# Electrical Simulation of Liquid Crystals

Introduction

Liquid Crystals (LCs) are state of matter intermediate between that of a crystalline and a liquid. The optical, mechanical, electrical and magnetic properties of LC medium are defined by the orientation order of the constituent anisotropic molecules. Due to the anisotropy of the electrical properties, the orientation of the LC molecules is effectively controlled by electric fields. As a result, LCs exhibit very specific electrooptical phenomena because of their large birefringence. All of these are important to the functional devices based on LCs, for example, flat panel displays that have been commercialized for decades.

The constituents of LCs are elongated or rod-like molecules and disk-like molecules. The average direction of the molecular long axes defines the director n, which gives the direction of the preferred orientation of LC molecules. The LC molecules reorient in externally applied electric fields because of their dielectric anisotropy. The electric energy of a LC depends on the orientation of the director in the applied electric field. Under a given electric field, the LC will be in the equilibrium state, where the total free energy is minimized.

Theory

The field-induced reorientation of the LC is now able to be calculated in Clever. The coupled equations governing the LC physics and the electrostatic potential are solved. The major variables are the voltage and the director. To calculate a director configuration of the LC, it is necessary to express the free energy of the system. Let (x,y,z) = (nx (x,y,z), ny (x,y,z), nz (x,y,z))be the director, where (x,y,z) is a point in the LC region. The Frank-Oseen free energy density is given as [1]:

f(x,y,z) = K11(∇ · )2 + K22( · ∇ × )2 + K33( × ∇ × )2
- (K22 + K24)∇ · (∇· + ×∇ × ) –q0 K22( · ∇ × )

where K11, K22, and K33 are the splay, twist, and bend elastic constants of the LC respectively. q0 is the chirality of the LC. The electric free energy density is given by:

g= ε0 ∆ε[n · (–∇ν)]2

where ε0 is the permittivity of vacuum, ∆ε = ε||– ε is the difference between the dielectric constant parallel and vertical to the director, and ν=ν(x,y,z)is the voltage at point (x,y,z). The Gibbs free energy density is then defined as:

F = f – g

The total free energy is the volume integration of the free energy density within the LC domain. The total free energy is minimized by a method based on a variational approach to the Oseen-Frank free energy formulation considering three elastic constants. The vector representation of the director field is used with the constraint
n2 = n2x + n2y + n2z = 1.

Simulation Examples

As an example, a twisted nematic (TN) geometry with the LC layer thickness of 3 µm is simulated. The elastic constants of the LC are K11=14.4×10-12N, K22=6.9×10-12N, and K33=18.3×10-12N. The anisotropic relative permittivity is ε//=10.7 and ε⊥=3.7. LC material parameters can be specified in the SetLC statement in Clever, as follows:

setLC epsParaDir=10.7 epsVertDir=3.7 splay=14.4e-12 twist=6.9e-12 bend=18.3e-12

The pretilt angles at the top and the bottom boundary of the LC layer are both 1˚. The initial twist angle is 90˚. These alignment conditions can be defined in the following commands.

LCbndaryT partition(0 10) rubAngle(0) tiltangle(1)
LCbndaryB partition(0 10) rubAngle(90) tiltangle(1)

where LCbndaryT and LCbndaryB denote the statement for the top and the bottom boundary, respectively. The rubAngle sets the rubbing angle with respect to the x axis and tiltangle specifies the tilt angle measured from the x-y plane.

The LC cell structure consisting of a LC layer sandwiched by two electrodes was created in Victory Process and exported with conformal tetrahedral mesh, as shown in Figure 1. The bias voltage on the electrode “Pix” was increased linearly from 0 to 6V in a 0.25V bias step through the following command in Clever:

Interconnect Capacitance \
domainboundarycondition=cyclic \
contact=”Pix” uservoltage=6 stevolt=0.25 strcture=”tn”

The solution was stored after each voltage step. The structure filename was generated in such a way that the current bias voltage is appended to the root file name “tn”. The director orientation of the TN-LC structure at 0 and 4V bias voltage is shown in Figure1(a) and 1(b), respectively. The arrows in the LC layer indicate the director orientation. Since Clever uses strong anchoring boundary condition for the LC simulation, the director orientation at the top and bottom boundaries will never change with the bias. Therefore, we can see as the bias is increased from 0 to 4V the director remains the same at the top and bottom boundaries while it becomes highly tilted in the middle of the LC layer.

 Figure 1. The TN-LC cell structure biased at (a) 0V and (b) 4V, respectively. Arrows indicate the orientation of the director.

The z component of the director nz was extracted along a line parallel to the z axis across the entire LC layer. The profile of nz at various bias voltages is shown in Figure 2. We can observe quantitatively how the director changes with increased bias. A nominal threshold voltage exists between 1.5V and 2V in this case. In order to identify the threshold voltage more clearly, the nz in the middle x-y plane (in terms of z axis) of the LC layer was extracted. The data was plotted as a function of the bias voltage with different twist angles of the director in Figure 3. We can see the transition region, i.e., the region where the nz goes from nearly 0 to nearly 1 becomes narrower and narrower with increased twist angle. Therefore the threshold voltage becomes more distinct. This is a well-known phenomenon in TN LC. Because of their steep transition, TNs with twist angle larger than 90˚ (known as super-TNs) are used to make multiplexed displays on passive matrices.

 Figure 2. The profile of the director component nz along the z direction with different bias voltages.

 Figure 3. The director component nz in the middle x-y plane of the LC layer as a function of bias voltages with different twist angles.

Another example is an in-plane switching (IPS) cell with two coplanar zigzag electrodes shown in Figure 4. The LC layer of the structure has a lateral size of 10µm×10µm and a thickness of 3m. The whole structure was created in Victory Process with conformal tetrahedral mesh. The LC material parameters are the same as those used in the TN example. The initial rubbing angle of the director is 90 with respect to the x axis. As the applied voltage at the “Pix” electrode is increased, the director inbetween two electrodes turns to the x direction gradually due to the increased electric field along the x axis. The director above electrodes twists in a much smaller amount because the electric field is almost vertical to the electrode surface.

 Figure 4. The IPS structure with zigzag electrodes.

Shown in Figure 5 is the contour plot of the x component of the LC director, nx, in a x-z plane located at y=7.5 m at the (a) 4V and (b) 6V bias voltage. Shown in Figure 6 is the contour plot of the nx in a x-y plane 1m to the electrode plane at the (a) 4V and (b) 6V. We can see clearly from these two figures how the director reorients with bias in different regions of the LC layer. The nx profile was extracted along a line parallel to the z axis across the LC layer at x=4.5 m and y=5 m, the middle point between two zigzag electrodes. The result is shown in Figure 7 where the electrodes are located in the left. The threshold voltage is estimated to be around 3V from the curve shape.

 Figure 5. The nx contour in a x-z plane at (a) 4V and (b) 6V.

 Figure 6. The nx contour in a x-y plane at (a) 4V and (b) 6V.

 Figure 7. The nx profile along the z direction with different bias voltages. The electrode plane is on the left side.

The capacitance of the LC cell between two electrodes can be obtained during the voltage ramping simulation. The calculation takes into account the anisotropic permittivity of the LC director. To save the capacitance, one needs to add the following command after the Interconnect statement:

save spice=”ips.net”

The data in the netlist file can be viewed in Tonyplot. The capacitance-voltage (CV) curve of the IPS cell is plotted in Figure 8. The threshold voltage is a bit smaller than 3V, in consistence with the director result shown in Figure 7.

 Figure 8. The capacitance-voltage curve of the IPS cell.

Conclusion

In summary, three-dimensional static simulation of the liquid crystal with user-definable material parameters can be carried out in Clever. All vector components of the LC director in the presence of the external electric field (applied voltage) are solved by the finite element method. The director data can be output at any stage of the simulation for visualization or post-simulation processing. The simulation on the TN and IPS structures has been demonstrated. Other LC geometries such as vertical alignment (VA) and fringe field switching (FFS) can be analyzed as well.

References

1. H. Mori et al, “Multidimensional Director Modeling Using the Q Tensor Representation in a Liquid Crystal Cell and Its Application to the π Cell with Patterned Electrodes”, Jpn. J. Appl. Phys. Vol. 38 (1999) pp. 135–146.