An electrowetting display pixel decoupling modeling method based on contact angle correction

Abstract

The application relates to the technical field of display devices, and proposes an electrowetting display pixel decoupling modeling method based on contact angle correction. The method comprises the following steps: establishing a two-dimensional model of an electrowetting display pixel; establishing a two-phase flow phase field model in the two-dimensional model, and adopting phase field variables to represent a water phase, an ink phase and an oil-water interface; establishing a fluid motion equation of the two-phase flow phase field model, wherein a static electric volume force term formed by an applied driving voltage is not set; determining a water phase equilibrium contact angle based on the applied driving voltage, correcting the water phase equilibrium contact angle according to a contact angle reference direction of the ink phase to obtain a corrected equilibrium contact angle; taking the corrected equilibrium contact angle as a wetting boundary condition of a hydrophobic insulating layer surface, and performing phase field evolution processing based on the wetting boundary condition to obtain a motion state of the oil-water interface. The application reduces the repeated introduction of the applied driving voltage, avoids the deviation caused by the inconsistency of the contact angle reference direction, and improves the prediction accuracy of the motion state of the oil-water interface.

Description

Technical Field

[0001] This application relates to the field of display device technology, and more specifically, to a method for decoupling modeling of electrowetting display pixels based on contact angle correction. Background Technology

[0002] Electrowetting display is a reflective display technology based on the dielectric wetting effect. Electrowetting display pixels typically consist of an aqueous phase, an ink phase, and a hydrophobic insulating layer at the bottom of the pixel cavity. Under an applied driving voltage, the three-phase contact state between the aqueous phase, ink phase, and hydrophobic insulating layer changes, causing the ink phase to spread or contract along the surface of the hydrophobic insulating layer, thereby altering the pixel aperture state and achieving display control. Since the movement process of the oil-water interface directly affects the pixel aperture ratio, response speed, and grayscale stability, it is necessary to model the evolution process of the oil-water interface within the electrowetting display pixel during driving waveform design, pixel structure optimization, and display performance analysis to obtain the movement trend of the ink phase and the interface response state under the applied driving voltage.

[0003] Existing methods for modeling electrowetting display pixels typically employ multiphysics coupling to describe the effect of applied driving voltage on the motion of the oil-water interface. In these methods, on one hand, the electrostatic field effect corresponding to the applied driving voltage is converted into an electrostatic volume force term in the fluid motion equations to participate in solving the fluid motion of the water and ink phases; on the other hand, the contact angle is determined based on the correspondence between the applied driving voltage and the contact angle, and this contact angle is used as the wetting boundary condition of the hydrophobic insulating layer surface. Thus, the influence of the applied driving voltage on the oil-water interface is simultaneously introduced into the model through both the electrostatic volume force term in the fluid motion equations and the contact angle boundary condition of the hydrophobic insulating layer surface. However, in the above modeling approach, the effect of the applied driving voltage is easily over-included, leading to redundant coupling between the electrostatic volume force term and the wetting boundary condition, thereby affecting the accuracy of modeling the motion state of the oil-water interface. Meanwhile, when using phase field variables to characterize the aqueous phase, ink phase, and oil-water interface, the contact angle reference direction corresponding to the phase field variables may not be consistent with the theoretical correspondence between the applied driving voltage and the contact angle. If the contact angle determined based on the aqueous phase is directly used as the wetting boundary condition, the evolution direction of the oil-water interface is likely to be inconsistent with the actual electrowetting process.

[0004] Therefore, there is an urgent need for a decoupling modeling method for electrowetting display pixels to reduce the repetitive effect of the applied driving voltage on the fluid motion equation and wetting boundary conditions, and to match the contact angle boundary conditions of the hydrophobic insulating layer surface with the contact angle reference direction in the phase field variables. Summary of the Invention

[0005] The purpose of this invention is to address the shortcomings of the prior art by providing a contact angle-based decoupling modeling method for electrowetting display pixels, thereby solving the problem of insufficient accuracy in modeling the motion state of the oil-water interface of electrowetting display pixels in the prior art.

[0006] To achieve the above objectives, the technical solution adopted by the present invention is as follows: This application provides a method for decoupling modeling of electrowetting display pixels based on contact angle correction, the method comprising the following steps: S1. Establish a two-dimensional model of the electrowetting display pixel. The two-dimensional model includes an aqueous phase region, an ink phase region located below the aqueous phase region, and a hydrophobic insulating layer located below the ink phase region. S2. Establish a two-phase flow phase field model in the two-dimensional model, and use phase field variables to characterize the water phase, ink phase, and the oil-water interface between the water phase and the ink phase. S3. Establish the fluid motion equations for the two-phase flow phase field model, and do not include electrostatic volume force terms formed by the applied driving voltage in the fluid motion equations; S4. Determine the water phase equilibrium contact angle based on the applied driving voltage, and correct the water phase equilibrium contact angle according to the contact angle reference direction of the ink phase to obtain the corrected equilibrium contact angle. S5. The modified equilibrium contact angle is used as the wetting boundary condition of the hydrophobic insulating layer surface, and the phase field evolution is performed based on the wetting boundary condition to obtain the motion state of the oil-water interface.

[0007] This application establishes an electrowetting display pixel model comprising an aqueous phase region, an ink phase region, and a hydrophobic insulating layer, and establishes a two-phase flow phase field model within this model region, enabling the aqueous phase, ink phase, and oil-water interface to be characterized through phase field variables. Based on this, this application centrally maps the effect of the applied driving voltage on the oil-water interface to the wetting boundary conditions on the surface of the hydrophobic insulating layer, and omits the electrostatic volume force term generated by the applied driving voltage in the fluid motion equations, thus preventing the applied driving voltage from simultaneously acting repeatedly on the two-phase flow model through both the electrostatic volume force term and the wetting boundary conditions, thereby reducing redundant coupling in the model. For the aqueous phase contact angle determined by the applied driving voltage, this application corrects it based on the contact angle reference direction of the ink phase in the phase field variables to obtain a corrected equilibrium contact angle, ensuring that the application direction of the contact angle boundary conditions matches the phase definition of the phase field variables. After modifying the equilibrium contact angle as the wetting boundary condition of the hydrophobic insulating layer surface, the oil-water interface can undergo phase field evolution under the constraint of this wetting boundary condition. This allows for the acquisition of an oil-water interface motion state that matches the actual electrowetting process while reducing redundant coupling, thereby improving the accuracy of modeling the motion state of the oil-water interface in electrowetting display pixels.

[0008] Furthermore, in step S1, the two-dimensional model also includes an electrode disposed on the top layer of the aqueous phase region and pixel walls disposed on both sides of the ink phase region. By setting the top layer electrode and pixel walls in the model region, the spatial boundaries between the aqueous phase region and the ink phase region in the electrowetting display pixel can be clearly defined, and the pixel walls can limit the lateral movement range of the ink phase, thereby providing clear geometric constraints for the subsequent oil-water interface evolution.

[0009] Furthermore, the side of the pixel wall facing the ink phase region and the surface of the hydrophobic insulating layer are configured as slip boundary conditions. This configuration provides a boundary condition basis for the movement of the three-phase contact lines at the solid boundary, reducing the restriction imposed by the non-slip wall surface on the movement of the three-phase contact lines, thereby more accurately characterizing the shrinkage or spreading process of the ink phase under varying contact angles.

[0010] Furthermore, in step S2, the phase field variables include a first phase field value corresponding to the ink phase and a second phase field value corresponding to the water phase. The first and second phase field values ​​are different, and the oil-water interface is characterized by the transition region between the first and second phase field values. By setting the first phase field value corresponding to the ink phase and the second phase field value corresponding to the water phase, the ink phase and the water phase can be distinguished in the same two-phase flow phase field model. The oil-water interface is characterized by the transition region between the two phase field values, so that the interface position does not need to be obtained through manual division, but can be continuously expressed with the change of phase field variables, which is beneficial for describing the dynamic evolution process of the oil-water interface.

[0011] Furthermore, in step S2, the phase field variables are evolved based on the Cahn-Hilliard equation to obtain the interface morphology of the oil-water interface as it changes over time. Evolving the phase field variables using the Cahn-Hilliard equation can describe the migration, deformation, and shrinkage processes that occur at the interface between the aqueous and ink phases over time. This method is suitable for expressing the interface state with continuous transition characteristics between two-phase fluids and is beneficial for obtaining the transient morphological changes of the oil-water interface under different driving conditions.

[0012] Furthermore, in step S3, the fluid motion equations are the Navier-Stokes equations. The external force terms in the Navier-Stokes equations include surface tension terms but exclude electrostatic volume force terms formed by the Maxwell stress tensor. Using the Navier-Stokes equations to describe the fluid motion of the water and ink phases effectively reflects the relationship between the fluid velocities, pressures, and interfacial interactions between the two phases. Retaining the surface tension term in the external force terms and excluding the electrostatic volume force term formed by the Maxwell stress tensor avoids the applied driving voltage repeatedly entering the model through both electrostatic volume forces and wetting boundary conditions, thereby reducing the impact of redundant coupling on the oil-water interface motion state.

[0013] Furthermore, in step S4, the aqueous equilibrium contact angle is determined based on the Young-Lippmann relation. Determining the aqueous equilibrium contact angle using the Young-Lippmann relation establishes a correspondence between the applied driving voltage and the change in contact angle. Thus, the effect of the applied driving voltage on the electrowetting process is first transformed into a change in contact angle, and then further used to set the wetting boundary conditions, ensuring that the voltage action corresponds to the change in the wetting state at the hydrophobic insulating layer.

[0014] Furthermore, in step S4, the correction process is a supplementary angle correction. After adopting the supplementary angle correction, the contact angle obtained based on the aqueous phase can be converted into a corrected balanced contact angle that adapts to the reference direction of the ink phase contact angle, so that the wetting boundary conditions are consistent with the phase definition of the phase field variables.

[0015] Furthermore, after step S5, an interface dataset for the oil-water interface is extracted based on the contour lines of the ink phase volume fraction, and the positions of the three-phase contact lines are determined according to the coordinate extrema in the interface dataset. Extracting the interface dataset from the ink phase volume fraction contour lines allows the interface positions to be obtained from the phase field evolution results without relying on manual image reading. Determining the positions of the three-phase contact lines based on the coordinate extrema in the interface dataset quantifies the movement process at the intersection of the aqueous phase, ink phase, and hydrophobic insulating layer, thereby improving the accuracy and automation of the interface motion state analysis.

[0016] Furthermore, based on the positions of the three-phase contact lines and the designed pixel width of the electrowetting display pixels, the aperture ratio response information of the electrowetting display pixels is obtained. This aperture ratio response information, obtained from the positions of the three-phase contact lines and the designed pixel width of the electrowetting display pixels, can convert the motion state of the oil-water interface into a quantitative indicator related to the display effect. This response information can be used to analyze the degree of ink phase shrinkage, pixel aperture changes, and the dynamic response process under the action of driving voltage, thus providing a basis for optimizing the driving waveform and evaluating the performance of the electrowetting display pixels.

[0017] Compared with the prior art, the beneficial effects of the present invention are as follows: (1) In this application, the electrostatic volume force term formed by the applied driving voltage is not set in the fluid motion equation of the two-phase flow phase field model. Instead, the effect of the applied driving voltage on the oil-water interface is expressed through the wetting boundary condition of the hydrophobic insulating layer surface. This reduces the repeated introduction of the applied driving voltage in the fluid motion equation and wetting boundary condition, and reduces the influence of the redundant coupling between the electrostatic volume force term and the contact angle boundary condition on the motion state of the oil-water interface, thereby improving the stability and accuracy of the electrowetting display pixel modeling process.

[0018] (2) This application determines the water phase contact angle based on the correspondence between the applied driving voltage and the contact angle, and corrects the water phase contact angle according to the contact angle reference direction of the ink phase to obtain the corrected equilibrium contact angle, so that the wetting boundary conditions applied to the surface of the hydrophobic insulating layer can match the phase definition of the phase field variable, and avoid the oil-water interface evolution direction deviating from the actual electrowetting process due to the inconsistency of the contact angle reference direction, thereby more accurately characterizing the shrinkage or spreading state of the ink phase under voltage.

[0019] (3) This application modifies the equilibrium contact angle as the wetting boundary condition of the hydrophobic insulating layer surface, so that the oil-water interface can move according to the contact angle change in the phase field evolution process, thereby obtaining the oil-water interface movement state; at the same time, the oil-water interface data can be extracted based on the contour lines of the ink phase volume fraction, and the position of the three-phase contact line and the aperture ratio response information can be determined, so that the ink phase movement process can be further transformed from the interface morphology into a quantifiable display response index, providing a basis for the analysis of the electrowetting display pixel structure and the optimization of driving conditions. Attached Figure Description

[0020] Figure 1 A flowchart illustrating a contact angle-based decoupling modeling method for electrowetting display pixels provided by this invention; Figure 2 This is a two-dimensional model structure diagram of the electrowetting display pixel provided by the present invention; Figure 3 This is a diagram showing the ink phase motion state without contact angle correction. Figure 4 This is a diagram showing the motion state of the ink phase after applying the corrected equilibrium contact angle; Figure 5 A comparison curve showing the change of the position of the three-phase contact line over time before and after contact angle correction.

[0021] Icons: 1-Top electrode; 2-Pixel cavity; 3-Pixel wall; 4-Hydrophobic insulating layer. Detailed Implementation

[0022] To make the implementation process of this invention clearer, a detailed description will be provided below in conjunction with the accompanying drawings.

[0023] This invention provides a decoupling modeling method for electrowetting display pixels based on contact angle correction. This method establishes a two-phase flow field model of the aqueous phase, ink phase, and oil-water interface in the electrowetting display pixel. It converts the effect of the applied driving voltage on the oil-water interface into wetting boundary conditions on the surface of the hydrophobic insulating layer 4, preventing the applied driving voltage from repeatedly entering the fluid motion equations as an electrostatic volume force term. Through this method, the motion state of the oil-water interface, the position of the three-phase contact line, and the aperture ratio response information can be obtained while reducing redundant coupling between the electrostatic volume force term and the contact angle boundary conditions. Figure 1 As shown, the method includes the following steps.

[0024] S1. Establish a two-dimensional model of the electrowetting display pixel. The two-dimensional model includes an aqueous phase region, an ink phase region located below the aqueous phase region, and a hydrophobic insulating layer 4 located below the ink phase region. Specifically, in the modeling process, a two-dimensional single-pixel profile model of the electrowetting display pixel is established in multiphysics simulation software. A two-phase flow phase field physical field and a transient study method with phase initialization are selected. The two-phase flow phase field physical field is used to describe the evolution process of the aqueous phase, ink phase, and oil-water interface. The transient study method with phase initialization is used to solve the motion process of the oil-water interface over time based on a given initial phase distribution.

[0025] like Figure 2 As shown, the two-dimensional model includes a top electrode 1, a pixel cavity 2, pixel walls 3, and a hydrophobic insulating layer (HIL) 4. The aqueous phase region and the ink phase region are located within the pixel cavity 2, with the ink phase region positioned below the aqueous phase region. The hydrophobic insulating layer 4 is positioned below the ink phase region and at the bottom of the pixel cavity 2, while the pixel walls 3 are positioned on both sides of the ink phase region. The top electrode 1 is represented in the two-dimensional model by the upper boundary line of the pixel cavity 2, representing the upper electrode boundary in the electrowetting display pixel, and is not considered as a solid computational domain with thickness in the two-phase flow solution. The pixel walls 3 are used to define the lateral movement range of the ink phase, and the hydrophobic insulating layer 4 is used to form the wetting boundary between the aqueous phase, the ink phase, and the solid wall. This method does not introduce electrostatic volume forces into the fluid motion equation by solving the electrostatic field Poisson equation; the applied driving voltage corresponding to the top electrode 1 is transformed into the wetting boundary conditions of the surface of the hydrophobic insulating layer 4 through a subsequent contact angle mapping relationship. The width of the two-dimensional model region is set to 160 μm, and the height is set to 52 μm; the geometric and material parameters of the model region are shown in Table 1.

[0026] Table 1 Summary of Model Parameters and Material Parameters During model initialization, the ink phase region covers the hydrophobic insulating layer 4, and the aqueous phase region is located above the ink phase region, forming an initial oil-water interface between the aqueous and ink phases. The position of the initial oil-water interface is determined by the thickness and width of the ink phase region. The aqueous phase, ink phase, and hydrophobic insulating layer 4 form a three-phase contact line at the edge of the ink phase region. Initially, the fluid velocity u is set to 0, and the pressure p is set to 0, so that the subsequent transient solution starts from the static flow field and the preset phase distribution. It should be noted that the ink thickness in Table 1 is the initial set thickness of the ink phase region before phase initialization. In this embodiment, the ink thickness is set to 4 μm, and the height of the pixel wall 3 is set to 5 μm, so that the initial ink phase region is lower than the top of the pixel wall 3. After phase initialization, the ink phase and aqueous phase are affected by the fixed contact angle boundary condition at the sidewall of the pixel wall 3, and the oil-water interface will bend slightly near the sidewall, and the three-phase contact line will move slightly upward along the sidewall of the pixel wall 3. The aforementioned height difference allows for a margin in the phase initialization process, preventing the initial oil-water interface from coinciding with the top edge of pixel wall 3, which could lead to unstable phase initialization and ensure that subsequent transient solutions can start from a stable initial phase distribution.

[0027] Figure 2 The contact angles in the equation are used to define the initial wetting state at different solid boundaries. Specifically, θ0 = 30° represents the initial contact angle of the bottom ink phase on the surface of the hydrophobic insulating layer 4, used to determine the initial wetting state of the ink phase on the surface of the hydrophobic insulating layer 4 before the applied driving voltage; θ = 130° represents the fixed contact angle boundary conditions of the top boundary and the upper wall of the pixel wall 3, used to define the phase field boundary state at non-bottom wall surfaces; θ = 60° represents the fixed contact angle boundary condition of the side of the pixel wall 3 facing the ink phase region, used to define the contact state and boundary evolution direction of the oil-water interface near the side of the pixel wall 3. When setting the boundary conditions, the top boundary of the pixel cavity 2 and the upper wall of the pixel wall 3 are set to no-slip boundary conditions, the lateral boundary of the aqueous phase region is set to open boundary conditions, and the side of the pixel wall 3 facing the ink phase region and the surface of the hydrophobic insulating layer 4 are set to Navier slip boundary conditions. The non-slip boundary is used to define the fluid velocity state at the top wall and the wall above the pixel wall 3. The open boundary is used to reduce the restriction of the lateral boundary of the aqueous phase region on the flow field evolution. The Navier slip boundary is used to reduce the constraint of the solid wall on the movement of the three-phase contact line, so that the position of the three-phase contact line can change along the side of the pixel wall 3 and the surface of the hydrophobic insulating layer 4, thereby more accurately characterizing the shrinkage or spreading process of the ink phase under the change of contact angle.

[0028] S2. A two-phase flow phase field model is established in the two-dimensional model, and phase field variables are used to represent the water phase, ink phase, and the oil-water interface between the water and ink phases. In the phase field model, the phase field variable Φ is used to distinguish the ink phase, water phase, and the interface region between them. The phase field variable Φ has a first phase field value corresponding to the ink phase and a second phase field value corresponding to the water phase. The first phase field value and the second phase field value are different. The phase field variable in the ink phase region approaches the first phase field value, and the phase field variable in the water phase region approaches the second phase field value. The oil-water interface is represented by a continuous transition region between the first phase field value and the second phase field value. In this model, the first phase field value is set to 1, and the second phase field value is set to -1, that is, Φ=1 represents the ink phase, and Φ=-1 represents the water phase. Since the oil-water interface is located in the transition region between the two phase field values, when directly using the phase field variable Φ to extract the oil-water interface, the contour line with Φ=0 can be taken as the center position of the oil-water interface. With the above-mentioned phase field variable settings, the aqueous phase, ink phase, and oil-water interface can be continuously expressed in the same model. The position of the oil-water interface does not need to be obtained by manual division, but can be automatically updated as the phase field variables evolve.

[0029] The phase field variables are evolved based on the Cahn-Hilliard equation to obtain the time-varying interface morphology of the oil-water interface. The Cahn-Hilliard equation can be expressed as: Where Φ is the phase field variable, t is time, u is the fluid velocity, γ is the mobility, Ψ is the chemical potential, and λ is the mixing energy density. The thickness parameter is denoted by . Using the Cahn-Hilliard equation, the phase field variable can form a continuous transition region with a finite thickness between the aqueous and ink phases, which migrates and deforms over time, thus describing the dynamic evolution of the oil-water interface.

[0030] S3. Establish the fluid motion equations for the two-phase flow phase field model, without including electrostatic volume force terms generated by the applied driving voltage in the fluid motion equations; both the water phase and the ink phase are treated as incompressible fluids, and the Navier-Stokes equations are used to describe the two-phase fluid motion. The continuity equation can be expressed as: The momentum conservation equation can be expressed as: Where ρ is the fluid density, u is the fluid velocity, p is the pressure, I is the unit tensor, μ is the dynamic viscosity, and F is the external force acting on the fluid per unit volume.

[0031] In existing fully coupled modeling methods, the external force F typically includes surface tension. ,gravity and the electrostatic volume force generated by the electrostatic field ,Right now: The electrostatic volume force is typically calculated using the potential distribution and Maxwell's stress tensor, and is used as a source term in the fluid motion equations. In this method, the electrowetting display pixels are microscale structures, and gravity has a relatively small impact on the oil-water interface evolution. Furthermore, to reduce the repeated introduction of the applied driving voltage into the fluid motion equations and wetting boundary conditions, an electric field solution process for providing the electrostatic volume force to the fluid motion equations is not established within the computational domain, and the electrostatic volume force term formed by the Maxwell's stress tensor is not set as an external force source term in the Navier-Stokes equations. After decoupling, the external force term can be expressed as: Therefore, the fluid motion equations are mainly used to describe the motion of the aqueous and ink phases under the influence of velocity, pressure, and surface tension. The applied driving voltage does not participate in the momentum equation as a global fluid volume force, but rather enters the phase field evolution process through the wetting boundary conditions of the subsequent hydrophobic insulating layer 4. This approach reduces redundant coupling between the electrostatic volume force term and the contact angle boundary conditions, and minimizes the interference caused by repeated voltage application on the oil-water interface motion.

[0032] S4. The aqueous phase equilibrium contact angle is determined based on the applied driving voltage, and then corrected according to the reference direction of the ink phase contact angle to obtain the corrected equilibrium contact angle. The correspondence between the applied driving voltage and the aqueous phase equilibrium contact angle is determined based on the Young-Lippmann relation. The Young-Lippmann relation can be expressed as: in, The equilibrium contact angle of the aqueous phase. The initial contact angle at the surface of the hydrophobic insulating layer 4 before the application of the external driving voltage. The vacuum permittivity, σ is the dielectric constant of the hydrophobic insulating layer 4, and σ is the interfacial tension coefficient between the aqueous phase and the ink phase. 4 represents the thickness of the hydrophobic insulating layer, and V represents the applied driving voltage.

[0033] Young-Lippmann relation obtained Using the aqueous phase as a reference, this model represents the equilibrium contact state of the aqueous phase at the solid wall surface under an applied driving voltage. In the phase-field model, the ink phase corresponds to the first phase field value, and the water phase corresponds to the second phase field value. The wetting boundary conditions on the surface of the hydrophobic insulating layer 4 are applied according to the reference direction of the phase field value corresponding to the ink phase field. This is derived from the Young-Lippmann relation. Unlike the contact angle reference direction of the hydrophobic insulating layer 4 surface in the phase-field model, if directly... As the wetting boundary condition of the surface of the hydrophobic insulating layer 4, it is easy to cause the contact angle application direction to be inconsistent with the phase definition of the phase field variable, so that the evolution direction of the oil-water interface is inconsistent with the shrinkage trend of the ink phase in the actual electrowetting process.

[0034] Based on this, this method is used for the equilibrium contact angle of the aqueous phase. Supplementary angle correction is performed to obtain the corrected balanced contact angle. The corrected balanced contact angle can be expressed as: in, This is a corrected equilibrium contact angle that matches the reference direction of the ink phase contact angle. Through the above correction process, the equilibrium contact angle obtained based on the aqueous phase can be converted into a contact angle that adapts to the reference direction of the ink phase phase field value, ensuring that the wetting boundary conditions on the surface of the hydrophobic insulating layer 4 are consistent with the phase definition of the phase field variable. In other words, the applied driving voltage is first converted into the aqueous phase equilibrium contact angle using the Young-Lippmann relationship. Then, by using supplementary angle correction, it is converted into a corrected balanced contact angle under the ink phase reference direction. . The dynamic wetting boundary angle of the surface of the hydrophobic insulating layer 4 is used to replace the fixed initial contact angle. It participates in the subsequent transient solution.

[0035] S5. The modified equilibrium contact angle is used as the wetting boundary condition for the surface of the hydrophobic insulating layer 4, and phase field evolution is performed based on the wetting boundary condition to obtain the motion state of the oil-water interface. According to the principle of minimizing free energy, the phase field variable Φ satisfies the wetting boundary condition at the solid wall. The wetting boundary condition can be expressed as: Where n is the unit normal vector of the wall. Here, Φ is the interface thickness parameter, and Φ is the phase field variable. To correct the balanced contact angle, by... By introducing the above wetting boundary conditions, the oil-water interface contact state on the surface of the hydrophobic insulating layer 4 can be constrained by the modified equilibrium contact angle, so that the wetting state change corresponding to the applied driving voltage acts on the phase field evolution process in the form of boundary conditions.

[0036] In the transient solution process, the Navier-Stokes equations are used to obtain the velocity and pressure fields of the aqueous and ink phases, the Cahn-Hilliard equations are used to update the spatial distribution of the phase field variable Φ, and the wetting boundary conditions on the surface of the hydrophobic insulating layer 4 are used to constrain the direction of change and contact state of the phase field variable at the solid wall. As time progresses, the phase field variable Φ forms a continuous transition region between the aqueous and ink phases, which corresponds to the oil-water interface. By extracting the distribution state of the phase field variable Φ at different times, the positional and morphological changes of the oil-water interface and the contraction or spreading process of the ink phase along the surface of the hydrophobic insulating layer 4 can be obtained. Thus, the influence of the applied driving voltage on the oil-water interface is concentrated and mapped into the wetting boundary conditions on the surface of the hydrophobic insulating layer 4, which can reduce the repeated action of the applied driving voltage on the two-phase flow model through the electrostatic volume force term and the contact angle boundary conditions. At the same time, due to The contact angle reference direction of the ink phase in the phase field variable has been corrected, and the wetting boundary conditions can be matched with the phase definition of the phase field variable. This reduces the impact of inconsistent contact angle reference directions on the evolution direction of the oil-water interface, making the obtained oil-water interface motion state more consistent with the actual motion process of the ink phase in the electrowetting display pixel.

[0037] The motion state of the oil-water interface includes the positional distribution of the oil-water interface, the morphology of the ink phase boundary, the displacement of the three-phase contact line, and the aperture ratio response calculated from the three-phase contact line displacement at different times. To quantify the motion state of the oil-water interface, after obtaining the transient solution results, the phase field evolution results are post-processed to obtain the position of the three-phase contact line and the aperture ratio response information. Specifically, at each output time, the global phase field distribution dataset is used... Contour data points representing the oil-water interface are extracted, and the interface dataset is obtained through interpolation. .in, This represents the global phase field distribution dataset within the model region. This represents the interface dataset formed by contour lines at the oil-water interface. When the phase field variable Φ is directly used to represent the two-phase distribution, Φ=1 corresponds to the ink phase, Φ=-1 corresponds to the water phase, and the center position of the oil-water interface is represented by contour lines with Φ=0. When the post-processing results of the simulation software use the ink phase volume fraction, the ink phase volume fraction is calculated from the phase field variable, and the position where the ink phase volume fraction is 0.5 corresponds to the center position of the oil-water interface. Therefore, in the post-processing of the results, the interface dataset is obtained by extracting contour lines where the ink phase volume fraction is 0.5. .

[0038] In obtaining the interface dataset After that, with Coordinate extrema retrieval was performed using this as the data source. Based on the shrinkage direction of the ink phase along the surface of the hydrophobic insulating layer 4, the extreme values ​​of the abscissa located near the surface of the hydrophobic insulating layer 4 in the retrieved interface dataset were obtained to determine the position of the three-phase contact line at the corresponding moment. When the right-side three-phase contact line is extracted using the boundary of one pixel as the origin of the horizontal coordinate, the maximum value of the corresponding horizontal coordinate in the interface dataset is taken as the position of the three-phase contact line. The position of the three-phase contact line is used to characterize the movement of the intersection of the oil phase, water phase, and hydrophobic insulating layer 4 over time. The position of the three-phase contact line can be expressed as: in, Let be the coordinates of the three-phase contact line at time t. This represents the ordinate of the surface of the hydrophobic insulating layer 4. This formula is used to locate the lateral position of the intersection of the oil phase, water phase, and hydrophobic insulating layer 4 from the oil-water interface dataset, thereby obtaining the displacement information of the three-phase contact line over time.

[0039] After determining the positions of the three-phase contact lines, the aperture ratio response information is obtained based on the positions of the three-phase contact lines and the pixel design width W of the electrowetting display pixels. The aperture ratio response information can be expressed as: Where AR(t) is the aperture ratio at time t, and W is the pixel design width of the electrowetting display pixel. This formula converts the three-phase contact line position into the pixel aperture ratio, enabling the oil-water interface motion state to be further converted into a quantitative result related to the display response. Through the above processing, the oil-water interface dataset, three-phase contact line position, and aperture ratio response information can be obtained sequentially from the phase field evolution results, thereby enabling quantitative analysis of the dynamic response process of the electrowetting display pixel. In the result processing module of the simulation software, an oil-water interface contour dataset is established based on the solved phase field distribution results. The two-dimensional plotting results are used to display the ink phase volume fraction distribution at different times, the contour dataset is used to extract the oil-water interface position, and the extreme value retrieval results are used to display the coordinates of the three-phase contact line. Thus, the ink phase morphology diagram, the three-phase contact line position curve, and the aperture ratio response curve are all obtained by post-processing the same phase field evolution results.

[0040] To illustrate the effectiveness of this modeling method, a comparative analysis was conducted on the oil-water interface motion state before and after contact angle correction, under the same two-dimensional model, material parameters, initial phase distribution, and applied driving voltage. Except for the different representation of the contact angle on the surface of the hydrophobic insulating layer 4, the geometric structure, material parameters, initial conditions, solution time steps, and post-processing methods remained consistent before and after contact angle correction.

[0041] Figure 3 The ink phase movement state without contact angle correction is shown. Figure 4The simulation illustrates the ink phase motion state after applying a modified balanced contact angle. During the simulation, ink shrinkage requires a certain amount of time. Figure 3 and Figure 4 This is a simulation diagram showing the effect of ink shrinking to the target equilibrium contact angle. From Figure 3 It can be seen that before contact angle correction, the ink requires approximately 5ms to shrink to the target equilibrium contact angle. From Figure 4 It can be seen that the ink can shrink to the target contact angle in less than 3ms. This indicates that after the contact angle is optimized, the ink phase can undergo a shrinkage movement corresponding to the actual opening process of the electrowetting display pixel under the wetting boundary conditions on the surface of the hydrophobic insulating layer 4, resulting in a faster ink response speed.

[0042] Figure 5 The diagram shows a comparison of the three-phase contact line positions before and after contact angle correction over time. The solid black line represents the change in applied driving voltage over time. The driving waveform includes two phases: the driving voltage gradually decreases from 0-10 ms and gradually increases from 10-20 ms. Under this driving voltage, the ink phase should exhibit a spreading followed by contraction trend. The blue line represents the curve of the three-phase contact line positions before contact angle correction over time, and the red line represents the curve of the three-phase contact line positions after contact angle correction over time. Figure 5 It can be seen that after adopting the corrected equilibrium contact angle, the position of the three-phase contact line shows a trend of first increasing and then decreasing. This trend reflects the displacement response of the ink phase, which first spreads and then contracts under the action of the applied driving voltage. This result shows that by correcting the equilibrium contact angle of the water phase corresponding to the applied driving voltage to a corrected equilibrium contact angle that matches the reference direction of the ink phase contact angle, and using this corrected equilibrium contact angle as the wetting boundary condition of the surface of the hydrophobic insulating layer 4, the influence of the mismatch of the contact angle reference direction on the evolution direction of the oil-water interface can be reduced. This makes the obtained oil-water interface motion state more consistent with the actual opening process of the electrowetting display pixel, and more accurately reflects the motion trend of the ink phase with the change of driving voltage.

[0043] Meanwhile, since this method does not include an electrostatic volume force term formed by the applied driving voltage in the fluid motion equation, the effect of the applied driving voltage on the oil-water interface mainly enters the phase field evolution process through the wetting boundary conditions on the surface of the hydrophobic insulating layer 4. Therefore, the simulation results above show that this method can match the contact angle boundary conditions with the contact angle reference direction in the phase field variables while reducing the repeated introduction of the applied driving voltage, and further obtain information on the ink phase motion state, the position change of the three-phase contact line, and the aperture ratio response, thus providing a modeling basis for the interface motion analysis, driving waveform design, and display response prediction of electrowetting display pixels.

[0044] The above are merely preferred embodiments of the present invention and are not intended to limit the present invention. Various modifications and variations can be made to the present invention by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A method for decoupled modeling of an electrowetting display pixel based on contact angle correction, the method comprising: The method includes the following steps: ​ S1. Establish a two-dimensional model of the electrowetting display pixel, the two-dimensional model including an aqueous phase region, an ink phase region disposed below the aqueous phase region, and a hydrophobic insulating layer disposed below the ink phase region. S2. Establish a two-phase flow phase field model in the two-dimensional model, and use phase field variables to characterize the water phase, ink phase, and the oil-water interface between the water phase and the ink phase; S3. Establish the fluid motion equations of the two-phase flow phase field model, and the fluid motion equations do not include electrostatic volume force terms formed by the applied driving voltage; S4. Determine the water phase equilibrium contact angle based on the applied driving voltage, and correct the water phase equilibrium contact angle according to the contact angle reference direction of the ink phase to obtain the corrected equilibrium contact angle. S5. The modified equilibrium contact angle is used as the wetting boundary condition of the surface of the hydrophobic insulating layer, and phase field evolution processing is performed based on the wetting boundary condition to obtain the motion state of the oil-water interface.

2. The contact angle correction based electro wetting display pixel decoupled modeling method according to claim 1, characterized in that: In step S1, the two-dimensional model further includes an electrode disposed on the top layer of the aqueous phase region and pixel walls disposed on both sides of the ink phase region.

3. The contact angle correction based electro wetting display pixel decoupled modeling method according to claim 2, characterized in that: The side of the pixel wall facing the ink phase region and the surface of the hydrophobic insulating layer are set as slip boundary conditions.

4. The contact angle correction based electro wetting display pixel decoupled modeling method of claim 1, wherein: In step S2, the phase field variables include a first phase field value corresponding to the ink phase and a second phase field value corresponding to the water phase. The first phase field value is different from the second phase field value, and the oil-water interface is characterized by the transition region between the first phase field value and the second phase field value.

5. The contact angle correction based electro wetting display pixel decoupled modeling method according to claim 4, characterized in that: In step S2, the phase field variables are evolved based on the Cahn-Hilliard equation to obtain the interface morphology of the oil-water interface as it changes over time.

6. The contact angle correction based electro wetting display pixel decoupled modeling method according to claim 1, characterized in that: In step S3, the fluid motion equation is the Navier-Stokes equation, and the external force term of the Navier-Stokes equation includes the surface tension term, but does not include the electrostatic volume force term formed by the Maxwell stress tensor.

7. The contact angle correction based electro wetting display pixel decoupled modeling method according to claim 1, characterized in that: In step S4, the aqueous phase equilibrium contact angle is determined based on the Young-Lippmann relation.

8. The contact angle correction based electro wetting display pixel decoupled modeling method according to claim 7, characterized in that: In step S4, the correction process is corner compensation correction.

9. The contact angle correction based electro wetting display pixel decoupled modeling method of claim 1, wherein: After step S5, the interface dataset of the oil-water interface is extracted based on the contour lines of the ink phase volume fraction, and the position of the three-phase contact line is determined according to the coordinate extrema in the interface dataset.

10. The contact angle correction based electro wetting display pixel decoupled modeling method according to claim 9, characterized in that: Based on the positions of the three-phase contact lines and the pixel design width of the electrowetting display pixel, the aperture ratio response information of the electrowetting display pixel is obtained.

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