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The selected representative non-metallic materials, namely monocrystalline silicon and acrylic polymers, are considered to be brittle materials. It is difficult to generate a crack-free surface on brittle materials using mechanical machining based on their intrinsically low fracture toughness40, which causes the surface to fracture because of crack propagation. Therefore, the most challenging part of vibration texturing for structural coloration on non-metallic materials is to generate a predictable grating geometry with a high aspect ratio (grating depth over spacing), while maintaining suitable cutting conditions for a good surface finish.
Elliptical vibration-assisted texturing typically utilizes a relatively large nominal cutting velocity, as shown in Fig. 4b. The scallops formed by the overlapping tool trajectories create conjunctive microstructures on the material surface30. Each vibration cycle generates a discrete surface feature during texturing. The microstructure generation rate is equal to the vibration frequency. Hence, this texturing process can potentially lead to the high-efficiency texturing of gratings using a high-frequency vibration tool. Because each grating feature is machined in a separate vibration cycle, the trajectory amplitudes must be carefully determined, to ensure that (1) the material is removed without damage to the surface and (2) a high aspect ratio can be achieved with minimal interference between the tool flank and workpiece.
Fig. 4 Illustration of elliptical (2D) vibration cutting with different nominal cutting velocities vc:
a Small vc; b Large vc.As shown in Fig. 4a, with the elliptical vibration of a diamond tool forming an overlapping trajectory, the uncut chip thickness can be significantly reduced, especially under a relatively small nominal cutting velocity41. With a decrease in the undeformed chip thickness, the material removal mechanism of the silicon undergoes a transition from a brittle regime to a ductile regime42, where the material is removed through plastic deformation to create a crack-free surface. Conversely, elliptical vibration texturing utilizes a tool return motion in each vibration cycle to generate high-aspect-ratio features with extremely small spacings, by avoiding interference between the tool flank and workpiece. Based on these assumptions, the tool vibration trajectory can be theoretically and experimentally evaluated to optimize the cutting conditions.
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The tool vibration trajectory affects the geometry and surface finish of the generated microstructures. The key geometries of the generated grating-type microstructures are depth h and spacing d. Additionally, the angle of attack θ characterizes the instantaneous cutting direction of the tool when it begins to engage the workpiece, as shown in Fig. 4b. This angle affects the tool–workpiece interaction, and should be well controlled to avoid the effects of interference between the tool flank and material surface. The relationship between the process parameters and microstructure geometries, as well as the angle of attack, can be described as
$$ \left\{ {\begin{array}{*{20}{l}} {d = \dfrac{{{v_{\text{c}}}}}{f},h = b - b\cos \left( {2\pi f{t_1}} \right)} \\ {\theta = \arctan \left( {\dfrac{{\sin \left( {2\pi f{t_1}} \right) \cdot b/a}}{{\cos \left( {2\pi f{t_1}} \right) + {\text{HVR}}}}} \right)} \end{array}} \right. $$ (1) where a and b are the elliptical vibration amplitudes in the cutting and depth-of-cut (DOC) directions, respectively; f is the vibration frequency; vc is the nominal cutting velocity; HVR is the horizontal velocity ratio; and t1 is an intermediate time instance. The HVR and t1 can be derived from the following equations:
$$ \left\{ {\begin{array}{*{20}{l}} {{\text{HVR}} = \dfrac{{{v_{\text{c}}}}}{{2\pi fa}}} \\ {\sin \left( {2\pi f{t_1}} \right) + \left( {{\text{HVR}} - 2\pi } \right) \cdot \left( {2\pi f{t_1}} \right) = 0} \end{array}} \right. $$ (2) When a circular diamond tool with a nose radius R is used, the textured gratings are slightly moon-shaped. Because the DOC is much smaller than R, the width w of the grating can be calculated as
$ {{w = 2}}\sqrt {{{2R}} \cdot {\rm{DOC}}} $ . The lateral overlap of neighboring gratings in the cross-feed direction may induce minor defects on the machined surface, as shown in Fig. 1d. The lateral overlap of the neighboring gratings can be avoided by setting the cross feed to be slightly larger than w. However, based on the inevitable irregularity of the workpiece surface, the actual DOC may vary with position, resulting in the lateral overlap of the neighboring gratings. We further analyzed the optical images of the machined surface using a fast Fourier transform, as shown in Fig. 5. In the cutting direction, the major peak represents the grating spacing at 1 μm; meanwhile, the noise in the low-frequency range indicates random defects, which do not exhibit a particular periodic distribution. In the cross-feed direction, no significant peak values can be identified, indicating that the cross-feed marks are not apparent. -
Examples of typical textured results in the ductile and brittle regimes are compared in Fig. 6a, b, which reveal a drastic difference in the surface quality. The cutting chips obtained during ductile- and brittle-regime texturing are also presented in Fig. 6c, d, respectively, which represent distinct material removal modes. In the ductile regime, the cutting chips are strip-shaped, verifying that each feature is machined in a separate vibration cycle. In the brittle regime, the cutting chips are block-shaped. This indicates that the material is removed via uncontrollable crack generation and propagation, resulting in unacceptable surface quality. In addition to the surface quality and cutting chips, the material removal modes affect the tool life. Tool wear in brittle-regime texturing was much more rapid than that in the ductile regime. Moreover, tool edge chipping can be observed in brittle-regime texturing. In ductile-regime texturing, the tool gradually becomes blunt owing to abrasion.
Fig. 6 Comparison of ductile- and brittle-regime machining.
a Ductile- and b brittle-regime textured microstructures on silicon. Chips obtained in the c ductile- and d brittle-regime texturing of silicon.The identified critical depth that separates the ductile regime from the brittle regime is plotted against the vibration amplitude a, grating spacing d, and HVR in Fig. 7a, b. A larger vibration amplitude a in the cutting direction is beneficial for an increase in the critical depth or ductile cutting range. For a fixed vibration frequency, the critical DOC sharply decreases with an increase in the nominal cutting velocity, but does not exhibit apparent dependence in conventional diamond machining. Additionally, for a fixed vibration frequency, when the nominal velocity is small, the critical DOC is significantly enhanced, similar to the case of conventional elliptical vibration cutting. The critical DOC converges to the case without vibrations with a larger velocity.
Fig. 7 Evaluation of process parameters.
a Identification of the critical ductile DOC. b Effects of the HVR on the critical ductile DOC. c Effects of the vibration amplitude in the cutting direction on the angle of attack. d Effects of the vibration amplitude in the cutting direction on the grating depth.The interactive effects of the vibration amplitude and nominal cutting velocity on the critical ductile DOC are presented in Fig. 7b. The experimental data for the critical ductile DOC were fitted using a power function of
$ \text{DOC}\text{}\;{=}\;\text{}\text{0.123}\text{}\; \times\; \text{}{\text{HVR}}^{{-}\text{0.24}} $ with an R-squared value of 0.6. The critical ductile DOC is inversely proportional to the HVR, whereas the grating depth is positively related to the HVR. Therefore, a set of contradictory conditions exists for balancing the enhanced critical depth and high aspect ratio of the gratings. The nominal cutting velocity is typically determined by the functional requirement (grating spacing), which cannot be set arbitrarily. Therefore, given a fixed structure spacing, an increase in the vibration amplitude a will enhance the ductile cutting range, but decrease the grating depth or its aspect ratio. Notably, when fabricating grating structures via elliptical vibration texturing, the critical DOC is not significantly improved compared to cutting without vibration, unlike conventional elliptical vibration cutting.To maintain the ductile-regime texturing of silicon during the entire cutting cycle, two conditions must be satisfied. First, interference between the tool flank and workpiece must be avoided to prevent impact-induced crack generation. Second, the generated structure depth must be maintained within the critical ductile DOC. These two conditions are further analyzed in Fig. 7c, d, based on Eq. 1. Fig. 7c presents the dependence of the angle of attack on the microstructure spacing and vibration amplitudes. The angle of attack θ increases with an increase in the microstructure spacing or a decrease in the vibration amplitude a. It should be kept smaller than the clearance angle α to avoid impact. For a clearance angle α = 10°, the vibration amplitude in the cutting direction should be greater than or equal to 2 μm, to ensure an angle of attack θ < 10°.
The elliptical trajectory also significantly affects the microstructure depth. As shown in Fig. 7d, the grating depth h increases with an increase in the grating spacing d and a decrease in the vibration amplitude a. From this perspective, a smaller vibration amplitude is preferred for a higher aspect ratio. However, the criteria for ductile-regime cutting must also be satisfied. In the lower-bound case, the grating depth should be kept smaller than the critical DOC (approximately 180 nm), as indicated by the dashed line in Fig. 7d.
The elliptical vibration trajectory should be optimized by considering both the angle of attack (to avoid interference) and critical DOC (to maintain ductile-regime cutting), to achieve the largest possible aspect ratio. According to this analysis, the elliptical vibration trajectory for the ductile-regime texturing of silicon was optimized as a = 2 μm and b = 1 μm.
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Intuitively, grating-type microstructures can also be generated using simple 1D cutting depth modulation. As shown in Fig. 8, 1D vibration texturing modulates the tool only in the DOC direction; however, in 2D vibration texturing, additional reciprocal motion is added in the cutting direction. In this study, we found that elliptical vibration-assisted texturing exhibited unique advantages in terms of enhanced control of microstructure geometries and surface finish compared to conventional 1D vibration-assisted texturing.
Simple cutting depth modulation in 1D vibration-assisted texturing will typically result in severely cracked surfaces, owing to the dependence of the grating depth on the nominal cutting DOC, as well as the high angle of attack, thus resulting in flank impact in each cycle. As shown in Fig. 8, the depth h of 1D vibration-textured gratings is sensitive to the changes in the DOC; hence, it is difficult to remain within the critical ductile-regime DOC and obtain repeatable results, considering the inevitable misalignment of the workpiece. Additionally, the angle of attack is significantly larger than that in the 2D case, leading to large tool–workpiece interference that may create a poor surface finish. Overall, the tool is excessively large to scrape into the intended pits in 1D vibration-assisted texturing.
However, in elliptical (2D) vibration-assisted texturing, additional vibration in the cutting direction enables the return motion of the tool, which significantly reduces the angle of attack. Additionally, the generated grating depth h is independent of the nominal DOC. Therefore, the process conditions are consistent, provided the grating spacing is fixed. As shown in Fig. 8, in 2D vibration texturing, the angle of attack is significantly reduced compared to that in the 1D case. It also decreases with a decrease in the grating spacing, which is beneficial for generating microstructures with extremely small spacing.
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The vibration-assisted texturing process can produce structurally colorized polychromatic images with a high resolution. The fabricated periodic grating-type microstructures can be used to disperse incident white light to provide structural coloration. As shown in Fig. 9a, the perceived color depends on the incident and observing angles. This relationship is governed by the following diffraction equation.
Fig. 9 Principle of structural coloration using vibration-assisted texturing.
a Principle of white-light dispersion. b Grating spacing modulation in elliptical vibration texturing.$$ d = \frac{{k \cdot \lambda }}{{\sin {\theta _{\lambda ,k}} + \sin {\theta _{in}}}} $$ (3) where λ is the light wavelength; θi and θλ,k are the incident and diffraction angles, respectively; and k is the diffraction order. According to Eq. 3, different light wavelengths have different diffraction angles for a given grating spacing. Therefore, the reflected or transmitted spectrum of incident white light can be dispersed to produce colorful visual effects. As indicated by Eq. 3, different light colors can have identical diffraction angles, by accurately controlling the grating spacing. To render a polychromatic image, the grating spacing can be determined at each position according to the desired image pattern under a given observation direction (fixed θλ,k). During vibration texturing, according to Eq. 1, pixelated patches of the gratings with various spacings can be created by dynamically modulating the nominal cutting velocity at each interval, as shown in Fig. 9b.
To determine the color appearance of a textured surface, we modified the scalar diffraction theory to calculate the spatial distribution of the light intensity first43−44. According to the scalar diffraction theory45, for one pixel in the diffraction element with n gratings, as shown in Fig. 10, the distribution of irradiance intensity on a far-field plane from the diffracting aperture can be expressed as
$$ I(\lambda ,\psi ) = \cos {\theta _{in}}{\left| {{\cal{F}}\{ U(y)\} } \right|^2} $$ (4) where I is the diffracted light intensity, ψ is the observation angle relative to the normal direction of the substrate plane, λ is the wavelength of light, and y is the lateral coordinate along the substrate surface, the direction of which is aligned with the nominal cutting direction. U(y) is the distribution of the complex amplitude of the light field that emerges from the diffractive aperture, which is described as
$$U(y)={{e}^{{}^{-4\pi ih(y)\sin {{\theta }_{in}}}\diagup{}_{{{\lambda }^{2}}}\;}}\left( 0\le y\le nd \right)$$ (5) where h(y) describes the surface profile of a one-pixel diffraction element with n gratings.
$ {\cal{F}} $ denotes the Fourier transform operator, and is defined as$$ {\cal{F}}\{ U(y)\} = \Delta x \cdot \int_{ - \infty }^\infty {U(y){e^{ - 2\pi iy}}{\text{d}}y} $$ (6) where Δx is the width of gratings in the cross-feed direction.
The diffracted light intensities I (λ) for different observation angles ψ can be derived from Eqs. 4–6. Further, the apparent color can be obtained through RGB color conversion, as46
$$ \left\{\begin{array}{c}X={\int }_{\lambda }^{}I\left(\lambda \right)\overline {x}\left(\lambda \right)d\lambda \\ Y={\int }_{\lambda }^{}I\left(\lambda \right)\overline {y}\left(\lambda \right)d\lambda \\ Z={\int }_{\lambda }^{}I\left(\lambda \right)\overline {z}\left(\lambda \right)d\lambda \end{array}\right., \;\;\lambda \in \left[\mathrm{380,780}\right] $$ (7) where
$ \overline {\text{x}} $ ,$ \overline {y} $ , and$ \overline {\text{z}} $ are the CIE 1931 2° standard color-matching functions with a Gaussian approximation47, 48. The obtained XYZ of each corresponding pixel is converted into the sRGB space using the MATLAB function xyz2rgb. Then, the sRGB values are scaled to a range of 0–255 for screen display. In our simulations, the light source was considered to be a daylight source, where the intensity distributions for different wavelengths were approximately uniform. A comparison of the experimental and simulated appearances of the structural coloration results is presented in Fig. 2, which validates the proposed simulation model. -
The experimental setup is presented in Fig. 11. An ultrafast 2D non-resonant vibration cutting tool, with a frequency bandwidth of up to 6 kHz and full stroke of up to 10 × 10 μm, was used to generate the controllable elliptical vibration trajectories49. A direct-drive linear stage and three-axis stage (Aerotech, USA) were used to perform the cutting and cross-feed motions, respectively. A level adjustment stage was used to control the inclination angle of the workpiece. Two types of workpieces were utilized: monocrystalline silicon (crystal plane of (100)) and acrylic polymer. A single-crystal diamond insert (Contour Fine Tooling, UK) was attached to the vibration tool with a nominal rake angle γ0 of −20°, clearance angle α of 10°, and nose radius R of 470 μm.
Fig. 11 Experimental setup.
a Experimental device. b Grooving schematic. c Optical image of grooving result. d Relationship between the cutting depth and groove width.Grooving experiments were performed to identify the characteristic critical cutting depth or ductile-to-brittle transition distance of silicon for the conventional and vibration-assisted diamond texturing methods. The detailed experimental parameters are listed in Table 1. The silicon workpiece was inclined at an angle of 2 μm/mm in the crystal direction of <110>. Different cutting velocities ranging from 0.2 to 4 mm/s were used to scratch the workpiece with a gradual increase in the DOC, because of the inclination angle, as shown in Fig. 11b. The machined grooves were observed using a digital microscope (RH-2000, Hirox, Japan). The captured surface morphology is presented in Fig. 11c, from which the distinct transition of the material removal states can be identified. According to the relationship between the cutting width w and DOC, as shown in Fig. 11d, the critical cutting depth in the ductile regime can be obtained to optimize the vibration trajectory. Finally, polychromatic image rendering on both the silicon and acrylic was performed using the optimized vibration trajectory.
No. a (μm) b (μm) vc (mm/s) 1 Without vibration 0.2, 0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4 2 0.5, 1, 2, 3, 4 1 0.2, 0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4 3 2 1 1.5, 2, 2.5, 3, 3.5, 4, 8 Tool vibration frequency f = 2000 Hz Table 1. Process parameters used in the experimental design for vibration trajectory optimization.
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Two types of optical systems were used to observe and record the structurally colored workpiece surfaces, after machining based on the different transparencies of silicon and acrylic. The silicon was measured in a reflection configuration, whereas the acrylic sample was characterized using a transmission setup. As shown in Fig. 12a, for the silicon surface textured with reflection gratings, the incident white light is on the same side as the digital camera. However, for the acrylic surface textured with transmission gratings, as shown in Fig. 12b, the incident white light is on the opposite side of the digital camera. However, for both optical systems, the optical axis of the digital camera was set to be perpendicular to the workpiece surface.
Structural coloration of non-metallic surfaces using ductile-regime vibration-assisted ultraprecision texturing
- Light: Advanced Manufacturing 2, Article number: (2021)
- Received: 14 May 2021
- Revised: 24 November 2021
- Accepted: 29 November 2021 Published online: 24 December 2021
doi: https://doi.org/10.37188/lam.2021.033
Abstract: Structural coloration stemming from microstructure-induced light interference has been recognized as a promising surface colorizing technology, based on its potential in a wide array of applications, including high-definition displays, anti-counterfeiting, refractive index sensing, and photonic gas and vapor sensing. Vibration-assisted ultraprecision texturing using diamond tools has emerged as a high-efficiency and cost-effective machining method for colorizing metallic and ductile surfaces by creating near-wavelength microstructures. Although theoretically possible, it is extremely challenging to apply the vibration-assisted texturing technique directly to colorize non-metallic and brittle materials (e.g., silicon and acrylic polymers) with high-quality, crack-free microstructures owing to the intrinsic brittleness of these materials. This study demonstrates the feasibility of direct texturing near-wavelength-scale gratings on brittle surfaces in the ductile regime to fabricate crack-free micro/nanostructures. The effects of tool vibration trajectories on the ductile-to-brittle transition phenomena were investigated to reveal the cutting mechanism of ductile-regime texturing and optimize the processing windows. Structural coloration on silicon and acrylic surfaces was successfully demonstrated by creating programmable and pixelated diffraction gratings with spacing values ranging from 0.75 to 4 μm.
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