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Study of a wheel-like electrorheological finishing tool and its applications to small parts JINGSHI SU,1,2 HAOBO CHENG,1,2,* YUNPENG FENG,1,2
AND
HON-YUEN TAM3
1
Joint Research Center for Optomechatronics Engineering, School of Optoelectronics, Beijing Institute of Technology, Beijing 100081, China Shenzhen Research Institute, Beijing Institute of Technology, Shenzhen 518057, China 3 Department of Mechanical and Biomedical Engineering, City University of Hong Kong, Hong Kong 999077, China *Corresponding author: [email protected] 2
Received 2 September 2015; revised 4 November 2015; accepted 10 December 2015; posted 15 December 2015 (Doc. ID 249377); published 21 January 2016
A wheel-like electrorheological finishing (ERF) tool for small parts polishing is proposed and thoroughly studied. First, the electrorheological polishing fluid is tested, and its properties suggest usability for electrorheological fluid-assisted finishing. Then, the mathematical removal model of the ERF tool is built employing the conformal mapping method and high-order multipolar moment theory. Finally, a micropattern of trough is fabricated on a slide glass (7 mm wide and 1 mm thick). The trough is 70 nm deep, and its flat bottom is 1.5 m wide (peak to valley of 3.16 nm and root mean square of 1.27 nm); the surface roughness finally achieves 0.86 nm. The results demonstrate the stable machining capability of the ERF tool for miniature parts. © 2016 Optical Society of America OCIS codes: (220.4610) Optical fabrication; (220.5450) Polishing. http://dx.doi.org/10.1364/AO.55.000638
1. INTRODUCTION Ultrafine, minisized optical components can be used across a broad range of applications in modern industry. For example, when the substrate for femtosecond laser processing is small (10 mm × 10 mm) and fragile [1], a 1 mm thick edge surface can be employed [2]. Very small parts are also used in microsensors [3] and so on. For laser devices, the subsurface damage of optical components becomes a key factor that determines the lifetime of optical parts [4], like components at the National Ignition Facility [5]. Accepted ways to suppress subsurface damage is to implement low-stress processing methods. One outstanding method for low-stress polishing is bowl feed polishing (BFP) [6]. During this polishing process, the workpiece is completely immersed in the polishing slurry for full-aperture fabrication of regular-shaped components. BFP can achieve an ultrasmooth surface with 0.2 nm roughness. Meanwhile, BFP possesses a low material removal ratio which is about several nanometers per minute. Therefore, it is often the last process to improve the overall surface fineness of a workpiece. Currently, to meet the diversified demands of optical components, more methods are being studied. Field-assisted finishing methods are also typical low-stress polishing methods, such as the magnetorheological finishing (MRF) [7] method and the electrorheological finishing (ERF) [8] method. MRF is a mature precision polishing technique capable of rapidly converging to the required surface figure avoiding subsurface damages. MRF utilizes the external 1559-128X/16/040638-08$15/0$15.00 © 2016 Optical Society of America
magnetic field to generate magnetic dipoles (carbonyl iron) which attract each other. This phenomenon leads to significant increases in the apparent viscosity and the yield stress of the magnetorheological (MR) fluid. The surface material can be removed under shear motion with stiffened MR fluid. The yield stress of the MR fluid exceeds 20 kPa under high magnetic field strength. For BK7 glass, the removal rate of 15 μm∕min was achieved under a magnetic field of 2.5 kG using MR polishing equipment made by the University of Rochester [9]. MRF is suitable for fabricating medium-sized components. Huang et al. [10] polished a 410 mm × 410 mm plane mirror and its figure error improved from 253.1 to 63.3 nm by peak-to-valley (PV) [root mean square (RMS) 43.0 to 8.23 nm]. The use of an ultrafine abrasive will significantly improve surface roughness (around 1 nm RMS with a nanoabrasive [11]). Another field-assisted finishing method is the ERF method. ER fluid is among the smartest or most intelligent materials; under an electric field its components produce the electroviscous effect (the so-called Winslow effect) to alter fluid viscosity and yield stress. It is composed of two phases: the disperse phase and the continuous phase. The disperse phase consists of small polarizable particles like alumina, silica, and starch. Generally, silicone oil is widely used as a continuous phase for its stability, low dielectric permittivity, low conductivity, and compatibility with various oil-soluble materials. When placed in an electric field, the apparent viscosity of the ER fluid increases by orders
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(Pa)
639
Shear stress of ER fluid
of magnitude with the electric field (AC or DC) and immediately returns to its initial status when the electric field is removed. The yield stress of the ER fluid is usually lower than −10 kPa. The problem is that it is impractical to increase the yield stress by improving the electric field strength excessively because high voltage leads to dielectric breakdown, so the electroviscous effect diminishes. This condition restricts the application of ERF in lapping and rough polishing. However, the low stress of the ER fluid makes it possible for nanoscale material removal. The mixture of soft polymers and fine abrasives is likely to further improve surface quality without leaving residue and subsurface damage which may lead to great damage in the laser components. In addition, designing and manufacturing electrodes to generate AC or DC electric fields is simpler and more flexible in adapting to various sizes and shapes of the workpiece than that of a permanent magnet and magnetic coil. Meanwhile, the response time of the electric field is a few milliseconds, which is better than the magnetic coil. From the perspectives of applicability and maneuverability, with the applications of microelectromechanical systems and laser devices getting more and more diversified, the ERF method has huge potential. Electrorheological (ER) polishing was first proposed by Kuriyagawa and Syoji in 1999, while a needle-like polishing tool for metallic dies was developed by Kuriyagawa et al. [12]. A rotating shaft was used as the anode, and the cathode was the workpiece itself. Kim developed a padless ultraprecision polishing tool [13] and described the behavior of particles in the fluid based on the dielectrophoresis theory [14]. Auxiliary electrodes were needed to produce an electric field with the metallic tool tip, so that the device would be capable of fabricating insulating components. Zhang et al. developed a theoretical model for the size of the polishing spot [15]. In their cases, to obtain a higher removal rate, the tool tips have to be extremely close to the workpiece surface (within 10 μm), so the top removal rate could achieve 0.4 μm∕min using a micron-sized diamond abrasive on borosilicate glass. Surface roughness on the nanometer scale could be achieved by using a micron-sized abrasive [13]. We did some research in the ER fluid study, and a needle-like integrated polishing tool which is more suitable for computer-assisted polishing was proposed [16]. Actually, a wheel-like polishing tool can provide larger linear velocity which is thought to be an efficacious way to improve finishing efficiency. In this study, we proposed a novel wheel-like integrated electrodes polishing tool. This device can provide a stable material removal footprint. The normalized removal feature of the tool head is calculated based on conformal mapping and linear electric multipolar moment theory. Finally, a trough with a 1.5 mm wide bottom is made by the ERF tool on BK7 slide glass.
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Shear rate
Fig. 1. fluid.
(S-1)
Shear stress property curve of the starch–silicone oil ER
because of its high removal capability. A HAAKE RV20 rheometer was used to test rheological properties of the ER fluid and the ER polishing fluid. This rheometer has a 0.5 mm gap between its electrodes. Figure 1 is the shear stress curve of the starch–silicone oil ER fluid (5 g of starch and 5 g of silicone oil) under different voltages applied on the electrodes in the rheometer. The fluid is a Newtonian-like fluid under zero electric field; its shear stress increases monotonously and linearly with shear rate, less than 200 Pa. However, its property changes remarkably after high voltage is applied. As the shear rate increases, the fluid shear stress reaches the yield point rapidly in a very short span of shear rate, and then it declines gently. The curve in Fig. 2 belongs to the ER polishing fluid which was a mixture of ER fluid and a small amount of ceria (0.5 g). Clearly, the ER polishing fluid possesses electrorheological characteristics as well. Under certain electric fields, the curve of the shear stress of the ER polishing fluid appears to level off after the yield point. As shear the rate grows, the shear stress increases slightly. Between the ER fluid and the ER polishing fluid there are a few differences. First, the antibreakdown property of the ER fluid is improved by adding ceria. Dielectric breakdown happened at 2.1 kV for the starch–silicone oil fluid; meanwhile, it is over 2.55 kV for the ER polishing fluid. Second, at identical voltage levels, the yield stress of the ER polishing fluid is smaller
2. ELECTRORHEOLOGICAL FLUID Our previous research has proved that the starch–siliconeoil-type ER fluid exhibits strong and stable electrorheological effects. However, in order to carry out optical fabrication by virtue of the ER effect, some abrasives like ceria must be added into the starch–silicone oil system, becoming the ER polishing fluid. The abrasive plays a significant role in glass polishing
Fig. 2. Shear stress property curve of the starch–silicone oil ER polishing fluid.
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Carbon brush Z axis
Slip ring
Turntable
Fig. 3. Body-centered cubic structure of the combination of starch and ceria.
than that of the ER fluid but is more stable. For instance, for the case of 1600 V, the ER fluid yields at 2000 Pa and the shear stress reduces to 1600 Pa at a shear rate of 300. For the ER polishing fluid, the shear stress is stable at around 1500 Pa during the increase of the shear rate. So in short, the shear stress of the ER polishing is smaller but is better on stability. We believe that the ER polishing fluid has a stable electrorheological property and shall be effective in optical fabrication. At present, the generally accepted interpretation of the ER effect is particle-chain theory. Dielectric particles in the electric field are polarized and consequently become strongly anisotropic. Two particles attract each other when aligned along the electric line. This phenomenon leads to the consequent transition from a condition of uniform distribution to that of a chain-like distribution. For the ER polishing fluid which contains ceria, the situation is more complicated. We can consider the combination of starch and ceria as a body-centered cubic (BCC) structure (Fig. 3) because of the disparity between starch and ceria in terms of particle size. As a high molecular polymer, starch particles have sizes of dozens of microns and ceria particles are just several microns. Therefore, the BCC structure appears to be relatively stable, with large starch particles attached to each other, while ceria particles fill in the gaps between the starch particles. We established the ER effect observing system to observe the ER effect visually. The transformation of the microstructure is shown in Fig. 4. Under an electric field, large starch particles attract head-to-tail in the direction of the electric field, forming fibrous structures which bridge the anode and cathode and with small ceria particles between the starch particles. 3. ERF TOOL DESIGN AND THE ERF MATHEMATICAL MODEL A. ERF Tool Design
An ERF platform including a wheel-like polishing tool and its carrier machine (see Fig. 5) is introduced. The two
Anode
X axis
Teflon
Y axis
Polishing shaft
Voltage source
Cathode
Fig. 5. Physical map of the experimental platform and ERF tool.
copper-made polishing wheels, which are 30 mm in diameter, are fixed in parallel onto the polishing shaft and separated from each other by Teflon of 2 mm thickness. The polishing wheels are used as the anode and cathode simultaneously. The polishing shaft is driven by a motor via bevel gear transmission. High voltage is supplied to the anode pass through the slip ring which is mounted on the shaft behind the polishing wheel, and the cathode is earthed. The sketch map and physical map of the tool head are shown in Fig. 6. For precision finishing, ERF employs a computer program to determine a schedule for varying the position of the rotating polishing wheel. Therefore, deterministic material removal from the workpiece surface is feasible. Components of the process are the ERF software and a four-axis PC-based computer numerical control platform employed to execute motion and programmable logic control. The machine, schematically shown in Fig. 5, finishes optical parts up to 100 mm in diameter. The active X, Y, Z, and turntable axes move simultaneously in synchronization and are responsible for taking the tool head toward the target position. The machine is located at the vibration-isolating foundation to suppress mechanical vibration disturbances. B. Calculation of the Electric Field Distribution in the Working Zone
From the microscopic perspective, the mutual action of ER particles is the key explanation of the high shear stress of the ER fluid. One significant factor that determines the particle interaction is the distribution of the electric field. As we all know, the electric field line is always orthogonal to the equipotential line, so the angle invariance and scale invariant feature of conformal mapping make it very suitable for simplifying the
Shaft
Cathode
Anode Electric Field line
Work piece
1 mm (a)
Fig. 4. Transformation of the particle status in the ER polishing fluid.
(b)
Fig. 6. (a) Sketch map and (b) physical map of the polishing wheel of the ERF tool.
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Ğ Intersect at infinity
a
Ğ
y a π w1
P
Symmetric plane of the electric field
Work piece
The tip of polishing wheel
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After conformal mapping, the electrodes have been converted onto axis. Then, suppose the following complex function:
v Z Plane
W Plane
Y lnW : x
u w1
Fig. 7. Schwarz–Christoffel cross-sectional map of the wheel-like electrode.
electric field generated by arbitrary electrodes. In this paper, we employ the conformal mapping method to solve the electric field. For a nonuniform electric field generated by polygonshaped (including open polygon) electrodes, we can use the Schwarz–Christoffel formula to transform the original shape of the electrodes onto the axis. The Schwarz–Christoffel [17] formula can be expressed as follows. Let P be the interior of a polygon Γ having vertices w1 …wn and turning angles α1 π…αn π in counterclockwise order. Then, Z Z jCje jβ W − u1 α1 W − u2 α2 …W − un αn dW A; (1) where complex A is an integral constant; jCj and e represent the scale and azimuth of rotation, respectively. In our practical issue, the electric field is rotationally symmetric, so the results are mapping on the rotational symmetry plane. Some approximations must be made for simplification, they are (1) the thickness of the plate electrodes is zero; (2) when conducting a calculation, it is considered that the size of the electrode extends infinitely because the extent of the electric field around the tip is much smaller than the electrode size; and (3) only half of the electric field is calculated because it is mirror symmetrical. The mathematical model of the tip of the polishing wheel is depicted in Fig. 7 (left) in which an open polygon can be abstracted. The three edges are denoted by dashed lines which can be regarded as electrodes. Obviously, there are two turning angles in P. The first and second turning angles are π and −π, respectively, so α1 1 and α2 −1. Besides, rotation of the last segment is not needed. Hence, β 0, and then W d Z jCj dW : (2) W 1 After integration of the complex variable function, iβ
Z jCjW ln W jAje jγ ;
(3)
where jAje ais the complex constant of integration. Let the verx −π R1 tex in plane Z occupy u1 −1, which is, ya ψ π in plane W . With another boundary condition that y 0 refers to ψ 0, the unknown coefficient could be solved as jCj πa . So we have jAj 0 a a x R cos ψ ln R; y R sin ψ ψ: (4) π π jγ
641
(5)
Let W satisfy the Cauchy–Riemann conditions, and we employ the real and imaginary parts to indicate the electric field line and the equipotential line. E w U ∕πW can be obtained by conformal mapping based on Eq. (5). Hence, the electric field in arbitrary coordinate P could be expressed as U 1 E z : (6) a W 1 C. ER Effect Theory and Calculation of the Force between Particles
Presently, researchers believe that the most accurate interpretation to explain particle attraction is particle polarization theory. Polarized ER particles attract nearby particles, so that all the ER particles will become members of chain structures. The popular dipole–dipole interaction model [13] is a simplified particle polarization model, where ER particles are regarded as dipoles for mathematical modeling. In this paper, to emulate the behavior of dielectric particles, the high-order multipolar moment theory is employed as the base theory of particle interactions. Different from the dipole– dipole interaction model, the high-order multipolar moment theory takes into account the high-order multipoles induced by the distorted electric field line surrounding the particles. Therefore, this model is closer to the actual case. A polarized particle deforms the nearby electric field and leads to a nonuniform electric field. Another particle that is located at a very small distance away is under the effect of the nonuniform electric field and produces high-order multipoles. Specifically speaking, the nonuniform electric field induces high-order multipoles such as quadrupoles and octupoles. Considering that the size of an actual ER particle is a micron grain, namely, very small compared to nonuniformity of the external DC electric field, we limit our attention to the axisymmetric geometry where linear multipoles adequately represent the induced field contributions. Also, the electric field is considered uniform at the microscale. Let two identical insulating dielectric particles of radius R and permittivity ε2 be suspended in the fluid of permittivity ε1 subjected to the z-directional uniform electric field of magnitude E 0 . The coordinate system origin is located at the center of particle a; particle b is set at the z axis with center-to-center spacing Ξ apart from particle a (refer to Fig. 8). As predicted, linear multipoles will be induced, and we can write down the function as below [18]: Ea E0
∞ X i 1pi b
i1
4πε1 Ξi2
(7)
;
∞ −1n i 1i 2…i 1 npb ∂n E a X ; ∂z n 4πε1 Ξi2n i1 i
n ≥ 1; (8)
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Fig. 9. Coordinate system of the effective force calculation.
Fig. 8. Dielectric particles of radius R and permittivity ε2 suspended in the fluid of permittivity ε1 subjected to a z-directional uniform electric field of magnitude E 0 . Particle b is set at the Z axis with centerto-center spacing Ξ apart from particle a.
pn a
4πε1 K n R 2n1 ∂n−1 E a ; n − 1! ∂z n−1
(9)
i where pi b and pa are the ith-order dipole moments of particle a and particle b, respectively; E a and ∂n E a ∕∂z n are the axial electric field and its derivatives at the center of particle a, respectively; K is the Clausius–Mossotti function defined as σp − σ1 ; (10) K n nσ p n 1σ 1
where σ p and σ 1 are the conductivities of the particle and fluid, respectively. The geometrical axial symmetry demands that n−1 pi : pi a b −1
(11)
By solving Eqs. (8)–(11) simultaneously, a series of multipole moments for each particle can be obtained. Solving the net force between two particles begins from the interaction between two aligned linear multipoles of order m and n, which are presented in Eq. (12). The net mutual interaction force between two particles is the double summation of all the F m;n terms from Eq. (13). Theoretically, the number of multipoles extends infinitely, but a practical calculation should be carried out by truncating the series at a finite number: F m;n −1n1 F mutual
m n 1!pm pn ; 4πε1 Ξmn2 m!n!
∞ X ∞ n m n 1!pm 1 X b pa : −1n1 mn2 Ξ m!n! 4πε1 m1 n1
(12)
(13)
D. General Prediction Model of the Material Removal Feature
During the polishing process, the stiffened ER fluid yields in a contact zone when it is moving relatively with the workpiece surface. The yield layer between the stiffened solid and the workpiece surface brings material shearing removal. The material is removed persistently and circularly with a continuous refreshing of the yield layer. The ratio of volume between starch and cerium oxide is around 50. Thus, it is necessary to consider starch as the primary member of particle chains. Furthermore, the electrodynamics of
the multidisperse liquid is very complicated. In order to simplify the problem, we focus on starch particles to discuss the model. Actually, in the working area the electric field lines are not parallel to the glass surface, as shown in Fig. 5. Thus, the interaction force model between particles needs to be modified. The coordinate system is depicted in Fig. 9, and the effective force refers to the angle between surface and the electric line and is expressed as 3 cos2 θ − 1 : (14) F m;n;θ F m;n · 2 Then, Fθ
XX
F m;n;θ
XX
F m;n ·
3 cos2 θ − 1 : (15) 2
Equation (15) is the final expression of the effective interaction force between two particles. With this expression, the effective particle interaction force at any working point beneath the polishing wheel can be calculated. This work mainly reviews the relationship between particle interaction and material removal. The strength of the flexible polishing die is the dominating factor in material removal. For ERF, the individual particle attractions directly determine the strength of the stiffened ER fluid, so we believe that the effective particle force distribution on the working area has direct correspondence to material removal. 4. EXPERIMENTS In order to eliminate interference caused by external conditions, the experiments were conducted in a laboratory at constant temperature (21°C); the polishing slurry was stirred well by an ultrasonic device for even distribution of the mixture. To test the proposed tool and verify the theoretical model, the experiments were carried out on BK7 glass, and the morphology of the spot was measured by a ZYGO interferometer with λ 632.8 nm. The ER polishing fluid was composed of 47.6 wt.% starch, 47.6 wt.% silicone oil, and 4.76 wt.% ceria. All of the parameters are listed in Table 1. Figure 10 shows the view of a polishing spot of 20 min of work. The shape of this polishing spot is an oval 3.4 mm long and 2.3 mm wide. AA 0 and BB 0 in the bottom illustration denote the profile of two orthogonal sections of the spot. Actually, AA 0 represents shear direction and BB 0 represents the Y axis. The BB 0 curve is axisymmetric and has a peak value in the middle and reduces to zero near the two edges. In the AA 0 direction, the hydrodynamic theory plays a primary role in the shear stress distribution; research on the hydrodynamic model for the ERF method will be further studied later. The normalized experimental and simulated profiles of the cross section of
Research Article Table 1.
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Parameters of the Polishing Experimentations
Parameter
Value
Mass fraction of CeO2 particles in fluid (wt.%) Mass fraction of starch particles in fluid (wt.%) Relative permittivity of silicone oil Conductivity of silicone oil (S/m) Conductivity of starch particle (S/m) Median diameter of starch particle (μm) Median diameter of CeO2 particle (μm) Tool rotation speed (rpm) Distance between anode and cathode (mm) Working distance (mm) Working voltage (kV)
4.76 47.6 2.7 1e − 13 3e − 3 74 7.5 600 2 0.6 4
the polishing spot are depicted in Fig. 11. The actual spot profile and the theoretical simulation profile (the normalized plot of the effective force distribution introduced above) are denoted by a solid line and a dashed line, respectively. The actual curve agrees well with the simulated one. The tool influence function (TIF) stability test, as a prerequisite for the practical use of the tool, was carried out. Figure 12 presents the map of an area with two spots. Spot (α) and spot (β) were taken under the conditions listed in Table 1, and between these two works there is a 10 h interval. The results indicate that the TIF of the tool is highly stable in terms of both the removal shape and the removal rate. The next experiment was micropattern fabrication. The specimen was a 1 mm thick, 7 mm wide BK7 slide glass (Fig. 13). The shape accuracy of the prepolished specimen is 13.92 nm in PV and 3.16 nm in RMS. The roughness of the initial surface is 3.46 nm in Ra. The fabrication process lasted 2 h, and other working conditions are listed in Table 1. Surface topography was captured using a ZYGO interferometer. The trough is relatively shallow near the two sides and has a deep, flat bottom in the central region. The depth of 70 nm demonstrates the capability of the tool for micromaterial removal. The PV is 5.06 nm at the BB 0 line. This indicates that
Fig. 10. Interferometer pattern of the polishing spot and its corresponding profile in two directions.
Fig. 11. Normalized cross-section profile of the experimental and simulated polishing spots.
Fig. 12. TIF stability. (a) Surface map of TIFs. (b) 3D model of TIFs. (c) Profile of the section line.
Fig. 13. Processing details of the micropattern.
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0.86 nm, as shown in Fig. 16. It is believed that the relatively low stiffness of the polishing pad leads to a smoother surface quality. 5. CONCLUSION
Fig. 14. profile.
Surface profile of the bottom of the trough and the initial
Fig. 15.
Surface roughness of the bottom of the trough.
This paper presents a wheel-like integrated electrodes ERF tool for fabricating miniaturized optical parts. Round, thin electrodes were employed as the polishing wheel, which provide a higher linear speed in the working zone. Particle interactions directly determine the fluid shear capacity and the material removal. In this paper, based on the linear high-order multipolar moment theory and the conformal mapping method, we built a model that could provide a qualitative and universal prediction about the cross-section profile of the polishing spot. The wheel-like ERF tool created an ellipse polishing spot of 3.4 mm × 2.3 mm, and the cross-section profile of the spot agreed well with the normalized effective particle force model. A micropattern of the flat-bottom trough was made. At the 1.5 mm wide bottom of the trough, the figure accuracy was 3.16 nm in PV and 1.27 nm in RMS. A surface roughness of 1.39 nm was obtained. After further improvement, the surface roughness achieved 0.86 nm. The results demonstrated the capability of the ERF tool for ultrafine machining of small parts. Funding. (120610).
City University of Hong Kong (CityU)
REFERENCES
Fig. 16.
Further improvement in the surface roughness.
the removal process is highly stable (i.e., PV/depth was about 0.075). The RMS at the trough bottom is 1.27 nm. The surface profile of the bottom of the trough and the initial profile are shown in Fig. 14. We focus on the 1.5 mm 6 mm area which is the bottom of the trough. The PV value reduces to 6.33 nm from 28.48 nm and the RMS value from 3.80 nm to 1.27 nm, respectively. The surface data indicates that the tool is capable of small-parts polishing. The surface roughness (Ra) of the specimen was measured with a Wyko NT1100 interferometer. The measured area in Fig. 15 belongs to the bottom of the trough. Surface roughness has been smoothed from 3.46 nm to 1.39 nm. Then, the voltage was reduced to 3 kV, and a 20 min smoothing procedure was performed on the specimen. Finally, the roughness achieves
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Research Article
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Study of a wheel-like electrorheological finishing tool and its applications to small parts JINGSHI SU,1,2 HAOBO CHENG,1,2,* YUNPENG FENG,1,2
AND
HON-YUEN TAM3
1
Joint Research Center for Optomechatronics Engineering, School of Optoelectronics, Beijing Institute of Technology, Beijing 100081, China Shenzhen Research Institute, Beijing Institute of Technology, Shenzhen 518057, China 3 Department of Mechanical and Biomedical Engineering, City University of Hong Kong, Hong Kong 999077, China *Corresponding author: [email protected] 2
Received 2 September 2015; revised 4 November 2015; accepted 10 December 2015; posted 15 December 2015 (Doc. ID 249377); published 21 January 2016
A wheel-like electrorheological finishing (ERF) tool for small parts polishing is proposed and thoroughly studied. First, the electrorheological polishing fluid is tested, and its properties suggest usability for electrorheological fluid-assisted finishing. Then, the mathematical removal model of the ERF tool is built employing the conformal mapping method and high-order multipolar moment theory. Finally, a micropattern of trough is fabricated on a slide glass (7 mm wide and 1 mm thick). The trough is 70 nm deep, and its flat bottom is 1.5 m wide (peak to valley of 3.16 nm and root mean square of 1.27 nm); the surface roughness finally achieves 0.86 nm. The results demonstrate the stable machining capability of the ERF tool for miniature parts. © 2016 Optical Society of America OCIS codes: (220.4610) Optical fabrication; (220.5450) Polishing. http://dx.doi.org/10.1364/AO.55.000638
1. INTRODUCTION Ultrafine, minisized optical components can be used across a broad range of applications in modern industry. For example, when the substrate for femtosecond laser processing is small (10 mm × 10 mm) and fragile [1], a 1 mm thick edge surface can be employed [2]. Very small parts are also used in microsensors [3] and so on. For laser devices, the subsurface damage of optical components becomes a key factor that determines the lifetime of optical parts [4], like components at the National Ignition Facility [5]. Accepted ways to suppress subsurface damage is to implement low-stress processing methods. One outstanding method for low-stress polishing is bowl feed polishing (BFP) [6]. During this polishing process, the workpiece is completely immersed in the polishing slurry for full-aperture fabrication of regular-shaped components. BFP can achieve an ultrasmooth surface with 0.2 nm roughness. Meanwhile, BFP possesses a low material removal ratio which is about several nanometers per minute. Therefore, it is often the last process to improve the overall surface fineness of a workpiece. Currently, to meet the diversified demands of optical components, more methods are being studied. Field-assisted finishing methods are also typical low-stress polishing methods, such as the magnetorheological finishing (MRF) [7] method and the electrorheological finishing (ERF) [8] method. MRF is a mature precision polishing technique capable of rapidly converging to the required surface figure avoiding subsurface damages. MRF utilizes the external 1559-128X/16/040638-08$15/0$15.00 © 2016 Optical Society of America
magnetic field to generate magnetic dipoles (carbonyl iron) which attract each other. This phenomenon leads to significant increases in the apparent viscosity and the yield stress of the magnetorheological (MR) fluid. The surface material can be removed under shear motion with stiffened MR fluid. The yield stress of the MR fluid exceeds 20 kPa under high magnetic field strength. For BK7 glass, the removal rate of 15 μm∕min was achieved under a magnetic field of 2.5 kG using MR polishing equipment made by the University of Rochester [9]. MRF is suitable for fabricating medium-sized components. Huang et al. [10] polished a 410 mm × 410 mm plane mirror and its figure error improved from 253.1 to 63.3 nm by peak-to-valley (PV) [root mean square (RMS) 43.0 to 8.23 nm]. The use of an ultrafine abrasive will significantly improve surface roughness (around 1 nm RMS with a nanoabrasive [11]). Another field-assisted finishing method is the ERF method. ER fluid is among the smartest or most intelligent materials; under an electric field its components produce the electroviscous effect (the so-called Winslow effect) to alter fluid viscosity and yield stress. It is composed of two phases: the disperse phase and the continuous phase. The disperse phase consists of small polarizable particles like alumina, silica, and starch. Generally, silicone oil is widely used as a continuous phase for its stability, low dielectric permittivity, low conductivity, and compatibility with various oil-soluble materials. When placed in an electric field, the apparent viscosity of the ER fluid increases by orders
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(Pa)
639
Shear stress of ER fluid
of magnitude with the electric field (AC or DC) and immediately returns to its initial status when the electric field is removed. The yield stress of the ER fluid is usually lower than −10 kPa. The problem is that it is impractical to increase the yield stress by improving the electric field strength excessively because high voltage leads to dielectric breakdown, so the electroviscous effect diminishes. This condition restricts the application of ERF in lapping and rough polishing. However, the low stress of the ER fluid makes it possible for nanoscale material removal. The mixture of soft polymers and fine abrasives is likely to further improve surface quality without leaving residue and subsurface damage which may lead to great damage in the laser components. In addition, designing and manufacturing electrodes to generate AC or DC electric fields is simpler and more flexible in adapting to various sizes and shapes of the workpiece than that of a permanent magnet and magnetic coil. Meanwhile, the response time of the electric field is a few milliseconds, which is better than the magnetic coil. From the perspectives of applicability and maneuverability, with the applications of microelectromechanical systems and laser devices getting more and more diversified, the ERF method has huge potential. Electrorheological (ER) polishing was first proposed by Kuriyagawa and Syoji in 1999, while a needle-like polishing tool for metallic dies was developed by Kuriyagawa et al. [12]. A rotating shaft was used as the anode, and the cathode was the workpiece itself. Kim developed a padless ultraprecision polishing tool [13] and described the behavior of particles in the fluid based on the dielectrophoresis theory [14]. Auxiliary electrodes were needed to produce an electric field with the metallic tool tip, so that the device would be capable of fabricating insulating components. Zhang et al. developed a theoretical model for the size of the polishing spot [15]. In their cases, to obtain a higher removal rate, the tool tips have to be extremely close to the workpiece surface (within 10 μm), so the top removal rate could achieve 0.4 μm∕min using a micron-sized diamond abrasive on borosilicate glass. Surface roughness on the nanometer scale could be achieved by using a micron-sized abrasive [13]. We did some research in the ER fluid study, and a needle-like integrated polishing tool which is more suitable for computer-assisted polishing was proposed [16]. Actually, a wheel-like polishing tool can provide larger linear velocity which is thought to be an efficacious way to improve finishing efficiency. In this study, we proposed a novel wheel-like integrated electrodes polishing tool. This device can provide a stable material removal footprint. The normalized removal feature of the tool head is calculated based on conformal mapping and linear electric multipolar moment theory. Finally, a trough with a 1.5 mm wide bottom is made by the ERF tool on BK7 slide glass.
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Shear rate
Fig. 1. fluid.
(S-1)
Shear stress property curve of the starch–silicone oil ER
because of its high removal capability. A HAAKE RV20 rheometer was used to test rheological properties of the ER fluid and the ER polishing fluid. This rheometer has a 0.5 mm gap between its electrodes. Figure 1 is the shear stress curve of the starch–silicone oil ER fluid (5 g of starch and 5 g of silicone oil) under different voltages applied on the electrodes in the rheometer. The fluid is a Newtonian-like fluid under zero electric field; its shear stress increases monotonously and linearly with shear rate, less than 200 Pa. However, its property changes remarkably after high voltage is applied. As the shear rate increases, the fluid shear stress reaches the yield point rapidly in a very short span of shear rate, and then it declines gently. The curve in Fig. 2 belongs to the ER polishing fluid which was a mixture of ER fluid and a small amount of ceria (0.5 g). Clearly, the ER polishing fluid possesses electrorheological characteristics as well. Under certain electric fields, the curve of the shear stress of the ER polishing fluid appears to level off after the yield point. As shear the rate grows, the shear stress increases slightly. Between the ER fluid and the ER polishing fluid there are a few differences. First, the antibreakdown property of the ER fluid is improved by adding ceria. Dielectric breakdown happened at 2.1 kV for the starch–silicone oil fluid; meanwhile, it is over 2.55 kV for the ER polishing fluid. Second, at identical voltage levels, the yield stress of the ER polishing fluid is smaller
2. ELECTRORHEOLOGICAL FLUID Our previous research has proved that the starch–siliconeoil-type ER fluid exhibits strong and stable electrorheological effects. However, in order to carry out optical fabrication by virtue of the ER effect, some abrasives like ceria must be added into the starch–silicone oil system, becoming the ER polishing fluid. The abrasive plays a significant role in glass polishing
Fig. 2. Shear stress property curve of the starch–silicone oil ER polishing fluid.
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Carbon brush Z axis
Slip ring
Turntable
Fig. 3. Body-centered cubic structure of the combination of starch and ceria.
than that of the ER fluid but is more stable. For instance, for the case of 1600 V, the ER fluid yields at 2000 Pa and the shear stress reduces to 1600 Pa at a shear rate of 300. For the ER polishing fluid, the shear stress is stable at around 1500 Pa during the increase of the shear rate. So in short, the shear stress of the ER polishing is smaller but is better on stability. We believe that the ER polishing fluid has a stable electrorheological property and shall be effective in optical fabrication. At present, the generally accepted interpretation of the ER effect is particle-chain theory. Dielectric particles in the electric field are polarized and consequently become strongly anisotropic. Two particles attract each other when aligned along the electric line. This phenomenon leads to the consequent transition from a condition of uniform distribution to that of a chain-like distribution. For the ER polishing fluid which contains ceria, the situation is more complicated. We can consider the combination of starch and ceria as a body-centered cubic (BCC) structure (Fig. 3) because of the disparity between starch and ceria in terms of particle size. As a high molecular polymer, starch particles have sizes of dozens of microns and ceria particles are just several microns. Therefore, the BCC structure appears to be relatively stable, with large starch particles attached to each other, while ceria particles fill in the gaps between the starch particles. We established the ER effect observing system to observe the ER effect visually. The transformation of the microstructure is shown in Fig. 4. Under an electric field, large starch particles attract head-to-tail in the direction of the electric field, forming fibrous structures which bridge the anode and cathode and with small ceria particles between the starch particles. 3. ERF TOOL DESIGN AND THE ERF MATHEMATICAL MODEL A. ERF Tool Design
An ERF platform including a wheel-like polishing tool and its carrier machine (see Fig. 5) is introduced. The two
Anode
X axis
Teflon
Y axis
Polishing shaft
Voltage source
Cathode
Fig. 5. Physical map of the experimental platform and ERF tool.
copper-made polishing wheels, which are 30 mm in diameter, are fixed in parallel onto the polishing shaft and separated from each other by Teflon of 2 mm thickness. The polishing wheels are used as the anode and cathode simultaneously. The polishing shaft is driven by a motor via bevel gear transmission. High voltage is supplied to the anode pass through the slip ring which is mounted on the shaft behind the polishing wheel, and the cathode is earthed. The sketch map and physical map of the tool head are shown in Fig. 6. For precision finishing, ERF employs a computer program to determine a schedule for varying the position of the rotating polishing wheel. Therefore, deterministic material removal from the workpiece surface is feasible. Components of the process are the ERF software and a four-axis PC-based computer numerical control platform employed to execute motion and programmable logic control. The machine, schematically shown in Fig. 5, finishes optical parts up to 100 mm in diameter. The active X, Y, Z, and turntable axes move simultaneously in synchronization and are responsible for taking the tool head toward the target position. The machine is located at the vibration-isolating foundation to suppress mechanical vibration disturbances. B. Calculation of the Electric Field Distribution in the Working Zone
From the microscopic perspective, the mutual action of ER particles is the key explanation of the high shear stress of the ER fluid. One significant factor that determines the particle interaction is the distribution of the electric field. As we all know, the electric field line is always orthogonal to the equipotential line, so the angle invariance and scale invariant feature of conformal mapping make it very suitable for simplifying the
Shaft
Cathode
Anode Electric Field line
Work piece
1 mm (a)
Fig. 4. Transformation of the particle status in the ER polishing fluid.
(b)
Fig. 6. (a) Sketch map and (b) physical map of the polishing wheel of the ERF tool.
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Ğ Intersect at infinity
a
Ğ
y a π w1
P
Symmetric plane of the electric field
Work piece
The tip of polishing wheel
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After conformal mapping, the electrodes have been converted onto axis. Then, suppose the following complex function:
v Z Plane
W Plane
Y lnW : x
u w1
Fig. 7. Schwarz–Christoffel cross-sectional map of the wheel-like electrode.
electric field generated by arbitrary electrodes. In this paper, we employ the conformal mapping method to solve the electric field. For a nonuniform electric field generated by polygonshaped (including open polygon) electrodes, we can use the Schwarz–Christoffel formula to transform the original shape of the electrodes onto the axis. The Schwarz–Christoffel [17] formula can be expressed as follows. Let P be the interior of a polygon Γ having vertices w1 …wn and turning angles α1 π…αn π in counterclockwise order. Then, Z Z jCje jβ W − u1 α1 W − u2 α2 …W − un αn dW A; (1) where complex A is an integral constant; jCj and e represent the scale and azimuth of rotation, respectively. In our practical issue, the electric field is rotationally symmetric, so the results are mapping on the rotational symmetry plane. Some approximations must be made for simplification, they are (1) the thickness of the plate electrodes is zero; (2) when conducting a calculation, it is considered that the size of the electrode extends infinitely because the extent of the electric field around the tip is much smaller than the electrode size; and (3) only half of the electric field is calculated because it is mirror symmetrical. The mathematical model of the tip of the polishing wheel is depicted in Fig. 7 (left) in which an open polygon can be abstracted. The three edges are denoted by dashed lines which can be regarded as electrodes. Obviously, there are two turning angles in P. The first and second turning angles are π and −π, respectively, so α1 1 and α2 −1. Besides, rotation of the last segment is not needed. Hence, β 0, and then W d Z jCj dW : (2) W 1 After integration of the complex variable function, iβ
Z jCjW ln W jAje jγ ;
(3)
where jAje ais the complex constant of integration. Let the verx −π R1 tex in plane Z occupy u1 −1, which is, ya ψ π in plane W . With another boundary condition that y 0 refers to ψ 0, the unknown coefficient could be solved as jCj πa . So we have jAj 0 a a x R cos ψ ln R; y R sin ψ ψ: (4) π π jγ
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(5)
Let W satisfy the Cauchy–Riemann conditions, and we employ the real and imaginary parts to indicate the electric field line and the equipotential line. E w U ∕πW can be obtained by conformal mapping based on Eq. (5). Hence, the electric field in arbitrary coordinate P could be expressed as U 1 E z : (6) a W 1 C. ER Effect Theory and Calculation of the Force between Particles
Presently, researchers believe that the most accurate interpretation to explain particle attraction is particle polarization theory. Polarized ER particles attract nearby particles, so that all the ER particles will become members of chain structures. The popular dipole–dipole interaction model [13] is a simplified particle polarization model, where ER particles are regarded as dipoles for mathematical modeling. In this paper, to emulate the behavior of dielectric particles, the high-order multipolar moment theory is employed as the base theory of particle interactions. Different from the dipole– dipole interaction model, the high-order multipolar moment theory takes into account the high-order multipoles induced by the distorted electric field line surrounding the particles. Therefore, this model is closer to the actual case. A polarized particle deforms the nearby electric field and leads to a nonuniform electric field. Another particle that is located at a very small distance away is under the effect of the nonuniform electric field and produces high-order multipoles. Specifically speaking, the nonuniform electric field induces high-order multipoles such as quadrupoles and octupoles. Considering that the size of an actual ER particle is a micron grain, namely, very small compared to nonuniformity of the external DC electric field, we limit our attention to the axisymmetric geometry where linear multipoles adequately represent the induced field contributions. Also, the electric field is considered uniform at the microscale. Let two identical insulating dielectric particles of radius R and permittivity ε2 be suspended in the fluid of permittivity ε1 subjected to the z-directional uniform electric field of magnitude E 0 . The coordinate system origin is located at the center of particle a; particle b is set at the z axis with center-to-center spacing Ξ apart from particle a (refer to Fig. 8). As predicted, linear multipoles will be induced, and we can write down the function as below [18]: Ea E0
∞ X i 1pi b
i1
4πε1 Ξi2
(7)
;
∞ −1n i 1i 2…i 1 npb ∂n E a X ; ∂z n 4πε1 Ξi2n i1 i
n ≥ 1; (8)
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Fig. 9. Coordinate system of the effective force calculation.
Fig. 8. Dielectric particles of radius R and permittivity ε2 suspended in the fluid of permittivity ε1 subjected to a z-directional uniform electric field of magnitude E 0 . Particle b is set at the Z axis with centerto-center spacing Ξ apart from particle a.
pn a
4πε1 K n R 2n1 ∂n−1 E a ; n − 1! ∂z n−1
(9)
i where pi b and pa are the ith-order dipole moments of particle a and particle b, respectively; E a and ∂n E a ∕∂z n are the axial electric field and its derivatives at the center of particle a, respectively; K is the Clausius–Mossotti function defined as σp − σ1 ; (10) K n nσ p n 1σ 1
where σ p and σ 1 are the conductivities of the particle and fluid, respectively. The geometrical axial symmetry demands that n−1 pi : pi a b −1
(11)
By solving Eqs. (8)–(11) simultaneously, a series of multipole moments for each particle can be obtained. Solving the net force between two particles begins from the interaction between two aligned linear multipoles of order m and n, which are presented in Eq. (12). The net mutual interaction force between two particles is the double summation of all the F m;n terms from Eq. (13). Theoretically, the number of multipoles extends infinitely, but a practical calculation should be carried out by truncating the series at a finite number: F m;n −1n1 F mutual
m n 1!pm pn ; 4πε1 Ξmn2 m!n!
∞ X ∞ n m n 1!pm 1 X b pa : −1n1 mn2 Ξ m!n! 4πε1 m1 n1
(12)
(13)
D. General Prediction Model of the Material Removal Feature
During the polishing process, the stiffened ER fluid yields in a contact zone when it is moving relatively with the workpiece surface. The yield layer between the stiffened solid and the workpiece surface brings material shearing removal. The material is removed persistently and circularly with a continuous refreshing of the yield layer. The ratio of volume between starch and cerium oxide is around 50. Thus, it is necessary to consider starch as the primary member of particle chains. Furthermore, the electrodynamics of
the multidisperse liquid is very complicated. In order to simplify the problem, we focus on starch particles to discuss the model. Actually, in the working area the electric field lines are not parallel to the glass surface, as shown in Fig. 5. Thus, the interaction force model between particles needs to be modified. The coordinate system is depicted in Fig. 9, and the effective force refers to the angle between surface and the electric line and is expressed as 3 cos2 θ − 1 : (14) F m;n;θ F m;n · 2 Then, Fθ
XX
F m;n;θ
XX
F m;n ·
3 cos2 θ − 1 : (15) 2
Equation (15) is the final expression of the effective interaction force between two particles. With this expression, the effective particle interaction force at any working point beneath the polishing wheel can be calculated. This work mainly reviews the relationship between particle interaction and material removal. The strength of the flexible polishing die is the dominating factor in material removal. For ERF, the individual particle attractions directly determine the strength of the stiffened ER fluid, so we believe that the effective particle force distribution on the working area has direct correspondence to material removal. 4. EXPERIMENTS In order to eliminate interference caused by external conditions, the experiments were conducted in a laboratory at constant temperature (21°C); the polishing slurry was stirred well by an ultrasonic device for even distribution of the mixture. To test the proposed tool and verify the theoretical model, the experiments were carried out on BK7 glass, and the morphology of the spot was measured by a ZYGO interferometer with λ 632.8 nm. The ER polishing fluid was composed of 47.6 wt.% starch, 47.6 wt.% silicone oil, and 4.76 wt.% ceria. All of the parameters are listed in Table 1. Figure 10 shows the view of a polishing spot of 20 min of work. The shape of this polishing spot is an oval 3.4 mm long and 2.3 mm wide. AA 0 and BB 0 in the bottom illustration denote the profile of two orthogonal sections of the spot. Actually, AA 0 represents shear direction and BB 0 represents the Y axis. The BB 0 curve is axisymmetric and has a peak value in the middle and reduces to zero near the two edges. In the AA 0 direction, the hydrodynamic theory plays a primary role in the shear stress distribution; research on the hydrodynamic model for the ERF method will be further studied later. The normalized experimental and simulated profiles of the cross section of
Research Article Table 1.
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Parameters of the Polishing Experimentations
Parameter
Value
Mass fraction of CeO2 particles in fluid (wt.%) Mass fraction of starch particles in fluid (wt.%) Relative permittivity of silicone oil Conductivity of silicone oil (S/m) Conductivity of starch particle (S/m) Median diameter of starch particle (μm) Median diameter of CeO2 particle (μm) Tool rotation speed (rpm) Distance between anode and cathode (mm) Working distance (mm) Working voltage (kV)
4.76 47.6 2.7 1e − 13 3e − 3 74 7.5 600 2 0.6 4
the polishing spot are depicted in Fig. 11. The actual spot profile and the theoretical simulation profile (the normalized plot of the effective force distribution introduced above) are denoted by a solid line and a dashed line, respectively. The actual curve agrees well with the simulated one. The tool influence function (TIF) stability test, as a prerequisite for the practical use of the tool, was carried out. Figure 12 presents the map of an area with two spots. Spot (α) and spot (β) were taken under the conditions listed in Table 1, and between these two works there is a 10 h interval. The results indicate that the TIF of the tool is highly stable in terms of both the removal shape and the removal rate. The next experiment was micropattern fabrication. The specimen was a 1 mm thick, 7 mm wide BK7 slide glass (Fig. 13). The shape accuracy of the prepolished specimen is 13.92 nm in PV and 3.16 nm in RMS. The roughness of the initial surface is 3.46 nm in Ra. The fabrication process lasted 2 h, and other working conditions are listed in Table 1. Surface topography was captured using a ZYGO interferometer. The trough is relatively shallow near the two sides and has a deep, flat bottom in the central region. The depth of 70 nm demonstrates the capability of the tool for micromaterial removal. The PV is 5.06 nm at the BB 0 line. This indicates that
Fig. 10. Interferometer pattern of the polishing spot and its corresponding profile in two directions.
Fig. 11. Normalized cross-section profile of the experimental and simulated polishing spots.
Fig. 12. TIF stability. (a) Surface map of TIFs. (b) 3D model of TIFs. (c) Profile of the section line.
Fig. 13. Processing details of the micropattern.
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0.86 nm, as shown in Fig. 16. It is believed that the relatively low stiffness of the polishing pad leads to a smoother surface quality. 5. CONCLUSION
Fig. 14. profile.
Surface profile of the bottom of the trough and the initial
Fig. 15.
Surface roughness of the bottom of the trough.
This paper presents a wheel-like integrated electrodes ERF tool for fabricating miniaturized optical parts. Round, thin electrodes were employed as the polishing wheel, which provide a higher linear speed in the working zone. Particle interactions directly determine the fluid shear capacity and the material removal. In this paper, based on the linear high-order multipolar moment theory and the conformal mapping method, we built a model that could provide a qualitative and universal prediction about the cross-section profile of the polishing spot. The wheel-like ERF tool created an ellipse polishing spot of 3.4 mm × 2.3 mm, and the cross-section profile of the spot agreed well with the normalized effective particle force model. A micropattern of the flat-bottom trough was made. At the 1.5 mm wide bottom of the trough, the figure accuracy was 3.16 nm in PV and 1.27 nm in RMS. A surface roughness of 1.39 nm was obtained. After further improvement, the surface roughness achieved 0.86 nm. The results demonstrated the capability of the ERF tool for ultrafine machining of small parts. Funding. (120610).
City University of Hong Kong (CityU)
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Fig. 16.
Further improvement in the surface roughness.
the removal process is highly stable (i.e., PV/depth was about 0.075). The RMS at the trough bottom is 1.27 nm. The surface profile of the bottom of the trough and the initial profile are shown in Fig. 14. We focus on the 1.5 mm 6 mm area which is the bottom of the trough. The PV value reduces to 6.33 nm from 28.48 nm and the RMS value from 3.80 nm to 1.27 nm, respectively. The surface data indicates that the tool is capable of small-parts polishing. The surface roughness (Ra) of the specimen was measured with a Wyko NT1100 interferometer. The measured area in Fig. 15 belongs to the bottom of the trough. Surface roughness has been smoothed from 3.46 nm to 1.39 nm. Then, the voltage was reduced to 3 kV, and a 20 min smoothing procedure was performed on the specimen. Finally, the roughness achieves
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