Nonlocal effects in models of liquid crystal materials
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Transcript of 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.
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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
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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.
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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.
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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
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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.
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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.
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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.
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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.
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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.
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• 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.
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• 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.
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• 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.
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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.
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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,
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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,
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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
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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)
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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.
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Constant field approxima:on
We now nondimensionalise and rescale,
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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
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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
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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.
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Constant field approxima:on, , ……..
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(If E>0 flexo pushes the director towards θ=0 and if E<0 towards θ=π/2)
Constant field approxima:on, , ……..
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(If E>0 flexo pushes the director towards θ=0 and if E<0 towards θ=π/2)
Constant field approxima:on, , ……..
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(If E>0 flexo pushes the director towards θ=0 and if E<0 towards θ=π/2)
Constant field approxima:on, ………
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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
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Nonlocal and dynamic effects
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A more realis:c voltage profile is a bipolar pulse
Nonlocal and dynamic effects
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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
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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
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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
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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
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The nonlocal region can be significant when elas:city increases
or when anchoring at z=d decreases
Nonlocal and dynamic effects
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Including flow can lead to overlaps (slower transients) and gaps (other solu:ons)
Summary
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• 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