of rubbing with a small tilt angle. The direction of rubbing and the resulting surface director are shown in Figure 2.5(c). The azimuthal di- rection of rubbing is indicated with the pretwistφ0, called the rubbing or alignment direction. The tilt angle at the rubbed surfaceθ0is named the pretilt.
A more detailed treatment of the alignment of liquid crystals at the surface will be given in the chapter on weak anchoring.
2.3 Electric and elastic properties
As a result of the uniaxial anisotropy, an electric field experiences a different dielectric constant when oscillating in a direction parallel or perpendicular to the director. The difference ∆ε = εk −ε⊥ is called the dielectric anisotropy. If the dielectric constant along the director εk, is larger than in the direction perpendicular to itε⊥, one speaks of positive anisotropy. The dielectric displacementDand the electric field Eare related to each other by a dielectric tensor of rank two. In case the director is parallel to thez-axis, the tensor relation can be written as
D = ε0εãE = ε0
ε⊥ 0 0 0 ε⊥ 0 0 0 εk
ãE, (2.1)
with ε0 = 8.85 10−12 F/m the dielectric permittivity of vacuum. Due to the anisotropy, the dielectric displacementDand the induced dipole moment are not parallel to the electric field, except when the directorn is parallel or perpendicular to the electric fieldE. Therefore, a torque
Γ =ε0∆ε
n.E n×E
(2.2) is exerted on the director. For materials with positive anisotropy, the director prefers to align parallel to the electric field. Liquid crystals with a negative anisotropy tend to orient themselves perpendicularly to the electric field.
In a configuration with an externally applied electric field, the macroscopic electrostatic energy per unit volume can be expressed as a function of the electric fieldEand the directorn:
fe= 1
2 DãE= ε0
2
∆ε nãE2
+ε⊥ EãE
. (2.3)
The reorientation of the liquid crystal director by an externally ap- plied electric field is one of the most exploited characteristic of liquid crystals. It is the base of operation for the so-called liquid crystal dis- plays (LCD’s).
The externally applied electric field E is often generated by elec- trodes on the bottom and/or top substrate. For optical devices trans- parent electrodes are used, made from Indium Tin Oxide (ITO). Ionic contamination of the liquid crystal material lowers the effective electric field that acts on the director. Since only squares of the electric field and the director are present in the electrostatic energy (2.3), the direction of the field ±Edoes not make a difference. Therefore, to avoid negative effects of ions moving in the liquid crystal such as image sticking and flicker [17–21], the applied voltages used in applications are always square waves with alternating positive and negative pulses. Also in this work, applied voltages are without exception assumed to be square waves.
A liquid crystal medium prefers a uniform director distribution. A variation of the director in space induces an increase of the free energy.
According to the elastic theory for liquid crystals, the distortion energy related to the variation of the director in space can be written as [3, 10]
fd= 1 2
hk11(∇ ãn)2+k22(nã ∇ ìn)2+k33(nì ∇ ìn)2i
, (2.4) with the three elastic constantsk11,k22andk33. This equation is known as the Oseen-Frank distortion energy. The three terms in the equation are related to distortion due to splay, twist and bend respectively as illustrated in Figure 2.6.
(a) splay (b) twist (c) bend
Figure 2.6:Twist, bend and splay distortion of the liquid crystal direc- tor.
Calculations of the equilibrium director distribution involve mini-
2.3 Electric and elastic properties 13
mizing the total free energy of the volume. In our approach, a constant voltage is applied to the electrodes, in which case the Gibbs free energy of the liquid crystal medium [22, 23]
F= Z
V
hfd(n)− fe
n,Ei
dV. (2.5)
must be minimized using the Euler-Lagrange equation. During my re- search I used different numerical tools to perform the necessary calcu- lations. For one-dimensional calculations with plane electrodes at top and bottom, I used the software tool Glue [24]. Glue was developed in the Liquid Crystals & Photonics Group of the Universiteit Gent. It performs calculations of the director distribution and the optical trans- mission for one-dimensional liquid crystal layers as a function of the applied voltage or wavelength of the incident light.
Two- or three-dimensional simulations have been performed using a dynamic three-dimensional Liquid Crystal Director Simulation tool MonLCD [25–29], developed by the Computer Modelling Group of University College London in the framework of the European project MonLCD (G5RD-CT-2000-00115). The method is based on an approach starting from the Oseen-Frank elastic distortion energy density and the electrostatic energy of the liquid crystal. It calculates the dynamic evo- lution of the liquid crystal director, which is governed by the three re- laxation equations
∂F
∂nδ
+ ∂
∂n˙δ
γ1 2
Z
V
˙ nàn˙àdV
!
=0, (2.6)
with à, δ = x,y,z and using the Einstein summation convention with respect toà. γ1 is the rotational viscosity of the liquid crystal mate- rial. A finite elements approach is used to solve the above system of differential equations for the director and the electric fields. A detailed description of the algorithm used for the relaxation and minimization of the total free energy and the calculation of the potential distribution are given in references [28–30]. A limited number of two-dimensional director calculations have also been performed using the commercial software tool 2dimMOS [31].
Table 2.2 gives an overview of the electrical and elastic parameters for the liquid crystals used for the experiments and simulations in this work [4, 10–12, 32].
Table 2.2:Electrical parameters of the used liquid crystals E7 ZLI-4792 6CHBT 5CB 5PCH
εk 19.6 8.3 12 19.7 17.1
ε⊥ 5.1 3.1 4 6.7 5
∆ε 14.5 5.2 8 13 12 .1
k11(pN) 12 13.2 8.57 6.4 8.5
k22(pN) 9 6.5 3.7 3 5.1
k33(pN) 19.5 18.3 9.51 10 16.2
γ1(mPa.s) 0.15 0.1232 0.083 0.081 0.123