FIGURE 4. VECTORS OF THE TENSIONS AND THE NOR-MAL LOAD AT THE CONTACT POINT.
these parameters were considered as the benchmark. Once the wire speed increases, the interval of time steps will decrease cor-respondingly because of the increase of the indentations within a unit time.
Another important consideration in the model is the varia-tion of the slope of the wire. During slicing process, the slopes of the wire segments may change because the depths of indentation cracks reduce the heights of the nodes locally. However, because of the tension of the wire, the slopes should keep decreasing from the left of the wire to the right, as shown in Fig. 3. Note that if a node is lower than the adjacent ones, the wire should not contact with it. A loop in the program was set to check if the slopes were decreasing.
SIMULATION RESULTS
The parameters and values obtained in the previous section are summarized in Table 2. The simulated time was 17,600 s.
Therefore, the total steps were 20,000. Figure 5(b) shows the simulation results when the slicing process reaches steady-state.
The bow angle of the wire is about 8.6◦, which is higher than the normal operation of 1◦∼ 5◦. The reason is because this model only considers the indentation cracks as the material re-moval mechanism. Other possible mechanisms such as plowing and erosion are not included.
Figure 6 shows the average normal loads and their standard deviations at different nodes. The results show that while the in-got approaches the wire before the steady-state condition, the av-erage normal loads at the edges are much higher than those at the
TABLE 2. PARAMETERS AND VALUES OF THE BENCH-MARK.
Wire tension Feed rate Time interval Element length T= 20 N f= 5µm/s dt= 0.88 s S= 1 mm
TABLE 3. SIMULATION RESULTS OF THE AVERAGE NOR-MAL LOADS AT THE CENTER OF THE WORKPIECE AND THE AVERAGE BOW ANGLES OF THE WIRE DURING t= 8800 ∼ 17600 s (THE LAST HALF OF THE SIMULATION).
THE WIRE TENSION IS 20 N.
Wire speeds Feed rates
TABLE 4. SIMULATION RESULTS OF THE AVERAGE NOR-MAL LOADS AT THE CENTER OF THE WORKPIECE AND THE AVERAGE BOW ANGLES OF THE WIRE DURING t= 8800 ∼ 17600 s (THE LAST HALF OF THE SIMULATION).
THE WIRE SPEED IS 10 m/s.
Wire tension Feed rates
2µm/s 5µm/s 8µm/s 10 N 0.0143 N 0.0502 N 0.0916 N
4.6◦ 15.9◦ 24.4◦
middle. Once it reaches steady state, the average normal loads are about the same values. However, the loads at the outermost edges have obvious lower average values than the others.
In order to study the performance of the slurry wire sawing process, other conditions were investigated by changing the pa-rameters in Table 2. Three feed rates, 2µm/s, 5µm/s, and 8 µm/s were studied. In addition, another wire speed, 20 m/s, was also employed by changing the interval of time steps to 0.44 s.
Therefore, the total time steps would become double compared to the wire speed of 10 m/s. The steady-state simulation results are listed in Table 3. In Table 4, another wire tension, 10 N, was studied.
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−100 −50 0 50 100
FIGURE 5. (a) INITIAL POSITION AND (b) STEADY-STATE SIMULATION RESULT WITH FEED RATE OF 5µm/s, WIRE TENSION OF 20 N, AND WIRE SPEED OF 10 m/s. THE DASH-LINE IS THE WIRE.
DISCUSSION
A numerical model and simulation results have been pre-sented in the previous sections. Based on this model, the slurry wire sawing process is discussed in this section. Figure 6(a) shows that the average normal loads at the edges of the cutting zone are much higher than those in the middle at the beginning of the slicing. The reason is because the wire did not reach its steady-state configuration yet. The wafer studied in this paper is in the shape of square. Therefore, the deviations of wire slopes at the edges of the cutting zone are much larger than others at the beginning of the slicing. However, once the wire reaches its balance configuration, the normal loads are even except those at the edges, as shown in Fig. 6(b). The loads at the edges
be-−60 −40 −20 0 20 40 60
FIGURE 6. AVERAGE NORMAL LOADS AND THEIR STAN-DARD DEVIATIONS OF (a) THE FIRST 1000 STEPS (t= 0 ∼ 880 s) AND (b) THE LAST 10000 STEPS (t= 8800 ∼ 17600 s) WITH FEED RATE OF 5µm/s, WIRE TENSION OF 20 N, AND WIRE SPEED OF 10 m/s. THE DASH-LINE IS THE WIRE.
come smaller than the others, which is opposite to the situation as shown in Fig. 6(a). From this discussion, it is obvious that the cutting edges are the force concentration at the start of slicing. It would be better to lower the feed rate or wire speed to avoid the rupture of the wire. However, lower feed rate or wire speed can increase the time to reach steady state of the slicing process.
Figure 7 shows the displacement at the middle of the wire with feed rates of 2µm/s and 5µm/s. The figure illustrates two
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0 0.5 1 1.5 2
FIGURE 7. HEIGHT VARIATION OF THE CENTER OF THE WORKPIECE WITH WIRE SPEED OF 10 m/s, WIRE TENSION OF 20 N, AND FEED RATE OF (a) 2µm/s and (b) 5µm/s.
different approaches to reach steady-state condition of the wire sawing process. The first one in Fig. 7(a) is overshooting. This is because the lower feed rate requires more time to reach steady state. The machining starts at the edges and propagates into the center of the workpiece. The deviations of the slopes at the mid-dle range are not sufficient to remove the material continuously and stably. Once the middle range starts to remove material ef-fectively, the height of the center of the ingot drops to the steady-state position. In contrast, the higher feed rate results in higher normal loads on the abrasives. A faster material removal propa-gates from the edges into the center of the workpiece. Therefore, the height of the center is not over the steady-state position dur-ing the slicdur-ing, as shown in Fig. 7(b).
In Table 3, it shows that the increase of feed rate will in-crease the average loads at the middle of the workpiece and the bow angles of the wire with the same wire speed and wire ten-sion. In addition, the average loads at the middle of the work-piece and bow angles of the wire will decrease with the increase of wire speed. It is expected that the higher wire speed will in-troduce more effective indentations within a unit time during the slicing process. Therefore, the loads and bow angles should not vary as much as those with lower wire speed. The lower average loads and bow angles indicate that the indentation cracks initiate immediately once the critical load reaches 0.03 N, and this keeps the lower loads on both the wire and the workpiece and the lower bow angles. Table 4 lists the results with another wire tension, T= 10 N, and the wire speed is 10 m/s. Comparing to the results in Table 3, it shows that the lower wire tension increases the bow angles. However, the average loads are only slight lower.
The simulation results show that the higher wire speed and wire tension can reduce bow angles, and the higher wire speed and lower wire tension can reduce the average normal loads. In addition to the wafer surface quality, another concern of the slic-ing process is the productivity. Therefore, the higher feed rate is desirable. However, the higher feed rate will increase the bow angle of the wire, which is considered as the risky index during slicing. If there is any sharp asperity in the cutting zone, it could result in the force concentration and rupture the wire. The higher wire speed and higher wire tension also make the wire more sen-sible to the sharp asperities or locally sticky conditions in the cutting zone.
CONCLUSIONS
In this paper, a preliminary numerical model was developed to study the local and gross slurry wire sawing process for sili-con wafers. In this model, the simulation results showed that the parameters including wire tension, wire speed, and feed rate af-fect the slicing process significantly. The increase of wire speed is suggested to lower the bow angle and normal loads. However, the limit of the wire speed depends on the material of the wire and the specification wire saw machine. In addition, slicing pro-cess will reach the steady-state slicing condition by two different approaches, overshooting or non-overshooting. This depends on the interaction of the slicing parameters. Although this model only considered the indentation cracks, the simulation results il-lustrate the complicated phenomena of slurry wire sawing pro-cess qualitatively.
ACKNOWLEDGEMENT
This project has been supported by National Science Coun-cil, Taiwan. Grant Number: NSC 100-2218-E-011-025.
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