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Abstract
X-ray mirrors, which are essential for constructing synchrotron radiation light sources, are highly required for full-range spatial wavelength errors. This paper investigated power-law non-Newtonian fluids and pointed out that both three-body removal and shear removal existed in the shear thickening polishing process. Subsequently, this paper calculates the shear force of the power-law non-Newtonian fluid polishing fluid in polishing the surface with different frequency errors. It establishes an MRR model of shear thickening polishing in the frequency domain by combining it with the Archard equation. Then, this model is also applied to optimize the polishing fluid formulation and processing parameters. Finally, the removal effect of the optimized polishing fluid on the mid-frequency ripple error is experimentally verified. On Ф50 mm monocrystalline silicon, the removal of mid-frequency ripple error with a spatial wavelength of 1 mm was achieved by shear thickening polishing technique while converging the surface roughness to 0.14 nm. Finally, the experimental results were applied to monocrystalline silicon with a length of 500 mm. This work provides a new research idea for the existing shear thickening polishing process. It provides theoretical and technical support for removing the mid- and high-frequency errors in high-precision X-ray mirrors.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Utilizing electromagnetic radiation emitted by the bending acceleration of charged particles close to the speed of light in the presence of a magnetic field, synchrotron light source boasts bright intensity, high collimation, high brightness, narrow pulse, high polarization, and wide continuous adjustable energy, which is one of the most important facilities for modern scientific research and high-end equipment manufacturing. The X-ray mirror is the key optical component in the beamline system, enabling beam deflection, beam splitting and nano-focusing, and is typically made from monocrystalline silicon. The quality and performance of X-ray mirrors have a direct impact on the construction of synchrotron radiation light sources. At present, the surface roughness of X-ray mirrors is usually better than 0.3 nm, and for the fourth generation of synchrotron radiation light source, the requirement is even higher, reaching 0.1 nm; the surface shape error of the mirror surface needs to be better than 2 nm; and the slope error needs to be better than 0.1µrad [1,2]. Typical X-ray mirrors processing methods includes bonnet polishing (BP), magnetorheological finishing (MRF), ion beam figuring (IBF), and elastic emission polishing (EEM). Zeiss process route, which is the mainstream process route for X-ray mirror manufacturing, adopts IBF for final precision improvement [3–5]. However, the error correction capability of IBF is closely related to the diameter of the ion beam [6]. For conventionally used diameter of ion beam (usually >2 mm), IBF has limited correction effects for surface error with spatial wavelength lower than 1 mm. To amend such surface error, smoothing and ultra-smoothing processes need to be introduced into the machining process.
Shear thickening polishing (STP) technology is an elastic domain polishing technology based on a non-Newtonian fluid polishing substrate, which has shear thickening properties and abrasive dispersion. Processing of the polishing slurry and workpiece contact part of the relative motion of the shear thickening phenomenon, so that the dispersed solid colloidal particles into a large number of “particle clusters” and the abrasive will be wrapped in them. This is equivalent to the formation of a “flexible fixed abrasive” in the machining position. Because of these unique shear thickening characteristics, shear thickening polishing is more conducive to removing the workpiece’s surface roughness and maintaining low-frequency surface shape than traditional polishing. In order to explore the application of shear thickening polishing, researchers have done a lot of theoretical and applied research on shear thickening technology. Yuan [7–9] proposed a shear thickening polishing method based on power-law non-Newtonian fluids. He discussed the shear thickening mechanism of the non-Newtonian fluid polishing slurry, proposed a material removal model based on the shear thickening mechanism, and experimentally verified the feasibility of shear thickening polishing. Li [10] proposed a new anhydrous shear thickening polishing (ASTP) based method to inhibit dissolution and improve the physical properties of KDP processing. He developed a material removal model for ASTP in KDP processing and conducted validation experiments of the predictive model, resulting in an ultra-smooth surface with a surface roughness of 1.37 nm. Zhou [11] developed a surface roughness prediction model for magnetic field enhanced shear thickening polishing based on macroscopic computational fluid dynamics and convolutional iteration theory. The maximum relative error of this prediction model is 9.1%. Ming [12] used a magnetorheological shear thickening polishing technique to polish spherical zirconia workpieces. He established the corresponding material removal rate theories to reduce the values of Ra and PV on the hemisphere from 131.8 nm to 19.9 nm and from 0.979µm to 0.372µm, respectively. He realized the non-destructive machining of hard and brittle materials by the shear thickening polishing technique. An adaptive shear thickening polishing method was proposed by the Leibniz Institute for Materials Engineering [13]. The institute utilized temperature variation to regulate the viscosity of the polishing slurry and conducted polishing experiments on LN crystals by the shear thickening polishing method. Finally, they obtained sub-nanometer nondestructive ultra-smooth surfaces. Shao [14] uses non-Newtonian fluid and flexible fibers to achieve the high consistency passivation of the cutting edge, resulting in a reduction of cutting edge surface roughness Ra from 118.00 nm to 9.35 nm; Zhang [15] uses chemical enhanced non-Newtonian ultrafine (CNNU) slurry to process NiP alloy aspheric optical mold to achieve sub-nanometer roughness on aspherical optical molds; Guo [16] restrains the generation of boundary steps of W–Ni–Fe alloy by balancing the mechanical and chemical effects during CMP processing. In summary, non-Newtonian fluid polishing has been widely used in ultra-precision machining for its high machining quality. However, at present, its main application areas are the reduction of surface roughness Ra of difficult-to-machine materials or metal materials. Most removal models are based on macro-scale predictions of removal rates during shear thickening and polishing, but their effectiveness in removing mid-frequency errors has received less attention.
This paper firstly analyzed the effect of shear thickening polishing on the removal of mid- and high-frequency error, pointing out that it has a good effect on the removal of high-frequency error but a poor effect on the removal of mid-frequency error. Then, power-law non-Newtonian fluids were investigated, and it was pointed out that both three-body removal and shear removal exist in the shear thickening polishing process. Subsequently, this paper calculates the shear force of the power-law non-Newtonian fluid polishing slurry in polishing the surface with different frequency errors, and establishes an MRR model of shear thickening polishing in frequency domain by combining with the Archard equation. Then, this model is also applied to optimize the polishing slurry formulation and processing parameters. Finally, the removal effect of the optimized polishing slurry on the mid-frequency ripple error is experimentally verified. Under the premise of ensuring the effective removal of high-frequency errors, the removal of mid-frequency ripple errors with a spatial wavelength of 1 mm by the shear thickening polishing technology is realized. This work provides a new research idea for the existing shear thickening polishing process. It provides theoretical and technical support for the mid- and high-frequency error removal process of high-precision X-ray mirrors.
2. Processing effect of STP
In order to verify the effect of shear thickening polishing on the removal of mid- and high-frequency errors from the workpiece, monocrystalline silicon polishing experiments were carried out using the laboratory's own CCOS polishing machine. The schematic is shown in Fig. 1. Rotational speed U and applied pressure P are determined based on preliminary a priori experiments to ensure that the shear thickening slurry works in the shear thickening stage. The polishing pads are made of pitch that is suitable for polishing monocrystalline silicon. The principle of single variable was ensured during the experiment to ensure that the diameter of the polishing pad, the rotational speed of the pad, the applied pressure, the feed rate, and the concentration of the abrasive as well as the ambient temperature and humidity were kept constant during the polishing process, and the processing time was recorded at the same time. The processing parameters are in Table 1:
Table 1. Experimental conditions.
The chemical-mechanical polishing slurry was formulated with ordinary SiO2 polishing slurry. The polishing slurry had a pH value of 8.5-9.5 and contained SiO2 abrasive with a rounded shape, an average particle size of 50 nm, and a content of 50 wt%. The shear thickening polishing slurry was formulated with silica/polyethylene glycol series. The shear thickening polishing slurry was non-Newtonian fluid. The SiO2 content was 50 wt%, with round particles averaging 50 nm in size. What’s more, the slurry contained 30 wt% polyethylene glycol and 2 wt% xanthan gum to thicken it. The remaining amount consisted of deionized water and a pH value adjuster to maintain a pH value of 8.5-9.5. During the preparation process, ensure that the mass ratio of abrasive in the two polishing slurry is the same.
Shear thickening polishing experiments were carried out on monocrystalline silicon after magnetorheological finishing. After processing for a certain time, the surface shape and roughness of monocrystalline silicon were measured respectively, and power spectral density (PSD) curves were drawn for analysis. Low- and mid-frequency surface shape errors were measured using a Zygo VeriFire Asphere laser wavefront interferometer with a resolution of 0.159 mm/pix for vertical interferometer measurement and 0.683 mm/pix for horizontal interferometer measurement. Measurement repeatability is less than RMS 0.15 nm for the vertical interferometer and less than RMS 0.3 nm for the horizontal interferometer. Roughness was measured using a Zygo NewView 700 white light interferometer. The PSD curves are shown in Fig. 2 and the roughness measurement results are shown in Fig. 3. The results show that after STP, the surface roughness of monocrystalline silicon has significantly converged from the initial Ra 0.34 nm to Ra 0.23 nm. However, it can be seen from the PSD curves of the workpieces that the convergence of the 1 mm mid-frequency ripple error in the PSD curves caused by magnetorheological finishing is not obvious. To eliminate mid-frequency errors due to magnetorheological finishing, SiO2 polishing slurry was used for normal chemical-mechanical polishing on monocrystalline silicon samples with the same processing parameters and time. The roughness measurement results are shown in Fig. 3(c), the surface roughness of monocrystalline silicon has deteriorated from Ra 0.23 nm to Ra 0.46 nm. However, the mid-frequency ripple error of 1 mm in the PSD curve disappears, as can be seen from Fig. 2.
Fig. 3. Surface roughness of the workpiece surface: (a) the initial roughness; (b) after STP; (c) after CMP.
The results of the two experiments show that ordinary chemical-mechanical polishing has a high removal efficiency of mid-frequency error, but the surface roughness convergence efficiency is low and the convergence effect is general, whereas shear thickening polishing has an obvious effect on roughness convergence, but it cannot effectively remove mid-frequency error generated in the process of sub-aperture polishing.
3. MRR model of STP in the frequency domain
3.1 Principle of shear thickening polishing
Chemical-mechanical polishing is a combination of chemical reaction and mechanical removal of the polishing method, the removal process is quite complex, and there are many factors affecting. The reaction process is sown in Fig. 4: first of all, the surface material of the workpiece and the oxidizing agent, catalyst, etc. in the polishing slurry come into contact with each other and have a chemical reaction; subsequently, a layer of relatively easy to remove the soft layer on the surface of the workpiece is generated; finally, the soft layer on the surface of the workpiece is removed under the mechanical action of the abrasive in the polishing slurry and the polishing pad. Chemical-mechanical polishing achieves effective removal of the workpiece surface under the combined influence of chemical and mechanical effects, and is an important method for the manufacture of high-precision optical components.
As an extension of the chemical-mechanical polishing method, material removal by power-law non-Newtonian fluid polishing is also achieved by a combination of chemical and mechanical action. The removal mechanism of power-law non-Newtonian fluid polishing is closely related to the rheological properties of the polishing slurry, which mainly affects the mechanical removal effect to change the surface quality of the workpiece. A three-body friction and wear model typically arises between the polishing pad, abrasive, and workpiece. Such a model develops after a soft layer appears due to the chemical reaction between the particles present in the polishing slurry and the surface of the workpiece. Due to the fluid properties of power-law non-Newtonian fluids, significant shear forces are produced during the polishing process resulting in shear removal of the soft surface layer on the workpiece [17]. Figure 5 shows the schematic diagram of the velocity distribution of the polishing slurry between the polishing pad and the surface of the workpiece. From the figure, it can be seen that the polishing pad carries out rotational movement around the rotation axis with a certain angular velocity, and the height between the polishing pad and the workpiece is about the thickness of the elastic lubrication film. Assuming that the linear velocity of the bottom surface of the polishing pad is U, the speed of the polishing slurry in the position near the polishing pad is approximately equal to the speed of the polishing pad, the speed of the position near the surface of the workpiece is 0. Thus, the speed of the polishing slurry creates different speed gradients in the range between the polishing pad and the surface of the workpiece, and generates shear stresses of different sizes depending on the size of the value of the speed change at different positions.
Figure 6 shows a schematic diagram of the principle of shear removal of atoms from the surface layer of a workpiece in ultra-smooth polishing. As the abrasive atoms move in close proximity to the surface atoms of the workpiece, a hydrolysis reaction occurs between them and new bonds are created. At the same time, differences in the chemical properties of the hydroxyl and surface atoms lead to changes in the electron cloud density between the surface and subsurface atoms. The binding of surface atoms to subsurface atoms is weakened relatively. Shear stresses are generated by the gradient change in the rotational speed of the polishing slurry between the polishing pad and the workpiece surface. Especially in close proximity to the workpiece surface position, due to the chemical reaction between the abrasive and the workpiece surface atoms, the chemical bond between the workpiece surface atoms and the subsurface atoms of the bond energy is weakened, and it is more likely to be sheared by the fluid shear force. And the size of the fluid shear force is related to the frequency of the error peaks on the surface of the workpiece, the greater the frequency of the error peaks, the greater the normal shear force of the abrasive acting on the error peaks, resulting in more efficient removal of error peaks of higher frequency by the abrasive. In addition, the fluid shear force will preferentially remove surface atoms at higher locations of the error peaks on the workpiece surface. This is because the smaller the distance between the polishing pad and the surface of the workpiece, the greater the shear force will be when the polishing pad speed is kept constant, making it easier to achieve shear removal of surface atoms.
3.2 Governing equations of fluid flow and MRR model of STP process
In order to study the fluid flow in the region between the polishing pad and the workpiece, a detailed analysis based on the basic theory of hydrodynamic lubrication will be carried out in this paper. To simplify the analysis process, the following assumptions were made:
- (1) The surface of the polishing pad is a smooth rigid body.
- (2) The shear thickening polishing fluid is a continuous polishing fluid, which is a homogeneous, isotropic and incompressible laminar flow.
- (3) The density of the polishing fluid does not change with time.
- (4) The motion of the fluid satisfies the basic laws of physics.
- (5) The abrasive are rigid spheres of uniform diameter.
- (6) The abrasive is uniformly dispersed in the shear-augmented phase (containing components such as the dispersed phase, dispersing medium, additives, etc.).
Considering the microcells near the top of the error peak, as shown in Fig. 7, according to Newton's second law, the mechanical differential equation in the x-axis direction of this microcell is obtained as [18]:
The equation can be simplified as follows:
According to the generalized Newton's law of internal friction, the following equation is the relationship between tangential and normal stresses in a fluid:
According to assumption 2: shear thickening polishing slurry is an incompressible fluid, then there is a fluid continuity equation:
Substituting Eq. (3) and Eq. (4) into Eq. (2), the mechanical equation of this microcell in the x-axis direction is:
The equation can be simplified as follows:
The flow of the polishing slurry in the x-direction is constant during the polishing process, so $\frac{{\partial {v_x}}}{{\partial z}} = 0$. Substituting $\frac{{\partial {v_x}}}{{\partial z}} = 0$ into Eq. (6), the equation can be simplified as follows:
The power-law non-Newtonian fluid viscosity's intrinsic equation can be expressed in Eq. (8) [19,20]:
where m is the consistency index and n is the viscosity index.Based on the actual working condition of shear thickening polishing, the boundary condition of the fluid can be obtained as:
The determination of the velocity component is by means of a regular perturbation expansion [19,20]. The expression for the velocity component is then obtained by:
By inserting Eq. (9) into Eq. (7) and integrating the boundary conditions of shear thickening polishing, a Reynolds equation for a power law non-Newtonian fluid in shear thickening polishing can be derived [21,22]:
The polishing pad experiences the same velocity change in both the x and y directions during shear thickening polishing. Thus, when disregarding the y-direction variance, Eq. (10) can be simplified to Eq. (11) [23].
Simultaneous integration on both sides of Eq. (11) and further simplification reduces the equation to the following:
If the workpiece surface error has a frequency of $\omega$, and the peaks and valleys of the error follow a sinusoidal function, the equation for the thickness of the polishing slurry during the polishing process is:
Calculating the derivative of Eq. (13) with respect to the error frequency $\omega$, the outcome is:
The polishing slurry thickness in the microcell $\varDelta h$ is:
Substituting Eq. (12) and Eq. (15) into (9), the velocity of the polishing slurry in the x-direction can be simplified as:
Calculating the derivative of Eq. (16) with respect to the microcell's position along the z-direction z, the velocity gradient at that position is:
By inserting Eq. (13) into Eq. (16) and differentiating Eq. (16) with respect to the error peak's angle $\theta$ of inclination, the outcome is:
Shear thickening polishing slurry, as a non-Newtonian fluid, is characterized as a pseudoplastic fluid. Its shear stress during flow is usually defined using the Herschel-Bulkley model with the equation [24,25]:
where k is the Herschel-Bulkley coefficient, n is the viscosity index, and ${\tau _z}$ is the yield stress, all three of which are intrinsic properties of shear thickening polishing slurry.Within $\left\{ \begin{array}{l} z > \frac{{{h_0}}}{3}\\ \theta \in \left[ {0,\frac{\pi }{2}} \right) \end{array} \right.$, there is
Therefore, $\tau $ increases as $\theta$ increases.
The material removal rate model quantitatively describes the processing performance of shear thickening polishing. The model for material removal rate is capable of unequivocally describing the processing performance of shear thickening polishing. Based on the model, it is possible to predict processing efficiency using the given parameters, thereby establishing a theoretical basis for actual processing. The process of shear thickening polishing generates positive pressure, shear stress, and relative velocity within the polishing area. The abrasive is exposed to external influences to remove material from the surface of the workpiece. The Archard equation is the material removal equation that extensively describes sliding wear. According to the Archard equation, the material removal rate during the polishing process can be described as [26]:
where ${\rho _w}$ is the density of the workpiece, Karc is the dimensionless Archard wear constant, Fall is the total normal force, S is the sliding distance, t is the machining time, and H is the hardness of the contact surface.According to Eq. (19), the normal shear stress can be found as:
The shear force of the fluid acting on the abrasive is the product of the shear stress and the action area, and the action area is the projected area of the abrasive indentation. And according to Hertz contact theory, the depth of abrasive indentation ${h_H}$ is [27–29]:
Thus, the diameter of abrasive indentation ${d_i}$ is:
According to the equation, ${d_i}$ increases as ${\tau _n}$ increases. The projected area of the abrasive indentation ${S_a}$ is:
Therefore, the normal shear force exerted on the error peak by a single abrasive is as follows:
When conducting three-body removal, it is assumed that the abrasives are uniformly distributed and have the same diameter. Under these conditions, the pressure load on the abrasives that come into contact with the workpiece can be calculated as:
where p is the contact pressure transmitted to the abrasives by the polishing pad and N is the total number of abrasives in contact with the workpiece.Therefore, the total force of the abrasives on the error peak is:
where kp is the percentage of abrasives actually involved in material removal, which decays exponentially with increasing viscosity [30,31].By substituting Eq. (26), Eq. (27) and Eq. (28) into Eq. (21) and simplifying, the material removal rate can be calculated as:
Since the inclination angle θ has a real physical meaning, it must satisfy $\theta \in \left[ {0,\frac{\pi }{2}} \right)$, the above equations have a unique solution. Therefore, the error frequency ω is mapped one-to-one with the inclination angle θ and can be solved for. Therefore, the equation can be simplified as:
Fig. 8. Material removal rates of shear thickening polishing in frequency domain with different viscosity index n.
As shown above, the blue curve is the removal rate of each frequency error using the viscosity index n = 1.0 (ordinary SiO2 polishing slurry) and the red and yellow curves are the removal rate of each frequency error using the shear thickening polishing slurry with viscosity index n = 1.5 and n = 2.0 respectively. The material removal rate of chemical mechanical polishing is defined as the acceptable removal rate. If the removal rate is lower than the removal rate of chemical mechanical polishing, the polishing slurry is considered to have insufficient error removal capability for the corresponding frequency, and if the removal rate is higher than the removal rate of chemical mechanical polishing, the polishing slurry is considered to have high removal capability. Therefore, in this paper, the intersection ν1 and intersection ν2 of the above figure are defined as the error removal cut-off frequency of polishing slurry 2 (n = 1.5) and polishing slurry 3 (n = 2.0), respectively.
3.3 Discussion of MRR model of shear thickening polishing in frequency domain
As shown in Fig. 9, α abrasives represent the abrasives involved in three-body removal of material and β abrasives represent the abrasives involved in shear removal of material. In normal chemical-mechanical polishing, meaning when n1 = 1, the polishing fluid is a Newtonian fluid and does not exhibit the shear thickening effect of a non-Newtonian fluid. The conventional SiO2 polishing fluid also has shear force during processing. However, the shear force is small, so the shear removal of the material is negligible, and the variation of the shear force with the frequency of the error is also negligible. Therefore, the α abrasives are in majority and there are more abrasives involved in material removal. In shear thickening polishing, according to the intrinsic equation for shear thickening slurry, an increase in the viscosity index n of the polishing slurry results in a more significant shear thickening of the slurry. The shear force exerted by the polishing slurry on the abrasive is no longer negligible. The magnitude of the change in shear force with the change in error frequency increases as the in the viscosity index n increases, resulting in a more sensitive shear removal rate of the material to the error frequency. The number of β abrasives gradually increased as the viscosity index n increased, while the total number of abrasives involved in material removal decreased somewhat. During the polishing process, a lubricating film is created between the polishing pad and the workpiece. The film's lubricating effect enhances the quality of the workpiece surface, albeit at the cost of reducing the material removal rate. The number of β abrasives gradually increased as the viscosity index n increased, while the total number of abrasives involved in material removal decreased somewhat. During the polishing process, a lubricating film is created between the polishing pad and the workpiece. The film's lubricating effect enhances the quality of the workpiece surface, albeit at the cost of reducing the material removal rate. As the polishing slurry's viscosity increases, it strengthens the lubrication between the abrasive and the workpiece. This reduces the probability of abrasive colliding with the workpiece and, consequently, decreases the proportion coefficient of the abrasive involved in material removal kp. This reduction in kp results in a decrease in the three-body removal of material effect during polishing. Additionally, due to the low spatial frequency of the error peaks, the polishing slurry exhibits an inadequate shear removal effect on the material. Combining the above two reasons, using shear thickening polishing with high viscosity results in a low material removal rate for mid-frequency ripple errors with a spatial wavelength of 1 mm.
Therefore, based on the MRR model of shear thickening polishing in frequency domain derived in section 3.2, to enhance the material removal rate of mid-frequency error through shear thickening polishing technology and achieve the proficient removal of high-frequency error while ameliorating the effect of getting rid of mid-frequency error, it is possible to optimize the composition of the polishing slurry by selecting the appropriate shear thickening polishing slurry viscosity and viscosity index. Therefore, achieving a balance between the three-body removal effect of the polishing slurry on the material and the shear removal effect is crucial. This allows for the effective removal of mid- and high-frequency errors in X-ray mirrors through shear thickening polishing.
4. Experiments and discussion
To investigate the impact of shear thickening polishing on the elimination of mid- and high-frequency errors from the workpiece, monocrystalline silicon polishing experiments were carried out using the laboratory's own CCOS polishing machine. The principle of keeping a single variable constant was employed during the experiment to fix the diameter of the polishing pad, the rotational speed of the pad, the applied pressure, the feed rate and the concentration of the abrasive, as well as the ambient temperature and humidity during the polishing process. The processing time was concurrently recorded. The processing object is three pieces of Ф50 mm monocrystalline silicon after magnetorheological finishing with similar initial surface shape and roughness. In magnetorheological polishing, the trajectory of the polishing wheel was a 1-mm pitch grating track. As a result, these workpiece surfaces had significant mid-frequency errors. According to the model, to eliminate the 1 mm mid-frequency error, the viscosity coefficient of the shear-thickening polishing solution was changed to enhance the three-body removal of the abrasive on the material. The processing parameters are as follows:
Polishing was conducted using a raster scanning method. The chemical-mechanical polishing slurry was an ordinary SiO2 polishing slurry identical to the one used in Section 2. The two shear thickening polishing slurry was formulated with silica/polyethylene glycol series. The SiO2 content was 50 wt%, with round particles averaging 50 nm in size. What’s more, the slurry contained 20-40 wt% polyethylene glycol and 0.5-2 wt% xanthan gum to thicken it. The remaining amount consisted of deionized water and a pH value adjuster to maintain a pH value of 8.5-9.5. The three polishing slurry have the same abrasive concentration.
Removal experiments were performed using #1 high viscosity shear thickening polishing slurry, ordinary SiO2 polishing slurry and #2 shear thickening polishing slurry with reduced initial viscosity. Every 20 minutes of polishing, the surface shape and roughness of the monocrystalline silicon were measured and PSD curves were plotted for analysis. Processing was stopped when the peak in the PSD curve at a spatial wavelength of 1 mm was eliminated or the convergence ratio was less than 1.2, and the total processing time was recorded. Figure 10, Fig. 11 and Fig. 12 shows PSD curves of three pieces of monocrystalline silicon before and after polishing, roughness Ra evolution with scanning times and roughness Ra before and after machining, respectively.
Fig. 10. Results of #1 STP: (a) PSD curves of the workpiece surface before and after #1 STP; (b) evolution of workpiece surface roughness during the #1 STP; (c) the initial roughness; (d) the final roughness.
Fig. 11. Results of CMP: (a) PSD curves of the workpiece surface before and after CMP; (b) evolution of workpiece surface roughness during the CMP; (c) the initial roughness; (d) the final roughness.
Fig. 12. Results of #2 STP: (a) PSD curves of the workpiece surface before and after #2 STP; (b) evolution of workpiece surface roughness during the #2 STP; (c) the initial roughness; (d) the final roughness.
The experimental results show that the high viscosity #1 shear thickening polishing slurry has a poor effect on the removal of mid-frequency error on the surface, as shown in Fig. 10, the peak decrease of the PSD curve is not obvious, and the peak convergence ratio of the PSD curve has been less than 1.2 after processing for 60mins, but the convergence ratio of the surface roughness is high, and the convergence ratio of the surface roughness Ra at the first time of processing is 1.59, and the surface roughness Ra converges to 0.16 nm after processing for 60mins. As shown in Fig. 11, ordinary SiO2 polishing slurry has the best effect on removing the surface mid-frequency ripple error among the three types of polishing slurry, and the processing time required to remove the peak of the PSD curve is 100mins, which is the shortest. However, the processing of monocrystalline silicon using ordinary SiO2 polishing slurry, the surface roughness convergence rate is slow, the first processing of the surface roughness Ra convergence ratio is 1.22, and the roughness convergence ratio tends to 1 after the processing time reaches 80mins, and after processing 100mins, the surface roughness Ra converges to 0.21 nm.
As can be seen from Fig. 12, the mid-frequency peak in the PSD curve after magnetorheological finishing can be eliminated after 160mins of processing using the modified #2 shear thickening polishing slurry, indicating that after lowering the initial viscosity, the #2 shear thickening polishing slurry has a better effect of removing the mid-frequency ripple error on the surface, and maintains the effect of the removal of the high-frequency error. The surface roughness Ra convergence ratio of the first processing is 1.46, and the roughness convergence ratio tends to 1 after the processing time reaches 100mins, and the surface roughness Ra converges to 0.14 nm after 160mins of processing.
5. Processing of X-ray mirrors by STP
Based on the investigation of the removal mechanism of shear thickening polishing technology in the previous sections, focusing on the characteristics and requirements of X-ray mirrors in the context of practical engineering applications, and concentrating on the related engineering problems, this section describes the effects and role of shear thickening polishing in the manufacturing process of X-ray mirrors with a length of 500 mm. The machining process is shown in Fig. 13. The processing parameters are same as Table 2.
Table 2. Experimental conditions.
The experimental results are shown in Fig. 14. PSD analysis was performed on the surface of the workpiece after STP polishing, as shown in Fig. 14(a). It can be found that the deterioration of the low frequency error of the workpiece is not obvious, while the mid- and high-frequency error of the workpiece converges significantly and obviously from 1mm−1 after STP. The roughness results before and after polishing are shown in Fig. 14(b) and 14(c). The homogeneous streaks produced by magnetorheological processing disappear, while the roughness converges from Ra 0.43 nm to Ra 0.25 nm with a convergence ratio of 1.72. It can be seen that the shear thickening polishing well meets the processing requirements of X-ray mirrors for mid- and high-frequency errors at spatial frequencies above 1 mm−1 and roughness of workpiece.
Fig. 14. Results of STP: (a) PSD curves of the monocrystalline silicon with a length of 500 mm before and after STP; (b) the initial roughness; (c) the final roughness.
6. Conclusion
This paper proposes a theoretical analysis model of the material removal rate in the frequency domain of shear thickening polishing. Based on this model, shear thickening polishing removal rate was analyzed for each frequency band error. Finally, the mid-frequency error is suppressed by shear thickening polishing by optimizing the polishing slurry formula and the processing technology parameters. The results are as follows:
- (1) By analyzing the physical properties of the shear thickening fluid, this paper points out that in the process of shear thickening polishing, there are both three-body removal and shear removal of materials;
- (2) The theoretical analytical model of shear thickening removal rate in frequency domain was established according to the Archard equation by analyzing the fluid motion of shear thickening polishing slurry. The concept of shear thickening polishing error removal cut-off frequency is proposed, and the error removal cut-off frequency is related to the viscosity coefficient of the shear thickening fluid;
- (3) The shear thickening polishing slurry and polishing parameters were optimized, and the removal effect of the optimized polishing slurry on the mid-frequency ripple error was experimentally verified. The mid-frequency ripple error with a spatial wavelength of 1 mm was removed by the shear thickening polishing technique under the premise of the effectively removing high-frequency error.
In summary, the three-body removal and shear removal of materials both exist in shear thickening polishing. By adjusting the ratio of the two removal effects, shear thickening polishing can be achieved to ensure efficient removal of high-frequency errors while well suppressing the mid-frequency errors on the surface of the workpiece, which provides a new research idea for the existing shear thickening polishing process, and provides theoretical and technical support for the machining process and fabrication of high-precision X-ray mirrors.
Funding
National Natural Science Foundation of China (51991371, 51835013).
Acknowledgment
The project was financially supported by the National Natural Science Foundation of China (51835013, 51991371).
Disclosures
The authors declare that they have no known conflict of interest.
Ethics approval and consent to participate. The authors declare that they comply with ethical standards.
Consent for publication. The authors declare that the consent to the publication of the relevant data is granted.
Authors’ contributions. The signed authors have all participated in the work of this paper.
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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