Liquid Crystals

Оглавление

Pawel Pieranski. Liquid Crystals

Table of Contents

List of Illustrations

List of Tables

Guide

Pages

Liquid Crystals. New Perspectives

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

1.3.6. Polarization state evolution through twisted director fields

1.3.6.1. Propagation parallel to the cholesteric helix: Mauguin regime

1.3.6.2. Propagation parallel to the cholesteric helix: circular Bragg reflection

1.3.6.3. Propagation perpendicular to the cholesteric helix

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. 1.5.1.1. Brief historical survey

1.5.1.2. Tunable and reconfigurable diffraction gratings

1.5.1.3. Lensing and beam steering

1.5.2. Elongated defects structures in frustrated cholesteric films. 1.5.2.1. Brief historical survey

1.5.2.2. Tunable and reconfigurable diffraction gratings

1.5.2.3. Waveguiding and beam steering

1.5.3. Regular optics from other topological structures. 1.5.3.1. Waveguiding along non-singular and singular disclination lines

1.5.3.2. Lasing from structures hosting point, line or loop defects

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 optics

1.8.2. Spontaneous singular geometric phase optics. 1.8.2.1. Tunable vortex generators from random umbilics in nematics

1.8.2.2. Tunable vortex generators from random disclinations in nematics

1.8.3. Directed self-engineered geometric phase optics. 1.8.3.1. Field-induced localization of umbilics in nematics

1.8.3.2. Self-induced singular beam shaping

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

1.9.1.1. Optical vortex coronagraphy: basic principles

1.9.1.2. A soft route to optical vortex coronagraphy

1.9.1.3. Multiple-star 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

3.2.3.1. Usual viscous terms

3.2.3.2. Leslie TM and TH terms

3.2.3.3. Akopyan and Zel’dovich TM and TH terms

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. 3.4.1.1. A bit of history

3.4.1.2. Principle of the experiment and theoretical prediction

3.4.2. Another static experiment proposed in the literature

3.4.3. Continuous rotation of translationally invariant configurations

3.4.3.1. TIC rotation velocity: a general formula

3.4.3.2. Experimental results with the compensated mixtures in the Leslie geometry

3.4.3.3. Experimental results with the diluted mixtures in the Leslie and mixed geometries

3.4.4. Drift of cholesteric fingers under homeotropic anchoring

3.4.4.1. Theoretical predictions

3.4.4.2. Experimental results

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. 3.5.2.1. Principle of the measurement

3.5.2.2. Theoretical predictions in the mixed planar-circular/homeotropic geometry

3.5.2.3. Experimental results

3.6. Conclusions and perspectives

3.7. References

4. Physics of the Dowser Texture

4.1. Introduction. 4.1.1. Disclinations and monopoles

4.1.1.1. Disclinations are ubiquitous

4.1.1.2. Monopoles are scarce

4.1.1.3. Disclinations and monopoles as topological defects

4.1.2. Road to the dowser texture. 4.1.2.1. Difficulties with nematic monopoles in the dowser texture

4.1.2.2. Generic experiments

4.1.2.3. Relative stability of the homeotropic and dowser textures

4.1.2.4. Persistence of the dowser texture

4.1.3. The dowser texture. 4.1.3.1. Order parameters of the dowser texture, the dowser field

4.1.3.2. Dowser texture as a natural universe of monopoles

4.1.3.3. Tropisms of 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

REMARK

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

REMARK

4.3.5. Summary and discussion of experimental results

4.4. Rheotropism. 4.4.1. The first evidence of the rheotropism. 4.4.1.1. Experiment

4.4.1.2. Theory of the rheotropism

4.4.2. Synchronous winding of the dowser field. 4.4.2.1. Experiment

REMARK

4.4.2.2. Interpretation in terms of the rheotropism

4.4.3. Asynchronous winding of the dowser field

4.4.3.1. DDC2

4.4.3.2. CDC1

REMARK

4.4.3.3. Asynchronous winding of dowser field with CDC2

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. 4.4.6.1. Experiment

4.4.6.2. Analysis

4.5. Cuneitropism, solitary 2π-walls. 4.5.1. Generation of π-walls by a magnetic field

4.5.2. Generation and relaxation of circular2π-walls

4.5.3. Cuneitropic origin of the circular2π-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

4.7.1.1. Effects due to the electric field inside the gap

4.7.1.2. Effects due to the electric field outside the gap, evidence of electro-osmosis

4.7.1.3. Electro-osmotic effects inside the gap

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

4.8.3.1. Generic experiment

4.8.3.2. Topological relationship between 2π-walls and monopoles

4.8.3.3. Conjecture on the rheotropic breaking of 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. 4.9.6.1. Gyrophilic behavior of dowsons d-

4.9.6.2. Gyrophobic behavior of dowsons d+

4.9.6.3. Gyrophobic behavior of dowsons d+

4.9.6.4. Gyrophilic behavior of dowsons d-

4.9.6.5. Stationary Cladis–Brand states

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

REMARK

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. 4.16.1.1. Uniform thickness h

4.16.1.2. Cuneitropic torque in a thickness gradient

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. 5.2.5.1. Director normal to the surfaces

5.2.5.2. Director parallel to the surfaces

5.2.6. Some misconceptions about optical rotation

5.3. Concluding remarks

5.4. Acknowledgments

5.5. References

List of Authors

Index

A

B

C

D

E

F

G

H

I, J

L

M

O

P

Q, R

S

T

U, V

W

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Отрывок из книги

Physics of Soft Matter, Field Director – Françoise Brochard-Wyart

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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 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.

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