Nonlocal effects in models of liquid crystal materials

Nonlocal effects in models of liquid crystal materials

Nigel Mo6ram

Department of Mathema:cs and Sta:s:cs

University of Strathclyde

(Ma6 Neilson, Andrew Davidson, Michael Grinfeld, Fernando Da Costa, Joao Pinto)

Introduc:on – liquid crystal materials

The liquid crystalline state of ma6er is an intermediate phase between the isotropic liquid and solid phases.

1 3rd June 2010 EPANADE – N.J. Mo6ram, University of Strathclyde

The material can flow as a liquid but retains some anisotropic features of a crystalline solid.

Introduc:on – liquid crystal phases

The liquid crystal can exhibit two types of order:

•  Orienta:onal order, where molecules align, on average, in a certain direc:on

•  Posi:onal order, where density varia:ons lead to a layered structure


The vast majority of liquid crystal based technologies use nema:c liquid crystal materials.

Introduc:on – the director

The average molecular orienta:on provides us with a macroscopic dependent variable which can be used to build a con:nuum theory of nema:c liquid crystals.


The main dependent variables will therefore be the director n and the fluid velocity v.

Other dependent variables can include the electric field E, the amount of order S and densi:es of ionic impuri:es.

Introduc:on – elas:city

One of the main differences between isotropic fluids and liquid crystals is their ability to maintain internal stresses, due to elas:c distor:ons of the director structure.


The presence of such distor:ons will be modelled through the inclusion of an elas:c energy.

Classic elas:c distor:ons include splaying, twis:ng and bending of the director.

Introduc:on – dielectric effect

•  Since each molecules contains small dipoles, or distributed charges, they are polarisable in the presence of an electric field.

•  This polarisability is different along the major and minor axes of the molecules.

•  The difference in permiYvi:es is measured by the dielectric anisotropy


In order to minimise the electrosta:c energy, a molecule, or group of molecules, will reorient to align the largest permiYvity along the field direc:on.

Introduc:on – flexoelectric effect

•  The dielectric effect can reorient liquid crystal molecules in one way only.

•  The flexoelectric effect has different effects depending on the direc:on of the electric field.


If molecules contain dipoles and shape anisotropy then different distor:ons are produced depending on the direc:on of the field.

Introduc:on – flow effects

•  Director rota:on and fluid flow are coupled, with director rota:on inducing flow and visa versa.

•  The viscosity is also dependent on the director orienta:on.


In total there are five independent viscosi:es in a nema:c liquid crystal.

(up to 23 viscosi:es in a smec:c liquid crystal)

Introduc:on – surface anchoring

•  The interac:on between liquid crystal molecules and the bounding substrates is an extremely important aspect of liquid crystal devices.

•  Surface treatments (mechanical and chemical) can induce the liquid crystal molecules to align parallel or perpendicular to the substrate normal.


The strength of this interac:on is measured by a surface anchoring strength

Introduc:on – liquid crystal displays

Standard liquid crystal displays consist of liquid crystal material sandwiched between electrodes, treated substrates and op:cal polarisers.


The applica:on of an electric field across the liquid crystal causes reorienta:on.

Introduc:on – liquid crystal displays

• When a field is applied the director reorients to align with the field.

• When the field is removed the surface anchoring dominates and the director structure relaxes to the original orienta:on.


•  This effect can change the transmission of light through the device.

• When this effect is pixellated (and with the addi:on of colour filters) a display can be produced.

Introduc:on – ZBD display

•  The Zenithal Bistable Device contains a structured surface which leads to two dis:nct director structures, one of which contains defects.


•  These two states are op:cally dis:nct. •  If we can switch between these two states we can maintain a sta:c image without the need to supply power.

Ver:cal Hybrid Aligned Nema:c (HAN)

Introduc:on – tV plots

•  If we apply a voltage pulse of V volts for τ milliseconds we can switch between the two states.


•  These plots are known as τV plots and are used to op:mise the device.

Ver:cal to HAN HAN to Ver:cal

A simplified model

•  Our model simplifies the complicated 2d structure and mimics the bistable surface with a surface energy which has two stable states.


A simplified model

• We now have an evolving 1d distor:on structure.

•  The director and electric field are func:ons of the distance through the device and :me.


Solving Maxwell’s equa:ons

The electric field must sa:sfy Maxwell’s equa:ons

The first of these introduces the electric poten:al U(z,t)

and the second, with an appropriate cons:tuta:ve equa:on, leads to,


Solving Maxwell’s equa:ons

The first term is the due to the dielectric effect and it is simply the orienta:on of the director that enters this term

the second is from the flexoelectric effect where gradients of the director orienta:on are important.

This equa:on can be solved to give,

where,


Director angle equa:on

The director angle θ(z,t) is governed by the equa:on,

where the leg hand side term derives from the dissipa:on due to rota:on of the director,

the K terms are due to elas:city

the E13 term is due to flexoelectricity the Δε term is due to the dielectric effect


Boundary condi:ons

At the upper surface (z=d) the director is (usually) assumed to be fixed,

whereas on the lower surface (z=0) the director angle obeys,

where the leg hand side term derives from the dissipa:on at the surface,

the K terms are from elas:c torques

the E13 term is due to flexoelectricity the W0 term is due to the bistable anchoring ( and have the same energy)


Constant field approxima:on

We first remove the nonlocal effect of the electric field and consider a simpler set of equa:ons

where E is now a constant electric field value.

The flexoelectric term in the boundary condi:on at z=0 is simply modifying the surface poten:al.

If E>0 this term pushes the director towards θ=0 and if E<0 towards θ=π/2.



We now nondimensionalise and rescale,



…leading to the following equa:ons

We can consider the linear stability of the ver:cal solu:on u=π/2 and find constraints on the stability which depend on the flexoelectric parameter.

Perhaps more interes:ng is an analysis of the sta:onary problem



We want to inves:gate the solu:on structure as we vary the electric field parameter η.

To do this we remove the field dependence in the interior equa:on using

so that



For σ=+1 we consider the phase plane defined by

and the intersec:on of the ini:al manifold

with the isochrone which is defined by the set of points

which sa:sfy

where is the first integral of the pendulum equa:on above.


Constant field approxima:on, , ……..


(If E>0 flexo pushes the director towards θ=0 and if E<0 towards θ=π/2)







Constant field approxima:on, ………


For sufficiently large β and κ


Nonlocal and dynamic effects

We now numerically solve the full equa:ons,

where,

with on on z=d

and on z=0




A more realis:c voltage profile is a bipolar pulse



If we apply such a pulse we obtain a more complicated τV diagram

Since Δε<0 we would assume that Ver:cal to HAN switching is easier. However, if V<0 flexo pushes towards HAN and if V>0 towards Ver:cal



Consider four different voltage values, for long pulse :mes, and look at the director profiles at points A, B, C, D during the applica:on of the voltage.


Start in the HAN state and apply pulse

Δε<0 pushes bulk to θ=0.

for V<0 flexo pushes to θ(0)=0

for V>0 flexo pushes to θ(0)=π/2


black red

green

blue

nega:ve V on posi:ve V on

H-‐>V

H-‐>V


Start in the Ver6cal state and apply pulse

Δε<0 pushes bulk to θ=0.

for V<0 flexo pushes to θ(0)=0

for V>0 flexo pushes to θ(0)=π/2


black red

green

blue


V-‐>H

The high voltage anomaly

We would expect the 80V case to behave as the 50V case.

We think the difference at z=d affects the field at z=0 through the nonlocal terms


black red

green

blue


H-‐>V V-‐>H

H-‐>V



The nonlocal region can be significant when elas:city increases

or when anchoring at z=d decreases



Including flow can lead to overlaps (slower transients) and gaps (other solu:ons)

Summary


•  Liquid crystal devices offer a rich source of interes:ng (mathema:cal and technological) problems.

• Most of these stem from the boundary condi:ons…

surface dissipa:on nonlocal terms

bistability elas:c torques

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Nonlocal effects in models of liquid crystal materials