Capillary force calculation method for driving liquid drops to move on wedge-shaped surface

Abstract

The invention discloses a capillary force calculation method for driving liquid drops to move on a wedge-shaped surface. The capillary force calculation method comprises the following steps: step 1, establishing a numerical model of the liquid drops moving on the wedge-shaped surface based on a lattice Boltzmann method of chemical potential; 2, real-time three-dimensional contact angle distribution is calculated based on the obtained density distribution data of the liquid drops in the movement process on the wedge-shaped surface; step 3, establishing capillary force calculation sub-models corresponding to the stages based on morphological evolution of a liquid drop contact line in the movement process; wherein the capillary force calculation sub-model is used for calculating evolution data of the net capillary force of the liquid drop along with time change in the whole movement process. The method is high in precision and firm in physical foundation, the calculation result is closer to the real physical process, the dynamic and non-uniform description capability is high, the complex conditions that the contact line form changes drastically, the contact line intersects with the boundary, the contact angle space is non-uniformly distributed and the like in the liquid drop movement process are effectively handled, and fine calculation of the dynamic capillary force is achieved.

Description

Technical Field

[0001] This invention relates to the fields of surface physics, microfluidics, fluid mechanics, and computer numerical simulation, specifically a method for calculating capillary force driving droplet motion on a wedge-shaped surface. Background Technology

[0002] The spontaneous directional motion of droplets on wedge-shaped surfaces holds significant promise for applications in microfluidics, enhanced heat transfer, and water collection. These wedge-shaped surfaces typically consist of a wedge pattern formed by hydrophilic regions and a surrounding hydrophobic background region, with the hydrophilic and hydrophobic regions exhibiting different static contact angles. The motion of droplets on these surfaces is primarily driven by non-equilibrium capillary forces induced by this geometric pattern. Therefore, developing a method for accurately and efficiently calculating the capillary forces acting on droplets on wedge-shaped surfaces is crucial for quantifying their motion mechanism, predicting their behavior, and optimizing surface design.

[0003] Currently, research on this topic mainly unfolds from three levels: experimental observation, numerical simulation, and theoretical analysis. These existing technologies have laid the foundation for understanding the phenomenon, but there are still shortcomings in achieving universal, efficient, and accurate capillary force calculation.

[0004] At the experimental research level, many studies have observed the velocity, displacement and morphological evolution of droplets by designing different wedge angles and wettability gradient patterns. For example, Alheshibri et al. (Alheshibri MH, et al. AppliedPhysics Letters, 2013, 102(17):174103) observed that droplets exhibited a "tadpole-like" motion on a wedge pattern; experiments by Chen et al. (Chen Y, et al. ACS Nano, 2020, 14(4):4654–4661) and subsequently Liu et al. (Liu M, et al. The Journal of Physical Chemistry C, 2019, 123(20):12736–12743) showed that increasing the wedge angle can improve the droplet transport speed; experiments by Wang et al. (Wang Y, Jin L. Surfaces and Interfaces, 2024, 46:104180) and Zhuang et al. (Zhuang J, Zheng J. Chemical Engineering Science, 2024, (283:119382) studied the enhancement effect of composite gradients on motion performance. These experiments qualitatively revealed the laws of parameter influence, but the experimental methods are mainly used for phenomenon observation and verification, and it is difficult to directly and conveniently provide complete quantitative data of transient capillary forces during motion, let alone use it as a universal calculation method to predict the force situation under different working conditions.

[0005] At the numerical simulation level, computational fluid dynamics or molecular dynamics can be used to simulate droplet motion on a full scale, providing detailed information on the flow and force fields. For example, Xu et al. (Xu B, Chen Z. MicrogravityScience and Technology, 2018, 30(4):571–579) used molecular dynamics simulations to study transport behavior under different gravitational fields; Nath et al. (Nath G, Ray B. Surfaces and Interfaces, 2024, 48:104235) used the lattice Boltzmann method to simulate the dynamics of evaporation; and Papadopoulou et al. (Papadopoulou E, et al. ACS Nano, 2019, 13(5):5465–5472) observed ultrafast transport phenomena at the nanoscale. However, these methods consume a lot of computational resources and are time-consuming. Although the results are relatively accurate, they are more like "simulation experiments" than "fast calculation methods", making it difficult to meet the application needs that require rapid evaluation or real-time analysis of a large number of parameter combinations.

[0006] At the theoretical modeling and analysis level, researchers have conducted a series of explorations to explain the driving mechanism. For example, Liu et al. (Liu M, et al. The Journal of Physical Chemistry B, 2020, 124(31):6905–6912) estimated the driving force by analyzing the staged morphology of droplets; in subsequent work, Liu et al. (Liu W, et al.ACS Omega, 2023, 8(18):16450–16458) further proposed a synergistic driving model that includes unwetting gradient force, surface tension difference induced by wedge structure, and Laplace pressure. Recently, Wang et al. (Experimental Thermal and Fluid Science, 2020, 114:110075) and Li et al. (ACS Applied Materials & Interfaces, 2021, 13(13):15857–15865) have attempted to provide a more quantitative description of the driving force by establishing a mechanical model and introducing a geometrically corrected wetting gradient force model, respectively. These theoretical works all highlight the importance of the difference between the advancing and retreating contact angles. However, these models typically require pre-assumptions or simplifications of the droplet contact line morphology or contact angle distribution (e.g., treating it as a constant value or a simple geometry), failing to fully couple the dynamic evolution and non-uniform distribution of the contact line during droplet movement. Furthermore, the calculation of capillary force is inherently highly dependent on the precise force balance of the local contact line. Therefore, existing theoretical models have limitations in terms of computational universality, accuracy, and dynamic description capabilities, and cannot yet form a reliable capillary force calculation method applicable to complex wedge-shaped surfaces and capable of handling dynamic processes.

[0007] In summary, while existing technologies have studied the motion of droplets on wedge-shaped surfaces from different perspectives, a gap remains in providing a method specifically designed for calculating dynamic capillary forces that combines physical accuracy with computational efficiency. Experimental observations cannot directly calculate forces, full-scale numerical simulations are too costly, and existing theoretical models suffer from insufficient accuracy and universality due to excessive simplification. This hinders a deeper quantitative understanding of this physical process and the precise design of related engineering projects. Summary of the Invention

[0008] The purpose of this invention is to address the common problems in existing methods for calculating the capillary force of droplets driven by wedge surfaces, such as insufficient accuracy, reliance on simplified assumptions about contact line morphology or contact angle distribution, and inability to handle dynamic non-uniform contact lines. This invention provides a new method for calculating the capillary force of droplets driven by wedge surfaces. This method boasts high accuracy and a solid physical foundation. It abandons simplified assumptions about contact line shape and contact angle distribution, directly obtaining key input parameters for capillary force calculation through real-time three-dimensional contact angle measurement, making the calculation results closer to the actual physical process. It exhibits strong dynamic and non-uniform descriptive capabilities. The proposed staged model effectively handles complex situations such as drastic changes in contact line morphology, intersection with boundaries, and spatial non-uniform distribution of contact angles during droplet motion, achieving refined calculation of dynamic capillary force.

[0009] The technical solution to achieve the objective of this invention is: The method for calculating the capillary force driving droplet motion on a wedge-shaped surface includes the following steps: Step 1: Establish a numerical model of the droplet motion on the wedge-shaped surface using the lattice Boltzmann method based on chemical potential; Step 2: Calculate the real-time three-dimensional contact angle distribution based on the density distribution data of the droplet moving on the wedge surface obtained from the numerical model; Step 3: Based on the morphological evolution of the droplet contact line during motion, the motion process is divided into three stages, and a capillary force calculation sub-model corresponding to each stage is established; wherein, the capillary force calculation sub-model calculates the evolution data of the net capillary force on the droplet during the entire motion process over time.

[0010] The equation for the lattice Boltzmann method based on chemical potential in step one is as follows: (1), Indicates at time Lattice position Up, along the discrete velocity direction The particle distribution function for i = 0, ..., N; This is the transformation matrix, used to linearly transform the distribution function to the velocity moment space, satisfying... and ; and These represent the velocity moment of the distributed function and the equilibrium state of the velocity moment of the distributed function, respectively. The flow field was established using the three-dimensional D3Q19 model. It is a diagonal matrix. ,in , Relaxation time is related to viscosity; The non-ideal force generated by the chemical potential gradient is the external force term. The lattice Boltzmann method based on chemical potential aims to simulate droplet motion systems containing both gas and liquid phases. To accurately describe the dynamic evolution of the droplet gas-liquid interface, a free energy functional is introduced to define the chemical potential, which is used to calculate the non-ideal forces during fluid phase transitions. The free energy functional includes a bulk free energy density term and an interface energy term related to the density gradient. The bulk free energy density is determined by an equation of state, which is selected from the van der Waals type or its modified form.

[0011] In step two, the real-time three-dimensional contact angle distribution is calculated based on the density distribution data of the droplet moving on the wedge-shaped surface obtained from the numerical model. Specifically: 2.1) Determine the three-phase contact point 'a' of the droplet on the first fluid calculation plane closest to the solid wedge surface, which is artificially set and fixed. For any given orientation along the droplet's circumference, i.e., with the droplet's centroid as the center, the positive x-axis direction (wedge opening direction) is 0°, and counterclockwise is the positive direction, slice the surface. Identify the adjacent liquid phase grid nodes and gas phase grid nodes at the gas-liquid interface in this given orientation, and perform interpolation calculations based on the coordinates and density values ​​of these two nodes to obtain the precise coordinates of the three-phase contact point. The formula used for the interpolation calculation is: (2), In equation (2), and These represent the coordinates of the liquid phase node and the gas phase node in the gas-liquid link, respectively. The average density of the gas and liquid phases; It is a unit direction vector pointing from the gas phase to the liquid phase; 2.2) Based on the three-phase contact point a and its adjacent contact points calculated in step 2.1), determine the contact line normal passing through point a, and define the normal plane perpendicular to this normal. 2.3) Locate the gas-liquid interface point b on the second layer plane within the defined normal plane; 2.4) Based on the geometric relationship between the three-phase contact point a, the gas-liquid interface point b, and the solid wedge surface within the normal plane, calculate the real-time contact angle at the slice orientation in step 2.1). The calculation formula is: (3), In formula (3) It is the contact angle. It is the x-coordinate of point a. It is the x-coordinate of point b.

[0012] In step 2.3), the method for locating the gas-liquid interface point b includes: projecting the three-phase contact point a onto the second layer plane to limit the search range, searching and determining the liquid phase point and the gas phase point in the normal direction within the search range, and then interpolating the coordinates of the gas-liquid interface point using formula (2).

[0013] In step three, based on the morphological evolution of the droplet contact line during motion, the motion process is divided into three stages, and capillary force calculation sub-models corresponding to each stage are established and calculated, specifically including: Determine which stage the droplet is in: 3.1) If the front contact line of the droplet extends within the wedge-shaped hydrophilic region and the rear contact line of the droplet is outside the wedge-shaped region, then the change in capillary force is calculated according to the corresponding capillary force calculation sub-model in the first stage. 3.2) If the contact line at the rear end of the droplet has entered the wedge region, the change in capillary force is calculated according to the corresponding capillary force calculation sub-model in the second stage; 3.3) If the droplet has completely entered the wedge region, the change in capillary force is calculated according to the capillary force calculation sub-model corresponding to the third stage.

[0014] In step 3.1), the capillary force is calculated based on the capillary force calculation sub-model corresponding to the first stage: The component of capillary force acting on a small arc segment of a continuous contact line in the predetermined direction of motion. Calculated by the following formula: (4), in, The surface tension coefficient is the gas-liquid surface tension coefficient. This is the static contact angle of the region where the infinitesimal arc segment is located. This represents the real-time dynamic contact angle at position x on the infinitesimal arc segment ds. The unit normal vector of the contact line of the infinitesimal arc segment. The unit vector is the preset direction of motion. Let be the arc length of the infinitesimal element; by integrating over the entire contact line, the net capillary force experienced by the droplet in this stage is obtained.

[0015] In step 3.2), the capillary force is calculated according to the corresponding capillary force calculation sub-model of the second stage: The contact line is divided into a leading edge arc segment located within the wedge-shaped hydrophilic region, a trailing edge arc segment located outside the wedge-shaped hydrophilic region, and lateral line segments located on the two sides of the wedge. The contribution of each sub-segment of the leading edge arc segment, trailing edge arc segment, and lateral line segment to the capillary force is calculated separately. The capillary force acting on these three contact line segments is used as... , and If it means: ,

[0016] ,

[0017] (5), in, The surface tension coefficient is the gas-liquid surface tension coefficient. The static contact angle within the wedge-shaped hydrophilic region. , , These represent the instantaneous contact angles on the trailing edge arc segment, the leading edge arc segment, and both sides of the droplet, respectively. The equilibrium contact angle is the angle between the two sides of the wedge that forms the heterogeneous surface region. The contribution of the lateral line segment needs to be considered in relation to the wedge angle. The geometric constraints project the capillary force on the lateral line segment onto the direction along the wedge-shaped opening, where the formula for the equilibrium contact angle is: (6), in The static contact angle within the wedge-shaped hydrophilic region. The static contact angle outside the wedge-shaped hydrophilic region. It is the geometric proportion factor of the area occupied by the hydrophilic region. It is the geometric scale factor of the area occupied by the hydrophobic region; The capillary forces calculated for each sub-segment of the leading edge arc segment, trailing edge arc segment, and lateral line segment are vector-synthesized in the direction of motion to obtain the total capillary force in the second stage, as shown in the formula: (7), in Indicates the wedge angle. , These are the angles between the positive x-axis direction and the capillary force direction of each infinitesimal arc segment on both sides of the droplet.

[0018] In step 3.3), the capillary force is calculated according to the capillary force calculation sub-model corresponding to the third stage: (8), in, The surface tension coefficient is the gas-liquid surface tension coefficient. It's a contact wire. It is the static contact angle within the wedge-shaped hydrophilic region. It is the real-time dynamic contact angle at position x on the infinitesimal arc segment ds. It is the unit normal vector of the contact line of the infinitesimal arc segment. The unit vector for the preset direction of motion.

[0019] The advantages of this technical solution lie in its high precision and solid physical basis. It abandons the simplistic assumptions about the shape of the contact line and the distribution of the contact angle, and directly obtains the key input parameters for capillary force calculation through real-time three-dimensional contact angle measurement. This makes the calculation results closer to the real physical process. It has strong dynamic and non-uniform description capabilities. The proposed staged model can effectively handle complex situations such as drastic changes in contact line morphology, intersection with the boundary, and non-uniform spatial distribution of contact angle during droplet motion, and realizes refined calculation of dynamic capillary force. Attached Figure Description

[0020] Figure 1 This is a flowchart of an embodiment of the present invention; Figure 2 This is a schematic diagram of droplet sliding on a solid wedge surface in an embodiment of the present invention; Figure 3 This is a schematic diagram of the real-time three-dimensional contact angle measurement method in an embodiment of the present invention, wherein (a) is a full view, (b) is a two-dimensional top view on the XY plane, and (c) is a two-dimensional side view in the normal plane; Figure 4 This is a diagram showing the contact angle measurement results in an embodiment of the present invention; Figure 5 The diagram shows the morphological evolution of the droplet sliding on the solid wedge surface in an embodiment of the present invention, wherein (a)-(f) are full views of the droplet at different time points, and (g) is a side view of the droplet motion; Figure 6 This is a schematic diagram for capillary force analysis in an embodiment of the present invention, wherein (a) shows the instantaneous contact angle in the XZ plane. and (b) The figure shows the instantaneous contact angle in the YZ plane. and Figures (c) and (d) show the configuration and capillary force distribution at various points on the contact line at different stages. Figure (c) shows stage 1, and Figure (d) shows stage 2. Figure 7 The following is a diagram showing the evolution of capillary force over time calculated in an embodiment of the present invention, wherein (a) is a capillary force diagram and (b) is a capillary force diagram for each sub-segment of stage 2. Detailed Implementation

[0021] The present invention will be further described below with reference to the accompanying drawings and embodiments, but this is not intended to limit the scope of the invention. Example

[0022] The method for calculating the capillary force driving droplet motion on a wedge-shaped surface includes the following steps: This embodiment of the method is used to accurately quantify the capillary force experienced by a droplet during spontaneous motion on a solid wedge-shaped surface. The method mainly includes: establishing a simulation environment and acquiring dynamic data, performing three-dimensional contact angle measurements, and constructing and applying a staged capillary force calculation model, such as... Figure 1 ; 1) Establishing the simulation environment and acquiring dynamic data: The multiphase flow lattice Boltzmann method (LBM) based on chemical potential is used to simulate the droplet motion process and obtain dynamic flow field data for subsequent force analysis. The execution flow of this method is as follows: Figure 1 As shown. The initial process settings are as follows: Figure 2 As shown, a three-dimensional computational domain (e.g., 600×256×200 grid units) is constructed. A wedge-shaped hydrophilic region is placed within this domain, surrounded by a hydrophobic background. The static contact angle of the hydrophilic region is... Set to 80°, static contact angle against a hydrophobic background. Set to 160°, wedge angle The angle is set to 30°. An initially spherical droplet (radius = 50 grid units) is placed at the tip of the wedge. Periodic boundary conditions are applied to the left and right sides of the computational domain, while other boundaries are non-slip walls.

[0023] The wettability of a wedge-shaped surface is achieved through chemical potential boundary conditions. For example... Figure 3 As shown, two layers of solid nodes (Z=2, Z=3 layers) are set on the surface of the solid wedge, as follows. Figure 3 As shown in (a). Different static contact angles are characterized by assigning specific chemical potential values ​​to these solid nodes, such as... Figure 3 (b) Figure 3 As shown in (c).

[0024] The Peng-Robinson equation of state is chosen to describe the bulk behavior of the fluid, and spatial scale transformation is used to sharpen the gas-liquid interface, thereby improving the resolution of subsequent contact angle measurements. The lattice Boltzmann equation is solved using the multiple relaxation time (MRT) collision operator to enhance numerical stability.

[0025] Run the simulation and record the dynamic data of the droplet's motion throughout the entire process, including but not limited to: the density field, velocity field, droplet center of mass position, and contact line position at each time step.

[0026] 2) Implement real-time three-dimensional contact angle measurement: to obtain the key input for capillary force calculation—the real-time dynamic contact angle of each point on the contact line. The data generated in step 1) needs to be post-processed.

[0027] The specific measurement process is as follows (in conjunction with...) Figure 3 principle): Determine the three-phase contact points: On the surface layer (Z=constant plane, such as...) Figure 3 (Z=2 layers) For any given azimuth angle around the droplet's centroid, on the gas-liquid interface (defined as an isosurface with density equal to the average density of the gas and liquid phases), find the liquid phase node and the nearest vapor phase node to the azimuth ray. Using the coordinates and density values ​​of these two points, calculate the precise coordinates of the three-phase contact point a at that azimuth angle using the linear interpolation formula (2).

[0028] Defining the normal plane: Using the contact point a and its two adjacent contact points, determine the normal to the contact line passing through point a, and then define the plane perpendicular to this normal, i.e., the normal plane γ (see...). Figure 3 a).

[0029] Locating the upper interface point: Within the normal plane γ, at the next higher layer (Z+ΔZ layer, such as...) Figure 3 On the gas-liquid interface of the middle Z=3 layer, search for and determine the point b located in the plane.

[0030] Calculating the contact angle: Within the normal plane γ, points a, b, and the solid wedge surface form a two-dimensional geometric relationship. Using this geometric relationship (e.g., calculating the angle between the line connecting points a and b and the normal to the solid surface), the real-time contact angle at that location can be calculated. (Right now ).

[0031] By repeating the above measurement process along the circumference of the contact line at regular angular intervals (e.g., 1°), the spatially non-uniform contact angle distribution covering the entire contact line range at each moment can be obtained. The contact angle results measured at different times are shown below. Figure 4 As shown.

[0032] 3) Construct and apply a staged capillary force calculation model: Based on simulation observations (such as...) Figure 5 The droplet motion process can be divided into three stages, such as... Figure 5 (a)- Figure 5 As shown in (g), corresponding capillary force calculation models need to be established for different stages, such as... Figure 6 As shown.

[0033] Stage 1: If the droplet's front contact line extends within the wedge-shaped hydrophilic region, and the droplet's rear contact line is outside the wedge region, then the change in capillary force is calculated according to the capillary force calculation model corresponding to Stage 1, such as... Figure 6 As shown in (c), the component of the capillary force acting on a certain infinitesimal arc segment ds on the contact line in the direction of motion (x direction). Calculated by the following formula: (4), in The gas-liquid surface tension coefficient is determined by simulation parameters. The static contact angle of the region where the infinitesimal arc segment ds is located: if it is located within the wedge-shaped hydrophilic region, = (e.g., 80°); if located in a hydrophobic background region, = (e.g., 160°). The real-time dynamic contact angle is located at position x on the infinitesimal arc segment ds. It is the unit normal vector of the contact line of the infinitesimal arc segment (pointing outwards from the droplet). Let be the unit vector along the wedge opening direction (i.e., the direction of motion). Integrating over the entire contact line C, we obtain the net capillary force for this stage: .

[0034] Phase 2: Divide the contact line into the leading edge arc segment within the wedge-shaped hydrophilic region, the trailing edge arc segment outside the wedge-shaped hydrophilic region, and the lateral line segments on both sides of the wedge. Calculate the contribution of each sub-segment to capillary force, such as... Figure 6 As shown in (d). The capillary force acting on these three contact segments is... , and If it means:

[0035]

[0036] (5), in, The surface tension coefficient is the gas-liquid surface tension coefficient. This is the static contact angle. , , These represent the instantaneous contact angles on arcs CF, ED, and DC, respectively. Figure 6 (a) and Figure 6 As shown in (b), The equilibrium contact angle is the angle between the two sides of the wedge that forms the heterogeneous surface region. The contribution of the lateral line segment needs to be considered in relation to the wedge angle. The geometric constraints project the capillary forces on the lateral line segments onto the direction along the wedge opening.

[0037] Total capillary force in stage 2 The vector sum contributing to each segment, with its component in the x-direction: (7), in Indicates the wedge angle. and These are the angles between the positive x-axis direction and the capillary direction of each infinitesimal arc segment of ED and CF, respectively.

[0038] The real-time contact angles corresponding to each segment obtained in step 2) are used as follows: Data, and known static contact angles 、 Surface tension coefficient wedge angle Substitute these values ​​into the formula (7) for the corresponding stage above to perform the calculation.

[0039] Total capillary force in stage 3 for: (8), in, The surface tension coefficient is the gas-liquid surface tension coefficient. It's a contact wire. It is the static contact angle within the wedge-shaped hydrophilic region. = α wedge (80°) It is the real-time dynamic contact angle at position x on the infinitesimal arc segment ds. It is the unit normal vector of the contact line on the infinitesimal arc segment. It is the unit vector in the x-direction.

[0040] By following steps 1) to 3), the capillary force evolution curve of the droplet from the start of its motion to its cessation can be calculated, as follows: Figure 7 As shown. Figure 7 (a) clearly demonstrates the overall trend of capillary force changes in the three stages, while Figure 7 (b) details the different contributions of capillary forces to the net force F2 on different contact segments in stage 2: the leading edge arc ED provides the main positive driving force, while the trailing edge arc CF has a reverse resistance, and the addition of lateral forces in the second stage further accelerates the droplet's motion. This precisely quantifies the mechanical effects of different parts of the contact line.

[0041] This example boasts advantages such as high computational accuracy, strong dynamic description capability, clear physical mechanism, and superior computational efficiency. It can accurately quantify the dynamic capillary force during the movement of droplets on a wedge-shaped surface, making it suitable for surface design and performance optimization in fields such as microfluidic chips, enhanced heat transfer, atmospheric water collection, printed electronics, and biomedical detection. It provides a reliable theoretical tool and design basis for related technologies.

Claims

1. A method for calculating capillary force driving droplet motion on a wedge-shaped surface, characterized in that, Includes the following steps: Step 1: Establish a numerical model of the droplet motion on the wedge-shaped surface using the lattice Boltzmann method based on chemical potential; Step 2: Calculate the real-time three-dimensional contact angle distribution based on the density distribution data of the droplet moving on the wedge surface obtained from the numerical model; Step 3: Based on the morphological evolution of the droplet contact line during motion, the motion process is divided into three stages, and a capillary force calculation sub-model corresponding to each stage is established; wherein, the capillary force calculation sub-model calculates the evolution data of the net capillary force on the droplet during the entire motion process over time.

2. The capillary force calculation method for droplet motion driven by a wedge-shaped surface according to claim 1, characterized in that, The equation for the lattice Boltzmann method based on chemical potential in step one is as follows: （1）， Indicates at time Lattice position Up, along the discrete velocity direction The particle distribution function for i = 0, ..., N; This is the transformation matrix, used to linearly transform the distribution function to the velocity moment space, satisfying... and ; and These represent the velocity moments of the distributed function and the equilibrium states of the velocity moments of the distributed function, respectively. The flow field was established using the three-dimensional D3Q19 model. It is a diagonal matrix. ,in , Relaxation time is related to viscosity; The non-ideal force generated by the chemical potential gradient is the external force term. The lattice Boltzmann method based on chemical potential introduces a free energy functional to define the chemical potential, which is used to calculate the nonideal forces in the fluid phase transition process; wherein, the free energy functional includes a bulk free energy density term and an interface energy term related to the density gradient; the bulk free energy density is determined by an equation of state, which is selected from the van der Waals type or its modified form.

3. The capillary force calculation method for droplet motion driven by a wedge-shaped surface according to claim 1, characterized in that, In step two, the real-time three-dimensional contact angle distribution is calculated based on the density distribution data of the droplet moving on the wedge-shaped surface obtained from the numerical model. Specifically: 2.1) Determine the three-phase contact point 'a' of the droplet on the first layer of fluid computational plane, which is artificially set and fixed near the solid wedge surface. For any given orientation along the droplet's circumference, i.e., with the droplet's centroid as the center and the positive x-axis direction as 0° (where the positive x-axis direction is the wedge opening direction, and counterclockwise is the positive direction), slice the surface. Identify the adjacent liquid and gas phase grid nodes at the gas-liquid interface in this given orientation, and perform interpolation calculations based on the coordinates and density values ​​of these two nodes to obtain the precise coordinates of the three-phase contact point. The interpolation calculation uses the following formula: (2)， In equation (2), and These represent the coordinates of the liquid phase node and the gas phase node in the gas-liquid link, respectively. The average density of the gas and liquid phases; It is a unit direction vector pointing from the gas phase to the liquid phase; 2.2) Based on the three-phase contact point a and its adjacent contact points calculated in step 2.1), determine the contact line normal passing through contact point a, and define a normal plane perpendicular to this normal. 2.3) Locate the gas-liquid interface point b on the second layer plane within the defined normal plane; 2.4) Based on the geometric relationship between the three-phase contact point a, the gas-liquid interface point b, and the solid wedge surface within the normal plane, calculate the real-time contact angle of the slice orientation in step 2.1). The calculation formula is: （3）， In formula (3) It is the contact angle. It is the x-coordinate of point a. It is the x-coordinate of point b.

4. The capillary force calculation method for droplet motion driven by a wedge-shaped surface according to claim 3, characterized in that, In step 2.3), the method for locating the gas-liquid interface point b includes: projecting the three-phase contact point a onto the second layer plane to limit the search range, searching and determining the liquid phase point and the gas phase point in the normal direction within the search range, and then interpolating the coordinates of the gas-liquid interface point using formula (2).

5. The method for calculating capillary force driving droplet motion on a wedge-shaped surface according to claim 1, characterized in that, In step three, based on the morphological evolution of the droplet contact line during motion, the motion process is divided into three stages, and capillary force calculation sub-models corresponding to each stage are established and calculated, specifically including: Determine which stage the droplet is in: 3.1) If the front contact line of the droplet extends within the wedge-shaped hydrophilic region and the rear contact line of the droplet is outside the wedge-shaped hydrophilic region, then the change in capillary force is calculated according to the corresponding capillary force calculation sub-model in the first stage. 3.2) If the contact line at the rear end of the droplet has entered the wedge-shaped hydrophilic region, then calculate the change in capillary force according to the corresponding capillary force calculation sub-model in the second stage; 3.3) If the droplet has completely entered the wedge-shaped hydrophilic region, the change in capillary force is calculated according to the capillary force calculation sub-model corresponding to the third stage.

6. The capillary force calculation method for droplet motion driven by a wedge-shaped surface according to claim 5, characterized in that, In step 3.1), the capillary force is calculated based on the capillary force calculation sub-model corresponding to the first stage: The component of capillary force acting on a small arc segment of a continuous contact line in the predetermined direction of motion. Calculated by the following formula: （4）， in, The gas-liquid surface tension coefficient, This is the static contact angle of the region where the infinitesimal arc segment is located. This represents the real-time dynamic contact angle at position x on the infinitesimal arc segment ds. The unit normal vector of the contact line of the infinitesimal arc segment. The unit vector is the preset direction of motion. Let be the arc length of the infinitesimal element; by integrating over the entire contact line, the net capillary force experienced by the droplet in this stage is obtained.

7. The capillary force calculation method for droplet motion driven by a wedge-shaped surface according to claim 5, characterized in that, In step 3.2), the capillary force is calculated according to the corresponding capillary force calculation sub-model of the second stage: The contact line is divided into a leading edge arc segment located within the wedge-shaped hydrophilic region, a trailing edge arc segment located outside the wedge-shaped hydrophilic region, and lateral line segments located on both sides of the wedge. The contribution of each sub-segment of the leading edge arc segment, trailing edge arc segment, and lateral line segment to capillary force is calculated respectively. The capillary force acting on the three contact segments—the leading edge arc segment, the trailing edge arc segment, and the lateral line segment—is used , and If it means: （5）， in, The gas-liquid surface tension coefficient, The static contact angle within the wedge-shaped hydrophilic region. , , These represent the instantaneous contact angles on the trailing edge arc segment, the leading edge arc segment, and both sides of the droplet, respectively. The equilibrium contact angle is the angle between the two sides of the wedge that forms the heterogeneous surface region. The contribution of the lateral line segment needs to be considered in relation to the wedge angle. The geometric constraints project the capillary force on the lateral line segment onto the direction along the wedge-shaped opening, where the formula for the equilibrium contact angle is: （6）， in The static contact angle within the wedge-shaped hydrophilic region. The static contact angle outside the wedge-shaped hydrophilic region. It is the geometric proportion factor of the area occupied by the hydrophilic region. It is the geometric scale factor of the area occupied by the hydrophobic region; The capillary forces calculated for each sub-segment of the leading edge arc segment, trailing edge arc segment, and lateral line segment are vector-synthesized in the direction of motion to obtain the total capillary force in the second stage, as shown in the formula: （7）， in Indicates the wedge angle, , These are the angles between the positive x-axis direction and the capillary force direction of each infinitesimal arc segment on both sides of the droplet.

8. The capillary force calculation method for droplet motion driven by a wedge-shaped surface according to claim 5, characterized in that, In step 3.3), the capillary force is calculated according to the capillary force calculation sub-model corresponding to the third stage: （8）， in, The gas-liquid surface tension coefficient, It's a contact wire. It is the static contact angle within the wedge-shaped hydrophilic region. It is the real-time dynamic contact angle at position x on the infinitesimal arc segment ds. The unit normal vector of the contact line of the infinitesimal arc segment. The unit vector for the preset direction of motion.

Images