Modeling of two-scale array microstructure and prediction of apparent contact angle based on WEDM

The core issue in the theoretical design of superhydrophobic surfaces is to elucidate the relationship between surface microstructure and wettability. In this paper, four two-scale array microstructure models with different shapes are proposed to predict the apparent contact angles (CAs) of EN-GJL-250 (Grey Cast Iron) surface under different wire-cut electrical discharge machining (WEDM) discharge parameters. The primary microstructure is the artificially designed semicircle, rectangular, sawtooth and square column array microstructure, and the secondary microstructure is the micro-nanostructure of the surface machined by WEDM. Firstly, the mechanical analysis of water droplets placed on the primary array microstructure is carried out by the force balance method. According to Newton’s third law, equations containing CA parameters are listed. Secondly, the finite element method (FEM) was used to analogue simulation the distribution of material surface temperature and flow field in the WEDM single-pulse discharge machining process. The surface equivalent geometric model of WEDM was established by a uniform arrangement of single pulse pits obtained by simulation, the surface roughness coefficients under different discharge parameters were calculated, and the influence of the surface roughness coefficients on the surface wettability was studied. The numerical results show that the CA is jointly determined by the surface roughness coefficient under different discharge parameters, and the shape and size parameters of the microstructure. Finally, semicircle, rectangular, sawtooth and square column array microstructures were fabricated on EN-GJL-250 surface by high-speed WEDM. Experimental results show: The CA of water droplets on the square column array microstructure is the largest, the predicted values of the CA were in good agreement with the experimental values, and the average relative error was 7.83%.


Introduction
Metallic materials are widely used in construction, ships, bridges, power and other industries. However, in the use process, metal materials due to corrosion, icing and other problems affect the performance of equipment [1,2], and even safety risks. The reasons for the above-hidden dangers are attributed to an important feature of solid surface -wettability [3][4][5], which is jointly determined by the chemical composition and microscopic geometric structure of material surface [6,7]. For a long time, researchers have shown great interest in how to control the micro-nanostructure of solid materials to control their wetting properties (such as superhydrophobicity). Superhydrophobic surface refers to a kind of special surface with CA greater than 150° and rolling angle less than 10°. The construction of the superhydrophobic surface on the metal substrate can effectively alleviate the problems such as non-corrosion resistance and easy icing in the use process, and can also endow it with special functions such as self-cleaning, oil-water separation, lubrication and drag reduction [8][9][10][11][12]. However, it is a fundamental problem to obtain and maintain superhydrophobic properties on metal surfaces: how to subjectively regulate the superhydrophobic properties of material interfaces? Since most metal materials have high surface energy [13] and show obvious hydrophilicity, it is more difficult to prepare superhydrophobic surfaces with metal as the substrate than with glass and low surface energy polymers.
The preparation of superhydrophobic surface includes the construction of rough microstructure and the reduction of surface free energy. At present, the methods for preparing micro-nanostructures on the metal surfaces include the template method [14], nano-spraying method [15], electrochemical machining method [16], etching method [17,18], sol-gel method [19], etc. However, these methods are complex operations or expensive equipment, which limits the industrial production of metal-based superhydrophobic surfaces. Wire-cut electrical discharge machining (WEDM) is a special machining method widely used in industry. It has the advantages of mature equipment, high process reliability, high automation, low processing cost and environmental protection. Importantly, WEDM is easier to fabricate metal surfaces with both micron and nano-composite structures, and the size of the microstructure can be controlled by adjusting the processing parameters, which provides the possibility of manufacturing multi-scale microstructures [20][21][22][23][24][25]. For example, Xiao et al. [26] processed the array microcolumn structure on the surface of aluminum alloy by WEDM, and then deposited nanoparticles on the microcolumn combined with an electrochemical method to directly obtain the superhydrophobic surface. Xiong et al. [24] constructed a micro-nanostructure by HS-WEDM, and then immersed the workpiece in stearic acid ethanol solution to prepare the superhydrophobic copper substrate surface. Wu et al. [27] prepared sinusoidal and rectangular microstructures on copper alloy surfaces by WEDM. It was found that due to the special material removal method of WEDM, the surface of the workpiece after machining has a multi-scale microstructure, which improves the hydrophobicity of the copper alloy surface. The maximum contact angles of sinusoidal and rectangular structures can reach 152.1° and 149.1°, respectively. Dong [28] prepared the micro-nanohierarchical structure on the surface of beryllium copper alloy by micro-EDM and carbon deposition. Oil treatment was then performed to enhance the surface strength. The prepared surface exhibits excellent superhydrophobic properties.
The preparation method of the superhydrophobic surface has been relatively mature, but the theoretical research is still lagging behind. The superhydrophobic properties of the prepared surface can only be measured by instruments. To obtain the ideal superhydrophobic surface, a series of size parameter experiments are often needed, which increases the cost of preparing the superhydrophobic surface. A theoretical model was established to predict the CA of the metal materials. By changing the shape and size of the surface microstructure, the surface wettability can be adjusted according to the actual needs, to guide the preparation of superhydrophobic surfaces [31,32]. Taking spherical microstructure as an example, Extrand and Moon [29] studied the influence of spherical structure parameters on surface wettability by force balance theory. The results show that decreasing the sphere radius is beneficial to increasing the droplet contact angle, and increasing the sphere radius breaks the static equilibrium between the droplet gravity and capillary action, and destroys hydrophobicity. Liu et al. [30] analyzed the influence of geometric parameters of array square columns on the transition between compound and non-compound wetting states of droplets from the perspective of energy. The results show that increasing the width and height of micropillars and decreasing the spacing of micropillars are beneficial to the formation of a stable Cassie wetting state. However, the existing theoretical design of micronanostructures on superhydrophobic surfaces is mainly aimed at the ordered microstructure at a single scale, and there are few composites (two-stage) micro-nanostructures on superhydrophobic surfaces. Establishing the CA prediction model for multi-scale microstructures helps to determine more practical and effective design criteria for hydrophobic structures [33]. Wang et al. [34] proposed a numerical calculation method for the water contact angle on the microstructure of the groove array by using the layered model. Firstly, the mechanical analysis of water droplets placed in the array microstructure was carried out, and then the wettability of WEDM surface morphology was evaluated. The established mathematical model can predict the contact angle under different geometric sizes. Dong et al. [35] proposed a numerical method combined with a nonlinear optimization algorithm to numerically calculate the contact angle, contact angle hysteresis and state transfer barrier of droplets.
To prepare the controllable superhydrophobic surface by WEDM, the CA prediction model of two-scale array microstructure was established, and the influence of microstructure shape and microstructure size parameters on the CA was analyzed. At the same time, the single-pulse discharge crater morphology under different discharge parameters of WEDM was simulated by FEM, and the geometric equivalent model of EN-GJL-250 WEDM surface was constructed to evaluate the wettability of WEDM surface morphology under different discharge parameters. Finally, the micro-structure cutting experiment was carried out on the EN-GJL-250 workpiece using WEDM technology to prepare the superhydrophobic surface. In this study, microstructure shape, peak current and pulse width were taken as input parameters, and CA was taken as an output parameter. The accuracy of the WEDM dual-scale array microstructure model in predicting CA was verified by experiments. According to the theoretical model, the controllable preparation of superhydrophobic surface by WEDM can be realized.

Apparent contact angle prediction model of primary array microstructure
In the process of studying the wettability of the metal surfaces, it is found that the CA of water droplets on the metal surfaces is related to the microstructure shape and size parameters. To explain the above problems, based on the Cassie-Baxter model, this section analyzes the force balance of water droplets on four different shape microstructures: rectangle, square column, sawtooth and semicircle. Here, the micro-nanostructure of the WEDM-machined surface is ignored when constructing the primary array microstructure. The geometry and size of four primary array microstructures are shown in Fig. 1. When water droplets drop on the microstructure, the original spherical droplets will deform under the combined action of gravity, support force and surface tension. Experiments show that the EN-GJL-250 material has hydrophobic properties after WEDM and low surface energy treatment. In other words, water droplets can maintain a stable spherical shape on the material surface. In view of this situation, a mathematical theoretical model is established to predict the CA. To simplify the calculation, the model is simplified as follows: 1. The array microstructures are uniformly distributed in the X-direction, and the size and shape of each microstructure are identical.
2. The droplet is spherically missing and the radius of the droplet does not change before and after the droplet falls on the surface of the workpiece. 3. Neglecting the effect of gravity load along the radial direction of the droplet. 4. The total volume of droplets remains constant and the reduced volume of droplets exists in the gap between microstructures. 5. The wetting state of the droplet on the microstructure is the Cassie-Baxter state. 6. The contact area between droplets and air in the microstructure gap is semicircular 7. A droplet drops onto a microstructure that is symmetric with one of the microstructures. 8. The internal cohesion of the droplet is constant.

Apparent contact angle prediction model for rectangular array microstructures
The dimensional parameters and force analysis of the spherical droplet on the microstructure are shown in Fig. 2, and the relationship between the parameters is given by Eqs.
(1)- (2): where R is the radius of ball-shaped droplets; R 1 is the threephase contact line radius of the droplet on micro-structure; is the angle between R and R 1 ; s is the CA of droplets; h is the distance between droplet center and microstructure plane; H is the distance between droplet tip and microstructure plane.
The model uses the force balance method to establish the equation that, the CA is determined based on the geometric parameters of the array microstructure. Water droplets are always in equilibrium on the array microstructure, and the forces applied on the droplets are analyzed mainly in the X and Z directions. The droplet is in a balanced state of four forces on the rectangular microarray structure, in which the four forces are: the support force of the microstructure pillar on the droplet F sl , the surface tension of the droplet at the microstructure gap F gl , the surface tension at the edge of the three-phase contact line F and the gravity of the droplet G . The relationship between the forces is given by Eq. (5).
In the rectangular array microstructure, T represents the length of a single period, 2n represents the total number of periods of the array microstructure, and the brackets represent the downward integration. The T and n are calculated according to Eqs. (6) and (7).
where a is the width of a single rectangular microcolumn; b is the gap width between rectangular microcolumns.

Calculation of the support force of the microstructure pillar on the droplet F sl
According to Newton's third law, the value of F sl is equal to the gravity of the liquid column on the rectangular microstructure.
To calculate the gravity of the liquid column on a single microstructure, it is necessary to obtain the contact area S s between the droplet and the solid part, as shown in Fig. 2. Calculate S s according to Eq. (8).
where S l is the total area of the gas-liquid contact part; S c is the total area of droplet contact with the metal surface. The contact plane between the droplet and the microstructure can be regarded as a circle and can be expressed by Eq. (9). Using the idea of calculus to calculate S l is more in line with the actual situation. The calculation formula is as follows: The S l can be calculated according to Eq. (13).
The S c can be calculated according to Eq. (14).
The average height of the liquid column on rectangular microstructure H can be calculated according to Eq. (15). Therefore, the F sl can be calculated according to Eq. (16).
where is the density of water; g is the acceleration of gravity.

Calculation of the surface tension of microstructure gaps on droplets F gl
Because the wetting state of the droplets on the microstructure surface is the Cassie-Baxter state, the shape of the droplets in the gap is similar to an arc. As shown in Fig. 2, is the angle between the arc tangent of the droplet in the microstructure gap and the Z direction. To determine F gl , the angle must be calculated first. Assume that the O (0,0) is the origin of the coordinate system, A ( k , p ) is the center of the arc, B (0,0), C ( b,0), and D ( b 2 , −v ) are three points on the arc, and r is the radius of the arc.
Combine the above three formulas, the r is calculated according to Eq. (20).
where v is distance between minimum point of droplet penetration gap and metal plane. The is calculated according to Eq. (21).
The action direction of the droplet surface tension is along the tangent direction of the droplet and the solid contact point at the gap. The two sides of the force projected in the X direction are balanced. As long as the force in the Z direction is obtained, the force F gl can be obtained. The crude line in the red range in Fig. 2 is the gas-liquid contact line. The gap force F gl−1 of the first cycle is calculated according to Eq. (22).
where gl is the gas-liquid surface tension coefficient. In the subsequent calculations, the distance of the array microstructure from the droplet center in the X-direction increases sequentially by one cycle T in length. The calculation equation is shown as follows (Eqs. (23) and (24)).
Therefore, the F gl can be calculated according to Eq. (25).

Calculation of the surface tension of three-phase contact line edge F
The semicircle length of the three-phase contact part L (excluding the contact line and both sides of the microstructure that overlap with the coordinate origin) is equal to the product of the radius of the three-phase contact circle and the corresponding central angle. The calculation equation is shown as follows (Eq. (26) and (28)).
Therefore, the F can be calculated according to Eq. (28).

Apparent contact angle prediction model for semicircle array microstructures
To theoretically explore the influence of the shape of microstructure on the CA under the same parameters, the following assumptions were made in this study: 1. Contact radius of a droplet on semicircular array microstructure ( R 1 ) is the same as that on rectangular microstructure. 2. The protruding semicircular microstructure is in a state of complete wetting.
The shape of microstructure is a single variable, and the basic size of semicircle microstructure is consistent with that of rectangular microstructure. The radius of the semi-circular microstructure is a 2 , the radius of the semi-circular gap is b 2 , and the depth of the microstructure is a+b 2 (far greater than the height v of the small droplet penetrating the gap). The direction of each force of a droplet on a semicircular array microstructure is shown in Fig. 3. The relationship between the forces is given by Eq. (30). Fgl' Fsl' Compared with the rectangular microstructure, the actual support area of droplets on the semicircular array microstructure pillar has changed, as shown in Fig. 3. The liquid pressure points to the inside of the microcolumn, and only the vertical component of the pressure is concerned here. The relationship between the parameters is given by Eqs.
where a 2 is the actual contact width of droplets on semicircular micropillars; b 2 is the gap width of a droplet on semicircular microstructure; is the angle between the end point of the arc at the microstructure gap connecting the center of the circle and the Z-direction.
At the same time, compared with the microstructure of the rectangular array, the changing parameters include angle . The is calculated according to Eq. (34).
where 0 is the intrinsic contact angle of the material surface.
Since none of the other parameters have changed, the CA of semicircular array microstructure is calculated according to Eq. (35).
at this time, which means that this array microstructure has ideal superhydrophobic performance. However, the contact shape between the droplet and the sawtooth array microstructure cannot completely change into linear contact. Therefore, under the premise of considering the special machining mode of WEDM, this study considers that the contact shape between the droplet and the sawtooth array microstructure is a semicircle, and the semicircle radius is much smaller than the bottom of the sawtooth array microstructure, as shown in Fig. 4. The relationship between the parameters is given by Eqs. (36) and (37): where r ′ is the radius of half circle at the top of sawtooth shape. The value of is related to the actual machining accuracy of the surface microstructure, and the value of is always less than 1; a 3 is the actual contact width of the droplets on sawtooth micropillars; b 3 is the gap width of the droplet on sawtooth microstructure. The direction of each force of the droplet on a sawtooth array microstructure is shown in Fig. 4. The relationship between the forces is given by Eq. (41).
The relationship between force and parameters on sawtooth array microstructure is the same as that on semicircular microstructure. The CA of the sawtooth array microstructure is calculated according to Eq. (42).

Apparent contact angle prediction model for sawtooth array microstructures
For special geometric microstructures such as sawtooth array microstructure, in the Cassie-Baxter state, the fraction of the contact area between the droplet and solid in the actual total contact area tends to 0, and the CA will reach the maximum 1 3

Apparent contact angle prediction model for square column array microstructures
The direction of each force of a droplet on a square column array microstructure is shown in Fig. 5.
The relationship between the forces is given by Eq. (43). (42) The supporting force of square column array microstructure pillar on droplet F sl can be calculated according to Eq. (44). where h + y is the average height of the water column on a single square column; a 2 is the solid-liquid contact area of a single square column; (a+b) 2 is the number of the square columns in the contact area of droplet and square column array microstructure.
The shape of the droplet on the microstructure surface is a circular defect, and its expression is x 2 + y 2 = R 2 . Based on Fig. 5, it is known that at the center of the droplet (i.e., when x = 0, y = R ), as x increases, the value of y changes appropriately. The calculation process of y is as follows: The F gl ′′ can be calculated according to Eq. (47).
The F ′′ can be calculated according to Eq. (48).
The CA of square column array microstructure is calculated according to Eq. (49).

Establishment of apparent contact angle prediction model for two-scale array microstructure
The research on the wettability of metal surface microstructure can be divided into two parts. The first is a small microstructure, size between tens of microns to hundreds of microns. When the size of the microstructure is appropriate, the wetting state of water droplets on the surface of the material conforms to the Cassie-Baxter model. The microstructure divides the contact position of water droplets and the material into gas-liquid and solid-liquid parts. The other part is the rough micro-nanostructure formed on the surface of the material after processing. The existence of a micronanostructure will increase the contact area between water droplets and the material surface. According to the Wenzel model, the presence of roughness can enhance the bulk wettability of materials.

WEDM single-pulse discharge surface topography simulation
First, COMSOL Multiphysics 5.3 (COMSOL Inc., Burlington, MA, USA), a commercial finite element software package, was used to simulate the thermal and flow field changes on the surface of the EN-GJL-250 material during the single-pulse discharge of WEDM. In this simulation model, the level set method is used to track the formation and growth process of single-pulse discharge pits, and the size characteristics of discharge pits at different discharge times are studied. Selection of simulation parameters: discharge voltage is 24 V, discharge current is 12 A, pulse width is 32 μs and duty cycle is 0.25. The total calculation time is 200 μs, the discharge duration is 100 μs, and the surface morphology of EN-GJL-250 material changes with time, as shown in Fig. 6. The simulation results show that in the process of forming the discharge pit, the molten material will first form a smooth depression at the center, then be pushed to both sides and continuously etched away, and at the same time, flanging protrusions will appear at the periphery of the pit. At 100 μs, the spark discharge time ended, but the size of the pits still tended to increase. Until 160 μs, the depth and depth of the pits began to remain stable, and the flanging protrusions were basically smooth, and there was no sharp phenomenon. The final appearance of the discharge pits is shown in Fig. 7A. To more intuitively analyze the variation law of crater morphology and feature size during the formation of single-pulse discharge craters, numerical simulations

Establishment of the geometric equivalent model of EN-GJL-250 WEDM surface
The surface morphology of WEDM is determined by numerous single-pulse discharge pits and discharge positions. In the actual discharge process, the discharge position has randomness and disorder, as shown in Fig. 8.
Calculating the roughness coefficient of the surface of EN-GJL-250 after WEDM is very important for establishing the CA prediction model based on the Wenzel theory. To facilitate the calculation, the following assumptions are made about the shape and arrangement of pits: 1. The size of all pits is the final size of the pits on the back surface after WEDM single-pulse discharge simulation, and the pits are evenly arranged on the processing plane.

The shape of a complete pit is a part of a ball.
3. It is considered that the plane of the workpiece before machining is smooth.
4. Water droplets can completely cover the processed surface.
After making the above assumptions, the surface morphology can be simplified to the standard microstructure of "pitsridges-protrusions", and thus the geometric equivalent model of EN-GJL-250 WEDM surface can be established, as shown in Fig. 9. Compared with the original plane, the contact area after processing has two parts, one is the blue concave part in Fig. 9 and the other is the yellow convex part in the Fig. 9.
The pits are evenly distributed on the machining plane, so the area change law of a pit is consistent with the overall change law. To facilitate the calculation, the wettability of a single pit in the model is studied. Figure 10a, b is a crosssectional view of the geometric equivalent model, and Fig. 10c is a top view of a single pit in the model.
The relationship between the parameters is given by Eq. (50): where a s is the height of the highest point from the bottom of the pit; h s is the distance between the initial plane and bottom of the pit; d p is the depth of single-pulse pits; h p is the where r p is the radius of a single pulse pit; r s is the length of residual radius after superposition of pits. According to Eq. (52), calculate the area S 2 of the blue ball cover.
where R c is the radius of the ball with the same size of the pit.
The R c can be calculated according to Eq. (53).
Combine Eqs. (51) and (52) to calculate the area S 2 of the blue ball cover: The S surface area of EN-GJL-250 before processing R c can be calculated according to Eq. (55).
The roughness coefficient of EN-GJL-250 surface is a multiple of the area of the rough surface after processing relative to the original smooth surface, and its value is equal to the ratio of the surface area of the rough surface to the surface area of the smooth surface. Calculate according to Eq. (55).
Under the condition of Wenzel wetting, is always greater than 1, and when the contact angle 0 < 90 • in the ideal state, For the geometric equivalent model of EN-GJL-250 WEDM surface, the CA w on the rough surface can be expressed as:

Establishment of prediction model of surface apparent contact angle of two-scale microstructure
After the surface of the workpiece is microstructured by WEDM, its surface is not absolutely smooth, and there are many discharge pits, as shown in Fig. 11. According to the Wenzel model, these micro-nanostructures increase the contact area of the solid-liquid interface and affect the contact angle of water droplets on the material surface. At the same time, when the workpiece surface is processed into a two-scale microstructure by WEDM, the water droplets do not completely conform to the Cassie-Baxter model, and there is Wenzel wetting state on the micro-nanostructure, so the length of the contact line of the water droplets on the microstructure will change, and the increasing ratio of the contact angle is related to . There is no solid surface at the gap, so it is considered that the value of the gap is constant. Considering the influence of the surface roughness coefficient, the size parameters of the microstructure become:  Fig. 10 (a, b)

Effect of discharge parameters on surface wettability in WEDM
By WEDM single-pulse discharge simulation, I p and T on can be adjusted to predict the size of discharge pits. Based on the geometric equivalent model of the WEDM surface and numerical calculation, the influence of WEDM discharge parameters on the CA of the micro-nanostructure under the Wenzel condition is obtained. Through the WEDM linear Fig. 12 shows the variation of the measured and predicted values of the static contact angle with the discharge parameters.
The results show that with the increase of discharge parameters, the roughness of the workpiece surface increases, the contact area between the water droplet and the rough surface of the workpiece increases, and the CA also increases. But water droplets are always in Wenzel wetting state, WEDM machined surface cannot reach the superhydrophobic state. Therefore, it is difficult to test the superhydrophobic performance of the WEDM surface without machining the microstructure. At the same time, by controlling discharge parameters, WEDM can effectively improve surface hydrophobicity.

Analysis of the apparent contact angle prediction model of the first-order array microstructure
Without considering the rough micro-nanostructure of WEDM, the microstructure of the first-order array is regarded as a smooth structure with hydrophobic properties. Without considering the rough micro-nanostructure  Figure 13 shows the curves of the CA s changing with a and b. When b = 200 μm, the curves in Fig. 13A show the effect of a on s . The results show that: adjusting the shape and size of microstructure can change the CA. At the same time, the increase of the solid part of the microarray structure leads to the decrease of the CA. From the perspective of model establishment, when the area of the solid part increases, the surface wetting state is more similar to the Young model, so s tends to decrease. In other words, the smaller the solid fraction is, the easier it is to obtain a larger CA. When a = 200 μm, the curves in Fig. 13B show the effect of b on s . With the increase of b , s increases first and then decreases. When b is in the range of 200-300 μm, the s reaches the maximum and the hydrophobic performance is the best.

Analysis of the apparent contact angle prediction model of the two-order array microstructure
In this paper, it is assumed that the droplets are in the Cassie-Baxter wetting state on the primary array microstructure and Wenzel wetting state on the secondary micro-nanostructure. Similarly, wc can be obtained by inputting the size parameters of the microstructure. There are many factors affecting wc . This paper studies the effect of the on wc . When a = 200 μm and b = 200 μm the curves in Fig. 14 show the effect of on wc . With the increase of , the rougher the surface of the material, the larger the solid-liquid contact area, so wc decreases. In addition, the larger the , the smaller the effect on wc , and the larger the effect of the size parameter on the contact angle. At the same time, since the solid-liquid contact area of the sawtooth microstructure is much smaller than the other three shapes, the effect of on the surface wc of the sawtooth microstructure is small.

Material and machine tool
EN-GJL-250 material has the characteristics of high strength, wear resistance and heat resistance. Among all cast iron materials, EN-GJL-250 is often used in the manufacture of some machine tool beds, columns and gears because of its anti-corrosion, shock absorption, and excellent casting properties. The percentage of chemical composition properties of the workpiece material is given in Table 1. In this  experiment, EN-GJL-250 with a size of 20 × 20 × 5 mm is selected as the workpiece to be processed. In this experiment, DK7632 high-speed wire electric discharge machine (HS-WEDM) was used to prepare surface microstructures of different shapes, as shown in Fig. 15a. The electrode wire is the molybdenum wire, the axial flushing method is adopted, and the working fluid is a mixture of water and emulsion. The machine tool adopts an electrode wire tensioning mechanism to reduce the probability of wire breakage during processing. Under the existing programming file, the machine tool can realize the function of self-processing. By changing the parameters on the control panel of the WEDM machine tool, the required processing conditions can be met.

Experimental design
In this experiment, four primary array microstructures of rectangle, semicircle, square column and sawtooth are designed, and their size parameters are consistent with those in the theoretical model, namely a = 200 μm, b = 200 μm. The surface topography of WEDM is formed by the overlapping of numerous discharge pits. These random discharge pits are defined as secondary microstructure. According to the simulation model of WEDM single pulse discharge, the morphology and size of discharge pits are affected by discharge parameters. In this experiment, the morphology and size of secondary microstructure are controlled by inputting different peak currents and pulse widths.
WEDM has a discharge gap in the process of machining discharge. If the cutting is performed according to the designed size, the actual size obtained will be smaller than the pre-designed size. Therefore, the cutting trajectory of the electrode wire should be planned first. Use WEDM to cut the narrow slit and measure the width of the slit with CCD. The average value is 260 μm, and the diameter of the electrode wire is 180 μm, so the discharge gap on one side is about 40 μm. The experimental machine tool is defined when editing the motion trajectory of the electrode wire. Before processing the micro-structure, the motion track of the electrode wire is drawn according to the size and shape of the micro-structure. The software UG11.0 is used to simulate the motion track of the electrode wire, and the G code is generated. The code is input into the machine tool to process the micro-structure. During the sequential processing of the experimental samples, it is very important to determine the position of the first spark discharge between the workpiece and the wire electrode. First, the initial gap between the electrode wire and the workpiece is controlled at 1 mm, and then the electrode wire is moved by 100 μm in turn to make the electrode wire gradually approach the workpiece until a spark is generated, as shown in Fig. 15b. Finally, the discharge parameters and the motion trajectory of the electrode wire are adjusted using the operation panel of the WEDM machine, and the normal processing status is shown in Fig. 15c.
The position of the central axis of the electrode wire, so the distance between the central axis of the electrode wire and the machining surface should be 130 μm. Design of experimental parameters: the interval ratio is 3:1, the feed rate is 4 m/min, and the machining characteristic is positive polarity. The experimental scheme is shown in Table 2.
The cut metal surface with array microstructure was cleaned in absolute ethanol, deionized water and acetone at room temperature for 10 min using an ultrasonic cleaner to  1  Straight cut  12  24  2  Rectangle  12  24  3  Square columns  12  24  4  Sawtooth  12  24  5  Semicircle  12  24  6  Semicircle  8  24  7  Semicircle  10  24  8  Semicircle  14  24  9  Semicircle  16  24  10  Semicircle  12  12  11  Semicircle  12  16  12  Semicircle  12  20  13 Semicircle 12 32 clean the oil and debris on the surface of the workpiece. The surface was quickly blown dry with a hair dryer, soaked in a 2% ethanol solution of stearic acid for 6 h, then taken out and baked in a constant temperature oven for 30 min, and the oven temperature was set to 60 °C.

Apparent contact angle
The CA of the droplet on the surface of the modified sample was characterized by an OCA20 (DATAPHYSICS, Germany) video optical contact angle measuring instrument, with a measurement range of 0-180°, an accuracy of ± 0.1°, and a resolution of ± 0.01°. In the wettability test, the droplet volume was 5 μL, and the recording time was 10 s. Three different measurement points were measured on the sample surface. The average value of the left and right contact angles of the same measurement point is taken as the measurement result of one measurement point. The average value of the measurement results of the three measurement points is used as the final data.

Surface texture
WEDM is a non-contact machining method. Due to the offset of the wire electrode, the discharge gap and the wire vibration, the actual shape and size of the microstructure will deviate from the designed shape. To further explore the relationship between the actual microstructure shape and CA on the surface of the EN-GJL-250 material, the actual microstructure shape and size were measured using a Sper-View W1 optical 3D surface profiler.

Micro morphology
The microstructure, morphology and chemical composition of the machined surface of EN-GJL-250 material were characterized by SU3500 Hitachi tungsten filament scanning electron microscope (SEM) equipped with energy dispersive spectrometer (EDS).
6 Results and discussion

Contact angle
According to the experimental design of WEDM (Table 1), a set of cutting experiments and contact angle measurement experiments were carried out. Figure 16 depicts the optical properties of surface water droplets after WEDM and low surface energy treatment of EN-GJL-250 material, photo.
The experimental data show that the static contact angle of the WEDM machined surface of EN-GJL-250 material ranges from 122 to 156.15°. Figure 16 (14) shows that the intrinsic contact angle measured on the surface of the EN-GJL-250 material is 67.72°, and it can be said that the surface interfacial energy of the EN-GJL-250 material is relatively high. Figure 16 (15) is an optical photo of the surface water droplets of EN-GJL-250 material cut by WEDM, but not chemically treated, indicating that WEDM processing of rough surfaces can achieve hydrophobic properties. Analysis reason: The rough microstructure formed by WEDM technology on the surface of the EN-GJL-250 material trapped a large amount of air, forming the "air cushion effect", thus reducing the contact area between water droplets and EN-GJL-250 material surface, preventing water droplets from wetting the surface, Fig. 16 Optical photo of water droplets on the surface of the experimental sample increasing the CA, to obtain a hydrophobic surface with good hydrophobicity. By comparing Fig. 16(NO.1), (15), it can be found that the surface hydrophobicity of the sample processed by WEDM is improved after surface energy treatment. Analysis reason: Under the action of dissolved oxygen in stearic acid ethanol solution, the surface of the EN-GJL-250 material is naturally oxidized, and the reaction is very slow due to the passivation of the oxide layer. However, the stearic acid dissolved in the ethanol solution provides a weak acid environment for the oxidation of the EN-GJL-250 materials, thus catalyzing the oxidation reaction process and releasing iron ions. Iron ions in the solution are rapidly captured by stearic acid molecules and chemically react to form carboxylate, which then forms a self-assembled film on the EN-GJL-250 matrix. Figure 17 shows the 3D surface topography and 2D crosssectional profile of the two-scale micro-structured surfaces with four different shapes prepared by WEDM. Due to the special material removal form of WEDM, the profile line of the processed microstructure cannot be a perfect curve. The blue curve in the 2D cross-sectional profile is the original surface morphology, and the red curve is obtained by filtering the profile curve with a three-dimensional spline filter. The cut-off wavelength is 80 μm, and the measurement of microstructure-scale parameters adopts the red cross section line. In the actual measurement process, the difference between the design dimensions a and b and the actual dimensions ranged from 85.7 to 102.3 μm, and the average value of these differences was 91.4 μm. The reason for this error is that there are factors such as electrode wire offset, discharge concentration and electrode wire vibration during the machining process of WEDM. The fluctuation of the error value is caused by the phenomenon of angular discharge and measurement error. Figure 18 shows SEM images showing the top regions of the array microstructures of four different shapes, with magnifications of 500 × , 1000 × and 4000 × for (a), (b) and (c), respectively. It can be seen from Fig. 17 that the WEDM machined surface of EN-GJL-250 material is mainly composed of randomly distributed pits, ellipsoid convex bodies and wrinkles. The molten crater is caused by the fact that the pulse voltage breaks down the working fluid and generates spark discharge in the process of WEDM, and a large amount of heat energy is instantaneously concentrated in Fig. 17 Three-dimensional surface profile of the experimental sample (NO. [2][3][4][5] the discharge microchannel, which melts a small amount of metal material on the working surface. The convex body is formed by the bubbles in the discharge channel exploding and flying out, and then fused and adhered to the processed surface and then condensed. The wrinkle is due to the overlapping of the discharge pits during the processing, resulting in the formation of wrinkled strip-shaped protrusions on the edges of the discharge craters. These randomly distributed discharge pits, protrusions and wrinkles can be well combined with stearic acid during low surface energy treatment, thereby reducing the surface free energy of the material. Figure 19 shows the comparison of contact angles of microstructures with different shapes under the same discharge parameters. The results show that: square column > semicircule > rectangular > sawtooth > linear cutting. Cause analysis: Fig. 18a, d, g, j shows that when the water droplets are in the Cassie-Baxter wetting state on the surface of the workpiece, the solid-liquid contact area  Fig. 20 The variation law of the model predicted value and experiment measured value of the CA of semi-circular array microstructures vary with T on of the square column array microstructure is the smallest, and the solid-liquid contact area of the rectangular array microstructure is the largest. By analyzing the contact angle prediction model, it can be seen that the larger the solid-liquid contact area is, the smaller the contact angle of water droplets is. However, because the solid-liquid contact of the sawtooth array microstructure is too small to maintain the spherical morphology of water droplets, the hydrophobic performance is the worst. Figure 14(NO.4) shows that water droplets are in Wenzel wetting state on serrated microstructures. At the same time, when the geometric size of the semicircular array microstructure is suitable, by controlling the discharge parameters, multilevel micro-nanostructures can be prepared on the surface of EN-GJL-250 material to obtain the superhydrophobic surface.

Effect of pulse width on apparent contact angle
When a = 200 μm, b = 200 μm and I p = 12 A, the curves in Fig. 20 show the effect of T on on CA. From the curve changes in Fig. 20, when the pulse width is equal to 20 μs, the CA of the semicircular array microstructure is the largest. When the pulse width increases, the contact angle first increases and then decreases. As the single-pulse discharge energy increases, the roughness of the microstructure surface increases slightly, resulting in the formation of secondary microstructure. At this time, the secondary microstructure forms the Cassie-Baxter model again, and the CA of the EN-GJL-250 material surface increases. In addition, it is also possible that due to the increase of the discharge energy, the microstructure gap increases, resulting in a slight increase in the CA. However, when the pulse width continues to increase, the discharge pits will become larger and deeper, and the secondary microstructure is closer to the Wenzel model. The liquid penetrates into the gaps between the secondary microstructure, resulting in the decrease of the CA of the EN-GJL-250 material surface.

Effect of pulse width on apparent contact angle
When a = 200 μm, b = 200 μm and T on = 12 μs, the curves in Fig. 21 show the effect of I p on CA. With the increase of the peak current, the energy of a single pulse is increasing, and the reason for the influence of pulse width on the CA is the same. Figure 22 shows the three-dimensional morphology of I p = 10 A and I p = 16 A, with Ra of 3.67 μm and 6.49 μm, respectively. It can be seen that with the increase of single pulse energy, the size, depth and number of pits on the surface of EN-GJL-250 material increase, resulting in changes in the state of droplets on the secondary microstructure, thus affecting the CA of droplets on the metal surface.

Conclusion
The following conclusions were derived from this study.
1. In this paper, based on the simulation method, the heat flow coupling model in the WEDM single-pulse discharge process is established. The simulation results show that the molten material will first form a smooth depression in the center and then be pushed to the sides and be removed continuously during the formation of discharge pits. Meanwhile, flanging bulges appear around the edges of the pits. Finally, with the change of discharge parameters, the size and morphology of single-pulse discharge pits also change. By establishing the geometric equivalent model of the surface machined by WEDM, it is found that it is difficult to achieve superhydrophobic properties of the surface machined by WEDM without any shape microstructure.
2. In this paper, four kinds of array microstructures suitable for WEDM machining are designed. Based on the force balance method, the CA prediction models of semicircle, rectangular, sawtooth and square column array microstructures are established respectively. The analysis model found that the smaller the solid-liquid contact area, the easier to obtain a larger CA; when a is less than 200 μm, b is in the range of 200-300 μm, the contact angle reaches the maximum. The surface roughness coefficient is introduced into the CA prediction formula by taking the micro-nanostructure of the WEDM machined surface as the secondary structure. The results show that with the increase of surface roughness coefficient, the surface hydrophobicity of the two-scale array microstructure is worse.
3. The superhydrophobic surface was prepared on EN-GJL-250 material by WEDM. The prediction model of the CA of the two-scale array microstructure was verified by experiments. The predicted CA values of the established mathematical model are in good agreement with the experimental values, and the overall trend is the same, with an average error of 7.83%. The surface superhydrophobic properties of four array microstructures with different shapes were compared, the results show that: square column > semicircule > rectangular > sawtooth. When the pulse width is 20 μs and the peak current is 10 A, the CA of the semicircular array microstructure is the largest.