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Evolution mechanism of scratch removal based on the implementation of magnetorheological finishing

Yongjie Li, Longxiang Li, Xingchang Li, Runmu Cheng, Hongda Wei, Ximing Liu, Yiren Wang, and Xuejun Zhang

Open Access Open Access

Abstract

Scratches on optical components induce laser damage and limit the increase in laser power. Magnetorheological finishing (MRF) is a highly deterministic optical manufacturing technology that can improve the surface roughness of optical components. Although MRF has exhibited significant potential for reducing subsurface damage and removing scratches, the principle and mechanism behind the scratch removal are not sufficiently understood. In this study, the theory of fluid mechanics is used to analyze the pressure, velocity, and particle trajectory distribution near a scratch. A physical model was developed for the differential removal of scratches at the bottom and surface of the optical components. The morphological evolution of the scratch was predicted during removal, and detailed experiments were performed to verify the effectiveness of the proposed model. The results indicate that scratches expand laterally rather than being completely removed. Furthermore, scratch removal efficiency is greater when the removal direction is perpendicular to the scratch rather than being parallel. This study offers an intrinsic perspective for a comprehensive understanding of the MRF technique used for scratch removal, which can be beneficial for removing scratches from aspherical optical systems.

© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

As laser fusion ignition facilitates the generation of clean energy, countries worldwide have begun to build high-power laser facilities; these include the National Ignition Facility in the United States, the SG series laser facilities in China, the Laser Mégajoule in France, and the GEKKO XII Laser in Japan [1]. The devices in these facilities are subjected to extremely high energy during operation, which requires high-quality optical components. However, scratches inevitably occur on optical components during processing. The three primary aspects of the damage caused by scratches can be summarized as follows: (1) transmittance of the optical changes; (2) local enhancement of the incident light field; and (3) the presence of impurity particles of the absorbent inside the scratch [2–5]. Recent studies have proposed a theory stating that scratches induce the highest photoluminescence intensity, decreasing the laser damage threshold [6]. The reduction in the damage threshold of optical components indicates that they are prone to laser damage. Therefore, the surface scratches of optical components must be efficiently controlled in high-energy laser systems.

Considerable efforts have been devoted to the research on controlling scratches on optical components. At present, four methods are predominantly used for scratch removal and repair. Considering the repair of damaged points and the removal of scratches as analogous processes, laser melting, chemical corrosion, and special processes are used for scratch removal, whereas micromachining is used for repairing scratches. The Lawrence Livermore National Laboratory pioneered the work on $C{O_2}$ laser for repairing damaged points. Raymond et al. used a 10.6 µm single-pulse $C{O_2}$ laser, with the power ranging between 17 and 37 W, to irradiate the local damaged point in 1 s by changing its morphology [7,8]. When the 351 nm and 8 J/$c{m^2}$ laser irradiated the repaired optics, the damage did not increase under 1,000 shots, confirming the effectiveness of the $C{O_2}$ laser in repairing the damaged point. Considering the limitation of the size of the repair, Isaac et al. proposed a galvanometer $C{O_2}\; $ laser repair method, wherein the repair size was expanded to 300 µm and the light source was a pulsed laser, which improved the quality of repair [9,10]. Philippe et al. applied the $C{O_2}$ laser melting method to remove scratches, investigated the performance differences of scratch removal under different laser parameters, including beam diameter and heating time, and determined the optimal parameters for removing scratches [11,12]; they successfully removed a 16-µm-deep scratch and increased the laser damage threshold of optical components from 4 to 16 J/$c{m^2}$. However, owing to the high temperature during laser repair, deformation occurs in the scratched area after repair, which affects the performance of optical components in severe cases.

Chemical corrosion is also used for scratch removal. In this method, optical components chemically react with acidic or alkaline solutions to remove laser damage precursors, such as scratches, impurities, and redeposited substances. Tayyab et al. investigated the effect of various HF corrosion processes on laser damage thresholds for fused quartz optics with scratches and determined that the laser damage threshold was inversely proportional to the width of the scratch [13]. Although they combined other processes to achieve efficient removal of scratches, the surface roughness of the optical components deteriorated under the action of HF solution [14]. Based on the idea of using a KOH solution to treat surface and subsurface damage after wafer polishing, Mathilde et al. proposed the use of a KOH solution in the scratch treatment of optical components and compared this with the use of HF solution after corrosion. The results indicated that both solutions passivated or eliminated surface scratches, improving the laser damage threshold; however, only the KOH solution reduced roughness [15]. Chemical corrosion scratch removal has been widely applied owing to its high efficiency and simple operation. However, the change in surface roughness and optimization of corrosion depth require further investigation.

Using special processes to remove scratches has potential advantages. Amanapu et al. proposed an abrasive-free chemical–mechanical polishing method to completely remove surface defects, including scratches [16]. Deng et al. used plasma chemical vacuum processing to remove surface scratches and subsurface damage caused by grinding and obtained a roughness surface of 0.6 nm [17]. Peng et al. used hydrodynamic polishing to remove scratches on the surfaces of 30 mm × 30 mm optical components [18] and increased the laser damage threshold from 29.78 to 45.47 $J/c{m^2}$. Although the use of these special processes can remove scratches effectively, the technology is not sufficiently mature, the removal efficiency is low, and the apertures of the optical components are small.

Micromachining has shown great application prospects in repairing damaged points. However, it has not yet been employed for scratch removal. Its principle is to process the damaged point into a specific microstructure to weaken the light-field enhancement and form the basis for controlling scratches. Lawrence et al. used a high-speed moving single-crystal diamond drill bit to manufacture the microstructure at the damaged point [19], repaired the damaged point of 0.14 mm, and increased the laser damage threshold to a value exceeding 13 $J/c{m^2}$. Cheng et al. manufactured a spherical structure at the damaged point using a micromilling cutter, and the laser damage threshold almost returned to a level similar to that of a scratch-free surface [20]. The application of this method for the removal of scratches requires further analysis.

Although the aforementioned methods can remove or repair scratches in specific situations, they focus on the local repair of optics and global removal of small-aperture optics. They cannot satisfy the global requirement of scratch removal for large-aperture optics. Magnetorheological finishing (MRF) is suitable for global scratch removal in large-aperture optics. However, the internal microscopic mechanism of scratch removal using this technique is not sufficiently understood. Shi et al. combined MRF with HF treatment to passivate scratches and shorten the processing time; the laser damage threshold was reduced by only 8.2% in comparison with scratch-free optical surfaces [21]. Catrin et al. investigated the ability of MRF to remove scratches under different parameters and reported that MRF can effectively remove scratches regardless of the parameters [22]. Ji et al. experimentally analyzed the law of scratch removal via MRF but did not provide a complete theoretical explanation [23]. Previous studies have predominantly focused on the ability of MRF techniques to remove scratches without investigating the underlying mechanisms of scratch removal. To adapt to the efficient global removal of scratches from large-aperture optical components, comprehensive research on the intrinsic microscopic mechanism of scratch removal using MRF is crucial.

In this study, we systematically studied the difference in efficiency and morphology changes of scratch removal by magnetorheological finishing in two directions. A differential removal model was established by combining fluid dynamics simulation and magnetorheological finishing theory. Based on the model, the difference in removal efficiency and morphology change process in the two directions is further explained. Experiments on scratch removal by magnetorheological finishing verify the effectiveness of the model.

2. Theory and simulation

2.1 Theory of MRF and computational fluid dynamics (CFD)

When the magnetorheological fluid follows the polishing wheel into the polishing area, a single ribbon protrusion is formed under the action of a gradient magnetic field, resulting in material removal from the workpiece. This process involves both continuous-phase fluids and discrete-phase particles. Therefore, the interaction between magnetorheological fluids and scratches should be examined based on fluid dynamics simulations.

MRF has been reported to remove material in the form of a shear force [24]. However, Kordonski et al. demonstrated that the general form of the Preston equation applies to MRF [25]. According to the Preston equation, material removal is associated with the relative speed and pressure between the polishing tool and workpiece. The influence of other parameters on the machining process can be expressed using a constant $k$:

$$MRR = kPV, $$
where k is the constant that represents the influence of other parameters; P denotes the relative pressure between the tool and workpiece; and V indicates the relative velocity between the tool and workpiece. The distributions of P and V at the scratch can reflect certain characteristics of the removal process without considering the constant k. According to the Navier–Stokes equation, the continuity of the flow field and the conservation of momentum can be expressed as follows [26]:
$$\frac{{\partial \rho }}{{\partial t}} + div(\rho {\boldsymbol u}) = 0;$$
$$\frac{{\partial (\rho u)}}{{\partial t}} + div(\rho u{\boldsymbol u}) = div(\mu \textrm{ }grad\textrm{ }u) - \frac{{\partial p}}{{\partial x}} + Su;$$
$$\frac{{\partial (\rho \upsilon )}}{{\partial t}} + div(\rho \upsilon {\boldsymbol u}) = div(\mu \textrm{ }grad\textrm{ }\upsilon ) - \frac{{\partial p}}{{\partial x}} + S\upsilon;$$
$$\frac{{\partial (\rho \omega )}}{{\partial t}} + div(\rho \omega {\boldsymbol u}) = div(\mu \textrm{ }grad\textrm{ }\omega ) - \frac{{\partial p}}{{\partial x}} + S\omega, $$
where $\rho $ denotes the density of the fluid; t indicates the time; u, $\upsilon$, and $\omega$ represent the velocities of the x, y, and $z$ components, respectively; P denotes the pressure; and $Su$, $S\upsilon$, and $S\omega$ indicate the generalized source terms. Typically, scratches on the wall features can generate Eddy currents. The $RNG$ and $k - \varepsilon$ models were adopted to describe the turbulent flow. The near-wall treatment was considered the standard wall function, and water was set as the continuous-phase fluid [27]. The pressure and velocity characteristics of a fluid in contact with a scratch can be obtained by solving Eqs. (2)–(5).

In addition to pressure and velocity distributions, the trajectory of the abrasive is another key factor in scratch removal. In the CFD simulation software, the trajectories of the discrete-phase particles can be solved by integrating the differential equation of the particle force under the integral Lagrange coordinate system. The force balance equation of the particle indicates that its inertial force is equal to other forces acting on it, such as virtual mass and pressure gradient forces. The equilibrium equation can be expressed in Cartesian coordinates as follows:

$$\frac{{d{u_p}}}{{dt}} = {F_D}(u - {u_p}) + \frac{{{g_x}({\rho _p} - \rho )}}{{{\rho _p}}} + {F_x}, $$
where u denotes the fluid velocity; ${u_p}$ indicates the particle velocity; $\rho$ represents the fluid density; ${\rho _p}$ denotes the particle density; and ${F_D}(u - {u_p})$ indicates the drag force per unit mass. Here, ${F_D}$ can be expressed as
$${F_D} = \frac{{18\mu }}{{{\rho _p}d_p^2}}\frac{{{C_D}{R_e}}}{{24}}, $$
where $\mu $ denotes the dynamic viscosity of the fluid; ${d_p}$ indicates the particle diameter; $Re$ represents the relative Reynolds number; and ${C_D}$ denotes the drag coefficient. According to the MRF theory [28], abrasive particles are carried by hydroxyl iron powder, and the size of the abrasive particles is substantially smaller than that of the iron powder. In this study, the iron powder particles were added as discrete-phase particles. When the density ratio of the particles to fluids is considerably greater than 1, the virtual mass and pressure gradient forces can be ignored. In this study, these two forces were ignored as the ratio of iron powder particles to water is substantially greater than 1. When selecting the physical model for the discrete-phase particles, only the erosion/accretion item was checked to observe the effects that may occur during the scratch removal process to better understand the interaction between the fluid and scratch.

2.2 Simulation and analysis at ${90^\circ }$

As the magnetorheological removal function is not circularly symmetrical, scratches exhibit directionality. MRF can remove scratches at different angles, ranging from ${0^\circ }$ to ${90^\circ }$. In this study, two extreme angles were investigated. Figure 1 depicts a schematic of the removal function and angle.

 figure: Fig. 1.

Fig. 1. Schematics of the (a) removal function and (b) removal angle.

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In the ${90^\circ }$ direction, the physical model of the magnetorheological fluid interacting with the scratches can be simplified to a two-dimensional (2D) model, as illustrated in Fig. 2. The width of the scratch is 20 $\mu m$ and the depth is 1 $\mu m$. To ensure model symmetry, the magnetorheological liquid column is also considered to be 20 $\mu m$.

 figure: Fig. 2.

Fig. 2. Two-dimensional geometric model for the computational fluid dynamics simulation.

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The continuous-phase fluid velocity at the inlet was set as 8 m/s. Iron powder particles with a diameter of 1 $\mu m$ were adopted as discrete phases, and the particle velocity was the same as the fluid velocity. During the actual execution of MRF, the particles are bound by a gradient magnetic field to form a liquid column that does not easily react with the fluid. Therefore, only one-way continuous–discrete phase coupling processes were considered in the simulation.

Figure 3 depicts the simulation results of the pressure distribution. One end of the fluid inflow direction is set to the proximal end of the scratch, whereas the end of the outflow direction is set to the distal end of the scratch. As indicated in Fig. 3, the pressure is greater at the distal end, and a high pressure value is observed at the bottom of the scratch. To better understand the pressure distribution in and around the scratch, the pressure curves at the distal edge and bottom of the scratch are extracted, as shown in Fig. 4.

 figure: Fig. 3.

Fig. 3. Simulation results of the pressure distribution.

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 figure: Fig. 4.

Fig. 4. Pressure curves of the (a) edge of the distal end and (b) bottom of the scratch.

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The curve indicates that (1) a low value of pressure appears at the distal edge of the scratch (dotted box in Fig. 4); and (2) the pressure distribution at the bottom of the scratch is not symmetrical, with high and low values of pressure near the distal and proximal ends of the scratch, respectively. Moreover, high values of pressure appear in the middle portion of the bottom of the scratch.

Figure 5 illustrates the simulation results of the velocity distribution. Water is a viscous fluid; The velocity is zero at the edge of the model at the point where it touches the boundary and it is higher in the middle position, consistent with the theory of fluid mechanics. The depth of the scratch is extended downward, and considering that the scratch causes a change in the fluid velocity in the y-direction, the velocity distribution in the y-direction is extracted (Fig. 5(b)). As indicated in the figure, the maximum velocity occurs at both ends of the scratch, whereas the velocity in the rest of the site remains uniform. Notably, the magnetorheological fluid flow is in the x-direction, with the y-direction perpendicular to the x-direction.

 figure: Fig. 5.

Fig. 5. Simulation results of the velocity distribution. (a) Velocity magnitude distribution. (b) Velocity distribution in the y-direction.

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Based on the pressure and velocity distributions, certain characteristics that occur during the ${90^\circ }$ removal of a scratch can be derived as follows: (1) the distal end of the scratch is removed first owing to the high pressure and velocity, and the side of the scratch expands macroscopically; (2) a diminished pressure level is observed at the distal end of the scratch, and the absence of an elevated velocity value leads to an anomalous material protrusion at this location. This is similar to the result reported in a previous study [27], adding to the credibility of this study; (3) the bottom of the scratch experiences a high pressure owing to the impact of the fluid, and if the particles successfully act on the bottom of the scratch, a significant portion may be removed.

Figure 6 depicts the erosion rate distribution of the particles on the scratch considering the discrete phase; the particle size is only a relative model representation and not the actual particle size. The figure indicates that the erosion rate is larger at the distal end, corresponding to the location where the impact is most intense. Furthermore, considering the clear presence of inertia when the particles encounter the scratch, they may not have made initial contact with the bottom of the scratch at the outset of the removal process. This may prevent the particles from completely touching the bottom when they begin to remove narrower scratches; instead, they follow the magnetorheological fluid to severely impact the distal end of the scratch.

 figure: Fig. 6.

Fig. 6. Schematic of particles impacting the scratches. (a) Particle trajectories and erosion rate distributions. (b) Partial enlarged view of the erosion rate.

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As the scratches were gradually removed, the particles reached the bottom completely. Considering that the magnetizing pressure and extrusion effect of the polishing wheel on the ribbon were not considered in the simulation, the simulation results do not reflect the actual scenario, and the scratch removal mechanism must be experimentally analyzed as well.

2.3 Simulation and analysis at ${0^\circ }$

The direction of removal at ${0^\circ }$ coincided with the direction of the scratch, and the length of the scratch was longer than that at ${90^\circ }$, indicating that it was easier for the particles to completely touch the bottom of the scratch owing to inertia. Therefore, we inferred that the particles could completely touch the bottom of the scratch. Notably, magnetorheological finishing is relatively stable over time. The removal rate of the optical element surface can be considered a constant during process. When the bottom of the scratch is more easily removed, the overall scratch removal rate is lower. Moreover, a 2D cross-section could not be used in place of a three-dimensional (3D) model simulation at ${0^\circ }$. Figure 7(a) depicts the schematic of the 3D simulation model. The arrow represents the direction of magnetorheological fluid flow. The simulation settings and scratch sizes are the same as those at ${90^\circ }$.

 figure: Fig. 7.

Fig. 7. Schematic of the model and simulation results. (a) Simulation model. (b) Pressure distribution on the x–y plane, z = 5 $\mu m$. (c) Pressure distribution curve extracted near the scratch. (d) Velocity distribution on the x–y plane, z = 5 $\mu m$.

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Figure 7(b) illustrates the pressure distribution perpendicular to the z-axis, wherein the pressure is lower at the scratch location; this is more evident from the extracted pressure distribution curve. Except at the two corners of the scratch, Fig. 7(d) indicates that the speed distribution remains uniform in the remaining positions. In terms of the pressure distribution, Fig. 7(c) indicates that the pressure is high on both sides, whereas it is low in the middle. This may cause particles to converge in the middle portion of the scratch, resulting in a higher removal rate in the middle of the bottom of the scratch and a lower removal rate on both sides, altering the morphology of the scratch.

Figure 8 depicts the schematic of the MRF technique employed to remove scratches at ${0^\circ }$. The forces on the polished area have been investigated and can be used in the analysis of scratch removal. Zhang et al. deduced the pressure on a polished area [29], expressed as

$$P = {P_d} + {P_m} = \frac{{2{\eta _0}Ux}}{{{Z^2}}} + V{\mu _0}M\nabla H, $$
where P denotes the pressure of the polishing area; ${P_d}$ indicates the fluid pressure; ${P_m}$ represents the magnetization pressure; ${\eta _0}$ denotes the initial viscosity of the magnetorheological fluid; U indicates the ribbon speed; x represents the transverse coordinate of the polished area; $Z1$ and $Z2$ denote the distances between the lowest point of the polishing wheel and the surface and bottom of the scratch, respectively; Z1 is the ribbon thickness of the magnetorheological polishing fluid. V indicates the volume of non-magnetic particles; M represents the magnetization of the magnetorheological fluid; ${\mu _0}$ denotes the vacuum permeability; and $\nabla H$ indicates the magnetic field gradient.

 figure: Fig. 8.

Fig. 8. Schematic of the magnetorheological finishing (MRF) technique at ${0^\circ }$ to remove scratches.

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The effect of the magnetizing pressure was disregarded, and only the force exerted by the magnetorheological fluid on the polished area was considered. When combined with the simulation results, the value of velocity U was nearly uniform. The transverse coordinates x were also approximately equal, and ${\eta _0}$ was maintained constant. The difference in pressure between the surface and bottom of the scratch was that $Z1$ and $Z2$ differed from those at the lowest point of the polishing wheel. As $Z1 < Z2$, we inferred that the pressure on the surface of the scratch was greater than that at the bottom of the scratch, resulting in a difference in the removal rate. This concurred with the qualitative simulation results.

For a scratch with a depth of only 1 $\mu m$, assuming that $Z1$ is 2 $mm$ and $Z2$ is 2.001 mm, the pressure ratio can be expressed as

$$\frac{{P1}}{{P2}} = \frac{{Z{2^2}}}{{Z{1^2}}} = 1.001, $$
where $P1$ and $P2$ denote the pressures on the surface and at the bottom of the scratch, respectively. Considering that this ratio is extremely small, the difference in the removal rate caused by the difference in pressure at ${0^\circ }$ is also small. From both simulation and theoretical points of view, the difference in the removal rate between the bottom and the surface of the scratch owing to pressure is not extremely high, rendering scratch removal more difficult. However, a previous study has reported that the flow capacity of the particles can be poor when the magnetorheological fluid particle size is in the same order of magnitude as the scratch size [30]. This results in an extremely low particle velocity at the bottom of the scratch. Therefore, at the initial stage of removing narrower scratches, the scratch removal rate is high, and the removal rate gradually reduces as the process advances.

3. Experimental results and discussion

A prerequisite for quantitative scratch removal analysis is the need for uniform and consistent scratches. Deeper scratches are created to gather more extensive datasets at varying depths. In this case, the traditional method of creating quantitative scratches using a scratch test instrument introduces brittle cracks [31], which affects the uniformity of the scratch size. In this study, we used a highly deterministic ion-beam etching technology to create uniformly sized scratches. Three 150 mm fused silica pieces were used, referred to as samples 1, 2, and 3. Table 1 lists the specific parameters of the scratches; the depth uncertainty of the scratches is ±50 nm.

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Table 1. Specific parameters of scratches.

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Figure 9 depicts the equipment used in the experiment, including an MRF machine with a processing capacity of 2 $m$ and a polishing wheel diameter of 360 mm. We used ZYGO NewView 9000 with a field of view of 0.867 $mm$ × 0.867 $mm$ and a pixel count of 1,000 × 1,000 to measure the size and morphology of the scratches after removal.

 figure: Fig. 9.

Fig. 9. Magnetorheological finishing (MRF) equipment used in the experiment.

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3.1 Experiment to remove scratches at ${90^\circ }$

One of the scratches on each piece is through the center of the element, and the other scratches are parallel to this scratch. During processing, the direction of the scratch is one of the directions of the coordinate system of the machining machine. It is only necessary to move the optics so that the direction of this coordinate system is perpendicular or parallel to the direction of the removal function. Then, the angle between the scratch and the removal function can be determined. The relationship between the depths of material removal and scratch removal was investigated. Five regions located at the scratch site were selected, with materials of different thicknesses uniformly removed from each region. Due to the high certainty of magnetorheological finishing, uniform thickness removal of the material can be achieved using grating trace machining. Table 2 lists the depth of material removal, depth of scratch removal, and amount of change in width. The change in width represents the difference between the scratch width after removing different thicknesses and the initial width.

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Table 2. Experimental data at ${90^\circ }$.

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Figure 10 presents the corresponding data. In the initial stages of removal, that is, when small amounts of material are removed, the removal rate of the scratch is rapid and almost linear. As the depth of material removal increases, the removal rate of the scratch gradually reduces, exhibiting a non-linear relationship, and scratch removal becomes difficult. The width of the scratch increases with the increase in the material removal; however, the rate of increase gradually reduces as the depth of removal increases.

 figure: Fig. 10.

Fig. 10. Relationship between the depth of material removal and the (a) depth of scratch removal and (b) the amount of change in width.

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Morphological changes in scratches at various removal depths can facilitate the understanding of the evolution of scratches. Figure 11 depicts the cross-sectional morphology of the scratches at each removal depth; each pixel represents 0.868 $\mu m$. The data for Fig. 11 is derived from the position where the machining is uniform for each removal. We used a white light interferometer to measure the cross-sectional morphology of each location. The most apparent phenomenon is that the scratch is initially extended at the distal end of the fluid impact, which is consistent with the simulation results. Second, the removal rate of scratches is slightly high in the initial stages. Additionally, as the morphology of the bottom does not change significantly, the material removal at the bottom of the scratch is weak. However, as the depth of material removal increases, the morphology at the bottom gradually changes. The scratch expands laterally and scratch removal becomes difficult. Combined with the simulation results, this phenomenon can be attributed to the existence of particle inertia.

 figure: Fig. 11.

Fig. 11. Morphology of scratches at different removal depths.

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Figure 12 depicts the physical model used to describe the specific process of the morphological evolution of scratch removal at ${90^\circ }$. When the scratch width is narrow, the particles do not completely touch the bottom in the initial stages of scratch removal, and only a small amount of particles act. This results in high scratch removal efficiency. The scratch widens as the removal progresses, increasing the number of particles touching the bottom, which in turn improves the flow capacity. This increases the removal rate at the bottom yet results in a smaller removal rate for the overall scratch removal. Eventually, the scratch evolves into a V-shaped morphology.

 figure: Fig. 12.

Fig. 12. Illustration of a physical model for the morphological evolution of scratch removal at ${90^\circ }$.

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3.2 Experiment to remove scratches with different widths

To investigate the characteristics of the MRF when removing scratches of different widths at ${90^\circ }$, three scratches with the same depth yet different widths were created on sample 2. The three areas were uniformly removed at different depths from the scratch location. The data obtained from the experiments are presented in Table 3.

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Table 3. Experimental data for the removal of scratches with varying widths.

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Figure 13 illustrates a comparison of the material and scratch removal depths for different widths. The following conclusions can be drawn from the visual inspection of Fig. 13: (1) the wider the scratch, the lower the removal rate; (2) each curve exhibits a tendency to reduce the removal rate. The wider the scratch, the less noticeable the tendency; (3) the scratch expands gradually, and the greater the initial width, the slower the rate of expansion.

 figure: Fig. 13.

Fig. 13. Comparison of scratch removal rates for different widths with respect to (a) the depth of scratch removal and (b) the amount of change in width.

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Figure 14 depicts the morphological evolution of scratches with varying widths. The data for Fig. 14 is derived from the position where the machining is uniform for each removal. We used a white light interferometer to measure the cross-sectional morphology of each location. As indicated in the graph, wider scratches are more easily removed from the bottom compared to narrower scratches, which elucidates the slower removal rate for wider scratches. Additionally, the result demonstrates the effect of particle inertia. Owing to manufacturing constraints, the bottom of the scratch is not entirely flat and contains certain “sharp corners.” These “sharp corners” do not exhibit significant changes during the initial stages of scratch removal and are removed only in the later stages. This indicates that particles cannot make contact with the “sharp corners” during the initial stages of scratch removal. We inferred that inertia is responsible for this phenomenon.

 figure: Fig. 14.

Fig. 14. Evolution of scratches with varying widths. (a) 20 $\mu m$, (b) 50 $\mu m$, and (c) 100 $\mu m$.

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The experimental results were consistent with the assumptions of the proposed model. When removing wider scratches, a majority of particles reached the bottom of the scratch during the initial stages of removal, resulting in a lower overall efficiency. When removing narrower scratches, only a small number of particles reached the bottom, and their poor mobility led to higher scratch removal efficiency.

3.3 Experiment to remove scratches at ${0^\circ }$

Similar to ${90^\circ }$, five different regions with varying removal depths were uniformly removed at ${0^\circ }$ to investigate the mapping relationship between the material removal and scratch removal depths along with the evolution pattern of the scratches. Table 4 lists the corresponding experimental data.

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Table 4. Experimental data at ${0^\circ }$.

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Figure 15 illustrates the removal trends. As indicated in the graph, the scratch removal efficiency is high at small depths of material removal. The removal efficiency gradually reduces as the depth of material removal increases. Additionally, the lateral expansion of the scratch progresses continuously with a tendency to decelerate.

 figure: Fig. 15.

Fig. 15. Relationship between the depth of material removal and the (a) depth of scratch removal and (b) the amount of change in width.

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According to the simulation and experimental results, two primary reasons existed for the change in the removal rate. One was the pressure difference between the bottom and surface of the scratch, which explained a few of the differences in the removal rate; however, this contribution was relatively small. Another reason was the change in the particle flow capacity at the bottom of the scratch. The diameter of the iron powder particles used in the experiment ranged between 1 and 4 µm, which was in the same order of magnitude as the scratch size; this indicated that only a few particles were sufficient to infuse into the scratch. During the initial phase of material removal, the particles situated at the bottom of the scratch did not attain high velocities, leading to a relatively gentle removal process. This was reflected in the increased scratch removal rate. As the scratch expanded laterally, the particles at the base of the scratch gained higher velocities, leading to increased removal at the base. This phenomenon was characterized by a decrease in the rate of scratch removal, which was consistent with previous analysis at sub–section 2.3.

Figure 16 depicts the morphology of the scratch during the removal. As the removal progresses, the scratch gradually evolves from a rectangle to a V shape with unique troughs at the edges. The shape results from the uneven removal at the bottom of the scratch. Despite the precise calibration of the angle between the removal and scratch directions, limitations in the calibration precision prevent achieving a theoretical ${0^\circ }$. A small angle exists between the removal and scratch directions, which can result in the non-uniform removal of material at the bottom of the scratch. The bottom of the scratch tilts to one side, and the uneven removal eventually leads to the evolution of the shape of the scratch. Notably, the positioning error when calibrating the coordinate system of the processing machine tool is within 1 mm and the diameter of the fused quartz element is 150 mm. It can be inferred that the angle error is within 1 degree.

 figure: Fig. 16.

Fig. 16. Morphology of the scratch at ${0^\circ }$ considering different removal depths.

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The unique troughs formed at the edge of the scratch were attributed to this small angle. The simulation results offer two premises: (1) when removing the scratch at ${0^\circ }$, particles tend to flow toward the scratch owing to the uneven pressure distribution; (2) during scratch removal at ${90^\circ }$, the low pressure at the distal edge of the removal causes an abnormal protrusion. This small angle is between ${0^\circ }$ and ${90^\circ }$. Both characteristics are assumed to exist in this angle. Based on the aforementioned two premises, the formation of anomalous troughs can be described using the physical model depicted in Fig. 17. The particles are blocked by protrusions in the process of scratch generation, and the accumulated particles result in significant material removal, creating the unique troughs. The arrows in Fig. 17(a) indicate the direction of inclination.

 figure: Fig. 17.

Fig. 17. Illustration of a physical model for the morphological evolution at ${0^\circ }$.

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3.4 Comparison of the characteristics of the scratch at the two angles

Figure 18 illustrates a comparison of the material removal depth, scratch removal depth, and the change in scratch width at the two angles. The following conclusions can be drawn: (1) the removal efficiency of ${90^\circ }$ is higher than that at ${0^\circ }$ when removing scratches of the same size; (2) the width variation is greater at ${90^\circ }$ than that at ${0^\circ }$ when removing same-sized scratches. In the initial stages of the removal at ${0^\circ }$, the overall scratch removal rate is large because of the poor fluidity and reduced removal rate at the bottom of the scratch. At ${90^\circ }$, the particles cannot completely touch the bottom of the scratch because of inertia, and the removal rate is reduced at the bottom of the scratch; The overall scratch removal rate is high.

 figure: Fig. 18.

Fig. 18. Comparative analysis of scratch removal at ${0^\circ }$ and ${90^\circ }$ in terms of (a) removal efficiency and (b) width variation.

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Consequently, no significant difference was observed in the removal rate between the two angles at the beginning. In the later stages of scratch removal, the lateral expansion of the scratch resulted in enhanced particle fluidity at ${0^\circ }$. This led to a minimal difference in removal rates between the surface and bottom at ${0^\circ }$, resulting in a lower overall removal rate for the scratch at a macro level. At ${90^\circ }$, particles could not make complete contact with the bottom, and the obstruction at the edges of the scratch further reduced fluidity compared to that at ${0^\circ }$. This resulted in a lower removal rate at the bottom than that observed at ${0^\circ }$, and on a macro scale, the scratch removal efficiency at ${90^\circ }$ appeared higher than that at ${0^\circ }$. The greater the material removal depth and the wider the scratch expansion, the more pronounced this difference.

4. Conclusions

In this study, fluid dynamics simulations and scratch removal experiments were performed to analyze the removal efficiency and morphological evolution of scratches using MRF at two angles. The fluid dynamics simulation model was established, and the velocity, pressure distribution, and particle trajectories were analyzed for both angles. In particular, the erosion rate distribution at ${90^\circ }$ was investigated. Furthermore, a physical model was proposed for differential removal to predict the morphological evolution and removal efficiency. The obtained results indicated that scratches gradually expanded to submillimeter orders and evolved into a V-shaped morphology in both cases. A material removal depth of 1 $\mu m$ could not remove a 1-µm-deep scratch, and residual depth remained. Additionally, the comparison of the difference in removal rate between the two angles indicated that the removal rate was higher at ${90^\circ }$ than that at ${0^\circ }$. Therefore, although complete scratch removal may be difficult, selecting an appropriate angle can significantly enhance the efficiency of reducing its visibility. The transformation of the scratch morphology at ${0^\circ }$ exhibited different characteristics, with anomalous troughs appearing near the residual scratches. The formation of these troughs was attributed to the small angle between the directions of removal and the scratch. However, further investigations are required to determine this small angle to utilize or avoid this feature. The determination of this small angle is impeded by the insufficient calibration accuracy of the equipment. In addition, future studies should focus on assessing the effect of MRF on removal of scratches at other angles and performing more accurate simulations by considering the effect of magnetic fields as well. In summary, the study findings provide both theoretical and experimental guidance for implementing MRF to efficiently remove scratches on large-aperture optical components.

Funding

National Key Research and Development Program of China (2022YFB3403405); Youth Innovation Promotion Association of the Chinese Academy of Sciences (2021215); National Natural Science Foundation of China (62275246, 62305334).

Acknowledgments

The authors would like to thank the anonymous referees for their valuable suggestions and comments that have helped improve the paper.

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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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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Figures (18)
Fig. 1.
Fig. 1. Schematics of the (a) removal function and (b) removal angle.

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Fig. 2.
Fig. 2. Two-dimensional geometric model for the computational fluid dynamics simulation.

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Fig. 3.
Fig. 3. Simulation results of the pressure distribution.

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Fig. 4.
Fig. 4. Pressure curves of the (a) edge of the distal end and (b) bottom of the scratch.

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Fig. 5.
Fig. 5. Simulation results of the velocity distribution. (a) Velocity magnitude distribution. (b) Velocity distribution in the y-direction.

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Fig. 6.
Fig. 6. Schematic of particles impacting the scratches. (a) Particle trajectories and erosion rate distributions. (b) Partial enlarged view of the erosion rate.

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Fig. 7.
Fig. 7. Schematic of the model and simulation results. (a) Simulation model. (b) Pressure distribution on the x–y plane, z = 5 $\mu m$. (c) Pressure distribution curve extracted near the scratch. (d) Velocity distribution on the x–y plane, z = 5 $\mu m$.

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Fig. 8.
Fig. 8. Schematic of the magnetorheological finishing (MRF) technique at ${0^\circ }$ to remove scratches.

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Fig. 9.
Fig. 9. Magnetorheological finishing (MRF) equipment used in the experiment.

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Fig. 10.
Fig. 10. Relationship between the depth of material removal and the (a) depth of scratch removal and (b) the amount of change in width.

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Fig. 11.
Fig. 11. Morphology of scratches at different removal depths.

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Fig. 12.
Fig. 12. Illustration of a physical model for the morphological evolution of scratch removal at ${90^\circ }$.

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Fig. 13.
Fig. 13. Comparison of scratch removal rates for different widths with respect to (a) the depth of scratch removal and (b) the amount of change in width.

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Fig. 14.
Fig. 14. Evolution of scratches with varying widths. (a) 20 $\mu m$, (b) 50 $\mu m$, and (c) 100 $\mu m$.

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Fig. 15.
Fig. 15. Relationship between the depth of material removal and the (a) depth of scratch removal and (b) the amount of change in width.

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Fig. 16.
Fig. 16. Morphology of the scratch at ${0^\circ }$ considering different removal depths.

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Fig. 17.
Fig. 17. Illustration of a physical model for the morphological evolution at ${0^\circ }$.

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Fig. 18.
Fig. 18. Comparative analysis of scratch removal at ${0^\circ }$ and ${90^\circ }$ in terms of (a) removal efficiency and (b) width variation.

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Tables (4)

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Table 1. Specific parameters of scratches.

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Table 2. Experimental data at 90 ∘ .

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Table 3. Experimental data for the removal of scratches with varying widths.

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Table 4. Experimental data at 0 ∘ .

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Equations (9)

Equations on this page are rendered with MathJax. Learn more.

M R R = k P V ,
∂ ρ ∂ t + d i v ( ρ u ) = 0 ;
∂ ( ρ u ) ∂ t + d i v ( ρ u u ) = d i v ( μ   g r a d   u ) − ∂ p ∂ x + S u ;
∂ ( ρ υ ) ∂ t + d i v ( ρ υ u ) = d i v ( μ   g r a d   υ ) − ∂ p ∂ x + S υ ;
∂ ( ρ ω ) ∂ t + d i v ( ρ ω u ) = d i v ( μ   g r a d   ω ) − ∂ p ∂ x + S ω ,
d u p d t = F D ( u − u p ) + g x ( ρ p − ρ ) ρ p + F x ,
F D = 18 μ ρ p d p 2 C D R e 24 ,
P = P d + P m = 2 η 0 U x Z 2 + V μ 0 M ∇ H ,
P 1 P 2 = Z 2 2 Z 1 2 = 1.001 ,
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