Liquid crystal (LC) materials, as a type of transmission medium, are widely used in display and nondisplay fields because of their capacity to modulate the outside signal and the control voltage.^[1] The uniform orientation of LC molecules in LC devices significantly influences the result of LC modulation. The alignment layer on the substrate surface determines the uniform orientation of LC molecules, i.e., the anchoring characteristics of the alignment layers to LC molecules.^[2] The LC molecule deformation in LC cells with different anchoring strengths is different even when the applied voltage is the same. At present, several LC display and nondisplay devices are based on weak anchoring.^[3–9] Therefore, the precise determination of anchoring energy on the substrate surface can ensure the accurate design of LC devices.

Anchoring treatment on the surface can be classified as either contact type or noncontact type. Contact type, that is, rubbing the substrate, is the main treatment currently used in industrial production. Noncontact type, including oblique evaporation^[10] and photoinduced alignment,^[11] is the major treatment used in laboratory research. To connect with industrial production, the anchoring effect caused by rubbing the substrate should be considered. Anchoring energy can be measured by using optical and electrical methods. The optical method has two main types: Freedericksz transition technology^[12,13] and optical waveguide technology.^[14] The former requires the electromagnetic coherence length to be approximately the same as the LC cell thickness, indicating that the Freedericksz transition can occur only when the surface has strong anchoring. The latter is a relatively high-precision measurement, but the experiment equipment and data processing are complicated, which can easily cause errors. The main electrical method is the high electric field technique.^[15,16] In the real measurement system, however, LCs may contain a small amount of conductive ion and flexelectric effect, which could affect the measurement result. An electrical measurement method to obtain the anchoring energy of the substrate surface is achieved by comparing the experimental C–U curve with a numerical simulation C–U curve based on Frank elastic theory.^[17,18] Ye et al.^[19] determined the influence of substrate anchoring on the capacitance of parallel alignment nematic and hybrid alignment nematic LC cells through theoretical analysis and numerical simulation. In the present paper, we propose an experimental design to measure the capacitance of twisted nematic (TN) LC cells and thus obtain the anchoring energy on the substrate surface using the alternating-current (AC) bridge.

In our proposed method, the anchoring energy is measured by comparing the C–U curves of experimental results and theoretical simulation. Two basic theories are involved, namely, the capacitance determination of LC cells on the basis of the AC bridge and the numerical simulation of the capacitance of LC cells on the basis of Frank elastic theory.

The schematic of the AC bridge is shown in Fig. 1, where S is the AC signal source, (i = x, 2, 3, 4) is a precision adjustable resistance, is the LC cell to be measured, is the precision fixed value capacitor, G is the galvanometer, and K is the switch.

Fig. 1.

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	Fig. 1. Schematic diagram of the AC bridge.

When the bridge reaches balance, points B and D have the same voltage amplitude and phase, that is to say, and , which meet the following relationship: 1 From Eq. (1), one can obtain 2 where φ′ is the phase difference between B and D, and ω is the angular frequency of AC signal.

The structure of the weak anchoring TN cell is shown in Fig. 2, with the thickness of l, and the alignment layers on the upper and lower substrates are rubbed in the x and y directions, respectively. The LC molecules are distorted by 90° from the lower to the upper substrate. The anchoring directions of the upper and lower substrates are and , respectively, in which the pretilt angles are and , and the pretwist angles are and . The LC director is determined by the tilt angle θ and twist angle ϕ, and is only a function of the spatial coordinate z. The electric field in the cell formed by the applied voltage U changes the orientation of the LC molecules, which can be determined on the basis of Frank elastic theory.

Fig. 2.

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	Fig. 2. The structure of a TN cell with a weak anchoring and coordinate system.

The free energy of the TN cell system per unit area is 3 where is the free energy density, and and are the surface anchoring energy per unit area of the lower and upper substrates, respectively. Their expressions are 4 5 6 where , , , , and are the splay, twist, and bend elastic constants of LC; is permittivity in the vacuum; , are the dielectric anisotropy of LC; and are the relative permittivity along the direction parallel and vertical to the molecule long axis of LC, respectively; P is the pitch induced by the chiral dopant; and ϕ are the tilt and twist angles of the LC director on the lower substrate; θ and ϕ are the tilt and twist angles of the LC director on the upper substrate; and are the polar and the azimuthal anchoring energy on the lower substrate; and and are the polar and the azimuthal anchoring energy on the upper substrate.

When the system reaches balance under an applied voltage, according to the variational principle, the Euler equations including the tilt angle θ, the twist angle ϕ, and the potential ϕ, can be deduced as follows:^[20] 7 8 9 Concurrently, the tilt θ and the twist angle ϕ of the LC director on the upper and lower substrates meet the following conditions 10 11 12 13 Substituting Eqs. (4)–(6) into Eqs. (7)–(13), we can obtain the equilibrium state equations and boundary conditions of the LC cell system 14 15 16 17 18 19 20

On the basis of the above equilibrium state equations and the boundary conditions, as well as the differential iteration method in Ref. [21], the distributions of the tilt and twist angles under different applied voltages and anchoring energies can be calculated, as shown in Figs. 3 and 4. In the calculation, LC E7 is considered with the following material parameters:^[22] pN, pN, pN, , and . The TN cell thickness is m, the pretilt angles on the upper and lower substrates are , the pretwist angles are and , and the LC pitch is . When the applied voltage (2 V) is beyond the threshold, the substrate anchoring energy coefficient significantly influences the tilt angle of the LC director. As the anchoring energy coefficient decreases (from strong to weak anchoring), the LC director near the substrate surface changes substantially, leading to the strong variation of cell capacitance. When the substrate anchoring is weak (10 J/m , the director tilt angle near the substrate surface increases, as the applied voltage is increased, thereby changing the TN cell capacitance.

Fig. 3.

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	Fig. 3. Director distribution under the applied voltage of 2 V for different anchoring energy coefficients: (a) tilt angle and (b) twist angle.

Fig. 4.

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	Fig. 4. Director distribution under the anchoring energy coefficient of 10 J/m for different applied voltages; the applied voltage increases from 0 V to 10 V with a voltage step of 1 V along the arrow direction: (a) tilt angle and (b) twist angle.

Once the tilt angle of the LC director at a voltage is determined, the TN cell capacitance can be expressed as^[23] 21 where and S is the substrate area. To compare conveniently the theoretical and experimental results, the reduced capacitance of the LC cell is introduced, and it can be expressed as 22 where ξ = z/l and is the capacitance of the TN cell without voltage application. After the tilt angle of the LC director under different voltages is determined, we can obtain the theoretically reduced capacitance curve of the TN cell with different substrate anchoring energy coefficients by using Eq. (22) (Fig. 6). The smaller the substrate anchoring energy coefficient (10 J/m ), the lower the applied voltage (0.8 V) to make the TN cell capacitance reach a stable value. The steepness of the C–U curve decreases as the substrate anchoring energy coefficient is increased. For the two situations of 5 J/m and 10 J/m , the two curves are almost coincident, indicating that the substrate anchoring energy coefficient 5 × 10 J/m has reached strong anchoring.

Fig. 5.

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	Fig. 5. Reduced capacitance of TN cell versus voltage for different anchoring energy coefficients.

Fig. 6.

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	Fig. 6. Reduced capacitance of TN cell versus anchoring energy coefficient for different voltages.

As indicated in Fig. 6, the TN cell capacitance remains constant with increasing anchoring energy coefficient regardless of the applied voltage (2, 3, or 4 V). When the substrate anchoring energy coefficient exceeds 5 × 10 J/m , the difference in TN cell capacitance is small. For the same anchoring energy coefficient, the increase in the effective dielectric constant in the TN cell induced by the large tilt angle of the LC director on the substrate surface due to the enhanced applied voltage leads to the increase in the TN cell capacitance.

The LC cell can be regarded as a capacitor, in which the LC is a dielectric medium. The change in LC molecular orientation changes the effective dielectric constant of the LC cell, thereby affecting the LC cell capacitance. Therefore, factors that change the LC director all reflect the change in the LC cell capacitance, including the anchoring energy on the substrate surface. Ye et al.^[24] designed the capacitance method to measure the flexelectric coefficient of the nematic LC because the flexelectric effect affects on the LC director.

In this study, we designed a method to measure the anchoring energy on the substrate surface using the AC bridge. The AC bridge is mainly composed of precision adjustable resistance, precision fixed value capacitor, and TN cell. By balancing the AC bridge, we can obtain the voltage and capacitance of the TN cell and the C–V experimental curve. Then, the anchoring energy on the substrate surface can be obtained by comparing with the numerical simulation C–V curve on the basis of Frank theory. Compared with other methods, this measurement does not need a complex optical path system, a high-precision instrument, and a harsh environmental condition. With only select precision adjustable resistance and precision fixed capacitor, the AC bridge can be assembled reasonably. Moreover measuring with high accuracy is easy, and the method can be applied to measure other performance parameters of LC devices, thereby expanding the means to measure the performance parameters of LC devices.