An introduction to the Defect Free Q -Tensor Approximation K.R. - - PowerPoint PPT Presentation

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An introduction to the Defect Free Q-Tensor Approximation

K.R. Daly, G. D’Alessandro & M. Kaczmarek

School of Mathematics University of Southampton

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 1 / 41

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Outline

1

Introduction to modelling nematic liquid crystals

2

A mathematical detour

3

Defect Free Q-tensor approximation in 2D

4

Defect Free Q-tensor approximation in 3D

5

Extensions

6

Conclusions

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 2 / 41

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Outline

1

Introduction to modelling nematic liquid crystals

2

A mathematical detour

3

Defect Free Q-tensor approximation in 2D

4

Defect Free Q-tensor approximation in 3D

5

Extensions

6

Conclusions

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 3 / 41

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What are liquid crystals?

Liquid crystals exists between the isotropic liquid and crystalline solid states. Nematic liquid crystals posses some orientational order due to the elastic interactions of the molecules but no positional order. The local average direction of the molecules is the director field. It is possible to move into and out of the liquid crystalline phase simply through changing the temperature.

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 4 / 41

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An example application: the electro-optic response

From a device point of view one of the most interesting properties of nematic liquid crystals is their interaction with an electric field.

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 5 / 41

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An example application: the electro-optic response

From a device point of view one of the most interesting properties of nematic liquid crystals is their interaction with an electric field. Specifically the liquid crystals will align in the presence of external electric fields.

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 5 / 41

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An example application: the electro-optic response

From a device point of view one of the most interesting properties of nematic liquid crystals is their interaction with an electric field. Specifically the liquid crystals will align in the presence of external electric fields. The optical properties, which related to the liquid crystal alignment, can be altered by applying an electric field.

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 5 / 41

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An example application: the electro-optic response

From a device point of view one of the most interesting properties of nematic liquid crystals is their interaction with an electric field. Specifically the liquid crystals will align in the presence of external electric fields. The optical properties, which related to the liquid crystal alignment, can be altered by applying an electric field.

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 5 / 41

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A simple model of liquid crystals: a vector

The orientation of the liquid crystal can be modelled in terms of a unit vector (Frank–Oseen and Ericksen–Leslie). ˆ n =   sin ϑ cos ψ sin ϑ sin ψ cos ϑ   .

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 6 / 41

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A simple model of liquid crystals: a vector

The orientation of the liquid crystal can be modelled in terms of a unit vector (Frank–Oseen and Ericksen–Leslie). ˆ n =   sin ϑ cos ψ sin ϑ sin ψ cos ϑ   . Models based on the vector representation have the advantage that they are physically intuitive.

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 6 / 41

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A simple model of liquid crystals: a vector

The orientation of the liquid crystal can be modelled in terms of a unit vector (Frank–Oseen and Ericksen–Leslie). ˆ n =   sin ϑ cos ψ sin ϑ sin ψ cos ϑ   . Models based on the vector representation have the advantage that they are physically intuitive. However, they can only model uniaxial liquid crystals.

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 6 / 41

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A simple model of liquid crystals: a vector

The orientation of the liquid crystal can be modelled in terms of a unit vector (Frank–Oseen and Ericksen–Leslie). ˆ n =   sin ϑ cos ψ sin ϑ sin ψ cos ϑ   . Models based on the vector representation have the advantage that they are physically intuitive. However, they can only model uniaxial liquid crystals. Furthermore, the inversion symmetry of liquid crystals would require “ˆ n = −ˆ n”, which is not possible for a non-zero vector.

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 6 / 41

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Limitations of the vector representation

To illustrate this problem we consider a rectangular domain of liquid crystal.

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 7 / 41

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Limitations of the vector representation

To illustrate this problem we consider a rectangular domain of liquid crystal. Under the application of an external force the liquid crystal is distorted.

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 7 / 41

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Limitations of the vector representation

To illustrate this problem we consider a rectangular domain of liquid crystal. Under the application of an external force the liquid crystal is distorted. If the external force is strong enough it can force defects to appear.

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 7 / 41

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Limitations of the vector representation

To illustrate this problem we consider a rectangular domain of liquid crystal. Under the application of an external force the liquid crystal is distorted. If the external force is strong enough it can force defects to appear.

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 7 / 41

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Limitations of the vector representation

To illustrate this problem we consider a rectangular domain of liquid crystal. Under the application of an external force the liquid crystal is distorted. If the external force is strong enough it can force defects to appear. This can cause conflicts in the vector representation of the liquid crystal alignment.

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 7 / 41

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The Q-tensor representation of the director field

We use a traceless symmetric tensor, Q, to describe the liquid crystal

• rientation.

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 8 / 41

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The Q-tensor representation of the director field

We use a traceless symmetric tensor, Q, to describe the liquid crystal

• rientation.

The direction of alignment of the liquid crystal is the eigenvector n of Q with largest eigenvalue.

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 8 / 41

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The Q-tensor representation of the director field

We use a traceless symmetric tensor, Q, to describe the liquid crystal

• rientation.

The direction of alignment of the liquid crystal is the eigenvector n of Q with largest eigenvalue. The scalar order parameter, i.e. the number that identifies whether the liquid crystal is in the isotropic (S = 0) or fully nematic (S = 1) phase, is given by S2 = Tr(Q2).

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 8 / 41

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The Q-tensor representation of the director field

We use a traceless symmetric tensor, Q, to describe the liquid crystal

• rientation.

The direction of alignment of the liquid crystal is the eigenvector n of Q with largest eigenvalue. The scalar order parameter, i.e. the number that identifies whether the liquid crystal is in the isotropic (S = 0) or fully nematic (S = 1) phase, is given by S2 = Tr(Q2). The Q-tensor naturally respects the inversion symmetry of the liquid crystal (n and −n are both eigenvectors with the same eigenvalue).

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 8 / 41

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The Q-tensor representation of the director field

We use a traceless symmetric tensor, Q, to describe the liquid crystal

• rientation.

The direction of alignment of the liquid crystal is the eigenvector n of Q with largest eigenvalue. The scalar order parameter, i.e. the number that identifies whether the liquid crystal is in the isotropic (S = 0) or fully nematic (S = 1) phase, is given by S2 = Tr(Q2). The Q-tensor naturally respects the inversion symmetry of the liquid crystal (n and −n are both eigenvectors with the same eigenvalue). The Q-tensor describes also biaxial liquid crystals and defects (where S = 0 and n is not defined).

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 8 / 41

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Q-tensor and vector director field

It is possible to express the Q tensor in terms of the vector director field n as Q =

• 3

2S

• ˆ

n ⊗ ˆ n − 1 3I

• .

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 9 / 41

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Q-tensor and vector director field

It is possible to express the Q tensor in terms of the vector director field n as Q =

• 3

2S

• ˆ

n ⊗ ˆ n − 1 3I

• .

This representation shows clearly that the Q-tensor is invariant with respect to sign reversal of n, ˆ n ⊗ ˆ n = (−ˆ n) ⊗ (−ˆ n) .

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 9 / 41

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The Q-tensor basis

It is possible to efficiently represent the Q-tensor using its components a

• n a basis set for n × n traceless symmetric tensors:

Qij =

• n

anT(n)

ij ,

with a = (a1, a2, . . .) .

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 10 / 41

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The Q-tensor basis

It is possible to efficiently represent the Q-tensor using its components a

• n a basis set for n × n traceless symmetric tensors:

Qij =

• n

anT(n)

ij ,

with a = (a1, a2, . . .) . There are five basis tensors in 3D (and, hence, a ∈ R5):

T(1) = 1 √ 6   −1 −1 2   , T(2) = 1 √ 2   1 −1   , T(3) = 1 √ 2   1 1   , T(4) = 1 √ 2   1 1   , T(5) = 1 √ 2   1 1   . DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 10 / 41

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The Q-tensor basis

It is possible to efficiently represent the Q-tensor using its components a

• n a basis set for n × n traceless symmetric tensors:

Qij =

• n

anT(n)

ij ,

with a = (a1, a2, . . .) . There are five basis tensors in 3D (and, hence, a ∈ R5):

T(1) = 1 √ 6   −1 −1 2   , T(2) = 1 √ 2   1 −1   , T(3) = 1 √ 2   1 1   , T(4) = 1 √ 2   1 1   , T(5) = 1 √ 2   1 1   .

There are only two in 2D (and, hence, a ∈ R2):

T(1) = 1 √ 2 −1 1

• ,

T(2) = 1 √ 2 1 1

• .

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 10 / 41

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The liquid crystal free energy

The alignment of the liquid crystal can be found by minimising its free

• energy. For example, in a relatively simple case this can be written in

non-dimensional units as F = ξ2 2 |∇a|2 − χaa · e + T0 2 |a|2 + 1 2|a|4 − √ 6

• p,q,r

Tr (TpTqTr) apaqar, where ξ2

0 is the (single) elastic constant,

χa is the coupling coefficient to the electric field e and T0 is the scaled temperature.

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 11 / 41

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The giants and the minnows

The key observation is that the terms of the free energy have extremely different sizes.

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 12 / 41

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The giants and the minnows

The key observation is that the terms of the free energy have extremely different sizes. In the case of the free energy F = ξ2 2 |∇a|2 − χaa · e + T0 2 |a|2 + 1 2|a|4 − √ 6

• p,q,r

Tr (TpTqTr) apaqar, we have ξ2

0 ∼ O(10−7), χa ∼ O(10−6), while T0 ∼ O(1).

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 12 / 41

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The giants and the minnows

The key observation is that the terms of the free energy have extremely different sizes. In the case of the free energy F = ξ2 2 |∇a|2 − χaa · e + T0 2 |a|2 + 1 2|a|4 − √ 6

• p,q,r

Tr (TpTqTr) apaqar, we have ξ2

0 ∼ O(10−7), χa ∼ O(10−6), while T0 ∼ O(1).

This property is independent of the details of the free energy.

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 12 / 41

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The giants and the minnows

The key observation is that the terms of the free energy have extremely different sizes. In the case of the free energy F = ξ2 2 |∇a|2 − χaa · e + T0 2 |a|2 + 1 2|a|4 − √ 6

• p,q,r

Tr (TpTqTr) apaqar, we have ξ2

0 ∼ O(10−7), χa ∼ O(10−6), while T0 ∼ O(1).

This property is independent of the details of the free energy. It is both a problem and an opportunity.

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 12 / 41

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The giants and the minnows

The key observation is that the terms of the free energy have extremely different sizes. In the case of the free energy F = ξ2 2 |∇a|2 − χaa · e + T0 2 |a|2 + 1 2|a|4 − √ 6

• p,q,r

Tr (TpTqTr) apaqar, we have ξ2

0 ∼ O(10−7), χa ∼ O(10−6), while T0 ∼ O(1).

This property is independent of the details of the free energy. It is both a problem and an opportunity. However, we should keep in mind that the terms are of similar size near

• defects. Hence, any result based on this difference in side will not be

valid near a defect.

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 12 / 41

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The liquid crystal equations

The equation for the minimum of the free energy has general form ∂a ∂τ = F(a) + ηL(a), η ≪ 1 where F(a) comes from the thermotropic part of the free energy, while ηL(a) represents the effect of the elastic and electrostatic energy.

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 13 / 41

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The liquid crystal equations

The equation for the minimum of the free energy has general form ∂a ∂τ = F(a) + ηL(a), η ≪ 1 where F(a) comes from the thermotropic part of the free energy, while ηL(a) represents the effect of the elastic and electrostatic energy. This difference in magnitude is indicative that there are two very different time scales in the problem. Hence, these equations are numerically stiff and computationally expensive to solve.

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 13 / 41

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The liquid crystal equations

The equation for the minimum of the free energy has general form ∂a ∂τ = F(a) + ηL(a), η ≪ 1 where F(a) comes from the thermotropic part of the free energy, while ηL(a) represents the effect of the elastic and electrostatic energy. This difference in magnitude is indicative that there are two very different time scales in the problem. Hence, these equations are numerically stiff and computationally expensive to solve. To overcome the computational effort required to solve numerically stiff equations we have used a perturbation technique to eliminate the fast time evolution.

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 13 / 41

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The liquid crystal equations

The equation for the minimum of the free energy has general form ∂a ∂τ = F(a) + ηL(a), η ≪ 1 where F(a) comes from the thermotropic part of the free energy, while ηL(a) represents the effect of the elastic and electrostatic energy. This difference in magnitude is indicative that there are two very different time scales in the problem. Hence, these equations are numerically stiff and computationally expensive to solve. To overcome the computational effort required to solve numerically stiff equations we have used a perturbation technique to eliminate the fast time evolution. This approach is valid away from defects.

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 13 / 41

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Outline

1

Introduction to modelling nematic liquid crystals

2

A mathematical detour

3

Defect Free Q-tensor approximation in 2D

4

Defect Free Q-tensor approximation in 3D

5

Extensions

6

Conclusions

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 14 / 41

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Fast and slow variables

Consider the system of equations ˙ x = ηx(1 − y), ˙ y = −y + x2, η ≪ 1 .

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 15 / 41

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Fast and slow variables

Consider the system of equations ˙ x = ηx(1 − y), ˙ y = −y + x2, η ≪ 1 . This system has two time scales:

1

The y variable evolves on a time O(1), while

2

the x variable evolves on a much slower time scale O 1 η

• .

Hence this system is stiff and is computationally expensive to solve.

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 15 / 41

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Fast and slow variables

Consider the system of equations ˙ x = ηx(1 − y), ˙ y = −y + x2, η ≪ 1 . This system has two time scales:

1

The y variable evolves on a time O(1), while

2

the x variable evolves on a much slower time scale O 1 η

• .

Hence this system is stiff and is computationally expensive to solve. However, if we are interested only on the slow time scale we can simplify the problem considerably.

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 15 / 41

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Removing the fast time scale

The fast y variable in ˙ x = ηx(1 − y), ˙ y = −y + x2, η ≪ 1 , (1) relaxes quickly to the instantaneous equilibrium y = x2. This defines an invariant manifold for equations (1) in the limit η = 0.

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 16 / 41

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Removing the fast time scale

The fast y variable in ˙ x = ηx(1 − y), ˙ y = −y + x2, η ≪ 1 , (1) relaxes quickly to the instantaneous equilibrium y = x2. This defines an invariant manifold for equations (1) in the limit η = 0. The slow x-dynamics is determined by substituting this relation in the first of equations (1): dx dτ = x − x3 , where τ = ηt is a “slow” time.

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 16 / 41

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Removing the fast time scale

The fast y variable in ˙ x = ηx(1 − y), ˙ y = −y + x2, η ≪ 1 , (1) relaxes quickly to the instantaneous equilibrium y = x2. This defines an invariant manifold for equations (1) in the limit η = 0. The slow x-dynamics is determined by substituting this relation in the first of equations (1): dx dτ = x − x3 , where τ = ηt is a “slow” time. This is equation is no longer stiff and can be easily integrated numerically (or analytically).

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 16 / 41

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Collapse to the η = 0 invariant manifold

1 2 3 6 x y η=10−2

t95 = 8.18 tm = 100.00

1 2 3 6 x y η=10−3

t95 = 5.63 tm = 1000.00

In both cases the dynamics collapses rapidly to the η = 0 invariant manifold.

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 17 / 41

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Geometrical point of view

It is possible, and instructive, to give a geometrical interpretation of this method. Consider the system ˙ x = 1 − x2 − y2 − ηy2, ˙ y = 1 − x2 − y2 + ηxy, η ≪ 1 .

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 18 / 41

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Geometrical point of view

It is possible, and instructive, to give a geometrical interpretation of this method. Consider the system ˙ x = 1 − x2 − y2 − ηy2, ˙ y = 1 − x2 − y2 + ηxy, η ≪ 1 . This is equivalent in polar coordinates to ˙ r = r(1 − r2), ˙ θ = ηr sin(θ), η ≪ 1 .

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 18 / 41

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Geometrical point of view

It is possible, and instructive, to give a geometrical interpretation of this method. Consider the system ˙ x = 1 − x2 − y2 − ηy2, ˙ y = 1 − x2 − y2 + ηxy, η ≪ 1 . This is equivalent in polar coordinates to ˙ r = r(1 − r2), ˙ θ = ηr sin(θ), η ≪ 1 . If η = 0 r = 1 is a stable solution for all values of θ.

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 18 / 41

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Geometrical point of view

It is possible, and instructive, to give a geometrical interpretation of this method. Consider the system ˙ x = 1 − x2 − y2 − ηy2, ˙ y = 1 − x2 − y2 + ηxy, η ≪ 1 . This is equivalent in polar coordinates to ˙ r = r(1 − r2), ˙ θ = ηr sin(θ), η ≪ 1 . If η = 0 r = 1 is a stable solution for all values of θ. If η ≪ 1, the solution point reaches quickly r ≃ 1 and then moves very slowly along the circle to the only stable equilibrium at θ = π.

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 18 / 41

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Tangent flow on the η = 0 invariant manifold

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 19 / 41

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Tangent flow on the η = 0 invariant manifold

The same procedure used in the previous example leads to the slow time equations d dτ x y

• =

−y2 xy

• DDK (Univ. of Southampton)

Intro to the DFQTA 31 March 2012 19 / 41

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Tangent flow on the η = 0 invariant manifold

The same procedure used in the previous example leads to the slow time equations d dτ x y

• =

−y2 xy

• = y

−y x

• .

The vector (−y, x) is tangent to the η = 0 invariant circle.

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 19 / 41

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Tangent flow on the η = 0 invariant manifold

The same procedure used in the previous example leads to the slow time equations d dτ x y

• =

−y2 xy

• = y

−y x

• .

The vector (−y, x) is tangent to the η = 0 invariant circle. Hence, in the slow dynamics, the solution point moves subject to a “force” tangent to the η = 0 invariant manifold.

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 19 / 41

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Observations

These examples are very ad hoc and should not be taken as a general case.

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 20 / 41

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Observations

These examples are very ad hoc and should not be taken as a general case. However, the method used to remove the fast time scale can be made formal, accurate and generic. In particular:

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 20 / 41

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Observations

These examples are very ad hoc and should not be taken as a general case. However, the method used to remove the fast time scale can be made formal, accurate and generic. In particular:

◮ We write the variables as a power expansion in η:

x = x0(τ) + ηx1(τ) + O(η2), y = y0(τ) + ηy1(τ) + O(η2).

◮ It is possible to determine the slow equations for (x0, y0) and their forcing

term tangent to the manifold in a self-consistent manner.

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 20 / 41

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Observations

These examples are very ad hoc and should not be taken as a general case. However, the method used to remove the fast time scale can be made formal, accurate and generic. In particular:

◮ We write the variables as a power expansion in η:

x = x0(τ) + ηx1(τ) + O(η2), y = y0(τ) + ηy1(τ) + O(η2).

◮ It is possible to determine the slow equations for (x0, y0) and their forcing

term tangent to the manifold in a self-consistent manner.

◮ The corrections (x1, y1) are determined by a forcing term normal to the

manifold and give an estimate of the error of the approximation.

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 20 / 41

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Observations

These examples are very ad hoc and should not be taken as a general case. However, the method used to remove the fast time scale can be made formal, accurate and generic. In particular:

◮ We write the variables as a power expansion in η:

x = x0(τ) + ηx1(τ) + O(η2), y = y0(τ) + ηy1(τ) + O(η2).

◮ It is possible to determine the slow equations for (x0, y0) and their forcing

term tangent to the manifold in a self-consistent manner.

◮ The corrections (x1, y1) are determined by a forcing term normal to the

manifold and give an estimate of the error of the approximation.

In other words, it is not only possible to eliminate the fast time scale, but also to determine when the approximation may fail.

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 20 / 41

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Outline

1

Introduction to modelling nematic liquid crystals

2

A mathematical detour

3

Defect Free Q-tensor approximation in 2D

4

Defect Free Q-tensor approximation in 3D

5

Extensions

6

Conclusions

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 21 / 41

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Finding the η = 0 invariant manifold

Using an appropriately restrictive geometry the Q tensor equations break down to two coupled equations of motion, ∂ta = η

• ∇2a + χ0e
• + a − a
• a2

1 + a2 2

• .

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 22 / 41

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Finding the η = 0 invariant manifold

Using an appropriately restrictive geometry the Q tensor equations break down to two coupled equations of motion, ∂ta = η

• ∇2a + χ0e
• + a − a
• a2

1 + a2 2

• .

We expand a as a power series in η, a = a0 + ηa1 + O(η2).

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 22 / 41

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Finding the η = 0 invariant manifold

Using an appropriately restrictive geometry the Q tensor equations break down to two coupled equations of motion, ∂ta = η

• ∇2a + χ0e
• + a − a
• a2

1 + a2 2

• .

We expand a as a power series in η, a = a0 + ηa1 + O(η2). Retaining only the leading order terms, the equilibrium solution of the equations of motion describe a manifold of solutions in the two dimensional space a0 = [a1,0, a2,0]. a0 − a0

• a2

1,0 + a2 2,0

• = 0 =

⇒ a2

1,0 + a2 2,0 = 1 .

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 22 / 41

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Finding the η = 0 invariant manifold

Using an appropriately restrictive geometry the Q tensor equations break down to two coupled equations of motion, ∂ta = η

• ∇2a + χ0e
• + a − a
• a2

1 + a2 2

• .

We expand a as a power series in η, a = a0 + ηa1 + O(η2). Retaining only the leading order terms, the equilibrium solution of the equations of motion describe a manifold of solutions in the two dimensional space a0 = [a1,0, a2,0]. a0 − a0

• a2

1,0 + a2 2,0

• = 0 =

⇒ a2

1,0 + a2 2,0 = 1 .

This manifold defines the scalar order parameter S to leading order, S2 = Tr(Q2) = a2

1,0 + a2 2,0 = 1 .

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 22 / 41

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The η = 0 invariant manifold

The equation a2

1,0 + a2 2,0 = 1

defines the η = 0 invariant manifold. The solution converges rapidly to a very small neighbourhood of this manifold. Once on the manifold the first order terms describe a slow motion along the manifold on the slow time scale of τ1 = ηt.

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Obtaining the equations for the slow dynamics

We substitute τ1 = ηt and the expansion a = a0(τ1) + ηa1(τ1) + O(η2) into ∂ta =η

• ∇2a + χ0e
• + a − a
• a2

1 + a2 2

• ⇐

⇒ η ∂a ∂τ1 = F(a) + ηL(a).

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 24 / 41

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Obtaining the equations for the slow dynamics

We substitute τ1 = ηt and the expansion a = a0(τ1) + ηa1(τ1) + O(η2) into ∂ta =η

• ∇2a + χ0e
• + a − a
• a2

1 + a2 2

• ⇐

⇒ η ∂a ∂τ1 = F(a) + ηL(a). Retaining the linear terms in η we obtain ∂F (a0) ∂a0

• a1 = ∂a0

∂τ1 − L (a0) .

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 24 / 41

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Obtaining the equations for the slow dynamics

We substitute τ1 = ηt and the expansion a = a0(τ1) + ηa1(τ1) + O(η2) into ∂ta =η

• ∇2a + χ0e
• + a − a
• a2

1 + a2 2

• ⇐

⇒ η ∂a ∂τ1 = F(a) + ηL(a). Retaining the linear terms in η we obtain ∂F (a0) ∂a0

• a1 = ∂a0

∂τ1 − L (a0) . This equation can be solved by considering the effects of the source term L both tangential (T) and orthogonal (N) to the manifold.

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 24 / 41

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Geometrical interpretation

Graphically, the components of L define the slow dynamics and the magnitude of the first order corrections.

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 25 / 41

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Geometrical interpretation

Graphically, the components of L define the slow dynamics and the magnitude of the first order corrections. The steady state solution is the one at which the tangent vector T = (−a2,0, a1,0) is orthogonal to L.

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The equations for the slow dynamics

Using a procedure similar to the one outlined in the toy examples we obtain that the equations for the slow dynamics are ∂a0 ∂τ1 = T

• −a2,0∇2a1,0 + a1,0∇2a2,0 − a2,0χ0e1 + a1,0χ0e2
• .

These equations are the “Defect Free Q-tensor approximation” (DFQTA) in two dimensions.

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Estimating the error

By projecting in the direction normal to the manifold we can find an estimate

• f the first order correction, the distance between the true solution and the

leading order order solution manifold. In the two dimensional case this can be expressed analytically in terms of the first order correction to the scalar order parameter. S1 =

• a0 · ∇2a0 + χa0 · e
• .

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An example - Photorefractive cell

The solid lines are equipotentials. The short blue lines represent the director

• field. Planar anchoring on top and bottom, periodic boundary conditions on

left and right.

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Example - Estimating the error

Approximation of error calculated from S1 and the difference between the approximate equations and the full stiff equations.

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Outline

1

Introduction to modelling nematic liquid crystals

2

A mathematical detour

3

Defect Free Q-tensor approximation in 2D

4

Defect Free Q-tensor approximation in 3D

5

Extensions

6

Conclusions

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3D case in a nutshell

The starting point for the three dimensional case is the tensor representation for a bi-axial nematic liquid crystal, Q = S

• ˆ

n ⊗ ˆ n − 1 3I

• + β (ˆ

m ⊗ ˆ m − ˆ w ⊗ ˆ w) . The equilibrium leading order solutions to the Euler Lagrange equations define a manifold of constant order parameter S and constant biaxiality β in the five dimensional component space: S = 3 + √9 − 8T0 4 β = Const . In most liquid crystal systems the free energy is such that the biaxiality is zero to leading order, i.e. β = 0. In this case the dimension of the first order manifold collapses to two (“cannellone to spaghetto” transition) simplifying the algebra significantly.

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3D Example

Using the three dimensional approximate equations we can calculate liquid crystal alignment in the absence of defects in less than 1% of the time used in a normal Q-tensor model.

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3D - Error estimation

The error in the scalar order parameter is calculated by projecting the first

• rder equations orthogonal to the η = 0 invariant manifold.

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Outline

1

Introduction to modelling nematic liquid crystals

2

A mathematical detour

3

Defect Free Q-tensor approximation in 2D

4

Defect Free Q-tensor approximation in 3D

5

Extensions Inclusion of a stationary defect The future

6

Conclusions

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Including defects by excision

The DFQTA models the liquid crystal alignment accurately and efficiently away from a defect.

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Including defects by excision

The DFQTA models the liquid crystal alignment accurately and efficiently away from a defect. The defect structure can either be approximated or calculated numerically using, for example, the unperturbed equations.

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Including defects by excision

The DFQTA models the liquid crystal alignment accurately and efficiently away from a defect. The defect structure can either be approximated or calculated numerically using, for example, the unperturbed equations. The liquid crystal alignment near the defect can then be patched to the bulk liquid crystal alignment.

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Patched grids

One conceptually simple method to include defect modelling is patched grids.

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Patched grids

One conceptually simple method to include defect modelling is patched grids. The full stiff equations are solved on a grid local to the defect core.

DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 36 / 41

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Patched grids

One conceptually simple method to include defect modelling is patched grids. The full stiff equations are solved on a grid local to the defect core. The DFQTA equations are solved on a coarse grid away from the core.

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Example - A +1/2 defect

To illustrate this we have modelled a simple +1/2 defect on a circular domain. The full stiff Q–tensor equations are solved near the defect. The DFQTA equations are solved away from the defect core.

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Example - A +1/2 defect

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Work to do

Track defects: their birth and death, and their motion.

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Work to do

Track defects: their birth and death, and their motion. Explore other ways to deal with defects.

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Work to do

Track defects: their birth and death, and their motion. Explore other ways to deal with defects. Include the liquid fluid dynamics in the model: this will allow us to study switching behaviour and motion of liquid crystals around particles.

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Work to do

Track defects: their birth and death, and their motion. Explore other ways to deal with defects. Include the liquid fluid dynamics in the model: this will allow us to study switching behaviour and motion of liquid crystals around particles. Consider also nanodoped liquid crystals.

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Outline

1

Introduction to modelling nematic liquid crystals

2

A mathematical detour

3

Defect Free Q-tensor approximation in 2D

4

Defect Free Q-tensor approximation in 3D

5

Extensions

6

Conclusions

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Conclusions

We have developed a fast and accurate algorithm for liquid crystal alignment away from defects. This algorithm exploits the huge difference in time scales in the dynamics of liquid crystals.

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Conclusions

We have developed a fast and accurate algorithm for liquid crystal alignment away from defects. This algorithm exploits the huge difference in time scales in the dynamics of liquid crystals. We are working at extending this to include defects, fluid dynamics and nanodoped liquid crystals.

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Conclusions

We have developed a fast and accurate algorithm for liquid crystal alignment away from defects. This algorithm exploits the huge difference in time scales in the dynamics of liquid crystals. We are working at extending this to include defects, fluid dynamics and nanodoped liquid crystals. For more information:

• K. R. Daly, G. D’Alessandro and M. Kaczmarek, An Efficient Q-Tensor-Based

Algorithm for Liquid Crystal Alignment away from Defects, SIAM J. Appl.

• Math. 70(8), 2844-2860 (2010)

http://www.personal.soton.ac.uk/dales/DFQTA/

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The end

Thank you!

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