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Modeling and analysis of the material removal profile for free abrasive polishing with sub-aperture pad
Journal of Materials Processing Technology 214 (2014) 285–294
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Journal of Materials Processing Technology journal homepage: www.elsevier.com/locate/jmatprotec
Modeling and analysis of the material removal profile for free abrasive polishing with sub-aperture pad Cheng Fan a,b , Ji Zhao a , Lei Zhang a,∗ , Yoke San Wong b , Geok Soon Hong b , Wansong Zhou a a b
College of Mechanical Science and Engineering, Jilin University, Changchun 130022, China Department of Mechanical Engineering, National University of Singapore, Singapore 117576, Singapore
a r t i c l e
i n f o
Article history: Received 16 February 2013 Received in revised form 3 September 2013 Accepted 9 September 2013 Available online 19 September 2013 Keywords: Free abrasive polishing Material removal profile Sub-aperture pad Material removal index
a b s t r a c t This paper addresses the problem of material removal in free abrasive polishing (FAP) with the subaperture pad both theoretically and experimentally. The effects of some polishing conditions upon the material removal are analyzed, including not only the process parameters, which refer to the normal force, angular spindle velocity and angular feed rate, but also the abrasive grain size, polishing slurry properties, topographical parameters of the sub-aperture pad, as well as tool path curvature. Based on the analysis, a model of material removal profile is proposed to facilitate more accurate polishing. First, by analyzing the contact among polishing pad, abrasive grain and workpiece surface in the micro level, the removal depth per unit length of the polishing path is derived, which is defined as the material removal index. Then, the distribution of this removal index can be obtained via modeling the pressure and relative sliding velocity in the contact region of polishing pad and workpiece. After that, the material removal profile can be calculated by integrating the material removal index along the tool path in the tool-workpiece contact region. To verify the effectiveness of the proposed model, a series of polishing experiments have been conducted. Experimental results well demonstrate that our model can accurately predict the material removal depth during the FAP. © 2013 Elsevier B.V. All rights reserved.
1. Introduction Free-form surfaces with high surface finish, integrity and form accuracy, such as precise aspheric and non-geometric optical lens and molds, have become key features of components increasingly required by optoelectronic and communication industries (Cheng et al., 2005). The polishing process, which is usually the final step of fabrication, is essential to the quality and duration of the part surfaces. Various types of polishing are used for optical fabrication, such as free-abrasive polishing (FAP), magneto-rheological finishing, abrasive slurry jet polishing, etc. FAP is considered as one of the most common and important operations in optical finishing. In a typical FAP process, a rotational sub-aperture polishing pad in the presence of liquid slurry which contains hard abrasive grains is pressed against the workpiece. Between the sub-aperture pad and the workpiece there forms an area of contact and the interactions between polishing pad, abrasive grains and workpiece result in the material removal in the contact area.
∗ Corresponding author at: College of Mechanical Science and Engineering, Nanling Campus, Jilin University, No. 5988, Renmin Street, Changchun 130022, China. Tel.: +86 18946686195. E-mail address: [email protected] (L. Zhang). 0924-0136/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jmatprotec.2013.09.010
The FAP process with the sub-aperture pad is usually used as an error compensation method to improve the form accuracy of the part surfaces if the material removal in the contact area is controllable. Jones (1977) first proposed to use the FAP process to fabricate the aspheric optical surface. By controlling the amount of material to be removed precisely, the mid-spatial frequency errors left after grinding or turning can be removed by sub-aperture polishing. Su et al. (1996) and Su and Sheen (1999) developed a planning strategy for the polishing process to remove an arbitrary profile with the precondition that the material removal depth in the contact area can be represented by a linear function of machining time. Therefore, it is essential to model and analyze the material removal of polishing in a quantitative manner. However, the fundamental mechanisms of material removal in FAP involve micro-fatigue, micro-crack, ploughing and cutting mainly depending on the state of the interactions between abrasives, polishing pad and workpiece, which have not been well understood (Brinksmeier et al., 2006). Thus, the research on the material removal rate (MRR) of FAP is still at an empirical level. The Preston’s equation is widely used to predict the MRR during the polishing process, which states that the MRR is determined by the product of the contact pressure, relative velocity, machining time and Preston coefficient which is related to the workpiece material, properties of polishing slurry and polishing pad, size of abrasive grains, etc. (Preston, 1927). Besides, some researchers tried to explain the mechanism and establish
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their own MRR models from the perspective of descriptive modeling. Yu et al. (1993) modeled the dynamic interaction between pad and workpiece. In this model, the polishing pad is characterized by a statistical asperity model and the effect of polishing pad surface roughness on the MRR is included. The contact behavior of a polishing pad with the workpiece surface was also investigated by Klocke and Zunke (2009) using an FE model. The predicted results show that the material removal in FAP can be described by the penetrating of the abrasive particles into the workpiece, and also the sliding or yielding after the penetration. A predictive model of MRR during the chemical-mechanical polishing was developed by Xie and Bhushan (1996) to predict the effects of grain size, polishing pad and contact pressure. Zhao and Chang (2002) presented a MRR model for chemical-mechanical FAP based on the elastic-plastic microcontact mechanics. In this model, the indentation depth of a grain into the workpiece is determined by the force equilibrium equation of the single grain. A similar MRR model was also developed by Lin (2007), which takes the contact deformation of the abrasive grains into consideration. Jin and Zhang (2012) proposed a statistical MRR model by assuming that there are two types of abrasive grains contributing to the material removal process, i.e., the grains rotating between the pad and workpiece and those embedded in the pad without rigid body motion. The previous research on the MRR focuses on the modeling of mechanical contact action among the pad, abrasive grains and workpiece surface. Due to the lack of basic understanding of the mechanism of the FAP process, these models are still regarded as half-empirical. The MRR distribution is not uniform in the contact region, mainly depending on distributions of the contact pressure and the relative sliding velocity between polishing pad and workpiece surface (Lee et al., 2013). The contact pressure distribution during the polishing was modeled by Roswell et al. (2006) based on the Hertizian contact theory. According to the investigation by Yang and Lee (2001), the contact area and the pressure distribution were affected by the equivalent radius of polishing tool and workpiece surface. In their work, the MRR distribution in the contact region was modeled based on the Preston’s equation, and the material removal depth in the contact area was calculated by the integral of local MRR. The MRR distribution was defined as the tool influence function (TIF) by Kim and Kim (2005). Based on the Preston’s equation, the TIF was modeled by assuming that the pressure distribution in the contact region is Gaussian. The Preston’s equation was also used by Cheung et al. (2011) to model the TIF, and the modelbased simulation was used to predict the generation of structured surfaces. These predictive models of MRR distribution and material removal depth are all based on the Preston’s equation. However, the Preston’s coefficient has to be determined by experiments before each polishing process, because the changes of polishing slurry, abrasive grains and polishing pad may result in different values of the Preston coefficient. Since the MRR models and the MRR distribution models have been studied independently, there has been
no study directly linking the abrasive grain size and pad properties to the material removal depth yet. In this paper, a novel model is proposed to predict the material removal depth of the workpiece surface. The material removal profile, which is defined as the material removal depth orthogonal to the polishing path, is used to describe the material removal in the FAP process. The effects of some polishing conditions upon the material removal are analyzed, including not only the process parameters, which refer to the normal force, angular spindle velocity and angular feed rate, but also the abrasive grain size, polishing slurry properties, topographical parameters of the sub-aperture pad, as well as tool path curvature. The remainder of this paper is organized as follows. In Section 2, the material removal index for FAP, defined as the material removal depth per unit length of the polishing path, is derived. In Section 3, the distributions of contact pressure and relative velocity are discussed and an approach is proposed to model the material removal profile. The validity of the proposed model for the removal profile is verified by experiments in Section 4. Findings are summarized in the concluding section. 2. Material removal index for free abrasive polishing In this paper, it is assumed that the material removal from the workpiece is primarily caused by abrasion wear by the abrasive grains in the slurry. Besides, the abrasive grains are spherical, with an average radius of Rabr and uniformly distributed in the polishing slurry. These assumptions have been substantiated by Zhao and Chang (2002), Xie and Bhushan (1996) and Lin (2007). The material removal index is defined as the material removal depth per unit length of the polishing path, which is different from the definition of MRR. 2.1. Contact between the polishing pad and workpiece surface Since the polishing pad is much rougher than the workpiece surface, the contact between them can be regarded as that between a rough surface and a smooth one, as is shown in Fig. 1. When the polishing pad is pressed on the workpiece surface under a constant load, the distance between the reference plane and the workpiece surface is hs . Yu et al. (1993) proposed a statistical model to characterize the polishing pad with random roughness. In this model, the Gaussian function was used to describe the distribution of the surface height of the polishing pad, which is given by ϕ(h) =
1 exp √ 2
−
h2 2 2
(1)
where is the standard deviation of the distribution of the pad surface. In the actual polishing process, the abrasive grains are rarely on the contact interface. Thus it is reasonable to assume that the
Fig. 1. Interactions between polishing tool, abrasive grains and workpiece.
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Assuming that the contacts between a grain and workpiece and between the grain and polishing pad are plastic, the interference w is given by Xie and Bhushan (1996) as w =
Hp (2Rabr − hs + h) Hw + Hp
(7)
Substituting Eqs. (6) and (7) into Eq. (5) yields A=
2Rabr
1.5
Hp (2Robr − hs + h) Hw + Hp
(8)
2.3. Number of abrasive grains participating in material removal
Fig. 2. Schematic of single abrasive abrasion.
entire contact load is sustained by the contact between pad and workpiece surface. The GW model (Greenwood and Williamson, 1966) is used to describe the contact between pad and workpiece surface. According to this model, the pressure between pad and workpiece is given by Pc =
4 ∗ Np Rp0.5 Epw 3
hmax
(h − hs )
1.5
ϕ(h) dh
(2)
h
where Rp is the average radius of the asperities, Np is the number of asperities per unit area, hmax is the maximum height of the pad surface and E* pw is the contact modulus of the polishing pad and ∗ = (1 − 2 )/E + (1 − 2 )/E . E and the workpiece given by 1/Epw p w p p w Ew are the Young’s modulus of the polishing pad and the workpiece respectively, and p and w are their Poisson’s ratio respectively. According to Eq. (2), the pressure Pc is a function of the separation distance hs . Thus, if the polishing pressure Pc is known, the separation distance hs can be calculated from Eq. (2).
As shown in Fig. 2, the abrasive grain is embedded between the pad and the workpiece. Rabr is the radius of the abrasive grain, and w and p correspond to the interference between grain and workpiece and that between grain and polishing pad, respectively. The volume of the material removed by a single abrasive grain is approximately equal to the product of its cross-sectional area immersed in the workpiece and the sliding distance, which gives (3)
where A is the cross-sectional area and L is the sliding distance. Assuming that the hard abrasive grain maintains its spherical shape, A can be expressed by 2 A = Rabr arcsin
a Rabr
− a(Rabr − w ),
w Rabr
,
(5)
where a can be expressed as
a=
2
2 − (R Rabr abr − w ) =
2Rabr w − 2w ≈
2Rabrw
(9)
˝ − l2 .
(6)
(10)
Rearranging Eq. (9) and substituting it into Eq. (10) gives
˝=
2/3
3d
(11)
3 4Rabr
When the applied load on the abrasive grain is small, the contact between abrasive grain and workpiece is elastic. According to the Hertizian contact theory, the indentation depth can be expressed as w =
3p 2
(4)
where a is the radius of the circular indentation on the workpiece surface. Compared with the grain radius Rabr , w is very small, and so is a/Rabr , thus Eq. (4) can be further linearized about a/Rabr as A ≈ aw 1 + O
4 3 3 R l = d 3 abr
The relationship between area density ˝ and the line density l is given by
2.2. The material removal of a single abrasive grain
V = AL
The number of the free-abrasive grains in the slurry directly participating in the abrasive wear process is one of the most important variables influencing the material removal. Previous studies assumed that the abrasive grains embedded in the workpiece surface and the asperities of the polishing pad participate in the material removal (Zhao and Chang, 2002; Jin and Zhang, 2012). Thus, the number of active grains is the product of the area density of the abrasive grains in the slurry and the real area of contact between polishing pad and workpiece. However, in the actual polishing process, the abrasive grains are rarely on the contact interface of the pad asperities and workpiece. Moreover, if the area density of grains is fixed, the MMR is independent of the abrasive grain size in these models, which is unreasonable. In order to overcome the deficiency of these models, the critical condition for the grains participating in material removal is described and derived. Furthermore, the effects of the pad properties on the number of active grains are considered, which is ignored by Xie and Bhushan (1996). Let the slurry concentration per unit volume be d , and the line density of the grain in the slurry be l. Then
4E ∗
Rabr ,
(12)
∗ where p is the average pressure acting on the grain and Eaw is the contact modulus of abrasive grain and workpiece given by ∗ = (1 − 2 )/E + (1 − 2 )/E . E and are Young’s modulus 1/Eaw a w a a a w and Poisson’s ratio of the grains, respectively. As the applied pressure increases, the elastic contact will change into plastic contact, and the critical value of the pressure p is determined by the hardness of the workpiece as p = Hw . Thus, the maximum interference w0 of the elastic contact is given by
w0 =
3H 2 w
4E ∗
Rabr .
(13)
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According to Eq. (8), the volume of material removed by a single grain across H is V0 =
2Rabr
Hp Hw + Hp
1.5
(2Rabr − hs + h)
1.5
dy.
(19)
Within the time interval dt, the overall volume of material removal at H can be expressed as V = dx dy dz = N1 V0 ,
(20)
where dz is the depth of material removal at H. Substituting Eqs. (18) and (19) into Eq. (20) yields Fig. 3. Schematic of polishing along a curved path.
wl = The material is removed from the workpiece surface in a plastic manner. By substituting Eq. (13) into Eq. (7), the critical separation distance he leading to plastic contact can be expressed by
Hw + Hp he = hs − 2 − Hp
3H 2
w 4E ∗
(14)
In the polishing process, the pad can only hold a certain portion of abrasive grains in the slurry. The material of the workpiece is assumed to be primarily removed by the abrasive grains attached to the polishing pad with the height of the pad surface higher than he . As the surface height distribution density of the polishing pad is ϕ(h) (see Eq. (1)), the number of abrasive grains per unit area at a height between h and h + dh is ˝ × ϕ(h). Thus, the number of grains participating in the material removal can be expressed as
N0 =
2Rabr vs ˝
va
1.5
Hp Hw + Hp
hmax
(2Rabr − hs + h) h2
1.5
ϕ(h)dh, (21)
where wl ≡
Rabr .
dz dy
(22)
is the material removal index for free-abrasive polishing defined as the depth of material removal per unit length along the polishing path. Eq. (21) can be further modified as wl =
2Rabr vs ˝
va
Hp Hw + Hp
1.5
hs
×
(2Rabr − hs + h)
1.5
hmax
he
(2Rabr )1.5 ϕ(h) dh
ϕ(h)dh + hs
(23)
hmax
˝ϕ(h)dh
(15)
h
where ϕ(h), ˝ and he are expressed by Eqs. (1), (11) and (14), and is the constant between 0 and 1 related the polishing pad’s ability to hold the abrasive grains. 2.4. Material removal index A contact patch in a certain shape is formed between the polishing pad and the workpiece when a normal load is applied on the polishing pad and the material removal takes place within the contact region. As shown in Fig. 3, point o is the center of the contact patch on the path and H is an arbitrary point on the x axis with the infinitesimal area dxdy. The direction of dy is along the tool path. For an infinitesimal time dt during which the tool is contacting with H, dt =
dy
va
,
(16)
where va is the feed rate of the tool moving along the tool path. During the time interval dt, the real contact area between polishing tool and H can be expressed as S = vs dx dt =
vs dx dy va
(17)
where vs is the relative sliding velocity between the tool and H. The number of grains per unit area participating in the material removal can be calculated by Eq. (15). Then, the total number of abrasive grains participating in the material removal of polishing during the time interval dt is
vs dx dy N1 = SN0 = va
hmax
˝ϕ(h) dh he
(18)
3. Modeling the material removal profile 3.1. Contact pressure It is assumed that process parameters, such as the normal polishing force, the angular spindle velocity, the feed rate and the curvature of the tool path, vary slightly along the tool path, and the curvature radius of the surface at the point o(P) is very large compared with the size of the sub-aperture pad. Thus, the contact region is approximately a circle with radius Rt and the average contact pressure can be expressed by Pc =
Fn Rt2
(24)
where Fn is the normal force on the pad. 3.2. Relative sliding velocity As shown in Fig. 4, the frame xoy is fixed at the point P and the plane xoy is the tangent plane of the surface at point P. The point O is the geodesic curvature center of the path, and oO = R which gives R=
1 , |kg |
(25)
where kg is the geodesic curvature of the path at point P. The frame XOY is set up at point O and the Y direction is parallel to the y direction. For the convenience of analysis, an arbitrary point in the frame xoy is described in the frame XOY by the polar coordinate form, and the corresponding coordinate transformation is given by
x = cos + R y = sin
(26)
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289
Fig. 4. Distribution of relative velocity when polishing along a curved path.
For the arbitrary point M in the contact circle, the velocity due to the tool rotation vm can be expressed as
vm = wp
x2 + y2 = wp
2 + 2 R cos + R2
And solution of cos can be obtained from Eq. (32) as cos =
(27)
(33)
And there is one solution of within [/2, ] which gives
where wp is the angular spindle velocity. The velocity due to the feed rate along the curved tool path is given by
va = wq
Rt2 − 2 − R2 . 2R
1 ( ) = − arccos
(28)
Rt2 − 2 − R2 2R
(34)
where wq is angular feed rate around the point O. The relative sliding velocity vs in the contact region is the composition of vm and va . Thus, the relative sliding velocity at M(x, y) can be expressed by vs =
v2a + v2m − 2va vm cos( − ) =
wq2 2 + wp2 ( 2 + 2 R cos + R2 ) − 2wq wp
2 + 2 R cos + R2 cos( − )
(29)
3.3. Material removal profile orthogonal to the tool path As illustrated in Fig. 4, for R > Rt , the bounding values of are
min = R − Rt and max = R + Rt . By substituting Eq. (34) into Eq. (30), the material removal profile can be calculated as
As shown in Fig. 4, when the polishing pad moves across point H along the curved tool path, the contact length between the pad and the workpiece at point H can be represented by the arc length L1 L2 centered at O with radius . The removal depth at point H on the surface can be calculated by integrating wl along the arc L1 L2 , which gives
h( ) =
2 3d
2
1.5 wp Rabr
1 ( )
4
2/3
Hp Hw + Hp
1.5
1 ( )
h( ) = 2
w1 d
(35)
Substituting Eq. (11), (23) and (29) into Eq. (35) yields
hs
(2Rabr − hs + h)
1.5
hmax
(2Rabr )1.5 ϕ(h) dh
ϕ(h) dh +
he
hs
wq2 2 + wp2 ( 2 + 2 R cos + R2 ) − 2wq wq
×
R − Rt ≤ ≤ R + Rt .
0
2 + 2 R cos + R2 cos( − ) d
(36)
0
1
h( ) =
w1 d
(30)
2
where 1 and 2 correspond to the starting angle and the ending angle of the arc L1 L2 . The contact circle can be expressed in the frame xoy by x2 + y2 = Rt2
4. Experimental verification 4.1. Experimental conditions
(31)
Substituting Eq. (26) into Eq. (31) yields Rt2 − ( cos + R2 ) − 2 sin2 = 0.
In Eq. (36), hs can be solved by Eq. (2) and Eq. (24) and he can be calculated by Eq. (14). The detailed calculation procedure for the material removal profile is shown in Fig. 5.
(32)
To verify the effectiveness of the theoretical model for the removal profile, a series of polishing experiments have been conducted on a 5-axis polishing machine which is composed of three
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C. Fan et al. / Journal of Materials Processing Technology 214 (2014) 285–294 Table 1 Polishing conditions. Item
Condition
Workpiece
Material: Glass ceramics, Hw = 8.3 GPa, Ew = 85.16 GPa, vw = 0.272 Material: CeO2 , d = 0.5% and 0.2%, Rabr = 0.83 m and 1.6 m, Ea = 249 GPa, va = 0.297, = 0.2 Polishing pad: Hp = 90 Mpa, Ep = 250 MPa, vp = 0.5, Rp = 0.03 mm, Np = 1.2 × 103 mm−2 , = 0.025 mm, hmax = 0.58 mm, Rt = 4 mm
Slurry Polishing tool
Fig. 5. Calculation steps for material removal profile.
translational axes (X-axis, Y-axis and Z-axis) and two rotational axes (B-axis and C-axis). The structure of the machine tool is shown in Fig. 6a. As shown in Fig. 6b, the piezoelectric force sensor is mounted on the end effector of the polishing spindle. It is used as a feedback sensor to control the normal polishing force. The workpiece is fixed on the rotational table. The set-up enables the polishing tool to move along a curved path with the normal polishing force varying from 1 N to 10 N and a specified feed rate. The
polishing spindle is mounted on the B-axis, with the angular speed adjustable from 209 rad/s to 2090 rad/s. The material removal profile is measured by Form Talysurf PGI 810 (Taylor-Hobson Ltd., UK) with a 0.3 mm tip radius stylus along a line perpendicular to the polishing path. The measurements are conducted under the waviness mode to suppress the high-frequency noise of the profiles, and the measured data are illustrated from Figs. 7–12. The workpiece material used in the experiments is glass ceramics. In order to get the physical properties of the workpiece, the nano-indentation experiments are conducted, where Hw = 8.3 GPa, Ew = 85.16 GPa and w = 0.272. An elastic polishing tool which is made of polyurethane is used as the sub-aperture polisher. The radius of the polyurethane sub-aperture pad Rt is fixed as 4 mm and the mechanical properties are Hp = 90 MPa, Ep = 250 MPa and
p = 0.5 (Zhao and Chang, 2002). The topographical parameters of the polyurethane sub-aperture pad are adopted according to the experimental measurements of Yu et al. (1993) for typical polyurethane pad surfaces, in which the average radius of asperity Rp = 0.03 mm, the asperity density Np = 1.2 × 103 mm−2 and the standard deviation of the distribution = 0.025 mm. The value of which describes the ability of the pad to hold abrasives is set according to Xie and Bhushan (1996), in whose work = 0.2 for the pad with medium hardness. The CeO2 grains are used as free abrasives in the slurry. The mean diameters of the grains selected are 0.83 m and 1.6 m, and the grain volume concentrations of the slurry are fixed as 0.5% and 0.2%. For a CeO2 grain in the slurry, Ea = 249 GPa and a = 0.297, as indicated by Kanchana et al. (2006). The detailed polishing conditions for the polishing experiments are
Fig. 6. Structure of experimental machine and polishing spindle. (a) Machine tool with five axes and (b) polishing spindle with force control.
C. Fan et al. / Journal of Materials Processing Technology 214 (2014) 285–294
Fig. 7. Comparison of material removal profiles for various normal polishing forces (wp = 209 rad/s, wq = 0.0015 rad/s, R = 6 mm, Rabr = 0.83 m, d = 0.5%). (a) Experimental profiles vs. simulation profiles and (b) theoretical profiles.
summarized in Table 1. The planning of process parameters in the experiments is shown in Table 2. 4.2. Results and discussion The mechanism of the FAP process is complicated and the influencing factors of the material removal depth involve the process parameters, properties of abrasive grains/slurry and polishing sub-aperture pad, and tool path curvature. Figs. 7–12 show the profiles of the material removal in different polishing conditions. The material removal profiles show good agreement between the experimentally measured results and the simulation results calculated by Eq. (36). The removal profiles are not symmetrical, as can be seen in both experiments and simulation results, which show deviation toward the center of the tool path curvature. This deviation is attributed to the longer dwelling time in the contact region near the curvature center of the tool path when polishing along a curved path. The experimental material removal profiles of the workpiece surface for various normal polishing forces Fn are shown Table 2 Process parameters in the experiments. No.
Fn (N)
wp (rad/s)
wp (rad/s)
R (mm)
Rabr (m)
d (%)
1 2 3 4 5 6 7
1.5 6 1.5 1.5 1.5 1.5 1.5
209 209 418 209 209 209 209
0.0015 0.0015 0.0015 0.0003 0.0015 0.0015 0.0015
6 6 6 6 5 6 6
0.83 0.83 0.83 0.83 0.83 1.60 0.83
0.5 0.5 0.5 0.5 0.5 0.5 0.2
291
Fig. 8. Comparison of material removal profiles for various angular spindle velocities (Fn = 1.5 N, wq = 0.0015 rad/s, R = 6 mm, Rabr = 0.83 m, d = 0.5%). (a) Experimental profiles vs. simulation profiles and (b) theoretical profiles.
in Fig. 7. Other experimental settings include wp = 209 rad/s, wq = 0.0015 rad/s, R = 6 mm, Rabr = 0.83 m and d = 0.5%. Fig. 7a shows the removal profiles obtained both experimentally and theoretically for Fn = 1.5 and 6 N, and Fig. 7b shows simulation results of the profiles for Fn varying from 1.5 N to 6 N. Both the simulation and experimental results show that the material polished increases as Fn grows. The effect of the normal polishing force Fn is reflected in the separation distance hs . A large Fn leads to a small hs , which causes more material to be removed by more abrasive grains participating in the process. According to the experimental results and theoretical predictions in this paper, the removal depth is approximately proportional to (Fn )0.65 . This result is consistent with the experimental finding by Shi and Zhao (1998). Their measurements indicate that the material removal rate is proportional to (Fn )2/3 . The angular spindle velocity wp also plays an important role in the removal profile. In Fig. 8a, the experimental settings are Fn = 1.5 N, R = 6 mm, wq = 0.0015 mm/s, Rabr =0.83 m, d = 0.5%, and wp = 209 rad/s or 418 rad/s. The comparison of material removal profiles also confirms that the experimental profiles are consistent with the theoretical ones. The theoretical material removal profile is calculated by varying the wp from 209 rad/s to 523 rad/s. These experimental and theoretical results indicate that the material removal depth increases as the angular spindle velocity wp increases. The effect of the angular feed rate wq on the material removal profile is shown in Fig. 9. And polishing settings are Fn = 1.5 N, R = 6 mm, wp = 209 rad/s, Rabr = 0.83 m, d = 0.5% and wq = 0.0015 rad/s or 0.0003 rad/s. Both the experimental and theoretical results in Fig. 9 indicate that the removal depth decreases as a result of the increasing angular feed velocity wq . A smaller angular feed rate leads to longer dwelling time at the points along the path, resulting in a larger amount of material removed from the workpiece surface. In the real polishing process, wp > >wq . Thus,
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C. Fan et al. / Journal of Materials Processing Technology 214 (2014) 285–294
Fig. 9. Comparison of material removal profiles for various angular feed rates (Fn = 1.5 N, wp = 209 rad/s, R = 6 mm, Rabr = 0.83 m, d = 0.5%). (a) Experimental profiles vs. simulation profiles and (b) theoretical profiles.
Fig. 10. Comparison of material removal profiles for various tool path curvatures (Fn = 1.5 N, wp = 209 rad/s, wq = 0.0015 rad/s, Rabr = 0.83 m, d = 0.5%). (a) Experimental profiles vs. simulation profiles and (b) theoretical profiles.
according to Eq. (36), the polished depth is proportional to wp , and inversely proportional to wq , which is also confirmed by the experimental results in Figs. 8 and 9. Fig. 10 shows the influence of the curvature radius of the polishing path R on the removal profile. In the experiments, the curvature radius R varies from 5 mm to 6 mm, while the other process parameters are fixed as Fn = 1.5 N, wp = 209 rad/s, wq = 0.0015 mm/s, Rabr =0.83 m and d = 0.5%. As shown in Fig. 10a, when the radius R decreases from 6 mm to 5 mm, the material removal depths increase from 5.93 m to 9.21 m by experimental data, and from 5.76 m to 8.16 m in theoretical calculation. The theoretical material removal profiles (see Fig. 10b) also indicate that a smaller curvature radius of the tool path corresponds to deeper removal. As indicated in Eq. (28), the feed velocity is the product of the angular feed velocity and the radius of the polishing path. As the angular feed velocity is kept constant, the feed velocity increases with the increasing of the radius of the polishing path. For the sub-aperture pad polishing along a tool path, a large feed rate leads to the reduction of processing time for a certain point on the tool path, which results in a reduction in the material removal depth. As shown in Fig. 4, the proposed model is developed when the curvature center O outside the contact circle. Thus, the boundary condition of the proposed model for the polishing path curvature is R > Rt . According to Eq. (36), the material removal depth is influenced by the radius of grains in the slurry. Experimental settings include Fn = 1.5 N, R = 6 mm, wq = 209 rad/s, wq = 0.0015 rad/s, d = 0.5% and Rabr = 0.83 m or 1.6 m. Fig. 11a shows the material removal profiles obtained both experimentally and theoretically for Rabr =0.83 m and 1.6 m, and the material removal profiles are consistent with those calculated by Eq. (36). Fig. 11b shows the theoretical material removal profiles as Rabr varies from 0.4 m to 3.2 m,
which indicates the removal depth increases with the increasing Rabr . According to simulation results in Fig. 13, the removal depth is approximately proportional to the mean radius of abrasive grains Rabr . This relation holds as it is the abrasive grains embedded in the pad/workpiece that abrade the surface. With the volume concentration fixed at a certain value, if the radius of abrasive grains is small, there are more grains in per unit volume of the slurry. However, a critical condition needs to be satisfied for the grains participating in the material removal, as indicated in Eq. (19). Thus, although there are more abrasive grains in the slurry, only a small portion of them have the ability to remove the material. On the other hand, although a big radius of abrasive grains corresponds to a small number of abrasive grains in the slurry, a large portion of abrasive grains satisfy the critical condition of material removal. Furthermore, the volume of material removed by a large grain is bigger than that by a small grain. As shown in Figs. 12 and 13, the volume concentration of the slurry can also influence the material removal. Moreover, according to Eq. (36), the depth of the removal profile is proportional to the (volume concentration)2/3 . The experimental results for various volume concentration, d = 0.5% and 0.2%, are shown in Fig. 12a, which are consistent with the theoretical ones. The polishing pad is one of the most significant components in the polishing system. The model developed in this paper reveals some insights into the mechanism of FAP in terms of the working principle of a polishing pad. In this study, two parameters of the polishing pad are discussed, namely, the standard deviation of pad asperity height and the average radius of pad asperities Rp . The effects of and Rp on the removal depth are shown in Fig. 14. The standard deviation of pad asperity height is a description of the pad roughness. A smaller means a smoother pad, which is
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293
Fig. 11. Comparison of material removal profiles for various mean radii of abrasive grains (Fn = 1.5 N, wp = 209 rad/s, wq = 0.0015 rad/s, R = 6 mm, d = 0.5%). (a) Experimental profiles vs. simulation profiles and (b) theoretical profiles. Fig. 12. Comparison of material removal profiles for various volume concentrations (Fn = 1.5 N, wp = 209 rad/s, wq = 0.0015 rad/s, R = 6 mm, Rabr = 0.83 m). (a) Experimental profiles vs. simulation profiles and (b) theoretical profiles.
capable of holding a greater number of abrasive grains. Thus it can be observed from Fig. 14 that the removal depth h decreases as increases, which is similar as described in the work on material removal rate by Jin and Zhang (2012). A similar finding is also presented by Xie and Bhushan (1996), which states that the material removal rate in CMP is proportional to the ()−0.3 . Compared with , the effect of Rp on the removal depth is much smaller, as shown in Fig. 14. As Rp increases from 0.01 mm to 0.1 mm, the removal depth only decreases a little.
The proposed model assumed that the abrasive grains are held by the polishing pad, and the GW model was used to characterize the contact between pad and workpiece, thus it cannot be used to predict the material removal of the traditional hard polishing disk, such as pitch disk. Another limitation is that the model and the planning of the experiments in this paper ignored the chemical
Fig. 13. Effects of the properties of slurry on the removal depth.
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Fig. 14. Effects of the properties of polishing pad on the removal.
effect of the polishing slurry on the material removal, which should be further discussed. 5. Conclusions In this paper, the material removal profile is used to characterize the material removal during the FAP process with the sub-aperture pad. A novel mathematical model of the material removal profile for the FAP process is developed, which successfully explains the effects of the properties of abrasive grains/slurry and topographical parameters of sub-aperture pad on the profile. According to the model, the removal depth is proportional to (normal polishing force)0.65 , the angular spindle veloctiy (volume concentration)2/3 and the radius of abrasive grains, and inversely proportional to the angular feed rate. Moreover, the removal depth decreases with the increasing deviation of pad asperity height and curvature radius of the path. Polishing experiments are conducted in which the normal polishing force, tool path curvature, angular spindle velocity, angular feed rate, abrasive grain size and volume concentration are varied. The experimental results agree with the theoretical calculations, which confirms the validity of the model in predicting the material removal profile and depth for the FAP process. The proposed model can be used to optimize the process parameters and improve the stability and determinacy of the polishing process. By controlling the influential parameters during the FAP process, the polished depth can be controlled to improve the form accuracy of the workpiece surface. Acknowledgements The authors are grateful to the financial support from Chinese National Program on Key Basic Research Project (973 Program) [Grant No. 2011CB706702] and Innovation Program of Jilin University for graduate student [Grant No. 20121078]. References Brinksmeier, E., Riemer, O., Gessenharter, A., 2006. Finishing of structured surfaces by abrasive polishing. Precision Engineering 30, 325–336.
Cheung, C.F., Kong, L.B., Ho, L.T., To, S., 2011. Modeling and simulation of structure surface generation using computer controlled ultra-precision polishing. Precision Engineering 35, 574–590. Cheng, H.B., Feng, Z.J., Cheng, K., Wang, Y.W., 2005. Design of a six-axis high precision machine tool and its application in machining aspherical optical mirrors. International Journal of Machine Tools & Manufacture 45, 1085–1094. Greenwood, J.A., Williamson, J.B.P., 1966. Contact of nominally flat surfaces. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 295 (1442), 300–319. Jin, X.L., Zhang, L.C., 2012. A statistical model for material removal prediction in polishing. Wear 274–275, 203–211. Jones, R.A., 1977. Optimization of computer controlled polishing. Applied Optics 16, 218–244. Kanchana, V., Vaitheeswaran, G., Svane, A., Delin, A., 2006. First-principles study of elastic properties of CeO2 ThO2 and PoO2 . Journal of Physics: Condensed Matter 18, 9615–9624. Kim, D.W., Kim, S.W., 2005. Static tool influence function for fabrication simulation of hexagonal mirror segments for extremely large telescopes. Optics Express 13, 910–917. Klocke, F., Zunke, R., 2009. Removal mechanisms in polishing of silicon based advanced ceramics. CIRP Annals – Manufacturing Technology 58, 491–494. Lee, H.S., Jeong, H.D., Dornfeld, D.A., 2013. Semi-empirical material removal rate distribution model for SiO2 chemical mechanical polishing (CMP) processes. Precision Engineering 37, 483–490. Lin, T.R., 2007. An analytical model of the material removal rate between elastic and elastic-plastic deformation for a polishing process. International Journal of Advanced Manufacturing Technology 32 (7–8), 675–681. Preston, F.W., 1927. The theory and design of plate glass finishing machines. Journal of the SOC of Glass Technology 11, 214–256. Roswell, A., Xi, F.J., Liu, G., 2006. Modelling and analysis of contact stress for automated polishing. International Journal of Machine Tools and Manufacture 46 (3), 424–435. Shi, F.G., Zhao, B., 1998. Modeling of chemical–mechanical polishing with soft pads. Applied Physics 67, 249–252. Su, Y.T., Horng, C.C., Sheen, J.Y., Hsiau, J.S., 1996. A process planning strategy for removing an arbitrary profile by hydrodynamic polishing process. International Journal of Machine Tools & Manufacture 36 (I1), 1227–1245. Su, Y.T., Sheen, J.Y., 1999. A process planning strategy for removing arbitrary and axially symmetric profile by a polishing process. International Journal of Machine Tools & Manufacture 39, 187–207. Xie, Y.S., Bhushan, B., 1996. Effects of particle size, polishing pad and contact pressure in free abrasive polishing. Wear 200, 281–295. Yang, M.Y., Lee, H.C., 2001. Local material removal mechanism considering curvature effect in the polishing process of the small aspherical lens die. Journal of Materials Processing Technology 116, 298–304. Yu, Y., Yu, C.C., Orlowski, M., 1993. A statistical polishing pad model for chemical–mechanical polishing. In: December 5–8, IEEE EDM. Washington, DC, pp. 865–868. Zhao, Y.W., Chang, L., 2002. A micro-contact and wear model for chemical–mechanical polishing of silicon wafers. Wear 252, 220–226.
Contents lists available at ScienceDirect
Journal of Materials Processing Technology journal homepage: www.elsevier.com/locate/jmatprotec
Modeling and analysis of the material removal profile for free abrasive polishing with sub-aperture pad Cheng Fan a,b , Ji Zhao a , Lei Zhang a,∗ , Yoke San Wong b , Geok Soon Hong b , Wansong Zhou a a b
College of Mechanical Science and Engineering, Jilin University, Changchun 130022, China Department of Mechanical Engineering, National University of Singapore, Singapore 117576, Singapore
a r t i c l e
i n f o
Article history: Received 16 February 2013 Received in revised form 3 September 2013 Accepted 9 September 2013 Available online 19 September 2013 Keywords: Free abrasive polishing Material removal profile Sub-aperture pad Material removal index
a b s t r a c t This paper addresses the problem of material removal in free abrasive polishing (FAP) with the subaperture pad both theoretically and experimentally. The effects of some polishing conditions upon the material removal are analyzed, including not only the process parameters, which refer to the normal force, angular spindle velocity and angular feed rate, but also the abrasive grain size, polishing slurry properties, topographical parameters of the sub-aperture pad, as well as tool path curvature. Based on the analysis, a model of material removal profile is proposed to facilitate more accurate polishing. First, by analyzing the contact among polishing pad, abrasive grain and workpiece surface in the micro level, the removal depth per unit length of the polishing path is derived, which is defined as the material removal index. Then, the distribution of this removal index can be obtained via modeling the pressure and relative sliding velocity in the contact region of polishing pad and workpiece. After that, the material removal profile can be calculated by integrating the material removal index along the tool path in the tool-workpiece contact region. To verify the effectiveness of the proposed model, a series of polishing experiments have been conducted. Experimental results well demonstrate that our model can accurately predict the material removal depth during the FAP. © 2013 Elsevier B.V. All rights reserved.
1. Introduction Free-form surfaces with high surface finish, integrity and form accuracy, such as precise aspheric and non-geometric optical lens and molds, have become key features of components increasingly required by optoelectronic and communication industries (Cheng et al., 2005). The polishing process, which is usually the final step of fabrication, is essential to the quality and duration of the part surfaces. Various types of polishing are used for optical fabrication, such as free-abrasive polishing (FAP), magneto-rheological finishing, abrasive slurry jet polishing, etc. FAP is considered as one of the most common and important operations in optical finishing. In a typical FAP process, a rotational sub-aperture polishing pad in the presence of liquid slurry which contains hard abrasive grains is pressed against the workpiece. Between the sub-aperture pad and the workpiece there forms an area of contact and the interactions between polishing pad, abrasive grains and workpiece result in the material removal in the contact area.
∗ Corresponding author at: College of Mechanical Science and Engineering, Nanling Campus, Jilin University, No. 5988, Renmin Street, Changchun 130022, China. Tel.: +86 18946686195. E-mail address: [email protected] (L. Zhang). 0924-0136/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jmatprotec.2013.09.010
The FAP process with the sub-aperture pad is usually used as an error compensation method to improve the form accuracy of the part surfaces if the material removal in the contact area is controllable. Jones (1977) first proposed to use the FAP process to fabricate the aspheric optical surface. By controlling the amount of material to be removed precisely, the mid-spatial frequency errors left after grinding or turning can be removed by sub-aperture polishing. Su et al. (1996) and Su and Sheen (1999) developed a planning strategy for the polishing process to remove an arbitrary profile with the precondition that the material removal depth in the contact area can be represented by a linear function of machining time. Therefore, it is essential to model and analyze the material removal of polishing in a quantitative manner. However, the fundamental mechanisms of material removal in FAP involve micro-fatigue, micro-crack, ploughing and cutting mainly depending on the state of the interactions between abrasives, polishing pad and workpiece, which have not been well understood (Brinksmeier et al., 2006). Thus, the research on the material removal rate (MRR) of FAP is still at an empirical level. The Preston’s equation is widely used to predict the MRR during the polishing process, which states that the MRR is determined by the product of the contact pressure, relative velocity, machining time and Preston coefficient which is related to the workpiece material, properties of polishing slurry and polishing pad, size of abrasive grains, etc. (Preston, 1927). Besides, some researchers tried to explain the mechanism and establish
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their own MRR models from the perspective of descriptive modeling. Yu et al. (1993) modeled the dynamic interaction between pad and workpiece. In this model, the polishing pad is characterized by a statistical asperity model and the effect of polishing pad surface roughness on the MRR is included. The contact behavior of a polishing pad with the workpiece surface was also investigated by Klocke and Zunke (2009) using an FE model. The predicted results show that the material removal in FAP can be described by the penetrating of the abrasive particles into the workpiece, and also the sliding or yielding after the penetration. A predictive model of MRR during the chemical-mechanical polishing was developed by Xie and Bhushan (1996) to predict the effects of grain size, polishing pad and contact pressure. Zhao and Chang (2002) presented a MRR model for chemical-mechanical FAP based on the elastic-plastic microcontact mechanics. In this model, the indentation depth of a grain into the workpiece is determined by the force equilibrium equation of the single grain. A similar MRR model was also developed by Lin (2007), which takes the contact deformation of the abrasive grains into consideration. Jin and Zhang (2012) proposed a statistical MRR model by assuming that there are two types of abrasive grains contributing to the material removal process, i.e., the grains rotating between the pad and workpiece and those embedded in the pad without rigid body motion. The previous research on the MRR focuses on the modeling of mechanical contact action among the pad, abrasive grains and workpiece surface. Due to the lack of basic understanding of the mechanism of the FAP process, these models are still regarded as half-empirical. The MRR distribution is not uniform in the contact region, mainly depending on distributions of the contact pressure and the relative sliding velocity between polishing pad and workpiece surface (Lee et al., 2013). The contact pressure distribution during the polishing was modeled by Roswell et al. (2006) based on the Hertizian contact theory. According to the investigation by Yang and Lee (2001), the contact area and the pressure distribution were affected by the equivalent radius of polishing tool and workpiece surface. In their work, the MRR distribution in the contact region was modeled based on the Preston’s equation, and the material removal depth in the contact area was calculated by the integral of local MRR. The MRR distribution was defined as the tool influence function (TIF) by Kim and Kim (2005). Based on the Preston’s equation, the TIF was modeled by assuming that the pressure distribution in the contact region is Gaussian. The Preston’s equation was also used by Cheung et al. (2011) to model the TIF, and the modelbased simulation was used to predict the generation of structured surfaces. These predictive models of MRR distribution and material removal depth are all based on the Preston’s equation. However, the Preston’s coefficient has to be determined by experiments before each polishing process, because the changes of polishing slurry, abrasive grains and polishing pad may result in different values of the Preston coefficient. Since the MRR models and the MRR distribution models have been studied independently, there has been
no study directly linking the abrasive grain size and pad properties to the material removal depth yet. In this paper, a novel model is proposed to predict the material removal depth of the workpiece surface. The material removal profile, which is defined as the material removal depth orthogonal to the polishing path, is used to describe the material removal in the FAP process. The effects of some polishing conditions upon the material removal are analyzed, including not only the process parameters, which refer to the normal force, angular spindle velocity and angular feed rate, but also the abrasive grain size, polishing slurry properties, topographical parameters of the sub-aperture pad, as well as tool path curvature. The remainder of this paper is organized as follows. In Section 2, the material removal index for FAP, defined as the material removal depth per unit length of the polishing path, is derived. In Section 3, the distributions of contact pressure and relative velocity are discussed and an approach is proposed to model the material removal profile. The validity of the proposed model for the removal profile is verified by experiments in Section 4. Findings are summarized in the concluding section. 2. Material removal index for free abrasive polishing In this paper, it is assumed that the material removal from the workpiece is primarily caused by abrasion wear by the abrasive grains in the slurry. Besides, the abrasive grains are spherical, with an average radius of Rabr and uniformly distributed in the polishing slurry. These assumptions have been substantiated by Zhao and Chang (2002), Xie and Bhushan (1996) and Lin (2007). The material removal index is defined as the material removal depth per unit length of the polishing path, which is different from the definition of MRR. 2.1. Contact between the polishing pad and workpiece surface Since the polishing pad is much rougher than the workpiece surface, the contact between them can be regarded as that between a rough surface and a smooth one, as is shown in Fig. 1. When the polishing pad is pressed on the workpiece surface under a constant load, the distance between the reference plane and the workpiece surface is hs . Yu et al. (1993) proposed a statistical model to characterize the polishing pad with random roughness. In this model, the Gaussian function was used to describe the distribution of the surface height of the polishing pad, which is given by ϕ(h) =
1 exp √ 2
−
h2 2 2
(1)
where is the standard deviation of the distribution of the pad surface. In the actual polishing process, the abrasive grains are rarely on the contact interface. Thus it is reasonable to assume that the
Fig. 1. Interactions between polishing tool, abrasive grains and workpiece.
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Assuming that the contacts between a grain and workpiece and between the grain and polishing pad are plastic, the interference w is given by Xie and Bhushan (1996) as w =
Hp (2Rabr − hs + h) Hw + Hp
(7)
Substituting Eqs. (6) and (7) into Eq. (5) yields A=
2Rabr
1.5
Hp (2Robr − hs + h) Hw + Hp
(8)
2.3. Number of abrasive grains participating in material removal
Fig. 2. Schematic of single abrasive abrasion.
entire contact load is sustained by the contact between pad and workpiece surface. The GW model (Greenwood and Williamson, 1966) is used to describe the contact between pad and workpiece surface. According to this model, the pressure between pad and workpiece is given by Pc =
4 ∗ Np Rp0.5 Epw 3
hmax
(h − hs )
1.5
ϕ(h) dh
(2)
h
where Rp is the average radius of the asperities, Np is the number of asperities per unit area, hmax is the maximum height of the pad surface and E* pw is the contact modulus of the polishing pad and ∗ = (1 − 2 )/E + (1 − 2 )/E . E and the workpiece given by 1/Epw p w p p w Ew are the Young’s modulus of the polishing pad and the workpiece respectively, and p and w are their Poisson’s ratio respectively. According to Eq. (2), the pressure Pc is a function of the separation distance hs . Thus, if the polishing pressure Pc is known, the separation distance hs can be calculated from Eq. (2).
As shown in Fig. 2, the abrasive grain is embedded between the pad and the workpiece. Rabr is the radius of the abrasive grain, and w and p correspond to the interference between grain and workpiece and that between grain and polishing pad, respectively. The volume of the material removed by a single abrasive grain is approximately equal to the product of its cross-sectional area immersed in the workpiece and the sliding distance, which gives (3)
where A is the cross-sectional area and L is the sliding distance. Assuming that the hard abrasive grain maintains its spherical shape, A can be expressed by 2 A = Rabr arcsin
a Rabr
− a(Rabr − w ),
w Rabr
,
(5)
where a can be expressed as
a=
2
2 − (R Rabr abr − w ) =
2Rabr w − 2w ≈
2Rabrw
(9)
˝ − l2 .
(6)
(10)
Rearranging Eq. (9) and substituting it into Eq. (10) gives
˝=
2/3
3d
(11)
3 4Rabr
When the applied load on the abrasive grain is small, the contact between abrasive grain and workpiece is elastic. According to the Hertizian contact theory, the indentation depth can be expressed as w =
3p 2
(4)
where a is the radius of the circular indentation on the workpiece surface. Compared with the grain radius Rabr , w is very small, and so is a/Rabr , thus Eq. (4) can be further linearized about a/Rabr as A ≈ aw 1 + O
4 3 3 R l = d 3 abr
The relationship between area density ˝ and the line density l is given by
2.2. The material removal of a single abrasive grain
V = AL
The number of the free-abrasive grains in the slurry directly participating in the abrasive wear process is one of the most important variables influencing the material removal. Previous studies assumed that the abrasive grains embedded in the workpiece surface and the asperities of the polishing pad participate in the material removal (Zhao and Chang, 2002; Jin and Zhang, 2012). Thus, the number of active grains is the product of the area density of the abrasive grains in the slurry and the real area of contact between polishing pad and workpiece. However, in the actual polishing process, the abrasive grains are rarely on the contact interface of the pad asperities and workpiece. Moreover, if the area density of grains is fixed, the MMR is independent of the abrasive grain size in these models, which is unreasonable. In order to overcome the deficiency of these models, the critical condition for the grains participating in material removal is described and derived. Furthermore, the effects of the pad properties on the number of active grains are considered, which is ignored by Xie and Bhushan (1996). Let the slurry concentration per unit volume be d , and the line density of the grain in the slurry be l. Then
4E ∗
Rabr ,
(12)
∗ where p is the average pressure acting on the grain and Eaw is the contact modulus of abrasive grain and workpiece given by ∗ = (1 − 2 )/E + (1 − 2 )/E . E and are Young’s modulus 1/Eaw a w a a a w and Poisson’s ratio of the grains, respectively. As the applied pressure increases, the elastic contact will change into plastic contact, and the critical value of the pressure p is determined by the hardness of the workpiece as p = Hw . Thus, the maximum interference w0 of the elastic contact is given by
w0 =
3H 2 w
4E ∗
Rabr .
(13)
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According to Eq. (8), the volume of material removed by a single grain across H is V0 =
2Rabr
Hp Hw + Hp
1.5
(2Rabr − hs + h)
1.5
dy.
(19)
Within the time interval dt, the overall volume of material removal at H can be expressed as V = dx dy dz = N1 V0 ,
(20)
where dz is the depth of material removal at H. Substituting Eqs. (18) and (19) into Eq. (20) yields Fig. 3. Schematic of polishing along a curved path.
wl = The material is removed from the workpiece surface in a plastic manner. By substituting Eq. (13) into Eq. (7), the critical separation distance he leading to plastic contact can be expressed by
Hw + Hp he = hs − 2 − Hp
3H 2
w 4E ∗
(14)
In the polishing process, the pad can only hold a certain portion of abrasive grains in the slurry. The material of the workpiece is assumed to be primarily removed by the abrasive grains attached to the polishing pad with the height of the pad surface higher than he . As the surface height distribution density of the polishing pad is ϕ(h) (see Eq. (1)), the number of abrasive grains per unit area at a height between h and h + dh is ˝ × ϕ(h). Thus, the number of grains participating in the material removal can be expressed as
N0 =
2Rabr vs ˝
va
1.5
Hp Hw + Hp
hmax
(2Rabr − hs + h) h2
1.5
ϕ(h)dh, (21)
where wl ≡
Rabr .
dz dy
(22)
is the material removal index for free-abrasive polishing defined as the depth of material removal per unit length along the polishing path. Eq. (21) can be further modified as wl =
2Rabr vs ˝
va
Hp Hw + Hp
1.5
hs
×
(2Rabr − hs + h)
1.5
hmax
he
(2Rabr )1.5 ϕ(h) dh
ϕ(h)dh + hs
(23)
hmax
˝ϕ(h)dh
(15)
h
where ϕ(h), ˝ and he are expressed by Eqs. (1), (11) and (14), and is the constant between 0 and 1 related the polishing pad’s ability to hold the abrasive grains. 2.4. Material removal index A contact patch in a certain shape is formed between the polishing pad and the workpiece when a normal load is applied on the polishing pad and the material removal takes place within the contact region. As shown in Fig. 3, point o is the center of the contact patch on the path and H is an arbitrary point on the x axis with the infinitesimal area dxdy. The direction of dy is along the tool path. For an infinitesimal time dt during which the tool is contacting with H, dt =
dy
va
,
(16)
where va is the feed rate of the tool moving along the tool path. During the time interval dt, the real contact area between polishing tool and H can be expressed as S = vs dx dt =
vs dx dy va
(17)
where vs is the relative sliding velocity between the tool and H. The number of grains per unit area participating in the material removal can be calculated by Eq. (15). Then, the total number of abrasive grains participating in the material removal of polishing during the time interval dt is
vs dx dy N1 = SN0 = va
hmax
˝ϕ(h) dh he
(18)
3. Modeling the material removal profile 3.1. Contact pressure It is assumed that process parameters, such as the normal polishing force, the angular spindle velocity, the feed rate and the curvature of the tool path, vary slightly along the tool path, and the curvature radius of the surface at the point o(P) is very large compared with the size of the sub-aperture pad. Thus, the contact region is approximately a circle with radius Rt and the average contact pressure can be expressed by Pc =
Fn Rt2
(24)
where Fn is the normal force on the pad. 3.2. Relative sliding velocity As shown in Fig. 4, the frame xoy is fixed at the point P and the plane xoy is the tangent plane of the surface at point P. The point O is the geodesic curvature center of the path, and oO = R which gives R=
1 , |kg |
(25)
where kg is the geodesic curvature of the path at point P. The frame XOY is set up at point O and the Y direction is parallel to the y direction. For the convenience of analysis, an arbitrary point in the frame xoy is described in the frame XOY by the polar coordinate form, and the corresponding coordinate transformation is given by
x = cos + R y = sin
(26)
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289
Fig. 4. Distribution of relative velocity when polishing along a curved path.
For the arbitrary point M in the contact circle, the velocity due to the tool rotation vm can be expressed as
vm = wp
x2 + y2 = wp
2 + 2 R cos + R2
And solution of cos can be obtained from Eq. (32) as cos =
(27)
(33)
And there is one solution of within [/2, ] which gives
where wp is the angular spindle velocity. The velocity due to the feed rate along the curved tool path is given by
va = wq
Rt2 − 2 − R2 . 2R
1 ( ) = − arccos
(28)
Rt2 − 2 − R2 2R
(34)
where wq is angular feed rate around the point O. The relative sliding velocity vs in the contact region is the composition of vm and va . Thus, the relative sliding velocity at M(x, y) can be expressed by vs =
v2a + v2m − 2va vm cos( − ) =
wq2 2 + wp2 ( 2 + 2 R cos + R2 ) − 2wq wp
2 + 2 R cos + R2 cos( − )
(29)
3.3. Material removal profile orthogonal to the tool path As illustrated in Fig. 4, for R > Rt , the bounding values of are
min = R − Rt and max = R + Rt . By substituting Eq. (34) into Eq. (30), the material removal profile can be calculated as
As shown in Fig. 4, when the polishing pad moves across point H along the curved tool path, the contact length between the pad and the workpiece at point H can be represented by the arc length L1 L2 centered at O with radius . The removal depth at point H on the surface can be calculated by integrating wl along the arc L1 L2 , which gives
h( ) =
2 3d
2
1.5 wp Rabr
1 ( )
4
2/3
Hp Hw + Hp
1.5
1 ( )
h( ) = 2
w1 d
(35)
Substituting Eq. (11), (23) and (29) into Eq. (35) yields
hs
(2Rabr − hs + h)
1.5
hmax
(2Rabr )1.5 ϕ(h) dh
ϕ(h) dh +
he
hs
wq2 2 + wp2 ( 2 + 2 R cos + R2 ) − 2wq wq
×
R − Rt ≤ ≤ R + Rt .
0
2 + 2 R cos + R2 cos( − ) d
(36)
0
1
h( ) =
w1 d
(30)
2
where 1 and 2 correspond to the starting angle and the ending angle of the arc L1 L2 . The contact circle can be expressed in the frame xoy by x2 + y2 = Rt2
4. Experimental verification 4.1. Experimental conditions
(31)
Substituting Eq. (26) into Eq. (31) yields Rt2 − ( cos + R2 ) − 2 sin2 = 0.
In Eq. (36), hs can be solved by Eq. (2) and Eq. (24) and he can be calculated by Eq. (14). The detailed calculation procedure for the material removal profile is shown in Fig. 5.
(32)
To verify the effectiveness of the theoretical model for the removal profile, a series of polishing experiments have been conducted on a 5-axis polishing machine which is composed of three
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C. Fan et al. / Journal of Materials Processing Technology 214 (2014) 285–294 Table 1 Polishing conditions. Item
Condition
Workpiece
Material: Glass ceramics, Hw = 8.3 GPa, Ew = 85.16 GPa, vw = 0.272 Material: CeO2 , d = 0.5% and 0.2%, Rabr = 0.83 m and 1.6 m, Ea = 249 GPa, va = 0.297, = 0.2 Polishing pad: Hp = 90 Mpa, Ep = 250 MPa, vp = 0.5, Rp = 0.03 mm, Np = 1.2 × 103 mm−2 , = 0.025 mm, hmax = 0.58 mm, Rt = 4 mm
Slurry Polishing tool
Fig. 5. Calculation steps for material removal profile.
translational axes (X-axis, Y-axis and Z-axis) and two rotational axes (B-axis and C-axis). The structure of the machine tool is shown in Fig. 6a. As shown in Fig. 6b, the piezoelectric force sensor is mounted on the end effector of the polishing spindle. It is used as a feedback sensor to control the normal polishing force. The workpiece is fixed on the rotational table. The set-up enables the polishing tool to move along a curved path with the normal polishing force varying from 1 N to 10 N and a specified feed rate. The
polishing spindle is mounted on the B-axis, with the angular speed adjustable from 209 rad/s to 2090 rad/s. The material removal profile is measured by Form Talysurf PGI 810 (Taylor-Hobson Ltd., UK) with a 0.3 mm tip radius stylus along a line perpendicular to the polishing path. The measurements are conducted under the waviness mode to suppress the high-frequency noise of the profiles, and the measured data are illustrated from Figs. 7–12. The workpiece material used in the experiments is glass ceramics. In order to get the physical properties of the workpiece, the nano-indentation experiments are conducted, where Hw = 8.3 GPa, Ew = 85.16 GPa and w = 0.272. An elastic polishing tool which is made of polyurethane is used as the sub-aperture polisher. The radius of the polyurethane sub-aperture pad Rt is fixed as 4 mm and the mechanical properties are Hp = 90 MPa, Ep = 250 MPa and
p = 0.5 (Zhao and Chang, 2002). The topographical parameters of the polyurethane sub-aperture pad are adopted according to the experimental measurements of Yu et al. (1993) for typical polyurethane pad surfaces, in which the average radius of asperity Rp = 0.03 mm, the asperity density Np = 1.2 × 103 mm−2 and the standard deviation of the distribution = 0.025 mm. The value of which describes the ability of the pad to hold abrasives is set according to Xie and Bhushan (1996), in whose work = 0.2 for the pad with medium hardness. The CeO2 grains are used as free abrasives in the slurry. The mean diameters of the grains selected are 0.83 m and 1.6 m, and the grain volume concentrations of the slurry are fixed as 0.5% and 0.2%. For a CeO2 grain in the slurry, Ea = 249 GPa and a = 0.297, as indicated by Kanchana et al. (2006). The detailed polishing conditions for the polishing experiments are
Fig. 6. Structure of experimental machine and polishing spindle. (a) Machine tool with five axes and (b) polishing spindle with force control.
C. Fan et al. / Journal of Materials Processing Technology 214 (2014) 285–294
Fig. 7. Comparison of material removal profiles for various normal polishing forces (wp = 209 rad/s, wq = 0.0015 rad/s, R = 6 mm, Rabr = 0.83 m, d = 0.5%). (a) Experimental profiles vs. simulation profiles and (b) theoretical profiles.
summarized in Table 1. The planning of process parameters in the experiments is shown in Table 2. 4.2. Results and discussion The mechanism of the FAP process is complicated and the influencing factors of the material removal depth involve the process parameters, properties of abrasive grains/slurry and polishing sub-aperture pad, and tool path curvature. Figs. 7–12 show the profiles of the material removal in different polishing conditions. The material removal profiles show good agreement between the experimentally measured results and the simulation results calculated by Eq. (36). The removal profiles are not symmetrical, as can be seen in both experiments and simulation results, which show deviation toward the center of the tool path curvature. This deviation is attributed to the longer dwelling time in the contact region near the curvature center of the tool path when polishing along a curved path. The experimental material removal profiles of the workpiece surface for various normal polishing forces Fn are shown Table 2 Process parameters in the experiments. No.
Fn (N)
wp (rad/s)
wp (rad/s)
R (mm)
Rabr (m)
d (%)
1 2 3 4 5 6 7
1.5 6 1.5 1.5 1.5 1.5 1.5
209 209 418 209 209 209 209
0.0015 0.0015 0.0015 0.0003 0.0015 0.0015 0.0015
6 6 6 6 5 6 6
0.83 0.83 0.83 0.83 0.83 1.60 0.83
0.5 0.5 0.5 0.5 0.5 0.5 0.2
291
Fig. 8. Comparison of material removal profiles for various angular spindle velocities (Fn = 1.5 N, wq = 0.0015 rad/s, R = 6 mm, Rabr = 0.83 m, d = 0.5%). (a) Experimental profiles vs. simulation profiles and (b) theoretical profiles.
in Fig. 7. Other experimental settings include wp = 209 rad/s, wq = 0.0015 rad/s, R = 6 mm, Rabr = 0.83 m and d = 0.5%. Fig. 7a shows the removal profiles obtained both experimentally and theoretically for Fn = 1.5 and 6 N, and Fig. 7b shows simulation results of the profiles for Fn varying from 1.5 N to 6 N. Both the simulation and experimental results show that the material polished increases as Fn grows. The effect of the normal polishing force Fn is reflected in the separation distance hs . A large Fn leads to a small hs , which causes more material to be removed by more abrasive grains participating in the process. According to the experimental results and theoretical predictions in this paper, the removal depth is approximately proportional to (Fn )0.65 . This result is consistent with the experimental finding by Shi and Zhao (1998). Their measurements indicate that the material removal rate is proportional to (Fn )2/3 . The angular spindle velocity wp also plays an important role in the removal profile. In Fig. 8a, the experimental settings are Fn = 1.5 N, R = 6 mm, wq = 0.0015 mm/s, Rabr =0.83 m, d = 0.5%, and wp = 209 rad/s or 418 rad/s. The comparison of material removal profiles also confirms that the experimental profiles are consistent with the theoretical ones. The theoretical material removal profile is calculated by varying the wp from 209 rad/s to 523 rad/s. These experimental and theoretical results indicate that the material removal depth increases as the angular spindle velocity wp increases. The effect of the angular feed rate wq on the material removal profile is shown in Fig. 9. And polishing settings are Fn = 1.5 N, R = 6 mm, wp = 209 rad/s, Rabr = 0.83 m, d = 0.5% and wq = 0.0015 rad/s or 0.0003 rad/s. Both the experimental and theoretical results in Fig. 9 indicate that the removal depth decreases as a result of the increasing angular feed velocity wq . A smaller angular feed rate leads to longer dwelling time at the points along the path, resulting in a larger amount of material removed from the workpiece surface. In the real polishing process, wp > >wq . Thus,
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Fig. 9. Comparison of material removal profiles for various angular feed rates (Fn = 1.5 N, wp = 209 rad/s, R = 6 mm, Rabr = 0.83 m, d = 0.5%). (a) Experimental profiles vs. simulation profiles and (b) theoretical profiles.
Fig. 10. Comparison of material removal profiles for various tool path curvatures (Fn = 1.5 N, wp = 209 rad/s, wq = 0.0015 rad/s, Rabr = 0.83 m, d = 0.5%). (a) Experimental profiles vs. simulation profiles and (b) theoretical profiles.
according to Eq. (36), the polished depth is proportional to wp , and inversely proportional to wq , which is also confirmed by the experimental results in Figs. 8 and 9. Fig. 10 shows the influence of the curvature radius of the polishing path R on the removal profile. In the experiments, the curvature radius R varies from 5 mm to 6 mm, while the other process parameters are fixed as Fn = 1.5 N, wp = 209 rad/s, wq = 0.0015 mm/s, Rabr =0.83 m and d = 0.5%. As shown in Fig. 10a, when the radius R decreases from 6 mm to 5 mm, the material removal depths increase from 5.93 m to 9.21 m by experimental data, and from 5.76 m to 8.16 m in theoretical calculation. The theoretical material removal profiles (see Fig. 10b) also indicate that a smaller curvature radius of the tool path corresponds to deeper removal. As indicated in Eq. (28), the feed velocity is the product of the angular feed velocity and the radius of the polishing path. As the angular feed velocity is kept constant, the feed velocity increases with the increasing of the radius of the polishing path. For the sub-aperture pad polishing along a tool path, a large feed rate leads to the reduction of processing time for a certain point on the tool path, which results in a reduction in the material removal depth. As shown in Fig. 4, the proposed model is developed when the curvature center O outside the contact circle. Thus, the boundary condition of the proposed model for the polishing path curvature is R > Rt . According to Eq. (36), the material removal depth is influenced by the radius of grains in the slurry. Experimental settings include Fn = 1.5 N, R = 6 mm, wq = 209 rad/s, wq = 0.0015 rad/s, d = 0.5% and Rabr = 0.83 m or 1.6 m. Fig. 11a shows the material removal profiles obtained both experimentally and theoretically for Rabr =0.83 m and 1.6 m, and the material removal profiles are consistent with those calculated by Eq. (36). Fig. 11b shows the theoretical material removal profiles as Rabr varies from 0.4 m to 3.2 m,
which indicates the removal depth increases with the increasing Rabr . According to simulation results in Fig. 13, the removal depth is approximately proportional to the mean radius of abrasive grains Rabr . This relation holds as it is the abrasive grains embedded in the pad/workpiece that abrade the surface. With the volume concentration fixed at a certain value, if the radius of abrasive grains is small, there are more grains in per unit volume of the slurry. However, a critical condition needs to be satisfied for the grains participating in the material removal, as indicated in Eq. (19). Thus, although there are more abrasive grains in the slurry, only a small portion of them have the ability to remove the material. On the other hand, although a big radius of abrasive grains corresponds to a small number of abrasive grains in the slurry, a large portion of abrasive grains satisfy the critical condition of material removal. Furthermore, the volume of material removed by a large grain is bigger than that by a small grain. As shown in Figs. 12 and 13, the volume concentration of the slurry can also influence the material removal. Moreover, according to Eq. (36), the depth of the removal profile is proportional to the (volume concentration)2/3 . The experimental results for various volume concentration, d = 0.5% and 0.2%, are shown in Fig. 12a, which are consistent with the theoretical ones. The polishing pad is one of the most significant components in the polishing system. The model developed in this paper reveals some insights into the mechanism of FAP in terms of the working principle of a polishing pad. In this study, two parameters of the polishing pad are discussed, namely, the standard deviation of pad asperity height and the average radius of pad asperities Rp . The effects of and Rp on the removal depth are shown in Fig. 14. The standard deviation of pad asperity height is a description of the pad roughness. A smaller means a smoother pad, which is
C. Fan et al. / Journal of Materials Processing Technology 214 (2014) 285–294
293
Fig. 11. Comparison of material removal profiles for various mean radii of abrasive grains (Fn = 1.5 N, wp = 209 rad/s, wq = 0.0015 rad/s, R = 6 mm, d = 0.5%). (a) Experimental profiles vs. simulation profiles and (b) theoretical profiles. Fig. 12. Comparison of material removal profiles for various volume concentrations (Fn = 1.5 N, wp = 209 rad/s, wq = 0.0015 rad/s, R = 6 mm, Rabr = 0.83 m). (a) Experimental profiles vs. simulation profiles and (b) theoretical profiles.
capable of holding a greater number of abrasive grains. Thus it can be observed from Fig. 14 that the removal depth h decreases as increases, which is similar as described in the work on material removal rate by Jin and Zhang (2012). A similar finding is also presented by Xie and Bhushan (1996), which states that the material removal rate in CMP is proportional to the ()−0.3 . Compared with , the effect of Rp on the removal depth is much smaller, as shown in Fig. 14. As Rp increases from 0.01 mm to 0.1 mm, the removal depth only decreases a little.
The proposed model assumed that the abrasive grains are held by the polishing pad, and the GW model was used to characterize the contact between pad and workpiece, thus it cannot be used to predict the material removal of the traditional hard polishing disk, such as pitch disk. Another limitation is that the model and the planning of the experiments in this paper ignored the chemical
Fig. 13. Effects of the properties of slurry on the removal depth.
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Fig. 14. Effects of the properties of polishing pad on the removal.
effect of the polishing slurry on the material removal, which should be further discussed. 5. Conclusions In this paper, the material removal profile is used to characterize the material removal during the FAP process with the sub-aperture pad. A novel mathematical model of the material removal profile for the FAP process is developed, which successfully explains the effects of the properties of abrasive grains/slurry and topographical parameters of sub-aperture pad on the profile. According to the model, the removal depth is proportional to (normal polishing force)0.65 , the angular spindle veloctiy (volume concentration)2/3 and the radius of abrasive grains, and inversely proportional to the angular feed rate. Moreover, the removal depth decreases with the increasing deviation of pad asperity height and curvature radius of the path. Polishing experiments are conducted in which the normal polishing force, tool path curvature, angular spindle velocity, angular feed rate, abrasive grain size and volume concentration are varied. The experimental results agree with the theoretical calculations, which confirms the validity of the model in predicting the material removal profile and depth for the FAP process. The proposed model can be used to optimize the process parameters and improve the stability and determinacy of the polishing process. By controlling the influential parameters during the FAP process, the polished depth can be controlled to improve the form accuracy of the workpiece surface. Acknowledgements The authors are grateful to the financial support from Chinese National Program on Key Basic Research Project (973 Program) [Grant No. 2011CB706702] and Innovation Program of Jilin University for graduate student [Grant No. 20121078]. References Brinksmeier, E., Riemer, O., Gessenharter, A., 2006. Finishing of structured surfaces by abrasive polishing. Precision Engineering 30, 325–336.
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