The Landau-de Gennes theory for nematic liquid crystals.

APALA MAJUMDAR Department of Mathematical Sciences

University of Bath, UK

Summer School on “Frontiers of Applied and Computational Mathematics”

9th-21st July 2018

Nematic Liquid Crystals

The nematic phase is the simplest liquid crystalline phase of matter with properties that are intermediate between a solid and a liquid.

The constituent molecules have

• no positional order (flow about freely) but

• tend to align along certain locally preferred directions i.e. exhibit long-range orientational ordering.

Nematic – Greek word for `thread’.

Important mathematical parameters -

Nematic liquid crystals are anisotropic liquids with preferred directions of

molecular alignment. The preferred alignment directions constitute the first set

of important parameters.

n (r)

n (r) : preferred direction

of orientation of the long

molecular axes.

Scalar order parameter “S”:

measure of the degree of alignment about preferred directions.

n

n n

S = 0; no alignment S = 1; perfect

alignment

S 0.5; typical liquid crystal.

The Oseen-Frank Theory for Nematic Liquid Crystals

• assume constant value of scalar order parameter

• describe preferred direction by a unit-vector field n(r)

• just two degrees of freedom to describe a three-dimensional unit-vector field

Limitations: • Cannot describe secondary direction of orientational ordering/preferred

alignment • Cannot describe higher-dimensional defects such as line and surface defects • In fact, powerful theorem by Hardt, Kinderlehrer and Lin states that Hausdorff

dimension of singular set is less than one even with elastic anisotropy

dVnnWnE ,][

22

42

2

3

2

2

2

1,

nntrKKnnK

nnKnKnnW

421423212;||;0,, KKKKKKKK

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; OF theory is a special uniaxial case with constant eigenvalues

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

0Q0λλλ321

I

3

1nn3λQ2λλλ;λλ

132

The Landau-de Gennes energy functional The physically observable configurations are modelled by minimizers of the Landau-de Gennes free energy functional subject to the imposed boundary conditions.

In the absence of any external fields and surface effects, the simplest form of the Landau-de Gennes energy is given by

The thermotropic potential : -

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

• dictates preferred phase of liquid crystal – isotropic/ uniaxial/ biaxial?

dVQ,QwQfQIB

0T c,,b,a

,,tr4

ctr

3

btr

2

aQf

**

2232

B

TT

cbaCQQQ

The Thermotropic Potential

0T c,,b,a

,,tr4

ctr

3

btr

2

aQf

**

2232

B

TT

cbaCQQQ

• Compute critical points of this quartic polynomial as a function of the temperature

ji

cbaC

i

i

i

i

i

i

i

i

jii

i

321B

3

1

23

1

2

3

1

3

3

1

2

321B

0,,f

0

),,(4

c

3

b

2

a,,f

All critical points are either uniaxial or isotropic for all temperatures.

The Thermotropic Potential

All critical points are either uniaxial or isotropic for all temperatures • Compute minimizers of bulk potential by looking at the critical points of the

thermotropic potential restricted to uniaxial tensors.

2

3Sn

InnSQ

cbaCSSS ,,9

c

27

2b

3

aSf

432

B

cbaCSSS ,,9

c

27

2b

3

aSf

432

B

Introduction to Q-tensor theory 2014

Nigel J. Mottram, Christopher J.P. Newton

Also see Professor Sir John Ball 2015 Lyon Notes. A.Majumdar 2010 In : European Journal of Applied Mathematics. 21, 2, p. 181-203

The Elastic Energy Density

• Make correspondence with Oseen-Frank energy density by using the substitution

3exp

InnSQ

OF

dVnntrkk

nnknnknknE

22

2422

2

33

2

22

2

11][

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

John Ball 2015 Lyon Lecture Notes. L.Longa et.al Liquid Crystals Volume 2, 1987 - Issue 6.

John Ball 2015 Lyon Lecture Notes. Davis & Gartland 1998 FINITE ELEMENT ANALYSIS OF THE LANDAU-DE GENNES MINIMIZATION PROBLEM FOR LIQUID CRYSTALS L.Longa et.al Liquid Crystals Volume 2, 1987 - Issue 6.

Key Steps in Proof • Coercivity

• Lower semicontinuity

• Non-empty admissible space

• Existence of minimizer from direct methods in the calculus of variations

J. Ball & A. Majumdar, 2010 Nematic liquid crystals: from Maier-Saupe to a continuum theory. Molecular Crystals and Liquid Crystals, 525, 1 - 11.

• Proof by explicit construction based on localized perturbation inside a unit ball

likkijlkjikkijkikjijkijkij

QQQLQQLQQLQQLQQw,,4,,3,,2,,1

,

likkijlkjikkijkikjijkijkij

QQQLQQLQQLQQLQQw,,4,,3,,2,,1

,

Boundary Conditions • Strong anchoring

• Weak anchoring

• recover strong anchoring as the anchoring coefficient “W” becomes very large

• Planar degenerate anchoring (where the eigenvectors are preferentially anchored on the surfaces)

xxQxQS

2

2SS

QQtrW

QI

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

Some problems with the Landau-de Gennes Theory: • Lack of physically relevant bounds for the order parameter; interpret Q as the second

moment of the probability distribution function for the molecular orientations

A. Majumdar, 2010 Equilibrium order parameters of liquid crystals in the Landau-de Gennes theory. European Journal of Applied Mathematics, 21, 181 - 203.

Some problems with the Landau-de Gennes Theory: • When do the Landau-de Gennes minimizers respect the second moment bounds?

• Need to consider minimizers of the thermotropic potential for low temperatures

3

),,(min

InncbasQ

0T c,,b,a

,,tr4

ctr

3

btr

2

aQf

**

2232

B

TT

cbaCQQQ

Some problems with the Landau-de Gennes Theory: • Lack of coercivity with 4 elastic constants, perhaps rectified with higher-order terms,

where do we stop?

• Need 4 elastic constants to match with Oseen-Frank theory.

• Validity for small length scales?

• Physically relevant values of the order parameter?

• Quantitative estimates for validity of Landau-de Gennes theory in terms of temperature, geometrical and material properties?

Schlieren texture with defects in a nematic sample.

(www.lci.kent.edu/defect.html )

The Isotropic-Nematic Transition Temperature

0

432

2232

**

B

c, T,b,αTTαa

a,b,cCQtrc

Qtrb

Qtra

Qf

c

ba

27

2

0;;33

2

min

QSn

Inn

C

BQ

• The special temperature

• The bulk energy minimizers – the isotropic state and a continuum of uniaxial states, both of which have equal energies

The Gradient – Flow Dynamic Model

Front propagation on a nematic droplet…

• Radial uniaxial Dirichlet condition

• Initial conditions with an interface

structure

• Look for solutions of the form

3ˆˆ

3

Irr

C

BQ

b

3ˆˆ,

IrrtrhQ

C

Bhthth

3,10,0

Front propagation by mean curvature as L 0

3ˆˆ,

IrrtrhQ

C

Bhthth

3,10,0

• Prove front propagation by mean curvature in an asymptotic limit

2

dt

d

Majumdar, A., Milewski, P. A. & Spicer, A. 19 Jul 2016 In : SIAM Journal on Applied Mathematics. 76, 4, p. 1296-1320 25 p.

Comments on proof

• Proof only works for initial conditions with an interface structure with suitably bounded weighted energy

)(1)(),(, RttRhRw

Key references (including concept of weighted energy): • Lia Bronsard and Robert V Kohn. Motion by mean curvature as the singular limit of Ginzburg-Landau

dynamics. Journal of differential equations, 90, 1991. • Lia Bronsard and Barbara Stoth. The singular limit of a vector-valued reaction-diffusion process. Transactions of the American Mathematical Society, 350, 1998. Technical differences due to additional term in evolution equation and the director field is a solution of the harmonic map equations and not the Laplace equation.

3ˆˆ,

IrrtrhQ

Majumdar, A., Milewski, P. A. & Spicer, A. 19 Jul 2016 In : SIAM Journal on Applied Mathematics. 76, 4, p. 1296-1320 25 p.

Numerical Examples

• Uniaxial initial conditions with and without an interface structure

Numerical Examples

• Biaxial initial conditions

• Symmetry-breaking initial conditions that depend on angular variable

Front propagation on a disc with 5D Q-tensors…

• Uniaxial Dirichlet conditions with a two-dimensional radial director field

• Can describe long-time behaviour in terms of two unit-vector fields

3311

Inn

C

BQ

b

• The corresponding Q-tensors are

Planar initial conditions; isotropic core at centre for all time

Non- planar initial conditions; perfectly ordered uniaxial state

22112313

232212

131211

QQQQ

QQQ

QQQ

Q

Numerical Examples on a Disc – Solving LdG Gradient Flow System

G Di Fratta et. al Half-integer point defects in the q-tensor theory of nematic liquid crystals. Journal of Nonlinear Science , 26, 2016.

1,0,00,cos,sin;sin,cos pmn

• Planar biaxial initial conditions

rQtrQt 1

,

Majumdar, A., Milewski, P. A. & Spicer, A. 2016 , SIAM Journal on Applied Mathematics. 76, 4, p. 1296-1320 25 p.

Numerical Examples on a Disc

• Planar symmetry-breaking biaxial initial conditions

rQtrQt 1

,

Numerical Examples on a Disc

• Non-Planar initial conditions

rQtrQt 2

,

Majumdar, A., Milewski, P. A. & Spicer, A. 19 Jul 2016 In : SIAM Journal on Applied Mathematics. 76, 4, p. 1296-1320 25 p.

Non-Planar Initial Conditions continued

rQtrQt 2

,

Eigenvalue evolution at origin

Loss of Interface Structure

rQtrQt 2

,

10

*logt

Majumdar, A., Milewski, P. A. & Spicer, A. 19 Jul 2016 In : SIAM Journal on Applied Mathematics. 76, 4, p. 1296-1320 25 p.

Gradient-Flow Dynamics with Forcing

Dirr, N. P. and Yip, N. K. 2006. Pinning and de-pinning phenomena in front propagation in heterogeneous media. Interfaces and Free Boundaries 8(1), pp. 79-109.

Joint work with Amy

Spicer, Nicolas Dirr and

Patrick Dondl.

Gradient-Flow Dynamics with Forcing

Joint work with Amy Spicer and Nicolas Dirr

Gradient-Flow Dynamics with Forcing

Joint work with Amy Spicer and Nicolas Dirr

This research is supported by

• EPSRC Career Acceleration Fellowship EP/J001686/1 and

EP/J001686/2.

• Royal Society Newton Advanced Fellowship

• Shanghai Jiao Tong University

• Oxford Centre for Nonlinear PDEs

• OCIAM Visiting Fellowship

Thank you for your attention!