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A method for measuring the geometrical topography of a Rockwell diamond indenter
View the table of contents for this issue, or go to the journal homepage for more 2010 Meas. Sci. Technol. 21 015307
(http://iopscience.iop.org/0957-0233/21/1/015307)
Yen-Liang Chen
1,2and Der-Chin Su
11Department of Photonics and Institute of Electro-Optical Engineering, National Chiao-Tung University, 1001 Ta-Hsueh Road, Hsinchu 30050, Taiwan, Republic of China
2Center for Measurement Standards/Industrial Technology Research Institute (ITRI), Bldg 16, 321, Kuang Fu Rd, Sec. 2, Hsinchu 30031, Taiwan, Republic of China
E-mail:[email protected](Der-Chin Su)
Received 21 July 2009, in final form 29 October 2009 Published 7 December 2009
Online atstacks.iop.org/MST/21/015307
Abstract
Firstly, a Rockwell diamond indenter tip is tested at different angles and positions by using a scanning white light interferometer, and several image data are obtained. Then, these data are merged together to form the associated geometrical topography with the image stitching method. Moreover, both the tip radius and the cone angle can be obtained with the
least-squares Gauss–Newton sphere and cone fitting algorithm. Its validity is demonstrated.
Keywords: white light interferometer, image stitching method, Rockwell diamond indenter, surface topography, Mirau interferometer
1. Introduction
Because hardness is a key characteristic parameter of materials, the hardness measurement is a very important technique for characterizing the mechanical properties of materials [1]. A blunt conical Rockwell diamond indenter is often used to estimate the hardness of materials. The measured results strongly depend on the geometrical topography of the indenter [2–6]. It is necessary to identify the geometrical topography of the indenter in advance. Several pieces of apparatus [7–9] have been developed to measure the geometrical topography of the Rockwell diamond indenter with the contact stylus method. Although they have good results, they are time consuming for three-dimensional (3D) measurement. In 2004, Takagi et al [10] developed an interference microscope to measure the radius and the cone angle of a Rockwell diamond indenter tip quickly. But the radius is measured with the reflected light from the surface; it may have some errors if its surface deviates from an ideal spherical one. Besides, an accurate rotation device is needed for measuring the cone angle. To overcome those drawbacks, we present an alternative and convenient optical method with a two-dimensional white light interferometer [11–13] and the use of the image stitching method without an accurate rotation
device. Because of the microscopic configuration of the two-dimensional white light interferometer, the measurable region is restricted and the overall topography cannot be obtained in a single measurement. It can be improved by measuring at different angles and positions. Several image data are obtained, and each set of data has a partial overlap with the neighboring one. Next, these data are merged together to form the associated topography with the image stitching method based on the iterative closest point (ICP) algorithm [14,15]. In addition, two crucial parameters, the tip radius at the center and cone angle along the outside area of the overall 2D topography, can be calculated by using the least-squares Gauss–Newton sphere and cone fitting algorithm [16]. The validity of this method is demonstrated. It has some merits such as simple nondestructive measurement, wide measurement range, easy operation, rapid measurement and low cost.
2. Principle
2.1. Scanning white light interferometer
A Mirau-type scanning white light interferometer (SWLI) is used in this method and its configuration is shown in figure 1. It consists of a white light source WS, a
Meas. Sci. Technol. 21 (2010) 015307 Y-L Chen and D-C Su
Figure 1.Schematic diagram of this method with a Mirau-type scanning white light interferometer. WS: white light source; BS: beam splitter; MO: microscopic objective; M: mirror; PA: precision actuator; P: microindenter tip; IL: imaging lens C: CCD camera.
beam-expander BE, a beam splitter BS1, a Mirau-type microscopic objective located with a precision actuator PA, a Rockwell diamond indenter tip P, an imaging lens IL and a CCD camera C. Here the Mirau-type microscopic objective also has a beam splitter BS2, a mirror M and a microscopic objective MO. The light beam coming from the WS is collimated by the BE. The collimated beam is reflected by the BS1 and enters a Mirau-type microscopic objective, in which it is divided by the BS2 into two parts: the reference beam and the test beam. They are focused on the M and the P, respectively; and both of them are reflected and come back along the original paths. Then, they pass through the BS1and the IL, and finally they enter the C. Hence, the interferogram recorded by the C can be approximated by a mean intensity I0(x, y) and a series of sinusoidal signals modulated by an envelope function as [17]
I (x, y, z)= I0(x, y){1 + γ (x, y, z) · cos[φ(x, y, z)]}. (1) The envelope function γ (x, y, z) is the fringe contrast, which varies much more slowly with the optical path difference (OPD) between two interfering beams than the phase difference φ(x, y, z). The basic principle of this method is that the maximum contrast occurs when the OPD is zero. The procedures for measuring the topography of the P are described as follows.
(1) Locate the P at a moderate azimuth angle and let the test beam be focused slightly higher than the P.
(2) Displace the PA step-by-step along the -z direction to vary the OPD and record the corresponding interferograms simultaneously by using the C. It is operated until the test beam is focused slightly lower than the P.
(3) Determine the positions of maximum contrast image points in every two-dimensional interferogram, and then
Figure 2.Geometrical relation of the tested blunt conical tip profile with α= 90◦.
record the data of their positions and the associated position of the PA in the z-axis.
(4) Combine the data recorded at different positions of the PA in procedure (3) together and the topography of the P can be obtained.
(5) Change the azimuth angle of the P and repeat the above procedures. Then, the associated topography of the P can be obtained similarly.
2.2. Measurement criteria for a Rockwell diamond indenter A Rockwell diamond indenter with cone angle θ and tip radius of curvature r is shown in figure2with azimuth angle α= 90◦ and it can be expressed as
z= ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ |x| + r secθ 2 cotθ 2, as|x| r cos θ 2, √ r2− x2, as|x| < r cosθ 2, (2)
where r > 0 and 0 < θ < π . Its slope varies according to x as |x| < r cosθ
2 and it becomes a constant as|x| r cos θ 2. The measurable solid angle depends on the numerical aperture NA of the MO and the associated slope of the tip. If the measurable range in a single test is L, then we have
dz dx x=L 2 − dz dx x=−L 2 2 tan sin−1N A 2 . (3)
Equation (3) can be rewritten as
r L
2 sin sin−12N A. (4) For easy measurement, the measured region should be kept horizontal. So, m measurements with different α by rotating the P around the y-axis, as shown in figure3(a), should be operated to obtain the overall information. If the measurement range|x| < r cosθ2 + L/2 and the overlapping width L/4 of any two neighbor measurements are chosen, then we have
2r cosθ
2 < (m− 1)L − (m − 1) L
4 < r(π− θ). (5) Here, m must be rounded up to an integer.
(c)
(d )
Figure 3.Image stitching procedures: (a) the tested tip at different α; (b) transformations of I1and Imto the coordinate systems of I2 and Im−1via O1and Om−1; (c) transformations of I2and Im−1to the
coordinate systems of I3and Im−2via O2and Om−2; (d) repeated
transformations to the coordinate system of Iiand a stitched
resultant image.
2.3. Image stitching for two adjacent images
The data in the overlapping regions of any two neighbor measurements need to be processed with the ICP algorithm [14, 15], and they become spatially matched to each other to be merged together. They are calculated in six degrees of freedom that include three rotations and three translations along the x-, y- and z-axes. For convenience, let the data of the overlapping regions of two adjacent images F{f1, f2, . . . , fk} and G{g1, g2, . . . , gk} be OF {of1, of2, . . . , ofn} and
OG{og1, og2, . . . , ogn}, respectively, so it is obvious that
OF ⊂ F , OG ⊂ G, and every element is a 3 × 1 vector with (x, y, z) data. The procedures for transforming the data of F to the coordinate system of G and to be merged into G are described as follows.
(1) Transform the data of OF and OG with respect to their centroids [14]: the centroids are ofc = (1/n)
n i=1ofi and ogc = (1/n)
n
i=1ogi, respectively; ofc and ogc are 3 × 1 vectors. So, the associated data of OF{of1, of2, . . . , ofn} and OG{og1, og2, . . . , ogn} to their centroids are changed to OA{oa1, oa2, . . . , oan} and OB{ob1, ob2, . . . , obn}, where oai = ofi− ofcand
obi = ogi− ogc.
(3) Merge together: let
R1= ⎡ ⎣q 2 0+ q12− q22− q32 2(q1q2− q0q3) 2(q1q3+ q0q2) 2(q1q2+ q0q3) q02+ q 2 2− q 2 1− q 2 3 2(q2q3− q0q1) 2(q1q3− q0q2) 2(q2q3+ q0q1) q02+ q32− q21− q22 ⎤ ⎦ (7) τ1 = ogc− R1· ofc. (8)
Then the intermediate data O1{o11, o12, . . . , o1n} can be obtained, where o1i = R1 · ofi + τ1, R1 is a 3 × 3 matrix; both τ1 and o1i are 3 × 1 vectors, respectively. If any element of τ1is larger than the setting threshold d, then OF{of1, of2, . . . , ofn} is replaced by
O1{o11, o12, . . . , o1n}. Next, the above procedures are operated repeatedly t times until all the elements of τt are smaller than d. Here, R = R1 · R2 · . . . · Rt and τ = τ1+ τ2+· · · + τt are the rotation matrix and the translation matrix, respectively. Finally, the data Ff1, f2, . . . , fk, which transformed the data of F to G’s coordinate system, are obtained, where fj = R·fj+τ , and j = 1, 2, . . . , k. The element fj is a 3× 1 vector with (x, y, z) data. Hence, the data of F and G have been merged together to the G’s coordinate system.
2.4. Image stitching for multiple images
In the test, there are m images (I1, I2, . . . , Im) with different α and m− 1 overlapping regions (O1, O2, . . . , Om1), as shown in figures3(a) and (b). For convenience, the coordinate system of Ii is chosen as a reference coordinate and other images should be transformed to it. Here i may be m/2 or (m + 1)/2. Images I1 and Im are firstly transformed to the coordinate systems of I2and Im−1via the associated overlapping regions
O1and Om−1respectively with the above procedures, as shown in figures 3(b) and (c). Then, the new merging images I2 and Im−1 are also transformed to the coordinate systems of
I3 and Im−2 via O2 and Om−2, respectively, as shown in figure3(c). These procedures are operated repeatedly until all the images are merged into one image, as shown in figure3(d).
3. Experiments and results
A 50× Mirau objective lens (Nikon) with NA = 0.55 was used to test a Rockwell diamond indenter (Kinik Company) with specifications of 200 μm± 20 μm tip radius and 120◦± 0.5◦ cone angle. The condition m= 4 was chosen according to
Meas. Sci. Technol. 21 (2010) 015307 Y-L Chen and D-C Su
Figure 4.Four measured image data with three overlapping regions.
(a)
(c) (b)
Figure 5.The measured results: (a) the resultant 2D topography in gray level and the profiles of (b) the dotted line and (c) the dashed line.
equation (5) and they were operated at α ≈ 70◦, 80◦, 100◦ and 110◦. To compensate the brightness differences arising from different α, the intensity of the WS is so adjusted that good quality images can be taken. Their results are shown in figure 4 in gray level, which represents the corresponding relative height. Each image is recorded with 480× 480 pixels and its dimension is L × L, where L = 104.2 μm. These
Figure 6.The orientated results of figure5(a) for indenter geometry characterization.
four image data were merged together with the conditions d = 0.01 μm and i = 2. The 2D topography with 1716 × 425 pixels was obtained and shown in figure5(a). The cross profiles of the dotted line and the dashed line in figure5(a) are calculated and depicted in figures5(b) and (c), respectively. In addition, the tip radius and the cone angle can be estimated with the least-squares Gauss–Newton sphere and cone fitting algorithm [16] and they are 198.2 μm and 119.6◦, respectively. The roundness value of the tip region is 4.3 μm.
For indenter geometry characterization, the stitching results in figure 5(a) are orientated onto the specifications given in the normative document [18] and they become the results as shown in figure 6. The data of the dotted line in figure 6 can be applied to depict the profile through the apex of the indenter tip. This profile was also measured by a commercial contact stylus instrument (ET4100, Kosaka Lab. Ltd). These two results are shown together in figure7(a) for comparison; the solid curve and the dotted curve represent
Figure 7.(a) Measured profiles through the apex of the indenter tip with this method (solid curve) and the contact stylus instrument (dotted curve); (b) their deviations.
(a) (b)
Figure 8.Geometries for calculating (a) the cone angle and (b) the tip radius of the P by using the least-squares fitting method with a stylus profiler.
the results of this method and the contact stylus instrument, respectively. Their deviations are shown in figure7(b) and the cross correlation function [19] is up to 99.96%. Moreover, the measured results of the contact stylus instrument are also processed with the least-squares fitting algorithm; the curves for calculating the cone angle and the tip radius can be depicted as shown in figures8(a) and (b), respectively. From these two curves, we can calculate their cone angle and tip radius to be 119.62◦ and 198.37 μm, respectively. It can be seen that both results are very consistent. According to ISO 6508-2 [18], the specifications for the shape of Rockwell indenters should be fixed as 200 μm ± 5 μm tip radius and 120◦ ± 0.1◦ cone angle. Based on these two measured results, we infer that the tested indenter may not be suitable for hardness calibration.
The area function [9] of the P is measured with this method, and the measured data (represented by dots) are depicted with the theoretical data (represented by a solid line) of an ideal Rockwell indenter in figure 9(a). Their radius deviations are plotted in figure9(b) and the deviations vary abruptly at the border region between the ball cap and the cone
frustum as Dai’s results [9]. Because the maximum deviation is less than 2.5 μm, it is still less than the radius deviation tolerance 5 μm listed in ISO 6508-3 [20].
4. Discussion
The image stitching method is operated in full six degrees of freedom [14] to minimize the mean square distance and compensate the angle error between two sets of matched points. Although the P is tested at different α, a commonly used rotation holder (±1◦) can be applied for tilting the indenter. The measured results in figure4are not in the same projection plane; however, the image features are similar in the overlapping region and two images can be matched very well to the nearest local minimum of a mean square distance. Because the data of I1, I3and I4are rotated with respect to the y-axis to map the data of I2in the image stitching process, the data of stitched result will expand in the z-axis.
From equation (4), we can understand that the tip radius should be larger than 181 μm under our experimental
Meas. Sci. Technol. 21 (2010) 015307 Y-L Chen and D-C Su
(a) (b)
Figure 9.(a) The area function of the P is depicted with the measured data (represented by dots) and the theoretical data (represented by a solid line); (b) their deviations.
conditions. Consequently, this method is suitable for measuring a Rockwell diamond indenter tip. The measurement resolution of these four results in figure4can be discussed as follows.
(1) Vertical measurement resolution. It depends on the scanning linearity of the PA. The PA with a closed-loop circuit is used in our tests and it has 0.15% linearity. So, the measurement resolution in our tests is z= 120 × 0.15%= 0.18 μm.
(2) Lateral measurement resolution. It depends on the diffraction limit of the microscopic system [21], and it can be expressed as
x = y ≈ 0.61λw
N A≈ 0.64 μm, (9)
where the central wavelength of the white light λw≈ 580 nm. For simplicity, a Rockwell diamond indenter tip is set as figure2where α= 90◦. Let the center of the sphere curvature be (0, 0, 0); then r is given as
r=x2+ y2+ z2. (10) The resolution of r can be written as
r=∂r ∂x · x +∂r∂y · y +∂r∂z · z = 1 r(|x| · x + |y| · y + |z| · z). (11) Substitute x, y and z into equation (11); then we have a best resolution r= 0.18 μm at (x, y, z) = (0, 0, r). θ can be determined by lines CE and DF in figure2, and their equations can be derived with the data of I1and I4. They can be expressed as z1= m1x1+ k1and z2= m2x2+ k2, respectively. Then we have θ= π − tan−1x1 z1 − tan−1x2 z2 . (12)
Consequently, the resolution of θ is derived and written as θ= 2 ·∂θ ∂x1 · x + 2 ·∂z∂θ 1 · z = 2 x2 1+ z21 (|z1| · x + |x1| · z). (13)
Substituting x and z into equation (13), the best resolution of cone angle can be evaluated as θ= 0.2◦where (x1, z1)≈ (L, 0).
5. Conclusion
In this paper, an alternative method for determining the tip topography of the Rockwell diamond indenter has been proposed by introducing the imaging stitching method into a scanning white light interferometer. Four image data at different angles and positions have been obtained with a scanning white light interferometer. Each result should have a partial overlap with the neighboring one for image stitching. Next, these data have been merged to form the associated topography with the image stitching method based on the iterative closest point (ICP) algorithm. Moreover, two crucial parameters, the tip radius at the center and cone angle along the outside area of the overall 2D topography, have been calculated by using the least-squares Gauss–Newton sphere and cone fitting algorithm. The validity of this method has been demonstrated. It has some merits such as nondestructive measurement, wide measurement range, easy operation, rapid measurement and low cost.
Acknowledgment
This study was supported in part by the National Science Council, Taiwan, ROC, under contract NSC95-2221-E009-236-MY3.
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