Theoretical modeling and analysis of material removal characteristics for KDP crystal in abrasive-free jet processing

Wei Gao; Qilong Wei; Jianwei Ji; Pengfei Sun; Fang Ji; Chao Wang; Min Xu

doi:10.1364/OE.27.006268

1. Introduction

Potassium dihydrogen phosphate (KDP), an excellent nonlinear single-crystal, can be used as a frequency converter or as a polarization electro-optical switch [1,2]. It is a key optical material in high-energy laser systems, such as the National Ignition Facility (NIF) in USA, Laser MegaJoule (LMJ) in France, and SG-III Laser Facility in China [3–5]. However, due to its high brittleness, soft texture, ready deliquescence, and sensitivity to temperature KDP crystal is regarded as very difficult to cut or polish, especially for high-quality requirements [6,7]. Currently, single-point diamond turning (SPDT) is used for precise KDP machining [8–10], but it can introduce microcosmic grooves, scratches, and cracks on the crystal surface, which produce a diffraction effect and stray light. Consequently, the surface quality does not meet the requirements of high-energy laser systems [11,12]. With the increasingly urgent need for ultra-precision KDP, many polishing methods, such as magnetorheological finishing (MRF), ion-beam figuring (IBF), and chemical mechanical polishing (CMP), have been proposed for enhancing surface accuracy in recent years.

As a flexible polishing technology, MRF has been widely applied in finishing other hard optical materials, such as fused silica, SiC etc [13–15]. Based on the water-solubility of KDP, the research on KDP MRF has been carried out using a non-aqueous magnetorheological fluid [16–18]. However, KDP MRF is limited by the issue of embedding of particles, which are very difficult to remove. The residual particles can cause secondary pollution and significantly decrease the threshold for laser damage to KDP crystal by absorbing a sufficient amount of energy to irreversibly modify the KDP surface structure [19]. This problem may be solved by applying IBF, but there is a high temperature gradient field for IBF processing, which would generate cracks or breaks [20,21].

CMP technology based on water-in-oil microemulsion circumvents the problem of embedding of particles [22,23]. However, large-sized KDP CMP is difficult to ensure a good flow property and removal uniformity due to the high viscosity of the polishing fluid at the KDP surface, which limits its broad application. The above methods have nevertheless demonstrated the feasibility of water dissolution polishing of KDP crystal, in spite of the aforementioned problems associated with MRF and CMP.

Abrasive jet polishing (AJP) is a non-contact deterministic precision manufacturing technique. Compared with other polishing methods, AJP has many advantages such as localized force, cooling of the debris [24,25]. As a flexible polishing method capable of processing any material, regardless of its properties, it has been widely used in polishing optical glass [26]. However, the AJP is not well-suited for KDP polishing because of the same issue to the embedding of particles, as described above for MRF.

Therefore, based on the water solubility characteristics of KDP crystal, we proposed a novel abrasive-free jet polishing (AFJP) method for KDP crystal with the purpose of improving surface quality without the embedding of particles [27]. This method makes use of a thermodynamically and kinetically stable microemulsion that contains nanometer range water droplets evenly dispersed in the non-aqueous carrier liquid. The sprayed out water nanodroplets can remove material through dissolution.

In our recent works, the feasibility of KDP AFJP and the trend of reducing surface roughness have been demonstrated [27]. On the other hand, this method can effectively reduce the deformed layer induced by mechanical stresses [28]. Hence, AFJP would seem to be a promising method for polishing KDP and for mitigating sub-surface defects of a KDP.

However, the removal mechanism of the KDP AFJP is not well clear. According to the AFJP experimental results, we found that the jet spot generated by AFJP was of an approximately Gaussian shape. In addition, the removal process of AFJP mainly involves nanodroplet impingement that plays a key role in processing of AFJP. Therefore, the analysis of the material removal characteristics will be useful for explaining the removal mechanism of the KDP AFJP. In this paper, nanodroplet impingement is firstly discussed and then a removal model of AFJP is proposed. Finally, experimental verifications are presented to evaluate the veracity of the model.

2. Principle and modeling of AFJP

2.1 Principle

The microemulsion is a system of water, oil, and a surfactant that exists as a single optically isotropic, thermodynamically and kinetically stable liquid solution. The droplet dispersion can be either of oil-in-water (o/w) or water-in-oil (w/o) type; in each case, the radii of the droplets are generally in the nanometer range [29]. In selecting an oil for KDP AFJP, it should be nonvolatile, or have low volatility, be nonflammable at room temperature with a high flash point, resistant to a potentially corrosive environment, unreactive towards KDP and the machine components, have low or no toxicity, and, lastly, not be capable of dissolving or adversely affecting the optical surface of a KDP crystal [16]. Ionic liquids (ILs) are organic salts that are liquids at room temperature. They are being increasingly studied as environmentally benign media or catalysts for chemical reactions and new style functional materials with promising applications in many fields, due to their unique and attractive physicochemical properties, including zero volatility, nonflammability, high chemical/thermal stability, and low toxicity [30,31].

Therefore, in this study, a common IL, 1-butyl-3-methylimidazolium hexafluorophosphate (bmimPF₆), has been used to prepare w/o microemulsions for KDP AFJP. The addition of a surfactant, such as Triton X-100 (TX-100), is critical for the creation of small-sized droplets, as it decreases the interfacial tension, i.e., the surface energy per unit area, between the oil and water phases of the emulsion.

In contrast to ordinary emulsions, which are kinetically stable but thermodynamically unstable and will undergo phase separation, microemulsions are thermodynamically stable and therefore do not require high inputs of energy or shear conditions for their formation. Therefore, TX-100/H₂O/bmimPF₆ microemulsions were prepared in two steps in this research, following a similar method as reported previously [32]. Some TX-100 was dissolved in bmimPF6 under magnetic agitation for some minute min, and then deionized water was dropped into the solution.

In a static state, the water contained in the droplets was separated from the KDP by the long chain surfactant without jetting, as shown in Fig. 1(a). Once this barrier was ruptured by the jet flow, the water in the droplets can contact and remove KDP at the impingement interface, as shown in Fig. 1(b). The microscale removal effects could not only keep smooth surfaces but also avoid KDP re-deposition. Since the jet velocity was relatively large (about 20 m/s), the contact time between a water droplet and KDP surface was very short. The shorter contact time results in the less material removal amount for each water droplet. Therefore, the dissolved KDP was miscible in a large number of nanoscale water droplets and did not concentrate on the KDP surface, which avoids the re-deposition of dissolved KDP and obtains controllable material removal.

Fig. 1 Sketch of (a) water droplet in microemulsion and (b) processing principle of the AFJP.

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In order to obtain appropriate water content in IL microemulsions, an experiment of compatibility tests was firstly investigated. Compatibility tests were conducted on a 10 mm × 10 mm KDP surface that had been marked with an NHT² nano-indentation system from CSM Instruments. The marked KDP was placed in two abrasive-free jet fluids, namely IL microemulsion containing certain percentages of water and the other is the polyethylene glycol (PEG)-200 also containing the same percentages of water, and then its surface was observed by means of a ZEISS Auriga scanning electron microscope (SEM) after soaking for 14 h.

Figure 2 shows the results of compatibility tests. For the PEG-200 water system that is a water/oil miscible fluid system, there were serious etching pits (Figs. 2(d) –2(f)) not only in the areas around the mark but also in the natural surface even if the water content is 0.04g/ml. In general, the areas around the mark are the easiest to be dissolved because here there are many defects and the chemical reactivity is high. However, the experimental results (Fig. 2(a)) suggested that dissolution around the mark or other areas on the KDP surface almost did not occur, since the long-chain surfactant coating on the water droplets directly avoids exposure of the KDP surface to water through a steric hindrance effect. However, there are still some etching pits with increasing the water content in IL microemulsion. Therefore, an appropriate water content that is 0.04 g/ml was used to this study.

Fig. 2 SEM images of KDP surface with an indentation mark after soaking in IL microemulsion containing (a): 0.04 g/ml, (b): 0.06 g/ml, (c): 0.12 g/ml; and PEG-200 system containing (d): 0.04 g/ml, (e): 0.06 g/ml, (f): 0.12 g/ml for 14 h.

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2.2 Modeling

The removal mechanism of AFJP involves dissolution in water droplets to complete material removal rather than the shear stress of abrasive particles. The jet impingement effects provide an impact force that can keep the sprayed out nanoscale water droplets contacting with KDP target surfaces. Nanodroplet impingement is crucial to AFJP, thus it is firstly discussed. Li et al [33] investigated normal impinging of nanoscale water droplets on the solid surface through molecular dynamics simulations, and found that there is not impinging breakup for nanoscale water droplets when the impinging velocity is small.

Figure 3 shows the sketch of deformation of a water droplet in compressing, restoring, and slipping process, respectively.

Fig. 3 Sketch of deformation of a water droplet in (a) compressing, (b) restoring, and (c) slipping process.

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As shown in Fig. 3(a), in deformation, the contact area between nanodroplet and solid surface is

A = π r_{x}^{2},

where r_x is the radius of contact region that is positively proportional to the deformation radius x of nanodroplet. It can be assumed as follows

r_{x} = \sqrt{C} (x - r_{0}),

where C is constant. Equation (1) can be expressed as

A = C π {(x - r_{0})}^{2} .

During deformation, the total materials removal can be expressed as

V = \int_{0}^{t_{m}} A \cdot S d t,

where S is the account of KDP dissolved by water per area per time in contact region. We assume that the movement of the frontier of the droplet is initialized with speed whose value is the same as that of impinging velocity. The speed of the frontier uniformly decelerates until it reaches 0 at the maximum spreading state. The average moving speed of the droplet frontier is

\bar{v} = v / 2

. Thus the total time t_m in deformation can be expressed as

t_{m} = d / \bar{v} = D_{m} - D_{0} / v

and deformation radius x of nanodroplet is

x = \bar{v} \cdot t + r_{0}

, where r₀ is the initial radius of droplets and v is the initial normal component of droplet’s impacting velocity. The distance that the frontier moves in the radial direction is

d = (D_{m} - D_{0}) / 2

. Equation (4) can be solved under the initial conditions t = 0 when x = r₀ and t = t_m when x = r_m, one has

V = 2 \int_{r_{0}}^{r_{m}} \frac{A \cdot S}{\bar{v}} d x,

As a result, the total materials removal is given by

V = \frac{1}{2} \frac{C π D_{0}^{3} S}{v} [\frac{1}{3} (β_{m}^{3} - 1) + β_{m} - β_{m}^{2}],

where β_m is the maximum spreading factor is defined as

β_{m} = D_{m} / D_{0}

, where D_m is the maximum spreading diameter or the maximum diameter of a droplet during deformation and D₀ is the initial diameter or the diameter of a droplet before impingement. This factor is necessary to describe the droplet deformation. Substituting

t_{m} = d / \bar{v} = D_{m} - D_{0} / v

to Eq. (6), the average material removal rate in deformation is given by

V_{t} = \frac{1}{6} C π D_{0}^{2} S {(β_{m} - 1)}^{2} .

According to the geometrical relation as shown in Fig. 3(a), the contact area has a relation of A<A_n, where $A_{n} = \frac{1}{3} π D_{0}^{2} β_{m}^{2}$ [33]. Substituting Eq. (3) into A<A_n, the constant C should meet $C < \frac{4}{3} {(\frac{β_{m}}{β_{m} - 1})}^{2} < \frac{4}{3}$ . According to the Eq. (7), the materials removal rate of a water nanodroplet increase with increasing its maximum spreading factor β_m in a parabola form.

Li et al [33] proposed an expression that implicates the maximum spreading factor with impingement velocity for the normal impinging of nanoscale water droplets on the solid surface as follows

\frac{3}{160} (β_{m}^{5} - 1) + \frac{γ}{η v} (\frac{1}{3} β_{m}^{2} + \frac{2}{3} \frac{1}{β_{m}} - 1) = \frac{D_{0} v ρ}{12 η} .

where γ, ρ, η are the coefficients of surface tension, mass density, and viscosity, respectively.

The dynamic behavior in the spreading of nanodroplet impingement on the solid wall is presented in Fig. 4, where γ = 11.06 mN/m [32], is the surface tension between water and IL, η = 0.851 mPa·s, and ρ = 0.996 g/cm³ is the viscosity and density of water, respectively. As shown in the inserted figure of Fig. 4, the maximum spreading factor showed good linear relationships at lower velocity.

Fig. 4 Plots of the maximum spreading factor as a function of the impingement velocity.

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The spreading water droplet restores with slipping as shown in Fig. 3(b). We assume that the process that spreading water droplets gradually restore under the action of a concentrated force is equivalent to that under the action of pressure as shown in Fig. 3(c). The restoring process is force equilibrium at lower restoring velocity, therefore the force equilibrium Eq. can be given by

γ \cdot 4 Δ {L^{'}}_{A C} = γ \cdot 4 c Δ L_{A C} = F,

where

Δ {L^{'}}_{A C}

is the length change of arc AC,

Δ L_{A C}

is the length change of line AC,

Δ {L^{'}}_{A C} = c Δ L_{A C}

(c≈1.1).

According to the geometrical relation as shown in Fig. 3(c), the length change of line AC has the relation of $Δ L_{A C} = \sqrt{L_{O C}^{2} + L_{O A}^{2}} - \sqrt{L_{O^{'} D}^{2} + L_{O^{'} B}^{2}}$ . By solving the Eq., the $Δ L_{A C}$ finally can be written as

Δ L_{A C} = \frac{D_{0}}{2} (\sqrt{β_{m x}^{2} + \frac{1}{β_{m x}^{2}}} - \sqrt{2}),

For the whole droplet, the force F can be regarded as a resultant force which roots in the pressure, thus the Eq. (9) can be rewritten as

γ \cdot 4 c Δ L_{A C} = P A^{'},

where A′ is the vertical projected area of a water droplet that is given by

A^{'} = π r^{2} = \frac{π}{4} D_{0}^{2} β_{m x}^{2}

.

Combining the Eq. (10), the relationship between the spreading factor and surface pressure can be expressed as follows

γ \cdot 8 c (\sqrt{β_{m x}^{2} + \frac{1}{β_{m x}^{2}}} - \sqrt{2}) = P \cdot π D_{0} β_{m x}^{2},

According to the Eq. (12), the spreading factor versus surface pressure at different diameter of water droplet can be shown in Fig. 5. It can be seen that the larger the diameter of the water droplet, the more sensitive it is to the pressure.

Fig. 5 Plots of the spreading factor as a function of the surface pressure.

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Finally, combining the Eq. (3), the slipping removal rate of a water droplet is given by

R (x) = \frac{1}{4} C π D_{0}^{2} S {(β_{m x} - 1)}^{2} .

where β_mx is spreading factor of water droplet during slipping that is described by Eq. (12).

The pressure also satisfies a certain linear relationship within a certain range. The pressure here is the pressure that the liquid gives to the water droplet, not is the pressure on the wall, but the pressure on the water droplet is less than the pressure on the wall. It can be seen from Fig. 5 that only a small pressure is required to cause a certain deformation of the water droplet so that the water droplet contacts the surface to achieve material removal.

3. Simulation and experiments

To demonstrate the predictability of the numerical model, and validate its effectiveness, a series of simulation and spot experiments were conducted under different polishing conditions and different materials.

3.1 Fluid field distribution

The removal mechanism of traditional AJP utilizes shear stress to complete material removal when abrasive particles are embedded in the surface under pressure [34]. According to the above theoretical analyses, the velocity and pressure play an important role in influencing the removal rate. Therefore, the distribution of velocity and pressure field distribution are very necessary for analyzing removal characteristics. The material removal mechanism of AFJP is different to that of traditional AJP, but the simulated velocity field distribution of AFJP obtained with ANSYS Fluent software is similar to that of traditional AJP.

3.2 Experimental setup

Jet spot experiments were conducted on precision equipment as shown in Fig. 6. The pressurization device was a gear pump that could be adjusted in the pressure range 0–1.5 MPa at flow rates of 0–2 L/min. The IL microemulsion used had a composition of 60 wt% bmimPF6 IL, 3 wt% (0.04g/ml) deionized water, and 37 wt% surfactant TX-100. Spot experiments were conducted a KDP crystal of 70 mm × 70 mm × 15 mm. Table 1 summarizes the impingement velocity of the jet center at different standoff distances and revolving speeds of the pump (RSP).

Fig. 6 Photographs of the experimental set-up.

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Table 1. Impingement velocity of the jet center at different standoff distances (units: m/s).

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4. Results and discussion

The simulations results of flow field distribution are shown in Fig. 7. The material removal mechanism of AFJP is different to that of traditional AJP, but the simulated velocity field distribution of AFJP obtained with ANSYS Fluent software is similar to that of traditional AJP (Fig. 7(a)). We can see that the flow crooks to two sides when the fluid jet impinges the workpiece wall. The whole impinging jet can be divided into three regions: free jet region, impingement region, and wall jet region. In impingement region, the jet undergoes significant bending and then there is a large pressure gradient as shown in Fig. 7(d). The axial velocity of the jet decreases rapidly in the impingement region and it is reduced to zero at the stagnation point. The radial velocity gradually increases due to the pressure gradient until reaching the maximum, and then it attenuates to zero quickly. The velocity profile on the wall is M-shaped as shown in Fig. 7(c). In addition, Peng [35] has demonstrated that small particles (<5 μm) follow the fluid streamlines very closely. The radii of the water droplets are in the nanometer range (<50 nm), therefore the trajectories of water droplets are dependent on the jet velocity field distribution as shown in Fig. 7(b).

Fig. 7 Simulation results of (a) flow field, (b) trajectories of water droplet, (c) velocity distribution on the workpiece and (d) pressure distribution on the workpiece for different impingement velocity.

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Figure 8 shows the 3D morphology features and 2D morphology features of jet spots generated with the IL microemulsion under different experimental conditions. The jet spot generated by the IL microemulsion was of an approximately Gaussian shape, with a smooth surface, and was free from traces of jet fluid flow. This suggested material removal with the IL microemulsion as an abrasive-free jet fluid to be controllable and selective.

Fig. 8 3D material removal characteristics and 2D morphology features of jet spots (time: 5 min, pressure: 0.5 MPa, nozzle: 1 mm).

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It can be seen from Eq. (7) that the impingement removal rate of the water droplet is proportional to the square of the impingement velocity. According to the characteristics of the jet, the removal of the jet center point is basically impingement removal. Therefore, the removal of the jet center point is in accordance with Eq. (7). Figure 9 shows the removal rate of the center point of the removal function as a function of the square of the center impingement velocity. It can be seen that the experiment and the theory are in good agreement, indicating the accuracy of the model.

Fig. 9 Experimental fitting results and theoretical calculation results.

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For slipping removal, we discuss the motion of a water droplet, as shown in Fig. 10.

Fig. 10 Schematic diagram of the water droplet removal process.

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According to Eq. (7), the impingement removal rate of the water droplet is proportional to the square of the impingement velocity. We assume $β_{m} - 1 = \sqrt{a} v_{y}$ , where a is constant. The Eq. (7) can be rewritten as

V_{t} = \frac{1}{6} C π D_{0}^{2} S a \cdot v_{y}^{2},

According to the illustrated geometric relationship, there is $v_{y} = v_{0} \cos θ$ , where θ varies from 0° to 90°.

The macroscopic impingement removal rate can be expressed as

R_{t} (x) = \frac{1}{6} N C π D_{0}^{2} S a v_{0}^{2} \cdot \cos^{2} θ .

where N is the number of impingement of water droplet applied by each point in the impingement area per unit time. Because the velocity of the water droplet is not constant in the process of slipping on the wall, the number of water droplet applied by each point in the wall area per unit time is also not a constant. Thus, the macroscopic removal rate is related to the slipping velocity on the sliding wall. We assume the number of water droplet applied by each point in the wall area per unit time conforms to the relationship of N = kv_x. The macroscopic slipping removal rate can be expressed as

R (x) = \frac{1}{4} k C π D_{0}^{2} S {(β_{m x} - 1)}^{2} \cdot v_{x} .

where k is constant, v_x is the radial velocity of water droplet which can be obtained by fluid simulation.

We assume that the pressure given by the fluid to the water droplet remains constant along the wall, then the total removal is the sum of the impingement removal and the slipping removal. We can convert the angle to coordinates and then normalize it to get the normalized removal function curve.

According to the trajectory of the water droplet (see Fig. (10)), assuming that the motion trajectory of the water droplet is circular, then there is ${(x - L)}^{2} + {(y - L)}^{2} = L^{2}$ . Further, there are $y = L - \sqrt{- x^{2} + 2 x L}$ , (0≤x≤L). According to the definition of the derivative, there is $y^{'} = \tan α = - \frac{L - x}{\sqrt{x (L - x)}}$ . Therefore, the $α = \arctan α = \arctan (- \frac{L - x}{\sqrt{x (L - x)}})$ , then according to the geometric relationship, $θ = α - \frac{π}{2}$ , the relationship between angle and x can be expressed as

θ = arc \tan (- \frac{L - x}{\sqrt{x (L - x)}}) - \frac{π}{2} .

Figure 11(a) shows the variation of cos²θ, indicating that the impingement removal of water droplet in the central impact region is linear. Figure 11(b) shows the normalized wall velocity curve distribution. It can be inferred that the material removal is greatest at the maximum tangential velocity of the wall.

Fig. 11 The variation law of (a) cos²θ, (b) normalized wall velocity.

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Figure 12 shows a comparison of the trend of normalized theoretical removal function and experimental removal function, showing good agreement with the experimental results.

Fig. 12 A comparison of the trend of normalized theoretical removal function and experimental removal function (In case of 10d-800 r/min).

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The removal rates play an important role in ultra-precision polishing. According to this model, the removal rates were affected by critical process parameters such as jet pressure, particle size and jet velocity etc. Next, we will systemically research this part by optimizing process parameters to obtain a fine surface quality.

5. Conclusion

In this paper, theoretical modeling and analysis of the material removal characteristics for KDP crystal in abrasive-free jet processing have been presented. The sprayed out water nanodroplets remove material through dissolution. The motion process of a water droplet can be divided into two processes, namely the compressing process and slipping process. Similarly, the removal action by water droplets could be divided into contact removal and slipping removal. The maximum spreading factor plays an important role in entire jet process. The materials removal rate of a water nanodroplet increase with increasing its maximum spreading factor in a parabola form. The microscale removal induced by jet impingement effects instead of shearing action dominates the material removal in the center of impingement. In other words, the contact removal dominates material removal in the center of jet region (Impingement region), while the slipping removal is predominant in another region. Based on the removal rate of a single water droplet, the model for macroscopic slipping removal rate was proposed, which associated with the impingement velocity and fluid pressure. Jet spot experimental results have proven that the proposed model can explain and describe the mechanism of KDP AFJP. Hence, AFJP would seem to be a promising method for polishing and cleaning KDP without a surface residue or sub-surface defects. In addition, the concept of abrasive-free jet polishing may also provide a reference for the jet polishing of other materials.

Funding

National Natural Science Foundation of China (Grant No. 51575501 and No. 61801451); National Science and Technology Major Project of the Ministry of Science and Technology of China (2017ZX04022001); Research Program of Department of Science and Technology of Sichuan Province (18YYJC0320).

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RSP (r/min)	5d*	10d**	15d**
500	10.62	9.31	7.21
800	14.23	12.47	9.66
1000	15.92	13.95	10.81

Theoretical modeling and analysis of material removal characteristics for KDP crystal in abrasive-free jet processing

Abstract

1. Introduction

2. Principle and modeling of AFJP

2.1 Principle

2.2 Modeling

3. Simulation and experiments

3.1 Fluid field distribution

3.2 Experimental setup

4. Results and discussion

5. Conclusion

Funding

References

Cited By

Figures (12)

Tables (1)

Equations (17)

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