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Simulation and experimental study on form-preserving capability of bonnet polishing for complex freeform surfaces
Precision Engineering 60 (2019) 54–62
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Simulation and experimental study on form-preserving capability of bonnet polishing for complex freeform surfaces
T
Xing Sua, Peng Jia, Yuan Jina, Duo Lia,*, David Walkerb,c,d, Guoyu Yub, Hongyu Lib,e, Bo Wanga,** a
Center for Precision Engineering, Harbin Institute of Technology, Harbin, 150001, China National Facility for Ultra Precision Surfaces, University of Huddersfield, LL17 0JD, UK c Department of Physics and Astronomy, University College London, WC1E 6BT, UK d Zeeko Ltd, Colville, Leicestershire, LE67 3WF, UK e Research Center for Space Optical Engineering, Harbin Institute of Technology, Harbin, 150001, China b
A R T I C LE I N FO
A B S T R A C T
Keywords: Complex freeform surface Bonnet polishing Form-preserving Contact analysis
Bonnet polishing has been widely used in structured surface generation, form correction and surface finishing. The inflated flexible tool is able to match the local surface to realize surface finishing. However, after surface finishing, the polishing induced form change is still not clear, especially for complex freeform surfaces with millimeter-level spatial periods. This paper presents a simulation and experimental study on form-preserving capability of bonnet polishing. The concept of form-preserving capability is discussed firstly. Subsequently, the relative surface speed and contact condition on ripple structures are calculated and analyzed. The results demonstrate that the contact condition of bonnet tool on structured surfaces has the main influence on formpreserving capability. Then, the influence of various factors on contact pressure difference is investigated. Compared with experimental results, the linear correlation between form change and the contact pressure difference is obtained, indicating that form-preserving capability can be strengthened by reducing contact pressure difference. Under optimal conditions, the form change can be controlled less than 20 nm, which proves the superior form-preserving capability of bonnet polishing.
1. Introduction High-end optical components are widely adopted in many advanced technical fields, including astronomical observation projects, high power laser systems and ultra-precision metrology systems. Stringent requirements on optical components put forward challenges to fabrication techniques. In the last decades, computer controlled optical surfacing (CCOS) techniques have been highly developed, such as magnetorheological polishing [1], fluid jet polishing [2], bonnet polishing [3] and ion beam figuring [4], which play increasingly significant roles in fabricating ultra-precision optics. Bonnet polishing, using an inflated flexible tool, is able to conform to the local surface with various curvature, which shows the strong ability for optical fabrication. Due to the importance of deterministic material removal in CCOS, many researchers focused on tool influence function (TIF) of bonnet polishing. Li et al. modeled TIF based on contact pressure distribution and relative surface speed, which was also used to predict edge TIFs for perfect edge control [5,6]. Zeng et al.
*
established an analytical model and revealed the effects of process parameters on material removal rate [7,8]. To obtain higher efficiency, Wang et al. developed a semi-rigid bonnet tool, which is reinforced with a stainless steel sheet [9]. Based on finite element analysis (FEA) and experimental measurement, Jiang et al. found that the TIF shape changed from Gaussian to concave when the offset increased above 1 mm [10]. Recently, a micro-analysis model based on micro-contact theory and wear theory was established. The model captured mutual interactions of slurry, pad and workpiece, which provided good insights into the material removal mechanism [11,12]. In the viewpoint of application, bonnet polishing offers superior performance on pre-polishing, surface finishing and form-correction polishing [13]. Walker et al. developed a fast production process based on bonnet polishing for aspherical prototype segment, in which the form error was improved from 30 μm PV to 136 nm RMS [14]. Through precession polishing with four steps, the surface texture can be improved to 0.5 nm Ra [15]. For finishing of X-ray molding dies, Beaucamp et al. introduced “continuous precession” bonnet polishing after fluid jet polishing to obtain 0.3 nm
Corresponding author. P.O. Box 413, Harbin, 150001, PR China. Corresponding author. E-mail addresses: [email protected] (D. Li), [email protected] (B. Wang).
**
https://doi.org/10.1016/j.precisioneng.2019.07.010 Received 4 April 2019; Received in revised form 29 July 2019; Accepted 30 July 2019 Available online 31 July 2019 0141-6359/ © 2019 Elsevier Inc. All rights reserved.
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material removal. Thus, the bonnet tool is expected to conform to surface structures to achieve uniform material removal. When uniform material removal is accomplished, the surface form generated by the first step can be kept. In this way, bonnet polishing plays the role in form-preserving polishing. However, surface structures can introduce disturbance to polishing process, such as relative surface speed and contact pressure distribution. The disturbance may cause non-uniform material removal of polishing. As a result, the generated surface form can be damaged to a certain extent, which is crucial for machining accuracy of the whole fabrication process. Regarding to nanofinishing of freeform surfaces, the specific conditions need to be clarified. For most cases, the freeform surfaces have millimeter-scale PV, such as knee joint implant [25,26] and hemispheric shell [27,28]. These kinds of surfaces requires a small tool, which can access local curved zones, shown in Fig. 2(a). To realize nanofinishing, the small t ool moves along with surface profiles with a constant offset. When the PV of the freeform surfaces ranges in micronscale [22], the large flexible tool can be used, shown in Fig. 2(b). Because of the flexibility and the small form PV, the large tool can contact with the local surface and move in a straight line. It should be pointed out that the complex freeform surfaces in this paper have specific characteristics of millimeter-level spatial periods and micron-level PV, which correspond to Fig. 2(b). One type of such freeform surfaces is continuous phase plate, with less than 3 μm PV and tens of millimeters spatial period [22]. Besides, the spatial period of complex freeform surfaces is usually smaller than polishing spot size, as shown in Fig. 1. In this case, the bonnet tool covers several peaks or valleys simultaneously, which may affect the contact condition and make the form-preserving polishing uncertain.
RMS surface texture [16]. Cheung et al. established a model-based simulation system to generate structured surface using bonnet polishing, showing good performance to predict the form error [17]. By control specific polishing tool orientation, Cao et al. reported a swing precess bonnet polishing (SPBP) method for generating freeform surfaces [18]. A model-based self-optimization method, which combined simulation of the machining process, error compensation and on-machine metrology, was developed and used to fabricate sinusoidal surfaces with over 20 mm spatial period [19]. However, bonnet polishing is not suitable to generate complex surface structures with spatial periods down to several millimeters, which requires a smaller spot size. In this aspect, reactive plasma machining techniques based on pure chemical etching, which have much smaller TIFs, show great strengths to fabricate complex freeform surfaces. Yamamura et al. investigated pulse width modulation controlled plasma machining method for high spatial resolution figuring, in which the full width at half maximum (FWHM) reaches 1.1 mm [20]. Arnold et al. developed plasma jet machining (PJM) to fabricate complex structures with millimeter-scale spatial period [21]. Atmospheric pressure plasma processing was used to fabricate larger aperture continuous phase plates with various spatial periods [22]. Nevertheless, the reactive plasma machining techniques deteriorate surface quality to a certain extent due to the pure chemical etching mechanism [23,24], which cannot meet optical finishing requirements. Thus, the post polishing needs to be conducted, which is used to smooth surface texture and preserve generated surface structures simultaneously. As mentioned above, the bonnet polishing has the ability of surface finishing, which is helpful to fabricate complex freeform surfaces cooperating with another sub-aperture fabrication method. But very little research work focuses on the form-preserving capability of bonnet polishing. In this paper, simulation and experimental work are conducted to study the form-preserving capability of bonnet polishing. The concept of form-preserving bonnet polishing is introduced and discussed firstly. Then polishing analysis model, including relative surface speed calculation and contact pressure simulation, is established. The polishing process on ripple structures under different conditions is analyzed. Subsequently, experimental results on ripple structures are presented and compared with the simulation results. At last, the relationship between the form-preserving capability and the contact pressure difference is summarized.
3. Modeling of bonnet polishing process Since Preston equation was proposed, it had been widely used to predict material removal of various polishing techniques [29]. The Preston equation is expressed as
Δz = k⋅p (x , y )⋅v (x , y )
(1)
where Δz is material removal rate, k is Preston coefficient related to substrate material, slurry and other polishing parameters, p (x , y ) is contact pressure on the substrate, and v (x , y ) is relative surface speed between polishing tool and substrate. In the surface finishing step, polishing parameters, such as feed rate and rotational speed of bonnet tool, are kept constant. And the Preston coefficient can be regarded as constant. Therefore, relative surface speed and contact pressure are the predominant factors for material removal, which need to be analyzed thoroughly.
2. Concept of form-preserving polishing using bonnet tool Fig. 1 shows the schematic diagram of form-preserving polishing using bonnet tool. The fabrication process of complex freeform surfaces with small structures can be divided into form generation and surface finishing. The form generation step requires an efficient fabrication method with small spot size, such as reactive plasma machining techniques. Then, bonnet polishing is adopted in the second step to smooth surface texture. The surface finishing is achieved through polishing
3.1. Calculation of relative surface speed Relative surface speed in contact zone depends on radius distance
Contact pressure Form generation Small size tool
Surface finishing Bonnet tool Peak Peak
Valley Spatial period Spot size
Fig. 1. Schematic diagram of form-preserving polishing. 55
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Fig. 2. Two types of nanofinishing on freeform surfaces. (a) small tool; (b) large tool.
Feed path
(a)
(b)
boundary condition is set as a pressure load. The edge of the base membrane is assigned as a fixed boundary. Since the displacement of polishing cloth surface is coupled to the base membrane, there is no additional constraint on polishing cloth. The substrate domain is set as a prescribed displacement for different offsets and the contact process is performed through substrate movement. In addition, symmetric constraints are set on the symmetric plane. To validate the model, the contact process on flat surface of a bonnet tool with 80 mm radius, polyurethane cloth, 1 bar inner pressure and 0.7 mm offset was simulated. As shown in Fig. 5(a), the contact pressure distribution shows M shape, with higher contact pressure in the outer region. This can be explained as follows. The central part of the bonnet tool has relatively uniform displacement and can match the substrate surface perfectly, which can be seen in Fig. 5(b). Thus, the contact pressure in the central area is mainly attributed to inner gas pressure. As for the outer region, the bonnet membrane bends and the non-contact region of the bonnet membrane can transmit additional force on the substrate, besides inner gas pressure. Hence, the contact pressure forms the M shaped distribution, which is similar to the results of “the bonnet warp” [7,8]. Through integrating the contact pressure, the total force on the substrate is obtained as 15.5 N. In the previous study, the total load on the substrate with the same kind of bonnet tool was measured using three OMEGA standard load cells and the measured total contact force was 17 N [30]. Comparing these two results, it can be found that the simulation error of the total contact force is about 8.8%, which proves the reliability of the model.
and rotational speed. Based on our previous study, the relative surface speed in the contact zone can be expressed as [5].
v (x , y ) = 2⋅cos ρ⋅(R − d )−1 s⋅(s − α )⋅(s − β )⋅(s − γ ) ⋅ω0 α = [(R −
x 2 + y 2 )2 + x 2 + y 2 ]1/2
β = (R − d )⋅cos(ρ)−1 γ = [y 2 + [(R − d )⋅tan(ρ) − x 2]]1/2 s = 0.5⋅α⋅β⋅γ
(2)
where ρ is precess angle, R is radius of bonnet tool, d is offset, ω0 is rotational speed around the axis of polishing tool. 3.2. Contact model between bonnet tool and substrate Fig. 3 shows the structure of the bonnet tool, including three main components: aluminum base, base membrane and polishing cloth. The aluminum base acts as a support framework and can be regarded as rigid body in the FEA model. The base membrane consists of a thin rubber layer and a fiber mesh, which can only sustain tension force. The polishing cloth is glued on the base membrane. With high-pressure air inflated, the base membrane and polishing cloth bulge with an extent of rigidity. When contacting with the substrate, the base membrane and polishing cloth are pressed on the local surface. As shown in Fig. 4, a FEA model was established using COMSOL software. To reduce computational time, a symmetric geometric model was built, including base membrane, polishing cloth and fused silica substrate. There are two physical interfaces in the model, membrane interface and solid mechanics interface. The membrane interface is adopted on the base membrane, in which only in-plane forces are calculated. The solid mechanics interface is added on the polishing cloth and substrate. These two interfaces are coupled together through the transfer of displacement parameters on the inner surface of polishing cloth. The contact pair is added to simulate the contact process. The outer surface of polishing cloth and the upper surface of substrate are assigned as destination surface and source surface, respectively. The linear elastic material model is defined. The material properties are provided by material suppliers, listed in Table 1. The base membrane surface sustains uniform pressure load, so the
4. Analysis of polishing process on ripple structures The practical interest is to achieve uniform material removal on freeform surface structures. As mentioned in Section 2, this paper focuses on freeform surfaces with millimeter-level spatial period and micron-level PV, such as continuous phase plate. In mathematical respect, such freeform surfaces are two-dimensional signals and can be described by the sum of Fourier series. Thus, sinusoidal surfaces are representative for freeform surfaces. Compared with complex freeform surfaces, the sinusoidal surfaces with certain spatial period and PV are more helpful for analyzing quantitatively. Therefore, sinusoidal ripple Precess angle
Polishing cloth Aluminium base
Aluminium base
Base membrane Base membrane
Polishing cloth Offset Substrate
Fig. 3. Structure of bonnet tool. 56
Spot size
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Fixed edge Base membrane
Polishing cloth Contact pair Substrate Fig. 4. FEA model of contact process.
Feed direction
Table 1 Material properties in contact model.
Z
Material
Density (kg/m3)
Elastic modulus (Pa)
Poisson's ratio
Base membrane Polyurethane Uninap Fused silica
0.40 × 103 0.42 × 103 0.53 × 103 1.21 × 103
9.0 × 108 1.2 × 107 2.1 × 106 7.0 × 1010
0.30 0.35 0.32 0.17
Ripple structure profile
structures can be used to investigate form-preserving capability for freeform surfaces. The contact process between the bonnet tool and ripple structures was analyzed, shown in Fig. 6. The ripple structure profile is expressed as
xtool
X
Fig. 6. Scanning process of bonnet tool on ripple structures.
(3)
z = (PV /2)⋅cos(x / T ⋅2π )
where PV is peak-valley (PV) value of ripple structures, T is spatial
Pa
(a) mm
(b) Fig. 5. Simulation results. (a) Contact pressure distribution; (b) Displacement in local contact region. 57
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color bar range was modified to show the ‘ripple pattern’ stress distribution more clearly in the enlarged region. Fig. 8(b) shows the contact pressure distribution on the substrate. The contact pressure difference at peaks and valleys is obvious, indicating that surface structures can lead to non-uniformity of the contact process. Fig. 9 shows the contact pressure profiles at four dwell positions of bonnet tool, which correspond to the scanning process of polishing. The dotted line represents the ripple structure profile, considered as fixed. The contact pressure at peaks and valleys is different and varies when the bonnet tool dwells at different positions. For peak A, when the bonnet tool dwells at one certain position, the contact pressure can be obtained and recorded as ppeak, i . The dwell positions are set to range from left to right relative to peak A, then a series of contact pressure at peak A is obtained. Notably, the range of dwell positions is required larger than the spot size of the bonnet tool. For example, when the spot size of the bonnet tool is 10 mm, the dwell positions are required to cover from −5 mm to 5 mm. The contact process outside of the range does not need to be simulated since there is no contact at peak A. Thus, a series of contact pressure at one certain peak or valley in the polishing process can be obtained to analyze the polishing process reasonably. Furthermore, the average difference of contact pressure at peaks and valleys is calculated by
Table 2 Simulation parameters of polishing process on ripple structures. No
1 2 3
Parameters of ripple structures
Polishing parameters
T (mm)
PV (nm)
Pressure (bar)
Polishing cloth
2.8, 5, 10, 15 2.8, 5, 10, 15 2.8, 5
2000
1.2
Polyurethane, Uninap
600, 1400, 2000, 4000 1400
1.2
Polyurethane, Uninap
0.3, 0.6, 0.9, 1.2
Polyurethane, Uninap
period of ripple structures. In practical cases, the bonnet tool scans on substrate and the contact condition on substrate varies in scanning movement. Thus, the contact process simulations were performed at different dwell positions ( x tool shown in Fig. 6). Table 2 lists the simulation parameters, including polishing processing parameters and ripple structure parameters. Besides, the rotational speed, bonnet radius and offset were 1000 rpm, 40 mm and 0.7 mm, respectively. 4.1. Relative surface speed on ripple structures Based on Equ. (2), the relative surface speed on flat surface and ripple structures was calculated. To calculate relative surface speed on ripple structures, the profiles of ripple structures were added on the offset value. Fig. 7(a) shows the normal relative surface speed on flat surface. Fig. 7(b) shows the result on ripple structure with 5 mm spatial period, corresponding to No 1 in Table 2. The difference of relative surface speed between the two surfaces was obtained, shown in Fig. 7(c). It can be seen that the deviation of relative surface speed induced by the ripple structure is much smaller compared with the normal relative surface speed, with a ratio less than 10−4. Therefore, the influence of relative surface speed on the uniformity of removal rate is much insignificant. The above results indicate that the deviation of relative surface speed induced by surface structures has a minor effect on form-preserving capability.
pdifference =
∑
(ppeak,
i
− pvalley, i )/ N
i = 1~ N
where ppeak, i is the contact pressure at peaks in the ith simulation, pvalley, i is the contact pressure at valleys in the ith simulation, N is the number of simulations at different dwell positions. Note that, pdifference is named as “contact pressure difference” concisely in the following text. 5. Contact pressure difference under various conditions According to Table 2, the contact process under different conditions is simulated firstly and then the contact pressure difference is calculated based on Equ. (4). Thus, the influence on contact pressure difference of the spatial period, PV and the inner pressure can be obtained, shown in Fig. 10–12. In these figures, the vertical coordinates represent the contact pressure difference, which is related with uniformity of material removal. The horizontal coordinates represent the value of the influence factor.
4.2. Contact pressure distribution on ripple structures Fig. 8 shows the simulation result of ripple structures No 1 in Table 2, with 5 mm spatial period and polyurethane polishing cloth. After contacting, the polishing cloth deforms and conforms to the surface form under inner pressure. As shown in Fig. 8(a), there is no gap between polishing cloth and substrate, indicating that the two objects are able to match totally. Thus, both peaks and valleys of the surface structure can be polished, which demonstrates the foundation of the form-preserving capability of bonnet polishing. On the other hand, the stress distribution in Fig. 8(a) shows a ‘ripple pattern’ consistent with the ripple structure. The black dotted line in Fig. 8(a) shows the direction of the ripple structure. Note that, the stress distortion induced by the ripple structure is relatively small, so that the
5.1. Influence of spatial period on contact pressure difference Fig. 10(a) shows the contact pressure distribution with polyurethane of No 1 in Table 2. The contact pressure profile on flat surface with the same parameters is also plotted. The deviation of contact pressure on ripple structures and flat surface is obvious. When the spatial period reaches 10 mm, the deviation becomes indistinctive at this dwell position. Based on Equ. (4), the contact pressure difference is obtained, shown in Fig. 10(b). The results using polyurethane and uniap are plotted in two types of point. The horizontal coordinates are
unit: mm/s
(a)
(4)
(b)
(c)
Fig. 7. Relative surface speed. (a) Flat surface; (b) Ripple structure surface; (c) Difference between (a) and (b). 58
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Pa
(a)
(b)
Contact pressure (kPa)
Fig. 8. Simulation result of contact process (a) von Mises stress distribution; (b) Contact pressure on substrate.
200
ppeak , i pvalley, i
0
ppeak, i+1 pvalley, i 1
p peak , i
0
2
pvalley, i
2
0
150
0
0
0
100
0
0
0
50
0
0
A
0 -10
-5
B
0
5
x (mm)
A
0
10
15 -10
-5
0
B
5
3
pvalley, i
3
0 A
0
10
p peak , i
15 -10
-5
0
B 5
x (mm)
x (mm)
A
0 10
15 -10
-5
B
0
5
10
15
x (mm)
Fig. 9. Contact pressure profiles at different dwell positions of bonnet tool.
2.8 mm, 5 mm, 10 mm, and 15 mm spatial periods, respectively. With the spatial period increasing, the contact pressure difference decreases regardless of polyurethane or uninap. This can be explained by ‘bridge effect’ of polishing cloth on ripple structures, in which the transverse shear effect becomes significant with smaller spatial periods, leading to higher contact pressure difference [[31]]. The contact pressure difference causes material removal difference at peaks and valleys, which weakens the form-preserving capability. When the spatial period is larger than 10 mm, the contact pressure difference with two polishing clothes is less than 5 kPa, much smaller than the normal contact pressure on flat surface. This indicates that the surface structures with large spatial periods can be polished with minor form change. In addition, the ratio of the contact pressure difference over the normal contact pressure is about 10−1–10−2, much larger than that of the relative surface speed in Section 4.1, which shows that the contact pressure difference is the major factor for form-preserving capability. It can also be found that the contact pressure difference using uninap is much smaller than that using polyurethane, which is attributed to different material elasticity. The elastic modulus of uninap is smaller, so it deforms more easily to match surface structures. Thus, a more uniform contact pressure distribution can be achieved, which provides a stronger form-preserving capability.
(a)
(b)
Fig. 11. Contact pressure difference with different ripple structures. (a) Polyurethane; (b) Uninap.
5.2. Influence of PV on contact pressure difference Fig. 11 shows contact pressure difference on ripple structures with different PV, corresponding to No 2 in Table 2. The results using polyurethane and uninap are presented in Fig. 11(a) and (b). The horizontal coordinates are 600 nm, 1400 nm, 2000 nm, and 4000 nm PV, respectively. One type of point represent the ripple structure with the same spatial period. Overall, the contact pressure difference increases linearly with increasing PV of ripple structures. For
(a)
(b)
Fig. 10. Simulation results with different spatial periods. (a) Contact pressure profiles with polyurethane; (b) Contact pressure difference. 59
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(a)
(b) Fig. 12. Contact pressure difference with different inner pressure. (a) Polyurethane; (b) Uninap.
cloth can conform to ripple structures totally, which is the fundamental reason for the form-preserving polishing result. On the other hand, the PV of the ripple structure decreases slightly after polishing, which is caused by removal non-uniformity at peaks and valleys, due to contact pressure difference. To evaluate the form change after polishing, the PV change of ripple structures after 1 μm polishing depth were obtained and summarized in Table 4. Table 4 includes experimental parameters, PV change after polishing and related simulation results. For experiments No 1 ~ 4, the initial PV of ripple structures varies within a relatively small range, which can be regarded as constant. Thus, the PV change is attributed to various spatial periods. It can be found that the PV change decreases with spatial periods increasing. In experiments No 2, No 5 and No 6, the spatial periods keep constant as 5 mm and the initial PV varies from 510 nm to 1210 nm. It can be seen that the PV change increases with the rising of initial PV. The influence of polishing cloth is presented in two experiments pairs (No 1, 7 and No 6, 8). Even though the initial PV does not keep equal, it can also be found that the PV change using polyurethane is obviously larger than that using uninap. For experiments No 8 ~ 10, the PV change fluctuates a little, indicating that the inner pressure has a negligible influence on removal uniformity as well as form-preserving capability. Even the initial PV reaches near 3 μm in experiments No 11 and 12, the PV change still keeps at a low level, which demonstrates great form-preserving performance. As listed in Table 4, the PV change corresponds to the contact pressure difference generally. The correlation between the PV change and the contact pressure difference is fitted linearly, shown in Fig. 14. The coefficient of determination is 0.90 adjusted R squared, indicating the goodness of the linear relationship. When the contact pressure difference is less than 3 kPa, the deviation between experimental results and fitting results is relatively large, which can be introduced by measurement error. However, the PV change keeps less than 20 nm in this zone, showing superior form-preserving capability. On the other hand, the PV change reaches 45 nm when the contact pressure difference rises to 8 kPa, which may not be acceptable in practice. Furthermore, an empirical formula is obtained to predict the PV change after bonnet polishing, expressed as Equ. (5). The T and PV denote the spatial period and PV of the ripple structure, same as that in Equ. (3). The general influence laws can be summarized as follows. The PV change is linearly proportion to the PV and exponentially inverse proportion to the spatial period. Comparing the two polishing clothes, the PV change and its increasing tendency using uninap is smaller than that using polyurethane.
ripple structures with larger PV, the deformation of polishing cloth at peaks becomes larger, causing higher contact pressure difference at peaks and valleys. The increase indicates that bonnet polishing will induce more form change for ripple structures with larger PV. Fig. 11(a) shows that the contact pressure difference using polyurethane increases rapidly with PV, reaching 70 kPa with the 2.8 mm spatial period. Moreover, the contact pressure difference using uninap keeps at a low level, less than 11 kPa, which offers a better choice to achieve formpreserving polishing. 5.3. Influence of inner pressure on contact pressure difference Fig. 12 shows the contact pressure difference with different inner pressures, corresponding to No 3 in Table 2. The results using polyurethane and uninap are presented in Fig. 12(a) and (b). For each polishing cloth, the contact pressure difference was obtained on ripple structures with 2.8 mm and 5 mm spatial periods. The horizontal coordinates are 0.3 bar, 0.6 bar, 0.9 bar, and 1.2 bar inner pressure, respectively. It is interesting to find that the inner pressure has a minor influence on the contact pressure difference. The change value is less than 1 kPa, which is much smaller than that induced by different PV and spatial periods. The results indicate that the polishing clothes can totally match surface structures even the inner pressure is at a low level, 0.3 bar. Note that, the contact pressure difference decreases slightly with increasing inner pressure. This can be explained as follows. In the contact process, the polishing cloth is pressed on ripple structures by inner pressure. The contact interface becomes closer at both peaks and valleys with higher inner pressure, which causes a slight decrease of contact pressure difference. Overall, the form-preserving capability is not affected by inner pressure. 6. Verification of form-preserving capability To validate the simulation analysis experimentally, fused silica substrate with ripple structures were prepared, and then polished on a Zeeko IRP 600 polishing machine. The surface forms before and after polishing were measured using Fisba μPhase 2 HR interferometer. The experimental parameters are listed in Table 3. Fig. 13 shows one ripple structure surface before and after polishing. It can be seen that the bonnet polishing does not damage the ripple structure. Based on the contact analysis above, the polishing Table 3 Parameters of bonnet polishing experiments. Bonnet radius (mm)
Rotational speed (rpm)
Precess angle (°)
Slurry
40
1000
15
Cerium oxide
⎧ 6.39·(0.596 +
Offset (mm)
PV change =
0.0386 ⋅ PV
⎨ 6.39·(0.432 + ⎩
) − 1.68, using polyurethane
e (0.324·T ) 0.0032 ⋅ PV e (0.136·T )
) − 1.68, using uninap (5)
0.7
60
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(a)
(b) Fig. 13. Ripple structure form. (a) Before polishing; (b) After polishing.
7. Conclusions
Table 4 Summary of polishing experiments and simulation on ripple structures. No
Spatial period (mm)
Initial PV (nm)
Inner pressure (bar)
Polishing cloth
PV change (nm)
Contact pressure difference (kPa)
1 2 3 4 5 6 7 8 9 10 11 12
2.8 5 10 15 5 5 2.8 5 5 5 15 15
480 510 470 709 1020 1210 490 1656 1550 1630 2640 2580
1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 0.9 0.3 1.2 1.2
polyurethane polyurethane polyurethane polyurethane polyurethane polyurethane uninap uninap uninap uninap uninap polyurethane
47 24 7 6 45 63 11 17 15 16 5.5 10
8 5 3 1.5 7.6 8.8 1.0 2.3 2.4 2.3 1.6 2.2
Experimental results Fitting results
60 PV change (nm)
In this paper, the form-preserving capability of bonnet polishing was studied based on simulation and experimental method. The simulation shows that the bonnet tool can deform and match surface structures with small spatial periods, which provides the foundation of form-preserving polishing. The experiments on ripple structures prove the form-preserving capability of bonnet polishing. It is found that form change after polishing depends on contact pressure difference at peaks and valleys linearly. The softer polishing cloth, larger spatial period and smaller PV can lead to lower contact pressure difference and less form change. Instead, the relative surface speed and inner pressure have a minor influence on form-preserving capability. Under specific conditions, the form change can be controlled less than 20 nm with low contact pressure difference, which demonstrates the excellent formpreserving capability. This work gives a comprehensive insight into the form-preserving capability of bonnet polishing, which implies its feasibility in practical optical fabrication. Based on the form-preserving capability, future work will focus on fabricating complex freeform optics with high efficiency and quality, cooperating with other subaperture figuring methods. Acknowledgments
45
The work is supported by National Natural Science Foundation of China (No. 51175123); National Science and Technology Major Project (No. 2013ZX04006011-205). The author would like to thank the China Scholarship Council (CSC) for its financial support in studying at University of Huddersfield, the National Facility for Ultra Precision Surfaces, North Wales, UK.
30 Equation a b R-Square (COD) Adj. R-Square
15 0
0
y = a + b*x -1.68 6.39 0.91317 0.90232
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2 4 6 8 Contact pressure difference (kPa)
Fig. 14. Correlation between form change and contact pressure difference.
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structured surfaces. Precis Eng 2017;50:361–71. [19] Cao Z-C, Cheung CF, Liu MY. Model-based self-optimization method for form correction in the computer controlled bonnet polishing of optical freeform surfaces. Opt Express 2018;26:2065–78. [20] Yamamura K, Takeda Y, Sakaiya S, Funato D, Endo K. High-spatial resolution figuring by pulse width modulation controlled plasma chemical vaporization machining. CIRP Ann 2016;42:508–11. [21] Arnold T, Boehm G, Eichentopf IM, Janietz M, Meister J, Schindler A. Plasma Jet Machining: a novel technology for precision machining of optical elements. Vakuum Forsch Praxis 2010;22:10–6. [22] Su X, Xia L, Liu K, Zhang P, Li P, Zhao R, et al. Fabrication of a large-aperture continuous phase plate in two modes using atmospheric pressure plasma processing. Chin Optic Lett 2018;16:102201. [23] Xin Q, Su X, Wang B. Modeling study on the surface morphology evolution during removing the optics surface/subsurface damage using atmospheric pressure plasma processing. Appl Surf Sci 2016;382:260–7. [24] Xin Q, Li N, Wang J, Wang B, Li G, Ding F, et al. Surface roughening of ground fused silica processed by atmospheric inductively coupled plasma. Appl Surf Sci 2015;341:142–8. [25] Sidpara AM, Jain V. Nanofinishing of freeform surfaces of prosthetic knee joint implant. Proc Inst Mech Eng B J Eng Manuf 2012;226:1833–46. [26] Sarkar M, Jain VK, Sidpara A. On the flexible abrasive tool for nanofinishing of complex surfaces. J Adv Manuf Syst 2019;18:157–66. [27] Chen M, Liu H, Su Y, Zheng F. Design and fabrication of a novel magnetorheological finishing process for small concave surfaces using small ball-end permanent-magnet polishing head. Int J Adv Manuf Technol 2016;83:823–34. [28] Liu H, Chen M, Yu B, Fang Zh. Configuration design and accuracy analysis of a novel magnetorheological finishing machine tool for concave surfaces with small radius of curvature. J Mech Sci Technol 2016;30:3301–11. [29] Preston F. The theory and design of plate glass polishing machines. J. Glass Technol. 1927;11:214–56. [30] Li H. Research on manufacturing mirror segments for an extremely large telescope: UCL. University College London); 2012. [31] Kim DW, Park WH, An HK, Burge JH. Parametric smoothing model for visco-elastic polishing tools. Opt Express 2010;18:22515–26.
[6] Li H, Walker D, Yu G, Sayle A, Messelink W, Evans R, et al. Edge control in CNC polishing, paper 2: simulation and validation of tool influence functions on edges. Opt Express 2013;21:370–81. [7] Zeng S, Blunt L. Experimental investigation and analytical modelling of the effects of process parameters on material removal rate for bonnet polishing of cobalt chrome alloy. Precis Eng 2014;38:348–55. [8] Zeng S, Blunt L. An experimental study on the correlation of polishing force and material removal for bonnet polishing of cobalt chrome alloy. Int J Adv Manuf Technol 2014;73:185–93. [9] Wang C, Wang Z, Wang Q, Ke X, Zhong B, Guo Y, et al. Improved semirigid bonnet tool for high-efficiency polishing on large aspheric optics. Int J Adv Manuf Technol 2017;88:1607–17. [10] Jiang T, Liu J, Pi J, Xu Z, Shen Z. Simulation and experimental study on the concave influence function in high efficiency bonnet polishing for large aperture optics. Int J Adv Manuf Technol 2018:1–7. [11] Shi C, Peng Y, Hou L, Wang Z, Guo Y. Improved analysis model for material removal mechanisms of bonnet polishing incorporating the pad wear effect. Appl Opt 2018;57:7172–86. [12] Shi C, Peng Y, Hou L, Wang Z, Guo Y. Micro-analysis model for material removal mechanisms of bonnet polishing. Appl Opt 2018;57:2861–72. [13] Walker DD, Beaucamp A, Brooks D, Freeman R, King A, McCavana G, et al. Novel CNC polishing process for control of form and texture on aspheric surfaces. Current Developments in Lens Design and Optical Engineering III. International Society for Optics and Photonics; 2002. p. 99–106. [14] Walker D. Faster production of high-quality telescope mirrors. SPIE Newsroom; 2013. [15] Walker DD, Freeman R, McCavana G, Morton R, Riley D, Simms J, et al. Zeeko/UCL process for polishing large lenses and prisms. Large lenses and prisms. International Society for Optics and Photonics; 2002. p. 106–12. [16] Anthony B, Yoshiharu N. Super-smooth finishing of diamond turned hard X-ray molding dies by combined fluid jet and bonnet polishing. CIRP Ann 2013;62:315–8. [17] Cheung CF, Lee WB, Ho LT, To S. Modelling and simulation of structure surface generationg using computer controlled ultra-precision polishing. Precis Eng 2011;35:574–90. [18] Cao Z-C, Cheung CF, Ho LT, Liu MY. Theoretical and experimental investigation of surface generation in swing precess bonnet polishing of complex three-dimensional
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Precision Engineering journal homepage: www.elsevier.com/locate/precision
Simulation and experimental study on form-preserving capability of bonnet polishing for complex freeform surfaces
T
Xing Sua, Peng Jia, Yuan Jina, Duo Lia,*, David Walkerb,c,d, Guoyu Yub, Hongyu Lib,e, Bo Wanga,** a
Center for Precision Engineering, Harbin Institute of Technology, Harbin, 150001, China National Facility for Ultra Precision Surfaces, University of Huddersfield, LL17 0JD, UK c Department of Physics and Astronomy, University College London, WC1E 6BT, UK d Zeeko Ltd, Colville, Leicestershire, LE67 3WF, UK e Research Center for Space Optical Engineering, Harbin Institute of Technology, Harbin, 150001, China b
A R T I C LE I N FO
A B S T R A C T
Keywords: Complex freeform surface Bonnet polishing Form-preserving Contact analysis
Bonnet polishing has been widely used in structured surface generation, form correction and surface finishing. The inflated flexible tool is able to match the local surface to realize surface finishing. However, after surface finishing, the polishing induced form change is still not clear, especially for complex freeform surfaces with millimeter-level spatial periods. This paper presents a simulation and experimental study on form-preserving capability of bonnet polishing. The concept of form-preserving capability is discussed firstly. Subsequently, the relative surface speed and contact condition on ripple structures are calculated and analyzed. The results demonstrate that the contact condition of bonnet tool on structured surfaces has the main influence on formpreserving capability. Then, the influence of various factors on contact pressure difference is investigated. Compared with experimental results, the linear correlation between form change and the contact pressure difference is obtained, indicating that form-preserving capability can be strengthened by reducing contact pressure difference. Under optimal conditions, the form change can be controlled less than 20 nm, which proves the superior form-preserving capability of bonnet polishing.
1. Introduction High-end optical components are widely adopted in many advanced technical fields, including astronomical observation projects, high power laser systems and ultra-precision metrology systems. Stringent requirements on optical components put forward challenges to fabrication techniques. In the last decades, computer controlled optical surfacing (CCOS) techniques have been highly developed, such as magnetorheological polishing [1], fluid jet polishing [2], bonnet polishing [3] and ion beam figuring [4], which play increasingly significant roles in fabricating ultra-precision optics. Bonnet polishing, using an inflated flexible tool, is able to conform to the local surface with various curvature, which shows the strong ability for optical fabrication. Due to the importance of deterministic material removal in CCOS, many researchers focused on tool influence function (TIF) of bonnet polishing. Li et al. modeled TIF based on contact pressure distribution and relative surface speed, which was also used to predict edge TIFs for perfect edge control [5,6]. Zeng et al.
*
established an analytical model and revealed the effects of process parameters on material removal rate [7,8]. To obtain higher efficiency, Wang et al. developed a semi-rigid bonnet tool, which is reinforced with a stainless steel sheet [9]. Based on finite element analysis (FEA) and experimental measurement, Jiang et al. found that the TIF shape changed from Gaussian to concave when the offset increased above 1 mm [10]. Recently, a micro-analysis model based on micro-contact theory and wear theory was established. The model captured mutual interactions of slurry, pad and workpiece, which provided good insights into the material removal mechanism [11,12]. In the viewpoint of application, bonnet polishing offers superior performance on pre-polishing, surface finishing and form-correction polishing [13]. Walker et al. developed a fast production process based on bonnet polishing for aspherical prototype segment, in which the form error was improved from 30 μm PV to 136 nm RMS [14]. Through precession polishing with four steps, the surface texture can be improved to 0.5 nm Ra [15]. For finishing of X-ray molding dies, Beaucamp et al. introduced “continuous precession” bonnet polishing after fluid jet polishing to obtain 0.3 nm
Corresponding author. P.O. Box 413, Harbin, 150001, PR China. Corresponding author. E-mail addresses: [email protected] (D. Li), [email protected] (B. Wang).
**
https://doi.org/10.1016/j.precisioneng.2019.07.010 Received 4 April 2019; Received in revised form 29 July 2019; Accepted 30 July 2019 Available online 31 July 2019 0141-6359/ © 2019 Elsevier Inc. All rights reserved.
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material removal. Thus, the bonnet tool is expected to conform to surface structures to achieve uniform material removal. When uniform material removal is accomplished, the surface form generated by the first step can be kept. In this way, bonnet polishing plays the role in form-preserving polishing. However, surface structures can introduce disturbance to polishing process, such as relative surface speed and contact pressure distribution. The disturbance may cause non-uniform material removal of polishing. As a result, the generated surface form can be damaged to a certain extent, which is crucial for machining accuracy of the whole fabrication process. Regarding to nanofinishing of freeform surfaces, the specific conditions need to be clarified. For most cases, the freeform surfaces have millimeter-scale PV, such as knee joint implant [25,26] and hemispheric shell [27,28]. These kinds of surfaces requires a small tool, which can access local curved zones, shown in Fig. 2(a). To realize nanofinishing, the small t ool moves along with surface profiles with a constant offset. When the PV of the freeform surfaces ranges in micronscale [22], the large flexible tool can be used, shown in Fig. 2(b). Because of the flexibility and the small form PV, the large tool can contact with the local surface and move in a straight line. It should be pointed out that the complex freeform surfaces in this paper have specific characteristics of millimeter-level spatial periods and micron-level PV, which correspond to Fig. 2(b). One type of such freeform surfaces is continuous phase plate, with less than 3 μm PV and tens of millimeters spatial period [22]. Besides, the spatial period of complex freeform surfaces is usually smaller than polishing spot size, as shown in Fig. 1. In this case, the bonnet tool covers several peaks or valleys simultaneously, which may affect the contact condition and make the form-preserving polishing uncertain.
RMS surface texture [16]. Cheung et al. established a model-based simulation system to generate structured surface using bonnet polishing, showing good performance to predict the form error [17]. By control specific polishing tool orientation, Cao et al. reported a swing precess bonnet polishing (SPBP) method for generating freeform surfaces [18]. A model-based self-optimization method, which combined simulation of the machining process, error compensation and on-machine metrology, was developed and used to fabricate sinusoidal surfaces with over 20 mm spatial period [19]. However, bonnet polishing is not suitable to generate complex surface structures with spatial periods down to several millimeters, which requires a smaller spot size. In this aspect, reactive plasma machining techniques based on pure chemical etching, which have much smaller TIFs, show great strengths to fabricate complex freeform surfaces. Yamamura et al. investigated pulse width modulation controlled plasma machining method for high spatial resolution figuring, in which the full width at half maximum (FWHM) reaches 1.1 mm [20]. Arnold et al. developed plasma jet machining (PJM) to fabricate complex structures with millimeter-scale spatial period [21]. Atmospheric pressure plasma processing was used to fabricate larger aperture continuous phase plates with various spatial periods [22]. Nevertheless, the reactive plasma machining techniques deteriorate surface quality to a certain extent due to the pure chemical etching mechanism [23,24], which cannot meet optical finishing requirements. Thus, the post polishing needs to be conducted, which is used to smooth surface texture and preserve generated surface structures simultaneously. As mentioned above, the bonnet polishing has the ability of surface finishing, which is helpful to fabricate complex freeform surfaces cooperating with another sub-aperture fabrication method. But very little research work focuses on the form-preserving capability of bonnet polishing. In this paper, simulation and experimental work are conducted to study the form-preserving capability of bonnet polishing. The concept of form-preserving bonnet polishing is introduced and discussed firstly. Then polishing analysis model, including relative surface speed calculation and contact pressure simulation, is established. The polishing process on ripple structures under different conditions is analyzed. Subsequently, experimental results on ripple structures are presented and compared with the simulation results. At last, the relationship between the form-preserving capability and the contact pressure difference is summarized.
3. Modeling of bonnet polishing process Since Preston equation was proposed, it had been widely used to predict material removal of various polishing techniques [29]. The Preston equation is expressed as
Δz = k⋅p (x , y )⋅v (x , y )
(1)
where Δz is material removal rate, k is Preston coefficient related to substrate material, slurry and other polishing parameters, p (x , y ) is contact pressure on the substrate, and v (x , y ) is relative surface speed between polishing tool and substrate. In the surface finishing step, polishing parameters, such as feed rate and rotational speed of bonnet tool, are kept constant. And the Preston coefficient can be regarded as constant. Therefore, relative surface speed and contact pressure are the predominant factors for material removal, which need to be analyzed thoroughly.
2. Concept of form-preserving polishing using bonnet tool Fig. 1 shows the schematic diagram of form-preserving polishing using bonnet tool. The fabrication process of complex freeform surfaces with small structures can be divided into form generation and surface finishing. The form generation step requires an efficient fabrication method with small spot size, such as reactive plasma machining techniques. Then, bonnet polishing is adopted in the second step to smooth surface texture. The surface finishing is achieved through polishing
3.1. Calculation of relative surface speed Relative surface speed in contact zone depends on radius distance
Contact pressure Form generation Small size tool
Surface finishing Bonnet tool Peak Peak
Valley Spatial period Spot size
Fig. 1. Schematic diagram of form-preserving polishing. 55
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Fig. 2. Two types of nanofinishing on freeform surfaces. (a) small tool; (b) large tool.
Feed path
(a)
(b)
boundary condition is set as a pressure load. The edge of the base membrane is assigned as a fixed boundary. Since the displacement of polishing cloth surface is coupled to the base membrane, there is no additional constraint on polishing cloth. The substrate domain is set as a prescribed displacement for different offsets and the contact process is performed through substrate movement. In addition, symmetric constraints are set on the symmetric plane. To validate the model, the contact process on flat surface of a bonnet tool with 80 mm radius, polyurethane cloth, 1 bar inner pressure and 0.7 mm offset was simulated. As shown in Fig. 5(a), the contact pressure distribution shows M shape, with higher contact pressure in the outer region. This can be explained as follows. The central part of the bonnet tool has relatively uniform displacement and can match the substrate surface perfectly, which can be seen in Fig. 5(b). Thus, the contact pressure in the central area is mainly attributed to inner gas pressure. As for the outer region, the bonnet membrane bends and the non-contact region of the bonnet membrane can transmit additional force on the substrate, besides inner gas pressure. Hence, the contact pressure forms the M shaped distribution, which is similar to the results of “the bonnet warp” [7,8]. Through integrating the contact pressure, the total force on the substrate is obtained as 15.5 N. In the previous study, the total load on the substrate with the same kind of bonnet tool was measured using three OMEGA standard load cells and the measured total contact force was 17 N [30]. Comparing these two results, it can be found that the simulation error of the total contact force is about 8.8%, which proves the reliability of the model.
and rotational speed. Based on our previous study, the relative surface speed in the contact zone can be expressed as [5].
v (x , y ) = 2⋅cos ρ⋅(R − d )−1 s⋅(s − α )⋅(s − β )⋅(s − γ ) ⋅ω0 α = [(R −
x 2 + y 2 )2 + x 2 + y 2 ]1/2
β = (R − d )⋅cos(ρ)−1 γ = [y 2 + [(R − d )⋅tan(ρ) − x 2]]1/2 s = 0.5⋅α⋅β⋅γ
(2)
where ρ is precess angle, R is radius of bonnet tool, d is offset, ω0 is rotational speed around the axis of polishing tool. 3.2. Contact model between bonnet tool and substrate Fig. 3 shows the structure of the bonnet tool, including three main components: aluminum base, base membrane and polishing cloth. The aluminum base acts as a support framework and can be regarded as rigid body in the FEA model. The base membrane consists of a thin rubber layer and a fiber mesh, which can only sustain tension force. The polishing cloth is glued on the base membrane. With high-pressure air inflated, the base membrane and polishing cloth bulge with an extent of rigidity. When contacting with the substrate, the base membrane and polishing cloth are pressed on the local surface. As shown in Fig. 4, a FEA model was established using COMSOL software. To reduce computational time, a symmetric geometric model was built, including base membrane, polishing cloth and fused silica substrate. There are two physical interfaces in the model, membrane interface and solid mechanics interface. The membrane interface is adopted on the base membrane, in which only in-plane forces are calculated. The solid mechanics interface is added on the polishing cloth and substrate. These two interfaces are coupled together through the transfer of displacement parameters on the inner surface of polishing cloth. The contact pair is added to simulate the contact process. The outer surface of polishing cloth and the upper surface of substrate are assigned as destination surface and source surface, respectively. The linear elastic material model is defined. The material properties are provided by material suppliers, listed in Table 1. The base membrane surface sustains uniform pressure load, so the
4. Analysis of polishing process on ripple structures The practical interest is to achieve uniform material removal on freeform surface structures. As mentioned in Section 2, this paper focuses on freeform surfaces with millimeter-level spatial period and micron-level PV, such as continuous phase plate. In mathematical respect, such freeform surfaces are two-dimensional signals and can be described by the sum of Fourier series. Thus, sinusoidal surfaces are representative for freeform surfaces. Compared with complex freeform surfaces, the sinusoidal surfaces with certain spatial period and PV are more helpful for analyzing quantitatively. Therefore, sinusoidal ripple Precess angle
Polishing cloth Aluminium base
Aluminium base
Base membrane Base membrane
Polishing cloth Offset Substrate
Fig. 3. Structure of bonnet tool. 56
Spot size
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Fixed edge Base membrane
Polishing cloth Contact pair Substrate Fig. 4. FEA model of contact process.
Feed direction
Table 1 Material properties in contact model.
Z
Material
Density (kg/m3)
Elastic modulus (Pa)
Poisson's ratio
Base membrane Polyurethane Uninap Fused silica
0.40 × 103 0.42 × 103 0.53 × 103 1.21 × 103
9.0 × 108 1.2 × 107 2.1 × 106 7.0 × 1010
0.30 0.35 0.32 0.17
Ripple structure profile
structures can be used to investigate form-preserving capability for freeform surfaces. The contact process between the bonnet tool and ripple structures was analyzed, shown in Fig. 6. The ripple structure profile is expressed as
xtool
X
Fig. 6. Scanning process of bonnet tool on ripple structures.
(3)
z = (PV /2)⋅cos(x / T ⋅2π )
where PV is peak-valley (PV) value of ripple structures, T is spatial
Pa
(a) mm
(b) Fig. 5. Simulation results. (a) Contact pressure distribution; (b) Displacement in local contact region. 57
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color bar range was modified to show the ‘ripple pattern’ stress distribution more clearly in the enlarged region. Fig. 8(b) shows the contact pressure distribution on the substrate. The contact pressure difference at peaks and valleys is obvious, indicating that surface structures can lead to non-uniformity of the contact process. Fig. 9 shows the contact pressure profiles at four dwell positions of bonnet tool, which correspond to the scanning process of polishing. The dotted line represents the ripple structure profile, considered as fixed. The contact pressure at peaks and valleys is different and varies when the bonnet tool dwells at different positions. For peak A, when the bonnet tool dwells at one certain position, the contact pressure can be obtained and recorded as ppeak, i . The dwell positions are set to range from left to right relative to peak A, then a series of contact pressure at peak A is obtained. Notably, the range of dwell positions is required larger than the spot size of the bonnet tool. For example, when the spot size of the bonnet tool is 10 mm, the dwell positions are required to cover from −5 mm to 5 mm. The contact process outside of the range does not need to be simulated since there is no contact at peak A. Thus, a series of contact pressure at one certain peak or valley in the polishing process can be obtained to analyze the polishing process reasonably. Furthermore, the average difference of contact pressure at peaks and valleys is calculated by
Table 2 Simulation parameters of polishing process on ripple structures. No
1 2 3
Parameters of ripple structures
Polishing parameters
T (mm)
PV (nm)
Pressure (bar)
Polishing cloth
2.8, 5, 10, 15 2.8, 5, 10, 15 2.8, 5
2000
1.2
Polyurethane, Uninap
600, 1400, 2000, 4000 1400
1.2
Polyurethane, Uninap
0.3, 0.6, 0.9, 1.2
Polyurethane, Uninap
period of ripple structures. In practical cases, the bonnet tool scans on substrate and the contact condition on substrate varies in scanning movement. Thus, the contact process simulations were performed at different dwell positions ( x tool shown in Fig. 6). Table 2 lists the simulation parameters, including polishing processing parameters and ripple structure parameters. Besides, the rotational speed, bonnet radius and offset were 1000 rpm, 40 mm and 0.7 mm, respectively. 4.1. Relative surface speed on ripple structures Based on Equ. (2), the relative surface speed on flat surface and ripple structures was calculated. To calculate relative surface speed on ripple structures, the profiles of ripple structures were added on the offset value. Fig. 7(a) shows the normal relative surface speed on flat surface. Fig. 7(b) shows the result on ripple structure with 5 mm spatial period, corresponding to No 1 in Table 2. The difference of relative surface speed between the two surfaces was obtained, shown in Fig. 7(c). It can be seen that the deviation of relative surface speed induced by the ripple structure is much smaller compared with the normal relative surface speed, with a ratio less than 10−4. Therefore, the influence of relative surface speed on the uniformity of removal rate is much insignificant. The above results indicate that the deviation of relative surface speed induced by surface structures has a minor effect on form-preserving capability.
pdifference =
∑
(ppeak,
i
− pvalley, i )/ N
i = 1~ N
where ppeak, i is the contact pressure at peaks in the ith simulation, pvalley, i is the contact pressure at valleys in the ith simulation, N is the number of simulations at different dwell positions. Note that, pdifference is named as “contact pressure difference” concisely in the following text. 5. Contact pressure difference under various conditions According to Table 2, the contact process under different conditions is simulated firstly and then the contact pressure difference is calculated based on Equ. (4). Thus, the influence on contact pressure difference of the spatial period, PV and the inner pressure can be obtained, shown in Fig. 10–12. In these figures, the vertical coordinates represent the contact pressure difference, which is related with uniformity of material removal. The horizontal coordinates represent the value of the influence factor.
4.2. Contact pressure distribution on ripple structures Fig. 8 shows the simulation result of ripple structures No 1 in Table 2, with 5 mm spatial period and polyurethane polishing cloth. After contacting, the polishing cloth deforms and conforms to the surface form under inner pressure. As shown in Fig. 8(a), there is no gap between polishing cloth and substrate, indicating that the two objects are able to match totally. Thus, both peaks and valleys of the surface structure can be polished, which demonstrates the foundation of the form-preserving capability of bonnet polishing. On the other hand, the stress distribution in Fig. 8(a) shows a ‘ripple pattern’ consistent with the ripple structure. The black dotted line in Fig. 8(a) shows the direction of the ripple structure. Note that, the stress distortion induced by the ripple structure is relatively small, so that the
5.1. Influence of spatial period on contact pressure difference Fig. 10(a) shows the contact pressure distribution with polyurethane of No 1 in Table 2. The contact pressure profile on flat surface with the same parameters is also plotted. The deviation of contact pressure on ripple structures and flat surface is obvious. When the spatial period reaches 10 mm, the deviation becomes indistinctive at this dwell position. Based on Equ. (4), the contact pressure difference is obtained, shown in Fig. 10(b). The results using polyurethane and uniap are plotted in two types of point. The horizontal coordinates are
unit: mm/s
(a)
(4)
(b)
(c)
Fig. 7. Relative surface speed. (a) Flat surface; (b) Ripple structure surface; (c) Difference between (a) and (b). 58
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Pa
(a)
(b)
Contact pressure (kPa)
Fig. 8. Simulation result of contact process (a) von Mises stress distribution; (b) Contact pressure on substrate.
200
ppeak , i pvalley, i
0
ppeak, i+1 pvalley, i 1
p peak , i
0
2
pvalley, i
2
0
150
0
0
0
100
0
0
0
50
0
0
A
0 -10
-5
B
0
5
x (mm)
A
0
10
15 -10
-5
0
B
5
3
pvalley, i
3
0 A
0
10
p peak , i
15 -10
-5
0
B 5
x (mm)
x (mm)
A
0 10
15 -10
-5
B
0
5
10
15
x (mm)
Fig. 9. Contact pressure profiles at different dwell positions of bonnet tool.
2.8 mm, 5 mm, 10 mm, and 15 mm spatial periods, respectively. With the spatial period increasing, the contact pressure difference decreases regardless of polyurethane or uninap. This can be explained by ‘bridge effect’ of polishing cloth on ripple structures, in which the transverse shear effect becomes significant with smaller spatial periods, leading to higher contact pressure difference [[31]]. The contact pressure difference causes material removal difference at peaks and valleys, which weakens the form-preserving capability. When the spatial period is larger than 10 mm, the contact pressure difference with two polishing clothes is less than 5 kPa, much smaller than the normal contact pressure on flat surface. This indicates that the surface structures with large spatial periods can be polished with minor form change. In addition, the ratio of the contact pressure difference over the normal contact pressure is about 10−1–10−2, much larger than that of the relative surface speed in Section 4.1, which shows that the contact pressure difference is the major factor for form-preserving capability. It can also be found that the contact pressure difference using uninap is much smaller than that using polyurethane, which is attributed to different material elasticity. The elastic modulus of uninap is smaller, so it deforms more easily to match surface structures. Thus, a more uniform contact pressure distribution can be achieved, which provides a stronger form-preserving capability.
(a)
(b)
Fig. 11. Contact pressure difference with different ripple structures. (a) Polyurethane; (b) Uninap.
5.2. Influence of PV on contact pressure difference Fig. 11 shows contact pressure difference on ripple structures with different PV, corresponding to No 2 in Table 2. The results using polyurethane and uninap are presented in Fig. 11(a) and (b). The horizontal coordinates are 600 nm, 1400 nm, 2000 nm, and 4000 nm PV, respectively. One type of point represent the ripple structure with the same spatial period. Overall, the contact pressure difference increases linearly with increasing PV of ripple structures. For
(a)
(b)
Fig. 10. Simulation results with different spatial periods. (a) Contact pressure profiles with polyurethane; (b) Contact pressure difference. 59
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(a)
(b) Fig. 12. Contact pressure difference with different inner pressure. (a) Polyurethane; (b) Uninap.
cloth can conform to ripple structures totally, which is the fundamental reason for the form-preserving polishing result. On the other hand, the PV of the ripple structure decreases slightly after polishing, which is caused by removal non-uniformity at peaks and valleys, due to contact pressure difference. To evaluate the form change after polishing, the PV change of ripple structures after 1 μm polishing depth were obtained and summarized in Table 4. Table 4 includes experimental parameters, PV change after polishing and related simulation results. For experiments No 1 ~ 4, the initial PV of ripple structures varies within a relatively small range, which can be regarded as constant. Thus, the PV change is attributed to various spatial periods. It can be found that the PV change decreases with spatial periods increasing. In experiments No 2, No 5 and No 6, the spatial periods keep constant as 5 mm and the initial PV varies from 510 nm to 1210 nm. It can be seen that the PV change increases with the rising of initial PV. The influence of polishing cloth is presented in two experiments pairs (No 1, 7 and No 6, 8). Even though the initial PV does not keep equal, it can also be found that the PV change using polyurethane is obviously larger than that using uninap. For experiments No 8 ~ 10, the PV change fluctuates a little, indicating that the inner pressure has a negligible influence on removal uniformity as well as form-preserving capability. Even the initial PV reaches near 3 μm in experiments No 11 and 12, the PV change still keeps at a low level, which demonstrates great form-preserving performance. As listed in Table 4, the PV change corresponds to the contact pressure difference generally. The correlation between the PV change and the contact pressure difference is fitted linearly, shown in Fig. 14. The coefficient of determination is 0.90 adjusted R squared, indicating the goodness of the linear relationship. When the contact pressure difference is less than 3 kPa, the deviation between experimental results and fitting results is relatively large, which can be introduced by measurement error. However, the PV change keeps less than 20 nm in this zone, showing superior form-preserving capability. On the other hand, the PV change reaches 45 nm when the contact pressure difference rises to 8 kPa, which may not be acceptable in practice. Furthermore, an empirical formula is obtained to predict the PV change after bonnet polishing, expressed as Equ. (5). The T and PV denote the spatial period and PV of the ripple structure, same as that in Equ. (3). The general influence laws can be summarized as follows. The PV change is linearly proportion to the PV and exponentially inverse proportion to the spatial period. Comparing the two polishing clothes, the PV change and its increasing tendency using uninap is smaller than that using polyurethane.
ripple structures with larger PV, the deformation of polishing cloth at peaks becomes larger, causing higher contact pressure difference at peaks and valleys. The increase indicates that bonnet polishing will induce more form change for ripple structures with larger PV. Fig. 11(a) shows that the contact pressure difference using polyurethane increases rapidly with PV, reaching 70 kPa with the 2.8 mm spatial period. Moreover, the contact pressure difference using uninap keeps at a low level, less than 11 kPa, which offers a better choice to achieve formpreserving polishing. 5.3. Influence of inner pressure on contact pressure difference Fig. 12 shows the contact pressure difference with different inner pressures, corresponding to No 3 in Table 2. The results using polyurethane and uninap are presented in Fig. 12(a) and (b). For each polishing cloth, the contact pressure difference was obtained on ripple structures with 2.8 mm and 5 mm spatial periods. The horizontal coordinates are 0.3 bar, 0.6 bar, 0.9 bar, and 1.2 bar inner pressure, respectively. It is interesting to find that the inner pressure has a minor influence on the contact pressure difference. The change value is less than 1 kPa, which is much smaller than that induced by different PV and spatial periods. The results indicate that the polishing clothes can totally match surface structures even the inner pressure is at a low level, 0.3 bar. Note that, the contact pressure difference decreases slightly with increasing inner pressure. This can be explained as follows. In the contact process, the polishing cloth is pressed on ripple structures by inner pressure. The contact interface becomes closer at both peaks and valleys with higher inner pressure, which causes a slight decrease of contact pressure difference. Overall, the form-preserving capability is not affected by inner pressure. 6. Verification of form-preserving capability To validate the simulation analysis experimentally, fused silica substrate with ripple structures were prepared, and then polished on a Zeeko IRP 600 polishing machine. The surface forms before and after polishing were measured using Fisba μPhase 2 HR interferometer. The experimental parameters are listed in Table 3. Fig. 13 shows one ripple structure surface before and after polishing. It can be seen that the bonnet polishing does not damage the ripple structure. Based on the contact analysis above, the polishing Table 3 Parameters of bonnet polishing experiments. Bonnet radius (mm)
Rotational speed (rpm)
Precess angle (°)
Slurry
40
1000
15
Cerium oxide
⎧ 6.39·(0.596 +
Offset (mm)
PV change =
0.0386 ⋅ PV
⎨ 6.39·(0.432 + ⎩
) − 1.68, using polyurethane
e (0.324·T ) 0.0032 ⋅ PV e (0.136·T )
) − 1.68, using uninap (5)
0.7
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(a)
(b) Fig. 13. Ripple structure form. (a) Before polishing; (b) After polishing.
7. Conclusions
Table 4 Summary of polishing experiments and simulation on ripple structures. No
Spatial period (mm)
Initial PV (nm)
Inner pressure (bar)
Polishing cloth
PV change (nm)
Contact pressure difference (kPa)
1 2 3 4 5 6 7 8 9 10 11 12
2.8 5 10 15 5 5 2.8 5 5 5 15 15
480 510 470 709 1020 1210 490 1656 1550 1630 2640 2580
1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 0.9 0.3 1.2 1.2
polyurethane polyurethane polyurethane polyurethane polyurethane polyurethane uninap uninap uninap uninap uninap polyurethane
47 24 7 6 45 63 11 17 15 16 5.5 10
8 5 3 1.5 7.6 8.8 1.0 2.3 2.4 2.3 1.6 2.2
Experimental results Fitting results
60 PV change (nm)
In this paper, the form-preserving capability of bonnet polishing was studied based on simulation and experimental method. The simulation shows that the bonnet tool can deform and match surface structures with small spatial periods, which provides the foundation of form-preserving polishing. The experiments on ripple structures prove the form-preserving capability of bonnet polishing. It is found that form change after polishing depends on contact pressure difference at peaks and valleys linearly. The softer polishing cloth, larger spatial period and smaller PV can lead to lower contact pressure difference and less form change. Instead, the relative surface speed and inner pressure have a minor influence on form-preserving capability. Under specific conditions, the form change can be controlled less than 20 nm with low contact pressure difference, which demonstrates the excellent formpreserving capability. This work gives a comprehensive insight into the form-preserving capability of bonnet polishing, which implies its feasibility in practical optical fabrication. Based on the form-preserving capability, future work will focus on fabricating complex freeform optics with high efficiency and quality, cooperating with other subaperture figuring methods. Acknowledgments
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The work is supported by National Natural Science Foundation of China (No. 51175123); National Science and Technology Major Project (No. 2013ZX04006011-205). The author would like to thank the China Scholarship Council (CSC) for its financial support in studying at University of Huddersfield, the National Facility for Ultra Precision Surfaces, North Wales, UK.
30 Equation a b R-Square (COD) Adj. R-Square
15 0
0
y = a + b*x -1.68 6.39 0.91317 0.90232
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