nonlocal effects in models of liquid crystal materials
DESCRIPTION
Nonlocal, differential equations, liquid crystals, bistable liquid crystal displays, modelling, theoryTRANSCRIPT
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
2 3rd June 2010 EPANADE – N.J. Mo6ram, University of Strathclyde
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.
3 3rd June 2010 EPANADE – N.J. Mo6ram, University of Strathclyde
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.
4 3rd June 2010 EPANADE – N.J. Mo6ram, University of Strathclyde
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
5 3rd June 2010 EPANADE – N.J. Mo6ram, University of Strathclyde
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.
6 3rd June 2010 EPANADE – N.J. Mo6ram, University of Strathclyde
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.
7 3rd June 2010 EPANADE – N.J. Mo6ram, University of Strathclyde
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.
8 3rd June 2010 EPANADE – N.J. Mo6ram, University of Strathclyde
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.
9 3rd June 2010 EPANADE – N.J. Mo6ram, University of Strathclyde
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.
10 3rd June 2010 EPANADE – N.J. Mo6ram, University of Strathclyde
• 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.
11 3rd June 2010 EPANADE – N.J. Mo6ram, University of Strathclyde
• 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.
12 3rd June 2010 EPANADE – N.J. Mo6ram, University of Strathclyde
• 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.
13 3rd June 2010 EPANADE – N.J. Mo6ram, University of Strathclyde
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.
14 3rd June 2010 EPANADE – N.J. Mo6ram, University of Strathclyde
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,
15 3rd June 2010 EPANADE – N.J. Mo6ram, University of Strathclyde
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,
16 3rd June 2010 EPANADE – N.J. Mo6ram, University of Strathclyde
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
17 3rd June 2010 EPANADE – N.J. Mo6ram, University of Strathclyde
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)
18 3rd June 2010 EPANADE – N.J. Mo6ram, University of Strathclyde
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.
19 3rd June 2010 EPANADE – N.J. Mo6ram, University of Strathclyde
Constant field approxima:on
We now nondimensionalise and rescale,
20 3rd June 2010 EPANADE – N.J. Mo6ram, University of Strathclyde
Constant field approxima:on
…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
21 3rd June 2010 EPANADE – N.J. Mo6ram, University of Strathclyde
Constant field approxima:on
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
22 3rd June 2010 EPANADE – N.J. Mo6ram, University of Strathclyde
Constant field approxima:on
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.
23 3rd June 2010 EPANADE – N.J. Mo6ram, University of Strathclyde
Constant field approxima:on, , ……..
24 3rd June 2010 EPANADE – N.J. Mo6ram, University of Strathclyde
(If E>0 flexo pushes the director towards θ=0 and if E<0 towards θ=π/2)
Constant field approxima:on, , ……..
25 3rd June 2010 EPANADE – N.J. Mo6ram, University of Strathclyde
(If E>0 flexo pushes the director towards θ=0 and if E<0 towards θ=π/2)
Constant field approxima:on, , ……..
26 3rd June 2010 EPANADE – N.J. Mo6ram, University of Strathclyde
(If E>0 flexo pushes the director towards θ=0 and if E<0 towards θ=π/2)
Constant field approxima:on, ………
27 3rd June 2010 EPANADE – N.J. Mo6ram, University of Strathclyde
For sufficiently large β and κ
(If E>0 flexo pushes the director towards θ=0 and if E<0 towards θ=π/2)
Nonlocal and dynamic effects
We now numerically solve the full equa:ons,
where,
with on on z=d
and on z=0
28 3rd June 2010 EPANADE – N.J. Mo6ram, University of Strathclyde
Nonlocal and dynamic effects
29 3rd June 2010 EPANADE – N.J. Mo6ram, University of Strathclyde
A more realis:c voltage profile is a bipolar pulse
Nonlocal and dynamic effects
30 3rd June 2010 EPANADE – N.J. Mo6ram, University of Strathclyde
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
Nonlocal and dynamic effects
31 3rd June 2010 EPANADE – N.J. Mo6ram, University of Strathclyde
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.
Nonlocal and dynamic effects
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
32 3rd June 2010 EPANADE – N.J. Mo6ram, University of Strathclyde
black red
green
blue
nega:ve V on posi:ve V on
H-‐>V
H-‐>V
Nonlocal and dynamic effects
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
33 3rd June 2010 EPANADE – N.J. Mo6ram, University of Strathclyde
black red
green
blue
nega:ve V on posi:ve V on
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
34 3rd June 2010 EPANADE – N.J. Mo6ram, University of Strathclyde
black red
green
blue
nega:ve V on posi:ve V on
H-‐>V V-‐>H
H-‐>V
Nonlocal and dynamic effects
35 3rd June 2010 EPANADE – N.J. Mo6ram, University of Strathclyde
The nonlocal region can be significant when elas:city increases
or when anchoring at z=d decreases
Nonlocal and dynamic effects
36 3rd June 2010 EPANADE – N.J. Mo6ram, University of Strathclyde
Including flow can lead to overlaps (slower transients) and gaps (other solu:ons)
Summary
37 3rd June 2010 EPANADE – N.J. Mo6ram, University of Strathclyde
• 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