Study of machining indentations over the entire surface of a target ball using the force modulation approach

A modified five-axis cutting system using a force control cutting strategy was to machine indentations in different annuli on the entire surface of a target ball. The relationship between the cutting depths and the applied load as well as the microsphere rotation speed were studied experimentally to reveal the micromachining mechanism. In particular, aligning the rotating center of the high precision spindle with the microsphere center is essential for guaranteeing the machining accuracy of indentations. The distance between adjacent indentations on the same annulus and the vertical distance between adjacent annuli were determined by the rotating speed of the micro-ball and the controllable movement of the high-precision stage, respectively. In order to verify the feasibility and effect of the proposed cutting strategy, indentations with constant and expected depths were conducted on the entire surface of a hollow thin-walled micro-ball with a diameter of 1 mm. The results imply that this machining methodology has the potential to provide the target ball with desired modulated defects for simulating the inertial confinement fusion implosion experiment.


Introduction
A normal inertial confinement fusion (ICF) implosion continuously compresses an imploding outer spherical shell towards the target ball center by an intense radiation field, incessantly * Author to whom any correspondence should be addressed.
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condensing and heating the fission fuel within the shell until meeting fire requirements [1][2][3]. Due to the inherent defects in the target surface, the asymmetric compression will trigger implosion in the weak part, which can break the target surface or result in performance failure from the generated Rayleigh-Taylor (RT) instability on the surface of the fuelshell [4,5]. Thus, RT instability has been a critical research field with considerable experiments and intensive theories due to its influence on ICF behaviors [6,7]. The defects and roughness of the target surface have been found to strongly affect RT instability [8][9][10]. The shapes and sizes of target defects are various and randomly distributed. In order to investigate the formation circumstances of RT instability, there is no doubt that it is an urgent need to fabricate nanostructures with modulated and controllable dimensions, which can simulate defects on a target surface, so as to analyze the ICF experiments. Most explorations regarding this subject focus on flat targets, where the evolution somewhat differs from a converging geometry [11][12][13]. For example, a hollow thin-walled microsphere around 1 mm in diameter is always regarded as a target ball. However, no effective processing technique exists for solving the issue of easy deformation when machining nanostructures on the surface of low-stiffness micro-balls. Therefore, fabricating high-precision nanostructures on the whole surface of a microsphere remains a challenge.
Nowadays, several processing approaches, such as laser fabrication [14][15][16], micro turning [17][18][19], nanomachining based on atomic force microscope (AFM) [20][21][22] and fast tool servo (FTS) [23][24][25] cutting, have been adopted to prepare nanostructures on microsphere surfaces. In 2000, some scholars prepared a template of microstructures by excimer laser processing. They obtained a sinusoidal microstructure with a period of 60 µm and amplitude of 1 µm through imprinting on a 280 µm × 280 µm area of the micro-ball surface [11]. However, due to the localization of the machining area and the complexity of the polymer processing mechanism, this method is restricted in the application of fabricating microstructures on an entire rotary surface. Microgrooves with an average width of 30 µm and a depth of 30 µm depth were achieved on the microsphere surface by utilizing a six-axis ultra-precision machining center [26]. Although the machining accuracy and the sizes of machined structures can reach the micron level using diamond micro turning, it is difficult to accurately control the expected depths at the nanoscale because of the motion and limited detection precision of the machine tool. The AFM tip-based nanomechanical machining technique [27,28], in which the cutting tip applies constant normal forces to the sample, can fabricate nanodots [29,30], nanowires [31,32], nanochannels [33,34], and square-shaped, circular, and triangular cavities on the microsphere surface [35]. Despite the fact that the AFM nanomachining approach is regarded as an effective method to machine microstructures on curved surfaces [36][37][38], the prominent disadvantages are its low efficiency with a machining speed below 1 mm min −1 , serious tool wear, simple nanostructures, and most importantly limited machining area as well as small depths with approximate 200 nm [39]. For the sake of solving the difficulties in machining large depths over the non-planar surfaces of hard-brittle materials, the force control FTS has been developed by learning from the AFM-based cutting strategy as well as the rigid body of the machine tool for realizing high precision and high efficiency fabrication. Differing from AFM tip-based nanomachining, the force control FTS supports sufficient stiffness and much longer servo motion for the purpose of machining large depths over large areas [40].
This paper proposes a novel approach using force modulation between cutting tool and workpiece to fabricate nanostructures with relatively large machined depths at different annuli on the whole surface of a brittle material target ball, which breaks through the traditional rigid principle of tool-based diamond cutting by displacement control. A distinct advantage of the proposed force modulation cutting strategy is similar to AFM-based nanomachining, in which a force sensor measures the normal force loaded onto the micro-ball, and a desired normal force can be loaded onto the cutting tip by controlling the displacement of PZT. As a result, nanostructures with consistent depths can be achieved by keeping the normal forces constant during scratching. Thus, with regard to the employed force modulation cutting method in this study, the sensitivity or measurement accuracy of the force sensor as well as the material removal state mainly contributes to the machining accuracy, and the stiffness of this system has no influence on the form accuracy of indentations. This paper is presented in the following sections. Section 2 briefly describes the establishment of a five-axis micromachining experimental setup based on a force feedback control system. In addition, the relationship between the normal forces and machined depths of the indentations as well as the processing technique, such as the alignment of the microsphere center, are analyzed.

Experimental setup
The modified five-axis micromachining system using load modulation is shown in figure 1(a). This machining system is composed of air floatation-driven high precision stages (X) and (Y) with a resolution and stroke of 100 mm and 1 nm, respectively, as well as a motor-driven coarse stage (Z) with a resolution of 0.5 µm. These stages were used to adjust the position of the cone tip relative to the micro-ball. Figure 1(c) reveals the scanning electron microscopy (SEM) images and the geometry of the conical diamond cone tip, which was employed in the indentation process. AFM scanning estimated that the cone angle was 60 • and the arc radii (r) of cone tip was 1 µm. During the machining process, the rotating speed of the micro-ball was precisely controlled by an airbearing rotary stage (C), with a radial rotation accuracy of 0.26 µm, to control the distances between adjacent indentations. A fine-tuning stage (LA1V-XY, NEWPORT Corporation, USA) with a dimensional resolution of 0.75 µm was fixed on the rotary platform to align the microsphere center with the rotating center and prevent low form accuracy from a serious eccentric error. During the indentation process, the cutting tip was aimed at the micro-ball center by a turbine worm rotating platform (OptoSigma Corporation, KSPA-986M, Japan), which has a rotating range and resolution of 360 • and 52.9 ′′ , respectively. The force-feedback control system also contains essential components, including a controller (UMAC, Delta Tau Data Systems, Los Angeles, CA, USA), a piezoelectric (PZT) actuator (PSt150/7/60VS12, COREM-ORROW, Harbin, China), and a force sensor of 50 g capacity (LSB-200, FUTEK, Irvine, CA, USA).
In this study, a hollow thin-walled polymer microsphere with a diameter and wall thickness of 1 mm and 30 µm, respectively, as shown in figure 1(b), was glued to a syringe needle. Both the applied force and the rotation speed of spindle   Figure 2(a) reveals the schematic of the proposed machining principle for the force feedback control system. The normal force loaded on the surface of the micro-ball through the tool tip was measured by a high-sensitivity force sensor. Depending on the loaded normal force (F N ) and the reference signal of the periodic sinusoidal force, the UMAC controller decides to extend or retract the PZT actuator to reach the feedback control requirements. Figure 2(b) indicates that the period of applied normal force was set at 1 s. During the first 0.5 s, the normal force was loaded from zero to the expected peak and then unloaded to zero. Afterwards, the normal force was kept at zero for an additional 0.5 s. Moreover, when the rotating speed of micro-ball was fixed, the shorter the period of normal force, the more indentations achieved. Although the computer program can adjust the period of normal force, the limited response frequencies of PZT and the force sensor as well as crowded indentations on one annulus, significantly influence the form accuracy of the machined indentations.

Methodology of indenting on the microsphere surface
In order to obtain consistent indentations on the same annulus of the microsphere during the continuous indentation process, the micro-ball rotated at a constant angular speed, and the cutting tool carried out reciprocating motions via the PZT actuator, ensuring the constant depth of each indentation. Obviously, the value of sinusoidal force amplitude determines the cutting depths of indentations. The angular speed of the micro-ball rotation determined both the number of indentations on the same annulus and the distance between adjacent indentations. In addition, the distance between different annuli was adjusted by a high precision moving stage Y and a coarse moving stage Z as well as a turbine worm rotating platform, which is described in the following section 2.4.

Alignment of the micro-ball center
If the cone tip coincides with the center of the microsphere during the whole indentation process, it guarantees the consistency of each indentation and the same distance between adjacent indentations, while considering the PZT's limited range of motion. Therefore, aligning the microsphere center with the rotating center of the air-bearing driven high precision spindle is absolutely essential. Firstly, for the coarse alignment, the 2D fine-tuning stage with the glued microsphere was installed on the rotary platform, which rotated a microsphere 360 • . Optical capture software monitored the microsphere's center position. For the fine alignment process, the 2D fine-tuning stage was used to adjust the position of the micro-ball. The details of the centering adjustment method are provided below. If the position of the micro-ball is always within this circular contour, the process of fine centering alignment is successfully completed. Otherwise, the above steps should be repeated.

Movement calculation for indenting on different annuli
For machining indentations at different annuli, the cone tip was rotated along the entire surface of the microsphere by a turbine worm rotating platform. However, it is difficult to make the cone tip coincide with the microsphere center when moving from one annulus to another, which results in the failure of geometries of machined indentations. Stated in the previous study [35], the largest rotation angle of the cutting tool is restricted by the moving capacity of the 2D high precision stage. With the improvement of micromachining tools, any desired rotation angle of the cutting tool can be implemented by the high-precision stages Y and Z and the turbine worm rotating platform in this study. The principle of accurate mathematic calculation for the relationship between the rotation angle (θ) of the cone tip and moving distances in Y and Z directions is illustrated in figure 3(b). Firstly, note that a vertical distance inevitably exists between the cutting tip and the rotating center of the turbine worm stage, defined as e in figure 3(a). In the upright direction of PZT, the cutting tool applies very little force when approaching the microsphere vertex. In the meantime, the current position of PZT is recorded. For machining on the first annulus, the cone tip was lifted and the required rotation angle (θ ′ ) was developed by the chosen vertical distance between the vertex of the microsphere and the desired machined annulus (∆h), as follows: where R is the radius of the microsphere. Based on the rotating angle (θ ′ ), the Y and Z stages move over distances l 1 and z 1 , respectively. When machining on the first annulus, the same normal force was loaded for approaching the surface of the microsphere and recording the extension value of PZT. Compared with the value recorded in the upright position of PZT, the increasing value of PZT extension (∆k 1 ) can be calculated. When indenting on the second annulus, the cone tip was adjusted to continue rotating another angle (θ ′′ ) depending on the desired vertical distance (∆h ′ ) between the first machined annuli and second one, as solved in the following equation: The previously mentioned steps were repeated, including moving a distance of l 2 and z 2 , obtaining the difference (∆k 2 ) between the upright position of PZT in the case of rotating θ ′′ . Thus, the value of h 0 and e can be derived in the form of: where θ ′ , θ ′′ , z 1 , z 2 , ∆k 1 , and ∆k 2 are known parameters. The moving distances l 1 and l 2 of the high-precision stage Y defined with respect to the tip rotating angles of θ ′ and θ ′′ , respectively, are described by the following equations (5) and (6), which consider the geometry relationship as shown in figure 3(a):

Fabrication by loading different normal forces
To evaluate the capability of the proposed five-axis micromachining system using a force modulation cutting strategy, the experiments of indenting by varying the loading normal forces were first conducted on the microsphere surface. The amplitudes of applied normal forces were set at 5, 10, 15, 17, 20, 23, 26, and 30 mN. And the rotation speed of the spindle was held constant at 3 • s −1 .
The AFM images with cross section measurements corresponding to the setting normal forces of 5, 10, 17, 23, and 30 mN are shown in figure 4. The indentation depths were evaluated to be 0.83, 1.2, 3.14, 3.58, and 4.01 µm. The relationship between the applied normal forces and cutting depths is illustrated in figure 6(a). Three regions of the map were clearly observed. In the case of machining forces from 5 to 15 mN, when the indenting depth was smaller than the arc radius of cone tip, approximately 1.5 µm, the spherical cap of the cutting tool played a critical role in machining. However, as the applied normal forces increased to the second part from 15 mN to the critical point of 17 mN, the indentation depths increased dramatically from 1.58 µm to 3.14 µm. This phenomenon occurred because the machined depths were significantly larger than the arc radii of the cone tip. As a result, the spherical cap and the cutting edge participated in the indentation process, which dramatically increased the machining depths. For the third part with cutting forces larger than 17 mN, the growth rate of the depths slowed. This might be attributed to the influence of material pile-up. The tool edges contributed more to the indentation depths.

Fabrication by varying the rotating speed of the micro-ball
To identify the influence of the microsphere's rotating speed on the topography of the machined indentations, the experiments were extended to make indentations by selecting rotating speeds from 1 • s −1 to 3 • s −1 with an increment of 0.5 • s −1 . The applied normal force was set at 10 mN.
The AFM images and the section profiles corresponding to rotating speeds of 1 • s −1 , 2 • s −1 , and 3 • s −1 are presented in figure 5. The depths were consistent and evaluated to be 1.4, 1.21 and 1.21 µm. As shown in figure 6(b), the rotating speed of the microsphere within a certain range, that is, increasing from 1 • s −1 to 3 • s −1 , has little influence on the depths of the machined indentations. The depths obtained from rotating speeds less than 1.5 • s −1 were almost unchanged. When the rotating speed increased to 2 • s −1 , the machined depth obtained was slightly smaller. As the rotating speed increased to 3 • s −1 , the depths became constant again. More importantly, it is evident from figure 7 that the shape of machined indentations differed from the shape of the cone tip, which suggests that the machined indentations tailed off in the direction opposite of the micro-ball rotation. This was mainly attributed to the fact that the micro-ball remained rotating during the entire loading and unloading process of tracking the sinusoidal normal force signal. That is to say, when the machining tip contacted the micro-ball surface, the microsphere kept moving. Furthermore, the faster the rotating speed of microsphere, the larger the tailing off of the machined indentations. That is the reason why the rotating speed should be controlled within 3 • s −1 to avoid unexpected nanostructures. In addition, the microsphere was made of a hard, brittle polymer. Thus, slight deformation or squeezing might occur during the material removal process. When the cutting tip left from the surface of the micro-ball, the material elastic recovery contributed to minor shape deviation.

Fabrication of indentations on the entire surface of a microsphere
In order to demonstrate the effectiveness of machining nanostructures over the entire microsphere surface using the proposed five-axis micromachining system, the indentations were conducted on different annuli of the micro-ball. Since the annuli on the microsphere have different radii, the circumferences of the annuli are different. However, the computer program ensured that the same number of indentations were made on each annulus. Therefore, the selected rotating speed should not be too slow, as to avoid the overlapping of adjacent indentations on the annulus with a smaller circumference due to the dense indentation. At the same time, considering the surface quality of the machined indentations, the rotating speed should not be too fast to cause tailing. To this end, the rotating speed was set at 2 • s −1 , that is, there were 180 indentations on each annulus, and the machining time took 180 s. According to the relationship between the normal forces and machining depths,  as shown in figure 6(a), the amplitude of normal force was set at 10 mN to prevent puncturing the hollow thin-walled microball. In the present study, 28 different annuli located on the entire surface of microsphere were selected for the indentation process. The vertical distance between adjacent annuli and the high precision stage movement were calculated by applying the method described in section 2.4.
As shown in figure 7, the overall optical image of the obtained indentations over the entire microsphere surface and the local image of the indentations on different annuli are marked from '1' to '10.' The following controllable vertical distances between adjacent annuli were achieved: 12 µm between '1' and '2', '3' and '4'; 32 µm for '4' and '5', '5' and '6'; 28 µm for '8' and '9', '9' and '10'. Cross-sections of two different positions were obtained by AFM scanning to assess the quality of indentations over the whole microsphere's surface (i.e. depth and consistency). There is no doubt that the indentation cutting depths were guaranteed to be constant at positions 1 and 2, where the depths were 1.4 µm and 2.1 µm, respectively, as indicated in figure 8. However, notice that the depths in both positions were different. A possible reason for this was that the surface material of micro-ball had a considerable effect on the machined depths, which means that the microsphere material was not uniform. In the future, according to the indentation depths, the distribution of material on the surface of the microsphere may be preliminarily judged by selecting processing indentations at different positions on the entire surface of the microsphere. The goal of machining indentations at different annuli over the entire microsphere surface using the proposed cutting strategy was achieved.

Conclusions
This paper investigated the possibility of machining indentations at different annuli over a hollow thin-wall microsphere surface with a 1 mm diameter using the proposed tip-based micromachining approach. The influence of loaded normal forces and micro-ball rotating speeds on the machined indentation depths were explored. The rotating speed of the microball determines the distance between adjacent indentations on the same. Based on the movement calculation, adjusting the high precision stage and the worm rotating platform based on the movement calculation can control the vertical distance between adjacent annuli. Most importantly, the microsphere center should be aligned with the gyration center of rotating stage, which is a key factor to ensuring the surface quality of machined indentations. Finally, the effectiveness of the proposed methodology of machining nanostructures on the entire surface of microsphere was validated successfully. Furthermore, these positive results are expected to overcome the difficulties of machining high precision 3D nanostructures on the target ball surfaces used in ICF experiments, which will promote the improvement of RT instability research.