCEP driven by an AC field and a linear electrophoresis driven by a DC field allows one to gain 3D spatial control of particle trajectories (Lazo & Lavrentovich 2013). Sasaki et al. (2015) demonstrated that LCEP combined with electrohydrodynamic convection forms chains of colloidal spheres that could transport an embedded micro-cargo through the LC medium. The asymmetry of the director field needed for LCEP propulsion can be produced by altering the shape of the colloidal particles, making them pear- or boomerang-like. Remarkably, the Sagués’ group (Hernàndez-Navarro et al. 2014, 2015a; Straube et al. 2018b) demonstrated a swarming behavior of anisometric particles driven by LCEP. Reconfigurable 2D lattices of clusters can also be designed and addressed dynamically (Hernàndez-Navarro et al. 2014). Finally, very recently, Sahu et al. (2020) discovered that metal-dielectric Janus particles placed in a nematic and setting perpendicular anchoring with a quadrupolar director field could be transported along different directions by changing the amplitude and frequency of the driving AC field.

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2.2.2. Liquid crystal-enabled electro-osmosis In the LCEP discussed so far, colloidal transport is enabled by specific director distortions introduced in an LC environment by the colloidal particles themselves. The approach is limited, as the colloid needs to be sufficiently large to distort the director and even then it can produce only a finite set of director patterns, some of which are of a quadrupolar symmetry that is not suitable for transport nor pumping. For example, as already discussed, a sphere with tangential alignment of the director (Figures 2.1(c), 2.3(c,d)) does not show any LCEP activity (Lavrentovich et al. 2010; Lazo et al. 2014). However, the director distortions do not have to be induced by the colloidal particles. The director distortions needed for charge separation and LCEK flows can be achieved in many other ways, from applying electric field or magnetic field, to patterning the director at the bounding surfaces of LC cells. The latter approach has been extensively studied recently (Culbreath et al. 2011; Murray et al. 2014; Ware et al. 2015; Guo et al. 2016). The most convenient and versatile approach to pattern the surface director is by photoalignment (Yaroshchuk & Reznikov 2012; Guo et al. 2014; Peng et al. 2016b; Peng et al. 2017a) based on plasmonic metamasks with nanoslits (Guo et al. 2016). When such a mask is illuminated with non-polarized light, the slits transmit a polarized optical field that is projected onto a photo-aligning layer which is usually a layer of azodye molecules capable of trans-cis isomerization. The direction of polarization of the transmitted beam is perpendicular to the long axis of the nanoslit. The locally polarized beam aligns the long axes of azodye molecules perpendicularly to the direction of polarization. The array of nanoslits can be prepared in any pre-designed geometry, which is transmitted into the pattern of alignment of azodye molecules. The azodye layer then transmits the orientation pattern onto the adjacent layer of the LC. In the presence of the electric field, the surface-imprinted pattern of molecular orientation enables spatial separation of charges and LCEO flows (Peng et al. 2015). The gradients and symmetry of the director field control the flow velocities and effects such as pumping and the transport of particles (Peng et al. 2015). Surface patterning of molecular orientation lifts the limitations on the type of particles that can be transported by LCEK, as these particles are not required to create the director distortions to support their transport. Figure 2.5(a–c) represents one-dimensionally periodic patterns of the director

nˆ = ( nx , n y ,0) in the photoaligned nematic cell (Peng et al. 2015). In the absence of

the electric field, the ions are distributed homogeneously throughout the sample. When the field E is applied along the x -axis, the material’s conductivity (or permittivity) anisotropy helps to separate the positively and negatively charged ions along the y -axis, as the charges move along the “guiding rail” of the director.

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Figure 2.5 illustrates the case of positive anisotropy of conductivity. In Figure 2.5(a), the charges accumulate in the regions with the “horizontal” director, nx = 1, n y = 0 ; the sign of the accumulated charges is defined by the polarity of splay deformations nˆ div nˆ . The charge density is proportional to E . The periodic bulk force f x = ρ E ∝ E 2 causes spatially periodic LCEO flows, that are directed either to the right or left, depending on the polarity of splay (Figure 2.5(d)). Other patterns of the director (Figure 2.5(b,c)) produce similar electro-osmotic flows (Figure 2.5(e,f)).

Figure 2.5. LCEO flows in the pre-designed photo-patterned surface alignment (Peng et al. 2015). (a)–(c) PolScope textures of a nematic cell with surface-imposed director patterns; “+” and “−” show the charges separated by the electric field; (d)–(f) corresponding LCEO flow velocity maps driven by an AC field directed along the horizontal x -axis; and (g) a tangentially anchored polystyrene sphere is transported by the induced LCEO flows driven by an AC electric field

If there is any inclusion in such a nematic cell with a patterned director, it will be transported to the right- or the left-hand side. For example, tangentially anchored

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polystyrene spheres, which do not show any electrophoretic activity under the AC electric field, can now be moved by LCEO effect (Figure 2.5(g)). The LCEO in patterned director fields has also been used to drive droplets of other fluids such as water and even gas bubbles (Peng et al. 2015). The LCEK directed by surface patterning does not impose any limitations on the properties of the “cargo” (such as separation of charges, polarizability or ability to distort the LC).

Figure 2.6. LCEK applications in microreactions. (a) PolScope image of the director field of a nematic LC with a (−½, ½) disclination pattern; the −½ core is marked by a triangle and the ½ core is marked by a semicircle; (b) corresponding LECK flow driven by an AC field; and (c) and (d) LCEK flows carry a water droplet (marked by a small arrow) in this pattern; the trajectory is shown by a curved arrow; the droplet is transported toward the core of the ½ disclination on the right-hand side and coalesces with another water droplet that is already trapped there

LCEK can be used in potential applications such as chemical reactions at microscale (Hernàndez-Navarro et al. 2013, 2015a), micromixing (Peng et al. 2015), sensing and sorting (Peng et al. 2016a, 2018). For example, the Sagués’ group demonstrated a microscale reaction (Hernàndez-Navarro et al. 2013, 2015a) by loading two water droplets with different reactants and driving them toward each other by LCEP. When the droplets coalesce, a chemical microreaction is triggered. Similar microreactions can be staged through photo-patterned LCEO flows. In the pattern with a defect pair (½, −½) (Figure 2.6(a)), LCEO flow is directed toward the

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core of the 1/2 defect (Figure 2.6(b)). Water droplets dispersed in such a patterned nematic would be driven toward the same defect core and coalesce there (Figure 2.6(c–d)). Thus, the 1/2 defect core serves as the pre-designed microreaction location. The droplets do not require any special conditions such as perpendicular anchoring to enable the propulsion by LCEO flows. LCs are known for their sensitivity to the physical and chemical features of the bounding substrates (Sonin 1995). Exploiting anisotropic interactions between an LC and an adjacent medium, various chemical and biological sensors have been demonstrated (Brake et al. 2003; Lin et al. 2011; Lowe & Abbott 2011; Carlton et al. 2013; Popov et al. 2016). By linking the sensitivity of LCs to surface properties with the LCEK flows, we can combine the functions of sensing and sorting in one microfluidic chamber (Peng et al. 2018). The pre-designed director patterns serve a dual purpose. First, a spatially varying director field sets preferred locations of the particles through elastic interactions between the director gradients of the underlying pattern and the director gradients induced by the anchoring at the particle’s surface (Peng et al. 2016a, 2017b). For example, particles with a dipolar symmetry of the director prefer the regions of splay, while particles with tangential anchoring and quadrupolar director move into the bend regions (Peng et al. 2017b). Second, the patterned director distortions cause charge separation and trigger LCEK flows when the electric field is applied. The induced flows move the particles with different surface properties toward the opposite ends of the sorting device. As an example, Figure 2.7(a,b) illustrates the separation of pure glycerol droplets with tangential anchoring (Volovik & Lavrentovich 1983) and glycerol droplets doped with a surfactant sodium dodecyl sulfate (SDS) that sets perpendicular anchoring. The droplets with SDS migrate to the splay region while pure glycerol droplets prefer to locate in the bend region. When an AC electric field is applied, the two types of droplets are separated since the LCEO flows in the splay and bend regions of the underlying director pattern are antiparallel and move the droplets to opposite ends of the sorting chamber. Besides SDS, other molecules, such as phospholipids, can be sorted by LCEO, as long as they induce a different surface alignment of an LC. For instance, the presence of a phospholipid 1,2-Dilauroyl-sn-glycero-3phosphorylcholine (DLPC) found in biological membranes can be sensed by the same LCEO effect, when DLPC is added to glycerol droplets (Peng et al. 2018, Figure 2.7(c,d)). Theoretical modeling of LCEK has progressed dramatically in the past few years (Lazo et al. 2014; Peng et al. 2015; Calderer et al. 2016; Tovkach et al. 2016; Conklin & Viñals 2017; Tovkach et al. 2017; Conklin et al. 2018a, 2018b). However, there are still some interesting problems awaiting analysis. For example,

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prior experiments and theory treat the director as being unaffected by the applied electric field. Obviously, this assumption is violated in real systems, as the electric field causes director distortions through dielectric, conductivity, flexoelectric and surface polarization mechanisms (Kleman & Lavrentovich 2003). The effect of ions, namely, their concentration, mobility and valency, is also of interest to explore; the ions can be produced either by adding salts to the LC or through the charge injection when the electric field is applied. Finally, most of the studies to date deal with a nematic LC as an anisotropic electrolyte. Other liquid crystalline media, such as cholesteric and smectic LCs, recently discovered ferroelectric nematics (Nishikawa et al. 2017; Mertelj et al. 2018; Chen et al. 2020), would be of interest to explore as anisotropic electrolyte since the director deformations in them and their response to the electric field are very different from the case of uniaxial apolar nematics.

Figure 2.7. Sorting of droplets with different surface properties by LCEK flows in patterned nematic cells. (a)–(c) Sorting of pure glycerol droplets and droplets of glycerol with an added SDS by the applied uniform AC electric field. The glycerol-SDS droplets with perpendicular anchoring prefer to stay in the splay regions since the director field there matches better the droplet-hedgehog geometry; the pure glycerol droplets with tangential anchoring reside in the regions of bend. LCEO flows move the nematic and the dispersed glycerol-SDS droplets in the splay regions to the left, while the pure glycerol droplets in the bend regions are moved to the right; (d)–(f) sorting of droplets of pure glycerol and glycerol with an added phospholipid DLPC by the applied AC electric field

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Recent exploration (Li et al. 2020) uncovered another distinct mechanism of LCEP, associated with the formation of solitary particles in a nematic under the action of an electric field (Li et al. 2018; Li et al. 2019). The soliton-induced LCEP is a two-step process. First, a colloidal particle that is usually LCEP inactive, such as a tangentially anchored sphere, develops a stable soliton-like “coat” of director deformations around itself that is asymmetric. Once the asymmetric soliton of the director field is formed, it becomes mobile because of the LCEP effect (Li et al. 2020). There is no doubt that the combination of LCEK with other effects, such as soliton formation (Li et al. 2020), linear electrophoresis (Lazo & Lavrentovich 2013), ICEP (Sahu et al. 2020), charge injection or peculiarities of the orientational order in media other than the uniaxial apolar nematics, would be actively studied in the future. 2.3. Controlled dynamics of microswimmers in nematic liquid crystals Dynamics of microorganisms such as flagellated bacteria in fluids fascinated scientists for centuries and continue to inspire cutting-edge research and innovation. The fluid in which the dynamics of microswimmers is explored is typically isotropic, such as water or a dilute polymer solution. Bacillus subtilis represents a typical example of a swimming microorganism. Bacillus subtilis has a rod-like body of a typical length of 5–10 μm. It propels in viscous fluids by rotating helicoidal appendages called flagella, which are composed of bundles of thin helical filaments (Purcell 1977; Lauga 2016). Swimming bacteria manifest a remarkable ability to sense and navigate their environment in search of nutrients. Flagella can steer a bacterium in a new direction by momentarily untangling the filaments and causing the bacterium to tumble (Lauga 2016). In a fluid with a homogeneous distribution of nutrients, alternating runs and tumbles of bacteria form a random trajectory reminiscent of an “enhanced” Brownian walk. The flows of the surrounding fluid created by the bacteria cause their interactions and collective dynamics (Koch & Subramanian 2011). In very dense dispersions, swimming bacteria could align parallel to each other, but this local ordering is not preserved on a global scale because of the activity triggered realignments and nucleation of topological defects (Dombrowski et al. 2004; Sokolov et al. 2007). Similar out-of-equilibrium patterns are met in many other systems, universally called “active matter” and defined as collections of interacting self-propelled particles (Ramaswamy 2010; Marchetti et al. 2013), each converting internally stored or ambient energy into a systematic movement (Vicsek et al. 1995; Toner et al. 2005; Sanchez et al. 2011, 2012; Nishiguchi et al. 2016). In order to extract a useful work from the chaotic dynamics of bacteria (or any other active matter) (Di Leonardo et al. 2010; Sokolov et al.

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2010), we need to learn how to control the spatial distribution of particles, geometry and the polarity of their trajectories. In real systems, microorganisms often move in crowded environments with some elements of order. This fact as well as the hope of finding the ways to control and guide the dynamics of microorganisms and to extract a useful work from this dynamics explain the growing interest in media with order, such as LCs (Figure 2.8). Early studies revealed that a uniformly aligned nematic guides the bacteria to swim along the overall director (Smalyukh et al. 2008; Kumar et al. 2013; Mushenheim & Abbott 2014; Zhou et al. 2014), which is not surprising, as the rod-like bacteria align parallel to the director to minimize the energy of surface anchoring and elastic distortions. In this section, we discuss how spatially varying director fields could command the dynamics of bacterial microswimmers such as B. subtilis. The medium represents a nematic phase of the so-called lyotropic chromonic liquid crystal (LCLC) that is not toxic to bacterial species (Woolverton et al. 2005; Figure 2.8(c)). The LCLCs are formed by elongated aggregates of plank-like polyaromatic molecules dispersed in water (Tam-Chang & Huang 2008; Lydon 2011; Park & Lavrentovich 2012; Collings et al. 2015); their non-toxic nature is explained by the absence of aliphatic chains (Woolverton et al. 2005). The most studied LCLC is disodium chromoglycate (DSCG) which forms a nematic phase when dispersed in water at concentrations roughly between 10 wt% and 17 wt%, depending on the temperature and ionic content (Tortora et al. 2010). This mechanism of B. subtilis propulsion based on the rotating flagella remains intact when the bacterium is placed in the nematic LCLC (Figure 2.8). Moreover, despite the significantly higher viscosity of the nematic LCLC medium as compared to pure water or to the isotropic phase of LCLC, reduction in the speed of bacteria is not significant; for example, Sokolov et al. (2015) reported a speed of 21 μm/s in pure water and 14 μm/s in the nematic LCLC. In dilute dispersions of bacteria, the LC director, either uniform or spatially distorted, serves as an “easy swimming” pathway for bacteria, as demonstrated not only for B. Subtilis (Zhou et al. 2014; Genkin et al. 2017; Zhou et al. 2017; Genkin et al. 2018; Dhakal et al. 2020; Koizumi et al. 2020; Turiv et al. 2020a), but also for other flagellated bacteria in LCLCs (Kumar et al. 2013; Mushenheim et al. 2014a, 2014b, 2015). This guiding effect of LCLCs on the low-density bacterial dispersions is similar to the effect of suppressing active flows in a system of active microtubules by interfacing them with a passive LC (Guillamat et al. 2016).

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Figure 2.8. Living liquid crystal comprised of swimming bacteria (extensile microswimmers) in a water-based lyotropic chromonic liquid crystal (Zhou et al. 2014). (a) Optical microscopy texture of a swimming bacterium in an LCLC, director waves created by the rotating flagella are seen as periodic modulations of the transmitted light intensity; (b) schematic of the extensile B. subtilis force dipole (dark red arrows), rotation sense of bacterial body and flagellum (orange arrows), and flows induced in the medium (blue arrows); and (c) schematic of a dilute bacterial dispersion ˆ in a uniform nematic LCLC; bacteria tend to swim along the local director n

In the concentrated solutions, called “living liquid crystals” or “living nematics”, the swimming regimes become very different from a simple following of the director, as the bacterial interactions with each other produce an active force that controls their collective dynamics, in particular, forcing the bacteria to form unidirectionally circulating swarms (Peng et al. 2016b; Koizumi et al. 2020) or to distribute non-uniformly in space (Peng et al. 2016b; Genkin et al. 2017; Genkin et al. 2018; Koizumi et al. 2020; Turiv et al. 2020a). The living LC is a particular experimental realization of an active matter (Zhou et al. 2014, 2017; Genkin et al. 2017, 2018; Koizumi et al. 2020; Turiv et al. 2020a) that behaves very differently from their dilute counterparts in which isolated bacteria do not interact with each other. The living LCs exhibit an important feature: their activity, which is defined by the swimming speed and concentration of bacteria, and their orientational order, which depends primarily on the LCLC surrounding, could be controlled independently of each other. The speed of bacteria could be controlled by the concentration of oxygen and nitrogen (Zhou et al. 2014; Sokolov et al. 2019), while

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the orientational order could be controlled by the pre-designed director field (Mushenheim et al. 2015; Peng et al. 2016b) or simply by changing the state of the LCLC to the isotropic or other phase. This tunability is important, as the coupling of activity and the orientational order is at the core of the current theoretical modeling of active matter (Aditi Simha & Ramaswamy 2002; Voituriez et al. 2005; Marchetti et al. 2013; Giomi et al. 2014; Thampi et al. 2014; Green et al. 2017; Doostmohammadi et al. 2018; Sokolov et al. 2019). As already indicated, active systems are intrinsically unstable with respect to orientational perturbations that often produce topological defects (Ramaswamy 2010; Sanchez et al. 2012; Marchetti et al. 2013). It is thus of particular interest to explore the interplay of spatially varying orientation and activity in living LCs. The orientation of the LCLC can be pre-designed to vary in space by plasmonic metamask photoalignment (Peng et al. 2017a), similarly to their thermotropic counterparts (Guo et al. 2016). One change in the fabrication process is a deposition of a protective LC elastomer coating that separates the water-soluble layer of azodyes from the water-based LCLC (Peng et al. 2017a). The orientational patterns of the director affect the individual and collective behavior of microswimmers very strongly, by controlling their distribution in space, rectifying their motion into a unipolar swimming along circular or rectilinear trajectories, and stabilizing the collective modes of propulsion against intrinsic activity-triggered instabilities such as undulatory trajectories and nucleation of topological defects that produce a spatio-temporal chaos (Zhou et al. 2014; Lavrentovich 2016). A simple example is shown in Figure 2.9(a), in which the LCLC director field is designed as defects of a topological charge +1 with different types of distortions, ranging from pure circular (Figure 2.9(a)) to spiral (Figure 2.9(d)) and to pure radial (not shown). The director field of all these structures writes nˆ 0 = ( cos θ , sin θ , 0 ) , y + θ 0 , m =1 and the phase θ 0 sets either a pure bend, x θ0 = π / 2 , pure splay, θ0 = 0 , or mixed splay-bend spiral, 0 < θ0 < π / 2 . In the pure bend or splay patterns, the bacteria swim along the director nˆ in a bipolar fashion (Figure 2.9(a–b)), similarly to the case of a uniformly aligned LCLC. Clockwise and counterclockwise swimming directions are of the same probability (Figure 2.9(c)). However, if the director is designed as a mixed splay-bend, e.g., with θ0 = 45o , which

where θ ( x , y ) = m tan −1

is a clockwise spiral (Figure 2.9(d)), the bacterial dynamics changes dramatically, producing a unidirectional counterclockwise circulation (Figure 2.9(e–f)). Importantly, this unidirectional circulation occurs only when the bacterial concentration exceeds some critical value; below this threshold, the bacteria swim individually parallel to the spiral director (Koizumi et al. 2020).

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The unipolar circulation in Figure 2.9(e,f) is different from the well-known circular swimming that many bacteria, including B. subtilis, experience in diluted water dispersions, while swimming as individuals near solid boundaries: this individual circling is caused by the interactions of the bacterium’s clockwiserotating head with the substrate (Lauga et al. 2006; Copeland & Weibel 2009; Perez Ipiña et al. 2019). The counterclockwise collective swimming in Figure 2.9(e,f) is driven by the bacterial interactions mediated by the underlying spiraling director, which overcome any possible effects caused by the head counterrotation.

Figure 2.9. Controlled dynamics of bacteria by the patterned LCLC. (a) Designed LC director field of circular +1 defect with pure bend deformation; (b) bacteria swim in the pure bend region following the curvilinear trajectory in bipolar directions; (c) probability of bacteria swimming in the clockwise and counterclockwise directions; (d) designed LC spiral pattern with +1 defect at the center; (e) bacteria swarm around the defect in a unipolar direction; (f) measured velocity map; and (g) calculated active driving force

The unipolar circulation in dispersions with a sufficiently high concentration is schematically explained in Figure 2.10 for a spiral vortex with θ 0 = 25° . In this vortex, collective motion is also biased toward the center of the pattern. Since no net force is applied to a self-propelled object, a swimming bacterium represents a moving hydrodynamic force dipole (Figure 2.8(b)) and produces two outward fluid streams coaxial with the bacterial body (Sokolov & Aranson 2012; Lushi et al. 2014; Zhou et al. 2014), as shown for bacteria 1 and 2 in Figure 2.10. The bacteria 1 and 2 are separated by a large distance, thus their streams do not overlap and they do not interact. These bacteria swim individually along nˆ 0 , as the finite surface

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anchoring at the body of the bacterium is the only guiding force (for a detailed discussion of the surface anchoring effects at the bacterial bodies, see Zhou et al. (2017)). Collective circular swimming occurs when the separation decreases to about 15–20 μm , which is the extension of the fluid perturbations around the bacteria (Koizumi et al. 2020), so that the bacteria start to interact hydrodynamically. Hydrodynamic interactions, mediated by the patterned director field, produce a net active force fa that triggers and drives the collective motion, as shown in the top part of Figure 2.10 for bacteria 3, 4, 5 and others nearby. Since each bacterium prefers to be parallel to nˆ 0 , and since nˆ 0 is pre-designed with splay and bend, the force dipoles of neighboring bacteria are titled with respect to each other. The tilted force dipoles cannot compensate each other and produce a net active force fa . In the example of Figure 2.10 with the spiral angle θ 0 = 25° , the active force fa is directed counterclockwise and toward the center of the vortex, as shown by the red arrows. This force triggers the transition from the individual to the collective counterclockwise circulation of the bacterial swarm and its condensation at the center of the pattern (Koizumi et al. 2020).

Figure 2.10. Transition from individual to collective motion caused by the increased bacterial concentration in a cell with a spiral vortex, θ 0 = 25° . In a diluted portion of the sample, bacteria 1 and 2 do not interact with each other and swim individually along the preimposed director. In the concentrated region, bacteria 3, 4, 5 and others nearby interact and produce the net active force fa shown by red arrows, which triggers clockwise circular swirling. Blue arrows depict hydrodynamic force dipoles

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Assuming that the bacteria are homogeneously distributed in the sample, the active force can be written as fa =α[ nˆ 0 ∇. nˆ 0 – nˆ 0 ×(∇× nˆ 0 )]; here α < 0 is the activity coefficient; the negative sign reflects the extensile character of B. subtilis (Peng et al. 2016b; Green et al. 2017). This is the same expression as the Simha-Ramaswamy active force in the case of a constant scalar order parameter (Aditi Simha & Ramaswamy 2002). Note also that the formula coincides with the expression for the flexoelectric polarization in distorted nematics formed by quadrupolar molecules (Prost & Marcerou 1977) and with an activation force shaping the response of LC elastomers with a patterned director field to stimuli such as heating/cooling (Babakhanova et al. 2018). The reason for all these similarities is that the force represents a derivative of a spatially varying quadrupolar entity such as a hydrodynamic dipole or ellipsoid of electric charges of quadrupolar symmetry (see Lavrentovich (2020) for more discussion). For θ0 = 45o (Figure 2.9(d)), the

{

}

active force simplifies to fa = f ar , f aϕ ,0 = {0, −α / r,0} when written in the cylindrical coordinates. The only non-zero component of the driving force is the azimuthal component following the counterclockwise direction (Figure 2.9(g)). Koizumi et al. (2020) extended the study to the vortices of an arbitrary spiral angle. The active force generalizes to fa = f ar , f aϕ ,0 = α {cos2θo , − sin 2θo ,0} / r

{

}

with a positive azimuthal component that forces a counterclockwise bacterial circulation for any spiral angle except 0 and 90o (Figures 2.9 and 2.10). When 0 < θ 0 < 45o , the radial component f ar = α cos 2θ o / r < 0 directs the bacteria toward the center of the vortex (Figure 2.10), while in the case of 45o < θ0 < 90o , the bacteria swarm is driven toward the periphery since f ar = α cos 2θ o / r > 0 . In other words, a predominant splay condenses the bacterial swirls toward the center, while the prevalence of bend expands them toward the periphery. In a 2D lattice of spiraling vortices (+1, −1) (Figure 2.11(a)), the bacteria gather into counterclockwise circulating swarms around m = 1 cores with θ0 = 45o , but avoid the m = −1 regions (Figure 2.11(b–c)). When the pattern contains a defect pair (+1/2, −1/2) (Figure 2.11(d)), the active force drives the bacteria from the −1/2 defect core toward the +1/2 core (Figure 2.11(e–f); Peng et al. 2016b). The neighborhood of the +1/2 defect is enriched with bacteria, while the −1/2 defect is deprived of them. Similar interplay of deformations and concentration of active units is observed in other living systems, namely, epithelial tissues (Saw et al. 2017; Turiv et al. 2020b) and arrays of neural progenitor cells (Kawaguchi et al. 2017).

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Figure 2.11. Bacteria swarming in the patterned 2D lattice of spiraling vortices and in a pattern with a disclination pair. (a) Designed director field of 2D lattice of spiraling vortices; (b)–(c) bacteria circulate counterclockwise around the +1 cores; (d) designed director field of the defect pair (+1/2, −1/2); and (e)–(f) bacteria move predominantly from the −1/2 core toward the +1/2 core

As discussed above, the director splay and bend play a different role in the collective dynamics of living LCs. This difference reveals itself profoundly when the director is patterned in a periodically alternating stripes of splay and bend (Figure 2.12(a)), resembling rows of letters “C”, nˆ 0 = nx , ny , nz =

(

)

{cos (π y / l ) , − sin (π y / l ) , 0} , where l is the period. Pure splay regions are located at y = 0, ± l , ±2l ,... and bend ones at y = ± l / 2, ±3l / 2,... , respectively. When the bacteria are suspended in an LCLC with such a pattern, they are condensed into polar jets that propagate unidirectionally along the splay regions toward the positive end of the x-axis, as shown by red arrows in Figure 2.12(b). If any bacterium happens to swim in the opposite direction, it enters the bend regions and is guided by the director along a U-turn trajectory and joins the polar jet in the neighboring splay region, as shown by the blue curved arrows in Figure 2.12(b). The polar transport is thus restricted to the splay regions, while the bend regions repel the bacteria and reverse the trajectories of accidental left-swimming bacteria. The bend regions help to quench the undulatory instability of the jets. The generated polar

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active flows can be used for directional cargo transport, as shown in Figure 2.12(d– g) for a single colloidal sphere and for a chain of colloids.

Figure 2.12. Bacteria polar jets in the alternating splay-bend strips and transport of particles by the active flow. (a) Designed director field of periodic alternating splay-bend strips in the form of the letter “C”; (b)–(c) measured bacteria swimming trajectories in the C-pattern; (d)–(e) transport of an individual particle driven by the bacterial jets; and (f)–(g) transport of a particle assembly by the active flow

2.4. Conclusion In this chapter, we first discussed the broad theme of electrically controlled dynamics of fluids and particles at microscales, covering the subjects of nonlinear LCEP (Lavrentovich et al. 2010; Lazo & Lavrentovich 2013) and LCEO (Peng et al. 2015) driven by a uniform electric field. In isotropic electrolytes, electrophoresis

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and electro-osmosis can only be triggered when the solid part of the system exhibits some peculiar properties, such as spontaneous formation of electric double layers or strong polarizability that leads to field-induced charges in the system. When the isotropic electrolyte is replaced with an anisotropic medium, namely, a nematic LC, the mechanisms become very different. The separation of charges in space in the presence of the electric field occurs because of the properties of the LC electrolyte, such as anisotropy of conductivity and permittivity and a spatially varying director field. As a result, LCEP and LCEO impose little limitations on the properties of the particles and interfaces. In LCEP, the particle does not need to be charged or polarizable and can be solid, fluid or even gaseous. The only requirement is that the particle creates a dipolar (or lower symmetry) director field around itself, which, in the case of spheres, can be achieved by perpendicular surface anchoring of the director. In LCEO, even these requirements are lifted, as the distortions needed for charge separation are created by surface treatments of the cell plates rather than by the particles themselves. When the electric field is applied to a pre-patterned LC, it generates clouds of charges at the director deformations, triggering the LCEO flows (Peng et al. 2015). Any object that is dispersed in the LC medium will follow the flow. LCEO lifts all the limitations on the nature of the “cargo”. The particle is transported along a controlled trajectory determined by the pre-designed molecular orientation. Importantly, since the velocities grow as the square of the electric field, both LCEP and LCEO can be driven by an AC electric field (Lavrentovich et al. 2010; Lazo & Lavrentovich 2013; Peng et al. 2015). The AC driving supports steady transport and avoids detrimental effects such as screening of the field and electrochemistry at electrodes. No mechanical parts and no external pressure gradients are needed. The flow polarity can be reconfigured either by changing the director patterns or the electric field direction (Peng et al. 2015). Since the charges are separated in the bulk of electrolytic LC medium rather than at the solid-liquid interfaces, the proposed approach eliminates the need for polarizable/charged interfaces. Both LCEP and LCEO might find applications in lab-on-the-chip and microfluidic devices of a new type. In the second part, we discussed how to control the dynamics of living microswimmers by using an anisotropic medium. The electrophoretically moved particles and self-propelling microorganisms appear to be very different at first sight but they show important similarities. In both cases, the propulsion is force-free. In the case of electrokinetics, the field acts on an electrically neutral system by separating charges in space and creating flow patterns near electrolyte-inclusion interface but while imposing no net force. The particle and the fluid move in opposite directions, and this produces either a pumping of a fluid when the particle

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is immobilized or a propulsion when the particle is freely suspended. In the case of flagellated bacteria, the rotation of flagellum pushes the fluid in one direction in order to propel the body in the opposite direction. As in the case of LCEP, anisotropy of the environment strongly influences the dynamics of microswimmers. Dilute dispersions with non-interacting bacteria show swimming parallel to the director field. Concentrated dispersions, in which the bacteria interact with each other hydrodynamically, show spectacular collective effects. In a uniformly aligned LCLC, the collective effects manifest themselves similarly to other active matter systems. As the activity is increased, the swimming trajectories change from rectilinear to undulating and then to a spatial-temporal chaos with nucleating topological defects (Zhou et al. 2014). The most interesting effects are observed when the underlying director is pre-designed to be spatially non-uniform by surface photoalignment. The patterned director rectifies the bipolar swimming of the bacteria along the non-polar director nˆ 0 = – nˆ 0 , into polar work-producing streams, that are either rectilinear (Turiv et al. 2020a) or in the form of unidirectionally circulating swarms (Peng et al. 2016b; Koizumi et al. 2020). In patterns with topological defects, the bacteria adapt their spatial distribution, heading toward defects of positive charge and avoiding negative charges. These pre-imposed LCLC patterns command the self-propelled bacteria in a number of ways, by controlling (1) geometry of trajectories; (2) polarity of locomotion; and (3) spatial distribution of bacterial concentration. Splay and bend play different roles in the collective dynamics. In the case of 1D periodical stripes of splay and bend, bacteria condense into polar jets in splay regions and are expelled from the bend regions. The polar jets show a thresholdless (meaning it occurs at any concentration of active units) unidirectional flow capable of carrying a microscopic cargo. In the pre-designed vortex patterns that cause a unidirectional circulation of bacterial swarms, prevalence of splay attracts the swarms to the center, while prevalence of bend repels them to the periphery. The demonstrated command of living LCs by a spatially varying anisotropic environment points toward a possible design of micromachines and microsystems with a well-controlled transport of active units. The acquired knowledge on the interplay of an orientational order and activity can be extended to other biological systems. Patterned LC elastomers can be used to produce human dermal tissues with pre-designed topological defects (Babakhanova et al. 2020; Turiv et al. 2020b). Templated LC polymer networks have been used as substrates to control the dynamics and organization of neural tumor cells (Rodriguez Sala et al. 2019; Jiang et al. 2020). It is expected that further exploration of the effect of orientational order on the active system will advance our ability to control the spatio-temporal behavior of microswimmers and cells, harness the energy of

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3

Thermomechanical Effects in Liquid Crystals Patrick O SWALD1 , Alain D EQUIDT2 and Guilhem P OY1 1

2

Laboratoire de Physique, ENS de Lyon, France CNRS, SIGMA Clermont, Clermont Auvergne University, Clermont-Ferrand, France

3.1. Introduction Any unconstrained thermodynamical system tends to spontaneously evolve in such a way as to homogenize its intensive variables. Let us take a concrete example. In a conducting fluid containing a solute, for instance, these variables are the temperature, the pressure, the velocity, the chemical potential of the solute and the electric potential. The thermodynamic equilibrium state of this system is reached when all these variables are constant and the free energy is minimal. However, these variables vary in space and time during the equilibration phase. Thus, a non-homogeneous distribution of temperature will result in the appearance of a heat flux. Similarly, a gradient of chemical potential will produce a flux of solute, while a gradient of electric potential and/or velocity will generate an electric current and/or a flux of momentum. To each of these irreversible phenomena is associated a transport coefficient, which enters into a transport law of the type Jk = γk Xk

Liquid Crystals, coordinated by Pawel P IERANSKI, Maria Helena G ODINHO. © ISTE Ltd 2021. Liquid Crystals: New Perspectives, First Edition. Pawel Pieranski and Maria Helena Godinho. © ISTE Ltd 2021. Published by ISTE Ltd and John Wiley & Sons, Inc.

[3.1]

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where Jk is the flux of the property k, γk is the associated transport coefficient and Xk is the gradient force that acts on this property. In the previous example, these laws were known for a long time and correspond to the Fourier law (1822) for the heat flux, to the Fick law (1855) for the solute diffusion, to the Ohm law (1827) for the electric current and to the Newton law of viscosity (1687) for the velocity when the fluid is Newtonian. The associated transport coefficients are the thermal conductivity, the chemical diffusion coefficient, the electric conductivity and the dynamical viscosity. Historically, these phenomenological laws were obtained empirically. However, it is possible to recover them by using the thermodynamics of irreversible phenomena (Prigogine 1967). This theory applies when the local equilibrium hypothesis is valid and supposes that there exist linear relations between the fluxes and the forces responsible for the transport phenomena. These general linear relations are important because they directly show that the transport phenomena can interfere with each other. This leads to crossed phenomena that are often smaller – and thus more difficult to measure – than the direct effects described above. For example, a temperature gradient may cause a concentration gradient (Soret effect or thermodiffusion) while a flux of heat can be generated by a difference in electric potential (a thermoelectric effect known as the Peltier effect). Inverse effects also exist, known as the Dufour effect in the chemical case and the Seebeck effect in the electric case. Many other effects of this type have been described in the literature such as the fountain effect in superfluid helium II. In that case, a temperature gradient inside a tube containing the superfluid generates a pressure gradient responsible for the expulsion of the superfluid out of the tube. Other examples, including electromagnetic and acoustical processes, are given in the reference book by de Groot and Mazur (1962). In this chapter, we describe two crossed effects observed in nematic and  The first effect cholesteric liquid crystals (LC) subjected to a temperature gradient G. couples the rotation of the molecules to the temperature gradient (thermomechanical [TM] effect), while the second one couples the flows to the temperature gradient (thermohydrodynamical [TH] effect). These effects do not exist in usual fluids, but are possible in LCs because of the existence of a long-range orientational order of the molecules, to which is associated a new hydrodynamical variable, the director n, which is the unit vector giving the mean orientation of the molecules at each point. In practice, the director can experience a torque and rotate on itself in the absence of any flow. This irreversible process is dissipative and described by the phenomenological law Γv = −γ1 ω 

[3.2]

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In this formula, of the same type as equation [3.1], Γv is a viscous torque, ω  is n 1 the rotation rate of the director equal to n × ∂ ∂t in the absence of flow and γ1 is a transport coefficient known as the rotational viscosity. This effect is described by the vertical blue arrow on the right in the general scheme in Figure 3.1. σ (s)

Fluxes

__>

− ζjki

νijkl

__>

− ξji

− Γ ( neq )

γ1

κij D

Forces

− j (q)

ζijk

__>

G

ξij

__>

ω

γ2 Figure 3.1. Direct and crossed effects in nematic and cholesteric liquid crystals subjected to a temperature gradient. The thermomechanical effect is described by the red arrows in solid line and the thermohydrodynamical effect corresponds to the red dashed-line arrows. σ (s) is the symmetric part of the non-equilibrium stress tensor, D  is the temperature gradient, Γ(neq) is the is the strain rate tensor, j q is the heat flux, G non-equilibrium torque and ω  is the director rotation rate

In nematic or cholesteric LCs, the Newton law for viscosity and the Fourier law for heat conduction can also be generalized for taking into account the particular symmetries of the phase. This leads to replace the viscosity and the thermal conductivity by a viscosity tensor νijkl and a thermal (or heat) conductivity tensor κij . In these systems, these two laws write in the form: (s)

σij = νijkl Dkl

[3.3]

and (q)

ji

= −κij Gj

[3.4]

(s)

where σij is the symmetric part of the non-equilibrium stress tensor, Dij = 12 (vi,j + vj,i ) is the strain rate tensor (by denoting the derivative with respect (q) to xj by a subscript j after a comma) and ji is the heat flux. These two equations describe the two direct effects represented by the left and middle vertical blue arrows in Figure 3.1.

 where N  = Dn/Dt − 1 (∇  ×v ) ×n 1 If there is a flow of velocity v , one must take ω  = n × N 2 is the corotational time derivative of the director.

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Besides these direct effects, the thermodynamics of irreversible phenomena teaches us that crossed effects can also exist. One of them is described by the green arrows in Figure 3.1. This crossed effect is well known in nematic and cholesteric LCs and couples the flows with the director rotation. This effect is described by another transport coefficient known as the rotational viscosity γ2 in the literature. This viscosity is very important and responsible in part for the “backflow” effects observed in numerous experiments (de Gennes and Prost 1995; Oswald and Pieranski 2005; Svenšek and Žumer 2001). The latter effect is even used in practice in, for instance, some bistable nematic LCD displays (Dozov et al. 1997). In this chapter, we focus on the two other crossed effects represented in Figure 3.1. The first one is the TM effect represented by the solid line red arrows. It couples the director rotation to the temperature gradient. The second one is the TH effect represented by the dashed-line red arrows. It couples the flows to the temperature gradient. These two effects were first predicted in undeformed cholesteric LCs by Leslie (1968) and later, independently, by Akopyan and Zel’dovich in 1984 and Brand and Pleiner in 1988 in deformed nematic LCs (Akopyan and Zel’dovich 1984; Brand and Pleiner 1988; Pleiner and Brand 1996). The general theory of these effects is given in Poy and Oswald (2018). In this paper, the calculations are extended to deformed cholesteric phases and the equivalence between the Akopyan and Zel’dovich model and the Brand and Pleiner model is demonstrated. We note right now that, for reasons to do with symmetry, these effects can only exist in systems that are not invariant under the reflection in a mirror. This is the case in the undeformed cholesteric phase of D2 symmetry2 in which no mirror symmetry exists because of the chirality of the phase, but not in the undeformed nematic phase of symmetry D∞h . In the nematic phase, TM and TH effects can only appear when the mirror symmetry of the director field is broken at the macroscopic level, i.e. when the phase is distorted. This is the reason why the transport coefficients ξij and ζijk describing these crossed effects in Figure 3.1 are reduced to two pure pseudoscalars (Leslie coefficients noted μ and ν in the following) in an undeformed cholesteric LC while the other coefficients – only present in deformed nematic or cholesteric phases – are linear functions of the director field distortions ni,j with proportionality coefficients that are pure scalars. We also emphasize that the cross-effect coefficients ξij and ζijk must satisfy the Onsager reciprocal relations (Onsager 1931a, 1931b; Wigner 1954) coming from the property of microscopic reversibility (Casimir 1945). These relations are respected in Figure 3.1 and we refer to Poy and Oswald (2018) for their demonstration. Finally, we recall that, according to the Curie principle, all the phenomenological equations

2 D∞ in the uniaxial approximation used here.

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must be invariant under the action of the symmetry group of the phase (D∞h for nematics, D∞ for cholesterics), undeformed or not, as explained in Poy and Oswald (2018). The goal of this chapter is to review the main works, theoretical, numerical and experimental, on the TM and TH effects in nematic and cholesteric LCs. The rest of this chapter is divided in four sections. In section 3.2, we recall the basic equations of the nematodynamics and we give the general expressions of the constitutive equations in the presence of a temperature gradient. In section 3.3, we summarize the main theoretical results obtained by Sarman and coworkers by molecular dynamics simulations in the case of the TM effect. The last two sections are essentially experimental and deal, respectively, with the TM effect (section 3.4) and the TH effect (section 3.5). Several experiments will be described in these sections, in particular the Éber and Janossy experiment that revealed for the first time the existence of a TM effect in a cholesteric LC. Finally, conclusions and perspectives are drawn in section 3.6. 3.2. The Ericksen–Leslie equations 3.2.1. Conservation equations In an ordinary fluid, the hydrodynamic variables are the density ρ, the velocity v and the internal energy per unit volume e. In a nematic or a cholesteric LC, a new variable exists, the director n. Each of these variables is associated with a conservation equation. The first one is the mass conservation equation, which reads in the incompressible limit  · v = 0 ∇

[3.5]

The next two equations derive from Newton’s second law. The first one describes the conservation of linear momentum. Also called Cauchy’s equation, it reads by neglecting body forces such as gravity: ρ

Dv  ·σ =∇ Dt

[3.6]

D ∂  is the material derivative with respect to time, = ∂t + v · ∇ In this equation, Dt  ∇·σ has for components σij,j and σ ≡ −P I+σ (eq) +σ (neq) is the total stress tensor, which can be decomposed into a pressure term −P I (with I the identity matrix), an

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equilibrium elastic stress tensor σ (eq) and a non-equilibrium stress tensor σ (neq) . The (eq) elastic stress tensor has for components σij = −nk,i (∂f /∂nk,j ) by denoting by f the elastic energy of expression    · n)2 + 12 K2 (n · ∇  × n + q0 )2 + 12 K3 n × (∇  × n) 2 f = 21 K1 (∇    · n(∇  · n) + n × (∇  × n) + 12 K4 ∇

[3.7]

Constants K1−4 are the usual Frank elastic constants and q0 is the equilibrium twist of the cholesteric phase (with q0 = 0 in the nematic phase and at the compensation temperature of the cholesteric phase, when it exists). The second equation (torque equation) describes the conservation of angular momentum and is obtained by applying the angular momentum theorem. It reads Γ(eq) + Γ(neq) = 0

[3.8]

In this equation, Γ(neq) is the non-equilibrium torque acting on the director and Γ is the equilibrium elastic torque3 of expression Γ(eq) = n × h where h = ∂f ∂f −δf /δn is the molecular field of components hi = dxd j ∂n − ∂n . Because the i,j i (neq) (neq) = n × f . With this notation, the torques are perpendicular to n, we can set Γ torque equation rewrites under the equivalent form

 (eq)

(I − n ⊗ n)(h + f(neq) ) = 0

[3.9]

where (a ⊗ b)ij = ai bj represents the dyadic product between two vectors and (I − n ⊗ n)b gives the component b⊥ of b perpendicular to n. The last equation is the heat equation. It comes from the energy conservation and reads by neglecting the term of thermal expansion ρ Cp

DT  · j (q) = σ (s) : D − Γ(neq) · ω  +∇ Dt

[3.10]

In this equation, A : B = Aij Bij represents the total contraction of the two second-order tensors A and B, Cp is the specific heat capacity at constant pressure,  is the rotation vector of the σ (s) is the symmetric part of σ (neq) and ω  = n × N

3 This torque must include the magnetic and electric torques if magnetic and electric fields are applied.

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 = Dn/Dt − 1 (∇  × v ) × n the corotational time derivative of the director with N 2 director. Solving equations [3.5], [3.6], [3.8] (or [3.9]) and [3.10] with adequate boundary conditions gives the fields n(r, t), v (r, t), P (r, t) and T (r, t). More precisely, we need to either know the velocity or the surface force at the boundary of the LC domain to solve the Cauchy equation. The latter condition reads σ · ν = fS

[3.11]

where ν is the unit vector normal to the boundary of the LC domain and directed outwards and fS is the surface force imposed on the boundary. For the torque equation, the associated boundary condition reads γS

  ∂n = (I − n ⊗ n) hS − Cν ∂t

[3.12]

where γS is a surface viscosity, hS = − δW δ n is the surface molecular field of ∂W components − ∂ni by denoting by W (ni , T ) the anchoring energy per unit surface ∂f . area and C is the surface torque tensor of components Cij = ∂n i,j Finally, the temperature or the heat flux must be specified on the boundary of the LC domain to solve the heat equation. In the next two sections, we give the complete expression of the bulk molecular field and we recall the constitutive equations compatible with the symmetries of the nematic or cholesteric phase. 3.2.2. Molecular field In static equilibrium, the torque equation [3.8] reduces to n × h = 0 where h = −δf /δn is the molecular field and f is the elastic energy given in equation [3.7], which depends on four constants Ki (i = 1 − 4) describing, in order, the splay, twist, bend and splay-twist distortions. These constants usually depend on temperature, so that the molecular field contains terms proportional to Ki (this part will be denoted  i (static thermomechanical terms denoted by by h(0) ) and terms proportional to ∇K h(T M ) ) when a temperature gradient is applied. i

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The former part is standard and reads in index form (Stewart 2004): h0i = (K1 − K2 )nj,ji + K2 ni,jj + (K3 − K2 )(nj nk ni,k ),j −(K3 − K2 )nj nk,j nk,i − 2K2 q0 ijk nk,j

[3.13]

where ijk is the Levi–Civita symbol and repeated indices are implicitly summed over. The reader will note that K4 does not enter into h(0) because the K4 -contribution to the free energy appears as a surface-like term in div(...) when K4 is assumed to be constant. As for the static thermomechanical contributions coming from the temperature variation of the elastic constants, they read (Dequidt et al. 2016): ⎧ h(ST M ) = (∇  · n)∇K  1 ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎪ M) ⎪  × n)(n × ∇K  2 ) + n × ∇(K  2 q0 ) ⎨ h(ST = (n · ∇ 2   ⎪ h(ST M ) = (n · ∇K  3 ) (∇  × n) × n ⎪ ⎪ 3 ⎪ ⎪ ⎪   ⎪ ⎩ h(ST M ) = −∇  · ∇K  4 ⊗ n − (∇K  4 · n) I

[3.14]

4

For completeness, we give the expressions of the magnetic and electric contribution to the free energy: ⎧ χa  M 2 ⎪ ⎪ ⎨ f = − 2μ0 (B · n) ⎪ E 1 ⎪  · n)2 ⎩ f = − ε0 εa (E 2

[3.15]

from which can be calculated the magnetic and electric contributions to the molecular  and/or an electric field E  are applied: field which add to h when a magnetic field B ⎧ χa ⎪  B  (n · B) ⎨ hM = μ0 ⎪ ⎩ hE = ε ε (n · E)  E 

[3.16]

0 a

Here, μ0 is the vacuum permeability, ε0 is the vacuum permittivity, χa is the magnetic anisotropy and εa is the dielectric anisotropy.

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3.2.3. Constitutive equations The constitutive equations are obtained by first calculating the irreversible entropy production. A straightforward calculation gives (de Gennes and Prost 1995; Oswald and Pieranski 2005) 

Dσ   (σ) ◦  Tσ ≡ T = σ (s) : D − Γ(neq) · ω  − j (σ) · G [3.17] +∇·j Dt In this expression, σ is the density of entropy, σ (s) is the symmetric part of the non-equilibrium stress tensor and j (σ) ≡ j (q) /T is the entropy flux. The next step consists of writing linear relations between forces and fluxes once they have been chosen. The details of these calculations and the complete  ω expressions of tensors νijkl , κij , ξij and ζijk defined in Figure 3.1 by taking G,  (σ) (neq) (s)   and D as forces and −j , −Γ and σ as fluxes, are given in Poy and Oswald (2018)4. From these expressions, the symmetric part of the non-equilibrium stress σ (s) and the non-equilibrium torque Γ(neq) can be calculated. Three contributions must be considered. The usual viscous contribution involving the Leslie viscosity coefficients αi (i = 1 − 6) (blue and green arrows in Figure 3.1), the Leslie contribution to the TM and TH effects (present in cholesterics only) and the Akopyan and Zel’dovich contributions to the TM and TH effects, which are linear in  n and appear in both deformed nematic and cholesteric LCs: ∇ ⎧ ⎨ f(neq) = f(v) + f(L) + f(AZ) [3.18] ⎩ σ (s) = σ (s,v) + σ (s,L) + σ (s,AZ) From these expressions, the complete non-equilibrium stress tensor can then be calculated by remembering that σ (neq) ≡ σ (s) +σ (a) , where σ (a) is the antisymmetry part of this tensor coming from the non-equilibrium torques of components 1 1 (a) (neq) (neq) σij = − ijk Γk = (δim δjl − δil δjm )nl fm 2 2

[3.19]

where δij is the Knonecker delta.

4 It must be noted that this choice of forces and fluxes, first used by Leslie (1968) and de Gennes and Prost (1995) for pragmatic reasons, is a bit strange as the fluxes do not all appear as a time derivative of a physical variable as in the classical theory based on microscopic equations (de Groot and Mazur 1962; Pleiner and Brand 1996). This choice is nevertheless possible and leads to the good constitutive equations providing the Onsager reciprocal relations and the Curie principle are applied correctly, which was not the case in Akopyan and Zel’dovich (1984), as noticed for the first time by Pleiner and Brand (1987, 1988). This is demonstrated in our paper (Poy and Oswald 2018).

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In the following, we give the result of these calculations for each contribution. 3.2.3.1. Usual viscous terms With the Leslie notations, the viscous force and the symmetric part of the viscous stress tensor reads: ⎧  − γ2 D · n ⎪ f(v) = −γ1 N ⎪ ⎪ ⎪ ⎪   ⎪ (s,v) ⎪ = α4 D + α1 n · D · n n ⊗ n ⎪ ⎨σ  α ¯ ⎪ + n ⊗ D · n + D · n ⊗ n ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ γ2 ⎪ ⎩  +N  ⊗ n) + (n ⊗ N 2

[3.20]

In this expression, the coefficients γ1 , α1 , α4 (which appears as an “ordinary” viscosity) and α ¯ represent the direct effects represented by the blue left and right arrows in Figure 3.1, while γ2 describes the crossed effect represented by the green arrows in this figure. This gives the following well-known expression for the complete viscous stress tensor:    ⊗ n + α3n ⊗ N  + α4 D σ (v) = α1 n · D · n n ⊗ n + α2 N  ⊗ D · n  + α6 N +α5 D · n ⊗ N

[3.21]

where α2 = (γ2 − γ1 )/2, α3 = (γ2 + γ1 )/2, α5 = (¯ α − γ2 )/2 and α6 = (¯ α + γ2 )/2. From these relations, we calculate γ1 = α3 − α2 and γ2 = α2 + α3 with α2 + α3 = α6 − α5 , an equality known as the Parodi relation (1970). This relation shows that the six viscosity coefficients αi , initially introduced by Leslie, are not independent because of the reciprocal Onsager relation that applies to the crossed effect “γ2 ”, represented by the green arrows in Figure 3.1. We underline that all the viscosities (including γ2 ) enter by construction into the expression of the complete viscous stress tensor. 3.2.3.2. Leslie TM and TH terms These terms are only present in cholesterics in which no mirror symmetry exists because of the chirality of the phase. According to Leslie, these terms read (Leslie 1968): ⎧ (L)  ⎪ (TM effect) ⎨ f = ν(n × G)  1  ⎪  + (n × G)  ⊗ n (TH effect) ⎩ σ (s,L) = μ n ⊗ (n × G) 2

[3.22]

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Here, ν is the TM Leslie coefficient5 and μ is the TH Leslie coefficient. From the previous equations, the Leslie contribution to the total non-equilibrium stress tensor can be calculated:      ⊗ n +μ2 n ⊗ (n × G)  σ (L) = μ1 n × G)

[3.23]

with μ1 = (μ + ν)/2 and μ2 = (μ − ν)/2. As for the viscosities, we note that the two coefficients μ and ν enter into the expression of the total stress tensor, while only ν enters into the expression of the thermomechanical force. Another important point is to note that μ and ν (and thus μ1 and μ2 ) are pseudoscalars that must vanish in a nematic phase for symmetry reasons. 3.2.3.3. Akopyan and Zel’dovich TM and TH terms These terms appear in both cholesteric and nematic LCs when the director field is  n are much more complicated. By using the distorted. These terms linear in ∇ Akopyan and Zel’dovich notations, they read: ⎧ 1  · n) G  + ξ2 (n · ∇  × n)(n × G)  ⎪ f (AZ) = (−ξ1 + ξ3 )(∇ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪  m · n − ξ3 m · G  ⎪ (TM effect) + (ξ3 − ξ4 ) (n · G) ⎪ ⎪ ⎪ ⎪   ⎪ ξ 6 ⎪  × n) n ⊗ (n × G)  + (n × G)  ⊗ n ⎪ σ (s,AZ) = − n · (∇ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪  ξ5 ξ7   ⎪  +G  ⊗ n) ⎪ − (∇ · n) (n ⊗ G + ⎪ ⎪ ⎪ 2 4 ⎪ ⎪ ⎪ ξ7 ⎪ ⎪  +m·G  ⊗ n) + (n ⊗ m · G ⎨ 2 [3.24]  ξ7 ξ8 ξ11 ξ12  ⎪ ⎪  ⎪ − + + ( n ⊗ d ·  n + d ·  n ⊗  n ) ( n · G) − ⎪ ⎪ 4 4 4 2 ⎪ ⎪ ⎪   ⎪ ξ ξ 7 12 ⎪  · n) (n ⊗ n) (n · G)  ⎪ − ξ10 − (∇ − ξ5 − ⎪ ⎪ 2 2 ⎪ ⎪ ⎪  ⎪ ξ11  ⎪  ⎪ (n ⊗ n)(n · (m · G)) − ξ7 − ξ9 − ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ξ11  ⎪  ⎪ (G ⊗ d · n + d · n ⊗ G) + ⎪ ⎪ ⎪ 4 ⎪ ⎩  m (TH effect) + ξ12 (n · G)

5 Wrongly called the Lehmann coefficient in the literature, as the Lehmann effect – which is the continuous rotation of cholesteric droplets subjected to a temperature gradient (Lehmann 1900) – is in great part due to effects other than the TM Leslie effect, as reviewed in Oswald et al. (2019a).

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where dij = ni,j and mij = 12 (ni,j + nj,i ). In these equations, the coefficients ξi (i = 1 − 4) are the thermomechanical Akopyan and Zel’dovich coefficients and the coefficients ξi (i = 5 − 12) are the thermohydrodynamical coefficients. Actually, the four TM coefficients can be associated with the four fundamental deformations of the director field. This is not so readily apparent when one looks at the previous equation. For this reason, we proposed in Oswald et al. (2017) to rewrite this force in the equivalent form:

f (AZ) =

4

ξ¯ig (i)

[3.25]

i=1

where we defined the splay, twist, bend and Gauss contributions to the thermomechanical effects as:    · n G  g (1) = ∇     × n n × G  g (2) = n · ∇        × n × n g (3) = n · G ∇

    · G  · n I − G  ⊗ n g (4) = ∇

[3.26] [3.27] [3.28] [3.29]

The correspondence between the new TM coefficients ξ¯1−4 and the ξ1−4 of Akopyan and Zel’dovich is as follows: ξ¯1 = −ξ1 − 12 ξ3 , ξ1 = 12 ξ¯4 − ξ¯1 , ξ2 = ξ¯2 − 12 ξ¯4 , ξ¯2 = ξ2 − 12 ξ3 , ξ3 = −ξ¯4 , ξ¯3 = − 12 ξ4 , ξ4 = −2ξ¯3 . ξ¯4 = −ξ3 , A similar procedure can also be applied to the eight TH coefficients ξ5−12 and leads to the equivalent form for σ (s,AZ) :

σ (s,AZ) = +

4      ζi (i) g ⊗ n + n ⊗ g (i) − ζ5 g (1) · n + ζ6 g (4) · n n ⊗ n 2 i=1

   ζ7   m  + ζ8 n · G G ⊗ d · n + d · n ⊗ G 2

[3.30]

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129

The correspondence between the new TH coefficients ζ1−8 and the ξ5−12 of Akopyan and Zel’dovich is as follows: ξ5 = ζ1 − 12 ζ4 ,

ζ1 = ξ5 + 12 ξ7 ,

ζ2 = −ξ6 + 12 ξ7 ,

− ζ3 = ζ4 = ξ7 , 1 2 ξ8

1 2 ξ11

− ξ12 ,

ξ6 = −ζ2 + 12 ζ4 , ξ7 = ζ4 , ξ8 = 2(ζ3 + ζ7 + ζ8 ),

ζ5 = ξ5 + 12 ξ7 − 12 ξ9 − ξ10 + 14 ξ11 − 12 ξ12 , ξ9 = 2ζ4 − 2ζ6 − ζ7 , ζ6 = ξ7 − 12 ξ9 − 14 ξ11 , 1 2 ξ11 ,

ζ7 = ζ8 = ξ12 ,

ξ10 = ζ1 − ζ4 − ζ5 + ζ6 + ζ7 − 12 ζ8 , ξ11 = 2ζ7 , ξ12 = ζ8 .

From the previous equations, the Akopyan and Zel’dovich contribution to the total non-equilibrium stress tensor and force can be obtained. Its general expression is very complex and will not be given here. A simplified expression can be found by assuming that the new TM coefficients are all equal (ξ¯1−4 ≡ ξ) and by setting ζ1−6 ≡ 2ζ − ζ  + ζ  , ζ7 ≡ ζ  − 2(ζ + ζ  ) and ζ8 ≡ ζ  for the new TH coefficients. With this choice, we find:   ⎧  ·∇  n ⎪ f(AZ) = ξ G ⎪ ⎪ ⎪ ⎪ ⎪  ⎪ ξ  ⎪ (AZ) ⎪  ⎪ = σ d · G ⊗  n −  n ⊗ d · G ⎪ ⎪ 2 ⎪ ⎪ ⎨

 1  ⊗ n + n ⊗ d · G  + (2ζ − ζ  + ζ  ) d · G ⎪ 2 ⎪ ⎪ ⎪

 ⎪ 1 ⎪ ⎪  +G  ⊗ d · n ⎪ − (2ζ − ζ  + 2ζ  ) d · n ⊗ G ⎪ ⎪ 2 ⎪ ⎪   ⎪ ⎪ ⎩  · n m + ζ G

[3.31]

The definitions of the coefficients ζ, ζ  and ζ  may feel a bit convoluted for now, but we will see later that, for some specific geometries, the terms in ζ  and ζ  do not contribute to the Navier–Stokes equation, thus motivating the convention adopted here. Again, we see that both TM and TH coefficients ξ1−12 enter into the expression of the total stress tensor, whereas only the TM coefficients ξ1−4 enter into the expression of the TM force. Last but not least, it is important to note that the ξi are true scalars, contrary to the Leslie coefficients μ and ν, which are pseudoscalars.

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3.2.3.4. Heat conductivity tensor The tensor of heat conductivity is given by ⊥ κij = κ⊥ δij + κ ni nj + {TM, TH terms}

[3.32]

⊥ where δij = δij − ni nj is the transverse Kronecker delta. In this expression, κ⊥ and κ are the thermal conductivities perpendicular and parallel to the director. To these usual terms add the TM and TH terms that are extremely small in comparison with the latter of the order of 10−12 − 10−11 in relative value (Dequidt et al. 2016). For this reason, they can be neglected and we shall not give them here. Their complete expression is however given in Poy and Oswald (2018).

3.3. Molecular dynamics simulations of the thermomechanical effect The previous Leslie and Akopyan and Zel’dovich theories of the TM effect are purely phenomenological and cannot help to understand its molecular origin. To achieve this, molecular simulations are necessary. They have all been performed by Sten Sarman and his coworkers. The first studies date back to the late 1990s. 3.3.1. Molecular models Although the efficiency of computers keeps on increasing, the molecular simulation of LCs with atomic resolution is still a challenge, especially when the goal is to compute transport coefficients. This is because: 1) the simulation of a liquid phase with long-range orientational order requires big systems with many molecules; 2) transport coefficients require long acquisition times, especially for viscous systems with slow relaxation like LCs. For this reason, molecular simulations of LCs are rather performed using simpler coarse grain models (Wilson 2005, 2007). The coarse grains need to have anisotropic interactions in order to produce liquid crystalline phases. The most common coarse grain model for LCs is the Gay–Berne potential (Gay and Berne 1981). This model is generic and simulated using reduced units. Sarman and co. generally used a simplified version of this potential, consisting only of a short-range repulsive interaction:

UGB (r12 , u1 , u2 ) = 4 (r12 , u1 , u2 )

σ0 r12 − σ(r12 , u1 , u2 ) + σ0

18 [3.33]

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131

where (r12 , u1 , u2 ) is the orientation-dependent strength of the interaction, while σ(r12 , u1 , u2 ) is the orientation-dependent diameter of the grain. σ0 is the diameter of the grain perpendicular to the grain axis u. The grains described by this potential are rigid ellipsoids of revolution, which can be oblate or prolate (Figure 3.2). They form discotic or calamitic nematic phases in a given range of temperature and number density, depending on their aspect ratio. In order to form cholesteric phases, the interactions have to include a chiral component. The simulations of Sarman use two types of cholesteric models due to (Memmer et al. 1995; Memmer and Kuball 1996; Memmer 1998, 2000): – The old simulations of the 2000s define molecules as rigid twisted strings of six oblate Gay–Berne ellipsoids (Sarman 2000, 2001) (see Figure 3.3). The long axis of the molecule is along the axis of the cholesteric helix, while the director is the average orientation of the small or intermediate axis of the molecules. This means that the phase is biaxial, with a strong nematic ordering of the long axes w  i along the helical axis (order parameter S1 ∼ 0.8) and a weaker cholesteric ordering of the intermediate axes ui perpendicular to the helical axis (order parameter S2 ∼ 0.5). The director is defined here using the orientations of ui not w  i . This does not correspond to the usual cholesterics used in experiments. – The newer simulations of the 2010s define molecules as single Gay–Berne ellipsoids plus a chiral potential (Sarman and Laaksonen 2013; Sarman et al. 2016): 7

σ0 Uchiral = −4 c (r12 , u1 , u2 ) (u1 · u2 ) (u1 × u2 ) · r12 r12 − σ(r12 , u1 , u2 ) + σ0 [3.34] where c is the strength of the chiral interaction. This model is closer to experimental cholesteric systems, with the director corresponding to the long axis of the ellipsoids.

u1

r12

u2

Figure 3.2. Gay–Berne ellipsoids illustrating the notations of equation [3.33]

3.3.2. Constrained ensembles The simulations are performed under periodic boundary conditions, so as to mimic an infinite medium without border effects. The simulation box is a fixed cuboid. For the simulation of nematic systems, the director is uniform and aligned with one of the axes. For the simulation of cholesteric systems, the box size has to be a multiple

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Liquid Crystals

of the cholesteric half pitch, often a full pitch. The match between the pitch and the box size is not obtained by varying the box size, but by varying the strength of the chiral interaction until the pressure becomes isotropic. This way, the cholesteric has its equilibrium twist. It should be noted that the small size of the simulation box limits the study to cholesterics of very small pitch and therefore systems that are much more chiral than those in the experiments (Sarman 2000).

Figure 3.3. Rigid twisted string of six oblate Gay–Berne ellipsoids featuring a cholesteric molecule in the older simulations

The eigenvector of the tensor order parameter Q associated with the greatest eigenvalue, where Q can be calculated directly from the grain axes ui : 3 Q= 2



N 1 1 ui ⊗ ui − I N i=1 3

 [3.35]

defines an effective director neff . In cholesteric systems, the average in the expression of Q is done after untwisting the helix by rotating the vectors ui by an angle −qzi around ez , the axis of the helix, q being its torsion. This allows us to define a single order parameter for the whole system, even in the cholesteric case (in this case it should be noted that neff does not represent the real director field of the cholesteric helix). In order to avoid the angular diffusion of the effective director, Sarman constrains its direction using Lagrange (or Gauss) multipliers, which represent an external torque applied to neff (Sarman and Evans 1993; Sarman 1996). The effective director neff is thereby held fixed, without any fluctuations. This method also allows us to impose a

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133

 or to impose a constant external torque τ constant angular velocity of the director, Ω, on the director and to let it be free to rotate. The simulations are performed at constant temperature using the same kind of constraint, which corresponds to isokinetic simulations. This means that there are no temperature fluctuations. Nevertheless, although the temperature T is constant and uniform, a fictitious  Q can be applied, which has the same effect as applying a temperature heat field F  Q ≡ 1 ∇T  ). By this mean, it is possible to study the effect of a gradient (in fact F T temperature gradient, while T is actually uniform (Schlacken 1987; Evans and Murad 1989; Sarman and Evans 1993). The way it works is the following: The equations of motion are perturbed by adding external forces and torques proportional  Q , so that energy is dissipated. The additional terms are chosen in such a way to F that the conserved quantities (linear and angular momentum) are not perturbed and  Q , namely the same as in the that the rate of dissipation per unit volume is j Q · F Q   presence of a temperature gradient ∇T = T F . It was shown that, in these circumstances, the system responds in terms of thermodynamic fluxes as if it was subjected to a temperature gradient. 3.3.3. Computation of the transport coefficients The simulations enable the computation of transport coefficients either from non-equilibrium simulations by imposing finite thermodynamic forces and measuring the thermodynamic fluxes, or from equilibrium simulations using the Green–Kubo formulas (Sarman 1999). The former method was most often chosen in the older simulations, because it requires less long simulation runs and is more computationally affordable. However, it is usually necessary to drive the system quite far from equilibrium and possibly out of the linear regime in order to measure a significant response. Indeed, small systems are the place of big fluctuations, which may hide the thermodynamic fluxes. In addition, equilibrium simulations give access to all the transport coefficients at once. This is why the latter method is chosen more and more in recent thermodynamic simulations. Sarman and his coworkers have computed the self-diffusion coefficients D and D⊥ , the heat conductivities κ and κ⊥ , the rotational viscosity γ1 , and other viscosity coefficients (Sarman and Evans 1993; Sarman 1994, 1995). In cholesteric systems, they also computed the Leslie coefficient ν in different ensembles (Sarman 2000, 2001; Sarman and Laaksonen 2013; Sarman et al. 2016). They found that better precision is achieved in the ensemble in which the temperature gradient is imposed and the director is fixed.

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Liquid Crystals

In this ensemble, for simulations that are long enough, in the steady-state slightly out of equilibrium, ν is computed as (Sarman 1999, 2000) ν=

1 T

1 T

−τz  ∂z T

lim lim 1 ∂z T →0 t→∞ T

[3.36]

where τz is the external torque required to keep the director fixed. In the same ensemble, but at equilibrium, the same coefficient is computed using the Green–Kubo formula: −V ν= kB T 2



∞ 0

τz (t) δjz(q) (0) dt

[3.37]

(q)

where δjz is the heat current fluctuation. Analogous formulas can be written for the other transport coefficients, either at equilibrium or out of equilibrium. 3.3.4. Analysis of the results The simulations of Sarman have first introduced and validated a way to perform molecular simulations in constrained ensembles, opening the route to the simulation of transport coefficients in LCs (Sarman 1995). In particular, the use of Lagrange (or Gauss) multipliers to keep the director fixed, and the mechanical analog of the temperature gradient in a constant temperature simulations have been key steps. The transport coefficients have been obtained from different ensembles, and using different methods, and they are consistent within the computation uncertainties. In particular, Sarman demonstrated numerically that the Leslie thermomechanical coefficient ν does exist in cholesterics. This is true even though the model is very simple, with molecules made up of rigid ellispoids. The old simulation models were far from the experimental molecules, whereas the newer simulations are more and more realistic, with more realistic shapes and longer cholesteric pitches. Therefore, the new simulations allow us to go beyond the purely hypothetical systems and to put numbers in place of reduced units. Yet, there is still no quantitative agreement between these simulations and the experiments. The rigid Gay–Berne models tend to underestimate γ1 and overestimate ν. It should be possible in the future to use even more realistic, multigrain flexible molecules (Daivis and Evans 1994). This would allow for simulating molecules with a specific chemical

Thermomechanical Effects in Liquid Crystals

135

structure, but this requires more computational power and more involved methods to impose the external constraints. The latest simulations have shown that ν increases with the spontaneous twist q0 of the cholesteric, so that qν0 is constant for these simple systems. This is true whatever the shape of the ellipsoids (discotic or calamitic) and even for achiral nematics doped with chiral impurities. Finally, it is worth nothing that the simulations have never been performed on deformed nematics to compute the thermomechanical coefficients of Akopyan and Zel’dovich. However, it seems that at least twisted nematics could be simulated in the same way. Sarman and co. also simulated uniform nematics under a temperature gradient and showed that the director experiences a torque (Sarman and Evans 1993; Sarman 1994; Sarman and Laaksonen 2014; Sarman et al. 2017, 2019). This torque is quadratic in  of the form ∇T Γ∇T = μT n · G  (n × G) 

[3.38]

 (in calamitic nematic and and tends to align the director perpendicular to ∇T  (in discotic nematic LC, μT > 0). This cholesteric LCs, μT < 0) or parallel to ∇T orientation is the one which minimizes the irreversible energy dissipation rate. To our knowledge, the only experimental estimate of the thermo-orientational coefficient μT has been given in Demenev et al. (2009); Trashkeev and Britvin (2011): |μT | ∼ 3 × 10−11 N/K2 . However, our own estimate realized by attempting to reorient with a temperature gradient the director in a sample treated for sliding planar anchoring suggests that this value is largely overestimated and certainly less than 10−13 N/K2 (Oswald 2019). It is clear that more experiments are necessary to settle this issue. 3.4. Experimental evidence of the thermomechanical effect This section is devoted to the experimental evidence of the TM effect in nematic and cholesteric LCs. The experiments can be categorized in two groups: the static experiments in which the director field is just distorted by the temperature gradient and the dynamic experiments in which the director constantly rotates under the action of the temperature gradient. The first experiments are delicate because the distortions caused by the non-equilibrium TM effects are extremely small and add to TM effects due to the temperature variations of the elastic constants. The Éber and Jánossy experiment belongs to this category. We will describe it in detail because it is the first experiment that revealed a non-equilibrium TM effect at the compensation

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Liquid Crystals

temperature of a cholesteric phase. The dynamic experiments are more spectacular because they show more directly the non-equilibrium TM effect. Three experiments of this type will be described. The first two will concern the continuous rotation of the cholesteric helix in samples treated for sliding planar anchoring and mixed anchoring. The third experiment will deal with the drift of cholesteric fingers and the formation of spirals in homeotropic samples. It must be emphasized that in all the experiments described in this section, flows were never observed in the samples in spite of careful observations. For this reason, we will neglect them in the calculations, and will focus only on the resolution of the torque equation, directly responsible for the phenomena described here. 3.4.1. The static Éber and Jánossy experiment 3.4.1.1. A bit of history As we already mentioned, it has long been believed that the Leslie TM effect was entirely responsible for the Lehmann effect (Lehmann 1900; Dequidt 2008; Oswald and Dequidt 2008b; Yamamoto et al. 2015; Ito et al. 2016). But this is wrong, as demonstrated by several experiments (Oswald 2012b; Poy 2017; Oswald and Poy 2018; Oswald et al. 2019b). For a review, see Oswald et al. (2019a). However, the first irrefutable experimental evidence of the Leslie TM effect in a cholesteric LC was reported by Éber and Jánossy (1982). This experiment was performed at the compensation temperature of a cholesteric phase, which raised a strong controversy with theorists, the latter claiming that the Leslie TM effect should disappear at this temperature (Pleiner and Brand 1987, 1988), while Éber and Jánossy found a non-zero effect at this temperature (Éber and Jánossy 1984, 1988). To make a decision about this issue, Padmini and Madhusudana (1993) conducted a new experiment to measure the electric analog νE of the thermomechanical coefficient ν in a compensated cholesteric mixture6. In doing this, they found that νE vanishes and changes sign at the compensation temperature, as proposed by theorists. As the same behavior should hold for the thermomechanical coefficient, they logically concluded that the Éber and Jánossy result was due to an artifact of measurement, thus putting a temporary end to the debate. Temporary, because we redid these two experiments and found the same results as Éber and Jánossy on one side (Dequidt and Oswald 2007b; Dequidt et al. 2008) and Padmini and Madhusudana on the other side (Dequidt and Oswald 2007a), raising again a contradiction. To solve it, we analyzed in more detail the results of the Padmini and Madhusudana experiment and showed that they could not be explained in terms of electromechanical coupling, but resulted from a flexoelectric effect

6 The existence of an analog electromechanical effect was predicted for the first time by de Gennes in 1074 (de Gennes and Prost 1994).

Thermomechanical Effects in Liquid Crystals

137

(Dequidt and Oswald 2007a). As for the Éber and Jánossy results, we realized that they were indeed compatible with the Leslie theory after noticing that the symmetry group of the cholesteric phase at the compensation temperature is D∞ (the phase remains chiral, not only at the microscopic scale, but also at the macroscopic scale), and not D∞h as in an usual nematic phase, in spite of the fact that the director field is unwound (Dequidt et al. 2008). 58 56

liqu idus sol idus

0.5 Isotropic 0.0 −1

q (μm )

T (°C)

54 52 Chol. (q < 0)

50

Chol. (q > 0)

48 46

−1.0 −1.5

(a) 0

−0.5

10

20 30 C (wt % CC)

40

50

(b) -12

-10

-8

-6

-4

-2

0

(T - Tsol) (°C)

Figure 3.4. (a) Phase diagram of the mixture EM + CC. Along the dashed line, the cholesteric is compensated. It is left-handed (q0 < 0) on the left of this line and right-handed (q0 > 0) on the right. (b) Equilibrium twist as a function of temperature in the mixture EM + 45 wt% CC. Reproduced from Oswald (2012b) with kind permission of The European Physical Journal

3.4.1.2. Principle of the experiment and theoretical prediction The original experiment by Éber and Jánossy was performed with a compensated cholesteric mixture. In such a mixture, there exists a temperature, called compensation temperature (Tc ), at which the equilibrium twist vanishes and changes sign. Mixtures of usual nematics with the chiral dopant cholesteryl chloride (CC) often have this property. A typical example is shown in the phase diagram of Figure 3.4 where the nematic LC used is an eutectic mixture (EM) composed of 57 wt% of 8CB (4-n-octylcynaobiphenyl) and 43 wt% of 8OCB (4-n-octyloxycynaobiphenyl). Note that similar phase diagrams are observed with the mixtures 8CB + CC7 or 8OCB + CC, also used by Éber and Jánossy and ourselves. Experimentally, the compensated mixture is introduced between two parallel glass plates treated for strong homeotropic anchoring. This sample is then placed inside a temperature gradient parallel to the glass plates. In our own experiments, the directional growth apparatus described in Oswald et al. (1993) was used to impose the temperature gradient. In this geometry, the cholesteric helix unwinds when its

7 We do not recommend this mixture as the CC easily crystallizes at the compensation temperature.

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Liquid Crystals

pitch becomes larger than the sample thickness. As a result, a band of homeotropic “nematic phase”, centered on the compensation temperature, forms in the sample. This region is bordered by cholesteric fingers, which are very clear under the microscope as shown in Figure 3.5 (for a review about the helix unwinding in homeotropic samples, see Oswald et al. (2000) and Oswald and Pieranski (2005)). G

Tc

Figure 3.5. Photo taken between crossed polarizers of a homeotropic sample subjected to a large temperature gradient. In the black band, the cholesteric phase is unwound because the pitch is larger than the thickness. This band is centered on the compensation temperature and is bordered by two regions filled with cholesteric fingers. The unwound zone is 465 μm wide. d = 40 μm and G = 51◦ C/cm. Reproduced from Dequidt et al. (2008) with kind permission of The European Physical Journal

In practice, the director field in the homeotropic “nematic” band is a little distorted because of the presence of the temperature gradient. Two effects add here. The first one is due to the temperature variations of the elastic constants Ki and the equilibrium twist q0 of the cholesteric phase. This equilibrium effect is described by the contribution hT M to the molecular field given in equation [3.14]. The second effect is due to the non-equilibrium Leslie, Akopyan and Zel’dovich thermomechanical forces f(L) and f(AZ) given in equations [3.22] and [3.24], respectively. At equilibrium, the torque equation reads n × (h0 + h(T M ) + f(L) + f(AZ) ) = 0

[3.39]

where h0 is the usual molecular field given by equation [3.13]. This equation is complicated, but can be simplified by noting that the director field is very little distorted. As a result, one can write that, to first order in distortion, n = (nx , ny , 1) by taking the x-axis along the temperature gradient and the z-axis perpendicular to the glass plates. In this limit, the previous equation becomes 

M) h0x + h(T + fx(L) + fx(AZ) = 0 x M) + fy(L) + fy(AZ) = 0 h0y + h(T y

[3.40]

Thermomechanical Effects in Liquid Crystals

which gives explicitly by noting that

∂ ∂y

= 0:

⎧ ∂ny (x, z) ∂ 2 nx (x, z) ∂ 2 nx (x, z) ⎪ ⎪ + K =0 + K3 ⎨ 2K2 q0 1 ∂z ∂z 2 ∂x2  ∂ 2 n (x, z) ⎪ ∂ 2 ny (x, z) ∂nx (x, z)  ⎪ y ⎩ Gνef f + K3 + K − 2q =0 2 ∂z 2 ∂x2 ∂z with νef f = ν + ξ2 q0 +

139

[3.41]

dK2 q dT .

In the previous equations, the second derivatives with respect to x can be neglected because the sample thickness d is always much smaller than the width of the nematic band (this can be checked a posteriori). Solving these equations by neglecting these derivatives gives (Dequidt 2008; Dequidt et al. 2008): ⎧  ⎞ ⎛ 2 ⎪ sin q0 (d − 2z) K ⎪ K Gν 1 1 z ⎪ 3 ef f ⎝ ⎪ ⎠   ⎪ nx = d − + ⎪ ⎪ K2 2K q d 2 2 2 0 ⎪ sin q d ⎨ 0 K3     ⎪ K2 K2 ⎪ ⎪ sin q (d − z) z sin q 0 0 ⎪ K3 K3 Gνef f ⎪ ⎪   ny = d ⎪ ⎪ K2 2K2 q0 ⎩ sin q d

[3.42]

0 K3

These formula generalize the solution given by Éber and Jánossy (1982) since they are still valid out of the compensation point Tc . In particular, they give back the spinodal limit for the nematic phase as nx and ny diverge when qd = π(K3 /K2 ) or d/p = K3 /(2K2 ) (Oswald et al. 2000; Oswald and Pieranski 2005). The solution can be linearized in q0 in the vicinity of the compensation temperature Tc (at which q0 = 0), which gives: nx =

Gνef f K2 d Gνef f q0 z(z − )(z − d) , ny = z(d − z) 3K32 2 2K3

[3.43]

This formula shows that measuring K3 and ny at Tc (where nx = 0) gives the dK2 q0 dq0 0 effective coefficient νef f = ν + dT |q0 →0 = ν + K2 dT . From this measurement 0 and that of the corrective term K2 dq dT , the value at Tc of the thermomechanical Leslie coefficient ν can be obtained.

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Liquid Crystals

3.4.1.3. Experimental results In practice, the distortion of the director field in the middle of the nematic band was obtained by measuring the phase shift ΦH between the extraordinary and ordinary components of a thin laser beam (5 μm in diameter at the waist), crossing the sample at this place. A straightforward calculation gives  ΦH = −

0 Gνef f 2 d K3

2

0 2 2 Gνef n2e − n2o f 2 ne − no kn d + d θ(sin ϕ)kd [3.44] o 240n2e K3 12n2e

In this formula, no and ne are the ordinary and extraordinary refraction indices, k = 2π/λ is the angular wavenumber of the laser, θ is the angle (very small experimentally, but impossible to exactly cancel) between the beam and the normal to the sample and ϕ is the azimuthal angle of the beam with respect to the temperature gradient. This equation shows that at normal incidence (θ = 0), ΦH is proportional to G2 and d5 , a result already given by Éber and Jánossy (1982). However, an additional term linear in G appears when the laser beam is slightly misaligned, but the term in G2 remains unchanged. Experimentally, ΦH was measured by using a rotating analyzer, a quarter-wave plate, a photodiode and a lock-in amplifier following the method of Lim and Ho (1978). A typical curve measured with the compensated mixture 8OCB + 50wt% CC is shown in Figure 3.6. This curve, obtained with a sample of thickness d = 100 μm, is well fitted by a parabola of type aH G − bH G2 , in agreement with equation [3.44]. Performing similar measurements with samples of different thicknesses showed that the fit parameter bH was proportional to d5 in agreement with equation [3.44]. From  0 2 νef f 0 was obtained. Finally, K2 and dq this measurement, a value of K dT were 3 measured. From these data, it was found that in the mixture 8OCB + 50wt% CC, ν = (2.8 ± 0.6) × 10−7 kg K−1 s−2 at Tc . Note that in these experiments, great care was taken to determine the uncertainties by using the maximum-likelihood method each time a quantity was measured. This value of ν is close to the value obtained before by Éber and Jánossy in the mixture 8CB + 50 wt% CC. This confirmed that the Leslie thermomechanical coefficient does not vanish at the compensation temperature of a cholesteric phase. 3.4.2. Another static experiment proposed in the literature The static Éber and Jánossy experiment was designed to measure the Leslie coefficient at the compensation temperature of a compensated mixture. In practice, other geometries could be considered to measure the thermomechanical coefficients, in particular the Akopyan and Zel’dovich coefficients in nematics. In this context,

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Akopyan and Zel’dovich, and later Poursamad, have proposed to impose a temperature gradient to a twisted nematic sample (Figure 3.7) and to measure the nz distortion of the director field (Akopyan and Zel’dovich 1984; Poursamad 2009). In that case, it can be shown by solving the torque equation in isotropic elasticity that, to the first order in nz : nz (z) =

0

 πz  d  πz  1 dK ξ¯2 − ξ¯4 Gz cos + sin G. 2K 2d π d K dT

8 6

(a)

(b)

4

bH (K −2cm 2)

ΦH (°)

−20

[3.45]

−40 −60 −80

slope 5

2

0.1

8 6 4 2

−100 −10

−5

0

5

5

10

6

7

8

d(μm)

G (°C / cm)

9

100

Figure 3.6. (a) Phase shift as a function of the temperature gradient when d = 100 μm. (b) Fit parameter bH as a function of the sample thickness showing that the d5 -dependence is well satisfied. 8OCB + 50 wt% CC, homeotropic samples. Reproduced from Dequidt and Oswald (2007b)

z

z=d y →

G

x →

n(x,z)

z=0

Figure 3.7. Geometry proposed by Poursamad to measure the Akopyan and Zel’dovich thermomechanical coefficients

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Liquid Crystals

As in the Éber and Jánossy experiment, two terms compete. The thermomechanical term, calculated by Akopyan and Zel’dovich and Poursamad, proportional to ξ¯2 − ξ¯4 and another term due to temperature variation of the Frank constant, neglected by these authors in their papers. At this level, it can be interesting to estimate these two terms. In order of magnitude dK dT ∼ 1 pN/K, K ∼ 3 pN which d dK −2 gives by taking d = 100 μm and G = 3000 K/m: πK . This dT G ∼ 3 10 ◦ corresponds to a tilt angle of the director of about 2 with respect to the horizontal plane in the middle of the sample. For the thermomechanical term, all depends on the order of magnitude of the ξ¯i ’s. According to Akopyan and Zel’dovich, these terms must be of the order of 10−11 N/K (Akopyan and Zel’dovich 1984). With this value, ¯ ξ¯4 Gd ∼ 0.2, which and the same as before for the other parameters, we calculate 2ξ2√−2K ¯ is considerable. It turns out that the ξi are much smaller experimentally, typically ranging between 10−15 and 10−14 N/K as we shall show later. This is why we think this experiment is not suitable for measuring the ξ¯i in usual nematics. Finally, we mention that a similar calculation was conducted for a twisted nematic phase subjected to a two-dimensional temperature gradient (Poursamad and Hakobyan 2008). 3.4.3. Continuous rotation of translationally invariant configurations While the Éber and Jánossy experiment is very elegant, it is very limited because it cannot be performed outside of the compensation temperature. In addition, this measurement is not direct since two TM effects are measured within the same time frame. Therefore, it was important to find alternative methods to eliminate these two problems. One of them was suggested by Leslie himself, who showed in his seminal paper (Leslie 1968) that the helix rotates at constant velocity in a temperature gradient when the director can freely rotate on the boundary of the cholesteric domain. This prediction was indeed confirmed by molecular dynamics simulations, as recalled in section 3.3. In this section, we describe two experiments in which translationally invariant configurations (TIC) are set into continuous rotation by application of a temperature gradient. In practice, the cholesteric phase is sandwiched between two parallel glass plates and the temperature gradient is applied along the normal to the glass plates. In the first configuration, the two plates are treated for sliding planar anchoring, so that the helix is slightly deformed and orients parallel to the temperature gradient. This geometry will be referred to as Leslie geometry in the following. In the second configuration, one plate is treated for sliding planar anchoring, while the other is treated for homeotropic anchoring. In this case, the helix still orients parallel to the temperature gradient, but it is highly deformed. This will be referred to as mixed geometry in the following, where we show how to calculate the rotation velocity of a TIC in all generality.

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3.4.3.1. TIC rotation velocity: a general formula In practice, the LC sample of thickness d is sandwiched between two parallel glass plates of total thickness h  d 8. The temperature gradient G is obtained by imposing a temperature difference ΔT between the two external top and bottom faces of the sample. As a result, the local gradient G in the LC is proportional to the ˆ = ΔT /h. We refer to Oswald and Dequidt (2008b) imposed temperature gradient G and Dequidt (2008) for a description of the setup used in experiments to impose the temperature gradient. With the z-axis vertical and oriented upwards, the components of the director in a TIC are given by ⎛

⎞ sin α cos φ n = ⎝ sin α sin φ ⎠ cos α

[3.46]

where the zenithal and azimuthal angles α and φ only depend on z since the system is supposed to be translationally invariant in the horizontal plane (x, y). In the following, we show that the director rotates at constant angular velocity ω when a temperature gradient is applied along the z-axis, provided that the anchoring is sliding on the two glass plates. This rotation is due to the non-equilibrium TM torque that must equilibrate with the elastic and viscous torques in the stationary regime according to the general torque equation [3.8]. Two methods can be used to calculate this velocity. The first, used by Leslie (1968) and ourselves in Dequidt (2008); Dequidt et al. (2008); Oswald and Dequidt (2008a) and Oswald (2012b) consists of directly solving the torque equation [3.8] subjected to the boundary condition 3.12. This is quite easy to do with Leslie geometry when (eq) (neq) α = π/2. In this case, the bulk torque equation reduces to Γz + Γz = 0 and reads explicitly ∂φ ∂ γ1 = ∂t ∂z



∂φ K2 − K2 q 0 ∂z



∂φ − νG + ξ¯2 G, ∂z

[3.47]

where the three contributions – elastic, viscous and thermomechanical – are easily recognizable. If the anchoring is sliding planar on the plates, this second-order differential equation must be solved with the boundary conditions γS (0)

∂φ = K2 ∂t



∂φ − q0 (0) ∂z

8 In all of our experiments h = 4 mm.

 [3.48]

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Liquid Crystals

at the bottom plate z = 0 and ∂φ γS (d) = −K2 ∂t



∂φ − q0 (d) ∂z

 [3.49]

at the top plate z = d, with γS being the rotational surface viscosity. ¯ Following Leslie (1968), we can look for a solution of the type φ = φ(z) + ωt. After substitution in the previous equations, we obtain

 ⎧ ¯ ∂ φ¯ ∂ ⎪ ¯2 ∂ φ G, ⎪ γ K − νG + ξ ω = q − K 1 2 2 0 ⎪ ⎪ ∂z ∂z ∂z ⎪ ⎪ ⎪ ⎪

 ⎨ ∂ φ¯ [3.50] − q0 (0) , γS (0)ω = K2 ⎪ ∂z ⎪ ⎪ ⎪

¯  ⎪ ⎪ ⎪ ∂φ ⎪ ⎩ γS (d)ω = −K2 − q0 (d) ∂z ˆ by denoting by where G is the local temperature gradient given by G = (κg /κ⊥ )G κg the thermal conductivity of the glass9. Integrating over the sample thickness, the first equation in [3.50] is written by using the two other equations and by assuming that the helix is slightly distorted (φ¯,z ≈ q0 ): d ω = − d 0

0

ν¯Gdz

γ1 dz + [γS (0) + γS (d)]

.

[3.51]

where ν¯ = ν − ξ¯2 q0 is the effective Leslie coefficient, which is measured experimentally. If the sample is very thin, the material constants are almost constant inside the sample. In this limit, the previous equation becomes simply by setting γ1 = γ1 + 2γS /d: ω=−

κg ν¯ ˆ G. κ⊥ γ1

[3.52]

where the values of the material constants are taken at the average temperature of the sample.

9 The temperature gradient G is obtained by writing that the heat flux across the glass plates and the sample is the same in the stationary regime.

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145

The same calculation could be made in the general case, when α = π/2, but it is much more complicated. For this reason, we prefer another method, proposed in Dequidt et al. (2016), which does not need to explicitly solve the torque equation to find ω. The starting point of this calculation consists of noticing that the elastic energy of   n, T )dV is constant during the rotation. the cholesteric phase F = V f (n, ∇ Mathematically, this condition reads dF = dt



d 0



  d ∂n ∂n ∂f ∂T  dz + (C · ν ) · −h · =0 + ∂t ∂T ∂t ∂t 0

[3.53]

Replacing h and C · ν in this equation by their expressions given in equation [3.9] 10 yields: and [3.12], after noticing that ∂T ∂t = 0 in a TIC 

d

γ1 0

d  d  ∂n ∂n ∂n ∂n ∂n = (f(L) + f(AZ) ) · · dz + γS · dz ∂t ∂t ∂t ∂t 0 ∂t 0

[3.54]

n ez × n), we obtain from this equation by using equation [3.46] Because ∂ ∂t = ω( and the expressions [3.22] and [3.24] of the Leslie and Akopyan and Zel’dovich forces

ω= where

1 (Iν + I2 + I3 ) Iγ

⎧  d ⎪ ⎪ ⎪ I = − G ν sin2 α dz, ν ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪  d ⎪ ⎪ ⎪ ⎪ G ξ¯2 φ,z sin4 α dz, ⎪ ⎨ I2 = 0  d ⎪ ⎪ ⎪ ⎪ G ξ¯3 φ,z cos2 α sin2 α dz, ⎪ I3 = ⎪ ⎪ 0 ⎪ ⎪ ⎪  d ⎪ ⎪ ⎪ ⎪ ⎩ Iγ = γ1 sin2 α dz + γS (0) sin2 α(0) + γS (d) sin2 α(d).

[3.55]

[3.56]

0

10 Note that here we must set the anchoring energy W equal to 0. This is required in order to have a constant rotation velocity, which is the basic assumption of this calculation.

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Liquid Crystals

In these formulas, the temperature gradient G depends on z in general. It is obtained by writing that the heat flux is the same in the sample and in the glass plates. Using equation [3.32], this gives: G=

κg ˆ G. κ⊥ sin2 α + κ cos2 α

[3.57]

Transmitted light (a. u.)

Equations [3.55]–[3.57] generalize to any TIC the formula [3.51], obtained in the Leslie geometry. They show that, in the general case, the rotation velocity of the helix only depends on the three non-equilibrium TM coefficients ν, ξ¯2 and ξ¯3 . A very important point is that the rotation disappears when these coefficients vanish. This shows that the static equilibrium TM contribution h(ST M ) to the molecular field cannot be responsible for the rotation of the helix in a TIC and contributes just to its deformation during the rotation. This is a major advantage with respect to the Éber and Jánossy experiment in which both effects – equilibrium STM and non-equilibrium TM – played the same role and should be separated. 140

100

60

20 0

500

1000 Time (s)

1500

Figure 3.8. Transmitted intensity between crossed polarizers. Crosses are experimental points and the curved solid line is the best fit to a sinusoidal law of period 100 s. Reproduced from Dequidt et al. (2008) with kind permission of The European Physical Journal

3.4.3.2. Experimental results with the compensated mixtures in the Leslie geometry The goal of these experiments was to directly evidence the TM effect predicted by Leslie in 1968. To perform them, a specific surface treatment of the glass plates, allowing a sliding planar anchoring of the LC molecules, was developed in ˙ collaboration with Zywo´ cinski (Oswald et al. 2008). This surface treatment – which

Thermomechanical Effects in Liquid Crystals

147

consists of the deposition by spin coating of a thin layer of a viscoelastic polymer11 – was fully characterized. In particular, its surface viscosity was measured (Oswald 2012a; Oswald and Poy 2013) and the problems of memorization that occur when the director stops rotating, or rotates very slowly, were described in Oswald (2014a). In sliding planar samples, a great number of defects can form since the anchoring direction is degenerated. In practice, most of the defects are ± 12 disclination lines, even if ±1 defects are sometimes observed. These defects usually nucleate in large number when the sample is cooled down from the isotropic liquid. As the anchoring is sliding, most of the defects annihilate (Oswald et al. 2008) but some of them can remain in the sample after annealing because they are pinned on a dust particle or a surface defect. In the following, we begin with a description of what happens in the samples far from the defects, in regions where the in-plane distortions of the director field are negligible. We will then study the director rotation in the vicinity of disclination lines and in the presence of a surface defect. Finally, we will show that it is possible to stop the rotation of the director by applying an electric field and we will describe the final state of the sample once the director has stopped rotating. 3.4.3.2.1. Rotation in a homogeneous sample The first experiment in the Leslie geometry was performed with a compensated mixture of 8OCB + 50 wt%CC (Dequidt et al. 2008). This choice was driven by the Éber and Jánossy experiment that showed the existence of a TM Leslie effect at the compensation temperature of this mixture. In order to measure the rotation velocity of the director, a 10-μm-thick sample was prepared. This sample was then placed in the setup, described in Oswald and Dequidt (2008b), to impose a temperature gradient and was observed in the polarizing microscope. By doing so, it was found that, at the compensation temperature, the transmitted intensity between crossed polarizers was oscillating in time in a sinusoidal manner (Figure 3.8), revealing that the director was rotating at constant velocity in the sample. For the first time, this showed the existence of the TM Leslie effect at the compensation temperature, in a direct way. The experiment was then performed at other temperatures from both sides of the compensation temperature (Oswald and Dequidt 2008a). An important point was the sense of rotation of the helix. It was determined by looking at the direction in which the polarizers should be rotated to maintain a constant intensity (another method will be described later). The result of these measurements is shown in Figure 3.9. As we can see, the sense of rotation is the same on both sides of the compensation temperature, counterclockwise (ω > 0) when the temperature gradient is directed downwards (ΔT < 0) and clockwise when the temperature gradient is directed upwards (ΔT > 0). The experiment also showed that ω is proportional to ΔT in

11 In our experiments, the polymer chosen was a polymercaptan, used as hardener for epoxy resins.

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Liquid Crystals

agreement with formula [3.51]. From these measurements, the coefficient ν¯ was deduced after the ratio κg /κ⊥ and the viscosity γ1 were measured. At Tc , it was found that κg /κ⊥ ≈ 7, γ1 ≈ 0.075 Pa s and γS ≈ 3.2 × 10−7 Pa s m, which led to ν¯(Tc ) = ν ≈ 1 × 10−7 kg K−1 s−2 . This value is compatible with that previously found in the Éber and Jánossy experiment. The curve shown in Figure 3.9(a) also suggests that ν¯ changes little with temperature. Indeed, the viscosity decreases when the temperature increases, which could explain, in large part, why the velocity increases when the temperature increases. This was checked in minute detail in the mixture EM + 50 wt% CC in which the two viscosities γ1 and γS were measured as a function of temperature. In this mixture it was found that, within experimental errors, ν¯ was approximately constant, of the order of 1×10−7 kg K−1 s−2 in a temperature interval of 10◦ C below the clearing temperature. This value is comparable with that found at the compensation temperature in the 8OCB + 50 wt% CC mixture (Oswald 2012b), studied previously. The independence of ν¯ with the temperature also suggests that the macroscopic Akopyan and Zel’dovich contribution to ν¯ is negligible in these mixtures in which q0 changes a lot with temperature. This point will be confirmed later. 5

3 2 ω (rad/min)

ω (rad/min)

4 3 2

1 0 −1 −2

1 (a) 0 56

57

58 T (°C)

(a)

(b)

−3

Tc 59

−40

−20

0 ΔT (°C)

20

40

(b)

Figure 3.9. (a) Angular velocity as a function of the sample temperature measured when ΔT = −40◦ C. (b) Angular rotation velocity as a function of the temperature difference measured at T = Tc . Mixture 8OCB + 50 wt% CC, d = 25 μm. Reproduced from Oswald and Dequidt (2008a)

More recently, we made similar measurements in diluted cholesteric mixtures12 (Oswald 2014b). In these mixtures, the pitch was chosen to be of the same order of magnitude as in the previously studied zcompensated mixtures at the clearing temperature. Although the pitch was similar (between 5 and 10 μm, in practice), we

12 In these mixtures, the concentration in weight of chiral molecules is less than 5%, whereas it is close to 50% in the compensated mixtures studied above.

Thermomechanical Effects in Liquid Crystals

149

found that ν¯ was much smaller (typically, 50 times smaller) in these mixtures than in the compensated mixtures. An example of systematic measurements with a diluted mixture of 7CB (4-n-heptylcyanopbiphenyl) and R811 (R-(+)-octan-2-yl 4-((4-(hexyloxy)benzoyl)oxy)benzoate) is shown in Figure 3.10. These data show that ν¯ is proportional to the concentration C of chiral molecules at low concentration. This result was expected since ν¯ must vanish when C → 0 and led us to define the Leslie thermomechanical power LTP =

ν¯ 2πC

[3.58]

by analogy with the helical twist power defined to be HTP =

q0 1 = . 2πC PC

[3.59]

-9

4x10

-9

4x10

1.2 wt%

−1 −1

(N m K )

−1

ν (N m−1K )

7CB + R811

3

3 2 0.6 wt% 1 (a)

0 -7

-6

-5

-4 -3 δT (°C)

(a)

-2

-1

2 1 (b)

0 0

0.0

0.2

0.4

0.6 C (wt%)

0.8

1.0

1.2

(b)

Figure 3.10. (a) Effective Leslie coefficient ν¯ as a function of temperature (δT = T − TChI ) measured in the diluted mixtures 7CB + 0.6wt% R811 and 7CB + 1.2wt% R811. (b) Average value of ν¯ as a function of the concentration of chiral molecules. Reproduced from Oswald (2014b)

Typical values of the HTP and the LTP are given in Table 3.1 for the mixtures 7CB + R811, 7CB + CC, MBBA + R811 and MBBA + CC, where MBBA is the N-(p-methoxy-benzylidene)-p-butylaniline, the first known LC with a nematic phase at room temperature. This table shows that the LTP crucially depends on the chiral molecule chosen, but also on the host nematic LC. For instance, the LTP of the R811 is always positive and three times larger in 7CB than in MBBA. In comparison, the LTP of the CC is smaller in absolute value, although still larger in 7CB than in MBBA. On the other hand, the LTP of the CC is positive in 7CB and negative in MBBA. These results show that there is no direct relationship between the HTP and

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Liquid Crystals

the LTP, i.e. between the spontaneous twist q0 and the effective Leslie thermomechanical coefficient ν¯. This appears more clearly still if we calculate the ratio R=LTP/HTP= ν¯/q0 , which is very different from one mixture to another (Table 3.1). This is not surprising as the HTP is a static property of the phase, whereas the LTP is dynamic in nature. LC Dopant LTP=¯ ν /(2πC) HTP=q0 /(2πC) R=¯ ν /q0

7CB R811 4.3 12.1 3.6

7CB MBBA MBBA CC R811 CC 1.2 1.5 -0.4 -2.9 10.2 -6,1 -4.2 1.5 0.6

Table 3.1. Average values of the LTP (in unit of 10−8 N m−1 K−1 wt%−1 ) of the HTP (in μm−1 wt%−1 ) and of the R-ratio (in fN K−1 )

The next step is to determine whether it is the microscopic term of Leslie ν or the macroscopic term of Akopyan and Zel’dovich −ξ¯2 q0 that is the more important in ν¯. The only way to answer this question is to measure the order of magnitude of the Akopyan and Zel’dovich TM coefficients ξ¯i . This problem will be discussed at the end of this section. But before this, we analyze the behavior of the defects in the Leslie geometry. 3.4.3.2.2. Rotation in the presence of disclination lines In the previous section, we mentioned that disclination lines are often present in samples. In practice, they are almost impossible to completely eliminate. Hence the question as to whether their presence could disturb the velocity measurements. To answer this question, we observed the rotation of the extinction branches of ±1/2 disclination lines that usually form in the samples and we simultaneously recorded the intensity transmitted between crossed polarizers at different places of the sample. In doing so, we observed that the director was rotating everywhere at the same velocity, as if the sample was homogeneous. An example of such measurements is shown in Figure 3.11. This photo was taken in a sample of the mixture EM+45 wt%CC at a temperature close to the compensation temperature under a temperature gradient ΔT = 40◦ C. In this case, the director was rotating clockwise so that the extinction branches of the -1/2 lines rotated clockwise while those of the +1/2 lines rotated counterclockwise13.

13 In practice, the sense of rotation of the director can be deduced from the observation of the sense of rotation of the extinction branches. Indeed, the sign of the topological rank of a disclination line can be easily determined by rotating together the polarizers, the branches rotating in the same direction if the rank is positive and in opposite directions if the rank is negative.

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We now calculate the rotation velocity of the director in the presence of disclination lines. In this case, angle φ between the director and the x-axis not only depends on z, but also on x and y, which makes the problem very complicated. To simplify the calculations, we assume first that the material constants do not depend on z and that K = K1 = K2 = K3 = −K4 (isotropic elasticity). With these hypotheses, the elastic energy reads simply   2  1 ∂φ 2  ∂φ 2  ∂φ + + K − q0 2 ∂x ∂y ∂z

Intensity

f=

[3.60]

1

time (s)

1

+1/2

-1/2

-1/2

+1/2

+1/2

3

2

+1/2

2

time (s)

Intensity

Intensity

-1/2

3

time (s)

Figure 3.11. Typical texture observed between crossed polarizers with the mixture EM+45 wt% CC at a temperature close to the compensation temperature when the two glass plates are treated for sliding planar anchoring. The three graphs show the average intensity measured as a function of time inside the three squares marked 1, 2 and 3 on the photo. ΔT = 40◦ C and d = 20 μm. The black bar is 100 μm long

152

Liquid Crystals

and the bulk and surface torque equations [3.47]–[3.49] become  2 ⎧  ∂  ∂φ ∂φ ∂ φ ∂2φ ⎪ ⎪ ⎪ ⎪ γ1 ∂t = K ∂x2 + ∂y 2 + ∂z ∂z − q0 − ν¯G ⎪ ⎪ ⎪ ⎪ 

⎨ ∂φ ∂φ in z = 0 γS =K − q0 ⎪ ∂t ∂z ⎪ ⎪ ⎪ 

⎪ ⎪ ∂φ ∂φ ⎪ ⎪ ⎩ γS in z = d = −K − q0 ∂t ∂z

[3.61]

Integrating the first equation over the sample thickness yields an equation for the d average angle φ¯ = d1 0 φdz by using the two boundary conditions: γ1

∂ φ¯ = KΔ⊥ φ¯ − ν¯G ∂t

where Δ⊥ = equation reads

∂2 ∂x2

+

∂2 ∂y 2

and γ1 = γ1 + 2γS /d. In polar coordinates (r, θ), this

 1 ∂  ∂ φ¯  1 ∂ 2 φ¯ =K r + 2 2 − ν¯G ∂t r ∂r ∂r r ∂θ ¯

∂φ γ1

[3.62]



[3.63]

An obvious solution is φ¯ = mθ + ωt with ω = − ν¯γG  . For integer or half-integer 1 m, this solution has no singularity in the plane θ = 0 and represents a disclination of topological rank m. This calculation shows that, in isotropic elasticity, the presence of a disclination line does not change the rotation velocity of the molecules, whatever the rank of the line. This is indeed observed experimentally with the ±1/2 lines. However, the situation is different for ±1 lines, as shown in the sequence of photos in Figure 3.12(a). If the extinction branches of the -1 line still rotate regularly, the branches of the +1 line rotate in a very irregular way while distorting heavily close to the core of the line. This effect is also visible in Figures 3.12(b) and (c), where we can see that the intensity measured close to the +1 line varies in an irregular way, contrary to that measured close to the -1 line. It turns out that this effect is due to the elastic anisotropy of the LC neglected so far in the calculations. This can be seen immediately by looking at how the energy of a disclination line of rank m varies during the rotation. To perform this calculation, we need the expression of the elastic energy per unit surface area of the sample. At the

Thermomechanical Effects in Liquid Crystals

153

compensation temperature, the director field is very slightly twisted along z and we can neglect the twist energy with respect to the splay and bend energies of expressions ⎧ 2  ⎪ ∂φ d K1 ∂φ ⎪ ¯ ⎪ ⎪ ⎨ f1 = 2 r2 cos(θ − φ) ∂θ + r sin(θ − φ) ∂r [3.64]  2 ⎪ ⎪ ∂φ d K ∂φ 3 ⎪ ⎪ sin(θ − φ) . − r cos(θ − φ) ⎩ f¯3 = 2 r2 ∂θ ∂r where φ is assumed not to depend on z. From these formulas, the energy Em =  2π  R (f¯ + f¯3 )rdθdr of a disclination line of rank m, core radius rc and external 0 rc 1 radius R can be calculated. By taking φ = mθ + ωt, we obtain: ⎧ K + K  R 1 3 ⎪ ⎪ m2 log ⎨ Em =1 = πd 2 rc [3.65]   R ⎪ K1 + K 3 K1 − K3 ⎪ ⎩ E1 = πd + cos(2ωt) log 2 2 rc 200

intensity (a.u.)

24 6

-1

8

150 100

+1 1

2

3

(b)

50

5

1 0

5

6

intensity (a.u.)

200

4

9

3 100

(a)

7

200 time (s)

300

400

150 100

(c)

50 0

7

8

9

100

200 time (s)

300

400

Figure 3.12. (a) Nine photographs taken at the compensation temperature showing the rotation of the four extinction branches of a pair of disclination lines of strength m = ±1. The two lines are 52 μm apart. Crossed polarizers, d ≈ 7.1 μm and G ≈ −70 000 K/m. (b and c) Intensity as a function of time recorded inside the white and black squares, respectively. On the top curve, the time at which each photo, numbered 1 to 9, was taken is indicated. EM + 45%CC. Reproduced from (Oswald 2012b) with kind permission of The European Physical Journal

This very simple calculation shows that for all disclination lines of rank m = 1 the energy does not change in time when the director rotates. By contrast, a +1 line

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Liquid Crystals

oscillates between a radial configuration of energy πdK1 ln(R/rc ) and a circular configuration of different energy πdK3 ln(R/rc ) if K1 = K3 . This change of energy is responsible for the irregular rotation of the extinction branches of the +1 defect.

0

π

π/5

4π/5

t = 0, π 2π/5

3π/5

Figure 3.13. Numerical simulation of the deformation of the extinction branches of a +1 line during rotation by an angle π of the director. The central picture shows that the extinction branches are mainly deformed close to the core of the line. The six smaller pictures show their aspect over time inside the dashed square (of side length 10 in units of L) drawn in the central picture. Time is indicated on each of picture in units of τ

This result can be shown in a more compelling way by numerically solving the torque equation around a +1 line. By setting φ = θ + ϕ(r, t), the torque equation is written in the form

∂ 2 ϕ 1 ∂ϕ + ∂r2 r ∂r



sin 2ϕ (1 + A cos 2ϕ) − A 2 r −

1+r

2

 ∂ϕ 2  ∂r

γ ∂ϕ ν¯G = 1 K1 K1 ∂t

[3.66]

3 −K1 where A = K K3 +K1 is the elastic anisotropy factor. By choosing as the length scale L = (K1 /¯ ν G)1/2 and as the time scale τ = γ1 /¯ ν G 14, the previous equation is written in the dimensionless form



∂2ϕ 1 ∂ϕ +   ∂r2 r ∂r



sin 2ϕ (1 + A cos 2ϕ) − A 2 r

 2  ∂ϕ 2 ∂ϕ 1+r −1=  ∂r ∂t

14 This is also the diffusion time L2 /D over distance L by setting D = K1 /γ1 .

[3.67]

Thermomechanical Effects in Liquid Crystals

155

where r ≡ r/L and t ≡ t/τ . This equation in ϕ(r , t ) was solved numerically with Mathematica by taking as boundary conditions ϕ(r , 0) = 0 and ∂r ϕ(r , t ) = 0 at r = rc and r = 100 (these two boundary conditions reflect the fact that the director is free to rotate on the core of radius rc and far from the core at r = 100). The core radius can be estimated by assuming that the director escapes along the z-axis inside the core, in order to suppress the singularity. In doing so, the molecules are homeotropically anchored on the plates in the core, which costs anchoring energy.  Minimizing the core energy with these assumptions yields rc ≈ 0.5 lp d where lp = K1 /Wa is the extrapolation anchoring length of the sliding planar anchoring. Typical values for the compensated mixture used here are K1 = 3 pN, K3 /K1 = 1.7 (A = 0.41) and Wa = 3 × 10−5 J/m2 (Oswald et al. 2008) and ν¯ = 10−7 kg K−1 s−2 , which gives, by taking G = 7 × 104 K/m (ΔT = 40◦ C) and d = 7.5 μm, L ≈ 20 μm, τ ≈ 10 s and rc ≈ 0.02 (in dimensionless unit). From this numerical solution, the aspect between crossed polarizers of the line can be calculated at different times, as shown in Figure 3.13. This simulation reproduces the observations fairly well. It shows that the extinction branches are deformed and rotate jerkily inside a disk of typical radius 5L. 3.4.3.2.3. Rotation in the presence of an anchoring defect In practice, it may happen that the polymercaptan layer dewets locally on the glass plates. This leads to a localized surface defect around which the extinction fringes of the director field form rings. One example is shown in Figure 3.14(a) where one can also see a -1/2 disclination line between crossed polarizers. This sequence of 10 photos, taken at a time interval of 10 s, shows that under the action of the temperature gradient, the rings continuously move inwards and collapse in the center of the defect, while the two extinction branches of the neighboring disclination line rotate at constant velocity. In addition, a time recording of the local intensity measured in the vicinity of the defects shows that the director rotates at constant velocity, as if the sample was homogeneous. This observation indicates that the surface defect, just like the -1/2 disclination line, does not change the rotation velocity of the director. To demonstrate this result, we solved the torque equation [3.63] for the average angle φ¯ with a surface defect placed at the origin. To model this defect, we assumed that the viscosity γ1 is a function of r of the type: γ1 + δγ1 e−(r/r0 )

2

[3.68]

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Liquid Crystals

where r0 represents the typical radius of the defect. Solving this problem with Mathematica yields ν¯G φ¯ = a − t b − νG





2  r2 r δγ1 r02 C + Γ 0, 2 − log + c log(r) b 4K r0 r02

− ν¯G

γ1 − b r2 . b 4K

[3.69]

2

3

4

5 (a)

(b)

Figure 3.14. (a) Typical texture observed between crossed polarizers in the presence of a surface defect and a -1/2 disclination line. Each photo is 450 μm wide. (b) Texture calculated numerically from the complete solution. Reproduced from Dequidt et al. (2008) with kind permission of The European Physical Journal

In this expression, C is the Euler constant, Γ the incomplete Euler function, and a, b and c are integration constants. In order that ϕ does not diverge at the center of the defect, we must take c = 0. Constant a corresponds to a phase shift. It may be removed by changing the time origin. In the following, we shall take a = 0. Finally, we must take b = γ1 to cancel the torque ∂φ/∂r at infinity. With this choice of the

Thermomechanical Effects in Liquid Crystals

157

constants, the solution reads 



2  r2 r δγ r2 ν¯G φ¯ = − t − νG 1 0 C + Γ 0, 2 + log γ1 γ1 4K r0 r02

[3.70]

This formula shows: 1) that the director rotates everywhere with velocity ω = −¯ ν G/γ1 , whatever the “strength” δγ1 of the defect, as if the sample was homogeneous; 2) that the director winds radially inwards around the defect, which explains the fringes observed between crossed polarizers around the defect, the number of which is proportional to δγ1 . To this solution, one can add the solution to the torque equation −θd /2 corresponding to the -1/2 disclination line visible in Figure 3.14(a), where θd is the polar angle measured from the core of this line. The texture between crossed polarizers calculated from the complete solution is shown in Figure 3.14(b). In this calculation, the “strength” δγ1 of the defect was chosen in order that the number of rings was approximately the same as in the experiment. 3.4.3.2.4. Rotation in the presence of an electric field In this section, we describe an alternative method to measure the effective Leslie coefficient ν¯ with an electric field. This measurement was performed in the Leslie geometry with the compensated mixture EM + 45 wt% CC. This LC is of positive dielectric anisotropy. The AC electric field (f = 10 kHz) was imposed by applying a voltage difference ΔV between two ITO electrodes deposited on the bottom plate at 2 mm apart (Figure 3.15(a)). As a result, the electric field is parallel to the glass plates and perpendicular to the edges of the insulating band separating the two electrodes. A calculation shows that the field is constant to better than 5%, given by E = π2 ΔV W , in the central part of width ∼ W/4 of the insulating band. For this reason, all measurements were performed in this region of the sample. Because the LC used is of positive dielectric anisotropy, its director tends to spontaneously align along the electric field. As a result, the electric field can be used to stop the rotation of the helix when a temperature gradient G is applied. From the measurement of the minimal electric field Estop , required to stop the rotation, the value of ν¯ was deduced. In practice, Estop was measured by recording the intensity between crossed polarizers as a function of time. Without an electric field, the intensity varies in a sinusoidal manner as shown in Figure 3.15(b). When the field is applied, the rotation becomes irregular, slowing down when the director is parallel to the field and accelerating when the director is perpendicular to the field (Figures 3.15(c) and (d)). As a result, the period of rotation Θ increases to finally diverge at Estop . In practice, Estop was measured by extrapolating to zero the curve of average angular velocity

158

Liquid Crystals

ω = 2π/Θ. Knowing Estop , the value of ν¯ was then calculated by numerically solving the torque equation in the static regime. top plate w intensity (a.u.)

ac er

35

→ ITO E

sp

sp

ac

er

ITO z

y x

bottom plate

25 20 (b)

15

ΔV

(a)

0

100

200 t (s)

300

400

30 intensity (a.u.)

35 intensity (a.u.)

30

30 25 20 15

(d) 0

100

200 t (s)

300

400

25 20 15 10

(c) 0

100

200 t (s)

300

400

Figure 3.15. (a) Schematic of the cell used to impose an electric field. (b–d) Transmitted intensity between crossed polarizers as a function of time when E = 0, 12 900, 17 000 V/m. EM + 45%CC. d = 7 μm, T = Tc . Reproduced from Oswald (2012b) with kind permission of The European Physical Journal

The procedure was as follows. In the static regime (when E > Estop ), the torque equation reads: ∂ ∂z

 1 ∂φ ∂φ K2 − K2 q0 − ε0 εa E 2 sin(2φ) = νG − ξ¯2 G ∂z 2 ∂z

[3.71]

∂φ This equation is subjected to boundary conditions ∂φ ∂z = q0 (0) at z = 0 and ∂z = q0 (d) at z = d. In this equation, q0 and K2 are functions of z because they depend on temperature. In the compensated mixture, K2 is, within experimental errors, a linear ∂K2 2 function of T so that one can write ∂K ∂z = K2T G with K2T = ∂T ≈ constant. After substitution in the previous equation and assuming ∂φ ∂z ≈ q0 , we obtain

K2

∂2φ ∂q0 ∂φ 1 + K2T G − K2 − q0 K2T G − ε0 εa E 2 sin(2φ) = ν¯G [3.72] ∂z 2 ∂z ∂z 2

Thermomechanical Effects in Liquid Crystals

159

The general solution of this equation has the form: 

z

q0 (u)du + ϕ(z)

φ(z) =

[3.73]

0

After substituting into equation [3.72], we obtain an equation for ϕ:

K2

 z  ∂2ϕ ∂ϕ 1 2 + K G ε E sin 2 q (u)du + 2ϕ(z) = ν¯G [3.74] − ε 2T 0 a 0 ∂z 2 ∂z 2 0

with ∂ϕ ∂z = 0 at z = 0 and z = d. In this equation, K2 , εa and the integral of q0 are known functions of z as their temperature variations are known. The value of ν¯ is obtained by setting E = Estop in this equation and by searching numerically for the largest value of ν¯ for which this equation has a solution. This method confirmed that ν¯ was constant between Tc and the melting temperature, of the order of 1.2 × 10−7 N/m/K, in good agreement with dynamical measurements. We end this paragraph by noting that π-walls can form and propagate in the sample when the electric field is larger than Estop . Each wall separates two zones in which the director is uniformly oriented and does not rotate. On the other hand, the wall propagation allows the director to rotate everywhere, even above Estop . From this point of view, these walls (or solitons) behave like dislocations in plasticity. Such walls are shown in Figure 3.16 when the temperature is equal to the compensation temperature. In this particular case, the director makes a constant angle with the electric field in each domain and rotates by ±π across each wall. The experiment showed that these walls propagate with a constant velocity that decreases when the electric field increases. → n ϕe → E ϕe + π

π−walls

→ E → n t=0

t = 64 s

→ n ϕe → E

A P t = 180 s

Figure 3.16. π-walls propagating when E/Estop ≈ 1.6. The intensity of the background is constant, which indicates that the director does not rotate out of the walls. T = Tc , d = 10 μm, Estop ≈ 2 × 104 V/m and G = 7 × 104 K/m. The white bar is 50 μmlong. Reproduced from Oswald (2012b) with kind permission of The European Physical Journal

160

Liquid Crystals

To understand this behavior and predict the velocity of a soliton, we must solve d the torque equation close to Tc for the average angle φ¯ = d1 0 φdz by taking into ¯ close to Tc , this equation reads account the electric field. Because sin(2φ) ≈ sin (2φ) by assuming isotropic elasticity: γ1

∂ φ¯ 1 ¯ = −¯ ν G + KΔ⊥ φ¯ − ε0 εa E 2 sin(2φ) ∂t 2

[3.75]

When G = 0, this equation has a solution of type √ φ¯ ≡ φ¯0 = ±2 arctan exp  where ξE =

2

ξE

 x

[3.76]

2K ε 0 εa E 2

is the electric coherence length. This equation describes a √ stationary π-wall of width ξE / 2 perpendicular to the x-axis. Note that here the x-axis can make an arbitrary angle with the electric field from which the angle φ¯ is ¯ ¯ referred. For this static solution, φ(−∞) = 0 and φ(+∞) = ±π. A similar solution still exists when G = 0 and E > Estop but now the wall propagates (Dauxois and Peyrard 2006). In this case, the solution must be searched in ¯ − V t) where V is the wall velocity along the x-axis. In the frame of the the form φ(x wall (X = x − V t), the torque equation rewrites in the from −V γ1

∂ φ¯ 1 ∂ 2 φ¯ ¯ − ε0 εa E 2 sin(2φ) = −¯ νG + K ∂X ∂X 2 2

[3.77]

¯

∂φ Multiplying this equation by ∂X and integrating between −∞ and +∞ gives, by ¯ ∂φ 2 ¯ ¯ noting that ∂X (±∞) = 0 and cos [φ(+∞)] = cos2 [φ(−∞)]:  

2  +∞ ¯ 2  ∞ K ∂ φ¯ ∂φ d ε0 εa E 2 cos2 φ¯ ¯ −¯ ν Gφ + −V γ1 dx = + ∂X 2 ∂X 2 −∞ −∞ dX

¯ ¯ = −¯ ν G[φ(+∞) − φ(−∞)]

[3.78]

This gives the general formula

V =

¯ ¯ ν¯G[φ(+∞) − φ(−∞)]  ¯ 2  +∞ ∂ φ γ1 −∞ ∂X dX

[3.79]

Thermomechanical Effects in Liquid Crystals

161

¯ ¯ For a soliton-like solution such that φ(X = −∞) = φ¯e and φ(X = +∞) = ¯ ¯ φe ± π (π-wall), where φe is the static solution obtained by equilibrating  the  νG thermomechanical torque with the electric torque: φ¯e = − 12 arcsin ε02¯ εa E 2 , this formula gives: V =

±π¯ νG  ¯ 2

 +∞ γ1 −∞

∂φ ∂X

[3.80] dX

At a small temperature gradient, the solution of equation [3.77] can be found in the form φ¯ = φ¯0 + φ¯1 + 2 φ¯2 + . . . where = ε0 νε¯aGE 2 is a small parameter (of the order of 0.2 in experiments) and φ¯0 is given in equation [3.76]. By using equations [3.80] and [3.76] and by limiting the calculation to first order in this gives: π¯ νG V =± 2γ1



K 1 + O( 2 ) ε0 εa E

[3.81]

Note that the sign of the velocity depends on the sense of rotation of the director across the wall. This formula predicts that the wall velocity decreases when the electric field increases, in agreement with observations. We can also compare the velocity of the wall measured experimentally, in the order of 0.8 μm/s when E = 3.2 104 V/m and G = 70 000 K/m, with the value of 0.94 μm/s calculated from the previous formula by taking K = (K1 + K3 )/2 ≈ 3 10−12 N, εa = 4.6, γ1 = 0.13 Pa s and ν = 1.2 × 10−7 N/m/K. These two values are close, which confirms the value of ν¯ given above. 3.4.3.3. Experimental results with the diluted mixtures in the Leslie and mixed geometries The experiments in the Leslie geometry just allow the measurement of the effective Leslie coefficient ν¯ = ν − ξ¯2 q0 . In the compensated mixtures, we suspect that the AZ term −ξ¯2 q0 coming from the macroscopic twist of the director field is negligible against the microscopic Leslie term ν coming from the chirality of the phase and the molecules. The situation is certainly different in diluted mixtures in which these two contributions could be of the same order of magnitude because of the small concentration of chiral molecules. To check this point, we measured the rotation velocity of a TIC in the Leslie and mixed geometries. According to our previous calculations, the rotation velocities must ˆ This is be different in these two geometries at equal external temperature gradient G. for two reasons: the first being that the local temperature gradient is different in the

162

Liquid Crystals

two configurations because of the anisotropy of the thermal conductivity15; and the second being that ω depends on ν and ξ¯2 in the Leslie geometry and on ν, ξ¯2 and ξ¯3 in the mixed geometry. Indeed, we recall that the rotation velocity in the Leslie geometry is given by ˆ ωL = − G

κg ν¯ κ⊥ γ 1

[3.82]

where ν¯ = ν − ξ¯2 q0 and by ˆ ωm = − G

κg ν¯I¯ν + ξ¯2 q0 I¯2 + ξ¯3 q0 I¯3 κ γ1 I¯γ

in the mixed geometry where we defined the new dimensionless integrals ⎧  d sin2 α ⎪ ⎪ ¯ν = 1 ⎪ I dz, ⎪ ⎪ d 0 1 − sin2 α ⎪ ⎪ ⎪ ⎪  d ⎪ ⎪ (q0 − φ,z sin2 α) sin2 α 1 ⎪¯ ⎪ I = dz, ⎪ 2 ⎨ q0 d 0 1 − sin2 α  d ⎪ ⎪ −φ,z sin2 α cos2 α 1 ⎪ ⎪ I¯3 = dz ⎪ ⎪ q0 d 0 ⎪ 1 − sin2 α ⎪ ⎪ ⎪  ⎪ ⎪ 1 d 2 ⎪ ⎪ ⎩ I¯γ = sin α dz. d 0

[3.83]

[3.84]

where α and φ are defined in equation [3.46] and = 1 − κ⊥ /κ is the relative anisotropy of thermal conductivity. These expressions were obtained from equation [3.51] and equations [3.55]–[3.56] by assuming that the material constants do not depend on z and by neglecting the surface viscosity γS . By eliminating ν¯ between equations [3.82] and [3.83], we obtain κ γ 1 ¯ κ⊥ ¯ ξ¯ = I¯2 ξ¯2 + I¯3 ξ¯3 = (I ω − I ν ωL ) ˆ γ m κ g q0 G κ

[3.85]

This quantity gives the order of magnitude of the Akopyan and Zel’dovich coefficients ξi if we assume that they are all equal: ξ = ξ2 = ξ3 = ξ4 . In that case, ¯ I¯2 − I¯3 ). ξ¯2 = −ξ¯3 = ξ/2 which gives ξ = 2ξ/(

15 For this reason, the rotation velocities must be different, even if the AZ contribution is negligible.

Thermomechanical Effects in Liquid Crystals

Constants

K1 (pN) K2 (pN) K3 (pN)

Values at TChI

0.96

0.84

1.37

κg κ

κg κ⊥

163

γ1 (Pa s)

5.1 7.4 0.0079

Table 3.2. Values at the transition temperature of the main physical constants of the liquid crystal CCN-37

1.2x10

-2

− ω (rad/s)

1.0

Leslie geometry mixed geometry typical error

0.8 0.6 0.4

-10

-8

-6 -4 T-TChI (°C)

-2

0

Figure 3.17. Angular velocity as a function of temperature measured in the Leslie and mixed geometries, with the mixture CCN-37 + 3 wt% CC. ΔT = 40◦ C and d = 10.7 μm. Reprinted from Oswald et al. (2017) with the kind permission of Taylor & Francis

To perform this experiment, the LC CCN37 (4α,4’α-propylheptyl-1α, 1 α-bicyclo-hexyl-4β-carbonitrile) was chosen because it dissolves much less of the polymercaptan used for the sliding anchoring than MBBA, or the cyanobiphenyls used before. Because of this property, the anchoring continues to slide much longer (up to 3 days, instead of just a few hours), which greatly facilitates the velocity measurements. Two mixtures were used: the mixture CCN37 + 3 wt% CC (mixture 1) and the mixture CCN37 + 0.166 wt% R811 (mixture 2). These concentrations were chosen so that the equilibrium pitch is the same in the two mixtures at the melting temperature (P = 60 μm corresponding to q0 = 105 m−1 ) and is larger than 4d, knowing that d ≈ 10 μm in these experiments. This last condition was required to avoid the fact that the TIC destabilizes into a banded texture in the mixed samples (Baudry et al. 1996), which would immediately stop the rotation. Figure 3.17 shows the evolution of the rotation velocities in mixture 1 as a function of temperature when ˆ = 104 K/m). As expected, the two textures rotate at different ΔT = 40◦ C (G CC CC velocities at all temperatures, with ωL ≈ −0.012 rad/s and ωm ≈ −0.0071 rad/s at the melting temperature. For comparison, we performed the same measurements R811 R811 with mixture 2 and found ωL ≈ −0.0032 rad/s and ωm ≈ 0 rad/s at the melting ¯ temperature. To calculate ν¯ and ξ defined in equation [3.85], we then measured the three elastic constants K1 , K2 and K3 , the conductivity ratios κg /κ and κg /κ⊥ ,

164

Liquid Crystals

and the viscosity γ1 at the melting temperature. These values are reported in Table δf 3.2. With these values, we numerically solved the differential equations δα = 0 and δf = 0, giving α(z) and φ(z) in the mixed geometry and numerically calculated the δφ ¯ ¯ ¯ integrals given in equation [3.84]: Iν = 0.66 and Iγ = 0.5, I2 = 0.19 and I¯3 = −0.1216. From these measurements, by taking the uncertainties17 into account, for mixture 1 we found: ν¯/q0 (CC) = 11.3 ± 1 fN K−1 , ξ¯ = −2.6 ± 1.3 fN K−1 , and for mixture 2: ν¯/q0 (R811) = 3 ± 0.9 fN K−1 ξ¯ = −1.9 ± 1.4 fN K−1 . These measurements show that the value of the measured Leslie coefficient ν¯ (or of the ratio ν¯/q0 ) depends on the chiral molecules, whereas the value of ξ¯ is the same for the two mixtures within the experimental errors. This is expected because ξ¯ is a nematic-like thermomechanical coefficient that should essentially depend on the LC used, here the CCN-37. It must be emphasized that ν¯ is only 3.8 times smaller in the mixture with R811, than in the mixture with the CC, although the concentration of R811 is 18 times smaller than that of CC. This means that the Leslie effect is 18/3.8 = 4.7 times stronger with the R811 than with the CC at equal concentrations. In other words, the LRP of the R811 is 4.7 times larger than the LRP of the CC, although the HTP of the R811 is 18 times larger than that of the CC. This again shows that the HTP and the LRP do not have the same origin, with the former corresponding to an equilibrium property, while the second is dynamic in nature. These results also show that in diluted mixtures ν and ξ¯2 q0 are of the same order of magnitude. This contrasts with the situation in compensated mixtures, in which the macroscopic term ξ¯2 q0 must be negligible with respect to ν, since it must be of the same order of magnitude as in diluted mixtures. This confirms our suspicion that ν¯ ≈ ν in the compensated mixtures studied before. Last but not least, we see that the experimental value of ξ, obtained from the previous value of ξ¯ by assuming that ξ = ξ2 = ξ3 = ξ4 , is of the order of -15 fN/K,

16 Note that the values of I¯2 and I¯3 are the same at the melting temperature in the two mixtures, as the latter has the same q0 at this temperature. We also mention an error in Oswald et al. (2017). In this paper, the value of Iξ ≡ I¯2 + I¯3 is wrong and must be replaced by 0.07 instead of 0.59. For this reason, the values of ξ¯ given in this paper are underestimated by a factor of 0.59/0.07 ∼ 8. Note that in this paper, we assumed ξ¯ = ξ¯1 = ξ¯2 = ξ¯3 = ξ¯4 . 17 We recall that 1 fN = 10−15 N.

Thermomechanical Effects in Liquid Crystals

165

which is about 103 times smaller in absolute value than the value given by Akopyan and Zel’dovich in their theoretical paper (according to Akopyan and Zel’dovich (1984), ξ ∼ 10−11 N/K). 3.4.4. Drift of cholesteric fingers under homeotropic anchoring It is well known that the helical structure of a cholesteric LC is frustrated in a homeotropic sample, and all the more so, the sample is thin (Brehm et al. 1974). In usual cholesterics, this frustration leads to the complete unwinding of the helix when the sample thickness d is smaller than a critical thickness dc of the order of P . In this case, the director is perpendicular to the glass plates everywhere, forming a homeotropic nematic phase. By contrast, a fingerprint texture forms when d > dc . This texture is composed of cholesteric fingers (CF), which are typically separated from one another by homeotropic nematics when dc < d < 1.2dc and adhere to each other when d > 1.2dc . Finally, isolated fingers can coexist with the homeotropic nematic when d = dc . In this section, we analyze the behavior of these isolated fingers when they are subjected to a temperature gradient. In practice, several types of fingers may form in the samples, referred as to CF1, CF2, CF3 and CF4 in literature (for a review, see Oswald et al. (2000), Oswald and Pieranski (2005) and Smalyukh et al. (2005)). In the following, we focus on the CF1s, which are the only ones in which the director field is continuous everywhere (Press and Arrott 1976), and we show that they can drift and form spirals under the action of the temperature gradient (Oswald and Dequidt 2008c). 3.4.4.1. Theoretical predictions The director field inside of a CF1 is shown in Figure 3.18. It can be numerically calculated by the minimization of the elastic energy. On this topic, we emphasize the important role of the elastic anisotropy, demonstrated for the first time in Lequeux et al. (1989). Indeed, isolated CF1s only form if the transition nematic→fingers is of the first order, which requires the elastic anisotropy to be large enough, according to the diagram shown in Figure 3.19 (Ribière et al. 1991). This is the case in most LCs, including those used in this work, in which K3 is larger than K1 and K2 (see Table 3.2, for instance). Gil and Thiberge were the first (Gil and Thiberge 1997) to propose that a Leslie-like electromechanical effect could be responsible for the of the CF1s when they are subjected to a DC electric field (Gilli and Gil 1994). We know today that this drift is not due to this effect, but to a coupling between the director field and the flows associated with the motion of flexoelectric polarization charges (Tarasov et al. 2003). However, the Gil and Thiberge model applies the thermomechanical case by replacing the electric field by the temperature gradient. In this model, the authors solve the torque equation in a perturbative way, for small values of the field, by using a Melnikov-type analysis. It turns out that this method is complicated and not general

166

Liquid Crystals

as it neglects the temperature variations of the material constants. For this reason, we propose another method (Dequidt 2008; Dequidt et al. 2016), which is completely general and easier to understand. z=d

z y x

z=0

Figure 3.18. Director field inside of a CF1 when q0 > 0 (right-handed cholesteric). The “nail representation” has been used to represent the director field. Tilted molecules are represented by nails, proportional in length to the director projection in the plane of the drawing

K

32

1.75 1.5

st(I) fir

r orde

(II)

ond

sec

2

gers

1.25 1

ers fing

n sitio tran

-o

tra rder

fin

ion nsit

0.75

ition

0.5 0.25 0

(III)

0.5

1

TIC

ns r tra

e

-ord

ond

sec

3

1.5

2

K 12

Figure 3.19. Nature of the frustration transition in the plane of the anisotropy parameters K12 = K1 /K2 and K32 = K3 /K2 . Reprinted from Phys. Rev. A, 44, 8198 (1991), Copyright 1999, American Physical Society

The starting point of the calculation consists of noticing that the total energy of a  n, T ) and the anchoring energy on the glass finger (including the elastic energy f (n, ∇ plates W (n, T )) does not change in time when the finger drifts perpendicularly to its

Thermomechanical Effects in Liquid Crystals

167

long axis at constant velocity V . This condition reads  ∂n δf ∂T dV −h · + ∂t δT ∂t V    ∂n δW ∂T  dS = 0. + (C · ν − hS ) · + ∂t δT ∂t S

 

dF = dt

[3.86]

Replacing h and C · ν − hS in this equation by their expressions given in equations [3.9] and [3.12] yields

 γ1 V

∂T ∂t

∂n ∂t

2



 dV + S

∂n ∂t

2



∂n fT M · dV ∂t V   δf ∂T δW ∂T dV + dS + S δT ∂t V δT ∂t

γS

dS =

If the finger propagates along the y-axis with velocity V , then = −V ∂T ∂y , with V = V (T M ) + V (T )

∂ n ∂t

[3.87] = −V

∂ n ∂y

and

[3.88]

where V (T M ) is the drift velocity due to thermomechanical cross-coupling and V (T ) is the drift velocity due to the variation of the free energy with temperature:

V (T )

1 Iγ



∂n f(T M ) · dV, ∂y V    δf ∂T δW ∂T 1 = − dV + dS , Iγ V δT ∂y S δT ∂y

V (T M ) = −

[3.89] [3.90]

with

 Iγ =

γ1 V

∂n ∂y

2



 dV +

γS S

∂n ∂y

2 dS.

[3.91]

Formula [3.90] shows that, in addition to the thermomechanical terms, the variation in temperature of both the elastic constants and the anchoring energy can

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Liquid Crystals

lead to a drift18. An important point is that this drift is due to the transverse gradient G⊥ = ∂T ∂y . The latter exists because of the anisotropy of the thermal conductivity of the LC. We conclude this section by giving a simplified expression of V (T M ) . It is obtained by neglecting the surface viscosity γS and the anisotropy of thermal conductivity, which is the same as assuming that the temperature gradient is little different from the ˆ κg ez . If G is not too large (linear regime), the spatial variations average gradient G κ of the material constants can also be neglected, and the director n can be replaced by the equilibrium solution n0 (linear regime). Under these assumptions, V (T M ) can be calculated by using the expressions of f(L) and f(AZ) given in equations [3.22] and [3.24]: ¯ ¯ ¯ ˆ κg ν Jν + ξ1 q0 J1 + ξ2 q0 J2 + ξ3 q0 J3 V (T M ) = −d G κ γ1 Jγ where the dimensionless integrals Ji are defined by  ⎧ 1 ∂n0 ⎪ J (n0 × ez ) · = dydz, ⎪ ν ⎪ ⎪ d  ∂y ⎪ ⎪ ⎪ 1 ⎪ ⎪  · n0 ) ez · ∂n0 dydz, ⎪ (∇ J1 = ⎪ ⎪ q0 d   ∂y ⎪ ⎪ ⎨ 1  × n0 ) (n0 × ez ) · ∂n0 dydz, (n0 · ∇ J2 = q d ∂y ⎪ 0 ⎪  ⎪   ∂n0 ⎪ 1 ⎪  × n0 ) × n0 · ⎪ (n0 · ez ) (∇ dydz, ⎪ ⎪ J 3 = q0 d ∂y ⎪ ⎪

2 ⎪  ⎪ ⎪ ∂n0 ⎪ ⎩ Jγ = dydz. ∂y

[3.92]

[3.93]

This formula generalizes a formula already given in Gil and Thiberge (1997); Oswald and Dequidt (2008c), where the thermomechanical coefficients ξ¯i were neglected 19. Several remarks are in order as follows. First, we note that q0 d in integrals J1 , J2 and J3 must be equal to q0 dc to observe isolated fingers. As a result, these integrals only depend on the elastic anisotropy and are independent of the sample thickness.

18 At this level, it must be noted that f and W are defined within additive constants f0 (T ) and W0 (T ) that are the only functions of T . It can be shown that f0 (T ) and W0 (T ) do not contribute to velocity. Idem for the term in ξ¯4 in f(T M ) on the condition that the finger does not contain singularities like disclination lines. 19 We mention that there is an error of sign in equation 3 of Oswald and Dequidt (2008c). On the other hand, the rest of the paper is correct, in particular the numerical value of the constant A calculated from this equation.

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Another important point is that the integrals Jν ,Jγ and Ji (1 = 1 − 3) do not change sign when q0 changes sign. As a result, the AZ contribution to v (T M ) must change sign when q0 changes sign, contrary to the Leslie contribution, the sign of which is given by the sign of ν (independent of q0 ). Finally, we note that V (T ) also changes sign when q0 changes sign, since ∂T ∂y changes sign when q0 changes sign. 3.4.4.2. Experimental results The drift of cholesteric fingers subjected to a temperature gradient has been observed in two very different systems: in a compensated mixture close to the compensation temperature and in a diluted cholesteric mixture close to a cholesteric-smectic A phase transition. In the first case, we will show that the drift is due to the Leslie thermomechanical coupling, while it is likely due to the divergence of the twist and bend constants in the second case. 3.4.4.2.1. Drift close to a compensation temperature The compensated mixtures are particularly well suited to observe the drift of CF1s for three reasons. First, the pitch changes with temperature. As a result, it is easy to fulfill the condition d = dc for a large range of thicknesses by just changing the sample temperature. Second, the Leslie coefficient is large, which is essential to observe a drift. Third, the experiment can be performed at two temperatures T1 < Tc and T2 > Tc , on either side of the compensation temperature, which allows the testing of the role of the sign of q0 on the drift velocity. In practice, short segments of CF1 form in homeotropic samples when d = dc . These segments have two necessarily different ends because of the absence of mirror symmetry in a cholesteric. One of them is rounded, marked with the minus sign on the pictures, while the other is pointed, marked with the plus sign20. Under the action of a temperature gradient, these segments drift perpendicularly to their axes, as shown in Figure 3.20. In this figure, the x-axis is oriented from the pointed tip to the rounded tip. With this choice, the experiments with the mixture 8OCB + 50 wt% CC show that V is positive (negative) when ΔT > 0 (ΔT < 0) at the two temperatures T1 and T2 , independent of the sign of q0 . The velocity is also found proportional to the temperature gradient (linear regime) and proportional to d, as shown in Figure 3.21. These results – particularly the fact that V does not change sign at Tc – show that the drift is mainly due to the Leslie thermomechanical coupling. This interpretation was confirmed by numerically calculating the constant  +∞  d Jν 1 −∞ dy 0 (ny nx,y − nx ny,y )dz A= =  +∞  d Jγ d dy (n2x,y + n2y,y + n2z,y )dz −∞

[3.94]

0

20 In the rounded tip, the twist is the same sign as q0 everywhere, whereas in the pointed tip, there is a small region in which the twist is of the opposite sign to q0 (Ribière and Oswald 1990).

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Liquid Crystals

from which a value of ν was found by taking V ≈ −Aν κ ˆ g Gd/γ1 κ : ν ≈ 1.3 × 10−7 kg K−1 s−2 at Tc . This value is very close to the values previously given, confirming that the drift is mainly due to the Leslie thermomechanical coupling. T = T2 > Tc

T = T1 < Tc

+

+ ΔT > 0 V>0

V>0 z y x

−

− q0 > 0

q0 < 0

Figure 3.20. Drift direction of CF1 segments observed in homeotropic samples of the mixture 8OCB + 50 wt% CC on both sides of the compensation temperature. Although q0 changes sign, the drift direction is always the same. Drawing inspired by Oswald and Dequidt (2008c)

T (°C) 54

56

58

60

v / dΔT (10−4 s−1°C−1)

5

v (μm s−1)

0.10

0.00

−0.10

Tc

4 3 2 1

(a)

(b) 0

−40

−20

0 ΔT (°C)

20

40

−0.4

−0.2

0.0 0.2 q0 (μm−1)

0.4

Figure 3.21. (a) Drift velocity as a function of the temperature difference ΔT , measured in a sample of thickness d = 10 μm at temperature T2 > Tc . (b) Ratio of the drift velocity over the temperature difference times the sample thickness as a function of the equilibrium twist or the temperature. , d = 10 μm; , d = 25 μm; , d = 40 μm. Reprinted from Oswald and Dequidt (2008c), Copyright 2008, American Physical Society

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0.20

4

0.15

3

0.10

2

0.05

1

0.00

V2 (Vrms)

vCF1 (μm/s)

It should also be noted that isolated CF1s can be stabilized by the application of an AC electric field, when the sample temperature is larger than T2 . In this case, the pitch becomes smaller than the thickness and the CF1s become unstable, growing from their ends while undulating until they invade the whole sample. The only way to stabilize the fingers again is to impose an electric field to favor the homeotropic nematic with respect to the fingers. The equilibrium is reached for a voltage denoted by V2 in literature. Of course, the larger the temperature above T2 , the larger V2 is and the thinner the CF1s are, whose width always remains close to 3.5P (Ribière and Oswald 1990). It is then possible to impose a temperature gradient to the fingers and to measure their drift velocity. Such measurements were performed in a 15-μm-thick sample of the mixture 8OCB + 50 wt% CC. The results, reported in Figure 3.22, show that the drift velocity strongly decreases when the temperature and the electric fields increase. This effect is expected since A must be roughly proportional to the width of the finger according to equation [3.94], a quantity close to 3.5P that strongly decreases when the temperature increases.

0 62

63

64

65

T (°C)

Figure 3.22. Drift velocity of the CF1s and coexistence voltage V2 as a function of temperature above the compensation temperature. Mixture 8OCB + 50 wt% CC, d = 15 μm, T2 = 61◦ C and ΔT = 30◦ C. Reprinted from Oswald (2008) with the permission of Taylor & Francis

We conclude this section by noting that the isolated segments of CF1 always destabilize in time by forming spirals. This evolution is inevitable (as shown in Pirkl and Oswald (1996); Oswald et al. (2000)) and is akin to what is observed with wave fronts in weakly excitable two-dimensional media (Mikhailov et al. 1994). Figure 3.23 shows two single and two triple spirals observed above and below the compensation temperature. The reader will note that these two types of spirals do not rotate in the same sense when T > Tc and ΔT > 0. This is due to the fact that the rounded tips lie in the center of the single spirals and on the outer border of the triple spirals, showing that the fingers are oriented in opposite directions in these two types of spirals. These images also show that far from their center, the spirals tend to

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Archimedian spirals of an equation in polar coordinates ρ(θ, t) =

P (θ + ωs t) 2π

[3.95]

where P is the pitch of the spiral and ωs is its angular rotation velocity related to the drift velocity by the relation V =

ωs P 2π

[3.96]

G>0

G>0

+

+

-

(a)

(b) G0

+

+

-

(c)

(d)

Figure 3.23. (a-b) Two single spirals rotating in the same direction observed in a 25 μm-thick sample at T = T1 = 56.4◦ C (a) and T = T2 = 58.9◦ C (b) when ΔT = 47.2◦ C; (c-d) Two triple spirals rotating in opposite directions at T = T2 = 60.8◦ C under a temperature difference ΔT = 36.8◦ C (c) and ΔT = −43.6◦ C in a 10-μm-thick sample. Mixture 8OCB + 50 wt% CC. Reprinted from Oswald and Dequidt (2008c), Copyright 2008, American Physical Society

This equation is general and gives the product ωs P. On the other hand, ωs (and, thus, P) depends on the anchoring conditions of the spiral at the center, which may change from one spiral to another. 3.4.4.2.2. Drift close to a cholesteric-smectic A phase transition The same experiment was conducted in the diluted mixture 8OCB + 1.14 wt% R811 (Oswald 2008). In this mixture, the Leslie coefficient is much smaller than in

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the compensated mixture because of the small concentration of chiral molecules and the Akopyan and Zel’dovich terms entering into the expression [3.92] of the drift velocity, which are not larger than the Leslie term according to our previous estimate of the ξ¯i . As a result, these fingers should not drift in a visible way. This is indeed what we have found experimentally, except very close to the smectic A phase transition where the CF1 spontaneously formed spirals, as shown in Figure 3.24. This result is very surprising as we know that the viscosity γ1 diverges at the smectic A-to-nematic phase transition (de Gennes and Prost 1995; Oswald and Pieranski 2006). One possibility to explain this observation with the “classical” thermomechanical model would be that the Leslie coefficient diverges faster than γ1 . To test this idea, measurements in the Leslie geometry were performed with the mixture 8CB + 1 wt% R811. It was found that the rotation velocity continuously decreases when approaching the smectic A phase, which clearly indicates that ν¯ does not diverge but remains approximately constant (Oswald 2016). These results show that this drift is not due to the classical thermomechanical coupling, but rather to the other mechanism described above, coming from the temperature variations of the free energy. The corresponding drift velocity V (T ) is given in equation [3.90]. This expression simplifies if the anchoring energy is very large and if the glass n ∂T conductivity is much larger than that of the LC. In these limits, ∂ ∂y and ∂y both tend to 0 on the glass plates, leading to  δf ∂T δT ∂y dydz (T ) = −   2 [3.97] V n dydz γ1 ∂ ∂y

100 μm

Figure 3.24. Spiral of CF1 observed between crossed polarizers close to the smectic phase. Mixture 8OCB + 1.14 wt% R811. ΔT = −40◦ C and d = 15 μm. In this experiment, an electric field was applied to stabilize the finger (V = 1.98 Vrms and f = 1 kHz). Reproduced from Oswald (2008) with the kind permission of Taylor & Francis

This formula shows that the drift velocity is proportional to the horizontal temperature gradient. By denoting the temperature at the bottom of the glass plate by T0 , it can be shown by using the heat equation, and by setting

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Liquid Crystals

ˆ κg [z + g(y, z)d] , that T (y, z) = T0 + G κ the equation

∂T ∂y

ˆ κg d ∂g , where g is the solution of =G κ ∂y



   · ∇g  + κa n0 · ∇g  + n0 · ez n0 = 0 ∇ κ⊥ d

[3.98]

with g = 0 at z = 0 and z = d. This shows that the horizontal gradient is proportional ˆ and is different from 0 as long as κa = 0. Due to this temperature gradient, one to G side of the finger is colder than the other21. For this reason, the finger tends to drift toward its hotter side to minimize its elastic energy. To some extent, one can say that the finger surfs on a heat wave that it creates itself. From the previous expression, one can estimate the order of magnitude of the drift velocity by orienting the finger in the same way as in Figure 3.20. This gives, within a numerical factor impossible to estimate by hand (Dequidt et al. 2016): ˆ κg ∂K κa q0 d , V (T ) ≈ −G κ ∂T κ⊥ γ1

[3.99]

where K is a combination of the elastic constants that diverges at the transition. This mechanism could explain the formation of spirals close to the transition. However, numerical simulations would be important to confirm this interpretation. We end this section by mentioning the existence of another thermomechanical effect observed during the growth of cholesteric fingers. As we mentioned before, CF1s grow from their ends while undulating when the confinement ratio is too large, typically more than 1, and the applied voltage (if any) is less than V2 . We have observed that during the growth, the ends of the fingers move at constant speed in a straight line when no temperature gradient is applied. By contrast, they follow circular trajectories when a temperature gradient is applied, the radius of which decreases when the temperature gradient increases. An example is shown in Figure 3.25. This phenomenon is observed at all temperature, but has not yet been explained. 3.5. The thermohydrodynamical effect In our experiments, we did not observe visible flows and, for this reason, we neglected them in our previous analyses. This is fully justified in the experiments on the Leslie geometry, when the director field is not distorted in the plane of the sample because, in that case, the solution to the full problem is v = 0 everywhere, when

21 This is compatible with the absence of mirror symmetry in cholesterics.

Thermomechanical Effects in Liquid Crystals

175

v = 0 on the boundaries of the LC domain. By contrast, flows should be present in the Leslie geometry close to the disclination lines, even if we have not yet detected them. However, the fact that we measured the same director rotation velocity in the homogeneous regions and close to the disclination lines clearly indicates that hydrodynamic flows interfere very little with the thermomechanical effect. In the following, we propose a method to measure the TH Leslie coefficient μ in a compensated mixture. We will then return to the mixed geometry, which has been used to estimate the TH Akopyan and Zel’dovich terms ξi (i = 5 − 12). Two experiments will be described, that both led to surprisingly large values for these coefficients. (a)

(b)

(c)

100 μm

Figure 3.25. Growth of a CF1 in the homeotropic nematic. The dashed lines show the trajectory of the rounded tip when ΔT = 0 (a), ΔT = 40◦ C (b) and ΔT = −40◦ C. The polarizer and analyzer are at 45◦ relative to each other. Mixture 8OCB + 1.14 wt% R811; d = 10 μm, T = 50◦ C, V = 2 Vrms < V2 = 4 Vrms. Reproduced from Oswald (2008) with the kind permission of Taylor & Francis

3.5.1. A proposal for measuring the TH Leslie coefficient μ: theoretical prediction We have seen how to measure the TM Leslie coefficient ν in a compensated cholesteric at the compensation temperature. A question that arises is whether it is possible to measure the associated TH term μ. We already mentioned that this term does not play any role in the Leslie geometry when the director field is not distorted in the plane of the sample. In this section, we show that the situation is different around a +1 disclination line. To simplify the problem, we assume that the director cannot rotate on the plates and in the bulk and is either oriented in a circular configuration (Figure 3.26(a)) or in a radial configuration (Figure 3.26(b)). Experimentally the circular configuration can be obtained by treating one plate with the polymercaptan to obtain a sliding planar anchoring and the other with a polyimide for planar alignment. The latter is then briefly pressed against a piece of velvet fabric kept in continuous rotation. In this way, the surface easy axis has the desired +1 circular geometry, which leads to the configuration described in Figure 3.26(a), after the sample has been filled with the LC. It is then enough to let

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Liquid Crystals

the sample rest for 1 day to memorize the circular anchoring on the plate treated with the polymercaptan and block the rotation of the director on this plate. This method cannot be used to prepare the radial configuration. In that case, a photoalignment technique must be used, such as the one described in Slussarenko et al. (2011), to which we refer for more details. z

z (a)

(b)

ez eθ r z

er

y

y

θ

x

x

Figure 3.26. Circular (a) and radial (b) configurations and system of polar coordinates (r, θ, z)

To calculate the flow around the +1 disclination line, we solved in polar  · σ = 0, where coordinates (r, θ, z) (see Figure 3.26(a)) the momentum equation ∇ σ ≡ −P I + σ (eq) + σ (neq) . In these equations, P is the pressure given by the  · v = 0. By assuming that the director field is imposed incompressibility condition ∇ by the anchoring conditions, we have n = (0, 1, 0) in the circular geometry and  · σ (eq) = 0 and the momentum n = (1, 0, 0) in the radial geometry. As a result, ∇ (v) (L)   equation becomes ∇P = ∇ · (σ + σ ), where σ (v) is the viscous stress tensor given in equation [3.21] and σ (L) the Leslie stress tensor given in equation [3.22]. To further simplify the problem, we assume that q0 = 0 (cholesteric at the compensation temperature) and that the viscosity, TM and TH coefficients αi , ν and μ are constant within the sample thickness. ∂ Under these assumptions, and remembering that ∂θ = 0 because of the axial symmetry, the momentum equation reads in the circular geometry as:



2 ⎧ ∂P vr ∂ vr 2 ∂vr ∂ 2 vz ∂ 2 vr ⎪ ⎪ − η 2, 2 + + = η + a ⎪ 2 2 ⎪ ∂r ∂r r ∂r ∂z ∂r∂z r ⎪ ⎪ ⎪ ⎪ 

⎨ 2 2 ∂ vθ ∂ vθ 1 ∂vθ μG vθ 0 = ηb 2 + η¯ − 2 + ηb 2 + , ⎪ ∂r r ∂r r ∂z r ⎪ ⎪ ⎪

2  ⎪ ⎪ ⎪ ∂ vz 1 ∂vz ∂ 2 vr 1 ∂vr ∂ 2 vz ⎪ ∂P = η ⎩ ; + + + + a ∂z ∂r2 r ∂r ∂r∂z r ∂z ∂z 2

[3.100]

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177

and in the radial geometry as: 

2 ⎧ ∂P vr ∂ 2 vr 1 ∂vr ∂ 2 vz ∂ vr ⎪ ⎪ − 2η + + η ¯ , = η + η a b ⎪ ⎪ ∂r ∂r2 r ∂r r2 ∂r∂z ∂z 2 ⎪ ⎪ ⎪ ⎪ 

⎨ ∂ 2 vθ ∂ 2 vθ 1 ∂vθ μG vθ 0 = ηc 2 + η¯ − 2 + ηa 2 − , ⎪ ∂r r ∂r r ∂z r ⎪ ⎪ ⎪

2 

2  ⎪ ⎪ ⎪ ∂ vz ∂ 2 vz 1 ∂vz ∂P ∂ vr 1 ∂vr ⎪ ⎩ + . = ηc + η ¯ + + 2η a ∂z ∂r2 r ∂r ∂r∂z r ∂z ∂z 2

[3.101]

In these expressions, ηa = 12 α4 , ηb = 12 (α3 +α4 +α6 ) and ηc = 12 (−α2 +α4 +α5 ) are the three Miesowicz viscosities (de Gennes and Prost 1995; Oswald and Pieranski 2005) and we set η = α1 − γ1 + 2(ηc + ηb − ηa ) and η¯ = 12 (ηc + ηb − γ1 ). As for the incompressibility condition, it reduces to: ∂vr ∂vz vr + + =0 ∂r ∂z r

[3.102]

An important point to note here is that only the TH Leslie coefficient μ enters into the problem. These equations immediately show that the solution is of the type vr = vz = 0 with P = constant. As for the orthoradial component vθ (r, z), it is given by solving the second equation in [3.100] for the circular configuration and in [3.101] for the radial configuration with the boundary conditions vθ (rc , z) = vθ (R, z) = vθ (r, 0) = vθ (r, d) = 0. Here, rc denotes the core radius of the disclination, R is its outer radius and d is the sample thickness. These equations can be solved with Mathematica. The resolution shows that a circular flow develops locally around the core of the disclination line (Figure 3.27). The velocity is proportional to μ and the temperature gradient G and passes through a maximum proportional to d at z = d/2 and r ≈ d/2. The senses of rotation are opposite in the circular and radial configurations. Finally, the velocity decreases at long distance (r > d) as vθ (r, z) = −

μG z(z − d) 2ηb r

[3.103]

in the circular geometry and as vθ (r, z) =

μG z(z − d) 2ηa r

in the radial geometry.

[3.104]

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Liquid Crystals

z (μm)

40

3 2

20

1

10 0

z (μm)

(a)

30

0

50

100

150

r (μm)

200

250

0

40

0

30

- 0.5

20

- 1.0

10 0

(b) 0

50

100

4

0

(c)

- 1.5

250

(d)

vθ (μm/s)

vθ (μm/s)

200

-0.5

3

-1.0

2

-1.5

1

0

150

r (μm)

100

200 r (μm)

300

400

- 2.0

0

100

200 r (μm)

300

400

Figure 3.27. (a, b) vθ -amplitude in the radial plane (r, z) and (c,d) vθ -profile along the r-axis at z = d/2. The blue curves have been calculated numerically and the yellow curves represent the asymptotic solutions given in equations [3.103] and [3.104]. (a and c) Circular geometry and (b and d) radial geometry

The orders of magnitude of the velocity fields in both configurations are shown in Figure 3.27. Simulations were performed by taking d = 40 μm, rc = 0.05 μm, R = 400 μm, G = 7 104 K/m (ΔT = 40◦ C), ηa = 0.042 Pa s, ηb = 0.024 Pa s, ηc = 0.1 Pa s, α1 = 0, γ1 = 0.075 Pa s and by assuming that μ ≈ ν = 2 10−7 kg s−2 K−1 . With these values, |vθMax | ≈ 3.4 μm/s in the circular geometry and |vθMax | ≈ 1.8 μm/s in the radial configuration. In practice, such flow should be experimentally detectable. 3.5.2. About the measurement of the TH Akopyan and Zel’dovich coefficients 3.5.2.1. Principle of the measurement The basic idea for measuring the order of magnitude of the TH coefficients ξi (i = 5 − 12) in a nematic phase was already given in the seminal paper by Akopyan and Zel’dovich (1984). Indeed, these authors note from the beginning of their paper that a flow should be induced by a temperature gradient in a mixed sample treated for planar unidirectional anchoring on one plate and homeotropic anchoring on the

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179

opposite plate (Figure 3.28(a)). The order of magnitude of the velocity v can be easily  · σ = 0. This equation considerably found by solving the momentum equation ∇ simplifies if one assumes that (1) n = {0, sin α(z), cos α(z)} in the (x, y, z) reference 22; (2) the temperature gradient is constant, equal to G 23 and frame, with α(z) = πz 2d (3) the material constants do not change with temperature in the sample thickness. By using the simplified expression of σ (AZ) given in equation [3.31], and by further assuming that the nematic phase behaves viscously as an isotropic liquid of viscosity η, we obtain:

η

 π 2  πz  ∂P ∂ 2 vy − ζ G sin = ∂z 2 2d d ∂y

[3.105]

z

z (a)

(b)

ez eθ r

y

z

y

er

θ

x

x

Figure 3.28. Mixed planar-homeotropic geometry. In (a), the anchoring is planar unidirectional on the top plate and in (b), it is planar circular

The reader will note that the TM term in ξ and the TH terms in ζ  and ζ” do not enter into this equation when α varies linearly with z. If the nematic phase is free to flow along the x-axis, then P = constant and the solution reads vy (z) = −

 πz  ζG sin 4η d

[3.106]

22 This is the case in isotropic elasticity on the condition that the flow does not destabilize the director field. 23 This consists of neglecting the thermal conductivity anisotropy.

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Liquid Crystals

As expected, the velocity is maximum in the middle of the sample and vanishes on the two glass plates. One will note that the velocity at z = d/2 is independent of the sample thickness because the TH and viscous stresses both vary as 1/d. This calculation assumes that the LC can flow freely along the x-axis, which is difficult to realize in practice. Indeed, the flow disappears as long as the LC is in contact with an obstacle (for instance, one of the spacers used to control the sample thickness). In that case, the TH stress is equilibrated by a pressure gradient and no flow occurs. To avoid this difficulty, Akopyan and coworkers proposed to perform the experiment in a sample treated for planar circular anchoring on one plate and homeotropic anchoring on the opposite plate (Akopyan et al. 1999, 2001, 2004). This geometry is represented in Figure 3.28(b). 3.5.2.2. Theoretical predictions in the mixed planar-circular/homeotropic geometry Because of the axial symmetry, calculations can be done in cylindrical ∂ coordinates by taking ∂θ = 0. To write the equations, we used Mathematica and assumed that (1) the director field is given by n = [0, sin α(z), cos α(z)], with α(z) = πz er , eθ , ez ); (2) G is constant; (3) the material 2d in the referential frame ( constants are independent of temperature and (4) the AZ stress tensor is given by the simplified expression [3.31]. However, we used the full expression of the viscous stress tensor given in equation [3.21], in which we made α1 = 0 because this term is usually very small in nematics (de Gennes and Prost 1995; Oswald and Pieranski 2005). In spite of the approximations, the equations are very long and will not be given here. We will just focus on the equation in vθ far from the core of the disclination (typically for r > d). In this region, vr and vz tend to 0, and so can be ∂ 2 vθ θ neglected as well as the terms in vrθ2 , ∂v ∂r and ∂r 2 , which gives

 πz  ∂ 2 vθ  πz  π ∂vθ + (ηb − ηc ) sin ηb + ηc + (ηc − ηb ) cos d ∂z 2 d d ∂z 2  πz  π Gζ − sin =0 2d2 d

[3.107]

The solution to this equation reads Gζ vθ (r, z) = ηb − ηc



πz − arctan 2d

√

   ηb tan πz √ 2d ηc

[3.108]

As expected, it can be checked that in the limit ηb → ηc = η, this equation gives back the result of equation [3.106]. This equation shows that far from the core of the

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181

disclination (in practice at distance r > d), a circular flow develops with a velocity in the center of the sample at z = d/2, given by 

  π ηb Gζ − arctan ηb − ηc 4 ηc

vθ (r, d/2) =

[3.109]

The reader will note that, as under unidirectional planar anchoring, the constants ξ, ζ  and ζ” do not enter into the solution far from the core of the disclination. This is not the case close to the core where the two terms in ζ  and ζ” are important. On the other hand, the TM term in ξ vanishes everywhere if the angle between the director and the z-axis varies linearly with z, as assumed in this calculation. The numerical solution to the full equations is shown in Figure 3.29 and is compared to the approximate solution given in equation [3.109]. In this simulation, we chose the same values for the material constants as in Figure 3.27, and we took ζ = ζ  = ζ” = 10−12 N K−1 . (v2r + v2z)1/2 (μm/s)

vr and vz contributions

40

z (μm)

0.03

(a)

30

0.02

20

0.01

10 0

0 0

50

100

r (μm)

150

200

vθ (μm/s) 0

vθ contribution

40

(b)

30

z (μm)

250

-0.1 -0.2

20

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Figure 3.29. Velocity field in the planar-circular/homeotropic geometry. (a) (vr2 +vz2 )1/2 amplitude and (b) vθ -amplitude in the (r, z)-plane. The arrows represent the projection of velocity field in the (r, z)-plane. (c) vθ as a function of r at z = d/2. The yellow curves represent the asymptotic solutions given in equation [3.109]

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Figure 3.30. Orthoradial component of the velocity vθ as a function of r at z = d/2 = 20 μm (a–e) and as a function of z at d = 100 μm when θm = 1 rad (f–j). (a and f) ξ = 0 and ζ = ζ  = ζ” = 0; (b and g) ζ = 0 and ξ = ζ  = ζ” = 0; (c and h) ζ  = 0 and ξ = ζ = ζ” = 0; (d and i) ζ” = 0 and ξ = ζ = ζ  = 0; (e, j), ξ = ζ = ζ  = ζ” = 0

As expected, these calculations confirm that a circular flow develops at distance r > d of the core of the disclination. An important point is that the velocity in the

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middle of the sample is proportional to G and ζ and is independent of r. The latter result contrasts with the numerical simulation of Akopyan et al. (2004), who found that the maximal velocity along z, vθmax , was proportional to r. We think that this result is wrong as we found a similar result to ours by numerically solving their own equations with Mathematica. We note that another variant of the experiment has been proposed in Akopyan et al. (2001). It consists of destabilizing, with an AC electric field, a sample treated for the circular configuration. If the applied voltage V is larger than the Fréedericksz critical voltage Vc , the director field distorts and reads in polar coordinates  in the

c form n = {0, sin α(z), cos α(z)} with α(z) = θm sin(πz/d) and θm ≈ 2 V V−V , c provided that V < 1.3 Vc . The same calculations as before shows that a circular flow develops around the core of the disclination. At distance r > d, the only non-zero component of the velocity is vθ . This component depends on all of the TH coefficients ζ, ζ  and ζ”, but also on the TM coefficient ξ, as shown in Figure 3.30. In this figure, we plotted vθ (r, d/2) as a function of r and vθ (r, z) calculated at r = 100 μm as a function of z for each contribution ξ, ζ, ζ  and ζ” and for the total contribution. In this calculation, we assumed that ξ = ζ = ζ  = ζ” = 10−12 N/K24 and we took θm = 0.8 rad and the same values as before for the other parameters. These calculations show that all of the terms contribute in a similar way and lead to typical velocities of the order of 0.15 − 0.3 μm/s.

3.5.2.3. Experimental results The experiment was performed by Akopyan and coworkers in the two mixed geometries depicted in Figure 3.29. In the two cases, the authors observe a flow with a maximal velocity proportional to the temperature gradient and of the order of 1 μm/s for G ≈ 1.4 105 K/m (Akopyan et al. 1997, 1999). This is in favor of a thermohydrodynamical effect with coefficients ζ, ζ  or ζ  of the order of 10−12 N/K, which is the value we have taken in our previous numerical simulations. However, the authors observe in the circular geometry of Figure 3.28 that the velocity increases linearly with the radius (see Figure 4 in Akopyan et al. (2001)), which is incompatible with our own numerical simulations. This is a real problem for which we have no solution to propose for the moment. In addition, these results suggest that the thermohydrodynamical coefficients ζ, ζ  or ζ  are much larger than the thermomechanical coefficient ξ in the order of 10−15 N/K, according to our experiments. This is surprising and for this we do not have any explanation. Note that an even larger value was given by Lavrentovich and Nastishin (1987) to explain the flows observed in flattened droplets deposited on a bath of glycerol and submitted to

24 Note that this value of ξ is much larger than that we have found experimentally (Oswald et al. 2017) in the order of 10−15 N/K.

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a temperature gradient25. The problem in this experiment is the presence of the Marangoni effects that can distort the measurements. The same problem occurs in the experiment by Choi and Takezoe (2016), showing the formation at the place where a laser beam is focused on a localized circular flow in a homeotropically oriented nematic layer with a free surface. In the paper by Choi and Takezoe, this effect is interpreted to be due to a Marangoni effect, but an alternative explanation was proposed by Zakharov and Maslennikov (2019) in terms of the thermohydrodynamical effect, provided the ζ-coefficients are large enough. 3.6. Conclusions and perspectives Until recently, it was thought that the main experimental evidence of the Leslie thermomechanical effect was the rotation of cholesteric droplets subjected to a temperature gradient, the so-called Lehmann effect. This standpoint is mistaken in our opinion, because other mechanisms exist – more efficient than the Leslie effect – to explain the Lehmann effect (for a review, see (Oswald et al. 2019a)). This is the reason why in this chapter we distinguished the Lehmann effect discovered experimentally by Lehmann in 1900, from the thermomechanical effect discovered theoretically by Leslie in 1968. If the latter does not fully explain the Lehmann effect, it nonetheless exists and is clearly measurable in cholesterics. We have shown this result by describing the static Éber and Jánossy experiment and the dynamic experiments on the rotation of the TICs under sliding anchoring. An important point to note is that the Leslie effect still exists at the compensation temperature of a cholesteric phase, at which the helix is unwound. The situation is more complex for the thermomechanical effect described by Akopyan and Zel’dovich in distorted nematics and cholesterics. These terms are associated with macroscopic distortions of the director field and are more difficult to measure. The comparative experiments with two different types of TICs have nonetheless proven their existence and have allowed us to estimate their order of magnitude. By contrast, the situation is much more confusing concerning the thermohydrodynamical effects predicted in both cholesterics and deformed nematics. At the moment, there is no experimental evidence of the Leslie thermohydrodynamical effect in cholesterics, but we proposed an experiment to evidence it at the compensation temperature. Concerning the thermohydrodynamical effect in a deformed nematic, only two experiments have been conducted, the first by Lavrentovish and Nastishin and the second by Akopyan and collaborators. In the

25 A mixed geometry is observed inside of the drops because the anchoring is planar on the glycerol and homeotropic at the interface with air.

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former, a droplet is deposited at the surface of a bath of glycerol. The problem in this experiment is the presence of Marangoni effects that can distort the measurements. For this reason, we prefer the Akopyan experiment realized between two glass plates treated for homeotropic and circular planar anchoring, respectively. In this case, there are no Marangoni effects, which greatly simplifies the predictions. In this experiment, circular flows are indeed detected by Akopyan et al. but the radial profile of the orthoradial velocity measured experimentally does not match with theory, according to our calculations. For this reason, we have some reservations about the value of the thermohydrodynamical coefficients found by these authors. These doubts are reinforced by the fact that this order of magnitude is not compatible with the one of the thermomechanical coefficients we have found experimentally – if we agree that all of the coefficients must be of the same order of magnitude, which is perhaps not the case. An alternative explanation for this disagreement could be that the thermohydrodynamical flows – which we neglected in our calculations – couple and slow down the rotation of the director in the mixed geometry under sliding planar anchoring, leading us to strongly underestimate the thermomechanical coefficients. For these reasons, we think that the experiments in mixed geometry need to be reproduced for confirmation in the future, in particular in the circular configuration when the director is fixed or free to rotate. In the latter case, it would also be important to numerically solve the full equations to determine if the circular flow disturbs the rotation of the director. It is clear that knowing the order of magnitude of the Akopyan and Zel’dovich coefficients would be important in the future to determine whether the thermohydrodynamical effect could be potentially useful for applications in microfluidics, as recently proposed by Zakharov and collaborators (Zakharov and ´ Vakulenko 2009; Zakharov et al. 2010; Sliwa and Zakharov 2020). The same issue arises in nonlinear optics where the thermohydrodynamical effect could induce flows and director field distortions, leading to a nonlinear optical response of the LC when the samples are illuminated by laser beams. This was shown by Akopyan and collaborators (Akopyan et al. 2000; Hakobyan et al. 2012; Hakobyan 2015), Poursamad et al. (2010, 2015), and Krimer and Residori (2007), who predicted molecular reorientation at intensities substantially lower than that needed for optical Fréedericksz transition in a dye-doped nematic sample. 3.7. References Akopyan, R.S. and Zel’dovich, B.Y. (1984). Thermomechanical effects in deformed nematics. Sov. Phys. JETP, 60, 953–958. Akopyan, R.S., Alaverdyan, R.B., Santrosyan, E.A., Chilingaryan, Y.S. (1997). Thermomechanical effect in a hybrid-oriented nematic liquid crystal. Tech. Phys. Lett., 23, 690–691.

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Physics of the Dowser Texture 1

2

Pawel P IERANSKI1 and Maria Helena G ODINHO2

Laboratoire de Physique des Solides, Paris-Saclay University, Orsay, France i3N/CENIMAT, NOVA School of Science and Technology, NOVA University Lisbon, Portugal

4.1. Introduction 4.1.1. Disclinations and monopoles In terms of the classification of topological defects proposed by de Gennes (1972) (see Figure 4.1), monopoles and disclinations in nematics are point (dimension δ = 0) and linear singularities (δ = 1), respectively, of the quadrupolar nematic order parameter Qαβ . 4.1.1.1. Disclinations are ubiquitous Disclinations, which appear in nematic droplets as floating threads (see Figure 4.2), are so ubiquitous and so characteristic (fingerprint-like) that the name “mésophase nématique”, referring to them through the Greek root νημα, was coined and used for the first time by Friedel (1922). In the case of the 5CB (4-cyano-4’-pentylbiphenyl) droplet shown in Figure 4.2, disclinations were generated by gentle stirring with a pipette tip. The series of four images taken at intervals of a few seconds shows that disclinations form loops (such as the ∞-like one marked with an arrow) that can split, shrink and finally collapse. The same behavior is observed with disclinations generated by a rapid isotropicLiquid Crystals, coordinated by Pawel P IERANSKI, Maria Helena G ODINHO. © ISTE Ltd 2021.

Liquid Crystals: New Perspectives, First Edition. Pawel Pieranski and Maria Helena Godinho. © ISTE Ltd 2021. Published by ISTE Ltd and John Wiley & Sons, Inc.

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nematic quench. This natural tendency that disclination loops have of collapsing, can be opposed by at least three methods, as follows (see Figure 4.3):

Figure 4.1. Classification of topological defects in systems with an order parameter, proposed by de Gennes (1972). d is the dimension of space in which the system exists, n is the number of components of the order parameter and δ is the dimension of the defects, which is 0 for points (monopoles) and 1 for lines (dislocations, disclinations or vortices). Adapted from de Gennes (1972)

a

b

c

d Figure 4.2. Spontaneous shrinking and collapse of disclination loops in a nematic droplet

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ˇ 1) they can be threaded on solid inclusions or fibers (Copar et al. 2016), where the repulsive interaction with surfaces opposes their collapse (see Figure 4.3(a)); 2) they can be stabilized or set into motion by the action of magnetic fields in twisted nematic cells (see Figure 4.3(b)); 3) they can be stabilized by the action of special anchoring patterns (see Figure 4.3(c)).














 



 Figure 4.3. Stable systems of disclinations. (a) Disclination loop threaded on a ˇ polymeric fiber (Copar et al. 2016; Cabeça et al. 2019). (b) Disclinations in a twisted nematic cell stabilized by a quadrupolar magnetic field (Srinivasan 2020). (c) Web of disclinations generated by patterned anchoring conditions (Wang et al. 2017) (courtesy of Wang and Yokoyama)

4.1.1.2. Monopoles are scarce In contradistinction to disclinations, monopoles are scarce (Hindmarsh 1995). They are absent in the images in Figure 4.2, which means that they were not generated like the disclinations by stirring. The occurrence of monopoles requires special geometries of the limit surfaces and of the anchoring conditions on them: 1) monopoles occur as companions of inclusions with homeotropic anchoring conditions (Poulin et al. 1997; Musˇeviˇc et al. 2006) (see Figure 4.4(a));

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2) they can be generated by the isotropic–nematic quench in capillaries with homeotropic anchoring conditions (Meyer 1972), (Williams et al. 1972), (Cladis and Brand 2003) (see Figures 4.4(b) and 4.5(f)); 3) they can be easily generated, set into motion and brought into collisions in the so-called dowser texture, as we will point out below (see also Pieranski et al. (2016a,b) and Pieranski and Godinho (2020)) (see Figure 4.4(c)).






 





Figure 4.4. Occurrence of nematic monopoles: (a) as companions of inclusions with homeotropic anchoring, (b) induced by homeotropic anchoring in capillaries, (c) as defects of the dowser texture

4.1.1.3. Disclinations and monopoles as topological defects The contrast between the ubiquity of disclinations and the scarcity of monopoles has topological reasons (Kleman and Lavrentovich 2006). Figure 4.5 shows that the director field on a circuit surrounding a disclination (Figure 4.5(a)) is mapped on just one meridian on the eastern hemisphere, which represents the space (real projective plane) of the nematic order parameter (Figure 4.5(b)). Therefore, to generate a disclination during the isotropic–nematic quench, it is enough to have three adjacent nematic-in-isotropic droplets with adequate orientations (Figure 4.5(c)). After the coalescence of droplets, the disclination must appear there.

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A similar bulk mechanism is very unlikely in the case of the monopole because the director field on a surface surrounding it (Figure 4.5(d)) is much more complex: it is mapped twice on the space of the nematic order parameter, once on the whole northern hemisphere and the second time on the whole southern hemisphere (Figure 4.5(e)). However, when the isotropic–nematic quench is realized in a capillary with homeotropic anchoring conditions, monopoles must appear between the adjacent domains with escaped director fields (Figure 4.5(f)).














  

 


















  





Figure 4.5. Topology of disclinations and monopoles

4.1.2. Road to the dowser texture 4.1.2.1. Difficulties with nematic monopoles in the dowser texture In principle, for the same reason as adequate boundary conditions, a monopole can exist in a nematic droplet maintained by a capillarity between two surfaces with homeotropic boundary conditions, as shown in Figure 4.6. However, in practice, a different configuration, shown in the Figures 4.6(a) and (c) arises after the introduction of the drop in the gap between the glass slide and the lens. Here, the homeotropic texture (represented by the black disc in Figure 4.6(c)) coexists with a distorted texture, also satisfying the homeotropic boundary conditions (colored and black interference fringes in polarized light), in which the director rotates by π between the limit surfaces. The homeotropic and distorted textures are separated by a disclination loop.

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The name of the distorted texture varied in past. It was called the “splay-bend state” by Cladis et al. (1987), the “H state” by Boyd et al. (1980), the “inversion wall” by Gilli et al. (1997), the “quasi-planar state” by Fazzio et al. (2001) or “flow-aligned” by Giomi et al. (2017). In Pieranski et al. (2016a), we proposed to call it “the dowser texture” because the director field lines in it have shapes similar to that of a Y-shaped wooden dowser’s tool. This name will be used throughout this chapter.   
 

    

   



 





 



  
 









 





 







  



   






 



  



 
 









Figure 4.6. The dowser texture. (a) Coexistence of the dowser and homeotropic textures. (b) Disclination at the border between the dowser and homeotropic textures. (c and f) Views in a polarizing microscope. (d and e) Nematic monopole in the dowser texture

As it is distorted, the dowser texture has the elastic energy per unit area of the order of felast =

K  π 2 h 2 h

[4.1]

so that it is energetically less favorable than the homogeneous homeotropic one. Therefore, usually its area shrinks for the benefit of the homeotropic texture (circle of radius r). The expansion of the homeotropic texture is stopped when the disclination reaches the border of the droplet, where it is repelled by an elastic interaction with the meniscus with homeotropic anchoring. Let us emphasize that after its expansion, the disclination does not disappear but remains in the vicinity of the meniscus, where its radius r is slightly smaller than the droplet’s radius R. The annular domain between r and R is filled with the residual dowser texture.

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In view of such observations made commonly during the preparation of homeotropic cells, the dowser texture was believed to be unstable for decades and little attention was paid to it. 4.1.2.2. Generic experiments This belief was first broken by Gilli et al. (1997) and more recently by Pieranski et al. (2016a) during experiments with disclinations threaded on polymeric fibers, where it was found that the residual dowser texture can expand at the expense of the homeotropic texture when the radius/thickness ratio r/h is small enough. The generic experiment started with the homeotropic texture filling the droplet, except for an annular domain of the residual dowser texture in the vicinity of the meniscus (see Figure 4.7(a)). An unexpected phenomenon occurred when the thickness h of the gap was increased to about 1.5 mm. The series of four pictures in Figure 4.7(b) shows that the peripheral disclination shrunk and coalesced with the captive disclination. As a result, the dowser texture filled the whole volume of the droplet. 







 
 

 
 

        




 

 

 


 
 
 





Figure 4.7. Generic experiment paving the way to the dowser texture. (a) Perspective view of the setup for studies of captive disclinations. (b) Surprising collapse of the peripheral disclination

This first observation of the expansion of the dowser texture was confirmed in a second experiment performed in the absence of the polymeric fiber. It is illustrated here, by a series of three pictures in Figure 4.8. No longer being hindered by the polymeric fiber, the peripheral disclination collapses into a point – a nematic monopole. 4.1.2.3. Relative stability of the homeotropic and dowser textures To explain how the expansion of the dowser texture is possible, we must take into account the elastic energy 2πrT of the disclination (of radius r and tension T) located,

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for topological reasons, at the frontier between the dowser and homeotropic textures (see Figure 4.6(a)). It has to be added to the elastic energy of the dowser texture in its volume π(R2 − r2 )h. The total energy Ftot ≈

K  π 2 (πR2 − πr2 )h + T 2πr 2 h

[4.2]

has a maximum at rcrit T /K ≈ 2 ≈2 h π /2

[4.3]

0

monopole Figure 4.8. Collapse of the peripheral disclination into a monopole

Usually, during the preparation of homeotropic samples, the typical thickness h = 100 mμ is small and the critical radius rcrit ≈ 2h is also small. Therefore, the radius of the homeotropic domain coexisting with the dowser texture is larger than the critical radius r > rcrit , so that the homeotropic domain expands and fills the droplet. As this implicit condition was always satisfied during the preparation of homeotropic samples, the belief in the instability of the dowser texture was founded on it. In the generic experiment with the captive disclination, the initial thickness was about 300 μm. For the volume πR2 h ≈ 10 mm3 , the corresponding radius of the drop was R = 2.5 mm. The corresponding point H in the graph of Figure 4.9(e) is located in the area of stability of the homeotropic texture. Subsequently, with V = const and increasing thickness h, the radius R of the droplet decreased along the hyperbolic trajectory (the dotted blue line from H to D). With the disclination remaining in the vicinity of the meniscus, the ratio r/h ≈ R/h decreased and crossed the critical value ≈ 2 (represented by the red line in Figure 4.9(e)), and the dowser texture started to expand.

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Figure 4.9. Three-stage road to the dowser texture. (a–d) Schematic representation of the method. (e) Corresponding trajectory on the R-vs-H diagram

4.1.2.4. Persistence of the dowser texture This first surprising observation was followed by a second: upon the reduction of the sample thickness along the red dotted trajectory from D to D’ in Figure 4.9(e), the critical line R/h ≈ 2 was crossed again in the opposite direction but, surprisingly, the dowser texture persisted until the gap thickness h was reduced to below ≈ 1 μm. In such a case, the monopole (Figure 4.5(c)) “blows up” into a disclination loop (Figure 4.5(f)) with a growing radius and the homeotropic texture is recovered. This observation of the surprising persistence of the dowser texture opened the door to experiments with it. We will point out below that the dowser texture is worthy of its name, not only because of the shape of the director field in it, but also because it can be used as a probe of fields, such as the velocity v of Poiseuille flows, electric fields E and thickness gradients g =∇h. 4.1.3. The dowser texture 4.1.3.1. Order parameters of the dowser texture, the dowser field In the table of topological defects proposed by de Gennes (Figure 4.1), nematic monopoles appear in the column “Nematics” as point defects of dimension δ = 0 of the tensorial order parameter Qαβ , with generally n = 5 components in the three-dimensional nematic phase (for uniaxial nematics, n = 3).

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The director field n(x,y,z) of the dowser texture in a thin nematic layer of thickness h can be expressed as n = (cos θ cos ϕ, cos θ sin ϕ, sin θ)

[4.4]

with the polar angle θ varying from −π/4 on the lower surface to π/4 on the upper surface. In the experiments discussed here, with the exception of the flow-induced homeotropic-dowser transition, the polar angle does not depend on x and y and conserves the same variation with z (in approximation of an isotropic elasticity) θ(z) =

πz h

[4.5]















 















 















Figure 4.10. Defects in systems with complex order parameters: (a) vortices in a superconductor, (b) dislocation in a Blue Phase crystal, (c) disclination in a SmC film, (d) umbilic in a nematic layer and (e) monopole in the dowser texture (adapted from Pieranski (2019))

On the other hand, the azimuthal angle ϕ of the director only depends on x and y coordinates (see Figures 4.10(e) and 4.11(c)), so that we can write n = cos θ(z)d + sin θ(z)ez

[4.6]

with ez = [0, 0, 1] and d = [cos ϕ(x, y), sin ϕ(x, y), 0]

[4.7]

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Figure 4.11. Nematic layers with degenerated anchoring conditions. (a) Film of 5CB floating on the surface of glycerin, investigated by Lavrentovich and Rozhkov (1988). The anchoring at the glycerin/5CB and air/CB interfaces is, respectively, planar and homeotropic. (b) Typical “schlieren” picture (courtesy of Lavrentovich). (c) The dowser texture between glass surfaces with homeotropic anchoring. (d) Typical “schlieren” texture

In this approximation, we can consider the dowser texture as a two-dimensional (2D) system with the unitary 2D vector order parameter d or, equivalently, with an unidimensional complex order parameter eiϕ . Taking this into account, we added the last column to the de Gennes’ table in Figure 4.1. Nematic monopoles can now also be seen as topological defects of the complex order parameter eiϕ and are therefore analogous to vortices or dislocations. Let us note that the unitary 2D vector field d = (cos ϕ, sin ϕ) – that we call the dowser field – is similar to the c field in Smectics C films or in a homeotropic nematic layer above the Fréedericksz transition, as well as in a nematic layer floating on glycerin (Lavrentovich and Rozhkov 1988). These analogies are summarized in Figures 4.10 and 4.11.

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Figure 4.12. Monopole–antimonopole pair generated in a dowsons collider. (a) View in a polarizing microscope. (b) Phase ϕ(x,y) and dowser d(x,y) fields around the pair. (c and d) Detailed phase and dowser fields of the monopole (d) and antimonopole (c)

4.1.3.2. Dowser texture as a natural universe of monopoles Since the discovery of its persistence, the dowser texture was extensively studied with the particular aim of generating and manipulating monopoles. A series of setups tailored for this purpose was developed (see section 4.2.1). They were used, among others, to generate monopoles, such as the pair shown in Figure 4.12(a). From the interference fringes (isogyres), we inferred the phase field ϕ(x, y) and the dowser field d(x, y) surrounding this pair (see Figure 4.12(b)). The phase field is coded using the colors defined in Figures 4.12(c) and (d), where the dowser field is represented by arrows in the same figures. Clearly, on the circuits surrounding these two monopoles, the defect of the phase is Δϕ = +2π and Δϕ = −2π as indicated. By convention, the monopole with Δϕ = −2π will be called an antimonopole. Our setups were also used to set monopoles and antimonopoles in motion on trajectories leading to collisions, which can result in the annihilation of monopole–antimonopole pairs (see Figure 4.13). By analogy with the hadron colliders, these setups are called dowsons colliders because monopoles and antimonopoles occurring in the dowser texture are dubbed, in short, dowsons d+ and d-. A typical result obtained with a dowsons collider is shown in Figure 4.13. On this image obtained by the superposition of 32 pictures taken at intervals of 10 s, the trajectories of dowsons d+ and d- appear as dotted lines. Yellow and blue arrows

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indicate the initial position dowsons. The collisions of pairs (d+,d-) are indicated by circles, which are, respectively, drawn with a solid line when annihilations occur, or with a dotted line when the colliding dowsons are passing by.

annihilation passing by

Figure 4.13. Collisions of monopole/antimonopole pairs in a dowsons collider. Superposition of images taken at intervals of 10 s. Yellow and blue arrows indicate the initial positions of monopoles and antimonopoles circulating in opposite directions. Collisions are indicated with circles drawn with a solid line when annihilation occurs or with a dotted line when monopoles are passing by

Today, because of these studies with dowsons colliders, monopoles can no longer be considered as scarce. On the contrary, they appear as topological defects that are very easy to generate and manipulate, as we will show in sections 4.8–4.10. In view of these achievements, the dowser texture appears today as a natural universe of nematic monopoles. 4.1.3.3. Tropisms of the dowser texture Experiments with dowsons colliders also allowed the discovery of remarkable properties of the dowser texture itself that arise from its symmetry C2v , which only contains three elements: the twofold axis C2 and two mirror planes passing through it: σ (see Figure 4.14(a)) and σ’ passing, respectively, through z and x axes. The group C2v is a subgroup of D∞h – the symmetry group of the cell filled with the isotropic phase or with the homeotropic texture (Figure 4.14(a)). The order parameters d or eiϕ resulting from the D∞h ⇒ C2v symmetry breaking are thus degenerated with respect to rotation around the z axis. For this reason, the dowser

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texture is expected to be very sensitive, in first order, to fields f. As we will see in the following sections, fields f, such as thickness gradient g =∇h, Poiseuille flow v or the electric field E exert on the dowser field d torques given by → − Γ = C d × f

[4.8]







 

    







 



  






 

 



 

     



   

 





  






 



Figure 4.14. Tropisms of the dowser texture. (a) Symmetry of the dowser texture and its order parameters ϕ and d. (b) Cuneitropism: alignment by the thickness gradient g =∇h. (c) Rheotropism: alignment by the Poiseuille flow v. (d) Electrotropism: alignment by the electric field E

Upon the action of these torques, the dowser field d tends to be aligned in the direction parallel (or antiparallel) to f. In other words, we can concisely say that the dowser texture possesses the following tropisms: – cuneitropism: alignment by thickness gradient g (see section 4.5); – rheotropism: alignment by Poiseuille flows v (see section 4.4); – electrotropism: alignment by electric fields E (see section 4.6).

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4.2. Generation of the dowser texture 4.2.1. Setups called “Dowsons Colliders” Several setups were developed as we progressed in studies of the dowser texture. They were tailored with the aim of accomplishing several tasks, starting with the necessary preparation of the dowser texture.   

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Figure 4.15. Generic setup called Double Dowsons Collider 1 (DDC1). (a) General view. (b) Up and down translation s(t) and flexion α(t) of the glass slide. (c) Radial flow driven by an upward translation of the slide. (d) Dipolar flow driven by the flexion α(t) of the slide. Remark: Radial and polar flows are shifted in phase by π/2

All of these setups have two elements in common (see Figure 4.15): a microscope glass slide (75×19×0.9 mm) and a lens (50 mm in diameter with the radius of curvature Rl =140 mm), both mounted on a vertical bench. The lens is fixed, while the glass slide is movable by means of precise translation stages in x, y and z directions.

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The motion in z direction allows the control of the thickness h of the slide/lens gap that, by capillarity, keeps the nematic drop in its center of minimum thickness h. 4.2.2. “Classical” generation of the dowser texture The scheme of the very first setup in Figure 4.15 shows that precision of the vertical motion s(t) is improved by means of a lever (24 cm in length) attached to the micrometric screw of the translation stage. Using this system, we can thus control the thickness h and generate the dowser texture using a three-stage process, discussed in section 4.1.2.1, which consists of following the path H→D→D’ indicated by the dotted lines in the graph in Figure 4.9(e). The initial thickness (minimal thickness at the center of the gap) and radius R of the nematic droplet at the point H do not matter, but are typically of the order of h ≈ 300 mμ and R ≈ 3 mm. Subsequently, the point D is reached by increasing the thickness to h≈1.5 mm and the collapse of the peripheral disclination (illustrated by schemes b to c in Figure 4.9, h = 1.4 mm) is initiated, as already discussed in section 4.1.2.1 (see Figure 4.8). Let us stress that the collapse of a large homeotropic domain encircled by the disclination loop is a very slow process that can last several hours. In the example of Figure 4.8, 11 hours were necessary for the peripheral disclination to collapse into the residual monopole situated in the center of the droplet. After the collapse of the peripheral disclination, the thickness is slowly reduced to about 10 mμ, and the final point D’ of the H→D→D’ path in Figure 4.9(e) is reached. 4.2.3. Accelerated generation of the dowser texture using the DDC2 setup The generation of the dowser texture can be shortened from hours to about one minute using another version of the Double Dowsons Collider, called DDC2, which will be described in section 4.4.3.1. The operational principle of this second method involves four stages: 1) Shreding: A vigorous streaming flow inside the droplet, driven by flexural vibrations of the flexible glass plate, stretches and shreds the unique homeotropic-in-dowser domain (r≈1.5 mm) into a multitude of small domains (see Figure 4.16), with radii ri ≈20 μm. The streaming flow also shifts the nematic droplet as a whole on the distance Δy. 2) Flow-induced homeotropic ⇒ dowser transition: After cessation of the streaming flow, the drop returns, by capillarity, to the initial position. The Poiseuille flow accompanying this translation reduces the radii of the small homeotropic-indowser domains; it is an evidence of a flow-induced homeotropic ⇒ dowser transition.

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3) Increase of the thickness: A subsequent increase in the gap thickness reduces radii ri below the critical value rc and the collapse of the homeotropic-in-dowser domains occurs rapidly. 4) Reduction of the thickness: Once all homeotropic-in-dowser domains have disappeared, the thickness of the gap can be reduced accordingly.

    



 

 


   

   


  
 

  
  
 
  


  


 


  
 

  




   
 


  



 
  

   




 

   



 Figure 4.16. Shortening of preparation of the dowser texture by a transitory application of a turbulent mixing

The second stage of this process is related to the recent works by Emeršiˇc et al. (2019) and Liu et al. (2019), who used microfluidic methods to study the relative stability of the homeotropic and dowser textures in the presence of a Poiseuille flow. Inspired by these works, we used our setup to explore the flow-induced homeotropic ⇒ dowser transition in more detail.

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4.3. Flow-assisted homeotropic ⇒ dowser transition 4.3.1. Experiment using the DDC2 setup Our experiments started by the generation of a unique homeotropic-in-dowser domain by means of a reduction of the gap slide/lens thickness to zero (see the series of three pictures in Figure 4.17). After its nucleation, the homeotropic-in-dowser domain is expanding. Once its radius r is large enough, the minimal thickness is increased to h = 60 μm. By means of this method, it is possible to generate H-in-D domains with different radii r.








Figure 4.17. Generation of a unique homeotropic-in-dowser domain by the reduction of the slide/lens gap to zero. (a) The radial dowser texture with the minimal thickness h close to zero, favoring nucleation of the homeotropic domain. (b and c) Expansion of the homeotropic domain

Subsequently, the H-in-D domains were submitted to slow streaming flows with velocities set to such values vc that the sizes rc of domains were conserved. One of these experiments is illustrated by the spatiotemporal cross-section in Figure 4.18(a). Initially, the size rc of the domain is relatively small and the slope vc of the domain’s trajectory in the (t,x) plane is also relatively small. Subsequently, the streaming flow is stopped and the size rc of the domain grows a little until the streaming flow is applied once again and adjusted to a new stationary value vc , which is larger than the first one. In the last growth-adjustment sequence visible at the end of the spatiotemporal cross-section in the Figure 4.18(a), vc is even larger. Results of several such experiments are represented by red crosses in the upper part of the plot 1/rc versus vc , as shown in Figure 4.18(c). They fit well to the linear law   vc 1 1 1− = rc rco vco with rco = 69 μm and vco = 3.9 μms−1 .

[4.9]

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Figure 4.18. Stability of homeotropic-in-dowser domains in Poiseuille flows. (a) Spatiotemporal cross-section showing the motion of a homeotropic-in-dowser domain. (b) Homeotropic and dowser textures submitted to a Poiseuille flow. (c) Stability diagram. Red crosses – experimental points extracted from the spatiotemporal cross-sections, such as the one in (a). Blue crosses – experimental data from figure 2G in Emeršiˇc et al. (2019). Solid red line – fit to the equation [4.9]. (5CB, h = 60μm)

Emeršiˇc et al. (2019), in their experiments with microfluidic channels, studied the flow-induced homeotropic-dowser transition in the complementary geometry of

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dowser-in-homeotropic domains. Their results, obtained with a channel of thickness h = 12 μm, are represented by blue crosses in the lower part of the plot in Figure 4.18c. They are fitted to the same linear law with rco = 10 μm and vco = 56 μ ms−1 . Let us note that the red line of the linear fit extends to the left part of the plot with negative values of velocity. So far, experimental points are missing, but at the end of the theoretical section below we will suggest how they can be obtained. R EMARK . The director field lines in the homeotropic texture submitted to hydrodynamic torques in the Poiseuille flow acquire a bow-like shape. The amplitude of this deformation is proportional to the flow velocity. For this reason, the deformed homeotropic texture was dubbed the bowser texture by Emeršiˇc et al. (2019) (see sections 4.3.2 and 4.14).



 








 









 
  

 



 






      







  
   
 




 



  
















Figure 4.19. Flow-assisted bowser-dowser transformation in a capillary. (a,a’) For v < vco , the bowser texture fills the capillary except for thin dowser boundary layers at lateral walls. (b) For v > vco , the flow-assisted bowser-dowser transformation occurs. The width of the dowser boundary layers grows. (c,c’) The capillary filled with the dowser texture

4.3.2. Flow-assisted bowser-dowser transformation in capillaries Liu et al. (2019) investigated the flow-assisted homeotropic-dowser transformation occurring in long straight capillaries with rectangular cross-sections, in detail. In the

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absence of flows, the ground state of the director field corresponds to the homeotropic texture, filling the capillary, except for the thin boundary layers of the dowser texture at lateral capillary’s walls. As the disclinations separating the dowser boundary layer from the bulk homeotropic texture are straight (see Figure 4.82(a)), this experiment corresponds to the case rc → ∞, i.e. 1/rc = 0, in Figure 4.18(c). When the flow velocity is smaller than the critical value vco , the homeotropic texture is deformed into the bowser texture, but disclinations remain in the vicinity of the lateral walls (see Figure 4.19(a)). For v > vco , an instability occurs: the dowser boundary layers become progressively broader (see (Figure 4.19(b)) until they coalesce and the whole capillary is filled with the dowser texture (see Figure 4.19(c)). Using, for example, a channel of thickness h = 50 μm, Liu et al. obtained vco = 9 μms−1 . 4.3.3. Flow-assisted homeotropic-dowser transition in the CDC2 setup Anticipating the description of the so-called Circular Dowsons Collider 2 (see Figure 4.28) that will be given in section 4.4.3.3, let us report on the flow-assisted homeotropic-dowser transition realized by means of this setup. CDC2 was tailored for the winding of the dowser texture and for experiments on the generation, motions and collisions of dowsons. For large amplitudes of the conical motion of the glass slide, CDC2 generates a stationary circular flow with the orthoradial velocity vmax (see Figure 4.20(b)), growing with the distance r from the center. The experiment illustrated in Figure 4.20 starts with a relatively large circular homeotropic-in-dowser domain located in the center of the nematic droplet (see Figure 4.20(a)). The circular streaming flow of amplitude vmax (r) generated in the droplet is thus parallel to the homeotropic/dowser interface of radius r (see Figure 4.20(b)). The subsequent evolution of the radius r(t) depends on the initial velocity vmax (t = 0) of the flow. Let us suppose that at t = 0 and r = r(0) (point t = 0 in Figure 4.20(h)), the velocity vmax = v(r(0)) is larger than the critical velocity vc (r) (red line in Figure 4.20(h)) defined by equation [4.9], so that the homeotropic-in-dowser domain starts to shrink. Let us suppose next, in the first approximation, that the velocity vmax (r) is proportional to the radius: vmax = ωr or alternatively, that 1/r = ω/vmax

[4.10]

In Figure 4.20(h), this variation is represented by the hyperbola drawn with a dotted red line. The stationary state is reached at the intersection labeled SS of the red and dotted lines, in which the radius rss of the homeotropic (bowser) domain is reduced with respect to r(0).

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Figure 4.20. Flow-assisted hemeotropic-dowser transition in the circular geometry of the CDC2 setup (see section 4.4.3.3). (a) Circular homeotropic-in-dowser domain expanding in the absence of flow. (b–e) Shrinking of the homeotropic-in-dowser domain induced by the circular streaming flow revealed by motion of a dust particle indicated by arrows. (f) Homeotropic-in-dowser domain expanding after cessation of the streaming flow. (g) Spatiotemporal cross-section extracted from a video along the line CS, defined in (a). (h) Theoretical explanation of the experiment

In the experiment depicted in Figures 4.20(a–f), the radius r ≈ 1.5 μm is 22 times larger than the critical radius rco = 69 μm. Therefore, the critical velocity vss = 22 μm/s measured in the stationary state SS is close to critical velocity vco (in the limit r → ∞). 4.3.4. Theory of the flow-assisted homeotropic-dowser transition The explanation of the linear dependence between the critical curvature 1/rc of the homeotropic/dowser interface and the flow velocity vc was first proposed by Emeršiˇc et al. (2019). More recently, Tang and Selinger (2020) developed this theory and presented it in detail. Here, we will analyze the flow-assisted homeotropic-dowser transition in slightly different terms.

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215

Let us start with the expression of the viscous torque per unit volume exerted by a shear flow on the director (see, e.g. de Gennes and Prost (1993) or section III.4 in Oswald and Pieranski (2005)). Γv = [−α2 sin2 θ + α3 cos2 θ]

dv dz

[4.11]

We will first suppose that the stationary deformations of the homeotropic and dowser textures, due to this torque, are so small that they can be neglected. This is true when the Poiseuille flow is slow enough. During the expansion of the dowser texture across the strip of width dx (see Figure 4.18(b)), the polar angle at the point P (in the upper half of the nematic layer) decreases from π/2 to πz/h and the torque Γz does the work (per unit volume) 

πz/h

Γv dθ =

w= π/2

dv (α2 − α3 )π(h − 2z) + (α2 + a3 )hsin(2πz/h) dz 4h

[4.12]

Knowing that the shear rate in the Poiseuille flow is dv/dz = −8vmax z/h2 , we obtain, by integration on z, the work per unit area  W = 2

h 2

0

W = vmax

 wdz = −vmax α2  πγ 6

1

−

γ2  π



π 1 + 6 π



 − α3

π 1 − 6 π

 or

[4.13] [4.14]

The factor 2 in equation [4.13] takes the same work done by the viscous torque in the lower half of the nematic layer into account. This work corresponds to an effective potential Uef f =-W (Tang and Selinger 2020), which has to be added to the total elastic energy given in equation [4.2]: 



F = Ftot + Uef f

 vmax  πγ1 K π2 γ2  = − − (πR2 − πr2 )h + T 2πr [4.15] 2 h2 h 6 π

The maximum of F’ is located at rco vmax 1 π2 K π2 K πγ1 =1− with = and vco = rc vco rco 2T h 2h 6 −

γ2 π



[4.16]

This expression justifies the linear law in equation [4.9], used above for the fit of experimental data.

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R EMARK . Equation 4.16 and the plot in Figure 4.18 remain valid for negative values of vmax under the condition that the dowser field d keeps its orientation parallel to the y axis. If an arbitrary direction of the dowser field was imposed, for example by an electric field, the linear term in equation [4.16] would be vmax · d/vco .

4.3.5. Summary and discussion of experimental results All experiments on the flow-assisted homeotropic-dowser transition were made with 5CB. As in this material |α3 | = ±2π/T ,

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Liquid Crystals

this winding process is synchronous with the excitation of frequency ωexc = 2π/T . The analogy with the winding of an elastic strip is not complete because in the more detailed view of the dotted rectangle (defined in Figure 4.25(a)) shown in Figure 4.25(c) the black isogyres have variable width and are not equidistant. The corresponding variation of the azimuthal angle ϕ with x can be expressed as: ϕ(x) ≈ 2π

 y y − δ sin 2π λ λ

[4.28]

g

d

y

c

0

g =
h

a b

Figure 4.25. Mechanical analogies of the wound up dowser texture. (a) Detailed view of the picture l in Figure 4.24. (b) Winding of an elastic strip. (c) detailed view of the dowser field. (d) Winding of a chain of pendula attached to an elastic string

This expression describes the state of equilibrium of a continuous wound up chain of pendula attached to an elastic wire and submitted to the gravity g. It is a solution of the equation K

d2 ϕ + lρg sin ϕ = 0 dy 2

[4.29]

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223

which describes the balance of the elastic and the gravitational torque (ρ is the mass per unit length of the chain, l is the length of pendula and g is the gravitational acceleration). In section 4.5, we will see that an analogous equation applies to the case of a dowser field wound up in x direction, for which the first term also expresses the elastic torque, while the second term is due to the cuneitropism: the torque (πK/h) sin ϕ tends to orient the dowser field in the direction of the thickness gradient. In the gap − → → between the spherical lens and the flat glass slide, the thickness gradient − g = ∇h is radial and directed outward. In the small rectangular dotted area, it is antiparallel to the x axis. R EMARK . As the wound up dowser field ϕ = ϕ(x, y) in Figure 4.25(a) is bidimensional and quite complex, the generalized elastic torque KΔϕ (see section 4.16.1) cannot be balanced by the cuneitropic torque alone. Therefore, the wound up dowser field will relax, i.e. unwind into the radial one (see Figure 4.21(b)), with the monopole in the center. The relaxation process is controlled by the third viscous torque γdϕ/dt contributing to the balance of torques. 4.4.2.2. Interpretation in terms of the rheotropism The winding of the dowser field cannot be due to the Poiseuille flows driven exclusively by the harmonic up and down translation s(t) = so sin(ωt) of the glass slide. Indeed, such flows are radial and have symmetry of revolution around the z axis, while the pattern of isogyres of the wound up dowser field is only symmetrical with respect to the (x,z) mirror plane. Starting from this striking disagreement, we also have to take the dipolar flows driven by the flexion of the elastic glass slide into account. Figures 4.15(a) and (b) show that the glass slide is clamped and attached to the translation stage at only one of its extremities. The flexion of the glass slide is due to the viscous stress σzz , in the squeezed or stretched droplet, which is proportional to dh/dt. The tilt angle α of the glass plate in the center of the droplet is therefore proportional to dh/dt. The tilt of the glass plate taken alone generates a dipolar flow represented in Figure 4.15(c). Here, it was calculated approximatively as a superposition of two radial flows: the first one diverging from a point source and the second one converging to a point well. The source and the well are located, respectively, at (0, yo ) and (0, −yo ). These radial flows are proportional to dα/dt ∼ d2 h/dt2 .

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2

2

2

1

1

1

0

0

0

1

1

1

T/8

2

2

1

0

1

T/4

2

2

2

1

0

2

1

2

2

2

2

1

1

0

0

0

1

3T/8 1

0

1

2

T/2

1

2

2

2

1

0

1

2

7T/8

2

2

3T/4

2

2

1

1

1

0

0

0

1

1

1

2

1

0

1

2

0

1

2

2

2

2

1

5T/8

2

1

0

1

2

2

1

0

1

2

Figure 4.26. Flows inside a nematic droplet driven by a harmonic up and down motion s(t) in the Double Dowsons Collider 1 (see Figure 4.15). Pictures labeled 0, T/8,...,7T/8 show the flow patterns at intervals of T/8. The picture in the center shows trajectories of molecules

Being proportional, as stated above, to respectively dh/dt and d2 h/dt2 , the radial and dipolar flows are thus shifted in phase by π/2. The sum of these radial and dipolar flows is illustrated by the series of eight pictures in Figure 4.26. The most significative is the picture in the center that shows local trajectories of molecules: they are elliptical anticlockwise and clockwise in the left and right halves of the drop, respectively. The rheotropic torques generated by such flows compete with the restoring elastic and cuneitropic torques. In areas where they are large enough, they can drive the winding of the dowser field. Otherwise, they generate only oscillations of the dowser field. Other details of calculations of the flow pattern and of its action on the dowser field can be found in Pieranski et al. (2017).

Physics of the Dowser Texture

225

4.4.3. Asynchronous winding of the dowser field In most experiments on generation, motions and collisions of nematic monopoles (which will be discussed later), it is more convenient to use setups operating at higher, acoustic frequencies with smaller amplitudes of the excitations. Two of them are schematically depicted in Figure 4.27. 






    

















 




























 

 





Figure 4.27. Setups tailored for the asynchronous winding of the dowser field. (a) Double Dowsons Collider 2 (DDC2). (b) Circular Dowsons Collider 1 (CDC1). (c and d) Approximative representations of the mean rheotropic torque γ(x, y). (e and f) Typical interference patterns of the dowser textures wound up in DDC2 and CDC1

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Liquid Crystals

4.4.3.1. DDC2 The first one, DDC2 in Figure 4.27(a), is almost identical to the DDC1 except for the mode of excitation. Here, two flexural modes of oscillation of the glass slide are excited simultaneously by means of a magnet fixed on it and of a coil positioned above the magnet. The fundamental flexural mode modulates the thickness h at the center of the gap and drives an alternating radial flow. The second flexural mode induces the tilt of the glass slide around the y axis passing through the center of the gap and drives an alternating dipolar flow. The frequency ωexc of excitation is intermediate between the eigenfrequencies ω1 and ω2 of the two flexural modes so that, as in DDC1, there is a phase shift of the order of π/2 between the radial and dipolar flows. The elliptical flows of small amplitudes driven by this means exert effective rheotropic torques Γ(x, y) having, respectively, clockwise and anticlockwise directions in the right and left halves of the nematic droplet (see Figure 4.27(c)). The pattern of the wound up pattern produced with DDC2 is shown in Figure 4.27(e); it is similar to the one in Figure 4.25(a). 4.4.3.2. CDC1 In the second setup, called the Circular Dowsons the Collider 1 (CDC1), two flexural modes are also excited by means of the magnet-coil system. However, as the mirror symmetry of the DDC2 is broken here (see Figure 4.27(b)), these flexural modes are different. The first one involves the torsion of the glass slide around the x axis, while the second one involves, as in DDC2, the tilt of the glass slide around the y axis. Both modes excite dipolar flows in orthogonal directions. These dipolar flows are shifted in phase approximatively by π/2 because the frequency of excitation ωexc is intermediate between the eigenfrequencies of the two modes. When the amplitudes of the two orthogonal dipolar flows are equal, the distribution Γ(x, y) of the effective rheotropic torque exerted on the dowser field now has the symmetry  of revolution around the z axis (see Figure 4.27(d)) so that it depends only on r = x2 + y 2 . The sign of the torque Γ(r) changes at r = rc . The dowser texture wound up in this device is shown in Figure 4.27(f). R EMARK . Dowser fields wound up in DDC2 and CDC1 contain monopoles labeled d+. They circulate on trajectories parallel to isogyres. As we will see in section 4.9.2, → − this motion is driven by a Lorentz-like force orthogonal to the local gradient ∇ϕ of the phase field. 4.4.3.3. Asynchronous winding of dowser field with CDC2 The symmetry of revolution of the dowser field wound up by the CDC1 is obviously not ensured by the configuration of this device, which is asymmetric but

Physics of the Dowser Texture

227

can only be approached by subtle adjustments of several parameters such as the mass and the length of the lever orthogonal to the glass slide as well as the frequency of excitation. Moreover, to reverse the direction of winding it is necessary to rebuild the setup shown in Figure 4.27(b) into a configuration symmetrical to it by the mirror reflection in the (x,z) plane. The Circular Dowsons Collider 2 shown in Figure 4.28 has the configuration tailored for the production of the circular winding. The conical motion of the glass slide is ensured by symmetries of the slide’s holder and of the electromagnetic excitation. Thanks to four semicircular cuts operated in a metallic plate, the slide’s holder, supported by two pairs of bridges, has two degrees of freedom: it can tilt around the x and y axes. 







 

   


 












 





N S



 


N S



 

Figure 4.28. Circular Dowsons Collider 2. (a) Perspective view of the glass slide’s holder with four magnets fixed on it. Four coils, positioned above the magnets, drive independent tilts around the x and y axes. Vertical motion s(t) is controlled by a translation stage (not shown). (b) Side view showing the tilt oscillation around the y axis. (c) Conical motion and vertical displacement of the glass slide

These two tilt motions are driven independently by two pairs of magnets and coils (X1-X2 and Y1-Y2). For a π/2 phase shift between AC currents with frequency

228

Liquid Crystals

f = 300 Hz applied to the pairs of X and Y coils, the resulting motion of the glass slide is conical. As in CDC1, the conical motion of the glass slide produces a rheotropic torque Γ(r) that winds asynchronously the dowser field. The winding process is illustrated by Figures 4.29 and 4.30. Figure 4.29(a) contains a series of 12 pictures taken at intervals of 0.8 s and shows the beginning of the winding process in detail. From the deformation of the four isogyres, which initially formed a maltese cross, we can infer that the sense of the winding is anticlockwise for r < rc and clockwise r > rc . The angular velocity ω = dϕ/dt of the winding can be determined from spatiotemporal cross-sections in Figure 4.29(b) extracted along circular lines AA’ and BB’ from the whole video containing 163 frames taken at intervals of 0.16 s. At the beginning of the winding, for 0 < t < 8.8s, only the conical excitation is applied and the angular velocity determined from the initial slopes of isogyres’ trajectories is ω = 0.069s−1 . 4.4.4. Hybrid winding of the dowser field with CDC2 We have found that the winding process is accelerated when the conical excitation of frequency fc = 300 Hz is associated with a harmonic up and down motion of the slide holder of a very low frequency, typically fz ≈ 77 mHz. The application of such a hybrid excitation mode appears in the spatiotemporal cross-sections of Figure 4.29(b) as a change of the isogyres trajectories’ slopes. The mean angular velocity determined that the new slope is ω = 0.48s−1 = 2πfz . 4.4.5. Rheotropic behavior of π- and 2π-walls Other features of the dowser field motions driven by the hybrid excitation are illustrated in Figure 4.30. The most important of them is the formation of 2π-walls, which are directly visible in three pictures of Figure 4.30(a) taken during the first period of the hybrid excitation. One can follow generation and evolution of these walls on the spatiotemporal cross-section in Figure 4.30(b). The time lines A, B and C are of particular interest. On lines A and C new isogyres are generated because the dowser field rotates in points A and C with extremal angular velocities. New isogyres generated on lines A and C converge to the line B but do not cross it because the dowser field does not rotate in the point B located on the circle of radius rc . In the vicinity of the line B, the isogyres are always associated in pairs, i.e. in π-walls. “Wavy” trajectories of the π-walls lead to their association into 2π-walls that split and reassociate into new 2π-walls, when the direction of the radial Poiseuille flow is reversed.

Physics of the Dowser Texture

 

229

 






 





 





Figure 4.29. Asynchronous winding of the dowser field with the Circular Dowsons Collider 2. (a) Pictures of the dowser texture taken at intervals of 0.8 s. (b) Spatiotemporal cross-sections extracted from a video along circular lines AA’ and BB’

The same behavior of π- and 2π-walls is visible in the spatiotemporal cross-section in Figure 4.24, a characteristic of the synchronous winding with the DDC1 setup. Here, new isogyres, generated on the time lines B and B’ converge toward the time line A on which for symmetry reasons the dowser field does not rotate. In the vicinity of

230

Liquid Crystals

the line A, wavy trajectories of π-walls lead them also to the association into 2π-walls that split and reassociate into new 2π-walls when the direction of the radial Poiseuille flow is reversed.








  

 









Figure 4.30. Hybrid winding of the dowser field with the Circular Dowsons Collider 2. (a) Pictures taken during the first period of excitation . (b) Spatiotemporal cross-section extracted along the line CS from 164 frames of a video

Physics of the Dowser Texture

231

4.4.6. Action of an alternating Poiseuille flow on wound up dowser fields 4.4.6.1. Experiment Deformation of the wound up dowser texture submitted to an alternating Poiseuille flow is easier to analyze in its stationary state, i.e. without the generation of new isogyres, which is achieved by an adequate adjustment of the excitation level. Let us consider the example shown in Figure 4.31 of the dowser field wound up in the DDC1. The picture of the interference pattern in Figure 4.31(a) has been taken in the absence of flows. On a long time scale of a few hours, the wound up dowser field (Figure 4.31a) relaxes elastically, i.e. unwinds by the collapse of isogyres’ loops. Therefore, on the time scale of 40 s of the experiment, this wound up dowser field can be considered quasi-static. It has been submitted to the Poiseuille flow generated during two periods (T = 20 s) of the up and down motion of the glass slide. The spatiotemporal cross-section extracted from a video along the dashed line AB defined in Figure 4.31(a) is shown in Figure 4.31(b). Here, trajectories of isogyres are straight before application of the flow at t = 0 s and become straight again after cessation of the flow at t = 40 s. This justifies the former assumption of the quasi-static state. 







































 























































Figure 4.31. Action of an alternating Poiseuille flow on a wound up dowser field. (a) Stationary wound up dowser texture. (b and c) Spatiotemporal cross-section of the wound up dowser texture submitted to an alternating Poiseuille flow. (d and e) Diverging and converging radial Poiseuille flows driven by a harmonic modulation of the gap thickness. (f and g) Detailed views of the interference patterns

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Liquid Crystals

 

  





























Figure 4.32. Simulation of the spatiotemporal cross-section. (a) Definition of the rheotropic torque. (b) Comparison of the experimental cross-section from Figure 4.31(b) with the simulation

In the presence of the alternating Poiseuille flow, trajectories of isogyres become wavy and one observes their association into π- and 2π-walls. The 2π walls are first formed during the first half-period when the Poiseuille flow is directed upward in the direction of the x axis. When the flow is reversed, the 2πwalls split into π-walls that reassociate into new 2π-walls. 4.4.6.2. Analysis In this experiment, the dowser field is submitted to four torques: the rheotropic, viscous, elastic and cuneitropic. The rheotropic torque has been introduced in section 4.4.1.2. When the Poiseuille flow v is parallel to the x axis (see Figure 4.31(c)), its expression is (see Figure 4.32(a)): Γrt = −αef f (vmax /h) sin ϕ with αef f = −2α2 h/π. The viscous torque acting on the dowser field is given by the integral (see section 4.16.2):  Γvisc = −

h/2 −h/2

γ1

∂ϕ ∂ϕ cos2 θdz = −γef f ∂t ∂t

[4.30]

with γef f = γ1 h/2. Similarly, the elastic torque is given by  Γelast =

h/2 −h/2

K

∂2ϕ ∂2ϕ cos2 θdz = Kef f 2 2 ∂y ∂y

[4.31]

Physics of the Dowser Texture

233

with Kef f = Kh/2. The cuneitropic torque, which will be discussed in more detail in the next section, is Γcunei = −(πKg)/h) sin ϕ. The balance of torques acting on the dowser field leads to the following equation of motion: ∂ϕ αef f vmax (t) Kef f ∂ 2 ϕ (πKg/h) − sin ϕ − = sin ϕ ∂t γef f ∂y 2 γef f γef f h

[4.32]

For t ≤ 0, the dowser field is considered quasi-static in the absence of the alternating Poiseuille flow so that ∂ϕ/∂t = 0 and this equation is reduced to the balance of the elastic and cuneitropic terms. In section 4.5, we will see that its solution ϕ(y, 0) = qy − Δϕ sin(qy)

[4.33]

describes a wound up phase field modulated by cuneitropism. The best fit with experimental data is obtained with Δϕ = 0.33. For t>0, this solution will be taken as an initial condition of equation [4.32], reduced to its viscous and rheotropic terms ∂ϕ = −C sin ϕ ∂t with C =

[4.34]

αef f vmax (t) γef f h

Results of its numerical integration have been used for calculation of the intensity of light transmitted between crossed polarizers I = Io sin2 [ϕ(y, t)]. The best fit with the experimental spatiotemporal cross-section in Figure 4.32(b) obtained with C = 0.25s−1 is plotted in Figure 4.32(c). 4.5. Cuneitropism, solitary 2π-walls 4.5.1. Generation of π-walls by a magnetic field During discussions of 2π-walls generated by the rheotropic winding of the dowser field, we have shortly mentioned that, beside the rheotropic torque, the cuneitropic torque also contributes to their structure. Here, we will deal with the simpler case of solitary 2π-walls generated first by means of a rotating magnetic field and shaped later by the cuneitropism alone.

234

Liquid Crystals



 



 





!









 !





 



 

 

 !



!





















 c

B









Figure 4.33. Winding of the dowser field by the magnetic field. (a) Initial radial dowser field. (b) Two π-walls generated by the magnetic field. (c and d) Configuration of the π-walls after rotation of the magnetic field by 2π and 5π. (e) Circular 2π-wall formed from 2 π-walls after suppression of the magnetic field. (f–h) Shrinking of the 2π-wall. (i–l) Detailed views of the 2π-wall. (b’–i’) Dowser fields in the π and 2π walls

A magnetic field B, parallel to the xy plane and applied to the radial dowser texture, deforms it into two π-walls as is shown in Figures 4.33(a) and (b). This deformation is due to the torque  Γmag =

h/2 −h/2

χa (n · B)(n × B)dz μo

[4.35]

Physics of the Dowser Texture

235

exerted by the magnetic field on the director n. As the thickness h is very small with respect to the magnetic coherence length ξm (see below), the z dependence of the director field inside the dowser texture is not affected by the magnetic field and can be expressed as a sum of two components, one parallel to the dowser field and the second one parallel to the z axis: n = n ≈ cos θd + sin θez

[4.36]

with θ = πz/h. After integration on z, we obtain: Γmag =

h χa h χa 2 (d · B)(d × B) = − B sin ψB cos ψB 2 μo 2 μo

[4.37]

with ψB corresponding to the angle between the dowser and magnetic fields. In equilibrium, the shape ψB (ξ) (see Figure 4.33b’) of the π-wall satisfying the balance of the elastic and magnetic torques: Kh ∂ 2 ψB h χa 2 − B sin ψB cos ψB = 0 2 ∂ξ 2 2 μo

[4.38]

is given by   − ξ ψB (ξ) = 2 arctan e ξB

[4.39]

with

ξB =

μo K 1 χa B

[4.40]

Let us mention that when the radial dowser field is submitted to the magnetic field, not only the two diametrically disposed π-walls are created. Due to the homeotropic anchoring conditions at the meniscus, the dowser field remains orthogonal to it. Therefore, at their junctions with the meniscus, the diametral π-walls split into pairs of peripheral walls parallel to the meniscus. These peripheral walls surrounding the whole droplet can be referred to as partial because the variation of the angle ψ across them is 0 < ΔψB < π/2.

Liquid Crystals







 





236



 
  










  





























Figure 4.34. Slow elastic collapse of the circular 2π-wall. (a) Spatiotemporal cross-section extracted from a video along the line CS defined in Figure 4.33(g). (b) Variation of the width ξc of the 2π-wall with its radius Rw . Dots – experimental results; continuous line – fit to the theoretical prediction given in equation [4.46]

4.5.2. Generation and relaxation of circular 2π-walls When the magnetic field starts to rotate slowly enough, the dowser field follows it and ΔψB starts to change (see Figures 4.33(c) and (d)). When after a whole 2π turn (Figure 4.33(c)) the magnetic field is suppressed, the dowser field relaxes into its radial configuration except for a circular 2π-wall that remains in the vicinity of the meniscus (see Figure 4.33(e)). To recover the ground state, with the radial dowser field filling the whole droplet, about 20 h are necessary. During this very slow relaxation, this solitary circular 2π-wall shrinks (see Figures 4.33(e–h)). Simultaneously, the width ξw of the wall decreases (see Figures 4.33(i–l)). An analogous behavior is observed when the slowly rotating magnetic field is suppressed after several whole 2π turns. For example, Figure 4.35 shows that four 2π-walls have been generated by this method. During the visco-elastic relaxation, these walls shrink simultaneously and collapse one after another. 4.5.3. Cuneitropic origin of the circular 2π-wall The ground state radial configuration of the dowser field and the circular 2π-wall have the same origin – the cuneitropism (from the latin word cuneus = wedge). This phenomenon, also referred to as “geometrical anchoring” by Lavrentovich (1992), takes place when the limit surfaces are not parallel but form an angle γ. In such a

Physics of the Dowser Texture

237

wedge geometry, cuneitropism is a manifestation of the dependence of the distortion energy of the dowser texture on its azimuthal orientation with respect to the thickness gradient g =∇h.




















Figure 4.35. Viscoelastic unwinding of the dowser field in a circular droplet can be seen as the shrinking and collapse of 2π-walls. Pictures taken at intervals of 80 min

To understand the coupling between the dowser field d and the thickness gradient g, let us consider first the dowser texture in a sample with a uniform thickness shown in Figure 4.36(a). Here, the director field n rotates by α = π between the limit plates separated by the distance h. In the approximation of the isotropic elasticity, the distortion energy per unit area is given by  Fdow =

h/2 −h/2

K  α 2 K dz = α2 2 h 2h

[4.41]

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Liquid Crystals

















































c

















Figure 4.36. Origin of the cuneitropism. (a and b) When h = constant, the distortion energy of the dowser texture does not depend on its azimuthal orientation. (c and d) In the wedge geometry, the distortion energy varies with the angle ψc between the dowser field d and the thickness gradient g

For symmetry reasons, this expression does not depend on the azimuthal orientation ψc of the dowser field d when the thickness is uniform. When the upper plate is tilted by a small angle γ (Figure 4.36(b)), the director rotates between the limit surfaces by the angle α = π − γ cos ψc and the energy of the dowser state becomes Fdow = (π − γ cos ψc )2

K K πK ≈ π2 − γ cos ψc 2h 2h h

[4.42]

As γ cos ψc = g · d, we can say that the dowser field d couples linearly with the thickness gradient g. The resulting torque given by Γc = −

πK dFdow πK =− γ sin ψc = d×g dψc 2h 2h

[4.43]

tends to align d along the gradient g. The shape ψc (ξ) of the 2π-wall (see Figure 4.33(i’)) results from the balance of the elastic and cuneitropic torques Kh d2 ψc πK − γ sin ψc = 0 2 2 dξ 2h

[4.44]

Physics of the Dowser Texture

239

By analogy with the π-wall discussed above, we have:  ξ ψc (ξ) = 4 arctan e− ξc

[4.45]

with  ξc = h

1 πγ

[4.46]

The width ξc of the 2π-wall should thus depend on the radius Rl because, in the sphere/plane geometry, the local thickness of the gap h as well as the tilt angle γ vary with the distance r from the center as  h(r) = hmin + Rl 1 −

r2 1− 2 Rl

 [4.47]

and γ(r) =

1 dh r  = dr Rl 1 −

r2 Rl2

[4.48]

This theoretical prediction fits the experimental results plotted in Figure 4.34(b) well. 4.6. Electrotropism 4.6.1. Definition of the electrotropism The rheotropism and cuneitropism discussed previously were due to linear couplings between the dowser field d and vectorial perturbations: the Poiseuille flow v and the thickness gradient g. Similarly, by definition, the electrotropism is a manifestation of a linear coupling of the dowser field d with the electric field E. We will see below that the coupling of the dowser field with the electric field E (parallel to the (x,y) plane) is in practice more complex than linear because the dowser field d is submitted to the torque ΓE = P × E

[4.49]

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Liquid Crystals

exerted by the electric field E on the polarization P, which is a sum P = Pf e + Pind (E)

[4.50]

of the spontaneous (flexo-electric) polarization (see section 4.6.2) Pf e = Pf e d

[4.51]

and of the anisotropic part of the polarization Pind induced by the electric field: Pind = (o a h/2)(d · E)d

[4.52]







 





 





Figure 4.37. Spontaneous flexo-electric polarization of the dowser texture

Therefore, the detailed expression of the electric torque ΓE = Pf e d × E + (o a h/2)(d · E)d × E

[4.53]

also contains the second-order term due to the dielectric anisotropy of nematics. When the dowser field d makes an angle ψE with the electric field E, the electric torque can also be written as ΓE = −Pf e sin ψE − (o a h/2)E 2 sin 2ψE

[4.54]

We will see in the following that in experiments with wound up dowser fields, it is possible to detect both terms of this expression.

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241

4.6.2. Flexo-electric polarization Before that, it is important to emphasize that the spontaneous polarization of the dowser texture was expected from symmetry considerations similar to those of Meyer (1969) who announced, for the first time, the concept of the flexo-electricity and namely the linear relationship between distortions of the director field in nematics and the dielectric polarization (see equation [4.56]). Indeed, as stated in section 4.1.3.3, the symmetry C2v of the dowser texture is so low that it allows for the existence of a spontaneous polarization Pf e parallel to the twofold axis C2 , that is to say parallel to the dowser field: Pf e = Pf e d (the subscript fe refers to the flexo-electric origin of the polarization). Using the explicit expression of the director field n = (sin θ cos ϕ, sin θ sin ϕ, cos θ) with θ = πz/h

[4.55]

and the de Gennes’ definition of the polarization density (de Gennes and Prost 1993) → − − → pf e = e1 n ∇ · n + e3 ( ∇ × n) × n

[4.56]

we can calculate the polarization per unit area of the dowser texture  Pf e =

h/2 −h/2

pf e dz = Pf e d

[4.57]

with Pf e =

π (e3 − e1 ) 2

[4.58]

Let us emphasize that this flexo-electric polarization of the dowser texture is independent of the gap thickness h. 4.6.3. Setup The setup used for detection and measurement of the flexo-electric polarization is depicted in Figure 4.38. It is essentially the DDC1 device described in section 4.2.1 in which the glass slide is equipped with one of the four systems of transparent ITO electrodes shown in Figure 4.38(c).

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Liquid Crystals

Figure 4.38. Setup. (a) General perspective view. (b) Cross-section of the sample. (c) Systems of ITO electrodes

4.6.4. The first evidence of the flexo-electric polarization The first evidence of the flexo-electric polarization of the dowser texture was obtained in experiments illustrated in Figure 4.39. They start with the synchronous winding of the dowser field discussed in section 4.4.2. After suppression of the excitation, the wound up dowser field reaches in a few seconds its quasi-equilibrium state with equally spaced black isogyres shown in Figures 4.39(a) and (d). Orientations of the dowser field indicated that these pictures have been detected by a transitory application of a divergent radial flow due to a small reduction of the gap thickness h. For example, in Figure 4.39(b) the orientation of the dowser field (white arrows) in enlarged isogyres is parallel to the Poiseuille flow (yellow arrows). Subsequently, using the one-gap system of electrodes, an electric field has been applied to the wound up dowser texture. The formation of 2π-walls well visible in Figures 4.39(c) and (e) unveils the presence of the flexo-electric polarization in both MBBA (c) and 5CB (e). Let us note a crucial difference between these two pictures: in 5CB, the dowser field d in enlarged isogyres separating the 2π-walls is parallel to the electric field E, while in MBBA it is antiparallel to E. The conclusion is that the sign of the flexo-electric polarization Pf e = (π/2)(e3 − e1 ) is positive in 5CB and negative in MBBA.

Physics of the Dowser Texture
















 

 

 






 






































243



Figure 4.39. The first evidence of the flexo-electric polarization in MBBA (a–c) and 5CB (d–e). (a and d) Wound up dowser texture in quasi-equilibrium. (c and e) 2π-walls induced by the electric field E. (b) Rheotropic detection of the wound up dowser field

4.6.5. Measurements of the flexo-electric polarization Values of the flexo-electric polarizations in MBBA and 5CB have been measured in experiments (see Figure 4.40) performed with a weak enough electric field to neglect the contribution of the second-order term due to the dielectric anisotropy a in equation [4.54]. They consist of measuring small deformations of the ground state radial configuration of the dowser field d submitted to an alternating electric field. The ground state radial dowser field observed in polarized monochromatic light (see Figure 4.40(b)) displays an interference pattern made of isogyres (maltese cross) and isochromes (rings indexed N = 3,4,...). Upon application of a sinusoidal voltage U (t) = Uo cos(2πf t) (f = 20 mHz) to the electrodes ITO1 and ITO2, we observe angular oscillations δα(t) of the black isogyres inside the ITO1/ITO2 gap. The orientation of crossed polarizers has been adjusted for a maximum intensity gradient dI/dα in the point P located in the middle of the ITO1/ITO2 gap (see Figure 4.40(d)). Oscillation δI(t) of the light intensity measured in P (see Figure 4.40(c)) is thus δI(t) = (dI/dα)δα(t). The intensity gradient dI/dα in P was determined from the plot I(α) in Figure 4.40(d). Knowing this, the angular oscillations of the dowser field δϕ(t) = −δα(t) have been plotted in Figure 4.40(e). The same measurements of δϕ(t) performed with 5CB are plotted in Figure 4.40(f). It is obvious that oscillations δϕ(t) of the dowser field have opposite signs in plots of Figures 4.40(e) and (f). The two schemes in Figures 4.40(g) and (h) show that in MBBA and 5CB the dowser field d is rotating, respectively, in anticlockwise and clockwise directions when, at t = 0, the electric field is antiparallel to the y axis.

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Using the definition of the electric torque P × E, we obtain a confirmation of the result discussed above: in MBBA and 5CB, the flexo-electric polarization Pf e is, respectively, antiparallel and parallel to the dowser field d (see Figures 4.40(g) and (h)). (



 

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Figure 4.40. Measurements of the flexo-electric polarization in MBBA and 5CB. (a and b) The ground state radial configuration of the dowser field observed in white and monochromatic light. (c) Intensity of the transmitted light measured in point P. (d) Plot of the intensity measured along the circular path A-P-B. (e and f) Angular oscillations δϕ of the dowser field inferred from variations of the intensity of light detected in point P. (g and h) Interpretation of the angular oscillations in terms of the electric torque Pf e ×E

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245

The two plots in Figures 4.40(e) and (f) have another interesting feature: oscillations of the dowser field are shifted by π/2 with respect to the applied voltage U(t), that is to say to the electric torque driving them. This means that the restoring elastic and cuneitropic torques are negligible so that the equation of motion can be simplified as follows: −

γ1 h dδϕ − Pf e Eo cos(2πf t) = 0 2 dt

[4.59]

At the low frequency f = 20 mHz of the applied voltage, the electric field in the gap of width lg is generated in the so-called conductive regime for which Eo = Uo /lg . Then, the equation of motion results in δϕ = δϕo sin(2πf t)

[4.60]

with δϕo =

Pf e U o γ1 hπf lg

[4.61]

and the flexo-electric polarization is given by Pf e =

δϕo γ1 hπf lg Uo

[4.62]

With lg = 1 mm, Uo = 5 V (MBBA) or 3 V (5CB), γ1 = 100 mPa·s measured by Oswald et al. (2013) and h = 31 μm measured from the interference pattern in Figure 4.40(b), we get Pf e ≈ −2.7pC/m or e3 − e1 = −1.7pC/m in MBBA

[4.63]

Pf e ≈ +4.2pC/m or e3 − e1 = +2.7pC/m in 5CB

[4.64]

and

In the case of MBBA, the value of e3 -e1 given in equation [4.63] agrees in sign and the order of magnitude with the result e3 -e1 = -3.3 pC/m obtained for the first time by Dozov et al. (1982). In the case of 5CB, the positive sign of e3 -e1 is contrary to the result e3 -e1 = −11 pC/m obtained by Link et al. (2001) from observation of 2π-walls. We will see below that electro-osmotic flows discussed in the following section contribute to the formation of the 2π-walls so they could be considered as a plausible reason for this disagreement.

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Liquid Crystals

4.7. Electro-osmosis 4.7.1. One-gap system of electrodes An experiment performed with 5CB and illustrated by Figure 4.41 provides evidence of electro-osmotic flows in a wound up dowser texture submitted to a strong electric field. The general view of the wound up dowser texture in quasi-equilibrium in Figure 4.41(a) unveils the cuneitropic splitting of the isogyres’ pattern into 2π-walls discussed previously in sections 4.4.1.1 and 4.5. It allows us to identify orientations of the dowser field (white arrows) in enlarged black isogyres where it is parallel or antiparallel to the x direction. This quasi-equilibrium wound up dowser field is then submitted to the electric field E = U/lg due to the voltage U(t) applied to the pair ITO1-ITO2 of electrodes separated by the gap of width lg = 0.3 mm. U(t) varies through time as follows: +0V⇒-19V⇒+19V⇒-19V⇒+19V⇒-19V⇒0V. The resulting evolution of the isogyres pattern is illustrated by the series of six pictures in Figure 4.41(b) as well as by the spatiotemporal cross-sections in Figures 4.41(c) and (e), which were extracted from a video along lines AB and CD defined in the picture labeled t1. 4.7.1.1. Effects due to the electric field inside the gap In agreement with observations reported in section 4.6.4, the picture labeled t2 in Figure 4.41(b) shows that the electric field E inside the gap generates 2π-walls separated by strips in which the dowser field d is parallel to E. In a step-like reversal of the electric field, these 2π-walls split into pairs of π-walls moving apart (in the picture t3) until their recombine into new 2π-walls in the picture t4. Motions of the π-walls during this splitting-recombination process are clearly visible in the spatiotemporal cross-section of Figure 4.41(c). The existence of well-defined π walls is due to a large intensity of the electric field for which the second-order dielectric anisotropy term in equation [4.54] becomes significative. Their motion is another evidence of the role played by the first-order electrotropic term. 4.7.1.2. Effects due to the electric field outside the gap, evidence of electro-osmosis Surprisingly, the isogyres’ pattern in Figure 4.41(b) is also affected by application of the voltage U(t) in areas outside the gap where the electric field vanishes. We will point out in the following that this is a manifestation of the electro-osmosis. By definition, electro-osmosis refers to flows driven by forces exerted by the electric field on mobile charges in electric double layers at walls of a channel filled with a liquid. Let us suppose that in the experiment illustrated by Figure 4.41, the

Physics of the Dowser Texture

247



    



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mobile charges at surfaces inside the gap are negative, as shown in Figure 4.42(c). The electric field E parallel to the x axis sets them in motion of velocity veo in the opposite -x direction.

- 

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Figure 4.41. Evidence of electro-osmotic flows. (a) General view of a quasi-equilibrium wound up dowser texture. (b) Formation and evolution of the π and 2π-walls in a strong alternating electric field. (c–e) spatiotemporal cross-sections taken along lines AB and CD defined in (a). (d–f) Numerical simulations (Pieranski and Godinho 2019a).

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Liquid Crystals

In a gap of infinite length, the flow would be plug-like v(z) = veo . However, as the gap has a finite width lg , the flow profile outside the gap must be parabolic, i.e. characteristic of Poiseuille flows. In virtue of the rheotropism, these Poiseuille flows of amplitude v(z = 0) = vP out outside the gap deform the equidistant isogyres’ pattern into 2π-walls separated by large strips in which the dowser field d is parallel to vP out . 4.7.1.3. Electro-osmotic effects inside the gap In the absence of external forces, the Poiseuille flows outside the gap must be accompanied by pressure gradients ∂p/∂x shown in the scheme in Figure 4.42(e). As the directions of Poiseuille flows are the same on the left and right sides of the gap, there must be a pressure difference Δp between the two extremities of the gap. The plug flow inside the gap is thus accompanied by an adverse Poiseuille flow of amplitude v(z = 0) = vP in driven by the pressure gradient Δp/lg . In summary, inside the gap the deformation of the dowser field is due to a simultaneous action of two torques, the electrotropic and rheotropic ones: Γ = Pf e d × E + αrt d × vP in

[4.65]

2 with αrt = − 2α π > 0 in 5CB.

From conservation of the global flow rate, we have Q=

2 2 vP out h = veo h − vP in h 3 3

[4.66]

ρ=

|vP in | |vP out |

[4.67]

with

we obtain vP in = −

3 ρ 3 ρ veo = − Ceo E 21+ρ 21+ρ

[4.68]

Ceo (< 0 in 5CB) is the coefficient relating the velocity of mobile charges in the double layer to the electric field acting on them. Finally, the global torque due to the electric field can be written as Γ = Papp d × E

[4.69]

Physics of the Dowser Texture

249

with Papp = Pf e − αrt

3 ρ Ceo 21+ρ

[4.70]

  

 

 



 



 







 




  




 











 


  



 

 

 





 







Figure 4.42. Poiseuille flows driven by electrosmosis in the one-gap system of electrodes. (a) Isogyres’ pattern deformed by application of the electric field. (b) Flow stream lines. (c) Flow profiles. (d) Cross-section of the nematic drop contained between the glass slide and the lens. (e) Pressure variation inside the drop

250

Liquid Crystals

In 5CB, the second term is positive. If it is large enough, the apparent polarization can be positive even if the true flexo-electric polarization is negative. 4.7.2. Two-gap system of electrodes The existence of electro-osmotic flows was confirmed in experiments with other systems of electrodes. In the system shown in Figure 4.38(c2), the two gaps, ITO1-ITO2 and ITO2-ITO3, can be connected either in series or in parallel. In the case of the connection in series (see Figure 4.43(a)), the electric fields E in the two gaps have the same x direction so that, by symmetry, the x components of the global electro-osmotic fluxes Qx satisfy the following equality: Qx (x, y) = Qx (−x, y)

[4.71]

In the case of the connection in parallel (see Figure 4.43(b)), the electric fields E in the two gaps have opposite directions so that, by symmetry, we have Qx (x, y) = −Qx (−x, y) and by consequence Qx (0, y) = 0

[4.72]

Due to the mirror symmetry of the droplet with respect to the (x,z) plane, we also have Qy (x, y) = −Qy (x, −y) and by consequence Qy (x, 0) = 0

[4.73]

Moreover, in the center (0,0) of the droplet where the thickness is minimal, the incompressibility condition takes the form ∂Qx ∂Qy + =0 ∂x ∂y

[4.74]

Using equations [4.72]–[4.74], we obtain: Q = (ax, −ay)

[4.75]

which corresponds to a flow with a stagnation point SP(0,0) (see Figure 4.43(b)). For these reasons, the isogyres’ pattern is equidistant in the stagnation point and the width ξ of the 2π-walls on the y axis decreases with the distance y from SP because the velocity |vyo | of the Poiseuille flow grows with y. Let us also note that the isogyre enlarged by this flow parallel to the y axis is squeezed, in the middle of a 2π-wall, in adjacent gaps where the electro-osmotic flow takes the orthogonal direction, i.e. parallel to the x axis.

Physics of the Dowser Texture




 










































































 






 
























Figure 4.43. Electro-osmotic flows in the two-gap system of electrodes. (a) Gaps connected in series. (b) Gaps connected in parallel

251

252

Liquid Crystals

4.7.3. Convection of the dowser field As stated above, in the case of gaps connected in parallel, the Poiseuille flow on the y axis is parallel to it in virtue of equation [4.75]. For this reason, it not only generates the 2π-walls but also convects them. Indeed, the viscous term in the equation of motion of the dowser field, i.e. of the phase ϕ (see section 4.16.2), also contains in principle, beside the time derivative, the convection term:  Γvisc z

= −γef f

→ − ∂ϕ − +→ v · ϕ ∂t

 [4.76]

→ − The isogyres are orthogonal to the y axis so that the gradient ϕ is parallel to it. − → In the series case, the Poiseuille flow along the x axis is orthogonal to ϕ so that the convection term vanishes. On the contrary, in the parallel case, the Poiseuille flow and → − the gradient ϕ are parallel so that the wound up dowser field is convected. All this is clearly visible in Figure 4.44 on spatiotemporal cross-sections taken along the line CS defined in Figure 4.43. In the series case (Figure 4.44(a)), trajectories of the isogyres are parallel to the time axis (except for the short splitting-recombination sequence due to the reversal of the voltage U). In the parallel case (Figure 4.44(b)), the trajectories of the isogyres are skewed and the angle ∂y/∂t they make with the time axis increases with the distance y from the stagnation point SP. The sign of this angle changes upon the reversal of the voltage U because the direction of the electro-osmotic flows v is reversed. 

























Figure 4.44. Convection of the dowser field. Spatiotemporal cross-sections along the line CS defined in Figure 4.43 (a) Gaps connected in series. (b) Gaps connected in parallel




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253

4.8. Dowser texture as a natural universe of nematic monopoles 4.8.1. Structures and topological charges of nematic monopoles In section 4.1.3.2, we stressed that the dowser texture is a natural universe of nematic monopoles because the +2π and −2π defects (called dowsons d+ and d-) of the dowser field are topologically equivalent to nematic monopoles. In other words, we could also say that dowsons are nematic monopoles embedded in the dowser texture. The concept of monopoles as singularities of vector fields arises from the 19th-century works of Poincaré (1886). Their existence in nematics was postulated in 1972 by Nabarro (1972). As we have seen in section 4.1, nematic monopoles have also been considered by de Gennes in the first classification of defects in systems with order parameters resulting from broken symmetries (de Gennes 1972). The general classification of monopoles in nematics is based on their topological charge defined by Kleman and Lavrentovich (2006) as the following integral over the surface of a sphere S 2 enclosing them: 1 N= 4π

 S2

dθdϕn · [∂θ n × ∂ϕ n]

[4.77]

It is very instructive to calculate the topological charge of the so-called radial and hyperbolic monopoles represented in Figure 4.45. They occur, for example, alternatively on the axis of homeotropic cylindrical capillaries shown in Figure 4.4(b). Experiments have shown that two such adjacent monopoles attract each other, come together and finally annihilate. One could think therefore, by analogy with electrostatic charges, that the topological charges of the radial and hyperbolic monopoles should be opposite and add to zero. Actually, this is a more subtle issue and we will discuss it below. Let us start with the radial monopole (see Figure 4.45 “radial”) with the director field expressed in spherical coordinates (r, θ, ϕ) as n(θ, ϕ) = [sin(θ)cos(ϕ), sin(θ)sin(ϕ), cos(θ)] Its topological charge calculated from equation [4.77] is N = +1.

[4.78]

254

Liquid Crystals













 o









 

 




Figure 4.45. Continuous transformation relating the radial and hyperbolic configurations of the nematic monopole

Similarly, the director field of the hyperbolic monopole (see “hyperbolic” in Figure 4.45) can be written as n(θ, ϕ) = [−sin(θ)cos(ϕ), −sin(θ)sin(ϕ), cos(θ)]

[4.79]

and its topological charge calculated from equation [4.77] is the same: N = +1. The equality of topological charges of the radial and hyperbolic hedgehogs was understood and emphasized by Kurik and Lavrentovich (1988) who showed that structures of the radial and hyperbolic hedgehogs are connected by a continuous transformation, represented in Figure 4.45, and which analytically can be expressed as → − n (θ, ϕ) = [sin(θ)cos(ϕ + ψo ), sin(θ)sin(ϕ + ψo ), cos(θ)]

[4.80]

with “the phase” ψo varying from zero (for the radial monopole) to π (for the hyperbolic monopole). Therefore, not only is there no difference in the topology of the radial and hyperbolic monopoles but also there exists a continuous set of structurally different monopoles having the same topological charge N = 1 (or N = -1). In fact, the set of structurally different monopoles is even larger than those defined by equation [4.80] because the symmetry with respect to the C∞ //z axis is not necessary, as noted by Nabarro (1972). In this context, the annihilation of a pair of the radial and hyperbolic monopoles occurring in capillaries appears to be paradoxical. To solve this paradox, let us note that the sign of the topological charge N, calculated from equation [4.77], is reversed upon the reversal n → −n of the director field. Therefore, if the sign of the director

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255

field in one of the monopoles of the pair is reversed, its topological charge becomes N = -1 and the sum of topological charges is zero as expected. Reversal of the director field in one of the monopoles of the pair is in fact a necessity, as we will point out below using the example of a pair (d+, d-) of dowsons in the dowser texture. 4.8.2. Pair of dowsons d+ and d- seen as a pair of monopoles Let us consider the dowser texture containing a pair of d+ and d- dowsons, i.e. a pair of the +2π and −2π disclinations of the two-dimensional dowser field d (see Figure 4.46). When they come in contact, their annihilation appears to be obvious from a topological point of view.






 














Figure 4.46. Pair of dowsons d+ and d- in the dowser texture. (a) View in polarized light; (b) The corresponding dowser field d

However, when the two defects are considered as monopoles in the three-dimensional director field n, their annihilation is more enigmatic in view of the former considerations. We must prove that the director field in one of the monopoles is reversed with respect to the configurations shown in Figure 4.45. For this purpose, let us consider the detailed views of the director field n inside the 2π wall connecting dowsons d+ and d- (see Figures 4.47(I) and (II)). In the first case (I) of the clockwise 2π wall (rotation of the dowser field along the x axis), the defect labeled +2π(d+) has the structure of the radial monopole in 3D (see

256

Liquid Crystals








  













+2



  




-2




 

+2


Figure 4.45), while the second −2π(d−) defect has the structure of the hyperbolic monopole (defined in Figure 4.45) with a reversed director field. For this reason, the total topological charge of this pair is zero.




















































 





























Figure 4.47. The +2π and −2π defects (dowsons d+ and d-) of the dowser field d as monopoles of the director field n. (a) Top views showing a pair of dowsons d+ and din the dowser field d. The 2π walls in I and II have opposite winding directions. As a result, positions of dowsons are exchanged. (b) Lateral views of the director field n in the (y,z) plane. (c and d) Lateral views of the director field in the (x,z) plane. (e and f) Identification of topological charges of monopoles corresponding to dowsons d+ and d-

In the second case (II) of the anticlockwise wall, both defects d+ and d- have the structure of the hyperbolic monopole in 3D, with different orientations. The monopole corresponding to the −2π(d−) defect has its C∞ axis oriented in the x direction, while the C∞ axis of the second monopole has the z orientation. Let us stress that the director field of the second monopole is reversed with respect to the first so that its topological charge is -1 and the total charge of the pair is zero again.

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We will come back to the discussion of the annihilation of pairs of monopoles in section 4.10 first but we must explain how monopoles are generated. 4.8.3. Generation of monopole–antimonopole pairs by breaking 2π-walls In section 4.1, we gave an explanation of the scarcity of monopoles in nematics. Namely, the director field on a sphere surrounding a monopole mapped on the space of the nematic order parameter (projective plane equivalent to one hemisphere) covers it twice; and such a complex configuration of the director field can hardly occur spontaneously during the isotropic ⇒ nematic transition. In section 4.8.3.2 we will show that, on the contrary, the configuration of the director field in 2π-walls in a wound up dowser texture is so close to that of monopoles that an adequate perturbation can result in the generation of monopoles. To be more precise, we must say the generation of monopole–antimonopole pairs because generation of a single monopole is precluded for topological reasons. 4.8.3.1. Generic experiment Before that, let us describe first a typical experiment, illustrated by Figure 4.48, in which such monopole–antimonopole pairs are generated. It starts by the synchronous winding of the dowser field in the DDC1. This process was described in detail in section 4.4.2. Here, the winding process is illustrated by a series of seven pictures (a– g). It starts from the quasi-static radial texture with the residual monopole located off the center. The direction of the 2π-wall indicates that the residual monopole, located at its extremity, is shifted to the right. Six subsequent pictures show that six new 2π-walls are generated by the up-anddown motion s(t) of the glass slide. The next and last move of the slide is directed downward, more rapid and has a larger amplitude so that the width ξrt of the 2π-walls is strongly reduced, as shown in the eighth picture (Figure 4.48(h)). The next two Figures 4.48(i) and (j) show the broken walls during relaxation after cessation of flows. Using the convention introduced in section 4.1.3.2, the monopoles and antimonopoles are labeled as +2π and −2π. Their identification results from the detailed analysis of the isogyres’ pattern. As an example, we show in Figure 4.48(k) a detailed view of the dotted rectangle from the picture 4.48(j). Orientations of the dowser field met along the anticlockwise circuit surrounding the extremity of the 2π-wall are redrawn in the picture (Figure 4.48(l)). Their rotation in the opposite clockwise direction is the fingerprint of the antimonopole -2π.

258

Liquid Crystals




























Figure 4.48. Generation of monopole–antimonopole pairs. (a) Radial dowser field. (b–g) Generation of six 2π-walls. (h) Thinning of 2π-walls by a rapid Poiseuille flow results in them breaking. (i–j) Relaxation of the dowser field after cessation of flows. (k) Identification of the dowser field along an anticlockwise circuit surrounding an antimonopole. (l) Corresponding clockwise rotation of the dowser field

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4.8.3.2. Topological relationship between 2π-walls and monopoles In Figure 4.49, we show another example of the generation of monopole–antimonopole pairs through the rheotropical thinning and breaking of 2π-walls. The generation process is resolved in time in pictures a1–a6. Picture b shows a detailed view of the dotted rectangle defined in a1. The dowser field d, drawn with white arrows, has been inferred from the pattern of isogyres. The four schemes in Figure 4.49(c) show perspective views of the director field in a dowser texture. They are labeled from 0◦ to 270◦ because their director field, when mapped on the sphere in d, forms meridians with these longitudes. For example, the director field from the scheme labeled 0° is mapped on the prime meridian 0◦ . Clearly, the director field from the (y,z) cross-section of one 2π wall covers the whole sphere once. As the space of the nematic order parameter corresponds to one hemisphere, the director field from the (y,z) cross-section of one 2π wall covers it twice, once on the whole North hemisphere and the second time on the whole South hemisphere. Let us recall that, as stated already in section 4.1.1.3, the director field on a sphere surrounding a monopole also covers the space of the nematic order parameter twice. In the present case of monopoles and antimonopoles located at extremities of a broken 2π-wall, we should rather consider the director field on a cylinder surrounding the monopole or the antimonopole. This property is illustrated once again in Figure 4.49(d) labeled “monopole”. In the case of the antimonopole, this is also true. The only difference is that during the mapping following the dotted circuit oriented West, the sphere of the order parameter is covered in the opposite east direction. We can thus say that the director fields (1◦ ) on a (x,z) cross-section of a 2π-wall and (2◦ ) on a cylinder surrounding a monopole (or antimonopole) are topologically equivalent. 4.8.3.3. Conjecture on the rheotropic breaking of 2π-walls During generation of pairs of monopoles (see Figures 4.48 and 4.49), segments of 2π-walls are replaced by a homogeneous dowser field texture oriented in the direction of the Poiseuille flow. From the point of view of the energy balance, on the one hand, the gain of the elastic energy is proportional to the segment’s length L of the suppressed wall of tension τrt : ΔEwall =-τwall L. On the other hand, there is an energy expense corresponding to the total energy of the two defects: ΔEM +AM =EM +EAM . Let us first estimate the energy gain. During the rheotropic generation process, the 2π-walls of the wound up dowser field are thinned by the Poiseuille flow of amplitude vo . From the balance of the elastic and rheotropic torques: Kh ∂ 2 ϕ 2α2 + vmax sin ϕ = 0 2 ∂y 2 π

[4.81]

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we get   y ϕ(y) = 4 arctan e− ξrt



[4.82]



 


























"





"

 "

!



" !

"


 




 

Figure 4.49. Topological relationship between 2π-walls and monopole–antimonopole pairs. (a) Time resolved generation of monopole–antimonopole pairs. (b) Detailed view of the dowser field in the dotted rectangle defined in the picture a1. (c) Perspective views of the director field inside the 2π-wall. (d) Mapping of the director field from the 2π-wall onto the space of the nematic order parameter

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with the thickness of the wall  ξrt =

π hK 4 −α2 vmax

[4.83]

which decreases with the flow velocity vo . The elastic energy per unit length of a 2π-wall of width ξrt or equivalently its tension τwall given by the integral  τwall =

∞ −∞

1 Kh 2 2



∂ϕ ∂x

2 dx = 2K

h ξrt

[4.84]

is inversely proportional to the wall width ξrt or, using expression [4.83], proportional √ to vmax . In conclusion, the energy gain from suppression of a segment of a 2π-wall √ grows as L vmax . The energy loss depends on the configuration of the generated monopoles, radial or hyperbolic, and given as (see the review article on point defects by Kleman and Lavrentovich (2006)): Erad = 8πR(K − K24 ) and Ehyp =

8π R(K − K24 ) 3

[4.85]

with R corresponding to the size of their volume. In statu nascendi, the size R is of the order of ξrt /2. Using equations [4.85] and [4.84], we finally obtain: ΔE = 2

8π ξrt hL (K − K24 ) − 2K 3 2 ξrt

[4.86]

The critical thickness corresponding to ΔE = 0 is then given as  ξc =

√ K 6 hL 8π K − K24

[4.87]

As the critical thickness decreases with the length L of the destroyed segment of the 2π-wall, we could think that the 2π-wall is unstable with respect to generation of dowsons pairs.

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This result is contrary to experimental facts reported above. However, we still do not know what the detailed mechanism is of the wall destruction→ dowsons’ pair generation process. If it is a continuous transformation with some energy barrier (per unit length of the wall) to overcome, the 2π-wall would not be unstable but only metastable with respect to the generation of dowsons pairs. 4.9. Motions of dowsons in a wound up dowser field 4.9.1. Single dowson in a wound up dowser field The energy balance given in equation [4.86] still holds after generation of the dowsons’ pair and, in this context, it has another meaning confirmed by observations: dowsons are moving apart because the energy ΔE decreases when the distance L between dowsons grows. When the distance L is much larger than the width ξwall of the wall, the dowsons d+ and d- of the pair become independent and their motion inside the wound up dowser field can be analyzed separately. For example, we show in Figures 4.50 and 4.51 the simplest case of one dowson d+ embedded in the dowser field wound up by means of the Dowsons Collider 2, discussed in section 4.4.3 and shown again in Figure 4.50(a). Due to the structure of the DDC2 device, the distribution of the rheotropic torque produced by it is antisymmetric with respect to the reflection in the (x,z) plane (see − → Figure 4.50(b)) and the phase gradient ∇ϕ on the x axis is orthogonal to it. In polarized light, this last feature of the wound up dowser texture is obvious in Figure 4.50(c): isogyres (which are lines of constant phase) are parallel to the x axis. The solitary dowson d+ is indicated in Figure 4.50(c) by an arrow, while in Figure 4.50(d) it is located in the dotted rectangle. As the picture in Figure 4.50(d) was taken 380 s after the picture in Figure 4.50(c) , it is obvious that the dowson is moving along the x axis. The motion of the dowson d+ is illustrated in more detail in Figure 4.51. First of all, the series of five pictures a–e shows successive positions of the dowson at t = 0, 380, 520, 620 and 700 s. These pictures also show that the local wavelength λ of the wound up dowser field varies with the x position of the dowson. These pictures were extracted from a time-lapse video containing 55 pictures taken at intervals of 20 s. Using all of them, the velocity v = dx/dt and the width ξwall = λ of the wall, terminated by the dowson, were measured and plotted as red crosses in Figure 4.51(f). Clearly, the velocity of the dowson depends on λ. Moreover, as on this log-log plot, the experimental points seem to be aligned on a line, they were fitted to the power law v ∼ λα with the best fit obtained for α=-1.24.

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Figure 4.50. Motion of a dowson d+ in a dowser field wound up in the Dowsons Collider 2. (a) Scheme of the Dowsons Collider 2. (b) Distribution of the rheotropic torque. (c and d) Isogyres pattern of a wound up dowser field with one dowson d+

4.9.2. The Lorentz-like force The motion of the dowson d+ is driven by the tension τwall of the wall to which it is attached. In the absence of flows and in quasi-equilibrium, the phase ϕ of a wound up dowser field must satisfy equation [4.81] reduced to the elastic torque alone. In two dimensions (x,y), it takes the form Kh ϕ = 0 2

[4.88]

In the vicinity of the x axis, its solution ϕ(y) =

2π y λ

involves the wavelength λ(y) resulting from the winding process.

[4.89]

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( m/s)











  



 

















( m)









Figure 4.51. Motion of a dowson d+ in a dowser field wound up in the Dowson Collider 2. (a–e) Pictures taken at t = 0, 380, 520, 620 and 700 s. (f) Plot of the dowsons’ velocity versus the width λ of the 2π-wall to which it is attached. Crosses – experimental data. Dotted line – the 1/λ dependence expected from equation [4.96]. Red line – the best fit to the power law v ∼ λα with α = −1.24

The elastic energy per unit length of the 2π-wall (or equivalently its tension τwall ) is given by the integral  τwall =

λ 0

1 Kh 2 2



∂ϕ ∂x

2 dx =

πKh 2π 2 λ

[4.90]

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Figure 4.52. Lorentz force on a vortex in superconductors and on a dowson in the dowser texture. (a) Phase singularity of a vortex or of a dowson. (b) Phase gradient due to the transport current in a superconductor or to the winding of the dowser field. − → → (c) Lorentz force proportional to ∇φ × − z

The corresponding force acting on the dowson is proportional to the phase gradient − → ∇ϕ and is orthogonal to it so that it can be alternatively expressed as → πKh − → − → τ wall = ∇ϕ × − m 2

[4.91]

→ where − m is a unit vector parallel (or antiparallel) to the z direction. In terms of the classification proposed by de Gennes (see section 4.1), dowsons are similar to vortices in two-dimensional superconductors because they both correspond − → to Δϕ = 2π defects of the phase field ϕ(x, y). In the presence of a current density j , vortices are submitted to the Lorentz force: − → → − Φo → − fL= j × c

[4.92]

− → → − → m of the exerted by the current density j on the quantified magnetic flux Φ o = Φo − vortex. Knowing that the current density is proportional to the phase gradient: → − → − j = qΨ2 ∇φ m

[4.93]

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we obtain: → − →  − → ( ∇φ × − m) f L = qΦo Ψ2 mc

[4.94]

We can now conclude that the force acting on a dowson in a wound up dowser texture, given by equation [4.91], is analogous to the Lorentz force exerted by the density current on vortices in superconductors, given by equation [4.94]. 4.9.3. Velocity of dowsons in wound up dowser fields During the motion of a dowson with velocity v, the driving tension of the wall τwall (or the Lorentz force), given by equation [4.91], is opposed by the force τvisc resulting from the viscous dissipation, which can be estimated as: τvisc = πγ1 hv

[4.95]

From the balance of forces, we obtain: v=

πK 1 2γ1 λ

[4.96]

The 1/λ dependence of v(λ) is plotted in Figure 4.51 with the dotted line. As mentioned above, the experimental data are better fitted with the power law v ∼ λ−1.24 . A plausible cause of this disagreement is that, as we point out below, the force fd acting on a dowson inserted in a wound up dowser field can differ from the Lorentz force ∼ 1/λ (equation [4.92]) calculated above. 4.9.4. The race of dowsons The second piece of evidence of the deviation from the fd ∼ 1/λ dependence came from an experiment called “the dowsons’ race” (see Figure 4.53), which starts by the rheotropic breaking (see section 4.8) of a large set of adjacent 2π-walls in a wound up dowser field, shown in Figure 4.53(a). By this means, we obtain two sets of dowsons d+ and d- aligned on two lines, as in Figure 4.53(b). Pulled by tension of walls to which they are attached, dowsons d+ and d- start, at t = 0 s, to move in opposite directions. Let us follow the motion of two adjacent dowsons d+ indicated by arrows and labeled d1 and d2 in Figure 4.53(b). At the beginning of the race, velocities of all dowsons d+ are the same in order that they remain aligned on one moving line. In particular, after 19 s of the race, dowsons d1 and d2 are still running side by side.

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Surprisingly, a dozen seconds later an instability occurs: the set of running dowsons is divided into two subsets, which could be called “slow” and “fast”. Therefore, in picture d, taken at t = 78 s, the slow dowson d1 is behind the fast dowson d2. In the next two pictures, taken at t = 183 s and 337 s, the gap between d1 and d2 becomes larger.












































Figure 4.53. The dowsons’ race. (a) Dowser texture wound up in the DDC2 device. (b) At t = 0, two sets of dowsons d+ and d- generated rheotropically (see section 4.8) start to move in the opposite direction. (c–f) Pictures taken at t = 19, 78, 183 and 337 s (collaboration with Hadjiefstatiou and Montagnat (2017))

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Figure 4.54. Race of dowsons. (a–g) Pictures taken at intervals of 20 s. (h) Spatiotemporal cross-section

Other features of the race are illustrated in Figure 4.54. The series of pictures a–g gives a more detailed view of dowsons d1 and d2. On the spatiotemporal crosssection extracted, along the strip defined in Figure 4.54(e), from a video recorded at rate of 1 frame/s, trajectories of dowsons d1 and d2 are well visible. Initially, they coincide but after about 50 s they separate. Velocity v2 of the dowson d2 is about three times larger than that of the dowsons d1, i.e. v1 . The trajectory of the dowson d2 with the velocity expected from the Lorentz force ∼ 1/λ alone, v2 = 2v1 , is also plotted with a dash dotted line. The explanation of the observed v2 = 3v1 dependence instead of the expected one, v2 = 2v1 , involves the calculation of the gain in the elastic energy due to the motion of slow and rapid dowsons. Four schemes in Figure 4.55 are useful for a straightforward understanding of the discussion. Three colors, gray, blue and yellow, represent three levels of the elastic energy density: E = 0 – gray, Eλ = (Kh/2)(2π/λ)2 – blue and Eλ/2 = (Kh/2)[2π/(λ/2)]2 – yellow. Displacement Δx of a slow dowson lowers the elastic energy by ΔEλ = −Eλ λΔx =

Kh 4π 2 Δx 2 λ

[4.97]

The force driving the motion of the slow dowson is thus

Fλ =

Kh 4π 2 2 λ

[4.98]

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269

x








x





Figure 4.55. Estimation of forces driving the motion of dowsons involved in the race. (a and b) Slow dowsons. (c and d) The most rapid dowsons

The rapid dowsons of width λ/2 are inserted between the 2π-walls of the slow dowsons. Therefore, there are two contributions to the gain of the elastic energy. From the scheme c in Figure 4.55(c), we have ΔEλ/2 = −Eλ/2 (λ/2)Δx =

Kh 4π 2 Δx = 2ΔEλ Δx 2 λ/2

[4.99]

which generates the force 2Fλ . The second contribution comes from the additional elastic relaxation in gray areas left by the displacement of the rapid dowson. Here, the gain in elastic energy is ΔE  = −(Eλ/2 − Eλ )Δx = −Eλ Δx

[4.100]

which generates the force Fλ . The total force acting on the rapid dowson is thus Fλ/2 = 3Fλ

[4.101]

so that, by virtue of equation [4.122], its velocity is three times larger than that of the slow dowson, as observed.

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4.9.5. Trajectories of dowsons observed in natural light Almost all experiments with the dowser texture were done with Dowsons Colliders equipped with crossed polarizers, as patterns of isogyres and isochromes are very helpful for interpretation of results. Isochromes allow us to measure the local thickness and visualize its gradients. Isogyres allow us to monitor the process of the phase winding or to identify the type of defect, d+ or d-, embedded in the wound up dowser field. In other words, because of the optical anisotropy of nematics, the phase ϕ of the complex order parameter eiϕ is an observable quantity. The importance and abundance of information that can be inferred from these interference patterns in polarized light becomes obvious when we deal with results of experiments with the dowser texture observed in non-polarized light, or if we examine results of experiments with vortices in superconductors imaged with a scanning tunneling microscope. In both cases, the phase in not an observable quantity and only the core of dowsons or vortices (Embon et al. 2017) can be distinguished. For example, Figure 4.13 was obtained by superposition of images taken without polarizers. In this composite picture, dowsons appear as blueish dots that form dotted lines corresponding to trajectories of dowsons d+ and d-. These trajectories lead to collisions of dowsons pairs indicated by circles drawn, respectively, with solid or dashed lines when annihilation or passing-by occurs. The second example of such a composite picture is given in Figure 4.56(a). Here, all trajectories of dowsons in the DDC2 setup start at the moment of generation of dowsons pairs. Annihilation of dowsons’ pairs, taking place on dashed lines, can be easily identified. The instability of the dowsons’ race discussed above is also well visible. The third example, given in Figure 4.56(c), concerns trajectories of dowsons in the DDC2 setup during the asynchronous winding of the dowser field. Arrows indicate their starting points at t = 0, corresponding to the beginning of the winding. The most striking feature of these trajectories is their spiral shape indicating that dowsons circulate around the maximum C of the winding angular velocity ω(x, y) and simultaneously move away from it. The second significative feature is that the dowson d- indicated by a larger arrow stays in the vicinity of its starting point for a long time, located at the maximum of the angular velocity ωmax . These those will be confirmed by those made with polarized light.

Physics of the Dowser Texture 













 

271

 
   


 
    













 


























 


 

 


 

 




   



 




  







 



  

  

 









Figure 4.56. Trajectories of dowsons in the DDC2 setup recorded in non-polarized light. (a) Generation and annihilation of dowsons’ pairs. The dowsons’ race instability is also well visible. (b) Trajectories of dowsons during the winding process. Note that the dowson d- indicated by a larger arrow stays in the vicinity of its starting point for a long time

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Liquid Crystals

4.9.6. Trajectories of dowsons observed in polarized light 4.9.6.1. Gyrophilic behavior of dowsons dIndeed, such “gyrophilic” behavior of dowsons d- is confirmed in the experiment illustrated by Figure 4.57. Here, two dowsons d- are located at the two extrema of the angular velocity and remain there during the winding. Other dowsons are going away and possibly annihilate by pairs, as indicated by the circles.














Figure 4.57. Gyrophylic behavior of dowsons d- during phase winding with the DDC2 setup. Two dowsons d- located in the vicinity of the extrema of the angular velocity of winding stay there. Their maltese cross-like isogyres patterns are rotating with angular velocities ωmin and ωmax . Other dowsons, going away, annihilate by pairs indicated by the circles

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Figure 4.58. Gyrophylic behavior of dowsons d- during the phase winding with the CDC2 setup. Beside the dowson d- trapped at the maximum of the angular velocity ωmax several other dowsons d- are trapped in the annular minimum ωmin of the angular velocity

Let us stress the isogyres’ patterns present, in these pictures, taken in polarized light allow us to easily identify dowsons d+ and d-. Moreover, they also allow for measuring the angular velocities ωmin and ωmax because the maltese cross-like isogyres of dowsons d- are rotating with these velocities. The gyrophilic behavior of dowsons d- is confirmed in the experiment carried out using the Circular Dowsons Collider 2 and illustrated by Figure 4.58. Here, beside the dowson d- trapped at the central extremum of the angular velocity ωmax several other dowsons d- are trapped in the second, annular extremum ωmin of the angular velocity. The maltese crosses of the central and the peripheral dowsons are rotating, respectively, in opposite anticlockwise and clockwise directions as expected. 4.9.6.2. Gyrophobic behavior of dowsons d+ The gyrophilic behavior of dowsons d- is confirmed once again in the experiment made using the CDC1 and illustrated by Figure 4.59. Here, at the beginning of the winding four dowsons d+ and three dowsons d- are embedded in the dowser field. The dowson d- located initially in the vicinity of the extremum ωmax is attracted to it and stays there as expected. This experiment also shows that two dowsons d+ located initially in the vicinity of the extremum ωmax have a gyrophobic behavior: they are going away during the winding. Other dowsons more distant from ωmax annihilate by pairs. The gyrophobic behavior of dowsons d+ is obvious in the experiment illustrated by Figure 4.60. Here, the dowser field contains only one dowson d+, which is called

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Liquid Crystals

residual because its presence is necessary for topological reasons inside the droplet with homeotropic boundary conditions. In Figure 4.60(a) it is located in the vicinity of the gap center. During the winding, Figures 4.60(b–d) show that new loop-like isogyres are generated at the maximum of the angular velocity of the winding which, as shown in the insert, is located in the gap center. Colloquially speaking, the dowson d+ is pushed away by expanding isogyres loops.


  




















Figure 4.59. Gyrophobic and gyrophylic behaviors of dowsons. (a) Four dowsons d+ and three dowsons d- embedded in the dowser field during winding. (b and c) Annihilation of two dowsons pairs. (d) The stationary Cladis–Brand 2 state with the dowson d- at the maximum of the angular velocity of winding, and two dowsons d+ orbiting around it

4.9.6.3. Gyrophobic behavior of dowsons d+ Explanation of the gyrophobic behavior of dowsons d+ is done graphically in Figure 4.61(I). When a dowson d+ (scheme a) is inserted in a defect-less wound up dowser field (scheme b) in the point with ϕins = 0, a 2π wall is created (scheme c). The detailed view of the inserted dowson shows that it has the same configuration as in (a). However, when the phase of the insertion point is ϕins = −π/2 (scheme f), the detailed view of the inserted dowson (scheme h) shows that its configuration

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275

changed with respect to the one in the scheme d. This means that the configuration of the dowson d+ changes when the insertion point moves in the direction of the phase gradient or, alternatively, if the wound up dowser field moves across the inserted dowson. Now, the elastic energy E(ϕins ) of the dowson d+ depends, via ϕins , on its configuration which in polar coordinates (r, ϕ) is given by d = [cos(ϕ + ϕins ), sin(ϕ + ϕins )]

[4.102]




















Figure 4.60. Gyrophobic behaviors of dowsons d+. (a) The residual dowson d+ located in the vicinity of ωmax . (c and d) During the winding it goes away as if it was pushed by isogyres generated at ωmax . (d) The stationary Cladis–Brand 1 state

On the basis of symmetry arguments, the function E(ϕins ) can be written as a Fourier series limited to cosine terms: E(ϕins ) = Eo + A cos(ϕins ) + B cos(2ϕins )

[4.103]

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Figure 4.61. Gyrophobic and gyrophylic behaviors of dowsons. (I) Dowson d+. (II) Dowson d-. (a) Dowser fields d and ϕ. (b–f) Defect-less wound up dowser field. (c–g) Wound up dowser fields with the dowson inserted in them. (d and h) Detailed views of the inserted dowson. Its configuration depends on the phase ϕins in the insertion point defined in (b) and (f)

In Figure 4.62, we plotted the pertinent part of this expression E(ϕins ) − Eo B = cos(ϕins ) + cos(2ϕins )] A A

[4.104]

with B/A=0.75 estimated from experiments discussed in Pieranski et al. (2016b). This function has two minima in which the configuration of the dowson d+ is close to the

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277

clockwise and anticlockwise circular ones. In conclusion, the gyrophobic behavior of d+ dowsons results mainly from the existence of an absolute maximum, which acts as an energy barrier to overcome when ϕins varies. At this absolute maximum, the dowser field has the outward radial configuration, which in 3D corresponds to the hyperbolic configuration of the monopole.


  





 



 

 












 












Figure 4.62. Variation of the energy of dowsons and of their configurations with the phase of the insertion point ϕins

4.9.6.4. Gyrophilic behavior of dowsons dThe case of the d- dowson is represented in Figure 4.61(II). When the insertion point varies from ϕins = 0 to ϕins = −π/2, the field d in the vicinity of the dowson rotates as a whole by π/4, which does not change the elastic energy. One could think therefore that the behavior of the dowson d- should be rather neutral with respect to changes of the phase ϕins = 0.

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In reality, it is more complex because, as shown in Figures 4.57–4.59, dowsons dcan be trapped in extrema of the angular velocity. So far no satisfactory explanation of this experimental fact has been found. 4.9.6.5. Stationary Cladis–Brand states The gyrophobic and gyrophilic behaviors of dowsons are at the origin of stationary states occurring during the winding of the dowser field. We call them Cladis–Brand 1 and Cladis–Brand 2 with the aim of emphasizing their similarity with an analogous stationary state, discovered previously in Smectic C free-standing films (Cladis et al. 1995). They have been already mentioned above but we show them again in Figure 4.63 with additional details, useful for discussion. During the asynchronous winding process in the CDC1 setup, the maximum of the angular velocity ω = dϕ/dt = ωmax is located in the center of the gap. We can consider this maximum as a phase source, that is to say a point in which the phase is created at the rate dϕ/dt = ωsource . During the winding, the phase growth rate ωsource decreases because the rheotropic torque is opposed by the growing elastic torque. This phase source is surrounded by a circle, drawn in the insert with a dashed line, on which ω = 0. Let us consider first the case of the CB1 state. It occurs when the dowser field at the beginning of winding contains only one gyrophobic dowson d+. As discussed previously, during the winding, the dowson d+ remains at the edge of the wound up dowser field and, pulled by the 2π-wall, is orbiting, with some period T, around the phase source. Such an orbiting dowson d+ acts as a phase sink absorbing the phase at the rate ωsink = −2π/T , which grows in time because the width λ of the wall decreases. The stationary state ϕ(x, y, t)CB1 is reached when ωsource + ωsink = 0

[4.105]

The spatiotemporal cross-section in Figure 4.63(c), extracted from a video along the line CS, gives a graphical demonstration of this equality. It shows that during one period T = 30 min four new isogyres are created at the source in the center and four isogyres are lost when the dowson d+ crosses the line CS. In the case of the CB2 state, four isogyres are created, every 21 minutes, at the center by one whole turn of the maltese-cross, and four isogyres are lost when one of the dowsons d+ crosses the line CS.

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Figure 4.63. Cladis–Brand stationary states. (a) C-B1 state: dowson d+ orbiting around the phase source in the center. (b) C-B2 state: two dowsons d+ orbiting around the dowson d- rotating in the center. (c) Spatiotemporal cross-section of C-B1. (d) Spatiotemporal cross-section of C-B2

4.10. Collisions of dowsons Collisions of dowsons have already been mentioned during discussions concerning Figures 4.13, 4.56(a), 4.57 and 4.59. In particular, Figures 4.13 and 4.56(a), obtained by superposition of images taken at intervals of a few seconds, showed trajectories of dowsons leading to collisions. Another example of colliding trajectories is shown in Figure 4.64. Figures 4.13, 4.56(a) and 4.64 show the two alternative outcomes of collisions: an annihilation indicated by circles drawn with a solid line or an avoidance indicated by circles drawn with a dashed line. Using the vocabulary of atomic or elementary particles physics, we can ask: What is the annihilation cross-section of dowsons’ pairs?

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Figure 4.64. Trajectories of dowsons leading to collisions of dowsons’ pairs obtained by superposition of pictures taken at intervals of 5 s. Starting points of dowsons d+ and d- are indicated with yellow and blue arrows, respectively. Collisions resulting in an annihilation of a dowsons’ pair are indicated by solid line circles, and those leading to an avoidance are indicated by dashed line circles

4.10.1. Pair of dowsons (d+,d-) inserted in a wound up dowser field In the search for the answer, let us consider two other examples of collisions, shown in Figure 4.65. These pictures, taken in polarized light, contain information about the dowser field, which is missing in images obtained without polarization. They suggest that the outcome of collisions depends on the distance of trajectories in terms of the phase difference Δϕ. When a pair of dowsons d+ and d- is inserted at points (x± , y± ) in a dowser texture wound up in y direction, the phase field can be expressed as ϕ(x, y) =

2π y + arctan λ



y − y+ x − x+



  y − y− + arctan − x − x−

[4.106]

When the two dowsons are far away, i.e. when |x+ − x− | λ, they move on linear trajectories defined y(t) = y+ and y(t) = y− . The linear distance between

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these trajectories is thus Δy = y+ − y−

[4.107]

while in terms of the phase the distance is Δϕ = 2π

Δy λ

[4.108]




























Figure 4.65. Collisions of dowsons’ pairs. (I) Avoidance, (II) Annihilation: (a) The dowser texture wound up in the CDC1 setup. (b–e) Images taken at intervals of 25 s

For example, in Figure 4.66(a–g), the phase difference Δϕ varies between 0 and 3π.

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Figure 4.66. Collisions of dowsons embedded in a wound up dowser field. Their distance in terms of the phase difference Δϕ varies from 0 in (a) to 3π in (g). The color code for the phase field is given in (h)

By comparing these pictures with those in Figure 4.65, we can say that in the case of the avoidance in Figure 4.65(I), the phase difference Δϕ is of the order of 3π/2 while in the case of annihilation in Figure 4.65(II), it is Δϕ ≈ π/2. 4.10.2. Cross-section for annihilation of dowsons’ pairs Statistical study of collisions of pairs of dowsons d+ and d- in wound up dowser textures lead us to the conclusion that their issue depends on the distance Δϕ of trajectories and, more precisely, there exists a critical phase difference Δϕc = π such that: 1) for |Δϕ| > Δϕc , the avoidance occurs; 2) for |Δϕ| < Δϕc the annihilation takes place; 3) for |Δϕ| = Δϕc the outcome of the collision is random. In other words, we can say that the annihilation cross-section is Δϕc = π in terms of the phase or, equivalently, Δyc = λ/2 in terms of the linear distance. R EMARK . Knowing from section 4.9.3 that the velocity of dowsons is inversely proportional to the wavelength λ, we can say that the linear annihilation cross-section of dowsons’ pairs is inversely proportional to their velocity: Δyc =

λ 1 ∼ 2 v

[4.109]

Let us stress that this rule is similar to that of absorption of neutrons by nuclei of atoms involving the de Broglie wavelength λ = h/(mv).

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4.10.3. Rheotropic control of the collisions outcome The rules, stated above, are valid only for collisions occurring in quasi-static or stationary wound up dowser fields with a homogeneous phase gradient → − | ∇ϕ| = 2π/λ. Now, we know from section 4.4.6 (see Figure 4.31) that Poiseuille flow splits a wound up dowser field into 2π-walls. We will show below that in the case of a dowson pair coming to a collision, this action of the Poiseuille flow can alter the issue of the collision. Let us consider the example of the dowsons’ pair in Figure 4.67. The phase field represented in Figure 4.67(a) has been calculated using equation [4.106] with (x+ , y+ ) = (−5, −π/4) and (x− , y− ) = (5, π/4) so that Δϕ = 2π

y+ − y− π = λ 2

[4.110]




















Figure 4.67. Action of a Poiseuille flow on the outcome of a collision. (a) Dowsons’ pair inserted in a quasi-static wound up dowser field. (b) Generation of 2π-walls by a Poiseuille flow v in x direction. (c) Generation of 2π-walls by a Poiseuille flow v in -x direction

In terms of the collision rules formulated above, an annihilation of this dowsons pair is granted. Application of the Poiseuille flow in the x direction perturbs the field ϕo (x, y) as shown in Figure 4.67(b): a narrow 2π-wall is now connecting the two dowsons. Obviously, pulled by this wall, the dowson will rapidly come into a collision and the annihilation will occur as expected. However, when the Poiseuille flow is applied in the -x direction, the wound up phase field is split into a different set of 2π-walls and such that the two dowsons are attached to two different 2π-walls (see Figure 4.67(c)). Pulled by these walls, the two dowsons will follow trajectories separated in space and

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the annihilation will be avoided. Experiments confirmed that a transient application of a Poiseuille flow with an appropriate velocity in an appropriate direction can determine the outcome of a collision. An example of such rheotropic control is shown in Figure 4.68. Initially (Figure 4.68(a)), dowsons d+ and d- are connected by a 2π-wall, which pulls on them and brings them into a collision. As the phase difference between the dowsons is about π/2, the outcome of the collision should be annihilation. However, application of a Poiseuille flow (see Figure 4.68(b)) splits the wound up dowser field into a set of narrow 2π-walls such that the two dowsons become connected to two different walls. During the subsequent evolution shown in Figures 4.68(c–f), the two dowsons, pulled by their walls, are passing by. The last three Figures 4.68(g–i) show that when the Poiseuille flow is suppressed, and the dowser field allowed to relax, the two dowsons will go apart.




















Figure 4.68. Action of a Poiseuille flow on the outcome of a collision. (a) Dowsons’ pair inserted in a quasi-static wound up dowser field. Their annihilation seems inevitable. (b) Generation of 2π-walls by a Poiseuille flow. (c–f) Motion of dowsons pulled by the 2π-walls to which they are attached. (g and h) Relaxation after cessation of the flow. The annihilation is avoided

In conclusion, the annihilating collision, which seemed to be inevitable in the Figure 4.68(a), is avoided.

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4.11. Motions of dowsons in homogeneous fields In the experiment discussed above, the motion of dowsons d+ and d- forming a pair was driven by the tension τ of narrow 2π walls generated by the Poiseuille flow. In these considerations, we implicitly assumed that the walls connected to dowsons reached their quasi-static configurations. One can ask what the motion of individual dowsons in an alternating Poiseuille flow v with a period much shorter than the characteristic time required for the formation of walls would be. In virtue of the formal analogy between the hydrodynamic and electric torques (see equation [4.8]), the same question concerns dowsons submitted to the action of a homogeneous alternating electric field E. In what follows, we will focus on this issue. From the experiment discussed above, we can infer that the motion of dowsons should depend on the direction of the electric field. Moreover, the dowser field surrounding dowsons d+ or d- can adopt a continuous set of configurations (a few examples of them are given in Figure 4.69). One can therefore ask whether the motion of dowsons d+ and d- in an electric field is only related to their topological charges ±2π or, on the contrary, if it also depends on their configuration. 





















  





 







Figure 4.69. Configurations of dowsons d+ and d- and their motions in an electric field. (a) Definition of polar coordinates. (b–g) Configurations of the dowson d+. (h–j) Configurations of the dowson d-. Direction of the force f indicated in schemes b-j are given by equation [4.115], with Pf e > 0. (k) Level representation of the dielectric energy density around the dowson d+ in its circular configuration

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All possible configurations of dowsons can be expressed as d = (cos ϕ, sin ϕ) with ϕ = mψ + φo

[4.111]

with angles ψ and ϕ defined in Figure 4.69(a). φo is an angular variable and m is the topological charge equal to +1 and -1 for dowsons d+ and d-, respectively. The interaction of dowsons with the electric field is expressed by the energy density per unit area F = −Pf e d · E

[4.112]

In a sector of the angular width dψ and length r, making the angle ψ with the x axis (see Figure 4.69(k)), the dielectric energy is thus δF = −Pf e d · E

r2 dψ 2

[4.113]

so that there is a force δf =

d(δF ) (cos ψ, sin ψ) = −Pf e d · Er(cos ψ, sin ψ)dψ dr

[4.114]

acting on the dowson. Using expression [4.111], we obtain the total force  f= −

2π 0

Pf e E sin(mψ + φo )(cos ψ, sin ψ)dψ or

f = Pf e Eπr(sin φo , −m cos φo )

[4.115] [4.116]

which depends on both the topological charge m and the configuration of dowsons determined by φo . The directions of the force f (indicated in Figure 4.69) correspond to the case of Pf e > 0. In particular, the force f acting on the dowson d+ in its radial outward and inward configurations (see Figure 4.69(b) and (e)) is, respectively, antiparallel and parallel to the electric field. The radial outward configuration, satisfying both the boundary condition and the action of the cuneitropism, can be reached through a spontaneous viscoelastic relaxation. This process that can last a few hours in a droplet with the radius in the

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range of a few millimeters, and can be made almost instantaneous by the application of a diverging Poiseuille flow accompanying the reduction of the slide/lens gap thickness (Figure 4.21 in section 4.4.1.1). By the same rheotropic mechanism, the converging Poiseuille flow, driven by an increase in the gap thickness, can force the radial inward configuration. This inward configuration is unstable so that it can return to the outward configuration by viscoelastic relaxation. Nevertheless, as the characteristic time of this relaxation process is of the order of several hours, the experiments testing the validity of equation [4.115] can be easily carried out. The results of experiments with the outward and inward radial configurations of the dowson d+ in MBBA are depicted in Figure 4.70. The spatiotemporal cross-sections in the last column show that the directions of motion are reversed between the two radial configurations. As they are also opposite to those in Figures 4.69(b) and (e), we can conclude, in agreement with the results discussed in section 4.6.5, that the flexo-electric polarization in MBBA Pf e is antiparallel to the dowser field d. The same experiments performed with 5CB directly agree with the theoretical schemes in Figures 4.69(b) and (e), because in 5CB, Pf e is parallel to the dowser field d. The zig-zag shaped trajectories of the dowson d+ in the spatiotemporal cross-section in Figure 4.70 also show that the direction of motion is instantaneously reversed upon the reversal of the electric field. Assuming that the velocity of motion is proportional to the driving force given in equation [4.115], this feature also agrees with theoretical predictions. 4.12. Stabilization of dowsons systems by inhomogeneous fields with defects 4.12.1. Gedanken experiment When any complex configuration of dowsons is allowed to relax in the presence of the cuneitropic torque (due to the plane/sphere geometry of the gap) alone, the ground state of the dowser field – the radial outward configuration with one dowson d+ in the center – is always reached. However, if the gap thickness h(x,y) had a more complex topography, such as the one represented in Figure 4.71(a), the cuneitropic torques would align the dowser field d along the thickness gradient g shown in Figure 4.71(b) and a square network of alternating dowsons d+ and d- would be stabilized by these means. Similarly, the same network of dowsons would be stabilized by an electric field E (Figure 4.71(b)), derived from the periodic potential U(x,y) represented in Figure 4.71(a). In such a periodic electric potential, stable positions of dowsons d+ and d- are located in extrema and saddle points, respectively.

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Figure 4.70. Motion of the dowsons d+ and of dust particles in electric field (MBBA)

Figure 4.71. Gadanken experiment: stabilization of a dowsons network by tropisms of the dowser texture. (a) Topography of the gap thickness h(x,y) or of the electric potential U(x,y). (b) Dowser field d oriented by the corresponding thickness gradient g = ∇h or electric field E = ∇U . (c) Simulation of the pattern of isogyres seen in polarized light

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4.12.2. Triplet of dowsons stabilized in MBBA by a quadrupolar electric field Such a periodic electric field cannot be realized by means of etched ITO electrodes. Therefore, inspired by the microfluidic study of Giomi et al. (2017), we used a much simpler system of four electrodes from Figure 4.38(c4), generating a quadrupolar electric field. The series of six pictures in Figure 4.72 shows that in the presence of such a quadrupolar electric field, the viscoelastic relaxation of a dowser field containing many dowsons leads to the stable triplet of dowsons shown in Figure 4.72(f). 




  



  

 









Figure 4.72. Triplet of dowsons stabilized by a quadrupolar electric field. (a) Initial configuration. (a–f) Relaxation through the annihilation of dowsons pairs. (f) Final triplet configuration stabilized by the electric field. In MBBA, the dowser field is antiparallel to the electric field

As expected, by analogy with the work of Giomi et al. (2017), the dowson d- is located in the stagnation point of the electric field (saddle point of the potential U). From the isogyres’ pattern in Figure 4.72(f), it is possible to infer that the dowser field d around this dowson d- is antiparallel to the electric field E. This is because, as we know from section 4.6, in MBBA the flexo-electric polarization P = Pf e d is antiparallel to d.

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4.12.3. Septet of dowsons in MBBA stabilized by a quadrupolar electric field Figure 4.73 shows that in a stronger electric field, beside the triplet state (Figure 4.73(b)), a septet of dowsons can also be stabilized by the quadrupolar electric field (Figure 4.73(c)). It contains the same triplet of dowsons as in Figure 4.73(b) and two additional pairs of dowsons, which are located in field-free areas above the two positive ITO electrodes. 






 


























Figure 4.73. Triplet and septet of dowsons. The septet configuration is stabilized by electro-osmotic flows, which are parallel to the electric field in MBBA

In the absence of the electric field, the annihilation of these two dowsons pairs is hindered by the electro-osmotic flows shown in Figure 4.73(a). In MBBA, they have the same direction as the electric field in the adjacent gap. 4.12.4. Dowsons d+ stabilized by corner singularities of the electric field A quadrupolar electric field with the stagnation point stabilizing the dowson d- can also be generated with the system of electrodes d6 shown in Figure 4.74(a) (see also Figure 4.38(d6)). It displays a pertinent difference with the system d4 : besides the central stagnation point, it also contains four point singularities of the electric field at corners of electrodes that are susceptible to stabilize dowsons d+. As expected, in an experiment realized with MBBA (see Figure 4.74(b)), the two dowsons d+ of the triplet configuration are located at sharp corners of negatively charged electrodes. In the same experiment realized with 5CB, the triplet of dowsons has a different orientation because the flexo-electric polarization Pf e in 5CB is parallel to the dowser field d, while in MBBA it is antiparallel to d.

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Figure 4.74. Stable configurations of a dowsons’ triplet (d+,d-,d+) in a quadrupolar electric field. (a) Electric field generated by the system of four ITO electrodes. It contains one stagnation point and four angular singularities. (b) Dowsons triplet in MBBA. The dowson d- is trapped in the stagnation point, while the two dowsons d+ are located at angular singularities of negatively charged electrodes. (c) The same experiment with a 5CB droplet. Here, the two dowsons d+ are located at angular singularities of positively charged electrodes

4.13. Dowser field submitted to boundary conditions with more complex geometries and topologies All of the experiments discussed so far in this chapter have been performed with nematic droplets held by capillarity in the center of the lens/slide gap (see Figure 4.75(a)). In the case of 5CB, the homeotropic anchoring condition at the nematic/air interface imposes the radial outward anchoring of the dowser field at the meniscus of nematic droplets considered as solid two-dimensional simply connected discs. We emphasized several times that the ground state of the dowser field compatible with these boundary conditions is radial, directed outward, and it must contain at least one dowson d+ (see Figure 4.75(b)). Studies of the dowser field in geometries other than the trivial one of a disc, have been initiated by the microfluidic studies of Sengupta (2013, 2015, 2018) and Giomi et al. (2017). This direction appears to be promising because in a single annular droplet, or in more complex networks of channels connected by n-junctions, the structure and topology of the dowser field d, imposed by anchoring conditions at lateral walls of channels, can be quite complex, as we will see below. 4.13.1. Ground state of the dowser field in an annular droplet Let us first consider an annular droplet (shown in Figure 4.75(c)) with the homeotropic anchoring of the director field n at its lateral menisci (see Figure 4.75(a)). It is contained between a lens and a glass slide with a hole in its center. As

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we already know, the resulting preferential orientation of the field d at the two menisci is “homeotropic outward”, i.e. orthogonal to them and oriented in outward directions (see Figures 4.75(f) and (f’)).  

  





 











 

























 

Figure 4.75. Dowser fields in a circular annular droplet. (a and b) Radial dowser field in a circular droplet with a dowson d+ in its center. (c–e) Clockwise and anticlockwise versions of the ground state of the dowser field in the annular droplet. There is a virtual dowson d+ in the hole of the annular channel. (f and f’) Homeotropic outward anchoring of the dowser field at the nematic/air menisci. (g) Transformation of the anticlockwise dowser field into the clockwise one by nucleation, motion and annihilation of a dowsons’ pair d+d-

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In contradistinction to the case of the circular droplet shown in Figure 4.75(b), the dowser field imposed by these boundary conditions of the annular droplet is defect-less, though there is a virtual dowson d+ inside of the hole. In the absence of dowsons pairs, the dowser field d in such a single annular droplet is still non-trivial because it must have, on average, either a clockwise or an anticlockwise curl (see Figures 4.75(d) and (e)). Figure 4.75(g) shows that the sign of the curl can be reversed by a three-stage process: (1) generation of a dowsons pair d+d-, (2) circular motion of dowson in opposite clockwise and anticlockwise directions (see Figure 4.75(g)) and (3) annihilation of the dowsons pair. 4.13.2. Wound up metastable states of the dowser field in the annular droplet The equilibrium states of the dowser field in the annular droplet must satisfy the requirement of vanishing the elastic torque, expressed as Kef f ϕ = 0

[4.117]

as well as the radial outward boundary conditions: ϕ(r2 , ψ) = ψ and ϕ(r1 , ψ) = (2n + 1)π + ψ at the outer and inner menisci of radii r1 and r2 . The phase field ϕ(r) in equilibrium can be thus expressed as ϕn (r, ψ) = ±(2n + 1)π

ln(r/r2 ) +ψ ln(r1 /r2 )

[4.118]

The two ground states shown in Figure 4.75(d) correspond to n = 0. Alternatively, these ground states can be seen as circular π-walls and the states with n > 0 can be seen as sets of 2n + 1 concentric π-walls. In contradistinction with the case of the circular droplet in which 2π-walls can shrink and finally collapse (when the configuration of the central d+ dowson is free to evolve, see Figure 4.35), the radial outward anchoring condition at the inner meniscus of the annular droplet hinders the shrinking of π-walls. For this reason, the states with n > 0, such as the ones in Figure 4.76, are metastable. The transitions between states with different n must be mediated by the three-stage process discussed above involving the generation, motion and annihilation of dowsons pairs. 4.13.3. Dowser field in a square network of channels, four-arm junctions More complex topologies of square or triangular networks of channels are shown in Figures 4.77 and 4.78.

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 Figure 4.76. Excited metastable state of the dowser field in an annular droplet. (a) The dowser field. (b) Isogyres pattern

In the case of a square network of identical channels (with a rectangular section), in Figure 4.77 we show two versions of the dowser field compatible with the homeotropic outward anchoring at lateral walls of channels. The first version in Figure 4.77(a) can be qualified as “microfluidic” because it is also compatible with the flow pattern indicated with large blue arrows. The directions of the dowser field are imposed by the rheotropic torques. It contains 11 dowsons dlocated at centers of cross shaped four-arm junctions of channels, as well as at certain three-arm junctions at the boundaries of the network (see section 4.13.6). There are also 12 d+ virtual dowsons located inside of the square shaped islands surrounded by channels. The total topological charge of the whole network is 12-11 = +1, as required by the homeotropic outward anchoring conditions at the external border of the network. The second version in Figure 4.77(a) is also compatible with the boundary conditions. However, its compatibility with a flow pattern would require the presence of sources and sinks. 4.13.4. Triangular network, six-arm junctions The dowser field in the triangular network of the channels shown in Figure 4.78 can also be qualified as microfluidic because it satisfies the anchoring conditions, as well as the orienting action of the flow pattern indicated with blue arrows. It is composed of six-arm junctions studied by Giomi et al. (2017). The flow pattern is such that the junctions contain two stagnation points, in which the dowsons d- are located.

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 Figure 4.77. Possible dowser fields in a square network of channels. (a) Microfluidic version compatible with rheotropism, conservation of flow and boundary conditions. (b) Static version only compatible with boundary conditions

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Figure 4.78. Dowser field in a triangular network of microfluidic channels. The number of six-arm junctions of channels is half that of islands. For the conservation of the total topological charge zero, there are two dowsons d- at each six-arm junction

4.13.5. Three-arm junctions The square network of channels shown in Figure 4.77 also contain three-arm junctions at its boundaries. The three different configurations of the dowser field inside of them are listed in Figure 4.79. 4.13.6. General discussion of n-arm junctions After detailed discussions of the dowser field inside the three-, four- and six-arm junctions, we are ready to draw general conclusions. For the sake of clarity, let us consider the example of the four-arm junction (see Figure 4.80). Its boundary is composed of eight segments. Four of them, drawn with solid lines correspond to either lateral walls of channels treated for homeotropic anchoring, or to nematic/air interfaces imposing the same anchoring conditions of the dowser field: orthogonal and directed outward.

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Figure 4.79. Dowser field in three-arm junctions. (a) Defect-less microfluidic version. (b) Microfluidic version with a dowson d- located in the stagnation point. (c) Static version with a dowson d+  
   


 
   





   







    


    





















Figure 4.80. Real, and virtual anchoring conditions of the dowser field along the boundary of the four-arm junctions. (a and a’) Presence of the dowsons d- is imposed by the −2π rotation of the dowser field along the whole boundary of the junction. (b and b’) Defect-less configuration of the dowser field is a result of null rotation of the dowser field along the whole boundary of the junction. (c and c’) Presence of the dowson d+ is imposed by the orthogonal outward anchoring along the whole boundary of the junction

Four other segments drawn with dotted lines can be considered as virtual boundaries. Rheotropic torques due to Poiseuille flows across them generate virtual

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boundary conditions for the dowser field: orthogonal with either outward or inward directions. The boundary of the junction in Figure 4.80(a) can be deformed continuously into the octagon shown in Figure 4.80(a’). Traveling clockwise along it results in the anticlockwise rotation of the dowser field. The topological charge of this junction is thus −2π, so that the dowser field contains at least one dowson d-. In the second case in Figure 4.80(b), as traveling along the whole boundary (simplified to an octagon in Figure 4.80(b’)) does not produces any phase defect, the topological charge of the dowser field is thus zero. In the last case, the topological charge resulting from boundary conditions is +2π so that the junction contains one dowson d+. Using the same method, the boundary of any other n-arm junction can be represented as a topologically equivalent 2n-gon with n real and n virtual sides, allowing the easy determination of the topological charge. In Figure 4.81, we show examples of the three-arm and six-arm junctions discussed previously. 4.14. Flow-induced bowson-dowson transformation Giomi et al. (2017) reported on microfluidic experiments realized with four-, sixand eight-arm junctions. Like in the experiments by Liu et al., upon the application of Poiseuille flows (see Figures 4.82(c) and (c’)), the homeotropic texture in the four-arm junction is continuously deformed by hydrodynamic torques into the bowser texture. The direction and the amplitude of the bow-like distortion is characterized by the vector b defined in Figures 4.82(c’) and 4.19(a’). In the limit of small distortions, we have b(x, y) ∼ vmax (x, y)

[4.119]

Therefore, as the Poiseuille flow in the four-arm junction contains a stagnation point, the bowser field b(x,y) contains a -2π defect, which can be called bowson b-. Let us remark that by contradistinction with the dowson d-, the bowson b- is not a true topological defect because it appears and disappears “softly” when the flow with a stagnation point is gently switched on and off. When the flow velocity vmax is larger than vco , the flow-assisted homeotropic-dowser transition occurs and the width of the boundary dowser layers grows (see section 4.3.2). After the coalescence of the dowser boundary layers (see Figures 4.82(d) and (d’)), a small domain of the bowser texture remains in the center

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of the junction. Finally, it collapses and the bowson b- is replaced by the dowson d(see Figures 4.82(e) and (e’)).









































Figure 4.81. Real, and virtual anchoring conditions of the dowser field along the boundary of the three- and six-arm junctions. (a–c) Dowser fields in three-arm junctions. (a’–c’) Topologically equivalent representation of boundaries. (d–f) Dowser fields in six-arm junctions. (d’–f’) Topologically equivalent representation of boundaries

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Figure 4.82. Flow-induced bowson-dowson transformation in the four-arm microfluidic junction. (a) Ground state: coexistence of the homeotropic texture with the dowser texture in boundary layers. (b) Cross-section of the channel in the ground state. (c and c’) Application of the Poiseuille flow. The homeotropic texture is continuously deformed into the bowser texture containing a -2π bowson located in the stagnation point of the flow. The width of the dowser boundary layers grows upon the flow-assisted homeotropic-dowser transition. (d and d’) Small bowser domain before its final collapse. (e and e’) The dowser texture with the dowson d- located in the stagnation point. Remark: This texture satisfies the anchoring conditions so that it is preserved after cessation of the flow

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In the six-arm junctions, the ground state homeotropic texture matches the lateral walls through the dowser boundary layers. Upon application of the Poiseuille flow, the homeotropic texture is continuously deformed into the bowser one and two bowsons b- appear “softly” in stagnation points. When the flow velocity vmax is larger than the critical velocity vco (given in equation [4.16]), the flow-assisted bowser-dowser transition increases the width of the dowser boundary layer. Finally, after the coalescence of the boundary layers and the collapse of the remaining bowser domains, two dowsons d- replace the bowsons b-.






















 

 











Figure 4.83. Sengupta instability of the dowson’s d- position in the four-arm junction. (a) Stable central position in slow flows. (b) For v > vcrit , the dowson d- leaves the central position and a narrow 2π-wall is created. (c) Detailed view of the 2π-wall. (d and e) Views of the instability in polarized light (courtesy of Anupam Sengupta)

4.15. Instability of the dowson’s d- position in the stagnation point Experiments with the four-arm junctions reported in Giomi et al. (2017) have shown that the position of the dowson d- in the stagnation point of the Poiseuille flow is unstable above certain critical flow rate (see Figure 4.83). To understand the origin of this instability, we have to examine the balance of forces acting on the dowson d-

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when it leaves the center of junction O that coincides with the stagnation point of the flow (see Figure 4.83). In the absence of flow, the central position of the dowson is stable because it minimizes its elastic interaction with the lateral boundaries of channels with orthogonal outward anchoring conditions. In a first approximation, the elastic − → restoring force is proportional to the deviation δ = (δx , δy ) from the center O: → − K f el ≈ −α (δx , δy ) w

[4.120]

where α > 0 is a numeric factor. In the presence of a flow with a stagnation point SP located in the center O, the convection of the distortion (discussed in section 4.7.3) generates a viscous force, which can be written as → − γ1 hvmax f visc ≈ β (−δx , δy ) w

[4.121]

where β > 0 is another numeric factor. The total force acting on the dowson d- is thus → − f tot ≈ (Cx δx , Cy δy )

[4.122]

with Cx = −α

K K γ1 hvmax γ1 hvmax −β and Cy = −α + β w w w w

[4.123]

For vmax > 0 (see Figure 4.83(b)), the coefficient Cx is negative so that the central position of the dowson d- is stable with respect to deviations δx along the x axis. Stability with respect to deviations δy depends on the sign of the second coefficient Cy . For flow velocities larger than the critical value given by vcrit =

α K β γ1 h

[4.124]

Cx is positive so that an instability occurs: the dowson d- leaves the center of the junction and moves along the y axis. Let us emphasize that due to its rheotropism, the dowser field d still remains aligned by the flow, except for a 2π-wall, which now connects the dowson to the stagnation point SP (see Figures 4.83(b) and (d)).

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4.16. Appendix 1: equation of motion of the dowser field In nematodynamics, the equation of motion of the director field is derived from the balance of the elastic, viscous, magnetic or electric torques (per unit volume) acting on the director field n(x,y,z,t): → − 3D − → → − 3D − →3D Γ elast + Γ 3D visc + Γ mag + Γ elec = 0

[4.125]

From the discussion of the dowser texture given in section 4.1.3.1, we know that the director field expressed as n = cos θd + sin θez with θ = πz/h

[4.126]

d(x, y, t) = [cos ϕ(x, y, t), sin ϕ(x, y, t), 0] and ez = [0, 0, 1]

[4.127]

with

has, in principle, only one degree of freedom: the azimuthal angle ϕ(x, y, t). Therefore, only the z components of the torques listed in equation [4.125] are involved in the motions of the dowser field. Moreover, the dependence of all of the components Γz on the z coordinate is set by the expression θ = πz/h. For these two reasons, it is convenient to consider the dowser texture as a two dimensional system and rewrite equation [4.125] as a balance of torques per unit area acting on the dowser field. 4.16.1. Elastic torque 4.16.1.1. Uniform thickness h In the approximation of the isotropic elasticity with K1 = K2 = K3 = −K4 = K and assuming that K13 = 0, the elastic torque per unit volume acting on the director field takes a simple form (see sections B.II.2 and B.II.3 in Oswald and Pieranski (2005)) → − 3D → → Γ elast = K − n × − n

[4.128]

With equation [4.126], its z component writes 2 Γ3D elast,z = K(nx ny − ny nx ) = K cos

 πz  h

ϕ

[4.129]

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After integration on z, we obtain the elastic torque per unit area: Γ2D elast =

Kh ϕ 2

[4.130]

To end this discussion, let us remark that the last K13 term of the elastic energy density K13 div(ndivn)

[4.131]

does not contribute to elastic torques in bulk. Its action on surfaces with strong anchoring conditions is negligible. 4.16.1.2. Cuneitropic torque in a thickness gradient In Pieranski et al. (2016a) and in section 4.5.3, we have shown that in samples without uniform thickness h, there is an additional elastic torque proportional to the − → → thickness gradient − g = ∇h: Γc =

πK d×g 2h

[4.132]

4.16.2. Viscous torques In nematodynamics, the viscous torque acting on the director n can be expressed as → − 3D ˆ Γ visc = −γ1 n × N − γ2 n × (n · A)

[4.133]

In the first term of this expression, N, defined as N=

→ − Dn 1 − Dn ∂n − → ω × n with = + (v · ∇)n Dt 2 Dt ∂t

[4.134]

represents the relative angular velocity of the director n with respect to the surrounding fluid of vorticity − → → − ω = ∇ ×v

[4.135]

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305

In the second term of the equation [4.133], the tensor Aij =

1 2



∂vi ∂vj + ∂xj ∂xi

 [4.136]

corresponds to the symmetric part of the velocity gradients tensor. In the absence of flows, the z component of the viscous torque [4.133] integrated on z has already been calculated in section 4.4.6.2:  Γvisc = −

h/2 −h/2

γ1

∂ϕ γ1 h ∂ϕ cos2 θdz = − ∂t 2 ∂t

[4.137]

A Poiseuille flow  − → v = 1−

z2 h2 /4

 [vxmax (x, y), vymax , 0]

[4.138]

generates an additional torque per unit volume → − → Γ visc = γ1 − n ×



 1→ − → − → − ˆ n × (− n · A) ω × n − γ2 → 2

[4.139]

Using equations [4.135] and [4.136], and after integration on z, we obtain three other contributions to the viscous torque per unit area. The first one γ2 − γ1 (vxmax sin ϕ − vymax cos ϕ) or π − −−−→ −→ 2α2 → Γrt = − d × vmax π

Γrt = −

[4.140] [4.141]

is identical to expression [4.24] of the rheotropic torque calculated in Pieranski et al. (2017) and already discussed in section 4.4.1.2, because γ2 − γ1 = 2α2 . This contribution is related to the gradient of the velocity v with respect to z. The second contribution  Γrot = γ1 h

1 1 + π2 3



1 2



∂vymax ∂vxmax − ∂x ∂y

 [4.142]

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Liquid Crystals

→ is related to the z component of the vorticity − ω . It vanishes in the bulk of flat and wide channels, because Poiseuille flows are irrotational, but can play a role in the vicinity of boundaries (e.g. lateral walls). The last contribution  ∂vymax ∂vxmax − ∂y ∂x     1 ∂vymax 1 1 ∂vxmax cos 2ϕ +γ2 hvmax + + π2 3 2 ∂x ∂y

→ − Γ BSP = γ2 h



1 1 + 2 π 3



1 sin 2ϕ 2



[4.143]

can also play a role in the vicinity of boundaries, as well as in stagnation points of microfluidic flows. 4.16.3. Magnetic torque The magnetic torque acting on the dowser field discussed and calculated in section 4.5.1, is given by Γmag =

h χa (d · B)(d × B) 2 μo

[4.144]

where χa = χ// − χ⊥ is the anisotropy of the magnetic susceptibility. 4.16.4. Electric torque The electric torque acting on the dowser field was discussed and calculated in Pieranski and Godinho (2019b) and in section 4.6.1. It is given by ΓE = Pf e d × E + (o a h/2)(d · E)d × E

[4.145]

where a = // − ⊥ is the anisotropy of the dielectric permittivity and Pf e is the flexo-electric polarization. 4.17. References Blinov, L. and Chigrinov, V. (1996). Electrooptic Effects in Liquid Crystal Materials. Springer-Verlag, New York. Boyd, G., Cheng, J., Ngo, P. (1980). Liquid-crystal orientational bistability and nematic storage effects. Appl. Phys. Lett., 36, 556–558.

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Cabeça, R., Godinho, M., Pieranski, P. (2019). Magnetic tweezers for captive disclination loops. The European Physical Journal Special Topics, 227(17), 2439–2454. Cladis, P. and Brand, H. (2003). Hedgehog-antihedgehog pair annihilation to a static soliton. Physica A, 326, 322–332. Cladis, P., Saarloos, W., Finn, P., Kortan, P. (1987). Dynamics of line defects in nematic liquid crystals. Phys. Rev. Lett., 58, 222–225. Cladis, P., Finn, P., Brand, H. (1995). Stable coexistence of spiral and target patterns in freely suspended films of smectic-c liquid crystals. Phys. Rev. Lett., 75, 2945. ˇ Copar, S., Seˇc, D., Aguirre, L., Almeida, P., Dazza, M., Ravnik, M., Godinho, M., Pieranski, P., Žumer, S. (2016). Sensing and tuning microfiber chirality with nematic chirogyral effect. Physical Review E, 93(3), 032703. Dozov, I., Martinot-Lagarde, P., Durand, G. (1982). Flexoelecrically controlled twist of texture in a nematic liquid crystal. J. Phys. Lett., 365, 356–369. Embon, L., Anahory, Y., Jeli´c, V., Lachman, E., Myasoedov, Y., Huber, M., Mikitik, G., Silhanek, A., Miloševi´c, M., Gurevich, A., Zeldov, E. (2017). Imaging of super-fast dynamics and flow instabilities of superconducting vortices. Nature Communications, 8, 85. ˇ Emeršiˇc, T., Zhang, R., Kos, Z., Copar, S., de Pablo, J., Tkalec, U. (2019). Sculpting stable structures in pure liquids. Sci. Adv., 5, eaav4283. Fazzio, S., Nannelli, F., Komitov, L. (2001). Sensitive methods for estimating the anchoring strength of nematic liquid crystals on Longmuir-Blodgett monolayers of fatty acids. Phys. Rev. E, 63, 061712. Friedel, G. (1922). Etats mésomorphes de la matière. Annales de physique, 18, 273. de Gennes, P.-G. (1972). Types de singularités permises dans une phase ordonnée. CRAS, 275, 319–321. de Gennes, P.-G. and Prost, J. (1993). The Physics of Liquid Crystals. Clarendon Press, Oxford. Gilli, J., Thiberge, S., Vierheilig, A., Fried, F. (1997). Inversion walls in homeotropic and cholesteric layers. Liq. Cryst., 23, 619. Giomi, L., Kos, Ž., Ravnik, M., Sengupta, A. (2017). Cross-talk between topological defects in different fields revealed by nematic microfluidics. Proceedings of the National Academy of Sciences, 114(29), E5771–E5777. Hadjiefstatiou, E. and Montagnat, L.-M. (2017). La course des monopôles nématiques, L2 Internship. Laboratoire de Physique des Solides, Université Paris-Saclay, Orsay. Hindmarsh, M. (1995). Where are the hedgehogs in quenched nematics? Phys. Rev. Lett., 75, 2502. Kleman, M. and Lavrentovich, O. (2006). Topological point defects in nematic liquid crystals. Phil. Mag., 86, 4117–4137. Kurik, M. and Lavrentovich, O. (1988). Defects in liquid crystals: Homotopy theory and experimental studies. Sov. Phys. Usp., 31, 196–223. Lavrentovich, O. (1992). Geometrical anchoring at an inclined surface of a liquid crystal. Phys. Rev. A, 46, R722. Lavrentovich, O. and Rozhkov, S. (1988). Strings with boojums at their ends: Topological defects of a new type in nematic liquid crystals. JETP Lett., 47(4), 254–258.

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Link, D., Nakata, M., Takanishi, Y., Ishikawa, K., Takezoe, H. (2001). Flexo-electric polarisation in hybrid nematic films. Phys. Rev. E, 65, 019701(R). Liu, Z., Luo, D., Yang, K.-L. (2019). Flow-driven disclination lines of nematic liquid crystals inside a rectangular microchannel. Soft Matter, 15(28), 5638–5643. Meyer, R. (1969). Piezoelectric effects in liquid crystal. Phys. Rev. Lett., 22, 918–921. Meyer, R. (1972). On the existence of even indexed disclinations in nematic liquid crystals. Phil. Mag., 27, 405–424. Musˇeviˇc, I., Škarabot, M., Tkalec, U., Ravnik, M., Žumer, S. (2006). Two-dimensional nematic colloidal crystals self-assembled by topological defects. Science, 313, 954–957. Nabarro, F. (1972). Singular lines and singular points of ferromagnetic spin systems and of nematic liquid crystals. J. de physique, 33, 1089–1098. Oswald, P. and Pieranski, P. (2005). Nematic and Cholesteric Liquid Crystals, Concepts and Physical Properties Illustrated by Experiments. CRC Press, Taylor and Francis, London, New York, Singapore. Oswald, P., Poy, G., Vittoz, F., Popa-Nita, V. (2013). Experimental relationship between surface and bulk rotational viscosities in nematic liquid crystals. Liqu. Cryst., 40, 734–744. Pieranski, P. (2019). Pierre-Gilles de Gennes: Beautiful and mysterious liquid crystals. Comptes rendus physique, 20, 756–769. Pieranski, P. and Godinho, M. (2019a). Electro-osmosis and flexo-electricity in the dowser texture. EPJE, 42, 69. Pieranski, P. and Godinho, M. (2019b). Flexo-electricity of the dowser texture. Soft Matter, 15, 1469–1480. Pieranski, P. and Godinho, M. (2020). On generation, motions and collisions of dowsons. Frontiers in Physics 7, Article 238. ˇ Pieranski, P., Godinho, M. and Copar, S. (2016a). Persistent quasiplanar nematic texture: Its properties and topological defects. Phys. Rev. E, 94, 042706. ˇ Pieranski, P., Copar, S., Godinho, M., Dazza, M. (2016b). Hedgehogs in the dowser state. EPJE, 39, 121. Pieranski, P., Hulin, J.-P., Godinho, M. (2017). Rheotropism of the dowser texture. EPJE, 40, 109. Poincaré, H. (1886). Sur les courbes définies par les équations différentielles. J. de math., 2(4), 151. Poulin, P., Stark, H., Lubensky, T., Weitz, D. (1997). Novel colloidal interactions in anisotropic liquids. Science, 275, 1770–1773. Sengupta, A. (2015). Topological microfluidics: Present and prospects. Liquid Crystals Today, 24(3), 70–80. Sengupta, A. (2018). Nematic liquid crystals and nematic colloids in microfluidic environment, PhD Thesis. Georg-August-Universität, Göttingen. Sengupta, A., Tkalec, U., Ravnik, M., Yeomans, J., Bahr, C., Herminghaus, S. (2013). Liquid crystal microfluidics for tunable flow shaping. Physical Review Letters, 110(4), 048303. Srinivasan, G. (2020). Disinclinaisons et parois dans les cellules nématiques torsadées, M1 Internship. Laboratoire de Physique des Solides, Université Paris-Saclay, Orsay.

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Tang, X. and Selinger, J. (2020). Minimisation principle for shear alignement of liquid crystals. Phys. Rev. E, 101, 032701. Ternet, D., Larson, R., Leal, L. (1999). Flow-aligning and tumbling in small-molecule liquid crystals: Pure components and mixtures. Rheologica Acta, 38(3), 183–197. Wang, M., Li, Y., Yokoyama, H. (2017). Artificial web of disclination lines in nematic liquid crystals. Nature Communications, 8(1), 1–7. Williams, C., Pieranski, P., Kleman, M. (1972). Nonsingular s=+1 screw disclination lines in nematics. Phys. Rev. Lett., 29, 90–92.

5

Spontaneous Emergence of Chirality Mohan SRINIVASARAO1 1

School of Materials Science and Engineering, School of Chemistry and Biochemistry, Center for Advanced Research on Optical Microscopy (CAROM), Georgia Institute of Technology, USA

5.1. Introduction This chapter deals with the spontaneous emergence of chirality or chiral structures from materials that are achiral. We will primarily focus on the appearance of chiral structures in the context of the simplest of the liquid crystalline phases, the nematic phase, where the constituent molecules possess long-range orientational order. In doing so, we intend to take a broader approach, where our observations are contexualized, taking into account the historical developments dealing with “chirality”. We will begin with a discussion of chirality since the early 1800s, highlighting the many contributions of French scientists in the 19th century to molecular chirality. Since the manifestations of “chirality” have often been through optics, and, in particular, to the rotation of a light beam’s plane of polarization, it would seem not only appropriate but also required that we consider the history of these ideas in some detail. Following these ideas, we shall discuss the appearance of chiral structures from achiral materials, in particular, from lyotropic nematic liquid crystals (LCs). A term that is often used to describe such appearances of chirality is “Chiral Symmetry Breaking”, a term that can be quite confusing, as well as, perhaps, being used rather Liquid Crystals, coordinated by Pawel PIERANSKI, Maria Helena GODINHO. © ISTE Ltd 2021. Liquid Crystals: New Perspectives, First Edition. Pawel Pieranski and Maria Helena Godinho. © ISTE Ltd 2021. Published by ISTE Ltd and John Wiley & Sons, Inc.

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incorrectly. We will spend a bit of time to note that this term, in principle, should not be used to describe the appearance of chiral structures from achiral materials. Then we will provide a discussion of the appearance of chirality or chiral structures from nematic LCs under various confinement geometries, from cylindrical to rectangular to square capillaries, and end with misconceptions dealing with optical activity. 5.2. Chirality: a historical tour It is reasonable to ask why should we be concerned with the historical aspects of chirality or handedness in this short chapter. I would argue that it is due to the elegance of the nature of science. Sir Hans Krebs, Nobel Laureate in Physiology or Medicine (1953) (Krebs 1970), provides an answer: “those ignorant of the historical development of science are not likely ever to understand fully the nature of science and scientific research”. Since the history of chirality, or molecular chirality, is particularly rich in beautiful experiments (and elegant theories), it would appear worthwhile to take a detour into the pages of history. The German philosopher Immanuel Kant is perhaps the first to have expounded on the property of handedness, during the latter part of the 18th century, dealing with a debate among the philosophers of the time, on the nature of space (Kant 1991). In referring to the two hands, which are mirror images of each other, Kant wrote sie können nicht kongruieren (they are not congruent), implying that mirror image objects that cannot be made exactly congruent are incongruent counterparts. Examples of such “incongruent counterparts” are asymmetric solid figures with identical shapes and sizes but opposite handedness, examples of which include snail shells, twining plants and right and left hands, among others. The science of molecular chirality encompasses a vast area of science (Wagnière 2007), with profound implications for biology (Mason 1991; Guijarro and Yus 2009). An article entitled “Chemistry at its most beautiful” published in the pages of Chemical and Engineering News, on the 25th of August 2003, presents the results of a survey of the journal’s readers on “the most beautiful experiments in the history of chemistry” (Freemantle 2003). At the top of list, in first place, was Louis Pasteur’s manual separation of sodium ammonium tartrate crystals into two sets that were mirror images of each other. A prominent historian of science noted the scientific significance of Pasteur’s discovery by stating: “the experiment was the effective beginning of stereochemistry and understanding molecules in three dimensions” (Freemantle 2003). Another prominent chemist and historian notes, “Pasteur’s separation of optical isomers opened an area of chemical structure particularly important to organic chemistry and biochemistry”.

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The now familiar narrative of Pasteur’s remarkable discovery starts when the 25-year-old Pasteur undertook a project to crystallize a number of different compounds, and he started his work on tartaric acid. It turns out that crystals of this organic acid are abundant in the sediments of fermenting wine. Often crystals of a second acid, called “paratartaric acid” or “racemic acid” (the etymological root of “racemic acid” comes from the Latin word racemus, meaning “cluster of grapes”) are also found in the sediments of wine barrels. At the time, it was noted that the chemical compositions of the two acids, tartaric and paratartaric, were identical. However, in solution, the two showed a striking difference; tartaric acid rotated a beam of polarized light passing through it to the right, while paratartaric acid did not rotate the plane of polarization. This, clearly, was quite puzzling and led Pasteur to wonder how this could be? Pasteur, at this point, was clearly convinced that the internal structure of the two compounds must be different, and that this difference might expose itself in the crystalline form. It must be noted that a number of experts in the developing field of crystallography had studied these compounds and noted that no difference could exist. Pasteur painstakingly examined the crystals and found the difference he was looking for. He found that all pure tartaric acid crystals looked identical but noticed that there were two types of crystals in paratartarate. One type was the mirror image of the other, a difference that Pasteur was looking for. Pasteur then proceeded to separate the left and right crystals manually under the microscope, to form two piles of crystals (Figure 5.1). He then proceeded to show that, in solution, one form rotated light to the left and the other to the right, the magnitude of the rotation, measured with a polarimeter shown in Figure 5.2, was nearly equal. This simple experiment showed that organic molecules with the same chemical composition can have spatially different forms, and with this work was launched the new science of stereochemistry.

Figure 5.1. Hemihedral crystals of sodium ammonium tartrates. Redrawn after Pasteur

Pasteur’s findings led him to realize that the molecules of tartaric acid had to be chiral, in effect, he had discovered molecular chirality. However, it should not escape the attention of the reader that Pasteur made his discovery of molecular chirality at a time when little was known about the chemical structure and atomic bonding, as well as the spatial arrangement, and we would have to wait until 1874

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Liquid Crystals

when Van’t Hoff and LeBel proposed, independently and almost at the same time, the asymmetric carbon atom as a basis for molecular chirality (Ball 2005). As noted by Van’t Hoff and LeBel, in such an atom the four substituents differ, and if the arrangement of the substituents on the carbon is not planar, two non-superposable mirror images are possible, thus introducing chirality (see Figure 5.3).

Figure 5.2. Schematic of a primitive polarimeter that is used to measure the optical rotation

Figure 5.3. Two mirror images of an amino acid, alanine

Stereochemistry, broadly viewed, deals with the study of the spatial arrangement of atoms that form the structure of the molecules and their manipulation, and is an important branch that deals with chirality or chiral molecules. Stereochemistry is a complex and rich discipline, and its language is just as rich and complex. When Pasteur discovered molecular chirality (chirality, as a term, was not in use at the time of Pasteur’s ground-breaking discovery), he recognized the need for a term to denote the phenomenon of handedness in chemistry and crystallography, and adopted the term dissymétrie (dissymmetry) and dissymétrique (dissymmetric) for the purpose. It should also be noted, at this point, that Pasteur did not coin the word dissymétrie, which appeared in the scientific literature in France well before Pasteur.

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It appears to have been coined by Frédéric Jacob Soret to describe a form of the crystals of aragonite (a form of CaCO3) as dissimétrique [sic], to indicate that the crystals lacked certain symmetry properties relative to the crystal axis. The terminology dealing with stereochemistry, as discussed in many texts, is often confusing or misused; however, we shall not be bogged down in such a discussion here. Nevertheless, these are important discussions, and hence the reader is referred to a number of excellent reviews on the nomenclature and terminology, and its egregious errors in the literature and in textbooks (Lowry 1964; O’Loane 1980; Gal 2013). William Thompson and Lord Kelvin introduced, in 1884, the terms “chiral” and “chirality” (Thomson and Lord Kelvin 1904): I call any geometrical figure, or group of points, chiral, and say that it has chirality, if its image in a plane mirror, ideally realized, cannot be brought to coincide with itself. “Chiral” and “chirality” are derived from the Greek word, “cheir”, meaning “hand”. Pairs of human hands are the most common examples of Kelvin’s geometrical type; they are “identical opposites”, mirror images of each other (Thomson and Lord Kelvin 1904). The word “enantiomorph”, closely associated with chirality, captures its Greek-derived meaning of “opposite form” (enantios morphe). Clearly, this is what Pasteur meant by “dissymétrie moléculaire”. Pasteur, in a lecture delivered to the Société Chimique de Paris in 1860, entitled Recherches sur la dissymétrie moléculaire des produits organiques naturels, speculated on the atomic arrangements in dextro- and levo-organic compounds that are optically active: Are the atoms of the right acid grouped on the spirals of a dextrogyrate helix, or placed at the summits of an irregular tetrahedron, or disposed according to some particular dissymmetric grouping or other? We cannot answer these questions. But it cannot be doubted that there exists an arrangement of the atoms in a dissymmetric order, having a non-superposable image, and it is no less certain that the atoms of the levo-acid realize precisely the inverse dissymmetric grouping to this. (Ramsay 1981) It was Van’t Hoff and LeBel in 1874 (Lowry 1964; Ball 2005) who provided the solution with the proposal of a tetrahedral carbon. Since the time of Kekulé (~1858), it was known that carbon atoms generally form bonds to four other atoms. Van’t Hoff suggested that the stick diagrams that were in use to depict organic molecular

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structures as an arrangement in space should be extended into the third dimension, out of the plane of the paper. Van’t Hoff’s proposal amounted to having the four bonds pointing to the corners of a tetrahedron. This, said Van’t Hoff, is how organic compounds acquire a handedness, resulting in the optical rotation. 5.2.1. Chirality and optics As is clear from the discussion thus far, optics played a key role in Pasteur’s discovery of molecular chirality. His discovery benefited immensely from the illustrious 19th century French traditions in the fields of crystallography, optics and theories of the arrangement of atoms in organic molecules. Hence it would seem beneficial to understand the key events that were instrumental for Pasteur’s discovery. As already described, Pasteur measured the rotation of the plane of polarization of light on passing through a chiral medium. George Wald (Wald 1957), in a paper titled, “The origin of optical activity” begins by stating: “no other chemical characteristic is as distinctive of living organisms as is optical activity”. Wald was referring to the phenomenon that has been studied since the early 1800s, when the French physicist Jean Baptiste Biot discovered that some substances of biological origin displayed optical rotation (Applequist 1987). Instruments designed to measure such optical rotations, referred to as polarimeters (key components are illustrated in Figure 5.2), have been used to study the interaction of polarized light with chiral or optically active materials. Some of the key events that led to the science of polarimetry and to the study of chirality (or optical activity) in various disciplines are described below: 1669

Erasmus Bartholinus discovers the double refraction of Iceland spar, but he was unaware of the polarizations of the beam split by double refraction.

1690

Christiaan Huygens provided a theory for the double refraction discovered by Bartholinus. He also showed that the two doubly refracted beams had different polarizations.

1808

Étienne-Louis Malus discovered that light, when reflected from glass and metallic surfaces, is polarized. According to Arago (1859), “[O]ne day in his house in the Rue d’Enfer, Malus happened to examine through a doubly refracting crystal (Iceland spar), the rays of the sun reflected from the windows of the Luxembourg Palace. Instead of two bright images which he expected to see, he perceived only one – the ordinary, or the extraordinary, according to the position which the crystal occupied before his eye. This singular phenomenon struck him much”. Malus deduced the law that bears his name, Malus’ Law, that the intensity of plane-polarized light

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317

transmitted by a linear analyzer is proportional to the cosine square of the angle θ between the polarization direction and the preferred direction of the analyzer: I/I0 = cos2θ. It is quite remarkable that the first quantitative expression was deduced with the human eye as the detection system! 1809

Francois Arago observed that light from the sunlit sky was partially polarized, and established the fact that the polarization maximum (vertically polarized) is located about 90° from the sun. He also discovered the presence of a point of zero polarization (Arago’s neutral point) at a position of 20°–25° above the antisolar direction. Optic axis

A R YG B V

A Figure 5.4. (Left) Rotation of plane of polarization of light in an optically active medium. (Right) Rotation for different colors by a plate of quartz 1 mm thick. Redrawn after Jenkins and White (1981)

1811

Francois Arago discovered the rotary polarization of quartz for a light beam traveling along the optic axis of the crystal. By now, it was known that when light propagates parallel to the optic axis of the crystal, there is no double refraction. However, Arago noted that, in the case of quartz, when a plane-polarized beam traveled along the optic axis, the plane of polarization rotated about the direction of the beam (see Figure 5.4 (left)), meaning it emerged vibrating in a different plane from which it entered the crystal. In a beautiful narration of Arago’s work, Kahr writes (Kahr and Arteaga 2012): “the white light, analyzed with a Rochon prism fashioned from doubly refracting Iceland spar crystals, produced pairs of coloured images with hues depending on the position of the prism relative to the input polarization. The colours were indifferent to the azimuth of the quartz plate. Arago had seen for the first–time optical rotation and optical rotatory dispersion.” This is schematically illustrated in Figure 5.4 (right) where

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different wavelengths are rotated to a different extent when the beam exits the crystal. Scientists of the time were fascinated by the blue color of the sky and were having trouble identifying the reasons for this color until John Strutt (also known as Lord Rayleigh) in 1871 provided an explanation based on molecular scattering, refuting Tyndall’s argument that the blue color of the sky is due to foreign matter in the atmosphere. The color of the sky, coupled with the polarization of the sky remained unexplained and John Tyndall, in 1869 (six decades after Arago’s discovery of the polarization of the sunlit sky), stated, “these questions constitute in the opinion of our most eminent authorities, the two great standing enigmas of meteorology” (Young 1982). 1812

Sir David Brewster discovers the relationship between refractive index and polarizing angle (Brewster’s Law).

1812

Jean Baptiste Biot repeats the experiments of Arago and demonstrates that the colors seen by Arago come about due to the direction of polarization of the light rotated and that each wavelength is rotated through different angles. Biot noted that this effect is similar to the dispersion of colors by refraction in a glass prism, but is due to optical rotation and is known as optical rotatory dispersion. Biot also discovered that there are two types of quartz, different only in how they rotate the plane of polarization of the incident polarized light beam.

1815

Biot, while studying quartz immersed in organic liquids, made a far-reaching discovery: that the liquids themselves were optically active. These included distilled oils of turpentine, lemon, sugar solutions and camphor. What was remarkable at that time was that, unlike quartz, these organic substances were optically active as liquids that did not exist in two forms of opposite handedness.

The discovery of optical activity in organic molecules by Biot posed two major questions, the answers to which were not obvious at the time: what is the nature of light that makes the phenomena of polarization and optical rotation possible, and what property of the organic molecules enables them to rotate the plane of polarization of the light? 1821 –1822

Fresnel, following Thomas Young’s notion that light is a transverse wave, implying that the direction of oscillation is perpendicular to the direction of propagation, provided the first theoretical interpretation of Malus’

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observations. Fresnel provided an explanation for the optical activity by considering that linearly polarized light can be thought to be composed of two circularly polarized rays, rotating in opposite directions – a left-circular and right-circular polarized ray – and the plane of polarization of the linearly polarized ray is rotated, should the two circularly polarized components travel in the medium at different velocities. In an effort to demonstrate this conjecture, Frensel constructed a three-piece prism, consisting of one piece of levorotatory quartz and two pieces of dextrorotatory quartz, cut in a way that light travels through the prism along the optic axes of all three parts. Linearly polarized light entering the prism was found to split into two circularly polarized rays of opposite sense when they exited the three-piece prism, thus demonstrating that the two components experience different velocities. It is remarkable that Fresnel’s conjecture of optical rotation as being equivalent to “circular double refraction” still forms the basis for modern theories of optical rotation. Fresnel also offered an alluring explanation for why the refractive index might be different for the two circularly polarized rays: “this may result from a particular constitution of the refracting medium or of its integral molecules which establishes a difference between the sense of right to left and that of left to right; such would be, for example, a helical arrangement of the molecules of the medium which would present opposite properties according as these helices were right-handed or left-handed” (italics added by the author for emphasis). Helices? One could ponder how Fresnel came to view certain matter to be present in a helical arrangement, as very little was known about the geometrical arrangement of molecules in matter at that time. Nature, of course, provides a number of examples of helices, such as in snail shells, the climbing habits of plants and whirlpools. In the case of quartz, Fresnel’s ideas, while wading into new areas of knowledge, also turn out to be correct, as schematically illustrated in Figure 5.5. Viewing along the optic axis of quartz-crystal model, one finds columns of silicon and oxygen atoms built up in spirals. The known chemical structure of crystalline quartz is SiO2. Modern X-ray diffraction has provided the structure of crystalline quartz, where the SiO4 tetrahedra are arranged along a helical axis, with three tetrahedral per turn. To propose “a helical arrangement of molecules” must have been natural to Fresnel, as it is a shape that can occur in two forms, right-handed and left-handed, being

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mirror images of each other, just as the two directions of the rotation of polarization are mirror images of each other.

Si O

x

x

O O

Si

Si

Optic axis

O

x

O Si

O

Figure 5.5. Spiral arrangement of silicon and oxygen atoms along the optic axis in quartz crystals. Redrawn after Jenkins and White (1981)

Almost two centuries after Fresnel’s remarkable experiment, Ghosh and Fischer (2006) demonstrated that a chiral liquid also splits a linearly polarized light beam into two circularly polarized beams of opposite handedness. Figure 5.6(a) depicts the experimental arrangement and the results. In order for the splitting of the beams to be large enough to be seen by the detector, Ghosh and Fischer used multiple refractions through prismatic cuvettes filled with optically active solutions (limonene, in the experiments), a scheme similar to that used by Fresnel. In Figure 5.7(b), the splitting of the beams of a 405 nm diode laser beam is clearly demonstrated and the two beams are indeed circularly polarized with opposite handedness.

Figure 5.6. Experiment of Ghosh and Fischer: (a) Schematic of the geometry used to image the double refraction from chiral fluid. (b) Intensity plots of a laser after passage through (I–IV) 8, 12, 16, and 20 interfaces, respectively. Image IV shows two well-separated profiles that are right and left circularly polarized. Reprinted with permission from Ghosh and Fischer, (2006)

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1845

Michael Faraday discovers the phenomenon that now bears his name, of an induction of optical rotation for a polarized light beam traversing an (isotropic) medium parallel to the magnetic field lines. Years prior to this discovery, Faraday also discovered electromagnetic induction. It is said that Einstein considered Faraday (along with Maxwell) responsible for the greatest change in the axiomatic basis of physics since Newton.

1848

Louis Pasteur describes the hemihedral crystals that are optically active, leading to the landmark discovery of “molecular chirality”.

1852

Sir George Stokes describes the four Stokes parameters that describe the polarization of electromagnetic radiation.

1864

James Clerk Maxwell writes a six-page memoir entitled “A dynamical theory of the electromagnetic field”, a development of a mathematical theory for the propagation of electromagnetic waves.

1869

The Scottish physicist, John Tyndall, performs brilliant experiments to demonstrate that the scattering of light by particles depends on the size of the particles, as well as being the first to realize that a cloud of fine particles polarizes the beam completely at a scattering angle of 90° from the propagating direction. Tyndall published a study of photochemical smogs under the title “On the blue colour of the sky, the polarization of skylight and on the polarization of light by cloudy matter generally”. This study would prove to be quite decisive to the subsequent analysis of scattered light by Lord Rayleigh. Specifically, Tyndall was the first to observe the bluish color of light upon the scattering of natural light.

1871

Following Tyndall’s work, not being satisfied with the explanation of the blue color of the sky, John Strutt (later known as Lord Rayleigh) provides an explanation of the blue color of the sky, deriving an expression for the wavelength dependence of the intensity of scattered sunlight.

1893

Lord Kelvin, in a lecture entitled, “On the molecular tactics of a crystal”, introduces the concept of chirality and the words “chiral” and “chirality”, words that did not exist at the time of Pasteur.

1911

In a paper entitled, “On the metallic colouring in birds and insects”, Albert A. Michelson (1927) observed that some scarab beetles possessed a metallic reflection and that the reflection was circularly polarized. He went on to suggest, correctly, that, “the effect must therefore be due to a ‘screw structure’ of ultra microscopic, probably of molecular dimension”. Neville

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Liquid Crystals

and Caveney (1969) pursued the origin of circularly polarized reflection in several scarab beetle cuticles, and found that a “helicoidal structure” is responsible for the color (selective reflection) and the handedness of the circular polarization, and pointed to the analogy of the scarab beetle cuticles to those of cholesteric liquid crystals or chiral nematic liquid crystals. 5.2.2. Chiral symmetry breaking and its misuse The general phenomenon of the spontaneous emergence of chiral structures or objects from achiral or “racemic” initial states is often referred to as spontaneous chiral symmetry breaking. As has been noted (Walba 2003; Ávalos et al. 2004), this term can be befuddling at best and incorrect at worst. This befuddlement arises because an object or a system with “chiral symmetry” is achiral (i.e. it is congruent with its mirror image) in the physics community, while chemists, utilizing point groups to describe congruence with its mirror image, however, use the term “reflection symmetry” to describe an achiral object. It seems useful at this point to quote from Mislow’s text (Mislow 2002) from the section entitled “Reflection Symmetry-Point Groups”: An object is said to have reflection symmetry if it is superimposable on its reflection or mirror image. Reflection symmetry is revealed by the presence of a rotation–reflection axis (also called mirror axis, improper axis or alternating axis) Sn: in a molecule having such an axis, reflection in a plane perpendicular to that axis followed by rotation by 360°/n will convert the object into itself. It is obvious, then, for the case where n = 1, the rotation–reflection axis is simply a mirror plane. Should an object lack reflection symmetry in its point group, then it is a chiral object. Hence, in this context, the phrase chiral symmetry most naturally suggests the antonym of reflection symmetry, i.e. a chiral object – precisely the opposite of what is meant by chiral symmetry in “chiral symmetry breaking”, as used in physics. Walba notes that from the perspective of chemists it will seem reasonable to use the term “achiral symmetry breaking” to describe the breaking of reflection symmetry, though this term is also rife with confusion. It has therefore been suggested by David Walba (2003) and others (Ávalos et al. 2004) that a compromise to bridge differences between different communities is to use the term “reflection symmetry breaking” as the least confusing terminology describing the phenomenon formerly known as chiral symmetry breaking. Barron (2009) provides an

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illuminating discussion on the use and misuse of the term “chiral symmetry breaking” and eventually settles on the term “mirror-symmetry breaking”. Herein, I use the term “spontaneous emergence of chirality” to denote the appearance of chiral structures from achiral systems. 5.2.3. Spontaneous emergence of chirality or chiral structures in liquid crystals A well-known system where reflection symmetry is broken in achiral LCs is the twisted nematic (TN) state, where the spontaneous reflection symmetry breaking is made possible by specific boundary conditions. Nematic LCs possess long-range orientational order, and are characterized by an average molecular orientation along a given direction referred to as the director, specified by a non-polar unit vector or ˆ (de Gennes 1974). Consider two uniaxial solid substrates, for example, director (n) mica, with the optic axis parallel to the plates, that provide strong “planar anchoring” for the nematic director, n̂. When a nematic LC is confined between two such plates that are parallel, it is found that the director aligns along the optic axis of the uniaxial substrate, resulting in a uniform achiral director structure, as shown in Figure 5.7. Now consider that one plate is rotated by 90° relative to the other, a chiral, uniformly twisted director configuration results. If the twist is low, coupled with high birefringence of the LC, then the plane of polarization of a linearly polarized light incident on the cell parallel to the optic axis of the first plate will rotate to follow the twist of the director, and will exit the second plate with its polarization along the optic axis of the second plate. In other words, the plane of polarization will have rotated by 90°, as if the light had transited through a chiral medium. Mica plate

Figure 5.7. Illustration of the director configuration of Mauguin twisted nematic cells reported in 1911. The optic axis is shown as red arrow. (Left) Before and (right) after the rotation of the top plate

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Liquid Crystals

In 1911, Mauguin reported on precisely this experiment using thin mica sheets as the solid substrates. While not much was known about the structure and dynamics of LCs in 1911, Mauguin nonetheless reasoned, quite correctly, that the director must be uniformly rotating through the cell, and established that the exiting polarization should rotate with the second plate. This is often referred to as the “wave-guiding mode” or the “adiabatic limit” of plane-polarized light in a TN cell with a 90o twist, and it occurs when Δnd >λ/2π, where Δn is the birefringence of the nematic fluid and d the thickness of the cell. This is also referred to as the Mauguin limit in the literature. Now consider an experiment that Mauguin did not report on in his 1911 article, where one takes the TN cell, with a 90o twist, and heats the system to its isotropic phase, and lets it cool back down to the nematic phase. In this case, in the isotropic phase, the system is achiral (still with the two plates at 90o) but when it cools to the nematic phase it must have a twisted structure, but since, in the nematic phase, n is equivalent to –n, domains of opposite handedness must form with equal probability. This is precisely what is observed in such an experiment, as can be seen from Figure 5.8.

Figure 5.8. Polarized light image of reverse twist domains in a sample of DSCG (disodium chromoglycate) between two parallel plates with an easy o axis of 90 to each other. The scale bar is 200 μm (Courtesy, Peter Collings)

When we observe these macroscopic domains by polarized light microscopy, we can easily observe the disclination lines that separate the heterochiral domains, though the output polarizaiton is identical for the pair of heterochiral domains. This is the simplest case of the spontaneous emergence of chirality from an achiral system. Another example of the emergence of chiral structures is during a temperature-driven anchoring transition (Amundson and Srinivasarao 1998; Zhou et al. 2002; Zhou et al. 2006); for example, a nematic LC confined between two plates where, at lower temperatures, the surfaces dictate a condition where the director is normal to

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the substrates (homeotropic alignment) and, at a higher temperature, adopts a director configuration parallel to the bounding substrates (planar alignment). In such a system, cycling through the anchoring transition temperature, say, from the high temperature planar state to a lower temperature homeotropic state, allows for the rotation of the director in two opposite directions [n = − n], thus leading to a 180° inversion wall, in this case, a pure twist wall. Such inversion walls have been observed in a number of experiments (Leger 1973; Ryschenkow and Kleman 1976). Figure 5.10 schematically illustrates the director configuration of an inversion wall consisting of 180° twist deformation along the x direction, the so-called Bloch wall (Kleman 1983), where d is the thickness of the Bloch wall and h is the sample thickness. Helfrich first theoretically described the director configuration of inversion walls formed due to the application of a magnetic field (Helfrich 1968). Such wall defects in nematic phases are usually unstable and collapse on themselves in a short time, but can be stabilized when confined by the bounding surfaces. Such a Bloch wall, observed between crossed polarizers, is shown in Figure 5.9, where the director configuration on either side of the wall is homeotropic and we can see the symmetric and parallel color bands with respect to the center yz plane of the wall when a white light source is used in the polarized light microscope. d

h

z y

x

xy

Figure 5.9. (Top) Schematic of a pure twist Bloch wall of width d. (Bottom) (left) A confocal image of a Bloch wall (dotted ellipse), and crossed polarized light images of a Bloch wall with (middle) white and (right) monochromatic illumination

The color sequence follows the Michel–Levy birefringence chart from either edge of the wall to the central plane. When a monochromatic light source is used, the wall between crossed polarizers shows interference fringes parallel to the wall. Should the Bloch walls be long enough, it is often observed that the wall is composed of two different handednesses, separated by what is referred to as a Neél wall, that mediates the difference in the two distinct handednesses of the Bloch walls, as illustrated in Figure 5.10. These structures are often not considered as

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Liquid Crystals

spontaneous emergence of chiral structures, but, in fact, these are clear examples of the emergence of a chiral structure from an otherwise achiral system.

Figure 5.10. POM images of Bloch walls of two handednesses, mediated by a Neel wall, along with schematic of the director configuration of a Neel wall

5.2.4. Spontaneous emergence of chirality due to confinement It is often convenient to study LCs in flat geometries, as it allows for easy manipulation of the director and the measurement of many characteristic properties using simple experiments. However, the confinement of LCs to curved geometry results in a much richer phenomenology. Curvature induces deformation of the director that costs energy. The terms describing the free energy of LCs are all of the order (∇n)2 to leading order. For this reason, Frank (1958), in his original description, uses the term “curvature elasticity” to describe the nematic elasticity. The presence of curvature can also result in competing effects of surface and bulk elastic constants which result in several interesting director configurations. The richness is further enhanced if the nematic fluid has significant anisotropy of its elastic constants, as is often the case with lyotropic nematic LCs. In this section, we discuss the spontaneous emergence of chiral structures from a class of LCs referred to lyotropic chromonic liquid crystals (LCLCs), as well as some lyotropic polymeric LCs. LCLCs have gained increasing attention in the last two decades as an interesting but still poorly understood class of lyotropic LCs (Attwood et al. 1986; Vasilevskaya et al. 1989; Lydon 1998a, 1998b, 2004). LCLCs consist of many dyes, drugs, nucleic acids, antibiotics and anti-cancer agents (Cox et al. 1971; Hartshorne and Woodard, 1973; Attwood et al. 1986; Vasilevskaya et al. 1989; Lydon 1998a, 1998b, 2004; Horowitz et al. 2005; Park et al. 2008). The main structural feature of the molecules that comprise a chromonic liquid crystalline phase is either a disk-like or plank-like rigid aromatic core surrounded by either ionic or hydrogen-bonding groups (Lydon 1998b). When these molecules are dissolved in water, they stack

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against each other into rod-like aggregates with polydisperse lengths, even at very low concentrations (Attwood et al. 1986; Vasilevskaya et al. 1989; Lydon 1998b, 2004). The π-π interactions among the molecules are thought to be the driving force for the aggregation of these molecules into rod-like shapes. (a) O

O

O Na OOC

O

(b)

SO3Na

N

OH

O O

COO Na

(c)

N OH

NaO3S

N

M

Figure 5.11. Molecular structures of (a) DSCG (disodium chromoglycate) and (b) (sunset yellow (SSY) FCF). (c) Schematic of two chromonic mesophases: nematic N and hexagonal (or columnar) M phases. (c) is redrawn after Lydon (1998b)

As the concentration reaches about 5–30 wt% of the dye in solution, a liquid crystalline phase forms (Attwood et al. 1986; Vasilevskaya et al. 1989; Lydon 1998b, 2004). There are two principal, well-characterized chromonic mesophases: a nematic N phase and a hexagonal M phase, as illustrated in Figure 5.11. The M phase is a fairly concentrated phase with a hexagonal array of molecular stacks, also referred to as the columnar phase. In the recent past, the study of LCLCs has led to a number of observations that have been quite surprising and unexpected, and has inspired re-examination of a number of studies dealing with nematic LCs. We will start our description of them with a rather intriguing list of new observations that lead to the emergence of chiral structures from achiral systems which are quite exciting, and illustrate the rich phenomenology arising, primarily, from the fact that these materials posses anisotropic elastic properties, coupled with confinement to curved geometries. As early as 1949, Onsager (1949) pointed out that a rod-like object in solution can form an ordered state, the so-called nematic phase (N), where the rods are parallel to each other, when their concentration exceeded some critical value.

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Tobacco Mosaic Virus (TMV) is a rod-like particle with a length of 3,000 Å and 180 Å in diameter, and is monodisperse. Monodisperse aqueous solutions of TMV form a nematic phase between the concentrations of 30 and 70 mg/ml in pure water. Onsager provided an explanation for the obersvations of Bawden, Pirie, Bernal and Fankuchen (Bawden et al. 1936), dealing with aqueous suspensions of TMV that phase separated into an isotropic phase (I) and a nematic phase (N). It should be noted that the N-phase emerges as spindle-like objects called “tactoids”, and such N-tactoids have been shown to exist for a number of other materials. In these earlier studies, there was no mention of chirality in the tactoids, although it might seem prudent to re-examine these systems in light of things that are discussed in the following pages. Recently, Tortora and Lavrentovich (2011), when studying solutions of disodium chromoglycate (DSCG, see Figure 5.11(a) for the chemical structure) in water with a finite concentration of polyethylene glycol (PEG), observed tactoids that displayed “chiral symmetry breaking”, meaning that the tactoids displayed both left-handed and right-handed twist structures, as illustrated in Figure 5.12. As already noted above, one should not use the term “chiral symmetry breaking” for these situations. Figure 5.13 illustrates a chiral tactoid that was observed by Srinivasarao (Srinivasarao and Berry 1992) during the preparation of monodomains of a polymeric nematic in solution, but, at that time, was not recognized for the spontaneous emergence of chirality. This appears to be the first observation of the emergence of chiral structures from an achiral nematic solution. Incidentally, such polymeric nematics have a large anisotropy in their elastic constants, with twist elastic constant being at least an order of magniture lower than either splay or bend (Desvignes et al. 1993).

P A a

b

c

d

Figure 5.12. Nematic tactoids in the biphasic N-Iso sample of a lyotropic chromonic liquid crystal formed by aqueous dispersion of disodium cromoglycate. (a) Polarizing microscopy texture of an achiral tactoid and (b) the corresponding director structure; (c) and (d) the same for a twisted director structure. The system contains no chiral additives (see Tortora and Lavrentovich (2011) for details)

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The rationale for the emergence of chiral structures in these tactoids is that the twist deformation replaces the energetically costly splay packing of the aggregates within the curved bipolar tactoids. This provides a simple prescription for creating chiral structures: that the material has a certain anisotropy of the elastic properties and that it has non-flat confinement! It is well known that nematic LCs, when confined to spherical droplets with planar anchroing, form what are known as bipolar droplets with a point defect at the poles of the sphere. In their studies dealing with droplets of LCLCs, Yodh and co-workers (Jeong et al. 2014) used solutions of Sunset Yellow (SSY) in water and found that the droplets have a twisted bipolar structure, a consequence of a very low twist elastic constant. This leads to a very large twist in the droplets, despite the lack of chirality in the system, and are similar to those observed when the droplets are made with LCs possessing intrinsic chirality. N

S

S

N

C

C

A PBZT

P

50 μm

Figure 5.13. Chemical structure of poly(benzlbisthiazole) (PBZT) and polarized optical image of a tactoid formed from PBZT solutions in methane sulfonic acid

5.2.5. Spontaneous emergence of chirality due to cylindrical confinement 5.2.5.1. Director normal to the surfaces In 1973, R.B. Meyer (Meyer 1973; de Gennes 1974) and Williams, Pieranski and Cladis (Williams et al. 1972) demonstrated that when a nematic fluid is confined to a cylindrical capillary, where the director is normal everywhere on the capillary surface, the director (n) escapes along the length of the capillary, therefore leading to a continous structure (plus a few point defects) and thus avoiding having a line defect throughout the capillary. This has been referred to as “escape in the third dimension” (or sometimes as an escaped radial) and has been experimentally verified over the course of five decades for many thermotropic LCs. Figure 5.14 illustrates such a situation where a well-known and well-studied thermotropic nematic LC, 5CB, is confined to a cylinder with homeotropic boundary conditions. We can clearly see the continous structure under crossed polarizers and a schematic of the director configuration is also shown in the figure. When such experiments are done with LCLCs the results are quite strikingly different with a new ground state.

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5CB, ER A P

Figure 5.14. POM image of the ER in a cylindrical capillary filled with 5CB, and schematics of the director configuration

Jeong et al. (2015) reported on such an experiment where the SSY solutions were confined to a cylindrical capillary with homeotropic boundary conditions. The initial configuration of the director was the well-known escape in the third dimension. However, this configuration quickly turned into a twisted configuration where the splay and bend deformations in the capillary were replaced by a twist (due to the low twist elastic constant). This results in a twisted and escaped radial configuration (referred to as TER, see Figure 5.15), where the nematic director is parallel to the cylindrical axis near the center; but to possess the homeotropic boundary condition near the capillary wall, the director escapes along the radius through bend and twist distrotions. This also leads to a panopoly of defects that will not be described in detail here, but has been discussed by Jeong et al. (2015). This TER structure is eventually replaced by a set of two singular lines forming a double helix (as shown in Figure 5.16) which remians stable for an extended period of time. It would appear that this structure, with two singular lines forming the double helix, is the new ground state for these lyotropic materials confined to a cylinder with homeotropic boundary conditions. DSCG

SSY A P

Figure 5.15. POM image of the twisted escaped radial texture (TER) in a cylindrical capillary for DSCG and SSY solutions in water, along with schematics of the director configuration

Spontaneous Emergence of Chirality

DSCG A P SSY

331

DSCG

100 μm SSY

Figure 5.16. Formation of the double helix from the TER structure. The white arrow in DSCG point to walls in the structure. Each image has the same polarizer directions and magnification

It appears that this report of a TER structure eventually leading to two line defects forming a double helical structure down the long axis of the cylinder, reported by Jeong et al., is not the first of its kind. Such a structure was reported by Jin-Hua Wang (1996), a student of R.B. Meyer. Wang observed such a structure in solutions of TMV in capillaries with homeotropic alignment, as early as 1996. That TMV forms a liquid crystalline phase was known as early as 1936, and led to the formulation of a theory for the phase separation by Lars Onsager in 1949, as noted above. In the experiments of Wang, TMV solutions in the nematic phase were confined to cylindrical capillaries with homeotropic boundary conditions. Wang also noted that the two bright “spiral lines” running through the entire length of the capillary were “clearly visible even without a microscope”. It was also noted that the lines did not lie on the surface of the quartz capillary, and that they were at a distance of 35 μm from the capillary walls. These observations are strikingly similar to those reported by Jeong et al. (2015), but for a system that is polymeric in nature. It should be mentioned that we have used another polymeric nematic material, poly benzyl glutamate with homeotropic boundary conditions, and the observations mimic what has been seen for LCLCs, TMV and also some micellar nematics (Dietrich et al. 2017). It appears that the observations described thus far, that the escape in the third dimension giving rise to a TER configuration, followed by the transformation of the TER to a system with two spiraling line defects down the length of the capillary, seems to be common for all nematic fluids that are lyotropic in nature, as most if not all have a very low twist elastic constant (Dietrich et al. 2020).

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Liquid Crystals

The presence of TER structures can be rationalized on the basis of a low twist elastic constant for this class of LCs. The free energy cost of splay, twist, bend and saddle-splay distortions of a nematic fluid is described by the well-known FrankOseen elastic free energy: 𝐹 = ∫ 𝑑𝑉[𝐾 (∇ ∙ 𝑛) + 𝐾 (𝑛 ∙ ∇ × 𝑛) + 𝐾 (𝑛 × ∇ × 𝑛) −(𝐾 + 𝐾 )∇ ∙ (𝑛(∇. 𝑛) + 𝑛 × ∇ × 𝑛)]

[5.1]

Jeong et al. (2015) performed numerical calculations to understand the rationale for the observation of TER structures using the known values for lyotropic LCs (Zhou et al. 2012). From their calculations they came to the conclusion that the TER director configuration lowers the elastic free energies of the “escape in the third dimension” or the escaped radial configuration, with a degenerate twist deformation, the twist being either left- or right-handed. They noted that total elastic free energy of the twisted (TER) configuration is smaller than that of the escaped radial configuration when K22< K22~0.27K, where K = K11=K33. So, the condition for observing a twisted structure in lieu of the escaped radial structure is that when the twist elastic constant is about an order of magnitude smaller than the corresponding splay and bend elastic constants, a condition that many lyotropic LCs satisfy. It should be noted, for cylinders with homeotropic boundary conditions, where nˆ is normal to the boundaries of the cylinder, the saddle-splay (K24) contribution to the total free energy takes on a constant value, and does not depend on what nˆ is in the bulk, and hence does not play a major role in determining the director configuration. 5.2.5.2. Director parallel to the surfaces Now we consider the case of confining lyotropic nematics in a cylinder with planar degenerated (gliding) boundary conditions. As already indicated, when nematic LCs are confined within flat boundaries with planar boundary conditions, the ground state is characterized by an average molecular orientation along a common direction referred to as the easy axis of the director, denoted by n. Curvature for nematics costs energy and the coupling of the curvature with the free energy can be appreciated when one notes that the elastic free energy expression of the n is of the order of (∇n)2. For this reason, the Frank free energy is often referred to as the curvature elasticity of nematic LCs (Frank 1958). Pairam et al. (2013) confined prototypical nematics, the thermotropic NLC 4-Cyano-4’-pentylbiphenyl (5CB), in a toroid with the director parallel to the surfaces everywhere, and demonstrated that the, achiral, axial state is unstable to the doubly twisted, chiral state that depended both on K22/K24 and the aspect ratio of the torus ξ =R0/ a, where R0 is the central ring radius and a the tube radius. They also demonstrated that the magnitude of the twist distortion in a toroid grew with

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decreasing aspect ratio due, in large part, to the K24 contributions to the overall free energy of the system, thus effectively screening the energy penalty of an increasing twist distrotion.

(a)

(b)

Figure 5.17. Schematic of the director configuration of (a) axial and (b) double twist

We now confine the LCLCs in a cylinder with planar boundary conditions and interrogate the resulting director configuration. In most experiments with conventional nematics, the director lies along the long axis of the capillary, the axial configuration, as illustrated in Figure 5.17(a). When such a configuration is observed under crossed polarizers, the intensity profile of the transmitted light beam is of the form: 𝐼 ∝ 𝑠𝑖𝑛 (2∅ ), where ∅p is the angle between the easy axis of the polarizer and the director. The observed intensity is a minimum when the director is either parallel or perpendicular to the incident polarization, leading to complete extinction of the light, with the maximum intensity being at an angle of 45° with respect to the incident polarization. This is exactly what is observed when one confines a thermotropic LC, say 5CB for example, with planar boundary conditions. However, when the experiments are repeated with solutions of DSCG, the observations point to a different result. Images under a polarized light microscope for two angular positions of a capillary filled with DSCG solutions are shown in Figure 5.18(a) and 5.18(b), corresponding to the long axis of the capillary being parallel and at 45° with respect to the incident polarization, respectively. It is quite clear in both cases that the expected extinction of the incident light is not achieved. Should one have an axial configuration, the image corresponding to Figure 5.18(a) would be one of complete extinction of light. Instead, incident light is transmitted by propagating through the sample and they appear to possess comparable intensities in both Figure 5.18(a) and Figure 5.18(b). This simple observation makes it abundantly clear that the plane of polarization of the linearly polarized incident light is rotated by the lyotropic LC, DSCG, and exits at an angle to the analyzer that is neither 0° nor 90°. As already noted above when dealing with Pasteur’s discovery of optical activity, this represents a classic signature of twists. A doubly twisted configuration would explain the observed images and is illustrated in Figure 5.17(b). The director of this doubly twisted configuration is axial at the center of the cylinder, twisting progressively as it approaches the surface.

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Liquid Crystals

P

A

100 μm

A P

100 μm

Figure 5.18. Crossed polarized microscopic images of DSCG in a cylindrical capillary

Validation of the hypothesis of a double twist can be done by carrying out Jones matrix simulation with a simple doubly twisted director ansatz, as such simulations quantify the change in the polarization state of the light as it traverses the sample. The doubly twisted director is specified by n = nr er + nθ eθ + nz ez , where er, eθ and ez are the orthonormal vectors in cylindrical coordinates, with nr = 0, nθ = ω (r/R), nz = (1 – nθ2)1/2. The twist parameter ω determines the amount of twist in the system; ω = 0 corresponds to an axial configuration, r is the radial distance from the center of the circular cross-section and R is the cylinder radius. The simulated optical microscopy textures conform closely to the experimental observations that light is indeed not extinguished with the capillary long-axis parallel to the polarization, of the incident polarization as well as reproducing all the aspects of the measured intensity profiles. These lend credibility to the idea that we must, indeed, have a doubly twisted director configuration in a cylindrical capillary. Based on the experimental observations and the Jones matrix simulations, we can then ask a number of questions that are of consequence: What drives the spontaneous twist deformation of the director field? How twisted is the structure of the double twist? Is there a size dependence of this twist? All of these questions are addressed in detail elsewhere (Davidson et al. 2015; Nayani et al. 2015), but we provide the essential arguments below. In order to understand the driving force for the doubly twisted configuration, we start by considering the contribution of the saddle-splay (K24) term in the free energy expression given by equation [5.1], where K11, K22 and K33 are the Frank elastic constants associated with splay, twist and bend bulk deformations. We ignore the splay-bend (K13) contribution as we are dealing with a case of planar anchoring,

Spontaneous Emergence of Chirality

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though the influence of K13 has been quite contentious (Lavrentovich and Pergramenshchik 1995). It should also be noted that several studies have neglected the contribution of saddle-splay under strong anchoring conditions, the rationale for which stems from the fact that the bulk integral of the saddle-splay can be reduced to a surface integral using Stoke’s theorem, and that the contribution of saddle-splay on the surface would be negligible in comparison to the cost of the anchoring violation. However, this argument is troublesome as the saddle-splay contribution can only be neglected if the director depends only on one Cartesian coordinate, and for strong anchoring, which is not the case in the experiments described (Kralj and Zumer 1995). For planar anchoring, when the confining boundaries are curved, the contribution of K24 cannot be neglected, which would indicate that planar anchoring may lay at an angle to the confining cylinder long-axis. Further, in geometries like cylinders where the two principal curvatures are different, the contribution of saddle splay plays a prominent role in determining the director configuration. For the case of planar anchoring, the saddle-splay term can align the director along the direction of the largest principal curvature. To better understand this, the K24 contribution to the free energy (per unit length) can be written as: 𝐹 = − (𝐾 + 𝐾 ) ∫ 𝑑𝑆(𝑘 𝑛 + 𝑘 𝑛 ), where k1 and k2 are the principal curvatures at a point on the surface and n1 and n2 are the director components along the corresponding directions. For a cylinder, n1 = nθ and n2 = nz hence, k1 = 1/R and k2 = 0. As a result, for the case of cylindrical geometry, the integral of 𝐹 is minimized when the director at the surface is along the eθ direction, thus driving the system to twist and provided that there is sufficient anisotropy of the elastic constants, the twisted structure is always stable. Using the same director ansatz outlined for the Jones matrix simulation, we obtain an expression in leading order of ω for the free energy per unit length: 𝐹 2𝜋𝐾

=

(𝐾 − 𝐾 ) 𝜔 𝜔 + + 4 𝐾

𝐾 𝜔 𝐾 2𝑛

[5.2]

It is, in fact, the sign of the quadratic term that determines whether the system adopts an axial configuration or breaks reflection symmetry, resulting in a chiral, doubly twisted director configuration. This theory is generic and would describe the reflection symmetry breaking of any nematic fluid. Provided K24 > K22, the director is always going to be twisted. Specifically, for LCLCs the low value of K22 almost always leads to the possibility of satisfying the twisting criterion. From the observation of twisted director configurations, it is self-evident that K24 for LCLCs studied must always be greater than K22. While the model is relatively simplistic (in terms of the ansatz used), the essential physics is captured with the criterion that K24 > K22 for a twisted structure.

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Liquid Crystals

For ascertaining the criterion for the twist, we can calculate the value of K24 by comparing the value of the twist parameter ω measured experimentally, and that obtained after minimizing the free energy expression with the experimental measurement of ω. The twist parameter is obtained by measuring the twist angle as the beam traverses the diameter of the cylinder. In order to do this, we used the Mauguin limit, where ∅