Download Bistable Nematic Liquid Crystal Device

Bistable Nematic Liquid Crystal Device
Deepak Aralumallige Subbarayappa, Daniel Johnson, Tyler Skorczewski,
Joseph Hibdon, Te-Sheng Lin, Daniel Cargill
Mentor: Linda Cummings, NJIT
July 30, 2009
1
Introduction
With the increasing use of liquid crystal-based displays in everyday life, led both by the
development of new portable electronic devices and the desire to minimize the use of printed
paper, Nematic Liquid Crystals (NLCs) are now hugely important industrial materials; and
research into ways to engineer more efficient display technologies is crucial. Modern electronic
display technology mostly relies on the ability of NLC materials to rotate the plane of polarized
light (birefringence). The degree to which they can do this depends on the orientation of the
molecules within the liquid crystal, and this in turn is affected by factors such as an applied
electric field (the molecules, which are typically long and thin, line up in an applied field),
or by boundary effects (a phenomenon known as surface anchoring). Most devices currently
available use the former effect: an electric field is applied to control the molecular orientation of
a thin film of nematic liquid crystal between crossed polarizers (which are also the electrodes),
and this in turn controls the optical effect when light passes through the layer. A schematic
of a typical pixel in such a device is shown in figure 1.
The main disadvantage of this set-up is that the electric field must be applied constantly
in order for the display to maintain its configuration - if the field is removed, the molecules
of the NLC relax into the unique, stable, field-free state (giving no contrast between pixels,
and a monochrome display). This is expensive in terms of power consumption, leading to
generally short battery lifetimes. On the other hand, if one could somehow exploit the fact
that the bounding surfaces of a cell affect the molecular configuration - the anchoring effect,
which can, to a large extent, be controlled by mechanical or chemical treatments - then one
might be able to engineer a bistable system, with two (or more) stable field-free states, giving
two optically-distinct stable steady states of the device, without any electric field required to
sustain them. Power is required only to change the state of the cell from one steady state
to the other (and this issue of “switchability”, which can be hard to achieve, is really the
challenging part of the design). Such technology is particularly appropriate for LCDs that
change only infrequently, e.g. “electronic paper” applications such as e-books, e-newspapers,
and so on.
Certain technologies for bistable devices already exist, and most use the surface anchoring
effect, combined with a clever choice of bounding surface geometry. The goal of this project
will be to investigate simpler designs for liquid crystal devices that exhibit bistability. With
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planar surface topography, but different anchoring conditions at the two bounding surfaces,
bistability is possible; and a device of this kind should be easier to manufacture. Two different
modeling approaches can be taken depending on what design aspect is to be optimized. A
simple approach is to study only steady states of the system. Such states will be governed by
(nonlinear) ODEs, and stability can be investigated as the electric field strength is varied. In
a system with several steady states, loss of stability of one state at a critical field would mean
a bifurcation of the solution, and a switch to a different state. Such an analysis could give
information about how to achieve switching at low critical fields, for example; or at physicallyrealistic material parameter values; but would say nothing about how fast the switching might
be. Speed of switching would need to be investigated by studying a simple time-dependent
PDE model for the system. We can explore both approaches here, and attempt to come up
with some kind of “optimal” design.
Figure 1: Schematic of a single cell in a liquid crystal display.
2
2
Steady state mathematical model
When modeling liquid crystals, the key dependent variables are normally its velocity field at
any point in space and time, v0 (x0 , t0 ); and the local average direction of its molecules, which is
described by a (unit) vector field n(x0 , t0 ), known as the director field (here and elsewhere we
will use a prime, 0 , to denote a dimensional quantity; unprimed variables are dimensionless).
Since flow effects will be negligible in any display device, we need only model the director
field. Stable equilibrium solutions for the director are those states that (locally) minimize a
suitably-defined total free energy of the system, J 0 , which will include both bulk and surface
contributions,
Z
Z
J0 =
W 0 dV 0 +
g 0 dS 0
In the presence of an electric field there are several contributions to the bulk free energy
W 0 that must be included: (i) an intrinsic ‘elastic’ energy We0 , which models the fact that
adjacent molecules would ideally like to be parallel (so that when they are not, this increases
the local free energy); (ii) a dielectric contribution Wd0 , which models the tendency of the
molecules (and thus the director field) to align parallel to an applied electric field E0 , and (iii)
a flexoelectric contribution, Wf0 , which arises since NLC molecules have a small permanent
electric dipole (which wants to align in the applied field), but are also slightly asymmetric. For
example, they might be pear-shaped, with a dipole along the axis of the pear; so that when a
field is applied, the pears all line up, causing a splaying-out of the director field towards the
fat bases of the pears. These three contributions to the total bulk free energy are as follows:
2We0 = K10 (∇0 · n)2 + K20 (n.∇0 ∧ n)2 + K30 ((∇0 ∧ n) ∧ n)2 ,
2Wd0 = −00 ⊥ E 0 · E 0 − 00 (|| − ⊥ )(n · E 0 )2 ,
Wf0
= −E 0 · (e01 (∇0 · n) + e03 (∇0 ∧ n) ∧ n),
where K10 , K20 , K30 are elastic coefficients, 00 is the dielectric permittivity of free space, || and
⊥ are permittivities parallel and perpendicular to the long axis of the molecule, and e01 , e03
are flexoelectric coefficients. The surface energy takes the general form
g0 = γ 0 −
A0
(n · p)2
2
modeling the anchoring effect referred to earlier; this surface energy is minimized when the
director is parallel to the vector p. Since surfaces can be treated to achieve different anchoring
conditions, we assume that both the anchoring direction p and the anchoring strength A can
be specified at each surface.
As a simple example we consider a planar cell with the bounding surfaces at z 0 = 0, z 0 = h0
and we look for the director field lying solely in the (x0 , z 0 )−plane and varying only in the
z 0 -direction,
n = (sin θ(z 0 ), 0, cos θ(z 0 )).
We also assume (as is common in liquid crystal modeling) that K10 = K30 = K 0 ; and finally,
that any applied electric field is uniform throughout the layer, E0 = E 0 (0, 0, 1). With no
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external electric field (E 0 = 0) only the elastic energy contributes to the bulk energy density,
so that
K0
W 0 = We0 = 0 θz20 ,
2h
and if we assume that θ = α is the preferred anchoring angle at the lower surface, while β is
the preferred angle at the upper surface, then the surface energy takes the form
gl0 = γ 0 −
gu0 = γ 0 −
A00
2
A01
2
cos2 (θ − α),
z 0 = 0,
cos2 (θ − β),
z 0 = h0 .
From this we have the total free energy J 0 given by
K0
J [θ] = 0
2h
0
Zh0
θz20 dz 0 −
A00
A0
cos2 (θ − α)|z 0 =0 − 1 cos2 (θ − β)|z 0 =h0 .
2
2
0
To find equilibrium solutions, we must use variational calculus. We suppose that θ(z 0 ) is
a minimizer of J 0 , and we look at the variations of J 0 [θ] as θ changes to θ + η.
J 0 [θ + η] = J00 + J10 + 2 J20 + O(3 ).
To obtain minima we set J10 = 0 and then use the fact that J20 > 0 at a minimizer to get the
stability criteria (J10 = 0 with J20 < 0 would be a local maximum of J 0 ). Doing so we obtain
the following boundary value problem, at zero field
θzz = 0
A0
θz = −
sin(2θ − 2α)
2
A1
θz =
sin(2θ − 2α)
2
in
0<z<1
on
z=0
on
z = 1.
The above equations are expressed in dimensionless form for ease of notation; lengths are
scaled by h, and surface energy coefficients by K/h. If more than one (stable) solution to
this problem exists, for suitable choices of parameters α, β, A0 , A1 , then we have a bistable
system, and the potential for a workable device, if we can find a way to switch between the
different stable states by application of an electric field.
We must therefore also investigate solutions and stability at nonzero electric field. Following through the same steps as above for this case with the more general expression for the
free energy leads to the following nonlinear boundary value problem
θzz = D sin 2θ
A0
F
θz = −
sin(2θ − 2α) − sin 2θ
2
2
A1
F
θz =
sin(2θ − 2β) − sin 2θ
2
2
4
in
0<z<1
(1)
on
z=0
(2)
on
z = 1,
(3)
with stability conditions (from consideration of the sign of the second variation, as described)
Z1
cos 2θ dz > 0,
(4)
0
(A1 cos(2θ − 2β) + F cos 2θ) > 0,
(5)
(A0 cos(2θ − 2α) − F cos 2θ) > 0.
(6)
Again these equations are dimensionless, and D, F are dimensionless dielectric and flexoelectric coefficients,
h2 E 2 0 (|| − ⊥ )
hE(e1 + e3 )
D=
, F =
.
2K
K
The strategy here would be to start with zero-field (F = 0); find two stable zero-field solutions,
and investigate how their stability changes as we vary the parameters D and F . If one solution
branch becomes unstable while the other remains stable, we must have a bifurcation, and the
solution would jump onto the only stable branch. Reducing the field to zero, a switch is
achieved. We discuss this briefly in section 4.
3
Time-dependent generalization of the model
Figure 2: Steady state solution without electric field.
As discussed in the Introduction, we may also need to consider time-dependent effects when
switching between states, since typically in applications the switching should be fast. The
nonlinear ODE boundary value problem (1)–(3) obtained above is here replaced by a nonlinear
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Figure 3: Switching between different director fields.
PDE plus boundary conditions. The appropriate generalization was given by Davidson &
Mottram [2], and replaces the ODE by a diffusive PDE
θt = θzz − D sin(2θ)
0 < z < 1,
(7)
with boundary conditions
A0
F
sin(2θ − 2α) − sin(2θ) on z = 0,
2
2
F
A1
sin(2θ − 2β) − sin(2θ) on z = 1.
= −
2
2
θz =
(8)
θz
(9)
We note also that this system is also useful in considering the second order ODE boundary
value problem (1)–(3) above, since solutions to the PDE will automatically approach stable
solutions of the ODE boundary value problem as time goes to infinity. In other words,
numerically-computed large-time solutions of the nonlinear diffusion equation should exactly
satisfy the ODE boundary value problem, plus its stability conditions. One motivation for
using this approach rather than directly solving the ODE is that the shooting method can be
very sensitive to the choice of initial slope. Very often the desired solution lies on the unstable
manifold of the slope space, and leads to a numerically ill-conditioned problem.
The PDE system (7)–(9) can be solved using the Crank-Nicolson method for the time
discretization, and applying Newton’s method for solving the nonlinear system at each time
step. This is discussed further in section 5.
4
Quasi-steady Analysis
We start with the steady state problem (1)–(3) for the director field, n, reproduced below,
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θzz = D sin 2θ
θz = −
0<z<1
A0
F
sin(2θ − 2α) − sin 2θ on z = 0
2
2
A1
F
sin(2θ − 2β) − sin 2θ on z = 1.
2
2
We want to investigate how the existence and stability of solutions to these equations change
as the electric field varies (that is, as the parameters D, F vary). We assume that the electric
field changes slowly and that the solutions associated with a particular field applied vary
slowly as well so that a small change in the electric field produces a small change in the
steady state solutions. The stability of the steady state solutions is checked by requiring that
stable solutions have positive second variation, J2 > 0. The equations that represent this
condition are (recall (4)–(6))
Z 1
D cos (2θ(z)) dz > 0
θz =
0
A1 cos (2θ − 2β) + F cos (2θ) > 0 at z = 1
A0 cos (2θ − 2α) − F cos (2θ) > 0 at z = 0.
After solving the governing equations and checking the stability conditions, a plot can be
made showing the stability of the solutions and the bifurcations as the electric field changes.
The ideal situation is illustrated in Fig. 4, The equations are solved using a shooting method
and the integral stability condition is checked using the midpoint rule. This analysis is done
for two situations. The first involves a single layer of liquid crystal, and the second involves
a two-layer system in which two different liquid crystals are used.
The results for the single layer case are plotted in Fig. 5. They show that as the (dimensionless) electric field becomes much stronger than the anchoring strength, all solutions to the
steady state equations tend to the solution where the director field aligns with the electric
field. While we do have bistability, in that there are two stable solutions to the ODE, there is
no evidence (at least for the case considered here) for switching between the two stable states.
We finally consider what could happen if, instead of one liquid crystal sandwiched between
two plates, two liquid crystals with different electric properties are between the plates. We
assume the liquid crystals are immiscible and that θ, θz are continous across the interface
separating the two layers. We further assume that one liquid crystal is much more electrically
active than the other, so much so that the field can be considered on in one region and off
in the other. This setup is shown in Fig. 6. The results of the previous analysis are shown
in Fig. 4. Again as the electric field increases relative to the anchoring strength, we see no
evidence for switching between two stable solutions.
5
Analysis of the time-dependent model
We now return briefly to the model introduced in section 3 to see if we can find switchable
solutions by this means. The governing equations and boundary conditions are as given in
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Figure 4: This figure represents the ideal result of the quasi-steady analysis. At zero applied
field there exist two stable solutions. If an electric field is applied in the positive direction
to the ‘top’ stable solution, this solution loses stability at some critical field strength, and
the director field must jump to the ‘lower’ stable solution. Conversely, applying an electric
field in the negative direction to the ‘lower’ solution will (at critical field strength) bump the
director field to the ‘top’ stable state.
Figure 5: Single liquid crystal results. The stable (o) and unstable (x) solutions to the steady
state equations are plotted against the ratio of electric field to anchoring strength. As the
electric field becomes much stronger than the anchoring strength, the only surviving solutions
involves the director field aligning with the electric field.
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Figure 6: Schematic of 2-layer liquid crystal model. The liquid crystal on top is considered
to be infinitely more electrically active than the liquid crystal on the bottom.
Figure 7: Results of quasisteady analysis for 2-layer liquid crystal device. The stable (o)
and unstable (x) solutions to the steady state equations are plotted against the electric field
normalized to anchoring strength. We see no evidence for switching between two stable states.
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Figure 8: Schematic of the domain of the liquid crystals with varying topography at the top
boundary.
(7)–(9). The strategy here is to first determine the stable steady solutions to the model (at
zero field), and then use one of these steady states as an initial condition in a time-dependent
model with nonzero field, to see if we can switch from one steady state to the other.
5.1
Steady state solution with no electric field
In the case of no electric field, D = F = 0 and the steady state is simply a linear function in
z. As shown in Fig. 2, two steady states are found, having distinct profiles. For definiteness
we chose parameters as A0 = A1 = 5, γ = 10, α = 0, β = π/4. The left-hand picture in
figure 2 shows the solutions θ as functions of z; clearly they are exactly linear. The right-hand
picture is a vector plot of the director field.
5.2
Switching between two steady states
It is very interesting that, using the strategy outlined above, we can switch between two steady
states via temporary application of an electric field. Here we show an example. Parameters
are again chosen as A0 = A1 = 5, γ = 10, α = 0, β = π/4; and we note that the electric
field parameters D and F are not independent, since the ratio F 2 /D must be a constant, Υ,
for any given device. As shown in Figure 3, we start from one steady state and increase the
strength of the electric field in time. After we turn off the electric field, the solution jumps
to the second steady state immediately. The red arrow indicates the evolution of the director
fields at different instants of time.
6
Variable Topography
Our final line of enquiry is to reevaluate the model proposed by Cummings & Richardson [1],
and adapt it to include the effect of variable topography at one of the boundaries as seen in
Figure 8. This changes the model from being one dimensional analysis to a two dimensional
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analysis. The variation at the top boundary will be given by Z 0 (x0 ) = h00 + h01 f (x0 ). We will
assume that the amplitude of the variations at the top boundary is small in comparison to
their wavelength; and for this analysis we assume equal anchoring strength A0 at both upper
and lower surfaces.
Our expressions for the bulk energy density W 0 and surface energy density g 0 are:
2
A0
n(x0 , 0) · p
2
2
A0
n(x0 , Z 0 (x0 ) · q ,
2
2
2We0 = K10 (∇0 · n)2 + K20 (n · ∇0 × n)2 + K30 (∇0 × n) × n ,
gL0 = γ 0 −
gU0 = γ 0 −
2Wd0 = −ε00 ε⊥ E 0 · E 0 − ε00 (εk − ε⊥ )(n · E 0 )2 ,
Wf0 = −E 0 · e01 (∇0 · n)n + e03 (∇0 × n) × n ,
(10)
(11)
(12)
(13)
where
q(x0 ) = Ψ(x0 ) sin(b) + Φ(x0 ) cos(b), 0, −Φ(x0 ) sin(b) + Ψ(x0 ) cos(b) ,
− 1
Φ(x0 ) = sin(cos−1 (Ψ(x0 ))),
Ψ(x0 ) = 1 + fx0 (x0 ) 2
p = [sin(a), 0, cos(a)] n(x0 , z 0 ) = sin(θ(x0 , z 0 )), 0, cos(θ(x0 , z 0 )) .
The equations can then be simplified
gL0 |z 0 =0 = γ 0 −
A0
sin2 (θ − a),
2
(14)
2
A0
Ψ(x0 ) cos(θ − b) + Φ(x0 ) sin(θ − b) .
2
We assume there is no applied electric field
gU0 |z 0 =Z(x0 ) = γ 0 −
Wf0 = Wd0 = 0,
⇒
E=0
(15)
and then with the equal-elastic-constants assumption used earlier, the total energy density
within the bulk is
K0 2
K10 ≈ K30 = K 0 ⇒ We0 =
θx0 + θz20 .
(16)
2
We nondimensionalize the problem via the following scalings:
x0 = L0 x,
A0 =
z 0 = h00 z,
K0
A,
h00
γ0 =
W0 =
K0
γ,
h00
K0
0 W,
h02
0
gU,L
=
K0
,
h00
(17)
and Z 0 (x0 ) = h00 H(x).
(18)
A
sin2 (θ − a),
2
(19)
Thus the non-dimensional equations are
gL |z=0 = γ −
gU |z=1+δf (x) = γ −
2
A
Ψ(x) cos(θ − b) + Φ(x) sin(θ − b) ,
2
11
(20)
where
− 1
Φ(x) = sin(cos−1 (Ψ(x))),
1 κ2 2
2
W =
θ + θz ,
2 δ2 x
and the non-dimensional parameters are
Ψ(x) = 1 + κ2 fx2
2
κ=
h1
L
δ=
h1
.
h0
(21)
(22)
(23)
So we want to take variations of
Z
Z
J=
W dV +
g dS,
where J is only a function of θ. As mentioned before we will let δ ∼ O(1) and assume that
κ 1, with θ(x, z; κ) = θ0 + κθ1 + κ2 θ2 + . . . .
The equation to be solved from the first variation and to O(1) (with respect to κ) is:
(θ0 )zz = 0,
A
sin(2(θ0 − a)),
2
A
z = H(x) : (θ0 )z = − sin(2(θ0 − b)),
2
x = 0, 1 : (θ0 )x = 0.
z = 0 : (θ0 )z =
(24)
(25)
(26)
(27)
We are looking for stable solutions to (24)–(27). We have
θ(x, z) = A0 (x)z + B0 (x),
where
A0 (x) =
A0 (x) = −
A
sin(2(B0 (x) − a))
2
A
sin(2(A0 (x)H(x) + B0 (x) − b)).
2
This gives two conditions:
cos(−b + a + A0 (x)H(x)) = 0 → Unstable,
sin(2B0 (x) − b − a + A0 (x)H(x)) = 0 → Two Stable Solutions.
The two stable solutions are, for k = 0, 1:
2A0 (x)
= (−1)k sin(b − a − A0 (x)H(x))
A
kπ − A0 (x)H(x) − b + a
B0 (x) =
.
2
In Figure 9 we have the stable solution for k = 0. We see that liquid crystals rotate about
π and match up with the geometry at the top and bottom boundaries. More specifically the
rotation of the liquid crystals follow the topography of the top domain along the x direction.
At the top of the boundary the isoclines are very similar to the top boundary and as we move
towards the bottom domain the amplitude of the isoclines become smaller and smaller due to
the flat boundary.
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3.5
1.2
z = H(x)
1
3
z
0.8
2.5
0.6
0.4
2
0.2
0
0
2
4
6
8
10
12
14
16
x
18
20
1.5
θ
Figure 9: The solution is obtained for a = b = π3 , and A = 5.
7
Variable Anchoring Strength
Here for completeness we record a model closely related to the above, where instead of allowing
variable topography at one of the bounding surfaces, we allow variable anchoring strength
along both surfaces z = 0 and z = h. So, the anchoring strength A is function of x and hence
again θ now depends on both x and z. We also allow a further generalization, in which the
electric field, while still constant, is assumed to be applied at some angle In the absence of
electric field: E = E(sin φ, 0, cos φ). With no field E = 0, and assuming periodicity of the
anchoring condition with a period L in the x direction, we have the following 2D nonlinear
boundary value problem
θxx + θzz = 0
0<z<1
θx = 0 on x = 0, L
A0 (x)
sin(2θ − 2α) on z = 0
2
A1 (x)
θz =
sin(2θ − 2β) on z = 1.
2
With an applied electric field the above equations are modified to
θz =
θxx + θzz = D sin(2θ − 2φ)
0<z<1
F
(cos(2θ − φ) + (ê1 − ê3 ) cos φ) = 0 on x = 0, L
2
F
A0 (x)
θz − (sin(2θ − φ) + (ê1 − ê3 ) sin φ) −
sin(2θ − 2α) = 0 on z = 0
2
2
F
A1 (x)
θz − (sin(2θ − φ) + (ê1 − ê3 ) sin φ) +
sin(2θ − 2β) = 0 on z = 1.
2
2
θx −
13
8
Conclusions
We started with a simple planar cell design, for which a realistic free energy can be written
down and minimized directly to obtain the governing equations. The result is a nonlinear
boundary value problem for the director field angle θ. The solution of this nonlinear boundary
value problem has two steady-state (stable) solutions, and we seek conditions under which we
can switch between these solutions by applying electric field. We investigated switching first
via the quasi steady analysis of section 4, where we analyzed the stability and bifurcation of
solutions with varying electric field. This quasi-steady analysis was supplemented by a brief
study of its time-dependent generalization in section 5.
Further, we considered the effect of variable topography, allowing the top boundary of the cell
to vary, which led to a two dimensional nonlinear boundary value problem as the governing
equations. For this variable topography case we carried out asymptotic analysis to determine
stable and unstable solutions. Finally, we formulated a model including the effect of variable
anchoring strength, as well as non-perpendicular electric field direction, which led to a similar
two dimensional nonlinear boundary value problem.
References
[1] Cummings, L.J., Richardson, G. Bistable nematic liquid crystal device with exoelectric
switching. Europ. J. Appl. Math. 17, 435-463 (2006).
[2] Davidson, A.J., Mottram, N.J. Flexoelectric switching in a bistable nematic device. Phys.
Rev. E. 65, 051710 (2002).
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