Simulation to Microtopography Formation of CBN Active Abrasives on a Honing Wheel Surface

: The microtopography of a honing wheel surface composed of active abrasive grains is the key factor affecting the honing characteristics, and control of it is a sufﬁcient condition to realize high-efﬁciency precision honing. Based on the magnetron sputtering method and phase ﬁeld method, a theoretical model of cubic boron nitride (CBN) coating formation on a honing wheel surface is established. The physical vapor deposition (PVD) discrete phase ﬁeld equation is solved by the ﬁnite difference method. A MATLAB program is compiled to simulate the formation process and micromorphology of the CBN coating on the honing wheel surface. A Taguchi method is designed to study the relationships of the sputtering time, substrate temperature, gas ﬂow rate, and reaction space with the number of active abrasives and the length, width, height, and size of the abrasives. The simulation results are highly similar to the scanning electron microscopy (SEM) examinations, which shows that the model can accurately and effectively simulate the abrasive morphology of the wheel surface under different process conditions and provide a theoretical basis for the prediction and control of the CBN wear morphology on a honing wheel surface. different reaction spaces. The ﬁtted equation of the length of abrasive grains is (cid:0) y = 1.34476 + 0.00441 x − (cid:0) 6.37153 × 10 − 6 (cid:1) x 2 (cid:1) ,


Introduction
With the increasing requirements of gears for automotive dual clutch automatic transmissions (DCTs), robot joint cycloid reducers, etc., it has become essential to reduce gear noise and improve the accuracy, load capacity, and life span of hardened gears [1,2]. Therefore, the development of coated tools [3][4][5][6] and CNC machines and the advancement of machining technologies such as gear honing [7], gear grinding [8], gear shaving [9], gear turning [10], and gear milling (faced-milled gear) [11,12] have become an even more urgent task. The gear honing technology, with the advantages of high machining efficiency, low cost, low residual stress, no burns on the tooth surface, and regular honing wheel pattern on the tooth surface that helps to reduce gear transmission noise, can be an alternative to grinding and has been applied to the finishing of hardened gears after quenching [6,13].
The key to achieving internal power honing machine processing is the deposition of superhard abrasive coatings such as CBN and diamond on steel substrates to produce highperformance (HP) honing wheels [6]. Compared with the corundum honing wheels made of polyester or ceramic bonds [14], CBN honing wheels can remove interfering workpiece material at higher cutting speeds (6-10 m/s) [15], larger honing forces (>400 N) [16], and more material removal (50-100 µm/flank) [17] without any grinding operation. Thus, it can substantially improve the material removal rate, machining accuracy, and surface quality of hardened gears [1,18]. The micromorphology of the honing wheel tooth surface, the teeth of honing wheels, as well as the movement of grain boundaries [41]. Compared with the traditional method of surface micro-morphology modeling by probabilistic statistics [42], the phase field method considers the various thermodynamic driving forces of long-range and short-range interactions. Thus, it is able to take into account the effects of complex internal and external factors such as physical fields, micro-morphology of grain boundaries, anisotropy, sputtering temperature, and gas flow on the grain growth process [43], making the model closer to reality.
Hence, the main purpose of the present study was to carry out phase field simulations to investigate the grain sputtering, coarsening, and final film formation during the microstructure evolution of magnetron sputtered CBN coatings. In order to verify the reliability of the present model, a comparison between experimental and simulation results was made by comparing the micromorphology and analyzing the intrinsic correlation and interaction mechanism of different processing parameters such as sputtering time, temperature, and gas flow rate with the number, size, and spacing of active CBN abrasive grains.

Simulation of the Micromorphology in the CBN Abrasive Growth Process
A honing wheel is used as a grinding tool for gear honing processes. The abrasive grains on the tooth surface of a honing wheel are small and have no fixed shapes, with irregular and uneven distribution, hindering the in-depth study of the morphology of the honing wheel tooth surface. Figure 1 shows the schematic diagram of the gear honing process. In the meshing area, a cell on the tooth surface of the honing wheel is taken arbitrarily, and micromorphological parameters such as the number, shape, size, and spacing of CBN abrasive grains can be obtained using SEM [44]. Under the slippery abrasion of the honing force, the active abrasive grains remove the material of the workpiece tooth surface along the contact traces l i , and the 3D micromorphology of the tooth surface after cutting can be observed by AFM (atomic force microscope) [45]. However, the use of SEM and AMF is not only costly and time-consuming for measurement, but also difficult to effectively identify the removal mechanism of active abrasive grains on the workpiece at different times. To solve the above problem, we carried out quantitative simulations of the micromorphology of CBN coating on the tooth surface of the honing wheel based on the magnetron sputtering principle and phase field method. CBN abrasive grains are polycrystalline materials, and the phase field method has significant advantages in simulating the sputtering, coarsening, and growth of grains on the teeth of honing wheels, as well as the movement of grain boundaries [41]. Compared with the traditional method of surface micro-morphology modeling by probabilistic statistics [42], the phase field method considers the various thermodynamic driving forces of long-range and short-range interactions. Thus, it is able to take into account the effects of complex internal and external factors such as physical fields, micro-morphology of grain boundaries, anisotropy, sputtering temperature, and gas flow on the grain growth process [43], making the model closer to reality.
Hence, the main purpose of the present study was to carry out phase field simulations to investigate the grain sputtering, coarsening, and final film formation during the microstructure evolution of magnetron sputtered CBN coatings. In order to verify the reliability of the present model, a comparison between experimental and simulation results was made by comparing the micromorphology and analyzing the intrinsic correlation and interaction mechanism of different processing parameters such as sputtering time, temperature, and gas flow rate with the number, size, and spacing of active CBN abrasive grains.

Simulation of the Micromorphology in the CBN Abrasive Growth Process
A honing wheel is used as a grinding tool for gear honing processes. The abrasive grains on the tooth surface of a honing wheel are small and have no fixed shapes, with irregular and uneven distribution, hindering the in-depth study of the morphology of the honing wheel tooth surface. Figure 1 shows the schematic diagram of the gear honing process. In the meshing area, a cell on the tooth surface of the honing wheel is taken arbitrarily, and micromorphological parameters such as the number, shape, size, and spacing of CBN abrasive grains can be obtained using SEM [44]. Under the slippery abrasion of the honing force, the active abrasive grains remove the material of the workpiece tooth surface along the contact traces li, and the 3D micromorphology of the tooth surface after cutting can be observed by AFM (atomic force microscope) [45]. However, the use of SEM and AMF is not only costly and time-consuming for measurement, but also difficult to effectively identify the removal mechanism of active abrasive grains on the workpiece at different times. To solve the above problem, we carried out quantitative simulations of the micromorphology of CBN coating on the tooth surface of the honing wheel based on the magnetron sputtering principle and phase field method.

Phase Field Method
Magnetron sputtering is a PVD method that ionizes a mixture of argon and nitrogen under the control of a circular magnetic field to produce argon and nitrogen ions that bombard a high-purity h-BN target at a high speed. The momentum is transferred to B Coatings 2021, 11, 540 4 of 19 and N atoms that are sputtered from the surface of the target to the substrate to prepare high-quality c-BN films [46,47]. To accurately describe the formation process of CBN wear abrasives on the honing surface, the Cahn-Hilliard equation is established for the evolution process of the gas mole fraction as a conserved field variable [48]: where F PC is the total free energy of the system; ∇F VD stands for the free energy function of the model; and ∇, V m , and P are the Laplace operator, the total mole fraction of argon in the reaction process, and the kinetic parameter equation, respectively. θ is the direction of local abrasive growth, within the range of (0, 2π). g is the free energy dissipation correction function, and λ is the derivative of the total free energy of the system with respect to the single well potential energy, λ = ∂F PC ∂ f (φ) . F PC is constructed by the field variable function f (r, t). It provides a symmetrical double well potential energy between the equilibrium gas phase and the solid phase, which can be expressed as [41] where φ is the phase field variable, and s is the entropy density of gas in the reaction process. ε and α are gradient energy coefficients. ∇θ is a phase field variable that has no practical physical significance and can only be used to distinguish the matrix phase from the precipitate phase. g(φ) and h(φ) are coefficient functions, which can be combined into coefficient Φ(r, t) 2 to reduce or eliminate the influence of disordered regions on abrasive growth orientation and abrasive boundary evolution. f (φ) is the double well potential energy function, which represents the change in potential energy in the different phase states in the reaction process [41] where f sol = 2( T T m − 1) [49] is a temperature dependent function in which T and T m are the microstructure evolution temperature and the phase transition temperature, respectively. The free energy in the solid phase consists of f (φ) multiplied by the polynomial step function p(φ), with p(0) = 0 and p(1) = 1. Based on the above analysis, the third and fourth terms of Equation (2) express the contribution of the growth process of ordered and disordered abrasives to the free energy during sputtering. The third term requires the orderly growth of abrasives, and the fourth term allows the dynamic evolution of abrasive boundaries in the process of abrasive growth.
The kinetic parameter equation P(|∇θ|) in the evolution process of an abrasive boundary controls the whole sputtering process through local amplification or reduction at the abrasive boundary and inside the abrasive, which reads as [41] The rotation angle τ θ during abrasive growth is controlled by µ and β, where τ θ = 1, µ = 10 3 , and β = 10 5 . In Equation (5), the abrasive growth direction and ion movement rate can be artificially controlled by selecting the interfacial dynamic coefficient β and the chemical potential energy µ of dimensionless parameter abrasives. The free energy function F VD in Equation (1) is composed of the field variable function f (r, t) and its gradient function, which provides a symmetrical double well potential energy between the gas phase and the solid phase and can be expressed as [50] where a is the interface gradient function of the surface tension, and the time and space evolution equations of the field variable function are controlled by the following dimensionless motion equations [50]: Equations (7) and (8) are used to show the growth and evolution of the solid field f (r, t) through the consumption of the energy of the gas field entering the simulation space. The free energy function is composed of field variables and their gradient functions. B and C are the polymerization and noise coefficients, respectively. B controls the gassolid conversion rate, which is capable of converting all incident gas ions into solids at the interface when it is large enough. C (∇ f ) 2 gη provides the surface relief through the uncorrelated Gaussian distribution η(r, t), and the parameter C controls the overall intensity of the noise. The diffusion coefficient D is modified in response to external force, and the magnitude and direction of the diffusion coefficient are the intensity and direction of the incident gas flux. The field variable f (r, t) describes the growth state of the thin film. It defines a solid region for film growth when f (r, t) ≈ 1, a vacuum environment or no solid film formation when f (r, t) ≈ −1, and a solid-gas interface when f (r, t) ≈ 0. The variable g(r, t) describes the flux density of the gas entering the system. When g(r, t) ≈ 0, no reactive gas enters the flux region, whereas when g(r, t) > 0, gas is transported to the system and sputtered on the coating surface.
This paper uses the magnetron sputtering vapor deposition method to prepare CBN coating on the tooth surface of a steel honing wheel, during which no chemical reaction occurs. Therefore, the system is an energy-conserving system, and the Cahn-Hilliard equation of diffusion, which is consistent with the evolution of conserved field variables, is represented as where x B (r, t) and F(x B , η k ) denote the state functions of the ion distribution in the system and the total energy of random ions, respectively, and η k is the anisotropic parameter. M is the interface dynamic parameter, which can be expressed as where V m , G m , and T(r, t) are the molar volume of the gas, the free energy of nucleation of the reaction gas, and the temperature field function of energy conduction in the reaction space, respectively.
In the process of vapor deposition, the substrate is heated to the required temperature, which plays an important role in the evolution of the surface morphology, phase nucleation, and microstructure in the growth of thin films. In addition, latent heat is released or absorbed by the film during vapor deposition and subsurface phase transition, which may lead to thermal wave and gradient changes in the system. Therefore, the influence of the temperature distribution and evolution must be considered in the modeling of solid growth during magnetron sputtering. To incorporate the temperature evolution into this study, the multiphase motion equation in Equation (10) is coupled with the temperature field function T through the heat conduction equation, which can be denoted as where ρ i , C P , and k T are the density of the reaction gas in the simulated space, the heat capacity, and the thermal conductivity of the mixed gas, respectively. L ik is the latent heat value from phase i to phase k, and f is the potential energy function of phase i.

Equation Solution and Parameter Determination
Equation (12) is a discretization form of Equation (2). The interfacial kinetic phase field parameter τ φ = 1 and the total free energy of the system are given as [51] In Equation (12), the latent heat value L = 2 × 10 9 J m 3 , which is selected according to the microstructure evolution energy at the melting temperature of the sputtering material in the system. According to reference [41], the dimensionless model parameters are ε = 0.0141, α = 0.0265, and s = 0.0176. The discretized phase field model can be solved using difference equations as follows: Thus, we can solve Equations (7) and (8) by applying the explicit first-order Euler method of Equation (13) and the central second-order finite difference method of Equation (14).
In the phase field simulation of the microstructure evolution of a CBN coating prepared by magnetron sputtering, the initial configuration of the theoretical simulation space is an involute tooth surface, and the coordinates of five grid points are generated with thickness in the Y direction, where f (r, 0) ≈ 1 and g(r, 0) = 0. This condition is assumed to provide a fixed substrate region below the constant boundary y = 0. When f (r, 0) = −1 and g(r, 0) = g 0 , the region is vacuum and contains the gas ion flux being sputtered. Outside the boundary of y = 32 µm, this condition can also provide a continuous source of sputtering gas ion flux during the sputtering process. In the substrate region, the initial condition for the formation of polycrystalline microstructures is φ(r, 0)= 1, and θ(r, 0) depends on the orientation of each abrasive in the region. In the vacuum region, φ(r, 0)= 0, and θ(r, 0) takes any value from (−π, π), according to Equation (5).
With the reasonable choice of dimensionless magnetron sputtering parameters in the above equation, it is possible to ensure the grain growth along the perpendicular direction during the simulated perpendicular magnetron sputtering and to provide realistic features for the micromorphology of the film. According to reference [50], the dimensionless spacetime motion Equations (7) and (8) are set to 1 and 0.1, respectively. The other dimensionless magnetron sputtering parameters are shown in Table 1. The noise coefficient C provides enough noise to the surface, allowing the formation of undulating features on the coating surface, but not enough to control the growth and stop the calculation after 1000 steps. The diffusion coefficient D should be smaller than the magnitude of the flux parameter A of the incident gas [50]. The initial value of the phase field density is g 0 = 1. The dimensionless space is discretized to 0.001, corresponding to the physical value of 1 nm. The minimum and maximum simulation areas of the CBN coating surface are 32 µm × 32 µm and 512 µm × 512 µm, respectively. To ensure the discretization of the simulation space, a parameter value between 1 nm and 100 nm is used to ensure the size of the PVD simulation domain and sufficient time to complete the evolution process of the abrasive evolution dynamics. The temperature for the microstructure evolution of the double well potential driven system is T ≈ 0.98T m , where T m is the phase transition temperature.

Simulation Experiment
Under the condition that the gas flow ratio of argon and nitrogen is 5:1, the random distribution function of argon ions and nitrogen ions ionized in the reaction chamber of the magnetron sputtering experimental platform is Seig = lam1 × Leig. The phase field model is solved by discretizing the distribution function, and the calculated results are transformed by inverse two-dimensional discrete cosine to obtain a continuous solution. Thus, we can model the process of grain deposition, coarsening, and final film formation of a magnetron sputtered CBN coating microstructure in VWM.
Taking the 13th group of process parameters in Table 2 as an example, 3D and 2D diagrams, cross-sectional diagrams, discrete particle energy diagrams, and the microscopic morphology of CBN abrasive grains on the tooth surface of the honing wheel under different sputtering times are shown in Figure 2. In the abnormal discharge stage, the target was at a negative potential, resulting in a self-bias effect, which accelerated the sputtering of nitrogen ions and argon ions that bombarded the target with high energy. Meanwhile, the magnetic lines further bonded the plasma in the reaction cavity to enhance the plasma density of the target and substrate, which facilitated the energetic ions to bombard more boron and nitrogen atoms sputtering in the substrate for migration, rearrangement, coalescence, and nucleation. The grains preferentially grew and nucleated along the (110) and (111) planes, forming the spherical islets shown in Figure 2a.
When t = 60 min, the plasma energy obviously increased, and CBN abrasives grew rapidly in a columnar shape at the nucleation point. However, the coating was discontinuous and thin, the surface roughness was too large, and the compactness was poor (Figure 2b). In the actual processing of the honing wheel with this shape, the rough and uneven convex summit increased the contact pressure, and the sharp abrasive grains were more prone to fracturing, breaking, or falling off when they were subjected to force, resulting in increased abrasive grain wear and a shortened tool life. When t = 120 min, the increase in the plasma energy was not obvious, and the abrasive growth in the height direction was faster than that in the width direction, resulting in slender columnar abrasives. Meanwhile, the denseness of the coating improved significantly, the layer thickness was discontinuous yet increased significantly, and the surface roughness became worse, which was somewhat different from the ideal coating morphology (Figure 2d). When t = 240 min, the grains grew faster in the width direction than in the height direction, and the grain shape transformed from slender to coarse. In the width direction, the large grains engulfed the small grains to improve the density of the coating; in the height direction, grains enlarged to the surface of the film to form a continuous coating, leading to a smaller surface roughness and a better overall cutting performance of the coating (Figure 2d).
An L25 (5 4 ) experiment was designed by the Taguchi method. Different values of the process parameters of the reaction space P 1 , substrate temperature P 2 , sputtering time P 3 , and gas flow rate P 4 were selected to generate maps of the 3D surface appearance. The number of active abrasives Q 1 , abrasive spacing Q 2 , height of active abrasives Q 31 , width of active abrasives Q 32 , length of active abrasives Q 33 , and the aspect ratio of abrasive grains Q 34 were calculated and measured, as shown in Table 2.

Process Parameters
Abrasive Morphology Parameters different sputtering times are shown in Figure 2. In the abnormal discharge stage, the target was at a negative potential, resulting in a self-bias effect, which accelerated the sputtering of nitrogen ions and argon ions that bombarded the target with high energy. Meanwhile, the magnetic lines further bonded the plasma in the reaction cavity to enhance the plasma density of the target and substrate, which facilitated the energetic ions to bombard more boron and nitrogen atoms sputtering in the substrate for migration, rearrangement, coalescence, and nucleation. The grains preferentially grew and nucleated along the (110) and (111) planes, forming the spherical islets shown in Figure 2a.
When t = 60 min, the plasma energy obviously increased, and CBN abrasives grew rapidly in a columnar shape at the nucleation point. However, the coating was discontinuous and thin, the surface roughness was too large, and the compactness was poor ( Figure  2b). In the actual processing of the honing wheel with this shape, the rough and uneven convex summit increased the contact pressure, and the sharp abrasive grains were more prone to fracturing, breaking, or falling off when they were subjected to force, resulting in increased abrasive grain wear and a shortened tool life. When t = 120 min, the increase in the plasma energy was not obvious, and the abrasive growth in the height direction was faster than that in the width direction, resulting in slender columnar abrasives. Meanwhile, the denseness of the coating improved significantly, the layer thickness was discontinuous yet increased significantly, and the surface roughness became worse, which was somewhat different from the ideal coating morphology (Figure 2d). When t = 240 min, the grains grew faster in the width direction than in the height direction, and the grain shape transformed from slender to coarse. In the width direction, the large grains engulfed the small grains to improve the density of the coating; in the height direction, grains enlarged to the surface of the film to form a continuous coating, leading to a smaller surface roughness and a better overall cutting performance of the coating (Figure 2d).
An L25 (5 4 ) experiment was designed by the Taguchi method. Different values of the process parameters of the reaction space P1, substrate temperature P2, sputtering time P3, and gas flow rate P4 were selected to generate maps of the 3D surface appearance. The number of active abrasives Q1, abrasive spacing Q2, height of active abrasives Q31, width of active abrasives Q32, length of active abrasives Q33, and the aspect ratio of abrasive grains Q34 were calculated and measured, as shown in Table 2.

Discussion
The significance level of the process parameters on the micromorphological parameters of the abrasive grains in Table 2 was analyzed, and the adequacy of the experimental results was verified. The results show that the significance levels between process parameters P1, P2, P3, P4, and CBN grain spacing were low, but the significance levels between

Discussion
The significance level of the process parameters on the micromorphological parameters of the abrasive grains in Table 2 was analyzed, and the adequacy of the experimental results was verified. The results show that the significance levels between process parameters P 1 , P 2 , P 3 , P 4 , and CBN grain spacing were low, but the significance levels between sputtering time and grain size, substrate temperature, gas flow rate, and grain size, reaction space and grain size, effective grain number, substrate temperature, sputtering time, and effective grain height were high.

Effect of Sputtering Time on Abrasive Size
As shown in Figure 3, the relationship between the sputtering time and the effective grain length, width, and height of CBN was relatively dispersed. After fitting the curves to the discrete points, it was found that the grain length, width, and height showed a more obvious growth trend with the reaction time. With the increase in sputtering time, the size of the grain width showed an oscillating trend. At 150 min, the grain width reached a peak of 1.905 µm for the first time, at 210 min it reached a trough value of 1.696 µm, and at the migration of grain boundaries. This trend was consistent with the actual test, i.e., film thickness increase was more difficult with increasing reaction time, but the len and width of the abrasive grains grew in different trends, resulting in a random distri tion of the shape and size of each abrasive grain.

Effect of Substrate Temperature and Gas Flow Rate on Abrasive Size
As shown in Figures 4 and 5, as the substrate temperature increased, the length a width of the abrasive grains showed a gradual increase, while the overall length of abrasive grains presented an oscillatory trend of increase. Namely, the length of abras grains gradually decreased at a temperature of 300 °C -330 °C , and then started to increa reaching the maximum length of 1.924 μm at a temperature of 390 °C , followed by anot decrease. At this point, there was a high nucleation density, clear grain shape, continuo and dense film, and better quality. Therefore, it can be concluded that with the increase

Effect of Substrate Temperature and Gas Flow Rate on Abrasive Size
As shown in Figures 4 and 5, as the substrate temperature increased, the length and width of the abrasive grains showed a gradual increase, while the overall length of the abrasive grains presented an oscillatory trend of increase. Namely, the length of abrasive grains gradually decreased at a temperature of 300 • C-330 • C, and then started to increase, reaching the maximum length of 1.924 µm at a temperature of 390 • C, followed by another decrease. At this point, there was a high nucleation density, clear grain shape, continuous and dense film, and better quality. Therefore, it can be concluded that with the increase in substrate temperature, the driving force of nucleation increases, the migration and nucleation rate becomes faster, and the overall microscopic shape and size of abrasive grains grows larger.
atings 2021, 11, x FOR PEER REVIEW 11 substrate temperature, the driving force of nucleation increases, the migration and nu ation rate becomes faster, and the overall microscopic shape and size of abrasive gr grows larger.      Figure 6 shows the size distribution and variation trend of the length, width, and height of the active abrasive grains in different reaction spaces. The fitted equation of the length of abrasive grains is y = 1.34476 + 0.00441x − 6.37153 × 10 −6 x 2 , which shows that with the increase of reaction space, the length of abrasive grains followed a trend of first increasing and then decreasing.

Effect of Reaction Space on Abrasive Size and Active Abrasive Number
Coatings 2021, 11, x FOR PEER REVIEW 0.92249x 0.02321 . The fitted image shows that the distribution of height was more c trated and uniform compared to the length and width of abrasive grains. As the r space enlarged, the height of the abrasive grains increased, distributed uniformly b 0.95 μm and 1.1 μm. In the simulated reaction space, the average height of activ abrasive grains was roughly the same as the height of actual abrasive grains, sugg that there was no abnormal growth of abrasive grains in the range of simulated p parameters.   The length of abrasive grains was distributed in the range of 1.4 µm-2.5 µm, and the majority of them were around 1.6 µm in length. The fitted equation for the width of abrasive grains is y = 1.49641 + 0.005353x − 2.32695 × 10 −5 x 2 + 2.87169 × 10 −8 x 3 , indicating that the width of abrasive grains was oscillating upward as the reaction space increased. The width of abrasive grains was concentrated in the range of 1.6 µm-2.0 µm, mainly fluctuating around 1.8 µm. The fitted equation for the height of abrasive grains is y = 0.92249x 0.02321 . The fitted image shows that the distribution of height was more concentrated and uniform compared to the length and width of abrasive grains. As the reaction space enlarged, the height of the abrasive grains increased, distributed uniformly between 0.95 µm and 1.1 µm. In the simulated reaction space, the average height of active CBN abrasive grains was roughly the same as the height of actual abrasive grains, suggesting that there was no abnormal growth of abrasive grains in the range of simulated process parameters. Figure 7 illustrates the relationship between the reaction space and the active abrasive grains. The simulations were performed with a magnetron sputtering power of 350 W, gas pressure of 0.4 Pa, nitrogen, argon flow rates of 30 cm 3 /min and 150 cm 3 /min, sputtering temperature of 390 • C, sputtering time of 210 min, and reaction spaces of 32 µm × 32 µm, 64 µm × 64 µm, 128 µm × 128 µm, 256 µm × 256 µm, and 512 µm × 512 µm. The results suggest that the number of active abrasive grains increased significantly with the increase in reaction space. The reason is that the number of boron and nitrogen atoms sputtering onto the substrate material per unit space under the same conditions was certain, and as the space became larger, the probability of nucleation within that zone grew, and the number of abrasive grains increased. Figure 6. Effect of reaction space on grain size. Figure 7 illustrates the relationship between the reaction space and the active sive grains. The simulations were performed with a magnetron sputtering power o W, gas pressure of 0.4 Pa, nitrogen, argon flow rates of 30 cm 3 /min and 150 cm 3 /min, tering temperature of 390 °C, sputtering time of 210 min, and reaction spaces of 32 32 μm, 64 μm × 64 μm, 128 μm × 128 μm, 256 μm × 256 μm, and 512 μm × 512 μm results suggest that the number of active abrasive grains increased significantly wit increase in reaction space. The reason is that the number of boron and nitrogen a sputtering onto the substrate material per unit space under the same conditions wa tain, and as the space became larger, the probability of nucleation within that zone g and the number of abrasive grains increased.  The fitted model for the growth of active abrasive number and reaction space is y = (0.00962 ± 0.00015)x (2.0531±0.00250) . The ratio of the regression sum of squares to residual sum of squares R 2 = 0.99999 and the residual sum of squares RSS = 88.3308, suggesting that the curve fit well for predicting the number of active abrasive particles involved in the honing process in different reaction spaces.

Effect of Substrate Temperature and Sputtering Time on the Grain Height
When preparing CBN films by magnetron sputtering, the higher the power, the higher the plasma density in the same reaction space, but too high a power can lead to target poisoning. At a sputtering power of 350 W [43], the reaction cavity just started to glow and the target material had not been poisoned. In addition, since the effect of power on the size of active abrasive grains is much lower than the effect of sputtering time and substrate temperature, the effect of power on the height of active abrasive grains was not considered. Figure 8 shows the contour plots of sputtering temperature and time on the height of abrasive grains under the process conditions of sputtering power 350 W, reaction space 32 µm × 32 µm, and gas flow rate 180 cm 3 /min. At a sputtering time of 155 min and a sputtering temperature of 300 • C, the height of CBN abrasive grains reached 1.15 µm; at a sputtering time of 180 min and a sputtering temperature of 324 • C, the average height of abrasive grains was 0.925 µm. This indicates that the sputtering time and temperature had a more significant effect on the height of abrasive grains. The coating thickness was greater than 1 µm at a sputtering time of 200-220 min and a sputtering temperature of 390-400 • C, allowing the preparation of CBN coatings with a maximum thickness of 1.178 µm. The higher the average height of active abrasive grains, the sharper the CBN abrasive grains, which helps to improve the material removal rate of the workpiece and enhance the efficiency of the honing process.
considered. Figure 8 shows the contour plots of sputtering temperature and time on the height of abrasive grains under the process conditions of sputtering power 350 W, reaction space 32 μm × 32 μm, and gas flow rate 180 cm 3 /min. At a sputtering time of 155 min and a sputtering temperature of 300 °C, the height of CBN abrasive grains reached 1.15 μm; at a sputtering time of 180 min and a sputtering temperature of 324 °C, the average height of abrasive grains was 0.925 μm. This indicates that the sputtering time and temperature had a more significant effect on the height of abrasive grains. The coating thickness was greater than 1 μm at a sputtering time of 200-220 min and a sputtering temperature of 390-400 °C, allowing the preparation of CBN coatings with a maximum thickness of 1.178 μm. The higher the average height of active abrasive grains, the sharper the CBN abrasive grains, which helps to improve the material removal rate of the workpiece and enhance the efficiency of the honing process.

Comparison of Simulation and Experimental Results
According to the preparation and measurement method of CBN coating in the literature [35], a test system consisting of an SP-6A magnetron sputtering experimental bench, 45 steel substrate material, a 99.99% purity CBN sputtering target, argon gas, nitrogen gas, and a field emission scanning electron microscope was built. The experimental parameters were set based on the simulation parameters and conditions, as shown in Table 3. A comparison of the experimental and simulation results of the micromorphology of the

Comparison of Simulation and Experimental Results
According to the preparation and measurement method of CBN coating in the literature [35], a test system consisting of an SP-6A magnetron sputtering experimental bench, 45 steel substrate material, a 99.99% purity CBN sputtering target, argon gas, nitrogen gas, and a field emission scanning electron microscope was built. The experimental parameters were set based on the simulation parameters and conditions, as shown in Table 3. A comparison of the experimental and simulation results of the micromorphology of the honing wheel tooth surface is shown in Figures 9 and 10. In Figure 9a,c, the distribution, aspect ratio, and number of the eight active abrasive grains A, B, C, D, E, F, G, and H in the simulated top view are all confirmed in the top view of the SEM shown in Figure 9b,d. Figure 10a,c,e,g are the main views of Figure 9. To facilitate an in-depth examination of the local details of the grains, partial enlargements of Figure 10b,d,f,h were drawn. Nine positions of I, J, K, L, M, N, O, P, and Q were selected in the SEM plots for comparison with the simulated plots, and the results showed that the height, grain shape, and size of CBN grains simulated by the phase field method were in good agreement with the experimental results.  Figure 9b,d. Figure 10a,c,e,g are the main views of Figure 9. To facilitate an in-depth examination of the local details of the grains, partial enlargements of Figure 10b,d,f,h were drawn. Nine positions of I, J, K, L, M, N, O, P, and Q were selected in the SEM plots for comparison with the simulated plots, and the results showed that the height, grain shape, and size of CBN grains simulated by the phase field method were in good agreement with the experimental results. The experimental results of the 22nd group of process parameters in Table 2 were selected to verify the validity of the simulation results of active grain size, and the distribution of grain size was plotted in Figure 11. The simulation results in Figure 11a show that among the 3505 active abrasive grains, the largest number of grains with similar size and the best consistency of abrasive grain size were found at an increasing density of 10. 17-12.70, with about 621 grains, accounting for 16.4% of the total number of grains. The aspect ratio of these grains was 0.94-1.12, and the height of the grains fell in the range of 1.07 μm-1.15 μm. Compared to the experimental results in Figure 11b, the total number of active abrasive grains was 127 less, with an error of 3.62%. The increasing density differed by 0.2-0.25, with an error of 2%. In the red area with the highest concentration of abrasive grains, the number of active abrasive grains was 43 more, with an error of 7.4%; The experimental results of the 22nd group of process parameters in Table 2 were selected to verify the validity of the simulation results of active grain size, and the distribution of grain size was plotted in Figure 11. The simulation results in Figure 11a show that among the 3505 active abrasive grains, the largest number of grains with similar size and the best consistency of abrasive grain size were found at an increasing density of 10.17-12.70, with about 621 grains, accounting for 16.4% of the total number of grains. The aspect ratio of these grains was 0.94-1.12, and the height of the grains fell in the range of 1.07 µm-1.15 µm. Compared to the experimental results in Figure 11b, the total number of active abrasive grains was 127 less, with an error of 3.62%. The increasing density differed by 0.2-0.25, with an error of 2%. In the red area with the highest concentration of abrasive grains, the number of active abrasive grains was 43 more, with an error of 7.4%; the difference in the aspect ratio of abrasive grains was 0.02-0.06, with an error of 2.2-5.7%; the difference in the height was 0.02 µm-0.06 µm, with an error of 1.8-5.9%. Similarly, comparing the size and number distribution of abrasive grains in other colored areas in Figure 11a,b, the errors of increasing density values were also small and within acceptable limits. The experimental results in Figure 11b indicate that among 3632 active abrasive grains, the largest number of grains with similar size and the best consistency of abrasive size occurred at an increasing density of 9.97-12.45, with about 578 grains, accounting for 15.9% of the total number, with an aspect ratio of 0.92-1.06 and a height of 1.01 µm-1.13 µm.
larly, comparing the size and number distribution of abrasive grains in other colored areas in Figure 11a,b, the errors of increasing density values were also small and within acceptable limits. The experimental results in Figure 11b indicate that among 3632 active abrasive grains, the largest number of grains with similar size and the best consistency of abrasive size occurred at an increasing density of 9.97-12.45, with about 578 grains, accounting for 15.9% of the total number, with an aspect ratio of 0.92-1.06 and a height of 1.01 μm-1.13 μm.  Based on the above analysis, conclusions can be drawn that the number, size, and spacing of CBN active abrasive grains on the tooth surface of the honing wheel simulated by the phase field method were consistent with those observed in the experiments. Moreover, the micromorphology of the abrasive grains at different locations on the tooth surface observed in the metallographic experiments was highly similar to that of the simulated ones, demonstrating the validity and correctness of the method. More importantly, the micromorphology of CBN active abrasive grains was simulated using the phase field model in physical and mathematical aspects, which helped to clarify the correlation between the process parameters of the CBN coating prepared by magnetron sputtering and the micromorphology parameters such as the number, size, and spacing of active abrasive grains affecting the honing process.  Based on the above analysis, conclusions can be drawn that the number, size, and spacing of CBN active abrasive grains on the tooth surface of the honing wheel simulated by the phase field method were consistent with those observed in the experiments. Moreover, the micromorphology of the abrasive grains at different locations on the tooth surface observed in the metallographic experiments was highly similar to that of the simulated ones, demonstrating the validity and correctness of the method. More importantly, the micromorphology of CBN active abrasive grains was simulated using the phase field model in physical and mathematical aspects, which helped to clarify the correlation between the process parameters of the CBN coating prepared by magnetron sputtering and the micromorphology parameters such as the number, size, and spacing of active abrasive grains affecting the honing process.

Conclusions
In this study, the magnetron sputtering method and phase field method were used to study the surface topography formation mechanism of active abrasive grains in a honing wheel. The relationships of the sputtering time, substrate temperature, gas flow rate, and reaction space with the number of active abrasive grains and the length, width, and height of the abrasives was investigated both by simulations and experiments. Based on the results, the main conclusions were drawn as follows: (1). By comparison of the simulation and experimental results, the method in this paper can effectively simulate the growth process of CBN abrasive grains, which grow from

Conclusions
In this study, the magnetron sputtering method and phase field method were used to study the surface topography formation mechanism of active abrasive grains in a honing wheel. The relationships of the sputtering time, substrate temperature, gas flow rate, and reaction space with the number of active abrasive grains and the length, width, and height of the abrasives was investigated both by simulations and experiments. Based on the results, the main conclusions were drawn as follows: (1). By comparison of the simulation and experimental results, the method in this paper can effectively simulate the growth process of CBN abrasive grains, which grow from a single spherical shape to a long strip at the nucleation point, and the coating thickness grows from discontinuous and thin to a continuous and dense coating. (2). Plasma energy (energetic ions) drive the migration, rearrangement, coalescence, and nucleation of boron and nitrogen atoms in the matrix, providing the driving force for the growth, coarsening, and film formation of CBN grains. In addition, a higher plasma energy would increase the growth and coarsening rate of columnar grains and improve the film quality. (3). The effects of sputtering time, substrate temperature, gas flow rate, and reaction space on the number and size of CBN abrasive grains are of high significance, and should be mainly controlled. Process parameters have a low level of influence on the shape and spacing of active abrasive grains, and should be controlled as a secondary factor. Therefore, the reasonable selection or optimization of process parameters can quantitatively control the number, size, and distribution state of abrasive grains, optimize the micromorphology of CBN coating on the tooth surface of honing wheels, and promote the processing performance of honing wheels.
(4). Despite the good agreement between the simulation and SEM experimental results, the model needs to be modified and improved in future studies due to the limitations of the selection of phase field parameters, solution accuracy, and computational volume of simulation. In future work, methods such as machine learning will be considered to optimize the parameters of the phase field model and improve the accuracy of the solution algorithm. In this way, it is possible to efficiently design the micromorphology of CBN grains on the tooth surface of honing wheels and produce high-performance honing wheels, thereby fundamentally improving the material removal rate, machining accuracy, and surface quality of hardened gears. Institutional Review Board Statement: Not applicable.