Modelling a planar bistable device on different scales Apala Majumdar Lecturer in Applied Mathematics EPSRC Career Acceleration Fellow University of Bath Analytical and Computational Paths from Molecular Foundations to Continuum Descriptions Workshop Isaac Newton Institute for Mathematical Sciences 19th March 2013

Research themes

• Continuum liquid crystal theories e.g. liquid crystal defects • Microscopic to macroscopic limits in liquid crystal science • Modelling of liquid crystal devices in industry • Related problems in nonlinear elasticity Nematic – Greek word for `thread’

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

• optical switches and sensors

• biological sensors

• liquid crystal elastomers  interdisciplinary research • mathematics, physics, engineering science, chemistry

Nematics continued….. Anisotropic physical properties : • birefringent fluids – strong coupling to incident light

• sensitive to external electric and magnetic fields Typical nematic deformations:

splay

twist

• small elastic moduli  interesting mechanical effects

bend

Display Applications Main area of work: modelling and optimization of bistable liquid crystal displays

• Collaborations with industry : Hewlett Packard • Basic working principle of any LCD : a transparent state and an opaque state

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

• lucrative for devices with static images for a long period of time

Bistable LCDs contd.: Where can mathematical modelling help? Ongoing collaborations with Hewlett-Packard • What induces bistability? Topological mechanisms proposed in A.Majumdar, C. J. P. Newton, J. M. Robbins & M. Zyskin, 2007 Topology and bistability in liquid crystal devices. Phys. Rev. E 75, 051703-051714.

• Structure of equilibria? • Optical properties? Inverse problems • Switching mechanisms Benefits for industry: save experimental effort, computational cost; well-defined routes for design of new devices!

Examples of bistable displays: • Planar bistable liquid crystal device

Tsakonas, Davidson, Brown, Mottram 2007

• Zenithally Bistable Nematic Device

Numerical modelling by Chris Newton, HP

Examples of bistable displays continued:

• Post Aligned Bistable Nematic Device (Hewlett Packard Laboratories)

Planar (P) light

Numerical modelling by Chris Newton Kitson and Geisow, Applied Physics Letters, 80,2002.

Tilted (T) dark

A Two-Dimensional Bistable Device Collaborators: Chong Luo, Radek Erban

Tsakonas, Davidson, Brown, Mottram 2007

• Micro-confined liquid crystal system: • Array of liquid crystal-filled square wells with dimensions between 20×20×12 microns and 80×80×12 microns. • Treat as a two-dimensional system.

Boundary Conditions : •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.

Tsakonas, Davidson, Brown, Mottram 2007

Chong Luo, Apala Majumdar and Radek Erban, 2012 "Multistability in planar liquid crystal wells", Physical Review E, Volume 85, Number 6, 061702

Bistability: two experimentally observed states Diagonal state: liquid crystal alignment along one of the square diagonals. Defects pinned along diagonally opposite vertices.

Tsakonas, Davidson, Brown, Mottram 2007

Rotated state: vertical liquid crystal alignment in the square interior. Defects pinned at two vertices along an edge.

Tsakonas, Davidson, Brown, Mottram 2007

Optical contrast? Theoretical and experimental optical textures: Tsakonas, Davidson, Brown, Mottram 2007

Theory:

Experiment :

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

Modelling in the Landau-de Gennes framework: • General continuum theory that can account for all nematic phases and physically observable singularities.

•Define macroscopic order parameter that distinguishes nematic liquid crystals from conventional liquids, in terms of anisotropic macroscopic quantities such as the magnetic susceptibility and dielectric anisotropy e.g. Q αβ

 1   G χ αβ  δαβ  3 

χ γ

γγ

   

 : magnetic susceptibility

• For a two-dimensional device, the Q – tensor order parameter is a symmetric, traceless 2×2 matrix.

 Q11 Q    Q12

   Q11 

Q12

No details about intermolecular interactions; molecular shape etc.

Modelling in the Landau-de Gennes framework contd. • Relate the Q-tensor to degree of orientational ordering and direction of preferred alignment :

δ ij    Q ij  s  n i n j   2  

n  cos θ , sin θ 

Ξ  x  Ω : s(x)  0 

A.Majumdar, 2012 The radial-hedgehog solution in Landau-de Gennes' theory for nematic liquid crystals. European Journal of Applied Mathematics, 23, 61 - 97. A. Majumdar, 2012 The Landau-de Gennes theory for nematic liquid crystals: Uniaxiality versus Biaxiality. Communications in Pure and Applied Analysis, 11, 1303 - 1337.

Landau-de Gennes energy • Strong anchoring or Dirichlet boundary conditions I LdG Q  

 |  Q|

2

1



2

ε

|Q|

2

1



2

dA

• Weak anchoring or surface anchoring energies I LdG Q  

 |  Q|

2

Ω



 W Q

11



1 ε

2

|Q|

2

1



,Q 12    g 1 , g 2 

2

dA

2

ds

Ω

ε

C

K

Ds

W 

W anc D K

• Include additional terms in presence of an external electric field

Strong anchoring I LdG Q  

 | Q |

2



1 ε

2

| Q |

2

1



2

Find local energy minimizers for admissible boundary conditions:

dA | g | 0

| g | 1

| g | 0

| g | 1

g  g1 , g 2 

| g | 1

Chong Luo, Apala Majumdar and Radek Erban, "Multistability in planar liquid crystal wells", Physical Review E, Volume 85, Number 6, 061702, 15 pages (2012)

| g | 0

| g | 1

| g | 0

Strong anchoring contd. For every admissible g, we find six different solutions : two diagonal and four rotated solutions.

Chong Luo, Apala Majumdar and Radek Erban, "Multistability in planar liquid crystal wells", Physical Review E, Volume 85, Number 6, 061702, 15 pages (2012)

Strong anchoring contd. For every admissible g, we find six different solutions : two diagonal and four rotated solutions.

Chong Luo, Apala Majumdar and Radek Erban, "Multistability in planar liquid crystal wells", Physical Review E, Volume 85, Number 6, 061702, 15 pages (2012)

Strong anchoring contd. For every admissible g, we find six different solutions : two diagonal and four rotated solutions.

Chong Luo, Apala Majumdar and Radek Erban, "Multistability in planar liquid crystal wells", Physical Review E, Volume 85, Number 6, 061702, 15 pages (2012)

Strong anchoring contd. How do we choose the best Dirichlet boundary condition : notion of optimal boundary condition?

Chong Luo, Apala Majumdar and Radek Erban, "Multistability in planar liquid crystal wells", Physical Review E, Volume 85, Number 6, 061702, 15 pages (2012)

Optimal boundary conditions continued :

Optimal interpolation between isotropic points at vertices | g | 0

| g | 1

| g | 1

| g | 0

| g | 1

g  g1 , g 2 

| g | 0

| g | 1

This variational problem admits 6 different optimal solutions, labelled as diagonal and rotated respectively. The corresponding traces are defined to be optimal boundary conditions.

Chong Luo, Apala Majumdar and Radek Erban, "Multistability in planar liquid crystal wells", Physical Review E, Volume 85, Number 6, 061702, 15 pages (2012)

| g | 0

Strong anchoring contd. Typical solution profiles and their defects

Optimal diagonal solution

Optimal rotated solution

Chong Luo, Apala Majumdar and Radek Erban, "Multistability in planar liquid crystal wells", Physical Review E, Volume 85, Number 6, 061702, 15 pages (2012)

Weak anchoring I LdG Q  

 | Q |

2

Ω



 W Q

11



1 ε

2

| Q |

2

1

, Q 12   g 1 , g 2 



2

dA

2

ds

Ω

The `g’ above is the optimal diagonal boundary condition.

Bifurcation diagram as a function of the anchoring strength.

Chong Luo, Apala Majumdar and Radek Erban, "Multistability in planar liquid crystal wells", Physical Review E, Volume 85, Number 6, 061702, 15 pages (2012)

• Multistability for large values of the anchoring coefficient `W’

Diagonal States

• Multistability for large values of the anchoring coefficient `W’

Rotated states

• Multistability for large values of the anchoring coefficient `W’

Rotated states

• Defect profiles for weak anchoring : no real defects or isotropic points

Weak rotated profile

Weak diagonal profile

• Switching characteristics of the planar bistable liquid crystal device :  dynamic model based on gradient flow approach

Order parameter profile inside domain – localized regions of reduced order near vertices.

 Make the anchoring strength `W’ different on different square edges (discussions with Professor Nigel Mottram)

• Switching from diagonal to rotated

Lattice-based molecular models • New lattice-based Landau-de Gennes model : might contain information about molecular shape anisotropy and intermolecular interactions • Defined by analogy with the well-known Lebwohl-Lasher model

U LL





sin

2

θ

i

 θ j

i j

• Equilibrium states : energy minimizers obtained by a Monte-Carlo algorithm

• Temperature-dependence in the acceptance criterion of the MC algorithm

Refined version of the Lebwohl-Lasher model • New lattice-based Landau-de Gennes model



Q i  Q 11 ( i ) , Q 12 ( i )



Find equilibrium states by a standard Monte Carlo algorithm.

How is the lattice-based Landau-de Gennes model different from the conventional Landau-de Gennes theory?

•

Lattice-based approach

 can introduce more tunable microscopic parameters • View lattice-based approach as a discretized version of conventional Landau-de Gennes theory.  Models solved by completely different numerical methods;  Compare conventional PDE-solvers with stochastic numerical methods (Monte Carlo, Molecular Dynamics etc.)

Some results in the lattice-based framework:

Diagonal state

Rotated state

Can reproduce experimentally observed bistability !

Switching?

Model a quasi-static evolution in the presence of an external electric field : rotated to diagonal switching .

Switching?

Model a quasi-static evolution in the presence of an external electric field : diagonal to rotated switching .

Joint work with Chong Luo and Radek Erban.

Off-lattice models •

Molecules have orientational and translational degrees of freedom

• Gay-Berne model: interaction energy between pairs of molecules depends on relative orientations of molecules and the distance between them.

Off-lattice models continued •

Hard Gaussian Overlap Model

•

Preliminary results

The Zenithally Bistable Nematic Device

Vertically Aligned State

Zenithally Bistable Nematic Device www.eng.ox.ac.uk

Hybrid Aligned State

Joint work with Peter Howell and Alexander Raisch (University of Oxford)

Modelling details

δ ij    Q ij  s  n i n j   2  

A third stable state : Hybrid-HAN state

The Hybrid-HAN state continued

Three competing equilibria: VAN, HAN, H-HAN Compare relative stabilities as a function of temperature, grating depth and cell height :

Switching between H-HAN and HAN by varying temperature?

Collaborators: Alexander Raisch, , Peter Howell

This research is supported by •

EPSRC Career Acceleration Fellowship EP/J001686/1.

•

OCCAM Visiting Fellowship, University of Oxford.

Thank you for your attention!