## Abstract

We analyze the non-uniform electric field distribution in an in-plane-switching blue phase liquid crystal (IPS-BPLC) cell and use the director model to simulate the electro-optical properties of an IPS-BPLC cell using a commercial simulator. The calculated results are in good agreement with the experimental data.

© 2012 OSA

## 1. Introduction

Polymer-stabilized blue-phase liquid crystal (BPLC) [1–6] is emerging as a viable material for use in display applications because it exhibits some attractive properties. (1) It has no threshold driving voltage. (2) It does not require any alignment layer. (3) Its gray-to-gray response time is in the sub-millisecond range. (4) The dark state is optically isotropic, and its viewing angle is wide and symmetric [7]. Previously, BPLCs were often observed in a very narrow temperature range (about 2°C) [8]. However when it is polymerized with a mixture of monomers, the stable temperature range has been extended to over 60°C [1], such that it makes BPLC a strong candidate for display applications. Accordingly, Samsung Co. demonstrated a BPLC display prototype at the 2008 SID exhibition.

The blue phase liquid crystal has orientational molecular order, yet flow like liquid. Up to three thermodynamically distinct phases are observed upon cooling from the isotropic liquid: BPIII, BPII, and BPI, respectively. The isotropic-to-anisotropic transition of BPLC is related to the electric-ðeld-induced birefringence known as the Kerr effect [9–11]. In the low field region, the conventional Kerr effect holds. As the electric field increases, the induced birefringence gradually saturates. A modified mathematical model called the extended Kerr effect [12] has been proposed to accurately describe the display performances. In the previous paper, we demonstrated that the director model can describe a BPLC cell driven with a uniform electric field very well [13].

In this paper, we use the director model to simulate and analyze the observed electro-optical properties of an IPS-BPLC cell. The BPLC cell is modelled by the stacking of a number of high pretilt nematic (HPN) liquid crystal layers. The results are in good agreement with the experimental results.

## 2. Theory

It is well known that the director is used to describe the molecular orientation of nematic liquid crystal (NLC). For a polymer-stabilized BPLC cell, liquid crystal molecules are constrained by polymer in crystal lattices. We consider the BPLC cell as a stacking of a number of NLC layers and can be simulated by the director model. It can describe the two facts of a BPLC cell: (1) the cell is optically isotropic without an external field and then becomes anisotropic when the field increased, (2) there is no threshold voltage.

In our previous paper, we simulate the electro-optical properties of a polymer-stabilized BPLC cell by strong anchoring HPN cells. In the simulation, the pretilt angle (θ) of the HPN cell is determined first. A BPLC cell driven by a uniform electric field is shown in Fig. 1(a)
. In a voltage-on state, the applied electric field (**E**) is parallel to the z-axis. The total free energy density (f) that includes elastic and electric field energy densities is given by [14]

**n**is the LC director;

*k*

_{11},

*k*

_{22}and

*k*

_{33}are splay, twist and bend elastic constants, respectively;$\Delta \epsilon $is the dielectric anisotropy, and E is the electric field. The first three terms can be estimated by one-constant approximation (

*k*= (

*k*

_{11}+

*k*

_{22}+

*k*

_{33})/3) because the BPLC cell shows isotropic optical properties at V = 0; and the BPLC cell produces the splay, twist, and bend deformations simultaneously under the action of the external electric field. The fourth term in Eq. (1) is the electric field energy density. Because

**E**is parallel to the z-axis, the electric field energy density is proportional to the square of the direction cosine of

**n**on the z-axis.

In the voltage-off state, **n** is considered to be randomly distributed in the bulk of the cell shown in Fig. 1(a). The average direction cosine on the z-axis can be calculated by

*V*

_{b}is the integral volume in the bulk of the BPLC cell, and

**z**is the unit vector on the z-axis. The average tilt angle (θ = 30°) of

**n**is then obtained, so the pretilt angle of the HPN cell can be set as 30° shown in Fig. 1(b).

Let us assume the LC material in Fig. 1(a) has positive dielectric anisotropy. When applying an electric field **E** to the BPLC cell, LC molecules begin to orient to the z-axis which results in a decreased refractive index ($\delta n$). For a vertical incident light beam passing through the BPLC cell, the phase retardation (${\varphi}_{bp}$) is related to $\delta n$ as

*d*is the thickness of the BPLC cell in simulation, λ is the wavelength of the incident light,

_{bp}*M*is the number of BPLC layers for 3-dimensional structure consideration, and ${\varphi}_{h}$ is the phase retardation in the layer. Although the phase retardation cannot be observed by applying a vertical field to a BPLC cell, one can measure ${\varphi}_{bp}$ by a Michelson interferometer due to the optical path difference (OPD) of incident light when applying a vertical electric field

_{z}**E**to the BPLC cell [13].

When in an IPS-BPLC cell, the applied **E** field is parallel to the substrate (we assume that the electric field is distributed uniformly in the area). LC molecules begin to orient to the y-axis which results in the effective refractive index change ($\Delta n$) when a vertical incident light beam passes through the BPLC cell. The phase retardation (${\varphi}_{bp}$) is related to $\Delta n$ as

*d*is the effective penetration depth of the electric field in the cell [4]. The transmittance of a BPLC cell is

_{eff}Now, we consider the difference between normal electric field **E** (simulation) in Fig. 2(a)
and parallel electric field **E** (IPS in experiment) in Fig. 2(b). Normal electric field **E** results in refractive index $\delta n$ and parallel electric field **E** results in$\Delta n$, so we can obtain the relationship of $\delta n$and $\Delta n$ as$\Delta n=3\cdot \delta n$ [15].

Next, we consider the substitution of the IPS-BPLC cell in Fig. 1(a) with multiple HPN cells. The IPS-BPLC cell with electrode spacing D is divided into M_{z} layers with each layer thickness (2*d _{h}*) in Fig. 3
. The applied voltage on each IPS-BPLC layer is

*V*=

_{h}*V*/

_{bp}*M*, because of the series connection. Here, the voltage multiplier M is the fitting parameter and M = 3M

_{z}[13].

Each isotropic BPLC layer can be expressed by two identical HPN cells. Each HPN cell has a thickness of *d _{h}* and applied voltage (

*V*). Then,

_{h}*d*and

_{h}*V*of an HPN cell are related to D and

_{h}*V*as

_{bp}We adjust the parameter M to fit the experimental data. Because the polymer network increases the anchoring energy, the magnitude of M indicates the influence of the polymer network. In this simulation, we assume that the polymer network is not distorted with the increase of the electric field.

## 3. Simulation and discussion

In this study, the experimental result is retrieved from Ref [7]. The BPLC is a mixture consisting of high birefringence cyanates, chiral dopants (Merck CB15 and R-1011) and monomers (EHA and RM257). The mixture has an intrinsic birefringence of 0.272 and 100°C clearing temperature. The BPLC mixture was used to fill in an in-plane-switching (IPS) cell. The cell gap is about 13*μm*, the electrode width is 5*μm*, and the electrode spacing is 10*μm*. The light source is an unpolarized 10-*mw* He-Ne laser (λ = 633*nm*). The voltage-dependent transmittance (V-T) curves are depicted in Fig. 4(a)
. Using the experimental data of Fig. 4(a) and Eq. (5), the phase retardation can be obtained as ${\varphi}_{bp}=2{\mathrm{sin}}^{-1}\left(\sqrt{T}\right)$. The experimental data of ${\varphi}_{bp}$ are depicted in Fig. 4(b).

In the simulation, a commercial simulator (TechWiz LCD) was used. To simulate the experimental results, we adjusted the value of *d _{bp}* = 3

*d*and the voltage multiplier M to fit the experimental results. Here,

_{eff}*d*is the effective penetration depth of the electric field and${\varphi}_{bp}$ is the total phase retardation of the IPS-BPLC cell with thickness

_{eff}*d*. The simulation results are shown in Fig. 4(b). A set of values,${d}_{bp}=7.0\text{\hspace{0.17em}}\mu m$,${d}_{eff}=3.2\text{\hspace{0.17em}}\mu m$ and M = 103, is found to fit the experimental results well.

_{eff}The effective BPLC cell thickness (${d}_{eff}=2.3\text{\hspace{0.17em}}\mu m$) obtained from the simulation is smaller than the real cell thickness ($d=13\text{\hspace{0.17em}}\mu m$). The fact that *d _{eff}* =

*d*/5 indicates that the penetration depth is about one-fifth the cell gap of the BPLC cell. This means that the

**E**field distribution is about one-fifth of the whole cell if we assume that the distribution of the

**E**field is uniform. The simulated results are consistent with the results in the paper reported by Professor Shin-Tson Wu’s Laboratory [4]. The other result is the voltage multiplier (M = 103). From the theoretical discussion, the result means that there are 103 vertical BPLC layers in the electrode spacing D ($=10\text{\hspace{0.17em}}\mu m$) such that the driving voltage is 103 times the value of

*V*. For 3-dim structure consideration, the layer thickness is D/M

_{h}_{z}= 3D/M = 291

*nm*. If we think of it as a lattice unit, the dimension is under the double twist cylinder (DTC) diameter of 300

*nm*such that the BPLC cell will be thermodynamically stable.

Although the simulation results fit the experimental data well at${d}_{eff}=2.3\text{\hspace{0.17em}}\mu m$ and M = 103, and we can obtain the effective penetration depth of the electric field in the IPS-BPLC cell and the layer thickness considered as the dimension of the BPLC lattice, the electrode spacing D ($=10\text{\hspace{0.17em}}\mu m$) of the IPS-BPLC cell is not equal to the electrode spacing ${d}_{bp}=7.0\text{\hspace{0.17em}}\mu m$ in the simulation. The fact that *d _{bp}* < D indicates the dilution of the liquid crystal by the polymer in the blue phase liquid crystal cell [13]. The unmoving polymer layers makes

*d*smaller than D.

_{bp}## 4. Conclusion

We have used the director model to successfully simulate the electro-optical properties of the IPS-BPLC cell, and the theoretical calculations match the experimental results well. We have also analyzed the distribution of the electric field in the IPS-BPLC cell. This is a valuable study which uses a simple model to simulate the electro-optical behaviors of the polymer-stabilized IPS-BPLC cell with a commercial simulator.

## Acknowledgment

The work is supported by the National Science Council of the Republic of China under Grant NSC 99-2221-E-027-049-MY3.

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