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Abstract
Infrared micro-lens arrays (MLAs) are widely used in advanced optical systems due to their advantages such as low focusing depth and high sensitivity. Elliptical vibration cutting (EVC) is a promising approach for the fabrication of MLAs on infrared brittle materials. However, the mechanism of ductile machining of MLAs prepared by EVC has not been fully elucidated so far. In this paper, based on the vibration intermittent cutting characteristics and the transient material removal state, a ductile machining model of MLAs on brittle material by EVC was established. This model effectively calculates the subsurface damage of the machined surface and realizes the prediction of the critical depth for ductile machining of MLAs. Furthermore, the concave micro-lenses were prepared on single crystal silicon by EVC and ordinary cutting (OC) to verify this model. The results demonstrated that EVC could significantly enhance the critical depth by approximately 4.3 times compared to OC. Microstructural surface damage predominantly occurs at the exit side of the tool cutting. This proposed model accurately predicts the actual critical depth, with an average error of about 7.5%. Additionally, elevating the amplitude in the depth of cut direction could increase the critical depth, but a larger amplitude would inhibit the increase of the critical depth. This study contributes to a better understanding of ductile machining of microstructure on brittle materials and facilitates the process optimization of MLAs fabrication using EVC.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Infrared micro-lens arrays (MLAs) have been widely used in infrared optical systems such as integral imaging [1], beam shaping [2], and optical sensing [3] due to their advantages such as low depth of focus, miniaturization, and good light uniformity. MLAs present a discontinuous microstructure surface composed of a plurality of periodically arranged lens units [4]. The substrate materials are generally brittle materials such as silicon, germanium, and zinc selenide [5,6]. The surface quality of MLAs directly affects whether they can play a better role in the corresponding application field, and the uneven surface will lead to imaging distortion and scattering effects [7]. Fabricating micro-structured surfaces on brittle materials with smooth and uniform is a challenge when the structural size of the designed surface is reduced to the micron or nanometer scale. At present, the methods that can be used to prepare MLAs include photolithography [2], hot embossing [8], femtosecond laser direct writing [9], etc., but they all have shortcomings such as uncontrollable precision and poor consistency between microstructure units. Diamond cutting is an effective method for preparing micro-structured surfaces due to its strong adaptability to machinable geometries and nanoscale surface roughness [10]. However, brittle fracture and excessive tool wear are prone to occur during brittle materials cutting, primarily due to their low fracture toughness [11]. To achieve a micro-structured surface with superior quality, it is imperative to uphold a ductile machining state throughout the cutting process.
Brittle material removal is categorized into two states: brittle and ductile. In the brittle state, the removal of material is achieved by crack initiation and extension. While the material is removed by plastic flow in the form of shear in the ductile state [12]. The transition from a brittle state to a ductile state can be attained at a smaller depth of cut (DOC), known as the critical ductile-brittle transition (DBT) depth [13]. Critical DBT depth and subsurface damage (SSD) depth are two essential critical thresholds for ductile machining [14]. Ductile machining can only be achieved when DOC is smaller than the critical DBT depth or when the SSD depth is less than 0. SSD depth is usually defined as the depth at which the crack tip penetrates the target surface. Bifano et al. [15] established a critical DBT model based on the Griffith fracture extension criterion [16], which focuses solely on the workpiece material properties. Further, Muhammad et al. [17] introduced a model utilizing material removal energy to forecast the critical DBT depth, considering material characteristics, machining parameters, and tool morphology. Zhang et al. [18] put forward a surface quality regulation method for diamond cutting of germanium with controlled DOC, and developed a critical DBT depth prediction model in different crystal orientations. Blackley and Scattergood [19] presented a quantitative analysis model for SSD depth by examining the tool morphology and fracture damage characteristics of materials. Yu et al. [20] proposed a method to predict the SSD depth by calculating the surface damage region of the machined surface to prepare damage-free micro-structured surfaces. Under the strict limitation of machining parameters, although diamond cutting can prepare silicon components in a ductile mode, the critical DBT depth is only more than 100 nm [21], and the micro-lens surface with a high aspect ratio is difficult to machine.
Elliptical vibratory cutting (EVC) technology has the potential to substantially enhance the machinability of brittle materials and is an effective method for preparing microstructures [22]. EVC utilizes intermittent high-frequency vibration to assist in material removal, which can significantly reduce cutting forces and heat, thereby improving the machinability of brittle materials. Shamoto et al. [23,24] conducted EVC experiments using a diamond tool, and confirmed the practicality of ductile machining of brittle materials like silicon and tungsten carbide. Zhang et al. [25] used EVC to carry out single crystal silicon cutting experiments. Compared with OC, the ductility machining ability has been greatly improved, the critical DBT depth has been increased by 12.5 times, and two high-precision microstructures of sinusoidal grids and dimples have been successfully prepared on silicon. Based on Muhammad’s model [17], Zhang et al. [26] proposed a cutting energy model suitable for EVC to predict the critical DBT depth in silicon cutting. Huang et al. [27] clarified the ductile machining mechanism of EVC by analyzing the geometric relationship of vibration cutting trajectory. It was found that the increase in the critical depth was attributed to the gap between the target surface and the transient cutting surface can withstand a certain amount of crack extension. The aforementioned prediction of critical depth is broadly suitable for cutting grooves and planes, but it is difficult to apply to micro-lens machining. Due to the micron-scale size and complex surface shape of the microstructure surface, the removal of the workpiece material is in a state of ductile and brittle coupling [28]. Micro-structured surface machining theoretically has a critical structural depth below which cracks will not propagate below the target surface, and a crack-free surface can be achieved [29]. Hence, the critical depth is determined from the perspective of SSD depth. Xing et al. [30] proposed a ductile machining model of EVC considering the ductile-brittle coupling state, using the SSD depth as a quantitative index to assess the feasibility of achieving a damage-free surface, and applied this model to fabricate an odd cosine structure on silicon. Liu et al. [31] proposed a self-tuning EVC and established a ductile machining model to ensure that transient cutting cracks do not extend to the target surface by adaptively adjusting the feed rate to meet the local shape change of the required surface. Although the above studies have achieved ductile machining of microstructures by predicting the SSD depth. However, there is no integration of the transient material removal state with the vibration trajectory to determine the critical structural depth for microstructure machining. In addition, ductile machining of the concave micro-lens prepared by EVC has not been modeled and analyzed.
Therefore, this paper presents a cutting depth-oriented ductile machining model for infrared MLAs prepared by EVC. The transient material removal characteristics are combined with the vibration trajectory to predict the critical depth of the concave microstructure surface. Furthermore, to validate the predictive model, a series of variable depth concave micro-lens cutting experiments are conducted on single crystal silicon. Finally, the variation of the critical depth of micro-lens for different cutting parameters was analyzed and discussed.
2. Theoretical analysis
2.1 Principle of preparing MLAs by EVC
Comprehending the machining mechanism of EVC constitutes a foundational procedure. The machining principle of MLAs prepared by EVC is shown in Fig. 1. To analyze the effect of DOC on the ductile machining of micro-lenses, one-cut forming is adopted here, i.e., concave micro-lenses are processed in one time, and the lens depth is equal to DOC. The dimple shape is generated by the elliptical cutting path envelope, as shown in Fig. 1(a). The width and length of micro-lenses are shown in Fig. 1(b) and (c). The coordinate system for the cutting path of tool tip σe(oe-xe,ye,ze) moves along the trajectory of the vibration center. The coordinate system σ0(o-x0, y0, z0) is the target contour fixed at the vibration center of the initial cutting cycle. d is the concave microstructure depth, R denotes the tool radius, and the cutting speed is vc.
Fig. 1. Process diagram of the concave micro-lenses prepared by EVC. (a) Envelope forming process of the dimple shape during EVC. The width (b) and length (c) of the machined micro-lens, respectively.
The target surface is discretized in the cutting direction, and each segment is separated by the vibration period δc. x0 is the position of the dimple in the feeding direction. As a result, the equation representing the actual target surface is given by:
In the case where the target surface takes the form of a spherical contour, the z0-axis aligns along the vibration axis, the y0-axis represents the tool cutting direction, and the x0-axis aligns the center axis of the dimple. The target surface equation is formulated as:
where, Ll is the length of the fabricated dimple, b is the mean-to-peak vibration amplitude in the DOC direction.Drawing upon the disseminated target contour, the tool is intermittently cut along the Y-axis, and the tool tip trajectory equation can be represented as:
The target contour is formed by the envelope of the elliptic vibration trajectory, and the center of the elliptic vibration trajectory moves along the cutting direction, then the steps for solving the elliptic vibration center trajectory are as follows: (1) The phase of the elliptical trajectory is determined by calculating the intersection of the tool cutting trajectory and the target contour in each vibration cycle. (2) The unit tangential vector at the intersection of elliptical vibration trajectory and target contour is obtained. Assuming that the two tangential vectors coincide, and the actual vibration center is Oe [ye0 (i), ze0 (i)].
2.2 Instantaneous undeformed chip thickness
Figure 2 provides a schematic depiction of the correlation between the movement of the tool and the vibration trajectory under a single cutting period. During the vibration cutting process, the actual DOC changes with the vibration trajectory every moment. The tool initiates machining from point A, which remains on the machined surface from the preceding vibration period, and reaches point B, point C in turn, and finally ends at point D. M represents the instantaneous tool position. The transient time of points A, C, D, and M can be expressed as tA, tC, tD, and tM, respectively.
Referring to the geometric relationship depicted in Fig. 2, the actual DOC apt at the instantaneous point M can be expressed as:
Fig. 2. Geometric depiction illustrating the relationship between tool and material in a single cutting cycle of EVC.
The angle ∠EDF between the direction of tool movement at point D and the z-axis is the same as the rake angle of the tool. Hence, tCcan be represented as:
The connecting line of point C with point D in the preceding cutting period is parallel to the rake surface of the tool, tDcan be defined as:
The nominal DOC in each cutting period is represented as:
In the preceding cutting period, the point in time when the tool passes through the point M’ is recorded as tM’, and the value of tM’ can be obtained by the Newton-Raphson method through Eq. (9).
Considering that the DBT depth for monocrystalline silicon is approximately 100 nm. In the actual machining of micro-lens, the primary material removal process involves a coupled mechanism of ductile and brittle behavior, characterized by plastic deformation and brittle fracture. This phenomenon is depicted in Fig. 3. To realize ductile machining of micro-lens, the transient removal state of material is analyzed along the cutting trajectory. This leads to the determination of the SSD depth and the critical depth required for ductile machining.
In the i-th cutting period, the contour equation along the x0-axis can be denoted as:
Here, $- {L_w}/2 + {l_X} < {x_0} < {L_w}/2 - {l_X}$.
The instantaneous undeformed cutting thickness uct in the feeding direction under each cutting cycle can be represented as:
here, $- \arccos [1 - {a_{pt}}(i)/R] < \theta < \arccos [1 - {a_{pt}}(i)/R]$.2.3 Ductile machining model of MLAs prepared by EVC
Workpiece material is removed by plastic deformation when the instantaneous uct is less than the critical DBT depth uc. When uct is greater than uc, crack initiation and propagation will commence, leading to the removal of brittle material through brittle fracture. uc can serve as the threshold for initiating crack formation, which can be expressed as [13]:
where, ψ is assumed to be 0.9 [30]. H represents the material hardness, while E denotes the material elastic modulus. KC represents the fracture toughness.The depths of crack initiation and lateral crack (see Fig. 4(a)) are described as [32]:
Fig. 4. Schematic diagram of SSD depth calculation. (a) Crack system in transient cutting process. (b) geometric relationship between target profile and crack extension.
In the cutting process, half-cutting tool angle α is denoted as [33]:
here, ς signifies the clearance angle of the tool. γ0 represents effective rake angle.By approximating the plastic deformation area with a circular shape. The plastic deformation area radius rp is formulated as [34]:
With the lateral crack depth Ch being approximately equal to the plastic deformation area radius (Ch = rp) [35]. The calculation of the cutting force is described as:
To predict the SSD depth under the target surface, as depicted in Fig. 4(b), it is necessary to determine the crack endpoint (x(i) n, z(i) n) at each position of the cutting cycle i as:
Hence, the SSD depth can be represented as:
To ensure the attainment of a damage-free surface on the target surface, it should be ensured that SSD satisfies SSDmax < 0 for each vibration cycle.
The critical depth of ductile machining of micro-lens prepared by EVC is calculated as shown in Fig. 5. Firstly, the target surface is discretized, the elliptic vibration centers at each position are determined, and the vibration cutting envelope locus is obtained. Then, apt and uct for a single cutting period are calculated. Further, the SSD at each location is calculated until all locations are calculated to obtain the subsurface damage distribution. Finally, by adjusting the critical depth d and iteratively calculating until the maximum SSD is less than 0, the critical depth dc for achieving ductile machining of micro-lens through EVC is obtained.
3. Experiments details
3.1 Materials and experimental setup
The microstructure preparation experiments were conducted on monocrystalline silicon (001). The workpiece dimensions were 25.4 mm in diameter and 3 mm in thickness. The main performance parameters of the materials are presented in Table 1. The cutting direction is along the crystal orientation <110 > of the single crystal silicon. In order to verify the predicted critical depth, a row of dimples with gradually increasing depths was prepared, the pitch of each dimple was 200 µm, and the depth difference between two adjacent dimples was 50 nm. Because it was a one-time cutting, the radius of curvature R is the radius of the tool.
Table 1. The material characteristics of monocrystalline silicon
3.2 Design of experiments
The experiment was conducted using a four-axis ultra-precision machine tool (Nanoform X, USA), the fixture loaded with monocrystalline silicon workpiece is sucked on a vacuum chuck, as depicted in Fig. 6. The 2-DOF vibrator (EL-50∑, Taga Electric Co., Ltd) is fixed on the z-axis. The vibration frequency of the vibrator is 41.6 kHz, and the vibration amplitudes in the cutting direction and the DOC direction are both 0 ∼ 4 µm. The micro-lenses are produced utilizing the slow tool servo (STS) technology. Each dimple is prepared in one-time cutting. The cutting path is generated using the CAM software DIFFSYS. The experiment adopts constant linear speed cutting, that is, the speed of the C-axis changes automatically with the feed process. DOC of each dimple gradually increases with the feed, and the increment is 50 nm each time. Due to the depth of the micro-lens used in practice is generally micron, and it is sufficient to determine the critical depth with an interval of 50 nm. The variable depth micro-lenses are prepared by EVC and OC. The experimental parameters for the fabrication of concave microstructure are detailed in Table 2. White light interferometry (Zygo Newview 9000) and scanning electron microscopy (SEM, JEOL JSM5500LV) were used to measure and characterize the surface morphology of micro-lenses.
Fig. 6. Diagram of the experimental setup. (a) Layout diagram of vibration device on ultra-precision machine tool. (b) Detail diagram of tool cutting the workpiece. (c) Schematic diagram of the dimple machining.
Table 2. The experimental parameters for the fabrication of microstructure
4. Results and discussions
4.1 Evolution of actual DOC in MLAs prepared by EVC
By substituting the machining parameters set in the experiment into Eq. (5), the actual DOC in the micro-lens preparation process using EVC can be ascertained, and the corresponding calculation outcomes are illustrated in Fig. 7. In the cutting process, the tool undergoes a series of vibratory cutting cycles along the cutting path, resulting in the formation of the dimple surface. The actual DOC apt gradually increases from 0 to its maximum value as the nominal DOC increases, reaching its peak at the deepest part of the dimple (the maximum nominal DOC). Subsequently, apt gradually reduces from its maximum to 0 as the nominal DOC increases. Notably, it can be found that apt in EVC is considerably smaller compared to the nominal DOC in OC within the identical cutting period.
Fig. 7. Actual DOC apt within a cutting cycle at different dimple depths. The variation in apt along the target profile for an amplitude a-b of 2-1 µm0-p when (a) d = 0.8 µm, d ≤ b and (b) d = 1.2 µm, d > b.
Since apt is directly correlated with both the position of the tool cutting trajectory and the dimple depths, the evolution of the actual DOC differs for various dimple depths, as shown in Fig. 7(a) and (b). This can be categorized into two cases: (a) when the dimple depth is less than the vibration amplitude b in the DOC direction, the actual DOC presents a pattern of initially increasing and subsequently decreasing within a single vibration cycle. From the starting cutting point A to point C, the actual DOC gradually increases to its maximum. During the transition from point C to point D, the actual DOC gradually decreases to 0. (b) when the dimple depth surpasses the vibration amplitude b, an abrupt rise in the actual DOC from point C to its peak, followed by a gradual decline, and becomes 0 abruptly when the tool moves to point D. It can be inferred that instantaneous cutting is more prone to producing cracks between point C and point D during a single cutting cycle.
4.2 Surface morphology and critical depth characterization of MLAs
According to the machining parameters in Table 2, the micro-lenses with variable depth processed by EVC are shown in Fig. 8. It is apparent that the size of the dimple increases along the feeding direction as the depth of micro-lens increases, leading to two distinct scenarios. These scenarios are classified as ductile and brittle machining, distinguishing between the distinct material removal modes on the micro-lens surface, respectively. Upon observation of the dimple surface, it can be noted that when the depth is less than 0.65 µm, the dimple surface remains smooth without any cracks. During this stage, the material removal mode on the machined surface is attributed to plastic deformation, and the micro-lens preparation process is called ductile machining according to the definition [14]. However, as the depth surpasses 0.65 µm, damages gradually appear on the dimple surface. At this point, the material removal mode is marked by a coupled state of plastic deformation and brittle fracture, and the micro-lens preparation process is termed as brittle machining. The initiation of subsurface damages is attributed to the length of crack propagation exceeding the target profile. As the depth continues to increase, the damage density gradually increases, and finally a large piece of material peels off. The structural depth at which the micro-lens surface machining mode is transformed from ductile machining to brittle machining is defined as the critical depth for ductile machining of microstructures. In this work, for micro-lens prepared by EVC, the critical structural depth dc for accomplishing ductile machining is determined to be 0.65 µm.
Fig. 8. Surface morphology characterization of micro-lens prepared by EVC with an amplitude a-b of 2-0.5 µm0-p. (a) Optical micrograph of the dimples. (b) 3-D surface topography of the dimples. (c) Cross-section of the dimples in (b).
To further illustrate the progression from ductile machining to brittle machining, the dimples near the critical depth in Fig. 8 are enlarged and compared with the micro-lens prepared by OC, the results are shown in Fig. 9. From Fig. 9(a), it is evident that at a dimple depth is 0.65 µm, there is no damage on the dimple surface, and the damage (indicated by black spots) begins to appear from a depth of 0.7 µm, and gradually increasing with further depth increment. In comparison, for OC (see Fig. 9(b)), the dimple surface with a depth of 0.15 µm is smooth without damage, and the damage begins to appear from a depth of 0.2 µm. The outcomes demonstrate that EVC can enhance the critical depth for ductile machining of micro-lens by about 4.3 times, thereby making a substantial improvement in the quality of the microstructure surface. This improvement is primarily attributed to the intermittent cutting of the workpiece by the tool along the elliptical trajectory in EVC. On the one hand, apt is less than the nominal DOC, resulting in a reduced instantaneous undeformed chip thickness uct and consequently a decrease in crack expansion depth. On the other hand, as depicted in Fig. 2, when the tool passes through point B, there is a certain distance of interval between the instantaneous cutting position of the tool and the target contour. The interval distance gradually increases with the continuous cutting of the tool until point D reaches the maximum. This distance acts as a buffer space for crack growth, providing a certain resistance against crack propagation.
Fig. 9. Detailed description of the critical depth of the dimples. (a) Dimples prepared by EVC. (b) Dimples prepared by OC.
Figure 10 characterizes the machining damage on the dimple surface at different depths. It confirms that the formation of damage initiates at a structural depth of 0.7 µm, which is consistent with the observations in Fig. 8 and Fig. 9. The damage enlargement diagram in Fig. 10 reveals the presence of parallel features in the surface damage. This observation suggests that when uct exceeds the critical DBT depth, cracks are generated along the cleavage plane. Notably, the angle between the cleavage plane and the material crystal plane (001) is near the angle (54.7°) between the crystal plane (001) and (111), as shown in Fig. 11(a). Consequently, the observed damage on the dimple surface results from cleavage fractures along the crystal plane (111). As the depth increases, the cleavage fracture phenomenon gradually intensifies. Additionally, Fig. 10 demonstrates a notable prevalence of cleavage fractures occurring at the exit side of the tool cutting, corresponding to the left side of the dimples. This behavior can be attributed to the cutting state of the tool during the entry and exit phases. Specifically, the tool cutting state on the entry side is downhill machining, while on the exit side, it is uphill machining. Under the same DOC, the distance between the instantaneous tool position and the target contour in uphill machining is smaller than that in downhill machining (see Fig. 11(b)). Consequently, it is more probable for damage to occur at the exit of the tool cutting.
Fig. 11. (a) Relationship diagram between the machined surface (100) and cleavage surface (111). (b) Comparison of damage between uphill and downhill machining.
To further characterize the surface topography of the dimples prepared by EVC at different depths, a consistent area at the bottom of the dimples was selected for measuring the surface roughness Sa by white light interferometer. As shown in Fig. 12, the surface roughness varies with increasing dimple depth. A noticeable trend is observed wherein the surface roughness deteriorates gradually with the increase in dimple depth. When the dimple depth is below the critical depth dc, the surface roughness changes relatively smoothly with increasing dimple depth. For instance, as the dimple depth increased from 0.4 µm to 0.6 µm, there was an increase in the surface roughness Sa from 4.881 nm to 6.163 nm, resulting in a relative growth rate of 6.415. However, when the dimple depth exceeds the critical depth, the surface roughness increases significantly. Indeed, a notable trend can be observed, where the surface roughness Sa experienced a rapid increase as the dimple depth expanded from 0.7 µm to 1.4 µm, escalating from 9.633 nm to 27.148 nm. The relative growth rate was 25.021. This sharp increase in surface roughness indicates a gradual deterioration in surface quality. This indicates that the ductile machining of the microstructure maintains a high surface quality, whereas brittle machining leads to inferior surface quality. The significant contrast in surface characteristics can be attributed mainly to the absence of cracks and pits during ductile machining, while brittle machining results in a plethora of cleavage fracture damages, ultimately leading to deteriorated surface roughness.
Fig. 12. The surface characteristics of the concave micro-lens prepared by EVC changes with the dimple depth.
4.3 Subsurface damage calculation and model verification
Along the cutting path, the SSD at the lowest end of the dimple is calculated as shown in Fig. 13. It can be found that SSD starts at 0 and then abruptly drops to a negative value. As the cutting depth increases, SSD gradually rises to its maximum value, followed by a gradual decrease until it reaches 0 again. This behavior is attributed to the variation in the nominal DOC (first increases and then decreases) along the cutting path during the micro-lens cutting. The maximum uct is less than the critical depth, resulting in SSD being 0 on both the entry and exit sides of the tool. This is because the nominal DOC is relatively small during this phase. However, as the maximum uct exceeds the critical depth as DOC increases, the crack propagation begins to expand. When the crack propagation length is less than the clearance between the instantaneous cutting position and the target contour, SSD becomes negative. As DOC continues to increase, cracks further expand until their length exceeds the clearance between the instantaneous cutting position and the target contour, resulting in the presence of cracks on the machined surface. As a result, SSD becomes positive and continues to increase. When SSD reaches a certain extent, it starts to decrease because the DOC gradually decreases. Furthermore, the maximum subsurface damage tends to occur on the exit side of the tool cutting, and the SSD distribution tends to shift towards the right side along the cutting path. This is primarily due to the exit side of the tool being associated with uphill machining, resulting in a smaller gap between the instantaneous cutting position and the target contour compared to the entry side under the same DOC.
The calculation of the SSD at each position on the dimple surface prepared by EVC can predict the theoretical damage distribution area on the dimple surface. The SSD is calculated for different dimple depths, and the outcomes are depicted in Fig. 14. It is evident that the region of damage distribution expands with the increase of dimple depth d. When d is below the critical depth dc for ductile machining, the entire surface of the dimple surface is formed by plastic deformation, indicating a relatively uniform surface with no damage. However, as d continues to increase, when d is greater than dc, the damage gradually appears from the bottom, and the dimple surface is formed by ductile-brittle coupled machining. The appearance of damage on the dimple surface leads to an enlargement of the damage distribution area. In addition, it can be found that the damage distribution area is biased towards the exit side of the tool cutting, so the exit side is more prone to damage. This observation aligns with the actual machining results depicted in Fig. 10.
Fig. 14. Prediction of the theoretical damage distribution area in the dimple surface with different depths.
Based on the model established in Section 2, theoretical calculations predict that the critical depth dc of the micro-lens prepared by EVC is 0.62 µm, with an amplitude a-b of 2-0.5 µm0-p and a cutting speed of 200 mm/min. The measured critical depth dc is found to be 0.65 µm, demonstrating good agreement with the predicted value. Notably, amplitude b significantly influences the DBT critical depth [25]. To further validate the accuracy of the model and explore the impact of varying amplitudes on the critical depth for ductile machining of micro-lens, critical depths for different amplitudes in the DOC direction were calculated, as illustrated in Fig. 15. The results demonstrate a high degree of agreement between the predicted outcomes and the experimental results, exhibiting an average error of 7.5%.
Fig. 15. The variation of critical depth under different amplitudes and the evaluation of model error.
In addition, the critical depth dc exhibits an initial increase followed by a decrease as the amplitude b increases. Specifically, with an increase in amplitude a-b increases from 2-0.25 µm0-p to 2-0.5 µm0-p, the critical depth dc raises from 0.45 µm to 0.65 µm. Conversely, when the amplitude a-b further increased from 2-0.5 µm0-p to 2-1 µm0-p, the critical depth dc decreases from 0.65 µm to 0.6 µm. This trend can be attributed to the limited range of pull-up movement of the tool during cutting when the amplitude b is relatively small. When the amplitude decreases continuously, the EVC process is close to one-dimensional vibration cutting. Consequently, increased friction between the tool and the workpiece results in a corresponding rise in cutting force, and the distance between the instantaneous tool position and the target topography is reduced, which is less conducive to ductile machining of microstructures. With an increase in amplitude b, the upward movement space of the tool expands, providing additional buffer space for accommodating crack expansion, and the cutting force decreases. Consequently, the critical depth increases. Nevertheless, when the amplitude b is too large, the tip slope of the vibration trajectory becomes steeper, and the chip pushed by the tool rake face becomes apparent, resulting in an elongated distance for the chip to be pulled in the DOC direction. This may cause tearing and peeling off of silicon atoms from the surface of the workpiece, limiting further increases in the critical depth. As a result, achieving the most effective ductile machining for the micro-lens cutting at a cutting speed of 200 mm/min can be accomplished using EVC with a vibration amplitude of 2-0.5 µm0-p.
5. Conclusions
In this work, a cutting depth-oriented ductile machining model was proposed to forecast the critical depth for achieving ductile machining of MLAs on brittle materials through EVC. Additionally, the transient material removal state and subsurface damage characteristics were analyzed. The conclusions are as follows:
- (1) The transient material removal mode of ductile machining of micro-lens on brittle material is plastic deformation, whereas brittle machining is characterized by a coupled state of plastic deformation and brittle fracture. Compared with OC, EVC can increase the critical depth dc by about 4.3 times.
- (2) The subsurface damage on the microstructure surface is closely related to the cleavage fracture along the crystal plane (111), and the damage predominantly occurs at the exit side of the tool cutting.
- (3) A critical depth prediction model for ductile machining of micro-lens was established, incorporating considerations of transient material removal state and target surface morphology. The results showed that the theoretical value aligned well with the experimental outcomes, exhibiting an average error of 7.5%.
- (4) The vibration amplitude in the DOC direction significantly influences the critical depth. Increasing the amplitude could increase the critical depth, but excessively large amplitudes could impede further critical depth enhancement.
In summary, the comprehensive analysis of the damage distribution characteristics and the critical depth for ductile machining of micro-lens on brittle materials prepared by EVC provides valuable insights for achieving high-precision and damage-free manufacturing of microstructure surfaces.
Funding
National Natural Science Foundation of China (52188102, 52225506); Program for HUST Academic Frontier Youth Team (2019QYTD12).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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