Measurement and simulation calculation of wire bow angle during the diamond wire saw process

The wire bow angle is an important factor that affects the shape precision of an ingot after the diamond wire sawing process. In this research, the wire bow angles of the inside and outside of an ingot were recorded with a high-speed camera. The effects of the processing parameters such as the wire tension force, feed speed, and wire velocity on the wire bow angles inside and outside the ingot were analyzed. A numerical simulation model of the wire bow in the wire sawing process is presented in this paper to describe the wire bow angle inside the ingot. It was shown that the wire bow angle inside the ingot was smaller than that of outside the ingot for all of the processing parameters. The wire bow angles improved with the increase of the feed speed and decrease of the wire velocity and the wire tension force. The results of the wire bow angle measurement of the inside ingot and the simulation calculation were similar for the process parameters.


Introduction
Diamond abrasive wire saw technology has been widely used to cut brittle and hard materials such as silicon, silicon carbide (SiC), and sapphire due to the higher efficiency, smaller kerf loss, and lack of size constraints [1]. During the cutting process, the contact force is found between the ingot and the diamond wire saw. Hence, the wire will bend to a certain extent and form a wire bow phenomenon, as shown in Fig. 1. The wire bow phenomenon during the wire sawing process will reproduce itself on the surface of the ingot, causing saw marks with the same shape as the wire bow to emerge on the wafer surface. The wire will break and cause material waste when the wire bow angle is too large. Additionally, the ingots have shape accuracy deviation because the contact position between the ingot and the wire saw cannot reach the set coordinate point.
Clark et al. [2] used a non-contact capacitive sensor to measure the deviation distance of the wire saw of an outside ingot and calculate the wire bow angle. Liu et al. [3] used a high definition digital camera to take pictures of the wire bow angle outside an ingot. The stability of the wire bow angle during the cutting process was ensured by controlling the feed speed and the wire velocity. Qin et al. [4] used the light projection method to measure the change in the wire saw projection position to control the wire bow angle. Wang et al. [5] applied the difference between the feed displacement and the saw slit depth to evaluate a diamond wire bow. In these detection methods, only the wire bow angle outside the ingot was measured.
Some scholars have established theoretical models of wire bow formation based on the principles of mechanics. Zhang et al. [6] divided a wire saw into a differential line for stress analysis during the cutting process, with the assumption that the wire saw was a continuum with some flexibility and elasticity in the process. Teomete [7] established a wire bow deflection model in which a uniform vertical load was applied in the middle of a wire saw. Liedke and Kuna [8] or Qiu et al. [9] derived the wire bow model for the wire sawing process description based on the macroscopic mechanical conditions.
Up to now, the accuracy of all the research models was verified by experimentally obtaining the wire bow angle of the outside ingot. The difference in the force method of the wire saw inside and outside the ingot will inevitably lead to the deviation in the accuracy of the model. Therefore, a study towards the wire bow angle inside and outside the ingot will be analyzed to improve the accuracy of the wire bow model. A high-speed camera was used to measure the wire bow angle during a cutting experiment for transparent crystal. A numerical model of the wire bow deflection was established based on the force of the experimental results. The experimental results of the wire bow angle inside the ingot and the simulation calculation results were compared.   Experimental design for a crystal sawn using a diamond wire saw. a Schematic representation and b experiment system two guide rollers was 486 mm. A piezoelectric dynamometer was assembled on the workbench, and the experimental specimen was affixed to the dynamometer and located in the middle of the two guide rollers to guarantee the stability when the specimen was sawn. The horizontal (F x ) and vertical force (F z ) are the force signals in the horizontal and vertical directions recorded by the piezoelectric dynamometer, respectively. The wire tension force (T) is set by the wire sawing machine, and the tension sensor is installed on the guide wheel to feedback the wire tension force. A pair of air cylinders can be used to adjust the tension force of the wire to maintain the set value. The wire saw ran with the wire velocity (v s ) between two spools and reciprocated the direction periodically, and the reciprocating cycle time was 120 s, as shown in Fig. 3. The ingot was fed with a feed speed (v f ) in the vertical direction to achieve the material removal.
A transparent K9 crystal was selected as the specimen material to make the wire bow angle inside the ingot measurable. The crystal was made up of 69.13% SiO 2 , 10.75% B 2 O 3 , 10.4% Na 2 O, and other oxides. The sawing length l of the crystal was 50 mm and the cutting depth was 30 mm in each experiment. The parameters of the wire sawing process are shown in Table 1.
Commercially available electroplated diamond wire was used for the sawing. A scanning electron microscopy (SEM) image of the surface topography of the wire is shown in Fig. 4. The diameter of the fixed diamond wire saw was 0.25 mm and the size of the diamond grit was 30-40 μm. The diamond wire saw had a length of approximately 10-16 grits/mm. To avoid the influence of wire wear on the experimental results, a new wire was adopted in each experiment.

Wire bow angle measurement
A high-speed camera (Phantom v2511) was used to take photographs of the sawing wire. Photographs were taken to record the shape of the wire saw in the completion of the wire saw cutting 30 mm depth of reaching a steady state. The resolution of the camera was 1280 × 800 pixels, the sample rate was 1 kHz, and the exposure time was 9 μs. A high-brightness lamp was installed behind the sawing machine to improve the brightness and clarity of the photos taken, as shown in Fig. 2b. The original photograph was magnified ten times and the contrast was adjusted. The intersection point between the sawing wire and the left side of the crystal in the photograph was set to the origin of the coordinates to establish the XOZ rectangular coordinate system, as shown in Fig. 5.

Force measurement
The forces during sawing were recorded by a piezoelectric dynamometer (Kistler 9119AA2, Kistler Co., Ltd. Switzerland). The sampling frequency was 1 kHz. The range of the dynamometer was 0-40 N. The resolution of horizontal F x and the vertical force F z was 0.04% and 0.12% of the dynamometer scale, respectively. The force measurement experiment detected the force signals of the five reciprocating cycles before the completion of the wire saw cutting 30 mm depth. Lifting wire is manually lifted to separate it from the specimen when the wire stopped moving.
The force signals of the whole measurement process were deducted from the value of the baseline, and then a low-pass cutoff frequency of 3 Hz was applied to determine the force value. The relationship between horizontal force and vertical force and sawing time is shown in Fig. 6. The

Simulation calculation model of the wire bow
The rectangular coordinate system was established with the highest point of the wire bow as the coordinate origin, as shown in Fig. 7. In the figure, l is the length of the ingot, and l a is the horizontal distance between the highest point and the cut-in point. l b is the horizontal distance between the highest point and the cut-out point.
The wire deflection z of any point on the wire in the cutting zone could be derived as [10]: The horizontal tension H could be expressed as: The horizontal distance l a and l b could be expressed as: The slope of any point in the cutting zone could be expressed as: The simulated calculated value of the wire bow angle of the inside of the ingot at the cut-in and cut-out positions could be derived as:

Experimental fitting result of the wire bow
The coordinates of 11 evenly divided discrete points of the wire saw on the left, inside, and right of the ingot in the picture were extracted, respectively. The wire saw was affected by the normal force and the tangential force in the cutting zone, and the wire bow curve was approximately to a quadratic function according to Eq. (1). Therefore, a quadratic polynomial was used to fit the discrete point coordinates on the wire saw of the inside ingot. The wire saw outside the ingot was tightened into a straight line with the wire tension force shown in Fig. 7, and the discrete point coordinates on the wire saw left and right of the ingot were therefore linearly fitted. Part of the length was taken to extract discrete points because the wire saw outside the ingot was linear.
The experimental fitting result of the wire bow is shown in Fig. 8. The fitting function of the wire saw was derived to obtain the slope, and then the wire bow angle was derived using the inverse trigonometric function. In the figure, 1 and

The effect of the wire saw reciprocating moves on the wire bow angle
The periodical reciprocate movement of the wire saw caused the cut-in position to change periodically on both sides of the ingot. The wire bow angle at the cut-in position was compared during the reciprocating motion, and a set of parameters was taken as an example, as shown in Fig. 9. The wire saw moved in a reciprocating direction has little influence on the wire bow angle for all of the processing parameters. Therefore, it is appropriate to analyze the experimental results of the positive direction movement of the wire saw.

The effect of the wire tension force on the wire bow angle
The effect of the wire tension force on the wire bow angle is shown in Fig. 10. In the figure,   Moreover, the tendency of the wire bow angle to change was gradually reduced with the increase of the wire tension force. The wire bow angles at the cut-in and cut-out positions were different with the increase of wire tension force, but the difference was only in the range of 0.05°. The wire bow angles at the cut-in and cut-out positions had the same variation.

The effect of the feed speed on the wire bow angle
The effect of the feed speed on the wire bow angle is shown in Fig. 11. With an increase in the feed speeds from 0.2 to 0.6 mm/min, the simulation calculation value of the wire bow angle inside the ingot in the cut-in position increased from 0.71° to 2.46°. The experimental values of the wire bow angle inside the ingot in the cut-in position increased from 0.50° to 2.37°, and the wire bow angle outside the ingot in the cut-in position decreased from 0.62° to 2.59°. The wire bow angle improved linearly with the increase of the feed speed.

The effect of the wire velocity on the wire bow angle
The effect of the wire velocity on the wire bow angle is shown in Fig. 12. With an increase in the wire velocity from 200 to 400 m/min, the simulation calculation value of the wire bow angle inside the ingot in the cut-in position decreased from 2.57° to 1.57°. The experimental values of the wire bow angle inside the ingot in the cut-in position decreased from 2.34° to 1.33°, and the wire bow angle outside the ingot in the cut-in position decreased from 2.60° to 1.47°. The tendency of the wire bow angle to change was gradually reduced with the wire velocity increase. In brief, the wire bow angles of the outside ingot were all larger than those of the inside for the wire tension forces in the range of 15 N to 35 N, the feed speeds in the range of 0.2 to 0.6 mm/min, and the wire velocity in the range of 200 to 400 m/min. The wire tension force and the feed speed had the greatest influence on the wire bow angle and the wire velocity had the least influence on the wire bow angle.
The variation trends of the wire bow angle for the experiment and the simulation calculation results were similar. The wire bow angles improved with the increase of the feed speed and decrease of the wire velocity and the wire tension force. The simulation calculation values were slightly Fig. 10 The effect of the wire tension force on the wire bow angle. a Cut-in position. b Cutout position Fig. 11 The effect of the feed speed on the wire bow angle. a Cut-in position. b Cut-out position larger than the experimental values because the wire saw was assumed to be an ideal string and the force load was a uniform load.

Discussion
According to the simulation calculation model of the wire bow in Fig. 7, the factors that affect the wire bow angle include wire tension force, tangential force, and normal force. Wang et al. [11] found that the sawing force increases approximately linearly with feed speed, and decreases nonlinearly with wire velocity during the sawing process. The reason is related to the volume of material removal per length of wire (q). According to the wire saw kinematics in the sawing process, the volume of material removal per length of wire (q) could be expressed as: where b is the slit width and w is the cutting length of the specimen. Here, the slit width b can be approximately equal to the diameter of the fixed diamond wire saw 0.25 mm, and the cutting length w is equal to sawing length l = 50 mm. When the width of the saw slit width and the cutting length of the specimen were constant, the removal volume per length of the wire saw had a linear relationship with the feed speed and an inverse proportional function relationship with the wire velocity. The effect of the volume of material removal per length on the wire bow angle of the inside ingot in cut-in position is shown in Fig. 13. The wire bow angles are found to be approximately proportional to the volume of material removal per length of wire, and the wire bow angle approximately reflected the volume of material removal per length of the wire. During the wire saw cutting process, the wire bow angles of the outside ingot were larger than those of the inside ingot for processing parameters to ensure the cutting ability of the wire saw.

Conclusion
1. The wire bow angle of the outside was larger than that of the inside in order to ensure the cutting ability of the wire saw during the whole wire saw cutting process.

The simulation calculation and experimental results
showed that the wire bow angles improved with the increase of the feed speed, the decrease of the wire velocity, and the wire tension force. 3. The experimental results showed that the wire bow angle is approximately proportional to the volume of material removal per length of wire, and the wire bow angle approximately reflected the volume of material removal per length of wire.