Periodic Electro-Optical Characteristics of PDLC Film Driven by a Low-Frequency Square Wave Voltage

Abstract

The electro-optical features of the PDLC films applied with a low-frequency square wave voltage were investigated. The transmittance curves indicated the double frequency of the applied voltage at 0–50 Hz, which resulted from the relaxation of an internal electric field polarized by ions in LC droplets. When the local electric field was reversed, the internal polarization electric field could be maintained and superimposed on the local electric field. The relaxation of the internal polarized electric field resulted in the relaxation of the transmittance. Furthermore, the transmittance curves changed with the frequency of the applied voltage.

1. Introduction

Polymer-dispersed liquid crystal (PDLC) is a composite material consisting of LC droplets randomly dispersed into an isotropic polymer matrix and represents a significant new class of electro-optical materials [1,2,3,4]. The electro-optical properties of PDLCs are dependent on the electrically controlled light scattering from the dispersed LC droplets. An alternating current (AC) voltage (frequency range of 10¹–10³ Hz) is typically used to prevent conductivity effects and typically applied to voltages from 20 to 100 V_rms to keep the devices transparent [5,6]. LC droplets in PDLCs can respond to an applied electrical field, aligning to maximize the dielectric constant. For LC with positive dielectric anisotropy, LC droplets can be in the direction of the applied electric field so that PDLC films can be switched from an opaque off-state to a highly transparent on-state [7,8]. The competition between the applied electric field and LC elastic and viscous torques governs the response time and switching voltage of PDLCs [9]. The response time of LC droplets is in the order of 10⁻³ s. Compared with conventional LCs and other heterogeneous systems, PDLCs have a new operating principle and dynamic characteristics [10].

In general, uncontrolled ions may occur in the LC during production [11,12,13], and almost all LCs contain free ions. For example, the ion concentration in LC E7 was estimated to be 10¹⁵/cm³ [14,15]. The free ions could decrease the dielectric constant and refractive index in LC droplets [16] and induce an internal polarization electric field (E_int) to affect the hysteresis and memory state in the PDLC films [8,17,18]. The voltage drop across the LC droplets is just a fraction of the voltage applied in a PDLC film because of a high voltage drop through the polymer matrix [17,19,20]. The electric field applied to LC droplets can be called the local electric field (E_loc) [20]. With a local electric field, the redistributed ions can accumulate on the surface of LC droplets to form E_int, which opposes E_loc [12,21]. E_int could play an essential role in the electro-optical response for PDLC devices. For example, E_int can screen the applied electric field (the charge screening effect) [12,14]. The PDLC films can be switched from transparent to scattering at a low frequency [14]. For frequencies below 100 Hz, the total electric field, i.e., the effective electric field (E_eff) in LC droplets, could be severely affected by the accumulated charge at the droplet/polymer interface [22]. E_int was investigated by observing the electro-optical response of a PDLC film to direct current (DC) and low-frequency sinusoidal wave-driving voltages [23]. It was revealed that E_int could align the LC droplets to produce electro-optical effects [23].

When investigating and applying the electro-optical effect of PDLCs, the AC square-wave voltage is usually used. In a period, the applied voltage changes polarity rather than amplitude, while it can be regarded as the voltage in a half-period as a DC voltage. However, the relationship between E_int and the frequency of the square wave voltage has not been investigated. In the work represented, we investigated the evolving curve of transmittance with time (T-t) with a square-wave voltage of 0–50 Hz. Moreover, depending on the characteristics of the T-t curves in different frequency ranges, the dynamic behavior and the electro-optical effect based on E_int were investigated. It was revealed that E_int can increase E_eff when the polarity of the AC square wave voltage is reversed.

2. Experiment and Method

2.1. Materials and Preparation

The LC used in our experiments was nematic LC E7 (Shijiazhuang Slichen Display Materials Co. Ltd., Shijiazhuang, China). The prepolymer in our experiments was composed of an adhesive (NOA65, Norland Products Inc., Cranbury, NJ, USA) and a photoinitiator (Irgacure 907, Shanghai Aladdin Biochemical Technology Co., Ltd., Shanghai, China) with a weight ratio of 98 to 2. The length, width, and gap of the LC cells (Hebei Yiya Electronics Co. Ltd., Shijiazhuang, China) were 3.0 cm, 2.0 cm, and 20 μm, respectively.

The γ-Fe₂O₃ nanoparticles coated with oleic acid (called MNPs, the same as below) were prepared by a chemically induced transition method, and MNPs were about spherical with an average size of about 10 nm [24]. The LC colloids containing MNPs were prepared as follows. First, the MNPs were added in hexane to form the pre-colloid. Then, the pre-colloid was added to the LC to form a mixture. Second, the mixture was heated to 60 °C to volatilize the hexane. Finally, the LC colloid was obtained with the MNPs’ mass fractions ϕ_m = 0.11%.

The PDLC films were prepared as follows. First, the LC E7 and LC colloids were mixed separately with the prepolymer at a weight ratio of 1 to 1 to form the PDLC mixtures. Second, the mixtures were heated to 85 °C, cooled to room temperature. Then, they were left for 24 h to form the two homogenous mixtures. Third, the heated mixtures (85 °C) were filled separately in LC cells by capillarity and cured with ultraviolet light (36.2 mW/cm² @ 325 nm) for 10 min at a 30-cm exposure distance. Consequently, the PDLC films undoped and doped MNPs were obtained, designated as the PDLC and M-PDLC samples, respectively.

2.2. Characterization and Measurement

The surface morphologies of the two samples were characterized by a scanning electron microscope (SEM, JSM-7100F, JEOL Ltd., Tokyo, Japan). Before the SEM analysis, the PDLC cells were punctured linearly and soaked in hexane for 48 h to extract the LC droplets. Then, the films were dried in a drying oven at 70 °C and plated with Au. The change in resistance with the frequency was measured using the precision LCR (E4980A, Agilent Technologies Inc., Beijing, China) (see Appendix A).

Figure 1 shows the schematic diagram of the electro-optical measurement. A light beam emitted from a He-Ne laser (1) (λ = 632.8 nm, Melles Griot Inc., Carlsbad, CA, USA) passed through an attenuator (2) (Zolix Inc., Beijing, China), a polarizer (3) (Thorlabs Inc., Newton, NJ, USA), a λ/4 wave plate (4) (λ = 633 nm, Thorlabs, Newton, NJ, USA), another polarizer (5) (Thorlabs Inc., Newton, NJ, USA), and a diaphragm (6) (Zolix Inc., Beijing, China), in turn, which provided a normal incident on a sample (7). The voltage applied to the sample (7) was a square-wave voltage with an adjustable frequency (0–50 Hz) controlled by a voltage amplifier (9) (Nantong Long Yi Electronic Technology Co., Ltd., Nantong, China). The distance between the sample (7) and a detector (8) (7ID230, SOFN Instruments Co., Ltd., Beijing, China) was 20 cm. The converted digital signals from a data acquisition card (10) (DAC, 6216, NI) were connected to the detector (8) and processed by a computer (11) in the LabVIEW environment. The applied square-wave voltage was 30 V_rms. Transmittance is defined as $T=I_t/I_i$, where $I_i$ and $I_t$ are the incident light and transmitted light intensities, respectively.

3. Results and Discussion

3.1. PDLC Sample with DC Applied Voltage

The evolution curve T-t of the PDLC sample with a 30-V DC voltage is shown in Figure 2. When the DC voltage was switched on and off, the transmittance increased rapidly to the maximum transmittance T_max and then gradually decreased to the off-state transmittance T₀, as illustrated in detail in insets (1), (2), and (3). This physical process was analyzed as follows.

When the DC voltage was off, LC droplets were randomly dispersed in the polymer matrix and the refractive index difference between the polymer and liquid crystal droplets (Δn) was the maximum value, with the off-state transmittance T₀. The local electric field relative to the applied voltage V and the film thickness d are expressed as [19,20]:

$$E_{loc}=\frac{V}{d}\left(\frac{3{\rho }_{LC}}{2{\rho }_{LC}+{\rho }_p}\right)$$

(1)

where ${\rho }_P$ and ${\rho }_{LC}$ are the polymer and LC resistivities, respectively. Bold E and italic E refer, respectively, to the vector and scalar of the electric field, the same as below.

When the DC voltage was switched on, the LC droplets could reorient towards the effective electric field $E_{\mathrm{eff}}$. It is more appropriate to discuss the reorientation as orientation force on LC droplets by field and not by voltage [19]. Therefore, the transmittance T changed with the effective electric field $E_{eff}$ and can be expressed as

$$T\propto E_{eff}=\left|E_{\mathrm{loc}}+E_{\mathrm{int}}\right|$$

(2)

In the effective electric field E_eff, the director of LC droplets tends to the direction of E_eff. If E_eff is high enough, the transmittance T can increase rapidly from T₀ to T_max; at the same time, E_int can be ignored because of the short time. Then, the maximum transmittance T_max corresponding to the maximum of the effective electric field $E_{eff,max}$ can be obtained:

$$T_{max}\propto E_{eff,max}\approx \left|E_{\mathrm{loc}}\right|$$

(3)

When E_loc is constant, E_int, which is formed by ions and is opposite to E_loc, will increase and E_eff will decrease gradually. As a result, the transmittance T decreases from T_max to T_min, i.e.,

$$T_{min}\propto E_{eff,min}=\left|E_{\mathrm{loc}}+E_{\mathrm{int},\max }\right|$$

(4)

when the maximum of internal polarization electric field, $E_{\mathrm{int},\max }=-E_{\mathrm{loc}}$ and $E_{eff,min}$ = 0. Consequently, $T_{min}=T_0$.

As shown in Figure 2, when the DC voltage was switched on, the transmittance T decreased rapidly, then gradually. The time from T_max to T₀, τ₀, was 2.046 s, and the time from T_max to 10% of T_max, τ₁₀, was 0.056 s, which is called the fast-response process, as shown in inset (1). The process corresponding to the time τ_10-0 (=τ₀ − τ₁₀ = 1.996 s) from 10% of T_max to T_min, as shown in inset (2), is called the slow-response process. When the time is more significant than τ₀, the corresponding process can be called the over-response process, the LC droplets return to their initial state, and T no longer changes with time.

When the DC voltage was switched off, the T-t curve was similar to what happened when the DC voltage was switched on, as shown in inset (3). These results show that E_int can reorient LC droplets [21]; when E_loc is off, the T_max can be expressed as:

$$T_{max}\propto E_{int, max}$$

(5)

When the DC voltage was switched on and off, the decreased processes from T_max to T_min corresponded to the E_int formation and depolarization [21]. The transmittance curves were consistent when the DC voltage was switched on and off, indicating that the E_int formation and depolarization times are the same.

3.2. PDLC Sample Applied with an AC Square Wave Voltage

When the applied AC square wave voltage frequency was 0.01–50 Hz, the LC droplets can reorient within the half-period (0.01–50 s). As shown in Figure 3, the transmittance period (T-period) corresponded to the half-period of the applied AC square wave voltage, i.e., the phenomenon of frequency doubling. From now on, the half-period of the applied voltage is expressed as T-period. The three frequency ranges of the T-period are addressed as follows.

3.2.1. Frequency Range 10–100 mHz

Within this frequency range, the T-period of the square wave voltage was 5 to 50 s and more than τ₀. As shown in Figure 3a–c, from 10 mHz to 100 mHz, the minimum transmittances $T_{min}^{\left(1\right)}$, $T_{min}^{\left(2\right)}$, $T_{min}^{\left(3\right)}$, and $T_{min}^{\left(4\right)}$ reached T₀, i.e., the LC droplets returned to the initial reorientation and the superscripts (1), (2), and (3), etc. indicate the numbers of the T-period. In addition, $T_{max}^{\left(1\right)}\approx T_{max}$ and ${\mathsf{\tau }}_{10}^{\left(1\right)}\approx {\mathsf{\tau }}_{10}$; $T_{min}^{\left(1\right)}/T_{max}^{\left(1\right)}$ does not change with frequency; $T_{max}^{\left(2\right)}>T_{max}^{\left(1\right)}$ and ${\mathsf{\tau }}_{10}^{\left(2\right)}>{\mathsf{\tau }}_{10}^{\left(1\right)}$. The measured data are listed in

Table 1. The data of the T-t curves of the PDLC film applied with the DC voltage and the AC.

	DC	10 mHz	20 mHz	100 mHz
${\displaystyle {\mathit{T}}_{\mathit{m}\mathit{a}\mathit{x}}^{\mathbf{\left(}\mathbf{1}\mathbf{\right)}}}$	16.81	17.21	17.21	17.21
${\mathit{T}}_{\mathit{m}\mathit{i}\mathit{n}}^{\mathbf{\left(}\mathbf{1}\mathbf{\right)}}$	0.41	0.42	0.42	0.44
${\mathit{T}}_{\mathit{m}\mathit{i}\mathit{n}}^{\mathbf{\left(}\mathbf{1}\mathbf{\right)}}\mathbf{/}{\mathit{T}}_{\mathit{m}\mathit{a}\mathit{x}}^{\mathbf{\left(}\mathbf{1}\mathbf{\right)}}$	0.024	0.024	0.024	0.026
${\mathsf{\tau }}_{\mathbf{10}}^{\mathbf{\left(}\mathbf{1}\mathbf{\right)}}$ (ms)	56	57	57	59
${\mathit{T}}_{\mathit{m}\mathit{a}\mathit{x}}^{\mathbf{\left(}\mathbf{2}\mathbf{\right)}}$		65.24	64.54	64.54
${\mathit{T}}_{\mathit{m}\mathit{i}\mathit{n}}^{\mathbf{\left(}\mathbf{2}\mathbf{\right)}}$		0.42	0.42	0.44
${\mathit{T}}_{\mathit{m}\mathit{i}\mathit{n}}^{\mathbf{\left(}\mathbf{2}\mathbf{\right)}}\mathbf{/}{\mathit{T}}_{\mathit{m}\mathit{a}\mathit{x}}^{\mathbf{\left(}\mathbf{2}\mathbf{\right)}}$ (×10−1)		0.064	0.065	0.068
${\mathsf{\tau }}_{\mathbf{10}}^{\mathbf{\left(}\mathbf{2}\mathbf{\right)}}$ (ms)		201	205	202
${\mathit{T}}_{\mathit{m}\mathit{a}\mathit{x}}^{\mathbf{\left(}\mathbf{2}\mathbf{\right)}}\mathbf{/}{\mathit{T}}_{\mathit{m}\mathit{a}\mathit{x}}^{\mathbf{\left(}\mathbf{1}\mathbf{\right)}}$		3.79	3.75	3.75

, and the experimental results are discussed as follows:

As shown in Figure 4a, the ion was uniformly distributed before applying $E_{\mathrm{loc}}^{\left(1\right)}$. Figure 4b shows that $E_{\mathrm{int}}^{\left(1\right)}$ can be formed entirely by applying $E_{\mathrm{loc}}^{\left(1\right)}$ because the time of the T-period was long enough. By the end of the first T-period, the internal depolarization electric field was maximized, and

$$E_{\mathrm{int},\max }^{\left(1\right)}=-E_{\mathrm{loc}}^{\left(1\right)}$$

(6)

$E_{\mathrm{eff}}^{\left(1\right)}=E_{\mathrm{loc}}^{\left(1\right)}+E_{\mathrm{int}}^{\left(1\right)}$. Therefore, $T_{min}^{\left(1\right)}=T_0$, according to Equation (4).

At the beginning of the second T-period, $E_{\mathrm{loc}}^{\left(1\right)}$ disappeared and $E_{\mathrm{loc}}^{\left(2\right)}$ was applied. In the meanwhile, $E_{\mathrm{int},\max }^{\left(1\right)}$ can be retained a moment because of relaxation and, as shown in Figure 4c, the effective electric field was maximized:

$$E_{eff,max}^{\left(2\right)}=\left|E_{\mathrm{loc}}^{\left(2\right)}+E_{\mathrm{int},\max }^{\left(1\right)}\right|=2E_{loc}^{\left(2\right)}$$

(7)

According to Equation (3), $E_{eff,max}^{\left(2\right)}=2E_{eff,max}^{\left(1\right)}$. Hence, $T_{\mathit{max}}^{\left(2\right)}$ was higher than $T_{\mathit{max}}^{\left(1\right)}$. When $E_{\mathrm{loc}}^{\left(2\right)}$ continued, $E_{\mathrm{int}}^{\left(1\right)}$ started depolarization until it was utterly depolarized. After $E_{\mathrm{int}}^{\left(1\right)}$ was inexistent, $E_{\mathrm{int}}^{\left(2\right)}$ began to form, as shown in Figure 4d. Then, the effective electric field in the second T-period can be expressed as:

$$E_{eff}^{\left(2\right)}=\left|E_{\mathrm{loc}}^{\left(2\right)}+E_{\mathrm{int}}^{\left(2\right)}\right|$$

(8)

When $E_{\mathrm{int}}^{\left(2\right)}\to -E_{\mathrm{loc}}^{\left(2\right)}$, $E_{eff}^{\left(2\right)}\to$ 0, with $T_{min}^{\left(2\right)}\to T_0$. With the maximum of the internal polarization electric field,

$$E_{int,max}^{\left(2\right)}=E_{loc}^{\left(2\right)}$$

(9)

Because $E_{loc}^{\left(1\right)}=E_{loc}^{\left(2\right)}$, from Equations (6) and (9), $E_{int,max}^{\left(2\right)}=E_{int,max}^{\left(1\right)}$. Similar to $E_{int,max}^{\left(1\right)}$, $E_{int,max}^{\left(2\right)}$ can be retained a moment in the third T-period. After applying with $E_{\mathrm{loc}}^{\left(3\right)}$, $E_{int}^{\left(2\right)}$ was depolarized gradually; then, $E_{\mathrm{int}}^{\left(3\right)}$ began to form, similar to Figure 4b. Consequently, the curves T⁽³⁾-t and T⁽²⁾-t were the same. There is the extra depolarization time of $E_{\mathrm{int}}^{\left(1\right)}$ in the second T-period; as a result, ${\mathsf{\tau }}_{10}^{\left(2\right)}>{\mathsf{\tau }}_{10}^{\left(1\right)}$. In addition, $T_{max}^{\left(2\right)}$ = $T_{max}^{\left(3\right)}$ = $T_{max}^{\left(4\right)}$, $T_{min}^{\left(2\right)}$ = $T_{min}^{\left(3\right)}$ = $T_{min}^{\left(4\right)}$, and ${\mathsf{\tau }}_{10}^{\left(2\right)}={\mathsf{\tau }}_{10}^{\left(3\right)}={\mathsf{\tau }}_{10}^{\left(4\right)}$ for the different frequencies, as shown in Figure 3a–c.

3.2.2. Frequency Range 0.5–2 Hz

Within this frequency range, the T-period was from 0.25 to 1 s and between τ₁₀ and τ₀, as shown in Figure 5a–c. The T-t curves within this frequency range were similar to 10–100 mHz. Moreover, $T_{max}^{\left(1\right)}\approx T_{max}$, $T_{min}^{\left(1\right)}\approx T_0$, and $T_{min}^{\left(1\right)}$ increased slightly with frequency. The characteristics can be explained as follows:

The fast-response was performed entirely in the first T-period, but the slow-response was not performed, i.e., $E_{\mathrm{int}}^{\left(1\right)}$ was not formed entirely due to the relatively short T-period. Therefore, $E_{int, max}^{\left(1\right)}$ was slightly less than $E_{loc}^{\left(1\right)}$, so that $E_{eff, min}^{\left(1\right)}\ne 0$ and $T_{min}^{\left(1\right)}$ was slightly higher than T₀. The maximum effective electric field of the second T-period was:

$$E_{eff,max}^{\left(2\right)}=E_{loc}^{\left(2\right)}+E_{int, max}^{\left(1\right)}\approx 2E_{loc}^{\left(2\right)}$$

(10)

The increases of $T_{max}^{\left(N\right)}$ and $T_{min}^{\left(N\right)}$ (N denotes the number of the T-period) with frequency can be related to the fact that the LC droplets could not entirely return to the initial random state. As with the frequency of 10–100 mHz, these T-t curves changed periodically from the second T-period and the frequency doubling remained.

3.2.3. Frequency Range 10–50 Hz

Within this frequency range, the T-period was from 0.01 to 0.05 s, less than τ₁₀. $T_{\mathit{min}}^{\left(N\right)}$ (N = 1, 2, 3, 4) was more than T₀, i.e., the LC droplets could not return to the initial random orientation at the end of each T-period, as shown in Figure 6. The T-t curves corresponded only to the fast response because the T-period was below τ₁₀, different from the over-response. Furthermore, $T_{max}^{\left(1\right)}$ increased with frequency and $T_{min}^{\left(1\right)}$ was more than 10% of $T_{max}^{\left(1\right)}$ for the three frequencies, i.e., $T_{min}^{\left(1\right)}>$ T₀. Besides, with the increase of frequency, $T_{max}^{\left(2\right)}$ decreased and $T_{min}^{\left(2\right)}$ increased. The measured data are listed in

Table 2. The data of the T-t curves of the PDLC film applied with an AC applied square voltage at 10 Hz, 20 Hz, and 50 Hz, respectively.

	10 Hz	20 Hz	50 Hz
${\mathit{T}}_{\mathit{m}\mathit{a}\mathit{x}}^{\mathbf{\left(}\mathbf{1}\mathbf{\right)}}$	18.96	19.58	20.44
${\mathit{T}}_{\mathit{m}\mathit{i}\mathit{n}}^{\mathbf{\left(}\mathbf{1}\mathbf{\right)}}$	2.82	8.92	19.53
${\mathit{T}}_{\mathit{m}\mathit{i}\mathit{n}}^{\mathbf{\left(}\mathbf{1}\mathbf{\right)}}\mathbf{/}{\mathit{T}}_{\mathit{m}\mathit{a}\mathit{x}}^{\mathbf{\left(}\mathbf{1}\mathbf{\right)}}$	0.149	0.455	0.955
${\mathit{T}}_{\mathit{m}\mathit{a}\mathit{x}}^{\mathbf{\left(}\mathbf{2}\mathbf{\right)}}$	78.32	77.48	72.64
${\mathit{T}}_{\mathit{m}\mathit{i}\mathit{n}}^{\mathbf{\left(}\mathbf{2}\mathbf{\right)}}$	35.69	53.78	68.20
${\mathit{T}}_{\mathit{m}\mathit{i}\mathit{n}}^{\mathbf{\left(}\mathbf{2}\mathbf{\right)}}\mathbf{/}{\mathit{T}}_{\mathit{m}\mathit{a}\mathit{x}}^{\mathbf{\left(}\mathbf{2}\mathbf{\right)}}$	0.456	0.695	0.939
${\mathit{T}}_{\mathit{m}\mathit{a}\mathit{x}}^{\mathbf{\left(}\mathbf{2}\mathbf{\right)}}\mathbf{/}{\mathit{T}}_{\mathit{m}\mathit{a}\mathit{x}}^{\mathbf{\left(}\mathbf{1}\mathbf{\right)}}$	4.131	3.957	3.554

. The experimental results are discussed as follows.

The resistance of the PDLC sample in this frequency range decreased with frequency (see Appendix A). Due to Equation (1), it can be assumed that ${\rho }_{\mathrm{p}}/{\rho }_{\mathrm{LC}}$ may decrease with frequency so that $E_{loc}^{\left(1\right)}$ and $T_{\mathit{max}}^{\left(1\right)}$ increase slightly with frequency. The T-periods were less than τ₁₀, so that the curves T-t can only be characterized by the fast-response and the maximum internal depolarized electric field of the first T-period:

$$E_{int, max}^{\left(1\right)}<E_{loc}^{\left(1\right)}$$

(11)

Thus, when applying $E_{\mathrm{loc}}^{\left(2\right)}$, the maximum effective electric field was:

$$E_{eff, max}^{\left(2\right)}=\left|E_{\mathrm{loc}}^{\left(2\right)}+E_{\mathrm{int},\max }^{\left(1\right)}\right|<2E_{loc}^{\left(2\right)}$$

(12)

Because $E_{loc}^{\left(2\right)}$ increased with frequency, $T_{\mathit{max}}^{\left(2\right)}$ in this frequency range was still more than that of 0.5–2 Hz. Different from the curves T-t in the frequency of 0.5–2 Hz, $T_{max}^{\left(\mathrm{N}\right)}$, $T_{min}^{\left(\mathrm{N}\right)}$, and $T_{min}^{\left(\mathrm{N}\right)}/T_{max}^{\left(\mathrm{N}\right)}$ (N = 1, 2, 3, …) increased with the number of the T-period. The process of this phenomenon is discussed for four cases, as follows.

(a)

In the first T-period

At the beginning of the first T-period, the ions were uniform; after applying $E_{\mathrm{loc}}^{\left(1\right)}$, the effective electric field was maximized. $T_{max}^{\left(1\right)}$ correspond to $E_{eff,max}^{\left(1\right)}=E_{loc}^{\left(1\right)}$. At the end of the first T-period, $E_{\mathrm{int}}^{\left(1\right)}$ was formed and opposite to $E_{\mathrm{loc}}^{\left(1\right)}$ and the effective electric field was minimized. $T_{min}^{\left(1\right)}$ corresponded to $E_{eff,min}^{\left(1\right)}=\left|E_{\mathrm{loc}}^{\left(1\right)}+E_{\mathrm{int},\max }^{\left(1\right)}\right|>0$, i.e., $E_{int,max}^{\left(1\right)}<E_{loc}^{\left(1\right)}$, and

$$T_{min}^{\left(1\right)}>T_0$$

(13)

Equation (13) shows that the LC droplets could not return to the initial random orientation and maintained a specific orientation at the end of the first T-period. This reserved orientation can be equivalent to being applied with an additional effective electric field, $E_{\mathrm{b}}^{\left(1\right)}$, which reoriented the LC droplets and did not change with the local electric field. Thus, $E_{\mathrm{b}}^{\left(1\right)}$ did not change with the reversal of the polarity of the local electric field and was inversely proportional to $E_{int}^{\left(1\right)}$. As the T-period got shorter, $E_{int}^{\left(1\right)}$ was close to 0 and $E_{\mathrm{b}}^{\left(1\right)}/E_{int}^{\left(1\right)}$ > 1. $E_{\mathrm{b}}^{\left(1\right)}$ was greater than the threshold electric field, a minimum electric field required to switch LC droplets, governed by the competition of electric and elastic torque in LC droplets [8,9].

(b)

In the second T-period

At the beginning of the second T-period, the remained $E_{\mathrm{int},\max }^{\left(1\right)}$ was not depolarized and in the same direction of $E_{\mathrm{loc}}^{\left(2\right)}$. As a result, the effective electric field was maximized, $T_{max}^{\left(2\right)}$ corresponded to $E_{eff,max}^{\left(2\right)}$, and

$$E_{eff,max}^{\left(2\right)}=\left|E_{\mathrm{loc}}^{\left(2\right)}+E_{\mathrm{int},\max }^{\left(1\right)}\right|+E_b^{\left(1\right)}$$

(14)

Because $E_{eff,max}^{\left(2\right)}>E_{eff,max}^{\left(1\right)}$, $\mathrm{then} T_{\mathit{max}}^{\left(2\right)}>T_{\mathit{max}}^{\left(1\right)}$.

At the end of the second T-period, $E_{\mathrm{int}}^{\left(1\right)}$ was depolarized and $E_{\mathrm{int}}^{\left(2\right)}$ was formed and opposite to $E_{\mathrm{loc}}^{\left(2\right)}$; consequently, the effective electric field was minimized, $E_{eff,min}^{\left(2\right)}$ corresponded to $T_{min}^{\left(2\right)}$, and

$$E_{eff,min}^{\left(2\right)}=\left|E_{\mathrm{loc}}^{\left(2\right)}+E_{\mathrm{int},\max }^{\left(2\right)}\right|+E_b^{\left(1\right)}$$

(15)

Because the T-period was less than τ₀, it was composed of the time depolarizing $E_{\mathrm{int}}^{\left(1\right)}$ and the time forming $E_{\mathrm{int}}^{\left(2\right)}$. In the first period, τ₀ consisted only of the time forming $E_{\mathrm{int}}^{\left(1\right)}$; hence, $E_{int, max}^{\left(2\right)}\approx E_{int, max}^{\left(1\right)}$ and $E_{eff,min}^{\left(2\right)}>E_{eff,min}^{\left(1\right)}$ and, as a result,

$$T_{min}^{\left(2\right)}>T_{min}^{\left(1\right)}$$

(16)

Because $T_{min}^{\left(2\right)}>T_0$, $\mathrm{then} E_{eff,min}^{\left(2\right)}\ne 0$, i.e., $E_{int,max}^{\left(2\right)}<E_{loc}^{\left(2\right)}+E_b^{\left(1\right)}$; a new equivalent electric field was regarded as $\Delta E_b$ and the total effective electric field was

$$E_b^{\left(2\right)}=E_b^{\left(1\right)}+\Delta E_b$$

(17)

Equation (17) means that the equivalent electric field had a transitive property.

(c)

In the third T-period

At the beginning of the third T-period, the remained $E_{\mathrm{int},\max }^{\left(2\right)}$ was not depolarized, was in the same direction of $E_{\mathrm{loc}}^{\left(3\right)}$, and the effective electric field was maximized, corresponding to $T_{max}^{\left(3\right)}$

$$E_{eff,max}^{\left(3\right)}=\left|E_{loc}^{\left(3\right)}+E_{int,max}^{\left(2\right)}\right|+E_b^{\left(2\right)}$$

(18)

Because $E_{int,max}^{\left(2\right)}\approx E_{int,max}^{\left(1\right)}$ and $E_b^{\left(2\right)}>E_b^{\left(1\right)}$, as a result, $E_{eff,max}^{\left(3\right)}>E_{eff,max}^{\left(2\right)}$ and $T_{\mathit{max}}^{\left(3\right)}$ > $T_{\mathit{max}}^{\left(2\right)}$.

At the end of the third T-period, $E_{\mathrm{int}}^{\left(2\right)}$ was depolarized and $E_{\mathrm{int}}^{\left(3\right)}$ was formed and opposite to $E_{\mathrm{loc}}^{\left(3\right)}$; consequently, the effective electric field was minimized, $E_{eff,min}^{\left(3\right)}$ correspond to $T_{min}^{\left(3\right)}$, and

$$E_{eff,min}^{\left(3\right)}=\left|E_{\mathrm{loc}}^{\left(3\right)}+E_{\mathrm{int},\max }^{\left(3\right)}\right|+E_b^{\left(2\right)}$$

(19)

In this T-period, $E_{\mathrm{int}}^{\left(2\right)}$ was depolarized, then $E_{\mathrm{int}}^{\left(3\right)}$ was formed, and $E_{int,max}^{\left(3\right)}\approx E_{int,max}^{\left(2\right)}$. Because $E_b^{\left(2\right)}>E_b^{\left(1\right)}$, $\left(E_b^{\left(2\right)}-E_{int,max}^{\left(3\right)}\right)>\left(E_b^{\left(1\right)}-E_{int,max}^{\left(2\right)}\right)$; as a result, $E_{eff, min}^{\left(3\right)}$ increased and then $T_{min}^{\left(3\right)}>T_{min}^{\left(2\right)}$.

(d)

In the fourth T-period

Similar to the third T-period, the effective electric field corresponding to $T_{max}^{\left(4\right)}$ was

$$E_{eff,max}^{\left(4\right)}=\left|E_{\mathrm{loc}}^{\left(4\right)}+E_{\mathrm{int},\max }^{\left(3\right)}\right|+E_b^{\left(3\right)}$$

(20)

$E_b^{\left(3\right)}>E_b^{\left(2\right)}$, the effective electric field corresponding to $T_{min}^{\left(4\right)}$ was

$$E_{eff,min}^{\left(4\right)}=\left|E_{\mathrm{loc}}^{\left(4\right)}+E_{\mathrm{int},\max }^{\left(4\right)}\right|+E_b^{\left(3\right)}$$

(21)

and $T_{max}^{\left(4\right)}>T_{max}^{\left(3\right)} and T_{min}^{\left(4\right)}>T_{min}^{\left(3\right)}$.

Consequently, the effective electric fields corresponding to $T_{max}^{\left(\mathrm{N}\right)}$ and $T_{min}^{\left(\mathrm{N}\right)}$ (N > 1) could be expressed, respectively, as

$$E_{eff,max}^{\left(N\right)}=\left|E_{\mathrm{loc}}^{\left(\mathrm{N}\right)}+E_{\mathrm{int},\max }^{\left(\mathrm{N}-1\right)}\right|+E_b^{\left(N-1\right)}$$

(22)

$$E_{eff, min}^{\left(N\right)}=\left|E_{\mathrm{loc}}^{\left(\mathrm{N}\right)}+E_{\mathrm{int},\max }^{\left(\mathrm{N}\right)}\right|+E_b^{\left(N-1\right)}$$

(23)

$T_{min}^{\left(N\right)}/T_{max}^{\left(N\right)}$ can be expressed as

$$T_{min}^{\left(N\right)}/T_{max}^{\left(N\right)}\propto \frac{E_{loc}^{\left(N\right)}+E_b^{\left(N-1\right)}-E_{int,max}^{\left(N\right)}}{E_{loc}^{\left(N\right)}+E_b^{\left(N-1\right)}+E_{int,max}^{\left(N-1\right)}}$$

(24)

From the results above, at the same frequency, $E_{loc}^{\left(\mathrm{N}\right)}$ was constant, $E_{int,max}^{\left(\mathrm{N}\right)}$ decreased, and $E_b^{\left(\mathrm{N}\right)}$ increased with the number of the T-period. Therefore, $E_{eff,min}^{\left(\mathrm{N}\right)}$ and $T_{min}^{\left(\mathrm{N}\right)}$ increased with the number of the T-period. As $E_b$ was higher than the electric field threshold, the response time to the local electric field decreased. Therefore, the $E_b$ increase will be more significant than the $E_{int,max}$ decrease with the number of the T-period. $T_{max}^{\left(N\right)}$ and $T_{min}^{\left(N\right)}$ increased with the number of the T-period. However, the increase of $T_{min}^{\left(N\right)}$ was greater than $T_{max}^{\left(N\right)}$; as a result, $T_{min}^{\left(\mathrm{N}\right)}/T_{max}^{\left(\mathrm{N}\right)}$ increased with the number of the T-period.

Because $\left(E_{int,max}^{\left(\mathrm{N}-1\right)}+E_b^{\left(\mathrm{N}-1\right)}\right)$ increased with the number of the T-period, $E_{eff, max}^{\left(N\right)}$ increased with the number of the T-period. $E_b^{\left(N\right)}$ did not change with the polarity reversal of the local electric field and was inversely proportional to $E_{int,max}^{\left(\mathrm{N}\right)}$. As the T-period got short, $E_{int}^{\left(1\right)}$ tended to 0 and $E_{\mathrm{b}}^{\left(1\right)}/E_{int}^{\left(1\right)}$ > 1. $\left(E_{int,max}^{\left(\mathrm{N}-1\right)}+E_b^{\left(\mathrm{N}-1\right)}\right)$ and $E_{eff,max}^{\left(\mathrm{N}\right)}$ tended to their maximums, i.e., in the mth period, $T_{max}^{\left(m\right)}$ corresponded to $E_{eff,max}^{\left(m\right)}$. In the nth period (n > m), $T_{max}^{\left(n\right)}=T_{max}^{\left(m\right)}$ and $T_{min}^{\left(n\right)}=T_{min}^{\left(m\right)}$, i.e., $T_{max}^{\left(n\right)}$ and $T_{min}^{\left(n\right)}$ were constant, as shown in the insets in Figure 6a–c.

In a word, the frequency doubling remained when the applied square wave voltage frequency was 0.01–50 Hz. However, the transmittance curves changed with the different frequency range. As the frequency was 10–100 mHz, from the second T-period, $T_{max}^{\left(\mathrm{N}\right)}$ and $T_{min}^{\left(\mathrm{N}\right)}$ were nearly identical, respectively. As the frequency was 0.5–2 Hz, $T_{max}^{\left(N\right)}$ and $T_{min}^{\left(N\right)}$ increased with frequency. As the frequency was 10–50 Hz, $T_{max}^{\left(\mathrm{N}\right)}$ and $T_{min}^{\left(\mathrm{N}\right)}$ increased with frequency and the number of the T-period.

3.3. MNPs’ Modulation in the M-PDLC Film

The experimental results showed that the curve T-t of the M-PDLC sample was similar to that of the PDLC sample, i.e., the frequency doubling was still available but the T-t curve was shifted, indicating the modulating effect of NMPs. The typical T-t curves are shown in Figure 7.

Observation of SEM (see Appendix B) shows that the microstructures of both samples, including the size and the distribution of the LC droplets, were almost identical. Consequently, the modulating effect of MNPs arose from the ionic adsorption effect of MNPs, instead of the microstructure [16]. The experimental results are explained as follows.

Figure 7a shows that the maximum transmittance of the M-PDLC sample, $T_{max}^{\prime }>T_{max}$, and the time from $T_{max}^{\prime }$ to 10% of $T_{max}^{\prime }$, ${\mathsf{\tau }}_{10}^{\prime }>$ τ₁₀, the minimum transmittance of the M-PDLC sample $T_{min}^{\prime }=T_{min}=T_0$. $T_{max}^{\prime }$ was identical when the DC voltage was switched on and off, indicating that the instantaneous maximum internal polarization electric field $E_{int,max}^{\prime }=E_{loc}^{\prime }$. The superscript $\prime$ denotes the corresponding parameters of the M-PDLC sample, the same as below.

As shown in Figure 7b, at 100 mHz, $T_{max}^{\left(\mathrm{N}\right)\prime }>T_{max}^{\left(\mathrm{N}\right)}$, ${\mathsf{\tau }}_{10}^{\left(\mathrm{N}\right)\prime }>{\mathsf{\tau }}_{10}^{\left(\mathrm{N}\right)}$, and $T_{min}^{\left(\mathrm{N}\right)\prime }=T_{min}^{\left(\mathrm{N}\right)}\left(=T_0\right)$. $T_{max}^{\left(\mathrm{N}\right)\prime }>T_{max}^{\left(\mathrm{N}\right)}$ can be attributed to the ionic adsorption by MNPs, which causes the ${\rho }_{\mathrm{LC}}$ to increase and the ${\rho }_{\mathrm{p}}/{\rho }_{\mathrm{LC}}$ to decrease. Consequently, the local electric field of the M-PDLC sample was $E_{loc}^{\left(\mathrm{N}\right)\prime }>E_{loc}^{\left(\mathrm{N}\right)}$, according to Equation (1). Because of the maximum transmittance corresponding to the maximum effective electric field and Equation (7), $T_{max}^{\left(\mathrm{N}\right)\prime }>T_{max}^{\left(\mathrm{N}\right)}$. The concentration of free ions in the LC droplets decreased after MNPs’ doping. The formation time of the internal polarization electric field $E_{int}^{\left(\mathrm{N}\right)\prime }$ was longer than $E_{int}^{\left(\mathrm{N}\right)}$. As a result, ${\mathsf{\tau }}_{10}^{\left(\mathrm{N}\right)\prime }>{\mathsf{\tau }}_{10}^{\left(\mathrm{N}\right)}$. Furthermore, $T_{min}^{\left(\mathrm{N}\right)\prime }=T_0$, indicating that within this T-period (2.5 s), $E_{int}^{\left(\mathrm{N}\right)\prime }$ can reach its maximum $E_{loc}^{\left(\mathrm{N}\right)\prime }$. Therefore, $T_{max}^{\left(N\right)\prime }$ will keep constant from the second T-period.

Figure 7c shows that $T_{max}^{\left(1\right)\prime }>T_{max}^{\left(1\right)}$ at 20 Hz. That was still caused by the increase in $E_{loc}^{\left(1\right)\prime }$. In addition, $T_{min}^{\left(1\right)\prime }>T_{min}^{\left(1\right)}$, which can be attributed to that the formation time of $E_{int}^{\left(1\right)\prime }$ being longer than $E_{int}^{\left(1\right)}$. Consequently, the maximum effective electric field $E_{eff, max}^{\left(1\right)\prime }>E_{eff, max}^{\left(1\right)}$. In addition, starting from the second T-period, $T_{max}^{\left(N\right)\prime }>T_{max}^{\left(N\right)}$ and $T_{min}^{\left(N\right)\prime }>T_{min}^{\left(N\right)}$. Thus $E_{eff, max}^{\left(\mathrm{N}\right)\prime }>E_{eff, max}^{\left(\mathrm{N}\right)}$ and $E_{eff, min}^{\left(\mathrm{N}\right)\prime }>E_{eff, min}^{\left(\mathrm{N}\right)}$. These experimental results may be related to the equivalent electric field $E_{\mathrm{b}}^{\prime }$ being dependent on the LC droplet orientation. According to Equations (23) and (24), $E_{loc}^{\left(N\right)\prime }>E_{loc}^{\left(N\right)}$, so that $E_{eff, max}^{\left(\mathrm{N}\right)\prime }$>$E_{eff, max}^{\left(\mathrm{N}\right)}$ and $E_{eff, min}^{\left(\mathrm{N}\right)\prime }>E_{eff, min}^{\left(\mathrm{N}\right)}$; as a result, $T_{max}^{\left(N\right)\prime }>T_{max}^{\left(N\right)}$ and $T_{min}^{\left(N\right)\prime }>T_{min}^{\left(N\right)}$.

The experiment results showed that the periodic change in transmittance tended to disappear when the frequency was above 100 Hz, but the transmittance increased to the maximum with time (see Appendix C). In addition, these results indicated that the internal polarized field was barely periodically changed due to the reduced T-period. However, the equivalent effective electric field can still increase with the number of T-period so that the LC droplets tended to the local field direction.

4. Conclusions

The electro-optic feature of the PDLC can be affected by an internal polarized electric field at 0–50 Hz, and the transmittance curves change with the different frequency ranges. When the local electric field was inverted, the relaxation of the internal polarized electric field resulted in the relaxation of the transmittance. As a result, the transmittance curves indicated the double frequency of the applied voltage. The internal depolarized electric field could offset the local electric field when the T-period was longer than the slow-response process. However, when the T-period time is only within the fast-response, the internal depolarized electric field cannot entirely form. Simultaneously, the LC droplets retained a specific orientation, equivalent to being applied with an additional effective electric field. The modulating effect of the PDLC doped with MNPs can prove the role of ions in the electro-optical response of PDLC.