3D electrohydrodynamic simulation of electrowetting displays

In the phase-field formulation, the moving interface of the oil and water is set as a tiny nonzero-thickness transition region [22]. The physical properties at the interface of the immiscible fluids could be described by functions within this region with the help of a continuous phase-field variable ϕ, which varies from −1 for water to 1 for oil. From the introduced volumetric fractions ${{\overline{V}}_{\text{water}}}=(1-\phi )/2$ and ${{\overline{V}}_{\text{oil}}}=(1+\phi )/2$, the physical quantities within the transition region are given as

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} \rho ={{\rho}_{\text{water}}}{{{\overline{V}}}_{\text{water}}}+{{\rho}_{\text{oil}}}{{{\overline{V}}}_{\text{oil}}} \\ \mu ={{\mu}_{\text{water}}}{{{\overline{V}}}_{\text{water}}}+{{\mu}_{\text{oil}}}{{{\overline{V}}}_{\text{oil}}} \\ \varepsilon ={{\varepsilon}_{\text{water}}}{{{\overline{V}}}_{\text{water}}}+{{\varepsilon}_{\text{oil}}}{{{\overline{V}}}_{\text{oil}}}, \\ \end{array}\end{eqnarray} \tag{ 1 }$$

where ρ, μ, and ε represent the density, viscosity, and dielectric constant of fluids, respectively. In the diffusive-interface picture, the evolution of the interface between oil and water is governed by the Cahn–Hilliard convection equation [26]

$$\begin{eqnarray}&&\frac{{\partial \phi }}{{\partial t}} + {\mathbf{u}} \cdot \nabla \phi = \nabla \cdot (M\nabla G),\end{eqnarray} \tag{ 2 }$$

where u represents the fluid velocity, M denotes the mobility (or diffusion coefficient), and G is the chemical potential. The mobility can be expressed as $M=\chi h_{\text{PF}}^{2}$, where χ is the characteristic mobility and ${{h}_{\text{PF}}}$ is the capillary width that scales with the thickness of the diffuse interface in PFM. The value of ${{h}_{\text{PF}}}$ should be determined empirically [22], and we adopt ${{h}_{\text{PF}}}={{d}_{\text{oil}}}$ in the present simulation, where ${{d}_{\text{oil}}}$ is the oil film thickness. The chemical potential, which is a partial differential of the total free energy with respect to ϕ, could be expressed as $G=\lambda [-{{\nabla}^{2}}\phi +\phi ({{\phi}^{2}}-1)/h_{\text{PF}}^{2}]$, where λ is the energy density parameter. In addition, λ and ${{h}_{\text{PF}}}$ are related to the oil–water interfacial tension through the relation: ${{\gamma}_{\text{ow}}}=2\sqrt{2}\lambda /3{{h}_{\text{PF}}}$.

The static electric field in the hydrophobic dielectric layer, hydrophilic grid, oil phase, and water phase, is assumed to be governed by the Laplace equation

$$\begin{eqnarray}&&\nabla \cdot \left({{\varepsilon}_{0}}{{\varepsilon}_{\text{r}}}\nabla V\right)=0,\end{eqnarray} \tag{ 3 }$$

where ${{\varepsilon}_{0}}$ is the vacuum permittivity, ${{\varepsilon}_{\text{r}}}$ is the relative permittivity of the numerical domains including both solid dielectrics and fluids, and V is the electric potential. Under the framework of PFM, ${{\varepsilon}_{\text{r}}}$ at the interface between oil and water is also treated as a spatially continuous function, as expressed in equation (1). Here, the assumption of leaky dielectric (no electric charge density) in equation (3) is adopted to simplify the electrostatic equation of the water phase. This assumption, however, would lead to an overestimation (or underestimation) of the electric field in the water phase (or the oil phase). To overcome this problem, we then assign an extremely high value to the dielectric constant within the bulk of water phase. With such a high dielectric constant, the water phase will behave much like a conductive liquid (i.e. the electric field E = −V becomes very small inside the water domain [27]). This nearly conductive behavior of the water phase is compatible with the general treatment in an EW theory, in which the water is usually set to be perfectly conductive and the electric field vanishes within the water [1].

To depict the dynamic movement of the immiscible two-phase flow, the transport of mass and momentum are governed by the incompressible Navier–Stokes equations

$$\begin{eqnarray}&&\rho \left(\frac{\partial \mathbf{u}}{\partial t}+\mathbf{u}\cdot \nabla \mathbf{u}\right)=-\nabla p+\nabla \cdot \mu (\nabla \mathbf{u}+\nabla {{\mathbf{u}}^{T}})+\rho \mathbf{g}+{{\mathbf{F}}_{\text{S}}}\\ &&\nonumber\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, +{{\mathbf{F}}_{\text{E}}},\nabla \cdot \mathbf{u}=0,\end{eqnarray} \tag{ 4 }$$

where ρ and μ are the density and viscosity of the fluids, which take the form as in equation (1). p, g, F_s, F_E respectively denote the pressure, the gravitational acceleration, the volumetric surface tension, and the volumetric electrodynamic force generated by an electric field. Since the Bond number is much smaller than 1 in our simulation problems, the gravitational force in equation (4) can be ignored [1, 28]. In the PFM, F_S can be calculated over the computational domain in terms of the chemical potential and phase-field variable by

$$\begin{eqnarray}&&{{\mathbf{F}}_{\text{S}}}=G\nabla \phi .\end{eqnarray} \tag{ 5 }$$

Obviously, F_S approaches zero except those at the diffusive thickness of the oil–water interface. The volumetric electrodynamic force F_E, a net effect of an applied electric field acting on the fluids, can be evaluated through the Korteweg–Helmholtz electric force density formulation [29]

$$\begin{eqnarray}&&{{\mathbf{F}}_{\rm{E}}} = {\rho _{\rm{f}}}{\mathbf{E}} - \frac{1}{2}{E^2}\nabla \varepsilon + \nabla \left( {\frac{1}{2}{\mathbf{E}} \cdot {\mathbf{E}}\frac{{\partial \varepsilon }}{{\partial \rho }}\rho } \right),\end{eqnarray} \tag{ 6 }$$

where ${{\rho}_{\text{f}}}$ is the free electric charge density. Since the oil and water are assumed to be dielectric liquids, ${{\rho}_{\text{f}}}$ is equal to zero within the bulk. The second term of equation (6), a contribution from the polarization of the fluids, vanishes everywhere except at the interface. The last term of equation (6), the electrostriction force density, can be ignored for incompressible flow. Additionally, by neglecting electrostriction, F_E can be expressed by the divergence of the Maxwell stress tensor T^M

$$\begin{eqnarray}&&{{\mathbf{F}}_{\rm{E}}} = \nabla \cdot {{\mathbf{T}}^{\rm{M}}}.\end{eqnarray} \tag{ 7 }$$

In component expression, $T_{ij}^{\text{M}}$ is written as

$$\begin{eqnarray}&&T_{ij}^{\text{M}}=\varepsilon {{E}_{i}}{{E}_{j}}-\frac{\varepsilon}{2}{{\delta}_{ij}}{{E}_{k}}{{E}_{k}},\end{eqnarray} \tag{ 8 }$$

where ${{\delta}_{ij}}$ is the Kronecker delta function, and i, j = x, y, z. By adding the electrodynamic force into the laminar phase field calculation in the simulation, we are able to depict the effect of the electric field on the fluids domain and thus to characterize the dynamic movement of the oil–water interface.

When solving the electrostatic field from the Laplace equation (3), the zero charge condition (i.e. n·D = 0) is adopted for all of the exterior boundaries with the exception of the electrode regions. For the electrode regions, as plotted in figure 4, we specify the voltage V as its applied value ${{V}_{\text{app}}}$ on the patterned electrode and $V=0$ on the grounded electrode.

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In the formulation of PFM, the boundary conditions for the solid surfaces (e.g. hydrophobic surface, hydrophilic grid, and top substrate) are considered as wetted walls, and along the surfaces we specify a wetted contact angle ${{\theta}_{\text{w}}}$, which is related to ϕ through

$$\begin{eqnarray}&&{\mathbf{n}} \cdot \nabla \phi = {\rm{cos}}\,{\theta _{\rm{w}}}|\nabla \phi |,\end{eqnarray} \tag{ 9 }$$

where n is the unit vector normal to the wall. Across the wetted wall, the mass flux is zero, and therefore the no-slip boundary condition (i.e. u = 0) is used to associate with the momentum equation (4). In addition, the boundary condition at the four outlets for a unit pixel is chosen as $p=0$.

The electrowetted contact angle in this model is treated as follows. When the water phase contacts the hydrophobic layer (i.e. the traditional EW effect occurs), the electrically induced force at the contact line, obtained from the surface integral of the Maxwell stress, is expressed as [15, 23–25]

$$\begin{eqnarray}&&{{f}_{\text{e}}}=\frac{1}{2}{{d}_{\text{hyd}}}{{\varepsilon}_{0}}{{\varepsilon}_{\text{hyd}}}E_{\text{hyd}}^{2},\end{eqnarray} \tag{ 10 }$$

where the subscript 'hyd' denotes the hydrophobic dielectric layer, and d and E represent the thickness and electric field, respectively. Since E vanishes inside the domain of the water phase, the difference of electric potential across the hydrophobic dielectric layer is equal to the external applied voltage (i.e. $V={{E}_{\text{hyd}}}{{d}_{\text{hyd}}}$) [24]. Consequently, in view of equation (10) the force balance at the contact line leads to the Lippmann–Young equation:

$$\begin{eqnarray}&&\text{cos} {{\theta}_{\text{e}}}=\text{cos} {{\theta}_{\text{hyd}}}+\frac{{{\varepsilon}_{0}}{{\varepsilon}_{\text{hyd}}}{{V}^{2}}}{2{{d}_{\text{hyd}}}{{\gamma}_{\text{ow}}}},\end{eqnarray} \tag{ 11 }$$

where ${{\theta}_{\text{e}}}$ and ${{\theta}_{\text{hyd}}}$ are the electrowetted contact angle and Young's contact angle of the hydrophobic dielectric layer measured from the water phase, respectively.

Therefore, once the electric field is determined in the voltage-on state by solving the Laplace equation in equation (3), the electrodynamic force generated at the interface can be substituted into the Navier–Stokes equation, equation (4), to drive the fluids. As a result, the topological changes of the oil–water interface with a zero-value contour of the continuous phase field can be traced through the Cahn–Hilliard convection equation (equation (2)).

The correctness of the numerical approach can be validated by comparing the simulation result of the basic electro–optic behavior of an EWD with that obtained from a physical model [30]. According to this model, the white area fraction (WAF) of a pixel, a measure of electro–optic switching behavior, can be defined by

$$\begin{eqnarray}&&\text{WAF}(V)=\left[1-\frac{{{A}_{\text{oil}}}(V)}{{{A}_{0}}}\right],\end{eqnarray} \tag{ 12 }$$

where ${{A}_{0}}$ is a normalization factor used to define the threshold value of the base area at zero voltage. ${{A}_{\text{oil}}}(V)$ represents the base area of the spherical oil cap covered on the hydrophobic surface with given applied voltage V, and it can be expressed as

$$\begin{eqnarray}&&{{A}_{\text{oil}}}(V)=\pi [1-\text{co}{{\text{s}}^{2}} \varphi (V)]{{\left\{\frac{{{{\overline{V}}}_{\text{oil}}}}{\frac{2}{3}\pi \left[1-\frac{3}{2}\text{cos} \varphi (V)+\frac{1}{2}\text{co}{{\text{s}}^{3}} \varphi (V)\right]}\right\}}^{2/3}},\end{eqnarray} \tag{ 13 }$$

where ${{\overline{V}}_{\text{oil}}}$ is the volume of oil film dosed into the pixel, and $\varphi (V)$ is the cap angle of the spherical oil cap in the presence of voltage. This angle obeys the equation

$$\begin{eqnarray}&&\text{cos} \varphi (V)=1-\frac{C{{V}^{2}}}{2{{\gamma}_{\text{ow}}}}=1-\eta ,\end{eqnarray} \tag{ 14 }$$

which is obtained from the Lippmann–Young equation by assuming that the oil film covers the entire pixel without applying voltage (i.e. $\text{cos} \varphi \left(0\right)=1$). Here, C is the capacitance per unit area (F m⁻²), and η is the EW number, which denotes the ratio between the electrostatic energy per unit area and the interfacial tension. The minus sign of the electrical term in equation (14) indicates that the angle $\varphi (V)$ is measured from the oil phase rather than from the water phase. In addition, it should be noted that the angle $\varphi (V)$ is introduced here so that one can intuitively associate the cap angle φ(V) with the base area of the oil droplet in the physical model.

In this simulation we adopt the non-patterned electrode in a 141 μm width pixel, as shown in figure 4(a). The contact angle of grid is set to ${{\theta}_{\text{grid}}}=90{}^\circ$ (measured from the water phase), which ensures that the oil film in the pixel is geometrically flat in the absence of voltage. In addition, from the self-assembly dosing process used in our experiments, the oil thickness is equal to the grid height of 6 μm.

Figure 5(a) shows the dependence of WAF on the applied voltage for the theoretical (curve) and simulation (symbol) results with different values of capacitance. It can be observed that when ${{A}_{0}}$ is given as the area of the external circle of the square optical element, i.e. ${{A}_{0}}=\pi w_{\text{pixel}}^{2}/2$ (see figure 3(a)), the numerical data points fit well into the curves obtained from the physical model. The inset of figure 5(a) displays the dependence between WAF and η in both the simulation and the physical model. This dependent trend has also been found in a series of experiments [31–33].

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To quantitatively characterize the optical range and limitation of an EWD with different C values, a phase diagram is illustrated in figure 5(b), where the threshold voltage ${{V}_{\text{th}}}$ is the voltage required to initiate the movement of the oil film and the critical voltage ${{V}_{\text{c}}}$ means the voltages required to reach $\varphi \left({{V}_{\text{c}}}\right)=90{}^\circ$ (i.e. $\eta =1$). It is found that the numerical data points in figure 5(a), plotted as symbols, fit well in the phase diagram. Moreover, the electro–optic response of an EWD can be sorted into three states: the black state (i.e. $\text{WAF}(V)=0$), the grayscale state (i.e. $0 <\text{WAF}(V) < \text{WAF}\left({{V}_{\text{c}}}\right),$ with the critical voltage ${{V}_{\text{c}}}$), and the optical saturation state (i.e. $\varphi \left({{V}_{\text{c}}}\right)\geq 90{}^\circ$). In the black state the electrostatically induced EHD force (which increases as applied voltage increases) is insufficient to break up the oil film. Subsequently, once the water touches the hydrophobic surface (i.e. the EHD force exceeds the surface tension force), the EW effect leading to a grayscale operation then begins.

In this section, the modeling of the switch process of an EWD is presented and the importance of the electrowetted contact angle on optical performance is discussed.

3.2.1. Switch-on process.

For practical EWDs, a patterned electrode is required to drive the oil film toward a preferred direction. The rounded pattern is commonly used to prevent electrical breakdown at the electrode edges [10]. Here, a semicircle notched electrode with a width of 70.5 μm is adopted in the simulation, as sketched in figure 4(b). In addition, the pixel width is set as 141 μm, and the thickness and the dielectric constant of the hydrophobic dielectric layer are ${{d}_{\text{hyd}}}=$2 μm and ${{\varepsilon}_{\text{hyd}}}=$6, respectively. The applied voltage V is given as 40 V, which is sufficient to break up the oil film according to the analysis in section 3.1.

In the simulation, we set the wetted contact angle ${{\theta}_{\text{w}}}=90{}^\circ$ at the hydrophilic grid, and thus the oil–water interface is initially flat in the confined pixel. Figure 6 shows the simulated evolution of the oil phase concentrated to the corner from a film to a droplet during the switch-on process of about 0.45 ms. Obviously, the choice of the patterned electrode determines the direction of movement of the oil phase toward the corner.

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The time history of the electric field distribution in the diagonal section of the numerical domain (see figure 7(a)) is illustrated in figure 7(b). This result enables us to distinguish the break-up and the EW stages in the switch-on process. At the beginning of the break-up stage (t = 0 ms), the water phase does not touch the hydrophobic surface and the largest magnitude of the electric field is found at the oil phase, as shown in figure 7(b). Hence, corresponding to the electric filed, a downward (toward the oil phase) volumetric electro-dynamic force F_E calculated through equation (7) appears at the oil–water interface, as shown in figure 7(c). In the EW stage (from t = 0.05 to 0.45 ms), which begins from the time the water phase is in contact with the hydrophobic surface, a strong electric field can be found in the dielectric layer. This contact gives rise to a significant change in the oil phase contact angle. The electric field in the vicinity of the contact line within the oil phase is found to be relatively high in both the vertical and horizontal directions, as respectively shown in figures 7(c) and (d).

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3.2.2. Switch-off process.

After the oil film is contracted to the pixel corner, the switch-off process begins as soon as the voltage is removed. In this simulation, the initial phase field for the switch-off process is set as the final phase field of the switch-on process at t = 0.45 ms. Figure 8 shows the evolution of the oil phase from a droplet back to a film. The value of response time for the switch-off process is ${{t}_{\text{off}}}$ ~ 5.6 ms. We also find that the spreading speed of the oil phase at the beginning of the stage (from t = 0 to 0.3 ms) is relatively high, which is consistent with the experimental observation [7].

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3.2.3. Effect of electrowetted contact angle.

Owing to the force balance at the contact line in equation (11), the contact angle of the water phase should be deviated from ${{\theta}_{\text{hyd}}}=160{}^\circ$ in the simulation of the EW stage. When the oil is completely concentrated in the patterned pixel corner, the contact angle of the water phase ${{\theta}_{\text{e}}}=125{}^\circ$ (i.e. the angle measured from the oil phase is $55{}^\circ$) is observed, as shown in figure 9(a). This angle agrees with the value calculated through the Lippmann–Young equation (=$123.4{}^\circ$) [1]. However, if we try to keep a constant contact angle (${{\theta}_{\text{hyd}}}=160{}^\circ$) at the hydrophobic surface, the contact angle of the water phase is found to be $158{}^\circ$ (i.e. the angle measured from the oil phase is $22{}^\circ$) as shown in figure 9(b). In addition, figure 9(c) shows the time evolution of WAF during the switch-on process, and it can be seen that the highest WAF for the case of electrowetted contact angle is larger than that for the case of constant contact angle. This tells us that a correct description of the contact angle has an influence on the accuracy of the numerical model when we measure the optical performance of an EWD, such as the response time or the transmittance.

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The hydrophilicity of the pixilating grid is a key factor that is associated with several types of defects observed in EWDs, such as oil dewetting patterns [10] and oil overflow [17]. To avoid these defects, a large number of tests should be conducted to collect sufficient data on the contact angles of the grid ${{\theta}_{\text{grid}}}$ that cause the defects. Such collection, however, is time-consuming and some angles cannot be easily obtained from experiments. Our simulation model provides a useful alternative so that a parametric analysis can be made. Here, we choose two pixel sizes ${{w}_{\text{pixel}}}$ = 141 and 282 μm with their respective oil thickness ${{d}_{\text{oil}}}$= 6 and 12 μm, and the other parameters are the same as those in section 3.2.

3.3.1. Oil dewetting patterns.

The influence of the contact angle of the hydrophilic grid ${{\theta}_{\text{grid}}}$ on EWD optical performance is demonstrated in figure 10, where the symbols (${{d}_{\text{oil}}}$, ${{w}_{\text{pixel}}}$, ${{w}_{\text{grid}}}$) represent the oil film thickness, the pixel width, and the grid width, respectively. It can be seen that WAF (solid circle) decreases as ${{\theta}_{\text{grid}}}$ increases for both pixel sizes when ${{\theta}_{\text{grid}}}> 95{}^\circ$. If we keep increasing ${{\theta}_{\text{grid}}}$, up to ${{\theta}_{\text{grid}}}=120{}^\circ$ for example, a hole is found in the center of the pixel (see the right inset of figure 10), which will cause a lower value of WAF. Furthermore, we notice that the dewetted oil droplets, i.e. the residue droplets, can still be found in the pixel if ${{\theta}_{\text{grid}}}=95{}^\circ$. This is attributed to the fact that when ${{\theta}_{\text{grid}}}>90{}^\circ$ the initial profile of the oil film in the pixel presents the concave shape instead of a flat or spherical one. Consequently, once a voltage is exerted, the oil film will be easily broken up at its center (i.e. the thinnest region), resulting in the occurrence of multiple droplets (see the middle inset of figure 10). The simulation result agrees with the experimental observation, shown diagrammatically in the photo in figure 2(a), in which the contact angle of the hydrophilic grid ${{\theta}_{\text{grid}}}$ is equal to $100{}^\circ$. In contrast, when ${{\theta}_{\text{grid}}}\leq 90{}^\circ$, the oil film is successfully contracted into a single droplet, thus giving relatively high values of WAF, shown in the left inset of figure 10. Therefore, the condition ${{\theta}_{\text{grid}}}\leq 90{}^\circ$ is suggested such that the occurrence of oil dewetting patterns in the pixels can be avoided and a better optical performance can be obtained for the EWD.

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3.3.2. Oil overflow.

The hydrophilicity of the grid also plays a critical role in the occurrence of oil overflow. To quantify this undesirable defect, we introduce a dimensionless parameter $\alpha \left(={{w}_{\text{mog}}}/{{w}_{\text{grid}}}\right)$, where ${{w}_{\text{mog}}}$ is the furthest distance the oil can move on a grid. When α is greater than 0.5, the oil phase crosses the border of a unit pixel and overflows to the adjacent pixels. In this simulation, the same parameters as those in section 3.3.1 are adopted.

Figure 11 presents the effect of ${{\theta}_{\text{grid}}}$ on α. We see that α increases as ${{\theta}_{\text{grid}}}$ increases for all of the simulation cases. With the setting of (${{d}_{\text{oil}}}$, ${{w}_{\text{pixel}}}$, ${{w}_{\text{grid}}}$) = (12, 282, 20), it is found that α exceeds 0.5 when ${{\theta}_{\text{grid}}}\geq 105{}^\circ$. This explains the experimental finding of oil overflow in figure 2(b), where the contact angle of the hydrophilic grid ${{\theta}_{\text{grid}}}=120{}^\circ$ was used. The simulation also indicates that the higher the oil volume is dosed into the pixel, the lower the contact angle of the grid required to avoid oil overflow. From the simulation, we suggest that the contact angle for the cases (6, 141, 20), (6, 141, 10), (12, 282, 20), and (12, 282, 10) to avoid oil overflow should be less than $130{}^\circ$, $95{}^\circ$, $105{}^\circ$, and $90{}^\circ$, respectively.

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It is expected that an oil droplet should recover the whole pixel area as soon as the applied voltage is removed. Some factors are thought to be associated with the oil non-recovery defect, such as the hydrophobicity of the dielectric layer [10, 32] and the oil thickness. The latter factor is addressed in this section from a theoretical viewpoint followed by a simulation.

Consider a spherical oil cap with volume ${{\overline{V}}_{\text{oil}}}$, whose base radius ${{R}_{0}}$, i.e. the radius of the circular interface between the cap and the solid surface, can be expressed as

$$\begin{eqnarray}&&{{R}_{0}}= {{\left[\frac{3{{{\overline{V}}}_{\text{oil}}}\text{si}{{\text{n}}^{3}} \varphi}{\pi \left(2-3\text{cos} \varphi +\text{co}{{\text{s}}^{3}} \varphi \right)}\right]}^{1/3}},\end{eqnarray} \tag{ 15 }$$

where φ is the contact angle of the spherical oil droplet, which is related to the Young's contact angle of hydrophobic dielectric surface ${{\theta}_{\text{hyd}}}$ through $\varphi =\pi -{{\theta}_{\text{hyd}}}$. Based on the self-assembled oil dosing process [34] that is conducted in our experiment, the volume of oil film dosed into the square shaped pixel should be equal to ${{\overline{V}}_{\text{oil}}}={{d}_{\text{oil}}}w_{\text{pixel}}^{2}$. Now let's assume that ${{R}_{0}}$ should be larger than a threshold radius ${{R}_{\text{th}}}={{w}_{\text{pixel}}}/2$ such that the oil film could successfully recover the pixel at zero voltage. The selection of ${{R}_{\text{th}}}$ is a result of the experimental observation. In view of the assumption and equation (15), a criterion for the oil recovery is found to be

$$\begin{eqnarray}&&{{d}_{\text{oil}}}\geq \frac{\left(2-3\text{cos} \varphi +\text{co}{{\text{s}}^{3}} \varphi \right)\pi {{w}_{\text{pixel}}}}{24 \text{si}{{\text{n}}^{3}}\varphi}.\end{eqnarray} \tag{ 16 }$$

Accordingly, we are able to construct a phase diagram for ${{\theta}_{\text{hyd}}}$ and ${{d}_{\text{oil}}}$ in figure 12(a), in which the dashed lines are obtained from equation (16) with two sizes of pixel. This phase diagram helps us to predict the recoverable condition in terms of oil thickness and contact angle of the hydrophobic dielectric layer. For example, if ${{\theta}_{\text{hyd}}}=160{}^\circ$ is adopted, the threshold oil film thicknesses for the pixel sizes ${{w}_{\text{pixel}}}=141$ and 282 μm will then be about 5 and 10 μm, respectively. This could explain why the thicknesses we adopted for the simulations reported in the previous sections are 6 and 12 μm with the pixel sizes of 141 and 282 μm, respectively. In addition, the oil non-recovery observed from our experiment as shown in the diagrammatical photo in figure 2(c), where the oil thickness ${{d}_{\text{oil}}}$ = 8 μm, the pixel size ${{w}_{\text{pixel}}}$ = 282 μm, and the contact angle of the hydrophobic layer ${{\theta}_{\text{hyd}}}=160{}^\circ$ were used, agrees with the theoretical prediction.

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In order to be compatible with the theoretical analysis, a simulation is performed in which we adopt 141 μm pixel and the non-patterned electrode. The contact angles of the grid and the hydrophobic dielectric layer are fixed at $90{}^\circ$ and $160{}^\circ$, respectively. To investigate how the oil thickness ${{d}_{\text{oil}}}$ influences the oil recovery, five different oil thicknesses ${{d}_{\text{oil}}}$= 3 ~ 7 μm are used. As shown in figure 12(a), when ${{d}_{\text{oil}}}\geq$ 5 μm, the oil in the 141 μm pixel (symbol o) is recoverable after the voltage is removed. However, when ${{d}_{\text{oil}}}\leq$4 μm, the oil non-recovery defect is found (symbol x). The time evolutions of WAF in the process of switch-on (solid line) and switch-off (dashed line) for ${{d}_{\text{oil}}}$ = 6 μm and ${{d}_{\text{oil}}}$ = 4 μm are shown in figures 12(b) and (c), respectively. The above result indicates that as the pixel size and the hydrophobicity of the hydrophobic layer are specified, there exists a critical value of oil thickness, above which a complete spreading can be achieved. This can also be experimentally verified as illustrated in figure 2(b), where part of the oil in the bottom pixels overflows into the upper pixels, leading to the oil non-recovery defect in the bottom pixels when the voltage is turned off. This oil non-recovery defect comes from the fact that the remaining volume of the oil in the pixel is lower than the critical value.

In this section, we demonstrate how a highly reliable EWD is realized. As stated above, the requirement for the contact angle of the grid to avoid both oil dewetting patterns and oil overflow is ${{\theta}_{\text{grid}}}\leq 90{}^\circ$. However, it is very challenging in a real test to let the hydrophilic grid adhere firmly onto the hydrophobic surface [10]. Therefore, the hydrophilic grid of contact angle ${{\theta}_{\text{grid}}}=90{}^\circ$ is suggested. As for the 282 μm pixel size, the thickness of the grid is chosen as 12 $\mu \text{m}$ so that the defect of oil non-recovery can be avoided. In what follows, the fabrication processes of the EWD are briefly introduced, and then the electro–optic performance of the EWD is presented in experimental and numerical terms.

3.5.1. Fabrication processes.

3.5.2. Electro–optic performance: simulation and experiment.

The electro–optic performance (η versus WAF) of the designed EWD is experimentally and numerically presented in figure 13. In the experiment, the threshold voltage ${{V}_{\text{th}}}$ is found to be 2.8 V, and thus the corresponding η is about 0.008. As shown in the inset of figure 13, the successful switch-on process without oil overflow and without dewetted droplets (i.e. the oil was fully contracted to the pixel corner from a film to a droplet) is observed in both the simulation and the experiment. To summarize, the agreement of the electro–optic performance between the experiment and the simulation in figure 13 lies in a correct description of the electro/dual phase coupling and the change in the apparent contact angle at the contact line in the simulation.

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