Pattern Formation in Confined Nematic Liquid Crystals

Apala Majumdar Department of Mathematical Sciences

University of Bath

1st CMM-Bath Workshop on Applied and Interdisciplinary

Mathematics 19th-21st March 2019

Liquid Crystals – what are they?

• Mesogenic phases of matter

• Intermediate between solids and liquids

decreasing temperature

Discovery..

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

Courtesy: Peter Palffy-Muhoray Lectures at Colorado - Boulder

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

Key word: anisotropy!!!

Courtesy: Images from Peter Palffy-Muhoray’s lectures at Colorado – Boulder (Physics Today 60 (9), 54 (2007))

Nematic Anisotropy to Applications …

• Best known as the working material of the multi-billion dollar liquid crystal display (LCD) industry.

New Applications and New Research

• New Materials and Colloid Science

• Photonics, Actuators, Security Applications

• Microfluidics and Biomimetic Systems

Liquid crystals in micron-scale droplets, shells, and fibers , J. Phys.: Condens. Matter,,(2017)

S. Mondal, A. Majumdar and I.M. Griffiths 2017 Nematohydrodynamics for Colloidal Self-Assembly and Transport Phenomena. Journal of Colloid and Interface Science. 528, p. 431-442.

Why Liquid Crystals - Rich Mathematical Landscape

Partial Differential Equations

Calculus of Variations

Scientific Computation

Algebra and Topology, Functional Analysis

Dynamical Systems

Questions that interest me…

The mathematics of

Orientational Order/Partial Order

Hierarchy of continuum theories and their predictions.

Material imperfections and defect-mediated structural transitions ( F. Duncan M. Haldane and J. Michael Kosterlitz 2016 Nobel Prize in Physics)

Microscopic to Macroscopic Derivations?

Applications.

Liquid Crystals are a fascinating playground for mechanics, geometry, modelling and analysis to meet physics and real-life

applications.

Real opportunity for new mathematics-driven approaches to new materials, optimal design, optimal performance and

efficient methodologies.

How do we mathematically model nematic liquid crystals?

Three popular “continuum” approaches -

• Oseen-Frank theory: assume that constituent molecules have a single direction of locally preferred alignment with “constant degree of orientational order”.

• Ericksen theory: assume that constituent molecules have a single distinguished direction with “variable degree of orientational order”.

• Landau-de Gennes theory: account for primary and secondary directions of alignment and “variable degrees of orientational order”.

n (r)

The Landau-de Gennes Theory

The Nobel Prize in Physics in 1991 was awarded to Pierre-Gilles de Gennes for "for discovering that methods developed for studying order phenomena in simple systems can be generalized to more complex forms of matter, in particular to liquid crystals and polymers“.

The Landau-De Gennes Theory

• 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.

• The Q – tensor order parameter is a symmetric, traceless 3×3 matrix.

22112313

232212

131211

QQQQ

QQQ

QQQ

Q

Five degrees of freedom.

Eigenvalues of the Q-tensor and LC Phases

0λ

ppλmmλnnλQ

3

1i

i

321

• isotropic – triad of zero eigenvalues

• uniaxial – a pair of equal non-zero eigenvalues

• biaxial – three distinct eigenvalues and two locally preferred directions of molecular alignment.

0Q0λλλ321

I

3

1nn3λQ2λλλ;λλ

132

The Landau-de Gennes Energy The physically observable configurations correspond to minimizers of the Landau-de Gennes energy subject to the imposed boundary conditions.

The thermotropic potential : -

• non-convex , non-negative potential with multiple critical points

• dictates preferred phase of liquid crystal – isotropic/ uniaxial as function of temperature?

The elastic energy density : -

𝐼 𝑄  =   𝑓𝐵 𝑄

𝐿  +  𝑤 𝑄, 𝛻𝑄  𝑑𝑉

𝑤 𝑄, 𝛻𝑄   =   𝐿|𝛻𝑄|2  𝑤 𝑄, 𝛻𝑄 = 𝐿1|𝛻𝑄|

2 + 𝐿2𝑄𝑖𝑗,𝑗𝑄𝑖𝑘,𝑘

𝑓𝐵 𝑄  =  𝐴

2𝑡𝑟𝑄2  − 

𝐵

3𝑡𝑟𝑄3  +  

𝐶

4𝑡𝑟𝑄2 2

𝐴 = α 𝑇  − 𝑇∗   α,  𝑏,  𝑐, 𝑇∗   >   0

Introduction to Q-tensor theory 2014 Nigel J. Mottram, Christopher J.P. Newton

𝑓𝐵 𝑄  = 𝐴

2𝑡𝑟𝑄2  − 

𝐵

3𝑡𝑟𝑄3  +  

𝐶

4𝑡𝑟𝑄2 2  

+  𝑀(𝐴, 𝐵, 𝐶) 𝐴 = α 𝑇  − 𝑇∗   α,  𝑏,  𝑐, 𝑇∗   >   0

The bulk potential can be explicitly written as a function of two eigenvalues (from the

tracelessness constraint) and we can compute the bulk minimisers explicitly as pairs of

minimising eigenvalues, depending on the temperature regime.

The Landau-de Gennes Euler Lagrange Equations

The physically observable configurations correspond to minimizers of the Landau-de Gennes liquid crystal energy functional subject to the imposed boundary conditions.

The Euler-Lagrange equations in the one-constant case :

• Energy minimizers are classical solutions of the Euler-Lagrange equations.

• Multiple solutions.

• Smooth and analytic (standard results in elliptic regularity)

Δ𝑄𝑖𝑗  =  1

𝐿  𝐴𝑄𝑖𝑗   −   𝐵 𝑄𝑖𝑝𝑄𝑝𝑗  − 

1

3𝑡𝑟𝑄2 𝛿𝑖𝑗   +   𝐶 𝑡𝑟𝑄

2 𝑄𝑖𝑗  𝑖, 𝑗 = 1,2,3

𝑤 𝑄, 𝛻𝑄   =   𝐿|𝛻𝑄|2 

Why is this a hard problem?

Δ𝑄𝑖𝑗  =  1

𝐿  𝐴𝑄𝑖𝑗   −   𝐵 𝑄𝑖𝑝𝑄𝑝𝑗  − 

1

3𝑡𝑟𝑄2 𝛿𝑖𝑗   +   𝐶 𝑡𝑟𝑄

2 𝑄𝑖𝑗  𝑖, 𝑗 = 1,2,3

• Qualitative properties of energy minimizers in asymptotic limits: can we recover structure of defect cores?

• Exact solutions of Euler-Lagrange equations

D. Henao & A.Majumdar 2012 Symmetry of uniaxial global Landau-de Gennes minimizers in the theory of nematic liquid crystals. SIAM Journal on Mathematical Analysis 44-5, 3217-3241 A.Majumdar and Y.Wang 2018 Remarks on Uniaxial Solutions in the Landau-de Gennes Theory. Journal of Mathematical Analysis and Application. Volume 464, Issue 1, Pages 328-353.

• Non-minimising solutions? G.Canevari, A.Majumdar and A.Spicer 2017 Order Reconstruction for Nematics on Squares and Regular Polygons: A Landau-de Gennes Study. SIAM J. Appl. Math., 77(1), 267293.

A Toy Problem: The Planar Bistable LC Device

• Micro-confined liquid crystal system.

• Array of nematic 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

Our work on this device….

• Y. Wang, G.Canevari and A.Majumdar 2018 Order Reconstruction for Nematics on Squares with Isotropic Inclusions: A Landau-de Gennes Study. Under review in SIAM Journal of Applied Mathematics.

• Giacomo Canevari, Apala Majumdar and Amy Spicer 2017 Order Reconstruction for Nematics on Squares and Regular Polygons: A Landau-de Gennes Study. SIAM J. Appl. Math., 77(1), 267293.

• Martin Robinson, Chong Luo, Patrick Farrell, Radek Erban, and Apala Majumdar 2017 From molecular to continuum modelling of bistable liquid crystal devices. Liquid Crystals. http://dx.doi.org/10.1080/02678292.2017.1290284

• H.Kusumaatmaja and A.Majumdar, 2015 Free energy pathways of a Multistable Liquid Crystal Device. Soft Matter 11 (24), 4809-4817.

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

• S.Kralj and A.Majumdar, 2014 Order reconstruction patterns in nematic liquid crystal wells. Proceedings of the Royal Society of London Series A - Mathematical Physical and Engineering Sciences, 470,20140276.

• C.Luo, A.Majumdar and R.Erban 2012 Multistability in planar liquid crystal wells. Physical Review E,

85,Number 6, 061702

Our Approach to this Toy Problem

• Step 1: Model Reduction Can rigorously prove using Gamma-convergence techniques (see below) that for sufficiently shallow three-dimensional wells, with particular surface energies on top and bottom that allow for planar anchoring and Dirichlet uniaxial conditions on lateral sides, (i) LdG energy minimizers have a fixed eigenvector in the z-direction and (ii) it suffices to study the variational problem on the square cross-section i.e. we can neglect the third dimension. Reduced a problem with five degrees of freedom to a problem with three degrees of freedom!

2

Golovaty, D., Montero, J.A. & Sternberg, P. J Nonlinear Sci (2015) 25: 1431. https://doi.org/10.1007/s00332-015-9264-7

• Restrict ourselves to:

• Dirichlet boundary conditions on square edges

• Uniaxial boundary conditions that minimize the bulk potential for low temperatures.

𝑆+ = 𝐵 + 𝐵2 − 24𝐴𝐶

4𝐶

𝑄 = 𝑞1(𝑥, 𝑦) 𝑛1⊗𝑛1  −  𝑛2⊗𝑛2 + 𝑞2(𝑥, 𝑦) 𝑛1⊗𝑛2  +  𝑛2⊗𝑛1   +𝑞3(𝑥, 𝑦) 2𝑒𝑧⊗𝑒𝑧 − 𝑛1⊗𝑛1  − 𝑛2⊗𝑛2

ons

qq6

,032

𝑓𝐵 𝑄𝑏  =  𝐴

2𝑡𝑟𝑄𝑏

2  − 𝐵

3𝑡𝑟𝑄𝑏

3  +  𝐶

4𝑡𝑟𝑄𝑏

2 2 

𝐴 = α 𝑇  − 𝑇∗ < 0

A Special Temperature

• Existence of unique solution branch with

compatible with the Dirichlet conditions on square edges which are equivalent to • Reduced a problem with five degrees of freedom to a problem with two

degrees of freedom!!!

𝐴  = −𝐵2

3𝐶< 0

2

𝐶1

𝐶3

𝐶2 𝐶4

𝑞1  =  𝑆+2=𝐵

2𝐶    𝐶1∪ 𝐶3 

𝑞1  =  −𝑆+2= −

𝐵

2𝐶    𝐶2 ∪ 𝐶4

 

𝑞2 = 0

𝑞3 = −𝑆+6  =  −

𝐵6𝐶

 

𝑄  = 𝑞1 𝑒𝑥⊗𝑒𝑥  −  𝑒𝑦⊗𝑒𝑦  + 𝑞2 𝑒𝑥⊗𝑒𝑦  +  𝑒𝑦⊗𝑒𝑥 −𝐵

6𝐶2𝑒𝑧⊗𝑒𝑧  −  𝑒𝑥⊗𝑒𝑥 − 𝑒𝑦⊗𝑒𝑦

12

21

-qq

qqQ

redΔ𝑄𝑖𝑗  =  

1

𝐿  𝑎𝑄𝑖𝑗   −     𝑐 𝑡𝑟𝑄

2 𝑄𝑖𝑗  𝑖, 𝑗 = 1,2

Special solutions with Two Degrees of Freedom at Special Temperature as critical points of ….

• Classical solutions of –

• Always have the existence of a solution branch with

The Small Limit and the new Well Order Reconstruction Solution

Uniqueness of solutions for sufficiently small squares from convexity of the Landau-de Gennes

energy

• Can construct a solution has two key properties:

• What does this mean for the actual Landau-de Gennes critical point

• Constant eigenframe

• Uniaxial diagonal cross with negative order parameter

• Ring of maximal biaxiality

xyqii

yxqi

0)(

),(0)(

1

2

yyxxzzxyyxyyxx

eeeeeeC

BeeeeqeeeeqQ 2

621

Well Order Reconstruction

Solution

• How do we construct a solution of the form

• Minimize the following functional on a quadrant of the rotated square

dAqC

BCq

LqqH

Q

2

2

4

22

2:][

• existence of minimizer , h(x,y), from direct methods in calculus of variations;

• extend minimizer to truncated square by odd reflection about the lines x=0 and y=0 (the rotated diagonals);

• this defines a function

Ha Dang, Paul C. Fife, L. A. Peletier, 1992 Zeitschrift für angewandte Mathematik und Physik ZAMP November 1992, Volume 43, Issue 6, pp 984-998

0),0()0,(

,;,

yqxq

yxyxq

ss

s

q=0

q=0 Q

The Well Order Reconstruction Solution (WORS)

22112211

26

),( nnnneeC

BnnnnyxqQ

zzSOR

We have defined a Landau-de Gennes critical point, labelled as the WORS

q=0

q=0

• exists for all ;

• the eigenvectors are constant and fixed and

• has the negatively-uniaxial diagonal cross

• The WORS is the UNIQUE critical point for

• The WORS solution is an UNSTABLE critical point for sufficiently large . Construct perturbation for which second variation is negative by using key idea from : M.Schatzman, Proceedings of the Royal

Society of Edinburgh: Section A Mathematics / Volume 125 / Issue 06 / January 1995, pp 1241-1275

Q

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. Canevari, G., Majumdar, A. and Spicer, A., 2017. Order reconstruction for nematics on squares and hexagons:a Landau-de Gennes study. SIAM Journal on Applied Mathematics, 77 (1), pp. 267-293.

SSSq

C

BCq

L

λΔq

22

2

3

2

nmB

CLO 1002

2

The large regime.

What can we say about critical points of the form at Look at minimizers of • For small , we recover the Well Order Reconstruction Solution. • For large , readily apply Ginzburg-Landau techniques to show that

• and the LdG critical points of the form above, converge (in an appropriately defined sense) to

yyxxzzxyyxyyxx

eeeeeeC

BeeeeqeeeeqQ 2

621

?03

2

C

BA

everywherealmost4

2

2

2

2

2

1C

Bqq

corners. fromaway ,0sin,cos

262

2

n

eeeeeeC

BInn

C

BQ

yyxxzz

Also see G. Canevari, Biaxiality in the asymptotic analysis of a 2D Landau-de Gennes model for liquid crystals, ESAIM: COCV 21 (2015), 101–137. I. Shafrir, On a class of singular perturbation problems (87 pages), in Handbook of Differential Equations, Section ``Stationary

Partial Differential Equations'', edited by Michel Chipot and Pavol Quittner, Elsevier/North Holland, 2004, p.~297--383.

The large regime.

• As ∞, the problem effectively reduces to

sin,cos2

,21

C

Bqq

0

D U 1 U2

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.

The large regime.

• As ∞, the problem effectively reduces to

sin,cos2

,21

C

Bqq

0

D U 1 U2

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.

112

ln~~

112

ln~

21

21

21

ssEE

ssE

UU

D

0

2

0

1

12

12cosech21

12

112coth21

n

n

n

ns

n

ns

A detailed bifurcation plot

From molecular to continuum modelling of bistable liquid crystal devices Martin Robinson, Chong Luo, Patrick E. Farrell, Radek Erban & Apala Majumdar. http://www.tandfonline.com/doi/abs/10.1080/02678292.2017.1290284

Unique solution for small

squares; six distinct solution

branches for large squares –

two diagonal and four rotated.

21 other critical points found using numerical deflation techniques for intermediate values of .

From molecular to continuum modelling of bistable liquid crystal devices Martin Robinson, Chong Luo, Patrick E. Farrell, Radek Erban & Apala Majumdar http://www.tandfonline.com/doi/abs/10.1080/02678292.2017.1290284 P. E. Farrell, Á. Birkisson, and S. W. Funke, “Deflation techniques for finding distinct solutions of nonlinear partial differential equations,” SIAM Journal on Scientific Computing, vol. 37, p. A2026–A2045, 2015.

dAqqCqqC

B

L

dAqqqqJ

22

2

2

1

2

2

2

1

22

2

2

2

121

2

||||:],[

Look for critical points (not just minimizers) of

• Adapt similar methods to hexagons or arbitrary polyhedra – observe generic trends

• Joint work with Lei Zhang (Peking), Yucen Han (Peking), Lingling Zhao (Dalian), Giacomo Canevari (Spain), Yiwei Wang (Chicago)

• What have we learnt? A specific problem with a rich solution landscape.

yyxxzzxyyxyyxx

eeeeeeC

BeeeeqeeeeqQ 2

621

• The blue points/lines correspond to S=0 i.e. they extend as negatively ordered uniaxial lines through a three-dimensional domain.

• These interfaces are globally stable for small systems.

• Solutions with interfaces are unstable for larger systems but only in certain directions control problems!

• Numerically, known that they act as transition states for larger systems.

• Add flow to the problem and study response and evolution of interfaces mean curvature flow with forcing terms!

• Bigger Picture?

• Lessons learnt

study of NLC interfaces in high-dimensional systems i.e. problems of a vectorial or tensorial nature;

Connections between entire solutions of the Allen Cahn equation and critical points of the Landau-de Gennes energy i.e. construct non-minimizing critical points with interfaces;

Location of interfaces determined by a Toda lattice new connections with integrability!

The Well Order Reconstruction Solution – specific example of a negatively-uniaxial interface separating the square into four quadrants.

M.del Pino et.al 2010 J. Funct. Anal., 258 (2), pp. 458-503.

• Stronger conjectures: Nematic defects are contained in special NLC interfaces which have a generic

structure!

Understanding the structure and location of NLC interfaces as a function of temperature, material constants much better understanding of NLC defects

Bringing together cutting-edge problems in geometric measure theory, nonlinear PDE

and integrability!

• See work by Majumdar, Henao & Pisante; Lamy et.al, Ignat et.al, Canevari

• Stronger conjectures: Dynamics of NLC interfaces mathematics of self-assembly!!

• Maria Crespo, Ian Griffiths, Apala Majumdar, Angel Ramos 2017 Solution landscapes in nematic microfluidics. Physica D: Nonlinear Phenomena, 351352, 1-13. https://doi.org/10.1016/j.physd.2017.04.004

• S. Mondal, A. Majumdar and I.M. Griffiths 2017 Nematohydrodynamics for Colloidal Self-Assembly and Transport Phenomena. Under review in Journal of Colloid and Interface Science.

• Connections to Smales’ open 7th problem.

• Nematodynamics with different boundary conditions open questions on Navier Stokes with added complexities from nemato-fluid coupling stresses!

Acknowledgments:

• EPSRC Career Acceleration Fellowship EP/J001686/1 and

EP/J001686/2

• Royal Society Newton Advanced Fellowship

• Visiting Fellowship, University of Oxford and Keble College

• The Institute of Mathematics and its Applications

• Chinese Academy of Sciences and Shanghai Jiao Tong University