nematic liquid crystals: introduction to the mathematics ... · kralj, s. and majumdar, a., 2014....
TRANSCRIPT
![Page 1: Nematic Liquid Crystals: Introduction to the Mathematics ... · Kralj, S. and Majumdar, A., 2014. Order reconstruction patterns in nematic liquid crystal wells. Proceedings of the](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0404127e708231d40be682/html5/thumbnails/1.jpg)
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
![Page 2: Nematic Liquid Crystals: Introduction to the Mathematics ... · Kralj, S. and Majumdar, A., 2014. Order reconstruction patterns in nematic liquid crystal wells. Proceedings of the](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0404127e708231d40be682/html5/thumbnails/2.jpg)
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
![Page 3: Nematic Liquid Crystals: Introduction to the Mathematics ... · Kralj, S. and Majumdar, A., 2014. Order reconstruction patterns in nematic liquid crystal wells. Proceedings of the](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0404127e708231d40be682/html5/thumbnails/3.jpg)
Liquid Crystals – what are they?
• Mesogenic phases of matter
• Intermediate between solids and liquids
decreasing temperature
![Page 4: Nematic Liquid Crystals: Introduction to the Mathematics ... · Kralj, S. and Majumdar, A., 2014. Order reconstruction patterns in nematic liquid crystal wells. Proceedings of the](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0404127e708231d40be682/html5/thumbnails/4.jpg)
History
Discovered by Reinitzer in 1888 : two melting points for cholesterol! !
Courtesy: Peter Palffy-Muhoray Lectures at Colorado - Boulder
![Page 5: Nematic Liquid Crystals: Introduction to the Mathematics ... · Kralj, S. and Majumdar, A., 2014. Order reconstruction patterns in nematic liquid crystal wells. Proceedings of the](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0404127e708231d40be682/html5/thumbnails/5.jpg)
History
Discovered by Reinitzer in 1888 : two melting points for cholesterol! !
Courtesy: Professor Sir John Ball 2015 Lyon Lecture Notes
![Page 6: Nematic Liquid Crystals: Introduction to the Mathematics ... · Kralj, S. and Majumdar, A., 2014. Order reconstruction patterns in nematic liquid crystal wells. Proceedings of the](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0404127e708231d40be682/html5/thumbnails/6.jpg)
Different Liquid Crystal Phases
Courtesy: Peter Palffy-Muhoray Boulder Lectures 2011
![Page 7: Nematic Liquid Crystals: Introduction to the Mathematics ... · Kralj, S. and Majumdar, A., 2014. Order reconstruction patterns in nematic liquid crystal wells. Proceedings of the](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0404127e708231d40be682/html5/thumbnails/7.jpg)
Nematic Liquid Crystals
Anisotropic rod-like molecules with directional properties
Long-range orientational ordering: molecules line up with one another
![Page 8: Nematic Liquid Crystals: Introduction to the Mathematics ... · Kralj, S. and Majumdar, A., 2014. Order reconstruction patterns in nematic liquid crystal wells. Proceedings of the](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0404127e708231d40be682/html5/thumbnails/8.jpg)
Nematics are ubiquitous!!
Biological molecules Macromolecules
Microparticles
![Page 9: Nematic Liquid Crystals: Introduction to the Mathematics ... · Kralj, S. and Majumdar, A., 2014. Order reconstruction patterns in nematic liquid crystal wells. Proceedings of the](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0404127e708231d40be682/html5/thumbnails/9.jpg)
Nematics are ubiquitous!!
Active nematics Cosmology
Courtesy: Peter Palffy-Muhoray Boulder Lectures 2011
![Page 10: Nematic Liquid Crystals: Introduction to the Mathematics ... · Kralj, S. and Majumdar, A., 2014. Order reconstruction patterns in nematic liquid crystal wells. Proceedings of the](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0404127e708231d40be682/html5/thumbnails/10.jpg)
Key word: anisotropy!!!
Soft materials : responses to external stimuli
Courtesy: Images from Peter Palffy-Muhoray’s lectures at Colorado - Boulder
![Page 11: Nematic Liquid Crystals: Introduction to the Mathematics ... · Kralj, S. and Majumdar, A., 2014. Order reconstruction patterns in nematic liquid crystal wells. Proceedings of the](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0404127e708231d40be682/html5/thumbnails/11.jpg)
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
![Page 12: Nematic Liquid Crystals: Introduction to the Mathematics ... · Kralj, S. and Majumdar, A., 2014. Order reconstruction patterns in nematic liquid crystal wells. Proceedings of the](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0404127e708231d40be682/html5/thumbnails/12.jpg)
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.
![Page 13: Nematic Liquid Crystals: Introduction to the Mathematics ... · Kralj, S. and Majumdar, A., 2014. Order reconstruction patterns in nematic liquid crystal wells. Proceedings of the](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0404127e708231d40be682/html5/thumbnails/13.jpg)
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
![Page 14: Nematic Liquid Crystals: Introduction to the Mathematics ... · Kralj, S. and Majumdar, A., 2014. Order reconstruction patterns in nematic liquid crystal wells. Proceedings of the](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0404127e708231d40be682/html5/thumbnails/14.jpg)
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
![Page 15: Nematic Liquid Crystals: Introduction to the Mathematics ... · Kralj, S. and Majumdar, A., 2014. Order reconstruction patterns in nematic liquid crystal wells. Proceedings of the](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0404127e708231d40be682/html5/thumbnails/15.jpg)
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.
![Page 16: Nematic Liquid Crystals: Introduction to the Mathematics ... · Kralj, S. and Majumdar, A., 2014. Order reconstruction patterns in nematic liquid crystal wells. Proceedings of the](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0404127e708231d40be682/html5/thumbnails/16.jpg)
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
![Page 17: Nematic Liquid Crystals: Introduction to the Mathematics ... · Kralj, S. and Majumdar, A., 2014. Order reconstruction patterns in nematic liquid crystal wells. Proceedings of the](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0404127e708231d40be682/html5/thumbnails/17.jpg)
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
![Page 18: Nematic Liquid Crystals: Introduction to the Mathematics ... · Kralj, S. and Majumdar, A., 2014. Order reconstruction patterns in nematic liquid crystal wells. Proceedings of the](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0404127e708231d40be682/html5/thumbnails/18.jpg)
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
![Page 19: Nematic Liquid Crystals: Introduction to the Mathematics ... · Kralj, S. and Majumdar, A., 2014. Order reconstruction patterns in nematic liquid crystal wells. Proceedings of the](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0404127e708231d40be682/html5/thumbnails/19.jpg)
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.
![Page 20: Nematic Liquid Crystals: Introduction to the Mathematics ... · Kralj, S. and Majumdar, A., 2014. Order reconstruction patterns in nematic liquid crystal wells. Proceedings of the](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0404127e708231d40be682/html5/thumbnails/20.jpg)
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
![Page 21: Nematic Liquid Crystals: Introduction to the Mathematics ... · Kralj, S. and Majumdar, A., 2014. Order reconstruction patterns in nematic liquid crystal wells. Proceedings of the](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0404127e708231d40be682/html5/thumbnails/21.jpg)
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
![Page 22: Nematic Liquid Crystals: Introduction to the Mathematics ... · Kralj, S. and Majumdar, A., 2014. Order reconstruction patterns in nematic liquid crystal wells. Proceedings of the](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0404127e708231d40be682/html5/thumbnails/22.jpg)
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
![Page 23: Nematic Liquid Crystals: Introduction to the Mathematics ... · Kralj, S. and Majumdar, A., 2014. Order reconstruction patterns in nematic liquid crystal wells. Proceedings of the](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0404127e708231d40be682/html5/thumbnails/23.jpg)
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)
![Page 24: Nematic Liquid Crystals: Introduction to the Mathematics ... · Kralj, S. and Majumdar, A., 2014. Order reconstruction patterns in nematic liquid crystal wells. Proceedings of the](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0404127e708231d40be682/html5/thumbnails/24.jpg)
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.
![Page 25: Nematic Liquid Crystals: Introduction to the Mathematics ... · Kralj, S. and Majumdar, A., 2014. Order reconstruction patterns in nematic liquid crystal wells. Proceedings of the](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0404127e708231d40be682/html5/thumbnails/25.jpg)
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.
![Page 26: Nematic Liquid Crystals: Introduction to the Mathematics ... · Kralj, S. and Majumdar, A., 2014. Order reconstruction patterns in nematic liquid crystal wells. Proceedings of the](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0404127e708231d40be682/html5/thumbnails/26.jpg)
Rotated state: vertical liquid crystal alignment in the interior.
Defects pinned at two vertices along an edge.
Tsakonas, Davidson,
Brown, Mottram 2007
![Page 27: Nematic Liquid Crystals: Introduction to the Mathematics ... · Kralj, S. and Majumdar, A., 2014. Order reconstruction patterns in nematic liquid crystal wells. Proceedings of the](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0404127e708231d40be682/html5/thumbnails/27.jpg)
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.
![Page 28: Nematic Liquid Crystals: Introduction to the Mathematics ... · Kralj, S. and Majumdar, A., 2014. Order reconstruction patterns in nematic liquid crystal wells. Proceedings of the](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0404127e708231d40be682/html5/thumbnails/28.jpg)
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.
![Page 29: Nematic Liquid Crystals: Introduction to the Mathematics ... · Kralj, S. and Majumdar, A., 2014. Order reconstruction patterns in nematic liquid crystal wells. Proceedings of the](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0404127e708231d40be682/html5/thumbnails/29.jpg)
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)
![Page 30: Nematic Liquid Crystals: Introduction to the Mathematics ... · Kralj, S. and Majumdar, A., 2014. Order reconstruction patterns in nematic liquid crystal wells. Proceedings of the](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0404127e708231d40be682/html5/thumbnails/30.jpg)
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
![Page 31: Nematic Liquid Crystals: Introduction to the Mathematics ... · Kralj, S. and Majumdar, A., 2014. Order reconstruction patterns in nematic liquid crystal wells. Proceedings of the](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0404127e708231d40be682/html5/thumbnails/31.jpg)
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
![Page 32: Nematic Liquid Crystals: Introduction to the Mathematics ... · Kralj, S. and Majumdar, A., 2014. Order reconstruction patterns in nematic liquid crystal wells. Proceedings of the](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0404127e708231d40be682/html5/thumbnails/32.jpg)
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
![Page 33: Nematic Liquid Crystals: Introduction to the Mathematics ... · Kralj, S. and Majumdar, A., 2014. Order reconstruction patterns in nematic liquid crystal wells. Proceedings of the](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0404127e708231d40be682/html5/thumbnails/33.jpg)
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.
![Page 34: Nematic Liquid Crystals: Introduction to the Mathematics ... · Kralj, S. and Majumdar, A., 2014. Order reconstruction patterns in nematic liquid crystal wells. Proceedings of the](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0404127e708231d40be682/html5/thumbnails/34.jpg)
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
![Page 35: Nematic Liquid Crystals: Introduction to the Mathematics ... · Kralj, S. and Majumdar, A., 2014. Order reconstruction patterns in nematic liquid crystal wells. Proceedings of the](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0404127e708231d40be682/html5/thumbnails/35.jpg)
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
![Page 36: Nematic Liquid Crystals: Introduction to the Mathematics ... · Kralj, S. and Majumdar, A., 2014. Order reconstruction patterns in nematic liquid crystal wells. Proceedings of the](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0404127e708231d40be682/html5/thumbnails/36.jpg)
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
![Page 37: Nematic Liquid Crystals: Introduction to the Mathematics ... · Kralj, S. and Majumdar, A., 2014. Order reconstruction patterns in nematic liquid crystal wells. Proceedings of the](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0404127e708231d40be682/html5/thumbnails/37.jpg)
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
![Page 38: Nematic Liquid Crystals: Introduction to the Mathematics ... · Kralj, S. and Majumdar, A., 2014. Order reconstruction patterns in nematic liquid crystal wells. Proceedings of the](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0404127e708231d40be682/html5/thumbnails/38.jpg)
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!