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Nano-abrasion machining of brittle materials and its application to corrective figuring
Precision Engineering 31 (2007) 47–54
Nano-abrasion machining of brittle materials and its application to corrective figuring Osamu Horiuchi a,∗ , Junichi Ikeno b , Hideo Shibutani b , Hirofumi Suzuki c , Yoshiaki Mizukami d a
Department of Production Systems Engineering, Toyohashi University of Technology, 1-1 Hibarigaoka, Tempaku, Toyohashi 441-8580, Japan b Graduate School of Science and Engineering, Saitama University, 255 Shimo-ohkubo, Saitama 338-8570, Japan c Faculty of Engineering, Kobe University, 1-1 Rokkodai, Nada-ku, Kobe 657-8501, Japan d Pharmaceuticals and Medical Devices Agency, 3-3-2 Kasumigaseki, Chiyoda-ku, Tokyo 100-0013, Japan Received 31 August 2005; received in revised form 4 February 2006; accepted 27 February 2006 Available online 5 September 2006
Abstract A method of ultraprecision abrasion machining named “Nano-abrasion machining” is proposed for optical finishing of brittle materials. The fundamental characteristics and its applicability for corrective figuring to improve form accuracy of optics of brittle materials are investigated. It is experimentally ascertained that the material removal rate and surface roughness are suitable for optical finishing. However, the cross-sectional profile of the machined spot that is dependent on the collision angle is a combination of V- and W-shape, which is unsuitable for the corrective figuring. Therefore, circular motion machining is introduced and a preferable profile with an axis-symmetric V-shape is realized. The machining method is applied to corrective figuring of optical glass of BK7. The NC program is generated with a computer program developed by modifying the scanning motion and the form accuracy is predicted. According to the simulation results, corrective figuring is performed. The flatness is improved from PV = 151 to 29 nm. From the experimental results, it is clarified that the nano-abrasion machining is applicable to corrective figuring of brittle materials. © 2006 Elsevier Inc. All rights reserved. Keywords: Blasting; Ultraprecision machining; Corrective figuring; Brittle material
1. Introduction Recently, great advances have been accomplished in ultraprecision machining. However, it is rather difficult to obtain a few nanometers form accuracy in ultraprecision grinding of brittle materials. To improve the form accuracy, corrective figuring is performed by local material removal cooperated with feedback of the form error map. The methods used for the local removal are polishing with a small polisher [1,2], elastic emission machining (EEM) [3], ion beam machining [4,5], plasma assisted chemical vaporization machining (PCVM) [6], magnetorheological finishing (MRF) [7] and so on. In case of polishing with a small tool, because of tool wear, it is rather difficult for the small tool to keep a constant material removal rate and surface roughness and it may introduce a problem if the workpiece is large. The
∗
Corresponding author. Tel.: +81 532 44 6708; fax: +81 532 44 6690. E-mail address: [email protected] (O. Horiuchi).
0141-6359/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.precisioneng.2006.02.005
methods of EEM, ion beam machining and PCVM need special and expensive apparatuses, respectively. MRF requires a high-precision and high-cost machine but this method seems a promising method in manufacturing. In this paper, to develop a new low-cost method, a method of ultraprecision abrasion machining named “Nano-abrasion machining [8,9]” is proposed for the local removal and its fundamental machining characteristics and applicability to corrective figuring of brittle materials, i.e. optical glass, are investigated. 2. Nano-abrasion machining Nano-abrasion machining is similar with water jet abrasive machining or liquid honing. In water jet abrasive machining and liquid honing, abrasive grits contained in water are generally ejected under a high pressure from a nozzle and collide on the surface of work materials at a high speed. Referring to the erosion tests done by Sheldon and Finnie [10,11], if the collision energy is low and the collision angle is shallow, the abrasion rate
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may decrease down to a few nanometers per minute and ductile mode abrasion with a few nanometers surface roughness may occur even for brittle work materials. Therefore this method is named “Nano-abrasion machining”. The proposed method has a great advantage that the machining accuracy is almost independent of the machine accuracy because the standoff from the nozzle tip to the collision point is far enough. A warp of the workpiece induced by holding does not affect the machining accuracy. Therefore this method is suitable for repeating a post-process measurement and corrective figuring. Additionally, the machining characteristics are stable because there is a small tool wear. 3. Fundamental experiments 3.1. Apparatus and method Figs. 1 and 2 show the schematic of experimental apparatus and the setup of nozzle and workpiece, respectively. In the machining chamber, the workpiece was mounted on an X–Y table driven by an NC controller. The machining liquid composed of pure water and small amount of abrasive grits was fed to a nozzle by a special screw pump (Heishin Ltd., 12NE20PM, Japan) under a high pressure. The maximum ejecting pressure and flow rate of the pump are 7 MPa and 0.28 m3 /h, respectively. A nozzle made of ceramic was used for wear-proof.
Fig. 1. Schematic of the experimental apparatus.
Fig. 2. Setup of nozzle and workpiece.
To investigate the fundamental characteristics of nanoabrasion machining, a series of experiments was performed without scanning of the nozzle against the workpiece. The work materials were optical glass of BK7, Zerodur and single crystal silicon. The flat glass workpieces were prepared by polishing with surface roughness Rz = 4 nm and the silicon wafer Rz = 2 nm. The machining liquid containing 1 wt% abrasive grains of white fused alumina (WA) was mainly used. The grain mesh sizes and mean diameters were #4000 (3.0 m), #8000 (1.2 m), #10000 (0.6 m) and #30000 (0.3 m). The diameter of nozzle was 1 mm and the ejecting pressures were 2, 4 and 6 MPa, then the velocities of jet flow were 61, 91 and 106 m/s, respectively. The collision angle was set at 30, 45, 60 and 90◦ . The standoff was 20 mm. The machining time was kept constant as 1 h. After the experiments, the machined surface was observed by an interferometer microscope to measure the depth profile and surface roughness. 3.2. Experimental results and discussion Fig. 3 shows an example of removal spot observed by the interferometer microscope. The removal spot looks like a crescent moon but the machined area is an oval with approximately 3 mm × 6 mm in size and contour map is almost symmetric with respect to the center line of jet flow, as shown in Fig. 3(a and b). The longitudinal profile along the centerline has an asymmetric V-shape as shown in Fig. 3(c) and the maximum depth was about 1.0 m. As shown in Fig. 3(d), the transverse profiles have two different shapes: parabolic and W-shape. The profile at a location before the deepest point is parabolic and after that it changes to W-shape whose central hill height gradually increases. The Wshape profile is considered due to a local reduction of collision angle and velocity because a part of jet flow collided on the work surface and then the reflected flow interfered with the other part of jet flow. Fig. 3(e and f) shows surface roughness measured at the deepest point where the material removal is enough and the roughness is considered independent of that of prepared surface. In this case, the surface roughness seems fine and uniform. Fig. 4 shows the relationship between ejecting pressure and maximum depth of removal spot for various experiments when the collision angle was 30◦ . The maximum depth, which directly relates to the material removal rate, increases together with the ejecting pressure. This result corresponds to the fact that the material removal rate increases with collision energy e and collision frequency f. Here, e = mν2 /2, where m is mass of an abrasive grain and ν is the velocity of collision. The collision frequency f means the number of collisions per unit time and is equal to the number of abrasive grains colliding per unit time, n. Therefore f = n = αw/m = αρq/m = αρAν/m, where α is weight content of abrasive grains in machining liquid, w the mass flow rate of machining liquid, ρ the density of machining liquid, q the volumetric flow rate of machining liquid and A is the cross-sectional area of nozzle. The machining liquid contained a constant wt% of abrasive grains. Consequently, under an ejecting pressure, the product of collision energy and collision frequency is constant even for different grain sizes or different m, as follows: ef = mν2 /2 × αρAν/m = αρAν3 /2.
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Fig. 3. Example of removal spot on machined surface: work material, BK7; abrasive, WA#30000; ejecting pressure, 2 MPa; collision angle, 30◦ ; standoff, 20 mm; machining time, 1 h.
Fig. 4. Variation of maximum depth of removal spot with various ejecting pressures when the collision angle is 30◦ . (a) BK7, (b) Zerodur and (c) single crystal silicon.
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Fig. 5. Variation of surface roughness with various ejecting pressures when the collision angle is 30◦ .
The maximum depth for grain size #10000 is comparable or slightly smaller than that for #30000. However that for #8000 is apparently greater and for #4000 is much greater than for #10000 or #30000. Therefore the maximum depth is not considered proportional to the above-mentioned product. There are small differences among the work materials: the maximum depth of Zerodur is slightly greater than that of single crystal silicon but smaller than BK7. All of the maximum depths are greater than 0.1 m that is in most cases the maximum form error to be corrected, therefore it is considered that the material removal rate of this method is large enough and applicable to corrective figuring. Fig. 5 shows the relationship between ejecting pressure and surface roughness for same conditions as Fig. 4. The obtained surface roughness commonly increased together with ejecting pressure. The surface roughness except for grain size #4000 varied Rz = 10–40 nm for BK7 while Rz = 5–20 nm for Zerodur and silicon. For grain size #4000, it varied Rz = 20–55 nm for BK7, Rz = 20–40 nm for Zerodur and Rz = 50–140 nm for silicon. Generally the surface roughness was supposed to increase together with the grain diameter. However the differences of surface roughness between different grain sizes except for #4000 were small. From this result, optical surface roughness of Rz = 5–20 nm could be obtained with finer abrasive grains and under a lower ejecting pressure. Fig. 6 shows the relationship between collision angle and maximum depth of removal spot. The maximum depth decreases
with an increase of collision angle. This tendency coincides with the theoretical prediction and experimental results of ductile mode erosion of brittle materials [10]. When the nozzle was inclined, the removal spot always looked like a crescent moon and the contour map and profiles were basically similar to those shown in Fig. 3. The machined area changed its form from oval to circular as the collision angle increased. When the nozzle was
Fig. 6. Variation of maximum depth of removal spot with ejecting pressures and collision angle: work material, BK7; abrasives, WA#10000; standoff, 20 mm; machining time, 1 h.
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Fig. 7. Removal spot obtained for collision angle of 90◦ : work material, BK7; abrasive, WA#10000; ejecting pressure, 2 MPa; standoff, 20 mm; machining time, 1 h.
set upright, i.e. the collision angle was 90◦ , the removal spot was circular and the profile had an axis-symmetric W-shape, as shown in Fig. 7. 4. Application to corrective figuring 4.1. Circular motion machining For a local material removal in corrective figuring, a removal spot with a circular plane figure and axis-symmetric V-shape profiles may be preferable because it is easy to estimate quantity of removal by a computer simulation and make the NC program. To obtain such a removal spot, it is necessary to modify the form of removal spot for collision angle 90◦ by moving the nozzle or workpiece adequately. In this study, a circular motion of workpiece was employed to realize such a removal spot. To predict the profiles of removal spot, computer simulations were performed for work material of BK7 at various circular motion radii. In the simulations, a half of the diametrical profile of removal spot shown in Fig. 8 was expressed with a 10-order function, h(r) = a0 + a1 r + a2 r2 + ··· + a10 r10 , where h(r) was depth of profile at a point apart by r from the center of removal spot. If the nozzle moves around the origin of x–y coordinates with radius R and angular velocity ω, the center of removal spot locates at (X, Y), where X = R cos ωt and Y = R sin ωt. The total depth ht at an arbitrary point (x, y) was calculated as follows: ht = h(r) dt, where r = (x − X)2 + (y − Y )2 . Then the machining was performed with circular motion. The results of simulations and experiments are shown in Fig. 9. From the results of the simulations (Fig. 9(a)), it is found that the V-shape profile of removal spot, which seems suitable
for corrective figuring, is obtained when the radius of circular motion is from 1.0 to 1.4 mm. Fig. 9(b) shows the machined profiles with circular motion of the workpiece. It was ascertained that the removal spot with V-shape is achieved when the radius of circular motion is 1.0 mm. In this condition, the profile of the removal spot at various ejecting pressures and circular motion velocities of nozzle was experimentally measured. The maximum depth of removal spot increased with the ejecting pressure but inversely decreased with the circular motion velocity, and the profiles had the V-shape. From the results, the profile of removal spot can be controlled by the ejecting pressure or circular motion velocity. 4.2. Simulation for corrective figuring The concept of corrective figuring with circular motions is illustrated in Fig. 10. The circular motion is performed around every machining point distributed with a constant step on the scanning path. To calculate the machining accuracy in corrective figuring, it is necessary to determine the velocities of circular motions around every machining point. In this study, the X-direction step and the Y-direction step were 0.54 mm equal to the pixel size of flatness measurement by the laser interferometer. The step size a is much smaller than the diameter of removal spot d = 10 mm and resultant material removal at a point is a sum of the material removals obtained by machining at the point and a number of neighboring points. The resultant material removal at a point is determined as follows. At first, the diametrical profile of removal spot with a circular motion velocity V¯ and under standard machining conditions was expressed by a function FV¯ (r), as shown in
Fig. 8. Diametrical profiles of removal spot used for circular motion simulation: work material, BK7; abrasives, WA#10000; ejecting pressure, 4 MPa; collision angle, 90◦ ; machining time, 1 h.
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Fig. 11. Diametrical profile of standard machining spot.
Fig. 9. Results of simulations and experiments for circular motion machining: work material, BK7; abrasives, WA#10000; ejecting pressure, 4 MPa; collision angle, 90◦ ; standoff, 20 mm.
Fig. 11. As the material removal was inversely proportional to the velocity of circular motion, a material removal FV (r) for a circular motion velocity V can be calculated by the equation FV (r) = (V¯ /V )FV¯ (r). Next, as illustrated in Fig. 12, a sum of the material removals by machining at the point 0 (i = 0) and the neighboring points on a line is calculated as follows; V¯ FVi (ia) = F ¯ (ia) (i = 0, ±1, ±2, ±3, . . .) . Vi V i
i
Then the resultant material removal is determined by taking account of all the two-dimensionally neighboring points: V¯ FVi,j (r)= F ¯ (r) (i, j = 0, ±1, ±2, ±3, . . .) Vi,j V j
i
where r =
j
i
i2 + j 2 a.
As it is difficult to predict the velocities of circular motion at neighboring points, they were assumed to be equal to that at the point: Vi,j = V0,0 . Moreover, if V0,0 is equal to the standard velocity V¯ , the standard resultantmaterial removal at the point ¯ is expressed as follows: D ¯ = j i FV¯ (r). D Finally, if D(X, Y) is the material removal required for correction at a point (X, Y), the velocity of circular motion at that point ¯ Now, it is possible to is determined: V (X, Y ) = (V¯ /D(X, Y ))D. generate the NC program for corrective figuring. In the above calculation, the diametrical profile of removal spot shown in Fig. 11 was expressed by two different ways. One was “Cone approximation” where the profile was assumed to be triangular and another was “High order function” where the profile was expressed by a high-order function. The pre-machined surface had an axis-symmetric sinusoidal convex with a height δmax and wavelength λ. Actually δmax was around 200 nm and there were several different wavelengths λ. The machining intended to make a flat surface. After the generation of NC program, the machining accuracy was predicted and evaluated by computer simulation. Fig. 13 shows an example of predicted machining accuracy. Here, Emax is the maximum form error to be obtained by machining and the ratio Emax /δmax is indicated on the vertical axis. As shown in this figure, the accuracies noted “High order function” are much higher than those of “Cone approximation” but some larger errors still
Fig. 10. Concept of corrective figuring with circular motions.
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ing points were equal to that at the point. Therefore the points with larger residual error were selected and their velocities were intensively corrected as follows: if the residual error was plus or minus, the velocity was decreased or increased. The accuracies noted “Velocity correction” are the results obtained by this correction, which are apparently higher than those of “High order function”. From this figure, it is clear that the machining accuracy depends on the ratio of wavelength of form error λ to the diameter of removal spot d. For example, a sinusoidal form error with 200 nm height can be reduced to several nanometers, if λ/d > 2. 4.3. Experiments of corrective figuring
Fig. 12. Summing the material removals obtained by machining at neighboring points.
remain for smaller λ/d. As illustrated in Fig. 13, the residual form error had a similar shape with the pre-machined surface, i.e. an axis-symmetric sinusoidal convex with a greatly reduced height. The larger errors were considered due to the above-mentioned assumption that the velocities of circular motion at neighbor-
Circular motion machining was applied to corrective figuring of optical glass of BK7. The work surface was pre-machined so that it had axis-symmetric sinusoidal concaves with 200 nm height and 20 mm wavelength. The corrective figuring was performed to make a flat surface of 15 mm × 15 mm square. The NC program generated by the above-mentioned computer simulation was used. In this experiment, the machining liquid containing 0.8 wt% abrasive grains of white alumina (WA) was used. The grain mesh size and mean diameter were #10000 and 0.6 m. The diameter of nozzle was 1 mm and the ejecting pressure was 4 MPa, then the velocity of jet flow was 91 m/s. The standoff was 20 mm. The radius of circular motion with collision angle of 90◦ was 1.0 mm and step interval was 0.53 mm. After the experiments, the machined surface was evaluated by measuring the surface roughness and flatness. Fig. 14 shows an example of experimental result obtained by corrective figuring of one scanning. The flatness was improved from PV = 151 to 29 nm. The surface roughness after machining was Ra = 1.53 nm and slightly increases compared with
Fig. 13. Machining accuracy predicted by computer simulation.
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Fig. 14. Experimental result of corrective figuring: work material, BK7; abrasives, WA#10000; ejecting pressure, 4 MPa; collision angle, 90◦ ; step interval, 0.54 mm; machining time, 8 h. (a) Before machining: flatness, PV = 151 nm; surface roughness, Ra = 1.49 nm and (b) after machining: flatness, PV = 29 nm; surface roughness, Ra = 1.53 nm.
Ra = 1.49 nm of pre-machined surface. From the experimental results, the nano-abrasion machining is applicable to corrective figuring of brittle materials. 5. Conclusions A method of ultraprecision abrasion machining named “Nano-abrasion machining” was proposed for optical finishing of brittle materials and its fundamental characteristics and applicability for corrective figuring were investigated. The results of this investigation are summarized below: 1. Fundamental machining characteristics of the nano-abrasion machining with respect to the work materials of optical glass of BK7, Zerodur and single crystal silicon were investigated. It was ascertained that the machined surface roughness and the material removal rate are sufficiently suitable for corrective figuring of brittle materials. 2. The cross-sectional profile of the machined spot that was dependent on the collision angle had a combination of Vand W-shape, which is unsuitable for the corrective figuring. Therefore circular motion of workpiece with the collision angle of 90◦ was introduced and a preferable profile with an axis-symmetric V-shape was realized. 3. A computer program developed to generate the NC program for figuring could predict the machining accuracy. 4. Nano-abrasion machining was applied to corrective figuring of optical grass of BK7. The flatness was improved from PV = 151 to 29 nm. From the experimental results, it was clarified that the nano-abrasion machining is applicable to corrective figuring of brittle materials.
Acknowledgments The authors would like to thank Mr. Y. Yoshidomi of Nikon Corporation for preparation of workpieces and Dr. S. Itoh of Aichi Industrial Technology Institute for flatness measurements. References [1] Aspden R, McDonough R, Nitchie FR. Computer assisted optical surfacing. Appl Opt 1972;11(12):2739–47. [2] Ando M, Negishi M, Takimoto M, Deguchi A, Nakamura N, Higomura M, et al. Super-smooth surface polishing on aspherical optics. In: Proceedings of SPIE, vol. 1720. 1992. p. 22–33. [3] Mori Y, Yamauchi K, Endo K. Elastic emission machining. Prec Eng 1987;9(3):123–8. [4] Wilson SR, Reicher DW, McNeil JR. Surface figuring using neutral ion beams. In: Proceedings of SPIE, vol. 966. 1988. p. 74–81. [5] Weiser M, Kubler C, Fiedler KH, Beckstette KF. Particle beam figuring – an ideal tool for precision figuring of optics. In: Proceedings of the sixth IPES. 1991. p. 209–14. [6] Mori Y, Yamamura K, Yamauchi K, Yoshii K, Kataoka T, Endo K, et al. Plasma CVM (chemical vaporization machining): an ultra precision machining technique using high-pressure reactive plasma. Nanotechnology 1993;4(4):225–9. [7] Pollicove H, Jacobs S, Richard M, Ruckman J. Deterministic process for manufacturing perfect surface. In: Proceedings of the ninth international conference on precision engineering. 1999. p. 36–41. [8] Horiuchi O, Naruse F, Ikeno J. Nano-abrasion machining of brittle materials. In: Proceedings of the 14th ASPE annual meeting. 1999. p. 505–8. [9] Horiuchi O, Yamaguchi S, Suyama A, Shibutani H, Suzuki H. Corrective figuring of optical glass by nano-abrasion machining. In: Proceedings of the 18th ASPE annual meeting. 2003. p. 427–30. [10] Sheldon GL, Finnie I. On the ductile behavior of nominally brittle materials during erosive cutting. Trans ASME J Eng Ind 1966;88:387–92. [11] Sheldon GL, Finnie I. The machining of material removal in the erosive cutting of brittle materials. Trans ASME J Eng Ind 1966;88:393–9.
Nano-abrasion machining of brittle materials and its application to corrective figuring Osamu Horiuchi a,∗ , Junichi Ikeno b , Hideo Shibutani b , Hirofumi Suzuki c , Yoshiaki Mizukami d a
Department of Production Systems Engineering, Toyohashi University of Technology, 1-1 Hibarigaoka, Tempaku, Toyohashi 441-8580, Japan b Graduate School of Science and Engineering, Saitama University, 255 Shimo-ohkubo, Saitama 338-8570, Japan c Faculty of Engineering, Kobe University, 1-1 Rokkodai, Nada-ku, Kobe 657-8501, Japan d Pharmaceuticals and Medical Devices Agency, 3-3-2 Kasumigaseki, Chiyoda-ku, Tokyo 100-0013, Japan Received 31 August 2005; received in revised form 4 February 2006; accepted 27 February 2006 Available online 5 September 2006
Abstract A method of ultraprecision abrasion machining named “Nano-abrasion machining” is proposed for optical finishing of brittle materials. The fundamental characteristics and its applicability for corrective figuring to improve form accuracy of optics of brittle materials are investigated. It is experimentally ascertained that the material removal rate and surface roughness are suitable for optical finishing. However, the cross-sectional profile of the machined spot that is dependent on the collision angle is a combination of V- and W-shape, which is unsuitable for the corrective figuring. Therefore, circular motion machining is introduced and a preferable profile with an axis-symmetric V-shape is realized. The machining method is applied to corrective figuring of optical glass of BK7. The NC program is generated with a computer program developed by modifying the scanning motion and the form accuracy is predicted. According to the simulation results, corrective figuring is performed. The flatness is improved from PV = 151 to 29 nm. From the experimental results, it is clarified that the nano-abrasion machining is applicable to corrective figuring of brittle materials. © 2006 Elsevier Inc. All rights reserved. Keywords: Blasting; Ultraprecision machining; Corrective figuring; Brittle material
1. Introduction Recently, great advances have been accomplished in ultraprecision machining. However, it is rather difficult to obtain a few nanometers form accuracy in ultraprecision grinding of brittle materials. To improve the form accuracy, corrective figuring is performed by local material removal cooperated with feedback of the form error map. The methods used for the local removal are polishing with a small polisher [1,2], elastic emission machining (EEM) [3], ion beam machining [4,5], plasma assisted chemical vaporization machining (PCVM) [6], magnetorheological finishing (MRF) [7] and so on. In case of polishing with a small tool, because of tool wear, it is rather difficult for the small tool to keep a constant material removal rate and surface roughness and it may introduce a problem if the workpiece is large. The
∗
Corresponding author. Tel.: +81 532 44 6708; fax: +81 532 44 6690. E-mail address: [email protected] (O. Horiuchi).
0141-6359/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.precisioneng.2006.02.005
methods of EEM, ion beam machining and PCVM need special and expensive apparatuses, respectively. MRF requires a high-precision and high-cost machine but this method seems a promising method in manufacturing. In this paper, to develop a new low-cost method, a method of ultraprecision abrasion machining named “Nano-abrasion machining [8,9]” is proposed for the local removal and its fundamental machining characteristics and applicability to corrective figuring of brittle materials, i.e. optical glass, are investigated. 2. Nano-abrasion machining Nano-abrasion machining is similar with water jet abrasive machining or liquid honing. In water jet abrasive machining and liquid honing, abrasive grits contained in water are generally ejected under a high pressure from a nozzle and collide on the surface of work materials at a high speed. Referring to the erosion tests done by Sheldon and Finnie [10,11], if the collision energy is low and the collision angle is shallow, the abrasion rate
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may decrease down to a few nanometers per minute and ductile mode abrasion with a few nanometers surface roughness may occur even for brittle work materials. Therefore this method is named “Nano-abrasion machining”. The proposed method has a great advantage that the machining accuracy is almost independent of the machine accuracy because the standoff from the nozzle tip to the collision point is far enough. A warp of the workpiece induced by holding does not affect the machining accuracy. Therefore this method is suitable for repeating a post-process measurement and corrective figuring. Additionally, the machining characteristics are stable because there is a small tool wear. 3. Fundamental experiments 3.1. Apparatus and method Figs. 1 and 2 show the schematic of experimental apparatus and the setup of nozzle and workpiece, respectively. In the machining chamber, the workpiece was mounted on an X–Y table driven by an NC controller. The machining liquid composed of pure water and small amount of abrasive grits was fed to a nozzle by a special screw pump (Heishin Ltd., 12NE20PM, Japan) under a high pressure. The maximum ejecting pressure and flow rate of the pump are 7 MPa and 0.28 m3 /h, respectively. A nozzle made of ceramic was used for wear-proof.
Fig. 1. Schematic of the experimental apparatus.
Fig. 2. Setup of nozzle and workpiece.
To investigate the fundamental characteristics of nanoabrasion machining, a series of experiments was performed without scanning of the nozzle against the workpiece. The work materials were optical glass of BK7, Zerodur and single crystal silicon. The flat glass workpieces were prepared by polishing with surface roughness Rz = 4 nm and the silicon wafer Rz = 2 nm. The machining liquid containing 1 wt% abrasive grains of white fused alumina (WA) was mainly used. The grain mesh sizes and mean diameters were #4000 (3.0 m), #8000 (1.2 m), #10000 (0.6 m) and #30000 (0.3 m). The diameter of nozzle was 1 mm and the ejecting pressures were 2, 4 and 6 MPa, then the velocities of jet flow were 61, 91 and 106 m/s, respectively. The collision angle was set at 30, 45, 60 and 90◦ . The standoff was 20 mm. The machining time was kept constant as 1 h. After the experiments, the machined surface was observed by an interferometer microscope to measure the depth profile and surface roughness. 3.2. Experimental results and discussion Fig. 3 shows an example of removal spot observed by the interferometer microscope. The removal spot looks like a crescent moon but the machined area is an oval with approximately 3 mm × 6 mm in size and contour map is almost symmetric with respect to the center line of jet flow, as shown in Fig. 3(a and b). The longitudinal profile along the centerline has an asymmetric V-shape as shown in Fig. 3(c) and the maximum depth was about 1.0 m. As shown in Fig. 3(d), the transverse profiles have two different shapes: parabolic and W-shape. The profile at a location before the deepest point is parabolic and after that it changes to W-shape whose central hill height gradually increases. The Wshape profile is considered due to a local reduction of collision angle and velocity because a part of jet flow collided on the work surface and then the reflected flow interfered with the other part of jet flow. Fig. 3(e and f) shows surface roughness measured at the deepest point where the material removal is enough and the roughness is considered independent of that of prepared surface. In this case, the surface roughness seems fine and uniform. Fig. 4 shows the relationship between ejecting pressure and maximum depth of removal spot for various experiments when the collision angle was 30◦ . The maximum depth, which directly relates to the material removal rate, increases together with the ejecting pressure. This result corresponds to the fact that the material removal rate increases with collision energy e and collision frequency f. Here, e = mν2 /2, where m is mass of an abrasive grain and ν is the velocity of collision. The collision frequency f means the number of collisions per unit time and is equal to the number of abrasive grains colliding per unit time, n. Therefore f = n = αw/m = αρq/m = αρAν/m, where α is weight content of abrasive grains in machining liquid, w the mass flow rate of machining liquid, ρ the density of machining liquid, q the volumetric flow rate of machining liquid and A is the cross-sectional area of nozzle. The machining liquid contained a constant wt% of abrasive grains. Consequently, under an ejecting pressure, the product of collision energy and collision frequency is constant even for different grain sizes or different m, as follows: ef = mν2 /2 × αρAν/m = αρAν3 /2.
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Fig. 3. Example of removal spot on machined surface: work material, BK7; abrasive, WA#30000; ejecting pressure, 2 MPa; collision angle, 30◦ ; standoff, 20 mm; machining time, 1 h.
Fig. 4. Variation of maximum depth of removal spot with various ejecting pressures when the collision angle is 30◦ . (a) BK7, (b) Zerodur and (c) single crystal silicon.
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Fig. 5. Variation of surface roughness with various ejecting pressures when the collision angle is 30◦ .
The maximum depth for grain size #10000 is comparable or slightly smaller than that for #30000. However that for #8000 is apparently greater and for #4000 is much greater than for #10000 or #30000. Therefore the maximum depth is not considered proportional to the above-mentioned product. There are small differences among the work materials: the maximum depth of Zerodur is slightly greater than that of single crystal silicon but smaller than BK7. All of the maximum depths are greater than 0.1 m that is in most cases the maximum form error to be corrected, therefore it is considered that the material removal rate of this method is large enough and applicable to corrective figuring. Fig. 5 shows the relationship between ejecting pressure and surface roughness for same conditions as Fig. 4. The obtained surface roughness commonly increased together with ejecting pressure. The surface roughness except for grain size #4000 varied Rz = 10–40 nm for BK7 while Rz = 5–20 nm for Zerodur and silicon. For grain size #4000, it varied Rz = 20–55 nm for BK7, Rz = 20–40 nm for Zerodur and Rz = 50–140 nm for silicon. Generally the surface roughness was supposed to increase together with the grain diameter. However the differences of surface roughness between different grain sizes except for #4000 were small. From this result, optical surface roughness of Rz = 5–20 nm could be obtained with finer abrasive grains and under a lower ejecting pressure. Fig. 6 shows the relationship between collision angle and maximum depth of removal spot. The maximum depth decreases
with an increase of collision angle. This tendency coincides with the theoretical prediction and experimental results of ductile mode erosion of brittle materials [10]. When the nozzle was inclined, the removal spot always looked like a crescent moon and the contour map and profiles were basically similar to those shown in Fig. 3. The machined area changed its form from oval to circular as the collision angle increased. When the nozzle was
Fig. 6. Variation of maximum depth of removal spot with ejecting pressures and collision angle: work material, BK7; abrasives, WA#10000; standoff, 20 mm; machining time, 1 h.
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Fig. 7. Removal spot obtained for collision angle of 90◦ : work material, BK7; abrasive, WA#10000; ejecting pressure, 2 MPa; standoff, 20 mm; machining time, 1 h.
set upright, i.e. the collision angle was 90◦ , the removal spot was circular and the profile had an axis-symmetric W-shape, as shown in Fig. 7. 4. Application to corrective figuring 4.1. Circular motion machining For a local material removal in corrective figuring, a removal spot with a circular plane figure and axis-symmetric V-shape profiles may be preferable because it is easy to estimate quantity of removal by a computer simulation and make the NC program. To obtain such a removal spot, it is necessary to modify the form of removal spot for collision angle 90◦ by moving the nozzle or workpiece adequately. In this study, a circular motion of workpiece was employed to realize such a removal spot. To predict the profiles of removal spot, computer simulations were performed for work material of BK7 at various circular motion radii. In the simulations, a half of the diametrical profile of removal spot shown in Fig. 8 was expressed with a 10-order function, h(r) = a0 + a1 r + a2 r2 + ··· + a10 r10 , where h(r) was depth of profile at a point apart by r from the center of removal spot. If the nozzle moves around the origin of x–y coordinates with radius R and angular velocity ω, the center of removal spot locates at (X, Y), where X = R cos ωt and Y = R sin ωt. The total depth ht at an arbitrary point (x, y) was calculated as follows: ht = h(r) dt, where r = (x − X)2 + (y − Y )2 . Then the machining was performed with circular motion. The results of simulations and experiments are shown in Fig. 9. From the results of the simulations (Fig. 9(a)), it is found that the V-shape profile of removal spot, which seems suitable
for corrective figuring, is obtained when the radius of circular motion is from 1.0 to 1.4 mm. Fig. 9(b) shows the machined profiles with circular motion of the workpiece. It was ascertained that the removal spot with V-shape is achieved when the radius of circular motion is 1.0 mm. In this condition, the profile of the removal spot at various ejecting pressures and circular motion velocities of nozzle was experimentally measured. The maximum depth of removal spot increased with the ejecting pressure but inversely decreased with the circular motion velocity, and the profiles had the V-shape. From the results, the profile of removal spot can be controlled by the ejecting pressure or circular motion velocity. 4.2. Simulation for corrective figuring The concept of corrective figuring with circular motions is illustrated in Fig. 10. The circular motion is performed around every machining point distributed with a constant step on the scanning path. To calculate the machining accuracy in corrective figuring, it is necessary to determine the velocities of circular motions around every machining point. In this study, the X-direction step and the Y-direction step were 0.54 mm equal to the pixel size of flatness measurement by the laser interferometer. The step size a is much smaller than the diameter of removal spot d = 10 mm and resultant material removal at a point is a sum of the material removals obtained by machining at the point and a number of neighboring points. The resultant material removal at a point is determined as follows. At first, the diametrical profile of removal spot with a circular motion velocity V¯ and under standard machining conditions was expressed by a function FV¯ (r), as shown in
Fig. 8. Diametrical profiles of removal spot used for circular motion simulation: work material, BK7; abrasives, WA#10000; ejecting pressure, 4 MPa; collision angle, 90◦ ; machining time, 1 h.
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Fig. 11. Diametrical profile of standard machining spot.
Fig. 9. Results of simulations and experiments for circular motion machining: work material, BK7; abrasives, WA#10000; ejecting pressure, 4 MPa; collision angle, 90◦ ; standoff, 20 mm.
Fig. 11. As the material removal was inversely proportional to the velocity of circular motion, a material removal FV (r) for a circular motion velocity V can be calculated by the equation FV (r) = (V¯ /V )FV¯ (r). Next, as illustrated in Fig. 12, a sum of the material removals by machining at the point 0 (i = 0) and the neighboring points on a line is calculated as follows; V¯ FVi (ia) = F ¯ (ia) (i = 0, ±1, ±2, ±3, . . .) . Vi V i
i
Then the resultant material removal is determined by taking account of all the two-dimensionally neighboring points: V¯ FVi,j (r)= F ¯ (r) (i, j = 0, ±1, ±2, ±3, . . .) Vi,j V j
i
where r =
j
i
i2 + j 2 a.
As it is difficult to predict the velocities of circular motion at neighboring points, they were assumed to be equal to that at the point: Vi,j = V0,0 . Moreover, if V0,0 is equal to the standard velocity V¯ , the standard resultantmaterial removal at the point ¯ is expressed as follows: D ¯ = j i FV¯ (r). D Finally, if D(X, Y) is the material removal required for correction at a point (X, Y), the velocity of circular motion at that point ¯ Now, it is possible to is determined: V (X, Y ) = (V¯ /D(X, Y ))D. generate the NC program for corrective figuring. In the above calculation, the diametrical profile of removal spot shown in Fig. 11 was expressed by two different ways. One was “Cone approximation” where the profile was assumed to be triangular and another was “High order function” where the profile was expressed by a high-order function. The pre-machined surface had an axis-symmetric sinusoidal convex with a height δmax and wavelength λ. Actually δmax was around 200 nm and there were several different wavelengths λ. The machining intended to make a flat surface. After the generation of NC program, the machining accuracy was predicted and evaluated by computer simulation. Fig. 13 shows an example of predicted machining accuracy. Here, Emax is the maximum form error to be obtained by machining and the ratio Emax /δmax is indicated on the vertical axis. As shown in this figure, the accuracies noted “High order function” are much higher than those of “Cone approximation” but some larger errors still
Fig. 10. Concept of corrective figuring with circular motions.
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ing points were equal to that at the point. Therefore the points with larger residual error were selected and their velocities were intensively corrected as follows: if the residual error was plus or minus, the velocity was decreased or increased. The accuracies noted “Velocity correction” are the results obtained by this correction, which are apparently higher than those of “High order function”. From this figure, it is clear that the machining accuracy depends on the ratio of wavelength of form error λ to the diameter of removal spot d. For example, a sinusoidal form error with 200 nm height can be reduced to several nanometers, if λ/d > 2. 4.3. Experiments of corrective figuring
Fig. 12. Summing the material removals obtained by machining at neighboring points.
remain for smaller λ/d. As illustrated in Fig. 13, the residual form error had a similar shape with the pre-machined surface, i.e. an axis-symmetric sinusoidal convex with a greatly reduced height. The larger errors were considered due to the above-mentioned assumption that the velocities of circular motion at neighbor-
Circular motion machining was applied to corrective figuring of optical glass of BK7. The work surface was pre-machined so that it had axis-symmetric sinusoidal concaves with 200 nm height and 20 mm wavelength. The corrective figuring was performed to make a flat surface of 15 mm × 15 mm square. The NC program generated by the above-mentioned computer simulation was used. In this experiment, the machining liquid containing 0.8 wt% abrasive grains of white alumina (WA) was used. The grain mesh size and mean diameter were #10000 and 0.6 m. The diameter of nozzle was 1 mm and the ejecting pressure was 4 MPa, then the velocity of jet flow was 91 m/s. The standoff was 20 mm. The radius of circular motion with collision angle of 90◦ was 1.0 mm and step interval was 0.53 mm. After the experiments, the machined surface was evaluated by measuring the surface roughness and flatness. Fig. 14 shows an example of experimental result obtained by corrective figuring of one scanning. The flatness was improved from PV = 151 to 29 nm. The surface roughness after machining was Ra = 1.53 nm and slightly increases compared with
Fig. 13. Machining accuracy predicted by computer simulation.
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Fig. 14. Experimental result of corrective figuring: work material, BK7; abrasives, WA#10000; ejecting pressure, 4 MPa; collision angle, 90◦ ; step interval, 0.54 mm; machining time, 8 h. (a) Before machining: flatness, PV = 151 nm; surface roughness, Ra = 1.49 nm and (b) after machining: flatness, PV = 29 nm; surface roughness, Ra = 1.53 nm.
Ra = 1.49 nm of pre-machined surface. From the experimental results, the nano-abrasion machining is applicable to corrective figuring of brittle materials. 5. Conclusions A method of ultraprecision abrasion machining named “Nano-abrasion machining” was proposed for optical finishing of brittle materials and its fundamental characteristics and applicability for corrective figuring were investigated. The results of this investigation are summarized below: 1. Fundamental machining characteristics of the nano-abrasion machining with respect to the work materials of optical glass of BK7, Zerodur and single crystal silicon were investigated. It was ascertained that the machined surface roughness and the material removal rate are sufficiently suitable for corrective figuring of brittle materials. 2. The cross-sectional profile of the machined spot that was dependent on the collision angle had a combination of Vand W-shape, which is unsuitable for the corrective figuring. Therefore circular motion of workpiece with the collision angle of 90◦ was introduced and a preferable profile with an axis-symmetric V-shape was realized. 3. A computer program developed to generate the NC program for figuring could predict the machining accuracy. 4. Nano-abrasion machining was applied to corrective figuring of optical grass of BK7. The flatness was improved from PV = 151 to 29 nm. From the experimental results, it was clarified that the nano-abrasion machining is applicable to corrective figuring of brittle materials.
Acknowledgments The authors would like to thank Mr. Y. Yoshidomi of Nikon Corporation for preparation of workpieces and Dr. S. Itoh of Aichi Industrial Technology Institute for flatness measurements. References [1] Aspden R, McDonough R, Nitchie FR. Computer assisted optical surfacing. Appl Opt 1972;11(12):2739–47. [2] Ando M, Negishi M, Takimoto M, Deguchi A, Nakamura N, Higomura M, et al. Super-smooth surface polishing on aspherical optics. In: Proceedings of SPIE, vol. 1720. 1992. p. 22–33. [3] Mori Y, Yamauchi K, Endo K. Elastic emission machining. Prec Eng 1987;9(3):123–8. [4] Wilson SR, Reicher DW, McNeil JR. Surface figuring using neutral ion beams. In: Proceedings of SPIE, vol. 966. 1988. p. 74–81. [5] Weiser M, Kubler C, Fiedler KH, Beckstette KF. Particle beam figuring – an ideal tool for precision figuring of optics. In: Proceedings of the sixth IPES. 1991. p. 209–14. [6] Mori Y, Yamamura K, Yamauchi K, Yoshii K, Kataoka T, Endo K, et al. Plasma CVM (chemical vaporization machining): an ultra precision machining technique using high-pressure reactive plasma. Nanotechnology 1993;4(4):225–9. [7] Pollicove H, Jacobs S, Richard M, Ruckman J. Deterministic process for manufacturing perfect surface. In: Proceedings of the ninth international conference on precision engineering. 1999. p. 36–41. [8] Horiuchi O, Naruse F, Ikeno J. Nano-abrasion machining of brittle materials. In: Proceedings of the 14th ASPE annual meeting. 1999. p. 505–8. [9] Horiuchi O, Yamaguchi S, Suyama A, Shibutani H, Suzuki H. Corrective figuring of optical glass by nano-abrasion machining. In: Proceedings of the 18th ASPE annual meeting. 2003. p. 427–30. [10] Sheldon GL, Finnie I. On the ductile behavior of nominally brittle materials during erosive cutting. Trans ASME J Eng Ind 1966;88:387–92. [11] Sheldon GL, Finnie I. The machining of material removal in the erosive cutting of brittle materials. Trans ASME J Eng Ind 1966;88:393–9.