The document discusses nonlocal effects in models of liquid crystal materials. It provides an introduction to liquid crystals, describing their properties and phases. Key aspects that continuum models must account for are discussed, including the director field, elasticity, dielectric and flexoelectric effects, flow effects, surface anchoring, and applications in liquid crystal displays. Nonlocal effects arise from long-range interactions between liquid crystal molecules.

1. 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)

2. Introduc:on
–
liquid
crystal
materials
The
liquid
crystalline
state
of
ma6er
is
an
intermediate
phase
between
the
isotropic
liquid
and
solid
phases.
The
material
can
flow
as
a
liquid
but
retains
some
anisotropic
features
of
a
crystalline
solid.
3rd
June
2010
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of
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1

3. 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.
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4. 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.
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5. 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.
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6. 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.
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7. 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.
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8. 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)
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9. 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
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10. 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.
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11. 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.
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12. 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.
Ver:cal
Hybrid
Aligned
Nema:c
(HAN)
• 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.
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13. Introduc:on
–
tV
plots
• If
we
apply
a
voltage
pulse
of
V
volts
for
τ
milliseconds
we
can
switch
between
the
two
states.
HAN
to
Ver:cal
Ver:cal
to
HAN
• These
plots
are
known
as
τV
plots
and
are
used
to
op:mise
the
device.
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14. 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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15. 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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16. 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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17. 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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18. 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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19. 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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20. 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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21. Constant
field
approxima:on
We
now
nondimensionalise
and
rescale,
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22. 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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23. 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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24. 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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25. Constant
field
approxima:on,
,
……..
(If
E>0
flexo
pushes
the
director
towards
θ=0
and
if
E<0
towards
θ=π/2)
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2010
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26. Constant
field
approxima:on,
,
……..
(If
E>0
flexo
pushes
the
director
towards
θ=0
and
if
E<0
towards
θ=π/2)
3rd
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2010
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of
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27. Constant
field
approxima:on,
,
……..
(If
E>0
flexo
pushes
the
director
towards
θ=0
and
if
E<0
towards
θ=π/2)
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28. Constant
field
approxima:on,
………
For
sufficiently
large
β
and
κ
(If
E>0
flexo
pushes
the
director
towards
θ=0
and
if
E<0
towards
θ=π/2)
3rd
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2010
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29. 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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30. Nonlocal
and
dynamic
effects
A
more
realis:c
voltage
profile
is
a
bipolar
pulse
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31. Nonlocal
and
dynamic
effects
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
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32. Nonlocal
and
dynamic
effects
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.
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33. Nonlocal
and
dynamic
effects
blue
H-­‐>V
Start
in
the
HAN
state
and
apply
pulse
green
Δε<0
pushes
bulk
to
θ=0.
H-­‐>V
red
for
V<0
flexo
pushes
to
θ(0)=0
black
for
V>0
flexo
pushes
to
θ(0)=π/2
nega:ve
V
on
posi:ve
V
on
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34. Nonlocal
and
dynamic
effects
blue
V-­‐>H
Start
in
the
Ver6cal
state
and
apply
pulse
green
Δε<0
pushes
bulk
to
θ=0.
red
for
V<0
flexo
pushes
to
θ(0)=0
black
for
V>0
flexo
pushes
to
θ(0)=π/2
nega:ve
V
on
posi:ve
V
on
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35. The
high
voltage
anomaly
blue
H-­‐>V
V-­‐>H
We
would
expect
the
80V
case
to
behave
as
green
the
50V
case.
H-­‐>V
We
think
the
difference
at
z=d
affects
the
field
red
black
at
z=0
through
the
nonlocal
terms
nega:ve
V
on
posi:ve
V
on
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36. Nonlocal
and
dynamic
effects
The
nonlocal
region
can
be
significant
when
elas:city
increases
or
when
anchoring
at
z=d
decreases
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37. Nonlocal
and
dynamic
effects
Including
flow
can
lead
to
overlaps
(slower
transients)
and
gaps
(other
solu:ons)
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38. 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…
nonlocal
terms
surface
dissipa:on
bistability
elas:c
torques
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