Nematic Liquid Crystals: Introduction to the

Mathematics and Physics

Apala Majumdar Department of Mathematical

Sciences University of Bath

Summer School on “Frontiers of Applied and Computational Mathematics”

9th-21st July 2018

A bit about myself…

• 2002 – 2006 Ph.D. in Applied Mathematics, University of Bristol CASE studentship with Hewlett Packard Laboratories Title: Liquid crystals and tangent unit-vector fields in polyhedral geometries Jonathan Robbins, Maxim Zyskin; Chris Newton (HP) • 2006 – 2012: University of Oxford Oxford Centre for Nonlinear Partial Differential Equations Oxford Centre for Collaborative Applied Mathematics Keble College, University of Oxford • 2012 – present: University of Bath, United Kingdom

(Visiting) affiliation with OCIAM (Mathematical Institute, University of Oxford) and Cluster Coordinator at Advanced Studies Centre, Keble College Oxford

Liquid Crystals – what are they?

• Mesogenic phases of matter

• Intermediate between solids and liquids

decreasing temperature

History

Discovered by Reinitzer in 1888 : two melting points for cholesterol! !

Courtesy: Peter Palffy-Muhoray Lectures at Colorado - Boulder

History

Discovered by Reinitzer in 1888 : two melting points for cholesterol! !

Courtesy: Professor Sir John Ball 2015 Lyon Lecture Notes

Different Liquid Crystal Phases

Courtesy: Peter Palffy-Muhoray Boulder Lectures 2011

Nematic Liquid Crystals

Anisotropic rod-like molecules with directional properties

Long-range orientational ordering: molecules line up with one another

Nematics are ubiquitous!!

Biological molecules Macromolecules

Microparticles

Nematics are ubiquitous!!

Active nematics Cosmology

Courtesy: Peter Palffy-Muhoray Boulder Lectures 2011

Key word: anisotropy!!!

Soft materials : responses to external stimuli

Courtesy: Images from Peter Palffy-Muhoray’s lectures at Colorado - Boulder

Nematics: unique properties continued…

Combination of

Orientational Order + Susceptibilities + Singularities

Novel mechanical, electro-magnetic and rheological responses

Wide array of applications

• Multi-billion dollar Liquid Crystal Display (LCD) industry

• optical switches and sensors, biological sensors

• liquid crystal elastomers

Display Applications

Key properties:-

• Anisotropic birefringent fluids – strong coupling to incident light

• Sensitive to external electric and magnetic fields .

Twisted Nematic Liquid Crystal Display – a

monostable liquid crystal display.

Bistable displays

Working principle:

• locally stable bright and dark states without an electric field

• power is needed to switch between distinct states but not to maintain them

• larger, higher resolution displays with much reduced power consumption.

Zenithally Bistable Nematic Device

www.eng.ox.ac.uk

Examples of bistable displays:

• Planar bistable liquid crystal device • Zenithally Bistable Nematic Device

Tsakonas, Davidson, Brown,

Mottram , Appl. Phys. Lett. 90,

111913 (2007)

Numerical modelling by

Chris Newton, HP

The Post Aligned Nematic Device

The Post Aligned Bistable Nematic (PABN) device (HP laboratories) – patterned lower surface and mixed boundary conditions stabilize multiple liquid crystal configurations.

Kitson and Geisow, Applied Physics Letters, 80,2002.

• Normal boundary conditions on top substrate.

• Tangent (planar) boundary conditions on bottom substrate and post surfaces.

The PABN cell is bistable – it supports two optically contrasting states with long-term stabilities.

Numerical modelling by Chris Newton (HP Labs)

Planar:

Low tilt around post

Bright state

Tilted:

Strong tilt around post

Dark state

Key questions from HP researchers: • What causes bistability in the PABN cell?

• Is there a well-defined correlation between geometry, boundary conditions and bistability/multistability?

Planar Tilted

Our approach: • Develop mathematical model for PABN cell

• Numerical simulations of PABN cell • Agreement of model with experimental data

Willman et.al , Journal of Display Technology, Vol 4, No. 3, September 2008

Why do mathematicians like these problems?

• Mathematically rich! Topology, Calculus of Variations, Partial Differential Equations, Bifurcation Theory, Dynamical Systems, Scientific Computation, Numerical Analysis…… • Can test theoretical predictions

• Link with applications

• Interdisciplinary

Theory+ Analysis

Simulations

Experiment

Applications

What can mathematics do for LC devices:

• Stable states structure multiplicity defects • Optical properties? Inverse problems • Switching mechanisms Estimates for switching times – relate to image control and refresh times?

See [1] A.Majumdar, C.J.P.Newton, J.M.Robbins and M.Zyskin, 2007 Topology and bistability in liquid crystal devices. Phys. Rev. E, 75, 051703--051714. [2] Raisch, A. and Majumdar, A., 2014. Order reconstruction phenomena and temperature-driven dynamics in a 3D zenithally bistable device. EPL (Europhysics Letters), 107 (1), 16002. [3] Kralj, S. and Majumdar, A., 2014. Order reconstruction patterns in nematic liquid crystal wells. Proceedings of the Royal Society of London Series A - Mathematical Physical and Engineering Sciences, 470 (2169), 20140276.

Different levels of modelling

Aim is to describe orientational order in a nematic liquid crystalline system; includes a description of the preferred directions and the degree of order about these directions

Different levels of modelling

Aim is to describe orientational order in a nematic liquid crystalline system; includes a description of the preferred directions and the degree of order about these directions

• Landau-de Gennes theory primary and secondary direction of preferred alignment (biaxiality) variable degree of order about preferred directions five degrees of freedom: three for preferred directions and two scalar

order parameter

• Ericksen Theory assume single direction of preferred alignment (uniaxial) variable degree of order about single distinguished direction three degrees of freedom: two for preferred direction in three dimensions

and one scalar order parameter

• Oseen-Frank Theory assume uniaxiality assume constant order two degrees of freedom in three dimensions

Different levels of modelling

Aim is to describe orientational order in a nematic liquid crystalline system at a continuum macrocopic level ; includes a description of the preferred directions and the degree of order about these directions

Courtesy: Professor Sir John Ball 2015 Lyon Lecture Notes

A Simple Model for a Nematic System:

The Planar Bistable LC Device

• Micro-confined liquid crystal system:

• Array of liquid crystal-filled square / rectangular wells with

dimensions between 20×20×12 microns and 80×80×12

microns.

• Surfaces treated to induce planar or tangential anchoring

Tsakonas, Davidson,

Brown, Mottram ,

Appl. Phys. Lett. 90,

111913 (2007)

Boundary Conditions :

Chong Luo, Apala Majumdar and Radek Erban, 2012 "Multistability in planar liquid crystal wells", Physical Review E, Volume 85, Number 6, 061702 Kralj, S. and Majumdar, A., 2014. Order reconstruction patterns in nematic liquid crystal wells. Proceedings of the Royal Society of London Series A - Mathematical Physical and Engineering Sciences, 470 (2169), 20140276.

Tsakonas, Davidson,

Brown, Mottram 2007

•Top and bottom surfaces treated to have tangent boundary conditions – liquid crystal molecules in contact with these surfaces are in the plane of the surfaces.

Bistability: two experimentally observed states

Tsakonas, Davidson,

Brown, Mottram 2007

Diagonal state: liquid crystal alignment along one of the diagonals.

Defects pinned along diagonally opposite vertices.

Also see Lewis, A., Garlea, I., Alvarado, J., Dammone, O., Howell, P., Majumdar, A., Mulder, B., Lettinga, M. P., Koenderink, G. and Aarts, D., 2014. Colloidal liquid crystals in rectangular confinement : Theory and experiment. Soft Matter, 39, pp. 7865-7873.

Rotated state: vertical liquid crystal alignment in the interior.

Defects pinned at two vertices along an edge.

Tsakonas, Davidson,

Brown, Mottram 2007

Optical contrast?

Theoretical and experimental optical textures:

Theory:

Experiment

:

Tsakonas,

Davidson,

Brown, Mottram

2007

Role of aspect ratios in optical properties? Joint work with Alex Lewis, Peter Howell.

Relevance to other LC-like systems

Experiments on viruses in confined and shallow 3D

chambers by Oliver Dammone et. al in Aarts Lab,

University of Oxford.

A toy 2D model • Model the geometry within a two-dimensional Oseen-Frank framework.

• Simplest continuum theory for nematic liquid crystals that describes the nematic state by a unit-vector field n. • Physical interpretation of n: describes the average locally preferred direction of molecular alignment at each point in space.

n (r)

A toy 2D model contd.

sin,cosn

• Oseen-Frank Energy Functional : •Two-dimensional vector field and use one-constant approximation (set

all elastic constants to be equal and K4 = 0)

dVnntrKKnnK

nnKnKE[n]

22

42

2

3

2

2

2

1

dA||KE

Ω

2

0

A toy Oseen-Frank 2D model

Lewis, A., Garlea, I., Alvarado, J., Dammone, O., Howell, P., Majumdar, A., Mulder, B., Lettinga, M. P., Koenderink, G. and Aarts, D., 2014. Colloidal liquid crystals in rectangular confinement : Theory and experiment. Soft Matter, 39, pp. 7865-7873.

• How do we describe the boundary conditions?

D

U1

U2

sin,cosn

Problem: point defects at corners and the energy blows-up at the corner defects!

dA||KE

Ω

2

A toy Oseen-Frank 2D model

Lewis, A., Garlea, I., Alvarado, J., Dammone, O., Howell, P., Majumdar, A., Mulder, B., Lettinga, M. P., Koenderink, G. and Aarts, D., 2014. Colloidal liquid crystals in rectangular confinement : Theory and experiment. Soft Matter, 39, pp. 7865-7873.

• How do we construct solutions – separation of variables

0

0);,1();,();,0(;1);0,( yfxfyfxf

2a

3a

4a

1a

A toy Oseen-Frank 2D model

Lewis, A., Garlea, I., Alvarado, J., Dammone, O., Howell, P., Majumdar, A., Mulder, B., Lettinga, M. P., Koenderink, G. and Aarts, D., 2014. Colloidal liquid crystals in rectangular confinement : Theory and experiment. Soft Matter, 39, pp. 7865-7873.

• Energy Calculations: compute the Oseen-Frank energy on a rectangle with aspect ratio , with small neighbourhoods of corners removed.

D U1 U2

denotes the rectangular aspect ratio: ratio of y-side to x-side.

A toy Oseen-Frank 2D model

Lewis, A., Garlea, I., Alvarado, J., Dammone, O., Howell, P., Majumdar, A., Mulder, B., Lettinga, M. P., Koenderink, G. and Aarts, D., 2014. Colloidal liquid crystals in rectangular confinement : Theory and experiment. Soft Matter, 39, pp. 7865-7873.

• Energy Calculations: compute the Oseen-Frank energy on a rectangle with aspect ratio , with small neighbourhoods of corners removed.

D U 1 U2

Role of Elastic Anisotropy

),(sin),,(cos yxyxn

• What happens if we bring elastic anisotropy into the picture •The Euler-Lagrange equations

dAsincosKsincosKE[n]2

3

2

1

yxxy

3

11

K

K

Asymptotics as 0

Lewis, A., Garlea, I., Alvarado, J., Dammone, O., Howell, P., Majumdar, A., Mulder, B., Lettinga, M. P., Koenderink, G. and Aarts, D., 2014. Colloidal liquid crystals in rectangular confinement : Theory and experiment. Soft Matter, 39, pp. 7865-7873. Alexander Lewis, 2015 PhD thesis, University of Oxford.

tohxYA ..12

exp2

11coshsin

1

• Outer Solution

• Inner Solution • Matching yields

dYYYXfXBX

sin),(1exp2lim

1

0

01

tohXYBf ..1expsin10

Asymptotics for small aspect ratio

Lewis, A., Garlea, I., Alvarado, J., Dammone, O., Howell, P., Majumdar, A., Mulder, B., Lettinga, M. P., Koenderink, G. and Aarts, D., 2014. Colloidal liquid crystals in rectangular confinement : Theory and experiment. Soft Matter, 39, pp. 7865-7873. Lewis, A., 2015 Ph.D. Thesis

D U 1 U2

• As 0, the energy is dominated by two boundary layers near the y-edges, with an interior uniform state.

• Asymptotics for the energy difference between the diagonal and rotated state as 0

3

11

K

K

This research is supported by

• EPSRC Career Acceleration Fellowship EP/J001686/1 and

EP/J001686/2.

• Professor Lei Zhang (Shanghai Jiao Tong) and Mr Lidong Fang

• Shanghai Jiao Tong University

Thank you for your attention!