Abstract

Single point diamond turning (SPDT) is highly versatile in fabricating axially symmetric form, non-axially-symmetric form and free form surfaces. However, inevitable microstructure known as turning marks left on the surface have limited the mirror’s optical performance. Based on chemical mechanical polishing (CMP) mechanism, smoothing polishing (SP) process is believed to be an effective method to remove turning marks. However, the removal efficiency is relatively low. In this paper, based on Greenwood-Williamson (GW) theory, the factors that limit removal efficiency of SP are discussed in details. Influences of process parameters (work pressure and rotational speed) are firstly discussed. With further analysis, surface spectral characteristics are identified as the inherent factor affecting further efficiency improvement. According to theoretical analysis, the removal efficiency of isotropic surface is nearly 1.8 times higher than anisotropy surface like surface with turning marks. A high efficiency turning marks removal process combining ion beam sputtering (IBS) and SP is proposed in our research. With removal depth exceeding 100 nm, the isotropic aluminum surface can be constructed by IBS so that the efficiency of SP process can be greatly improved. Though deteriorated by IBS, the surface roughness will be rapidly reduced by SP process. Finally, experiments are conducted to verify our analysis. A 3.7 nm roughness surface without turning marks is achieved by new method while direct SP can only reach roughness of 4.3 nm with evident turning marks. Experimental results show that removal efficiency nearly doubled which matches well with the theoretical analysis. Our research not only can be used as a high efficiency turning marks removal and surface quality improvement method but also can be a new method for high precision aluminum optics fabrication.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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    [Crossref]

2020 (1)

2019 (1)

Z. Zhu, S. To, W. L. Zhu, P. Huang, and X. Zhou, “Cutting forces in fast-/slow tool servo diamond turning of micro-structured surfaces,” Int. J. Mach. Tools Manuf. 136, 62–75 (2019).
[Crossref]

2018 (1)

W. Zhu, F. Duan, X. Zhang, Z. Zhu, and B. Ju, “A new diamond machining approach for extendable fabrication of micro-freeform lens array,” Int. J. Mach. Tools Manuf. 124, 134–148 (2018).
[Crossref]

2015 (4)

2014 (3)

2013 (2)

A. Beaucamp and Y. Namba, “Super-smooth finishing of diamond turned hard X-ray molding dies by combined fluid jet and bonnet polishing,” CIRP Ann.- Manuf. Tech. 62(1), 315–318 (2013).
[Crossref]

J. E. Harvey, “Parametric analysis of the effect of scattered light upon the modulation transfer function,” Opt. Eng. 52(7), 073110 (2013).
[Crossref]

2012 (1)

H. Gong, F. Z. H. Fang, and X. T. Hu, “Accurate spiral tool path generation of ultraprecision three-axis turning for non- zero rake angle using symbolic computation,” Int. J. Adv. Manuf Technol. 58(9-12), 841–847 (2012).
[Crossref]

2011 (2)

Z. Z. Li, J. M. Wang, X. Q. Peng, L. T. Ho, Z. Q. Yin, S. Y. Li, and C. F. Cheung, “Removal of single point diamond-turning marks by abrasive jet polishing,” Appl. Opt. 50(16), 2458–2463 (2011).
[Crossref]

J. D. Nelson, A. Gould, C. Klinger, and M. Mandina, “Incorporating VIBE into the precision optics manufacturing process,” Proc. SPIE 8126, 812613 (2011).
[Crossref]

2010 (1)

2009 (3)

2008 (1)

2007 (2)

P. Dumas, C. Hall, B. Hallock, and M. Tricard, “Complete subaperture pre-polishing & finishing solution to improve speed and determinism in asphere manufacture,” Proc. SPIE 6671, 667111 (2007).
[Crossref]

D. D. Walker, A. Baldwin, R. Evans, R. Freeman, S. Hamidi, P. Shore, X. Tonnellier, S. Wei, C. Williams, and G. Yu, “A quantitative comparison of three grolishing techniques for the Precessions™ process,” Proc. SPIE 6671, 66711H (2007).
[Crossref]

2005 (3)

P. Dumas, D. Golini, and M. Tricard, “Improvement of figure and finish of diamond turned surfaces with magneto-rheological finishing,” Proc. SPIE 5786, 296–304 (2005).
[Crossref]

H. Y. Wu, W. B. Lee, C. F. Cheung, S. To, and Y. P. Chen, “Computer simulation of single-point diamond turning using finite element method,” J. Mater. Process. Technol. 167(2-3), 549–554 (2005).
[Crossref]

J. C. Stover, “Light scatter metrology of diamond turned optics,” Proc. SPIE 5878, 58780R (2005).
[Crossref]

2004 (2)

K. Qin, B. Moudgil, and C. W. Park, “A chemical mechanical polishing model incorporating both the chemical and mechanical effects,” Thin Solid Films 446(2), 277–286 (2004).
[Crossref]

A. Cordero-Dávila, J. González-García, M. Pedrayes-López, L. A. Aguilar-Chiu, J. Cuautle-Cortes, and C. Robledo-Sanchez, “Edge effects with the preston equation for a circular tool and workpiece,” Appl. Opt. 43(6), 1250–1254 (2004).
[Crossref]

2002 (1)

M. T. Tuell, J. H. Burge, and B. Anderson, “Aspheric optics: smoothing the ripples with semi-flexible tools,” Opt. Eng. 41(7), 1473–1474 (2002).
[Crossref]

1987 (1)

J. I. Mccool, “Relating Profile Instrument Measurements to the Functional Performance of Rough Surfaces,” Asme. J. Trib. 109(2), 264–270 (1987).
[Crossref]

1975 (1)

1966 (1)

J. A. Greenwood and J. B. P. Williamson, “Contact of Nominally Flat Surfaces,” Proc. R. Soc. Lond. A 295, 300–319 (1966).
[Crossref]

Aguilar-Chiu, L. A.

An, H. K.

Anderson, B.

M. T. Tuell, J. H. Burge, and B. Anderson, “Aspheric optics: smoothing the ripples with semi-flexible tools,” Opt. Eng. 41(7), 1473–1474 (2002).
[Crossref]

Baldwin, A.

D. D. Walker, A. Baldwin, R. Evans, R. Freeman, S. Hamidi, P. Shore, X. Tonnellier, S. Wei, C. Williams, and G. Yu, “A quantitative comparison of three grolishing techniques for the Precessions™ process,” Proc. SPIE 6671, 66711H (2007).
[Crossref]

Beaucamp, A.

A. Beaucamp and Y. Namba, “Super-smooth finishing of diamond turned hard X-ray molding dies by combined fluid jet and bonnet polishing,” CIRP Ann.- Manuf. Tech. 62(1), 315–318 (2013).
[Crossref]

Burge, J. H.

Chen, Y. P.

H. Y. Wu, W. B. Lee, C. F. Cheung, S. To, and Y. P. Chen, “Computer simulation of single-point diamond turning using finite element method,” J. Mater. Process. Technol. 167(2-3), 549–554 (2005).
[Crossref]

Cheung, C. F.

Z. Z. Li, J. M. Wang, X. Q. Peng, L. T. Ho, Z. Q. Yin, S. Y. Li, and C. F. Cheung, “Removal of single point diamond-turning marks by abrasive jet polishing,” Appl. Opt. 50(16), 2458–2463 (2011).
[Crossref]

H. Y. Wu, W. B. Lee, C. F. Cheung, S. To, and Y. P. Chen, “Computer simulation of single-point diamond turning using finite element method,” J. Mater. Process. Technol. 167(2-3), 549–554 (2005).
[Crossref]

Church, E. L.

Cordero-Dávila, A.

Cuautle-Cortes, J.

Dai, Y. F.

C. Y. Du, Y. F. Dai, H. Hu, and C. L. Guan, “Surface roughness evolution mechanism of the optical aluminum 6061 alloy during low energy Ar+ ion beam sputtering,” Opt. Express 28(23), 34054–34068 (2020).
[Crossref]

L. Zhou, Y. F. Dai, X. H. Xie, and S. Y. Li, “Frequency-domain analysis of computer-controlled optical surfacing processes,” Sci. China Ser. E-Technol. Sci. 52(7), 2061–2068 (2009).
[Crossref]

Du, C. Y.

Duan, F.

W. Zhu, F. Duan, X. Zhang, Z. Zhu, and B. Ju, “A new diamond machining approach for extendable fabrication of micro-freeform lens array,” Int. J. Mach. Tools Manuf. 124, 134–148 (2018).
[Crossref]

Dumas, P.

P. Dumas, C. Hall, B. Hallock, and M. Tricard, “Complete subaperture pre-polishing & finishing solution to improve speed and determinism in asphere manufacture,” Proc. SPIE 6671, 667111 (2007).
[Crossref]

P. Dumas, D. Golini, and M. Tricard, “Improvement of figure and finish of diamond turned surfaces with magneto-rheological finishing,” Proc. SPIE 5786, 296–304 (2005).
[Crossref]

Evans, R.

D. D. Walker, A. Baldwin, R. Evans, R. Freeman, S. Hamidi, P. Shore, X. Tonnellier, S. Wei, C. Williams, and G. Yu, “A quantitative comparison of three grolishing techniques for the Precessions™ process,” Proc. SPIE 6671, 66711H (2007).
[Crossref]

Fang, F.

F. Fang, K. T. Huang, H. Gong, and Z. Li, “Study on the optical reflection characteristic of surface micromorphology generated by ultra-precision diamond turning,” Opt. Lasers Eng. 62, 46–56 (2014).
[Crossref]

Fang, F. Z. H.

H. Gong, F. Z. H. Fang, and X. T. Hu, “Accurate spiral tool path generation of ultraprecision three-axis turning for non- zero rake angle using symbolic computation,” Int. J. Adv. Manuf Technol. 58(9-12), 841–847 (2012).
[Crossref]

F. Z. H. Fang, X. D. Zhang, and X. T. Hu, “Cylindrical coordinate machining of optical freeform surfaces,” Opt. Express 16(10), 7323–7329 (2008).
[Crossref]

Freeman, R.

D. D. Walker, A. Baldwin, R. Evans, R. Freeman, S. Hamidi, P. Shore, X. Tonnellier, S. Wei, C. Williams, and G. Yu, “A quantitative comparison of three grolishing techniques for the Precessions™ process,” Proc. SPIE 6671, 66711H (2007).
[Crossref]

Golini, D.

P. Dumas, D. Golini, and M. Tricard, “Improvement of figure and finish of diamond turned surfaces with magneto-rheological finishing,” Proc. SPIE 5786, 296–304 (2005).
[Crossref]

Gong, H.

F. Fang, K. T. Huang, H. Gong, and Z. Li, “Study on the optical reflection characteristic of surface micromorphology generated by ultra-precision diamond turning,” Opt. Lasers Eng. 62, 46–56 (2014).
[Crossref]

H. Gong, F. Z. H. Fang, and X. T. Hu, “Accurate spiral tool path generation of ultraprecision three-axis turning for non- zero rake angle using symbolic computation,” Int. J. Adv. Manuf Technol. 58(9-12), 841–847 (2012).
[Crossref]

González-García, J.

Gould, A.

J. D. Nelson, A. Gould, C. Klinger, and M. Mandina, “Incorporating VIBE into the precision optics manufacturing process,” Proc. SPIE 8126, 812613 (2011).
[Crossref]

Greenwood, J. A.

J. A. Greenwood and J. B. P. Williamson, “Contact of Nominally Flat Surfaces,” Proc. R. Soc. Lond. A 295, 300–319 (1966).
[Crossref]

Guan, C. L.

Hall, C.

P. Dumas, C. Hall, B. Hallock, and M. Tricard, “Complete subaperture pre-polishing & finishing solution to improve speed and determinism in asphere manufacture,” Proc. SPIE 6671, 667111 (2007).
[Crossref]

Hallock, B.

P. Dumas, C. Hall, B. Hallock, and M. Tricard, “Complete subaperture pre-polishing & finishing solution to improve speed and determinism in asphere manufacture,” Proc. SPIE 6671, 667111 (2007).
[Crossref]

Hamidi, S.

D. D. Walker, A. Baldwin, R. Evans, R. Freeman, S. Hamidi, P. Shore, X. Tonnellier, S. Wei, C. Williams, and G. Yu, “A quantitative comparison of three grolishing techniques for the Precessions™ process,” Proc. SPIE 6671, 66711H (2007).
[Crossref]

Harvey, J. E.

J. E. Harvey, “Integrating optical fabrication and metrology into the optical design process,” Appl. Opt. 54(9), 2224 (2015).
[Crossref]

J. E. Harvey, “Parametric analysis of the effect of scattered light upon the modulation transfer function,” Opt. Eng. 52(7), 073110 (2013).
[Crossref]

Ho, L. T.

Hu, H.

Hu, X. T.

H. Gong, F. Z. H. Fang, and X. T. Hu, “Accurate spiral tool path generation of ultraprecision three-axis turning for non- zero rake angle using symbolic computation,” Int. J. Adv. Manuf Technol. 58(9-12), 841–847 (2012).
[Crossref]

F. Z. H. Fang, X. D. Zhang, and X. T. Hu, “Cylindrical coordinate machining of optical freeform surfaces,” Opt. Express 16(10), 7323–7329 (2008).
[Crossref]

Huang, K. T.

F. Fang, K. T. Huang, H. Gong, and Z. Li, “Study on the optical reflection characteristic of surface micromorphology generated by ultra-precision diamond turning,” Opt. Lasers Eng. 62, 46–56 (2014).
[Crossref]

Huang, P.

Z. Zhu, S. To, W. L. Zhu, P. Huang, and X. Zhou, “Cutting forces in fast-/slow tool servo diamond turning of micro-structured surfaces,” Int. J. Mach. Tools Manuf. 136, 62–75 (2019).
[Crossref]

Hui, C. S.

P. Wang, T. Suet, and C. S. Hui, “Improvement of the diamond turned surface texture by bonnet polishing process,” Acta. Optica. Sinica. 35(3), 0322001 (2015).
[Crossref]

Ju, B.

W. Zhu, F. Duan, X. Zhang, Z. Zhu, and B. Ju, “A new diamond machining approach for extendable fabrication of micro-freeform lens array,” Int. J. Mach. Tools Manuf. 124, 134–148 (2018).
[Crossref]

Kim, D. W.

Kim, S. W.

Klinger, C.

J. D. Nelson, A. Gould, C. Klinger, and M. Mandina, “Incorporating VIBE into the precision optics manufacturing process,” Proc. SPIE 8126, 812613 (2011).
[Crossref]

Lee, W. B.

H. Y. Wu, W. B. Lee, C. F. Cheung, S. To, and Y. P. Chen, “Computer simulation of single-point diamond turning using finite element method,” J. Mater. Process. Technol. 167(2-3), 549–554 (2005).
[Crossref]

Li, Q.

Li, S. Y.

Li, Z.

F. Fang, K. T. Huang, H. Gong, and Z. Li, “Study on the optical reflection characteristic of surface micromorphology generated by ultra-precision diamond turning,” Opt. Lasers Eng. 62, 46–56 (2014).
[Crossref]

Li, Z. Z.

Mandina, M.

J. D. Nelson, A. Gould, C. Klinger, and M. Mandina, “Incorporating VIBE into the precision optics manufacturing process,” Proc. SPIE 8126, 812613 (2011).
[Crossref]

Mccool, J. I.

J. I. Mccool, “Relating Profile Instrument Measurements to the Functional Performance of Rough Surfaces,” Asme. J. Trib. 109(2), 264–270 (1987).
[Crossref]

Moudgil, B.

K. Qin, B. Moudgil, and C. W. Park, “A chemical mechanical polishing model incorporating both the chemical and mechanical effects,” Thin Solid Films 446(2), 277–286 (2004).
[Crossref]

Namba, Y.

A. Beaucamp and Y. Namba, “Super-smooth finishing of diamond turned hard X-ray molding dies by combined fluid jet and bonnet polishing,” CIRP Ann.- Manuf. Tech. 62(1), 315–318 (2013).
[Crossref]

Nelson, J. D.

J. D. Nelson, A. Gould, C. Klinger, and M. Mandina, “Incorporating VIBE into the precision optics manufacturing process,” Proc. SPIE 8126, 812613 (2011).
[Crossref]

Nie, X. Q.

Park, C. W.

K. Qin, B. Moudgil, and C. W. Park, “A chemical mechanical polishing model incorporating both the chemical and mechanical effects,” Thin Solid Films 446(2), 277–286 (2004).
[Crossref]

Park, W. H.

Pedrayes-López, M.

Peng, X. Q.

Qin, K.

K. Qin, B. Moudgil, and C. W. Park, “A chemical mechanical polishing model incorporating both the chemical and mechanical effects,” Thin Solid Films 446(2), 277–286 (2004).
[Crossref]

Robledo-Sanchez, C.

Shi, F.

Shore, P.

D. D. Walker, A. Baldwin, R. Evans, R. Freeman, S. Hamidi, P. Shore, X. Tonnellier, S. Wei, C. Williams, and G. Yu, “A quantitative comparison of three grolishing techniques for the Precessions™ process,” Proc. SPIE 6671, 66711H (2007).
[Crossref]

Stover, J. C.

J. C. Stover, “Light scatter metrology of diamond turned optics,” Proc. SPIE 5878, 58780R (2005).
[Crossref]

Suet, T.

P. Wang, T. Suet, and C. S. Hui, “Improvement of the diamond turned surface texture by bonnet polishing process,” Acta. Optica. Sinica. 35(3), 0322001 (2015).
[Crossref]

To, S.

Z. Zhu, S. To, W. L. Zhu, P. Huang, and X. Zhou, “Cutting forces in fast-/slow tool servo diamond turning of micro-structured surfaces,” Int. J. Mach. Tools Manuf. 136, 62–75 (2019).
[Crossref]

H. Y. Wu, W. B. Lee, C. F. Cheung, S. To, and Y. P. Chen, “Computer simulation of single-point diamond turning using finite element method,” J. Mater. Process. Technol. 167(2-3), 549–554 (2005).
[Crossref]

Tonnellier, X.

D. D. Walker, A. Baldwin, R. Evans, R. Freeman, S. Hamidi, P. Shore, X. Tonnellier, S. Wei, C. Williams, and G. Yu, “A quantitative comparison of three grolishing techniques for the Precessions™ process,” Proc. SPIE 6671, 66711H (2007).
[Crossref]

Tricard, M.

P. Dumas, C. Hall, B. Hallock, and M. Tricard, “Complete subaperture pre-polishing & finishing solution to improve speed and determinism in asphere manufacture,” Proc. SPIE 6671, 667111 (2007).
[Crossref]

P. Dumas, D. Golini, and M. Tricard, “Improvement of figure and finish of diamond turned surfaces with magneto-rheological finishing,” Proc. SPIE 5786, 296–304 (2005).
[Crossref]

Tuell, M. T.

M. T. Tuell, J. H. Burge, and B. Anderson, “Aspheric optics: smoothing the ripples with semi-flexible tools,” Opt. Eng. 41(7), 1473–1474 (2002).
[Crossref]

Walker, D. D.

D. D. Walker, A. Baldwin, R. Evans, R. Freeman, S. Hamidi, P. Shore, X. Tonnellier, S. Wei, C. Williams, and G. Yu, “A quantitative comparison of three grolishing techniques for the Precessions™ process,” Proc. SPIE 6671, 66711H (2007).
[Crossref]

Wang, J. M.

Wang, P.

P. Wang, T. Suet, and C. S. Hui, “Improvement of the diamond turned surface texture by bonnet polishing process,” Acta. Optica. Sinica. 35(3), 0322001 (2015).
[Crossref]

Wei, S.

D. D. Walker, A. Baldwin, R. Evans, R. Freeman, S. Hamidi, P. Shore, X. Tonnellier, S. Wei, C. Williams, and G. Yu, “A quantitative comparison of three grolishing techniques for the Precessions™ process,” Proc. SPIE 6671, 66711H (2007).
[Crossref]

Williams, C.

D. D. Walker, A. Baldwin, R. Evans, R. Freeman, S. Hamidi, P. Shore, X. Tonnellier, S. Wei, C. Williams, and G. Yu, “A quantitative comparison of three grolishing techniques for the Precessions™ process,” Proc. SPIE 6671, 66711H (2007).
[Crossref]

Williamson, J. B. P.

J. A. Greenwood and J. B. P. Williamson, “Contact of Nominally Flat Surfaces,” Proc. R. Soc. Lond. A 295, 300–319 (1966).
[Crossref]

Wu, H. Y.

H. Y. Wu, W. B. Lee, C. F. Cheung, S. To, and Y. P. Chen, “Computer simulation of single-point diamond turning using finite element method,” J. Mater. Process. Technol. 167(2-3), 549–554 (2005).
[Crossref]

Xie, X. H.

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Yin, G. J.

Yin, Z. Q.

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D. D. Walker, A. Baldwin, R. Evans, R. Freeman, S. Hamidi, P. Shore, X. Tonnellier, S. Wei, C. Williams, and G. Yu, “A quantitative comparison of three grolishing techniques for the Precessions™ process,” Proc. SPIE 6671, 66711H (2007).
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Z. Zhu, S. To, W. L. Zhu, P. Huang, and X. Zhou, “Cutting forces in fast-/slow tool servo diamond turning of micro-structured surfaces,” Int. J. Mach. Tools Manuf. 136, 62–75 (2019).
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Z. Zhu, S. To, W. L. Zhu, P. Huang, and X. Zhou, “Cutting forces in fast-/slow tool servo diamond turning of micro-structured surfaces,” Int. J. Mach. Tools Manuf. 136, 62–75 (2019).
[Crossref]

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Acta. Optica. Sinica. (1)

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[Crossref]

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F. Fang, K. T. Huang, H. Gong, and Z. Li, “Study on the optical reflection characteristic of surface micromorphology generated by ultra-precision diamond turning,” Opt. Lasers Eng. 62, 46–56 (2014).
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Figures (13)

Fig. 1.
Fig. 1. Schematic diagram of the SP process.
Fig. 2.
Fig. 2. Contact of mirror and pad surfaces.
Fig. 3.
Fig. 3. Removal rate variation with pad pressure (a) standard gaussian distribution of pad asperity heights (d is the average distance between two surfaces) (b) removal rate variation with average contact distance d.
Fig. 4.
Fig. 4. Periodicity of different surfaces (a) isotropous surfaces, (b) anisotropic surface (turning marks), (c) amplifying images of red dotted areas, (d) sinusoidal error characteristic.
Fig. 5.
Fig. 5. Comparison of removal efficiency of isotropous and anisotropic surfaces.
Fig. 6.
Fig. 6. Schematic diagram of the IBS process.
Fig. 7.
Fig. 7. Data fitting of roughness with different IBS removal depth and corresponding typical morphology.
Fig. 8.
Fig. 8. Roughness evolution of IBS-SP process (a) initial SPDT surface, (b) after IBS process, (c) after first SP, (d) after second SP.
Fig. 9.
Fig. 9. Roughness evolution of direct SP of SPDT surface (a) initial SPDT surface, (b) after first SP, (c) after second SP, (d) after third SP, (e) after fourth SP.
Fig. 10.
Fig. 10. PSD analysis of direct SP of SPDT surface.
Fig. 11.
Fig. 11. Roughness evolution of IBS-SP process (a) after IBS, (b) after first SP, (c) after second SP, (d) after third SP, (e) after fourth SP.
Fig. 12.
Fig. 12. PSD analysis of IBS-SP process.
Fig. 13.
Fig. 13. Comparison of convergence rate.

Tables (3)

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Table 1. Parameters of SPDT process

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Table 2. Parameters of IBS process

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Table 3. Parameters of SP process

Equations (29)

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R v = K V A ,
A 2 a 3 3 r ,
p r = L A r ,
A r = π N A 0 R F 1 ( d ) ,
L = 4 3 N A 0 E s p R 1 2 F 3 2 ( d ) ,
F n ( d ) = d ( z d ) n ϕ ( z ) d z ,
1 E s p = 1 υ p 2 E p + 1 υ w 2 E w ,
F w = B e π a 2 ,
F p = P r π b 2 ,
R v = 2 3 r 2 C 3 2 ( σ R ) 3 4 ( E s p B e ) 3 2 K V .
C = 4 3 π ( F 3 / 3 2 2 ( d ) F 1 ( d ) ) .
R e = R v p R v v .
m 0 = A V G ( z 2 ( x ) ) ,
m 2 = A V G [ ( d z ( x ) / d x ) 2 ] ,
m 4 = A V G [ ( d 2 z ( x ) / d x 2 ) 2 ] ,
R = 0.375 ( π / m 4 ) 1 / 2 ,
σ = ( 1 0.8968 / α ) 1 / 2 m 0 1 / 2 ,
α = ( m 0 m 4 ) / m 2 2 ,
z ( x ) = A e sin ( k 1 x ) ,
k 1 = 2 π f x cos ( θ ) ,
m 0 = 0 L A e 2 sin 2 ( k 1 x ) d x L = A e 2 2 A e 2 sin ( 2 k 1 L ) 4 L k 1 ,
m 2 = A e 2 k 1 2 k 1 2 m 0 ,
m 4 = k 1 4 m 0 ,
R v = 0.353 π 3 8 r 2 C 3 2 A e 3 2 ( E s p B e ) 3 2 K V k 1 3 2 .
R e = R 0 k 1 3 2 ,
R e = R 0 ( 2 π f x ) 3 2 cos 3 2 ( θ ) ,
R 0 = 0.353 π 3 8 r 2 A e 3 2 ( E s p B e ) 3 2 K V ( C p 3 2 C v 3 2 ) .
f c = 3 2 ln 10 π d 6 σ .
P S D ( ω i ) = [ E ( ω i ) ] 2 Δ ω .

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