Near the electrodes, the vertical component of the electric field is strong. Figure 5.8(a) shows the director and the potential distribu- tion in case the dielectric layer is absent and the liquid crystal layer is deposited directly on top of the hexagonal electrodes. The plot shows that without the dielectric layer, the director above the electrodes tilts to angles of almost 90◦. When the applied voltage is removed, a tilt of 90◦ can result in the formation of domains separated by a defect line if the director in the adjoining regions relaxes in opposite directions.
Figure 5.8:Distribution of the director and the potential in thexz-plane (y = 0 in Figure 5.2) at t1 = 500 ms when driving configuration C2
is applied. (a) without the dielectric layer (the liquid crystal layer is positioned directly above the electrodes, plotted area: 20.4ì2.1àm), (b) the device as described above with a dielectric layer of 1.3àm inserted between the electrodes and the liquid crystal layer (plotted area: 20.4× 3.4àm).
Furthermore, the non-uniformity in the optical thickness of such a layer prevents its intended use as wave plate. In order to favor lateral
rotation instead of tilting, a dielectric layer is inserted between the elec- trodes and the liquid crystal layer. This layer shields the liquid crystal from the regions above the electrodes with strong vertical electric fields.
Figure 5.8(b) shows that a more homogeneous distribution of the liquid crystal director is achieved when a dielectric layer is included. In the following, a dielectric layer is always included unless specified other- wise.
a) Simplified model for the influence of the dielectric layer
To illustrate the effect of the thickness and dielectric constant of the di- electric layer on the electric field in the liquid crystal, a simplified two- dimensional model with isotropic media is used, shown in Figure 5.9.
An isotropic dielectric layer with dielectric constant εd is sandwiched between two infinite isotropic media with dielectric constantεlc. Since the director tends to orient along the electric field,εkis used for the liq- uid crystal layer. In the middle of the dielectric layer, electrodes lines are present with alternating potential as in the in-plane switching mode of liquid crystals.
+V /20 +V /20
-V /20 -V /20
liquid crystalồlc
dielectricumồd
z
x
d0
w g
A A’
d
Figure 5.9:Simplified two-dimensional configuration which allows an analytic approximation of the field for the in-plane switching mode.
Electrode stripes in a dielectric layer with thickness 2dobetween liquid crystal media.
Analytical calculation of the potential distribution in this configu- ration is not possible, but a simplification is possible by approximating the potential between two neighboring electrodes along thex-axis by a linear function as represented in Figure 5.10. In this case, separation of variables [130] can be used to solve the potential problem. The obtained
5.3 Director simulations 97
+V /20
-V /20
w g
x Potential
Figure 5.10: Potential variation along the x-axis in Figure 5.9 if the potential between neighboring electrodes is approximated by a linear function.
expression of the potential distribution is z<do : V(x,z) =
X∞ n=1,3, ...
hAnenπzw+g +Bne−nwπz+gi
sin nπx w+g
! (5.1a)
z>do : V(x,z) =
X∞ n=1,3, ...
Cne−nπzw+g sin nπx w+g
!
(5.1b) with
An = αn(εd−εlc) (5.2a)
Bn = αne2nwπdo+g (εd+εlc) (5.2b)
Cn = 2αne2nwπdo+g εd (5.2c)
αn =
4V0 w+g sin
nπg 2(w+g)
gπ2n2
(εd−εlc)+e2nπw+gdo (εd+εlc)
(5.2d)
Note that for z = 0, this expression is independent of the dielectric constants εd andεlc andV(x,0) results in the function plotted in Fig- ure 5.10. From the potential distributionV(x,z), the electric field can be calculated asE=−∇V.
The potential and electric field distribution, calculated in the dotted rectangle of Figure 5.9, is shown in Figure 5.11. The background color shows the potential distribution, with indication of the equipotential lines. The thick dashed lines represent the electric field lines. The width and spacing of the electrodes correspond with the width and spacing of the hexagonal electrodes:w= √
3a= 5.2àm and g=b=5àm. The
thickness of the dielectric layer is chosen the same as in the simulated hexagonal device of Figure 5.6:do =1.3àm. For the liquid crystal layer, only a layer with thicknessd=2.1àm is represented in Figure 5.11. In Figure 5.11(a), the liquid crystal layer is positioned directly on top of the electrodes (do = 0àm). In Figures 5.11(b) and 5.11(c), the dielectric layer is inserted.
(a) without dielectric layer
(b) with dielectric layer (εd=3.5,εlc=19.6)
(c) with dielectric layer (εd=19.6,εlc=3.5)
Figure 5.11: Analytic approximation of the electric field lines (dashed) and equipotential lines (full) in the in-plane switching configuration for isotropic media. (a) liquid crystal directly deposited on the electrodes without dielectric layer (plotted area: 20.4ì2.1àm), (b) with a dielectric layer between the liquid crystal and the electrodes (plotted area: 20.4× 3.4àm), (c) with a highεd dielectric layer between the liquid crystal and the electrodes (plotted area: 20.4ì3.4àm).
The field lines which start or end at an electrode in Figure 5.11(a) are oriented perpendicularly to the surface. In the liquid crystal layer, this will lead to high tilt angles of the director. If the dielectric layer is inserted between the liquid crystal and the electrodes, the electric field at the bottom surface of the liquid crystal is no longer perpendicular. In Figure 5.11(b),εd =3.5 andεlc =εk=19.6 is used as an approximation of the simulated device. Figure 5.11(c) represents a hypothetic case in whichεdandεlcare interchanged compared to Figure 5.11(b).
Since |εd−εlc| is large, the change in direction of the electric field
5.3 Director simulations 99
at the interface is quite large. Nevertheless, closer inspection of Fig- ures 5.11(b) and 5.11(c) reveals that the direction of the electric field at the bottom surface of the liquid crystal layer looks similar, althoughεd andεlc were interchanged. Therefore, the electric field at the bottom of the liquid crystal layer is further inspected. From (5.1a), the electric field just above the interface between the two materials can be calcu- lated as
E(x,do+) = X∞ n=1,3, ...
−nπ
w+gCne−nπw+dog cos nπx w+g
! 1x
+ X∞ n=1,3, ...
nπ
w+gCne−nπw+dog sin nπx w+g
!
1z. (5.3)
The magnitude|E|and the directionθeof the electric fieldE(x,do+) just above the interface are plotted in Figure 5.12 as a function of the lateral positionx for the three cases of Figure 5.11 and an applied voltage of 5 V. The direction of the field is given by its angleθewith the positive x-axis. The plot starts and ends above the centers of neighboring elec- trodes, illustrated by the lineAA0in Figure 5.9. The location of the two halve electrodes below the line AA0 is indicated with the thick black lines on thex-axis in Figure 5.12.
Between the electrodes, the three curves of the directionθeas a func- tion of the lateral positionxare almost identical. On the electrodes, the electric field ford0 =0àm is perpendicular to the surface as expected.
If a dielectric layer is inserted between the electrodes and the liquid crystal, the direction of the electric field just above the interface is not heavily influenced by the dielectric constant εd. The two plots with a dielectric layer in Figure 5.12 show an almost linear variation from +90◦ to -90◦, but the strength of electric field in the liquid crystal layer decreases when loweringεd.
In Figure 5.13 the magnitude|E|and the directionθeare plotted for different values of the thickness do for an applied voltage of 5 V and the dielectric constantsεd = 3.5 and εlc = εk = 19.6. From the plots it is clear that the thicknessdo influences mainly the magnitude of the electric field. Once a dielectric layer is inserted, the direction of the electric fieldθeis hardly modified.
From Figures 5.12 and 5.13 follows that a low value of the thickness doand a high value of the dielectric constantεdare favorable for a high electric field in the liquid crystal layer.
Figure 5.12: The angleθewith thex-axis (top) and the magnitude|E| (bottom) of the electric fieldE(x,do+) just above the interface between the dielectric and the liquid crystal layer for the three cases of Fig- ure 5.11.
The influence of the widthwof the electrodes and the gapgis visi- ble in Figure 5.14, which was calculated using an applied voltage of 5 V, a thicknessdo =1.3àm and dielectric constantsεd=3.5 andεlc=εk. In the different plots the gap gbetween the electrodes is varied from 1 to 4àm, but the distancew+gis kept constant at 10àm. The direction and strength of the electric field above the electrodes is hardly influenced, but in the center between the electrodes the strength of the the electric field increases for smaller gaps. Therefore, a small value of the gapgis favorable.
In the simulated hexagonal device of the previous section, a rather thick dielectric layer with a low dielectric constant is used. This seems unfavorable, since it reduces the strength of the electric field in the liq- uid crystal. The motivation to use a material with these properties was the easy and fast processing of the polymer BCB used in the exper- iments. The minimal reachable thickness by spinning without addi- tional process steps is about 1àm.
b) Mirror plane perpendicular to the average electric field
As explained in section 5.2, the orientation of the mirror planes de- pends on the driving configuration. Due to symmetry, the local elec-
5.3 Director simulations 101
Figure 5.13: The angleθewith thex-axis (top) and the magnitude|E| (bottom) of the electric fieldE(x,do+) just above the interface between the dielectric and the liquid crystal layer for different thicknesses of the dielectric layer,εd=3.5 andεlc=εk=19.6.
tric field in the equipotential mirror plane (containing the centers of the hexagonal electrodes at the same potential level in Figure 5.4) should be parallel with the plane and thus perpendicular to the average horizon- tal electric fieldEavof the driving configuration. To achieve a consistent director distribution, it is important that the director orientation does not follow the local electric field lines, but is determined by the elastic forces exerted by the liquid crystal director away from the plane.
In Figure 5.15, the director and potential distribution are shown in the equipotential mirror plane for different device configurations at t0 = 250 ms with driving configurationC1 applied. In Figure 5.15(a), the liquid crystal is positioned directly on top of the electrodes as in Figure 5.8. The director above the electrodes is almost perpendicular to the surface due to the strong vertical electric field in the mirror plane, only between the electrodes the director is oriented along the average electric field.
Introducing the dielectric layer in Figure 5.15(b) shows that the di- rector becomes perpendicular to the mirror plane as desired. Since the director is perpendicular to the observed plane, only the white tips of the cones which represent the director are visible. Increasing the electrode width at the expense of the gap between them, increases the
Figure 5.14: The angleθewith thex-axis (top) and the magnitude|E| (bottom) of the electric fieldE(x,do+) just above the interface between the dielectric and the liquid crystal layer for different widths of the gap g between the electrodes, keeping the distance between the center of neighboring electrodes w+ g constant. (do = 1.3 àm, εd = 3.5 and εlc=εk=19.6)
strength of the electric field between the parallel rows of electrodes at equal potential that rotates the director in the plane parallel to the sur- faces. Larger hexagons with reduced spacing are also favorable for a homogeneous director. They lower the horizontal variation of the po- tential in the equipotential mirror plane as shown in Figure 5.15(c) and thus weaken the horizontal electric field parallel in this plane. There- fore, the director is encouraged to follow the average horizontal electric fieldEav, perpendicular to the mirror plane.
For a dielectric layer with increased dielectric constant, the electric fields in the liquid crystal are stronger. As a result, the local director will follow closer the local electric field instead of the average electric field Eav. Inside the equipotential mirror plane, the horizontal variation of the potential is larger as can be seen in Figure 5.15(d). At the same time, the electric fields away from the equipotential mirror plane are stronger and less along the average electric field. As a result, the director in the mirror plane follows less good the average director.
5.3 Director simulations 103
Figure 5.15: Distribution of the potential and the director in the equi- potential mirror plane (xzof Figure 5.2), att0 = 250 ms with driving configuration C1 applied. (a) without dielectric layer (plotted area:
20.4ì2.1 àm), (b) with a dielectric layer (d0 = 1.3 àm, plotted area:
20.4ì3.4 àm), (c) with the same dielectric layer but larger electrode dimensions (plotted area: 23.3ì3.4àm), (d) with a dielectric layer with higher dielectric constant (plotted area: 20.4ì3.4àm)