Bonnet Polishing of Microstructured Surface

Abstract

Microstructured surfaces have been adopted in various and wide applications. Different types of microstructures made of ductile materials can be generated by cutting process, for example, turning and milling with specified diamond cutters. However, these processes generally are not capable to handle with hard and brittle materials which are called difficult-to-machine materials. Computer-controlled ultra-precision polishing with bonnet provides an enabling solution to generate microstructures due to its feasible influence function. With proper machining parameters, specified shape of the tool influence function is hence obtained, and then with aid of tool path planning, microstructured surface topography is generated, especially for those difficult-to-machine materials. In this chapter, research work for generating microstructured surface by computer-controlled ultra-precision bonnet polishing is presented. The material removal characteristics and tool influence function of bonnet polishing are explained, and a multi-scale material removal model and a surface generation model were developed. Surface generation of microstructures by single precess polishing and swing precess polishing is explained in details. A series of simulation and real polishing experimental studies are undertaken based on a seven-axis ultra-precision freeform polishing machine. The generated microstructured surfaces with various patterns have been analyzed. The research results have demonstrated that the proposed bonnet polishing provides an enabling and effective approach for generating microstructured surfaces.

1 Introduction

Microstructured surfaces are usually possessing specially designed functional textures used in the development of high-precision applications such as optical (Beaucamp and Namba 2013), biomedical (Charlton and Blunt 2008), and automotive components (Cheung et al. 2010). Different scales of structures are found in various applications, for example, structured surfaces with a width of 2.0 mm and depth of 0.2 mm were intentionally machined (Zhou et al. 2006); 500-μm shark skin-like structures reduce wall shear stress (Bechert et al. 2000); pore surface can enhance the load capacity and stiffness of the fluid film between the seal-mating rings (Wakuda et al. 2003). Currently, three-dimensional (3D) structured surfaces are manufactured by different methods such as electroforming and laser machining, micro-milling or micro-grinding processes, etc.(Johansen et al. 2000; Chen et al. 2016; Shen et al. 2014; Schaller et al. 1999; Egashira et al. 2014). However, it is hard to control the form error and surface texture at different areas of such complex 3D structured surfaces by using these methods.

There are various types of computer-controlled polishing (CCP) used for the fabrication of precision components, such as CCP with small and large rotating tools (Beckstette et al. 1989), water jet polishing (Fähnle et al. 1998; Zhu et al. 2009), reactive or nonreactive ion beam polishing (Johnson and Ingersoll 1983; Sudarshan 1995; Zhao et al. 1990; Carter et al. 1993), etc. During the past few decades, much research has been performed on the modeling and simulation of surface generation in CCP processing (Schinhaerl et al. 2008a, b; Xi and Zhou 2005), as well as the modeling of polishing mechanics (Gee 1996). However, most of the previous research has been focused on the reduction of surface roughness and figuring of aspheric and freeform surfaces. At present, multi-axis computer-controlled ultra-precision polishing (CCUP) can be used for machining freeform surfaces with high form accuracy and good surface roughness. This is particularly true for machining difficult-to-machine and ferrous materials which are not amenable by using other ultra-precision machining technologies such as single-point diamond turning and ultra-precision raster milling.

So far, much research work has been undertaken on the study of surface generation in computer-controlled ultra-precision polishing and polishing mechanics (Cheung et al. 2011; Cao et al. 2016; Evans et al. 2003; Zeng and Blunt 2014). Currently, CCUP is found to be used to reduce surface roughness and figuring of aspheric and freeform surfaces (Namba et al. 2008; Shiou and Ciou 2008; Kumar et al. 2015). However, the use of CCUP for the generation of complex 3D structured surfaces for functional applications has received relatively little attention. Stout and Blunt (2001) stated that there is great demand in structured engineered parts. In order to reduce the manufacturing cost and find out more potential structures, it is necessary to explore more manufacturing processes to generate different 3D structured surfaces. There is a need for the development of a new and controllable polishing process to generate complex 3D structured surfaces to meet the demand for functional applications.

As a typical process of computer-controlled polishing, bonnet polishing is a computer-controlled sub-aperture polishing process that actively controls the position and orientation of the bonnet which is the spinning, inflated, and membrane tool as it sweeps through the polished surfaces (Walker et al. 2002a, b), as shown in Fig. 1. The polishing bonnet is covered with the polishing pad, and the slurry is dragged by the porous polishing pad into the interface between the pad and workpiece. Bonnet polishing was originally invented at London’s Optical Sciences Laboratory and was then further developed by Zeeko Ltd. It has been exploited for commercial production of the IRP robotic polishing system (Walker et al. 2001). Bonnet polishing is particularly suitable for machining difficult-to-machine materials which are not amenable to using other ultra-precision machining technologies such as diamond cutting process. Polishing of such materials with sub-micrometer form accuracy and surface finish in the nanometric range is complex and multi-scale in nature. Bonnet polishing also has the advantage of high polishing efficiency, mathematically tractable influence function, and flexibly controllable spot size with variable tool hardness (Bingham et al. 2000).

Fig. 1

Schematic illustration of bonnet polishing

In this chapter, research work for generating microstructured surface by computer-controlled ultra-precision bonnet polishing is based on single precess polishing and swing precess polishing. The material removal characteristics and tool influence function of bonnet polishing will be first explained, and multi-scale material removal model and a surface generation model were developed. Then surface generation of microstructures by single precess polishing and swing precess polishing is explained in details. A series of experimental studies are followed to demonstrate the microstructures generation by bonnet polishing, and the results are discussed, in order to better understanding the bonnet polishing process. A brief summary is provided at the end of the chapter. The proposed methods can be potentially used for fabricating microstructured surfaces with high surface finish, especially for difficult-to-machine and ferrous materials.

2 Material Removal Characteristics in Bonnet Polishing

In bonnet polishing, the material removal characteristics represent the distribution of the material removal rate across the size of the polishing tool. The material removal characteristics are referred to be the tool influence function (TIF), and they are assessed in terms of width, maximum depth, and volumetric material removal rate (Schinhaerl et al. 2007). Bonnet polishing involves forcing a spinning, inflated bonnet, covered with the polishing pad, against the polished surfaces flooded with a liquid slurry of abrasive particles. The slurry is dragged by the porous polishing pad into the interface between the pad and workpiece. The material removal in bonnet polishing is accomplished by the interactions between the polishing pad, workpiece, and abrasive particles. The mechanism of material removal is a complex process, which is affected by various parameters such as tool radius, precess angle, polishing depth, head speed, tool pressure, polishing time, polishing cloth, slurry concentration, particle size and material properties of particle and workpiece, etc. To better understand the pad-abrasive-workpiece contact mechanics and polishing mechanisms in bonnet polishing, a series of experimental and theoretical studies are undertaken to investigate the material removal characteristics in bonnet polishing first in this section, and this also helps to explain some common experience.

2.1 Experimental Design

An experiment is designed and aims to investigate the interactions among the polishing pad, workpiece, and abrasive particles. Three samples (A1, A2, A3) made of nickel copper (NiCu) were prepared by the Moore Nanotech 350FG using single-point diamond tooling and then polished on a Zeeko IRP 200 ultra-precision freeform polishing machine. Sample A1 was polished without water and abrasive particles, sample A2 was polished using pure water without abrasive particles, and sample A3 was polished using a slurry comprising 2.066 vol.% of Al₂O₃ abrasives with an average size of 3.22 μm. All these samples were polished under the identical polishing parameters as shown in Table 1, and they are measured by a Zygo Nexview 3D optical surface profiler and HITACHI TM3000 tabletop scanning electron microscope. With the consideration of the importance of dwell time map for the surface generation by bonnet polishing, the effect of polishing time on surface generation for various materials was studied in the other experiment. Three samples made of different materials of steel, optical glass (BK7), and NiCu were polished using a slurry comprising 2.066 vol.% of Al₂O₃ abrasives with an average size of 13.12 μm. All experiments were conducted on the Zeeko IRP 200 ultra-precision freeform polishing machine using the different polishing time of 60, 120, and 180 s, and the other parameters can be seen in Table 1.

Table 1 Polishing parameters used in the experimental studies

2.2 Results and Discussion

2.2.1 Pad-Abrasive-Workpiece Interactions

Table 2 shows the experimental results for studying interactions between the pad, workpiece, and particles and indicates that sample A3 has the highest material removal rate, while the amount of material removal of sample A2 is smaller than that of sample A3, and sample A1 has the lowest material removal. It is also found that the polishing pad not only contributes to the material removal but also generates microscale scratches on the polished surface as shown in the scanning electronic microscopy (SEM) photographs in Table 2. The outcomes of these experiments can be summarized as follows:

• (i) The interaction between the pad and the polished surface decreases the material removal rate for the dry bonnet polishing process.

• (ii) The functions of the abrasive slurry in bonnet polishing including transport of abrasive to a loose abrasive process, flushing or the transport of the debris away from the abrasive process, culling in the contact area, mechanical lubrication of the abrasive contacts, etc.

• (iii) The material removal in bonnet polishing is shared by the polishing pad and the abrasives trapped in the pad-workpiece interface, and the amount of material removal by the polishing pad is much smaller than that by the abrasive particles.

• (iv) The material removal associated with plastic deformation of the polishing pad could produce the scratches and hence damage the surfaces being polished.

Table 2 Experimental results for studying interactions (pad, workpiece, and particles)

Figure 2 shows the surface topography of polishing pad measured by Alicona IFM G4 optical 3D measurement device. The geometry and the mechanical properties of the tallest asperities of the pad surface may play a dominant role in pad scratching. To obtain mirror surface finish without pad scratches, abrasive wear occurred in bonnet polishing should be dominated by plastic removal mode of abrasive particles, while the material removal caused by the polishing pad should be mitigated through flatting the asperities (Saka et al. 2010; Hutchings 1993; Kim et al. 2013), controlling the polishing depth and adopting appropriately the polishing pad owned low pad hardness.

Fig. 2

The three-dimensional surface topography of the polishing pad

2.2.2 Effect of Polishing Time

Since the surface generation by bonnet polishing is dominated by the influence function of the polishing tool instead of the geometry of the cutting tool itself, the surface generation mechanism of bonnet polishing is quite different from that of other ultra-precision machining processes such as single-point diamond turning and raster milling (Cheung et al. 2011). The tool influence function (TIF) affected by various factors is commonly regarded as a tool that is used in calibration, prediction, or form correction in the polishing process. With the data of the tool influence function, the polishing tool can be commanded where it should stay longer or shorter for removing more or less materials from the surface, respectively. In the second experiment, Fig. 3 shows that the removal volume increases linearly with increasing polishing time for all cases, and this infers that the material removal rate is constant when using only polishing time as a variable parameter while keeping other parameters constant. This infers that bonnet polishing is a relative and cumulative polishing process for various materials and the surface generation of bonnet polishing is a linearly cumulative effect of dwell time together with the constant material removal rate for identical polishing condition.

Fig. 3

The effect of polishing time on the surface generation

3 Modeling and Theoretical Investigation of Material Removal

It is well known that the surface generation of the polishing process can be regarded as the convolution of the influence function and the dwell time map along the pre-specific tool path. Hence the determination of the material removal characteristics and an optimized tool path generator is of paramount importance for modeling and simulation of the surface generation in bonnet polishing. A multi-scale material removal model is developed by the study of the contact mechanics, kinematics theory, and abrasive wear mechanism. Then the polishing tool path is planned based on the desired surface integrity of the optical surface to be generated using the predicted data of the material removal characteristics. Finally, the surface generation is simulated based on the developed multi-scale material removal model and polishing tool path planning.

Based on the experimental results of previous section, the material removal caused by the polishing pad is much less than that caused by the abrasive particles. Hence, it is assumed that material removal occurs primarily as a result of abrasive wear of the surface by the abrasive particles in the slurry. From the view of the mechanical behavior, the basic model of the material removal characteristics of bonnet polishing process can be described by Eq. 1 as shown in Fig. 4.

$$ {\displaystyle \begin{array}{r} MRR\left(x,y,t\right)=N\left({k}_{ac},{V}_c,t,{R}_p,{R}_a,{\sigma}_z\right)\cdot \\ {}E\left(\eta, P\left(x,y,{R}_b,d,\omega, Y,\nu, \varphi, {\eta}_1,{\eta}_2\right),V\left(x,y,S,\varphi, {R}_b,d\right),{H}_w,\beta \right)\end{array}} $$

(1)

Fig. 4

Schematic illustration of the multi-scale mechanisms affecting the material removal characteristics in bonnet polishing

where MRR(x, y, t) is material removal at position (x, y) during the polishing time t; the term N(k_ac, V_c, t, R_p, R_a, σ_z) is the spatial distribution of active abrasive particles participated in the material removal which represents the effect of the volume fraction V_c, the polishing time t, the radius of the abrasive particle R_p, the pad asperity radius R_a, the standard deviation of asperity heights σ_z, and the coefficient related to the particle size distribution and the hydrodynamics condition k_ac; and the term E(η, P(x, y, R_b, d, ω, Y, ν, φ, η₁, η₂), V(x, y, S, φ, R_b, d), H_w, β) is the volume removed by a single particle that describes the effect of the pressure distributionP(x, y, R_b, d, ω, Y, ν, φ, η₁, η₂), the velocity distribution V(x, y, S, φ, R_b, d), the hardness of polished workpiece H_w, the semi-angle of a cone particle β, and a volume fraction of a wear groove removed as a wear debris η.

The slurry particles involved in material removal are those that are embed in the surface of the compliant polishing pad, and they are dragged across the polished surface by the relative velocity between two surfaces, and the active number of these particles is generally related to the distribution of particle size, the hydrodynamics condition between the polishing pad and the workpiece, as well as the surface topography of the polishing pad and the target surface. To simplify the theoretical modeling, the pad-particle-workpiece contact is assumed to be solid-solid contact neglecting the effect of the fluid flow, and the particle size is assumed to be constant which is the same as the average size of the particle distribution.

In the present study, statistical theories were used to model the surface topography of polishing pad assessed by the pad asperity radius R_a and the standard deviation of asperity heights σ_z (Kim et al. 2014; Greenwood and Williamson 1966). For relatively soft pad and low abrasive concentration, the active number of abrasive particles tends to be proportional to the real contact area and the slurry concentration (Fu et al. 2001; Luo and Dornfeld 2001, 2003). As a result, the active number of abrasive particles can be expressed by Eq. 2 as follows:

$$ N\left({k}_{ac},{V}_c,t,{R}_p,{R}_a,{\sigma}_z\right)={K}_{ac}\frac{V_ct}{R_p^2}{\left(\frac{R_a}{\sigma_z}\right)}^{1/2} $$

(2)

The effective contact area between the polishing pad and the entrained particle is approximately equal to $\pi {R}_p^2$ (Luo and Dornfeld 2001), and hence the force applied on an abrasive, W_p, can be expressed by Eq. 3:

$$ {W}_p=\pi {R}_p^2P\left(x,y,{R}_b,d,\omega, Y,\nu, \varphi, {\eta}_1,{\eta}_2\right) $$

(3)

In the bonnet polishing process, since the abrasive particles are sufficiently small and numerous in the contact region, the load carried by each tends to be below the critical value needed to cause cracking. Below this critical value, a hard abrasive particle would cause plastic deformation only, and the wear is undertaken by plastic processes without brittle fracture. Abrasive particles may roll and/or slide over the surface which involve in three distinct modes of plastic deformation including cutting, plowing, and wedge formation (Challen and Oxley 1979; Hutchings 1992). The transition from one mode to another and the relative efficiency of each mode may depend on the attack angle of particle, the normal load, the hardness of the material, and the state of lubrication (Hokkirigawa and Kato 1988). In the polishing process, lubrication of the slurry can lead to more particle cutting and reduce the couple necessary for particle rotation by lowering the friction between the particles and the surface (Bingley and Schnee 2005). In this study, an abrasive particle, assumed to be a cone of semi-angle β, is dragged across the surface in plastic contact which flows under an indentation pressure H_w. Since the depth of indentation, δ_p, is much smaller than the radius of the abrasive, the volume of wear debris produced by the cone particle per unit time can be expressed by Eq. 4 as follows:

$$ {\displaystyle \begin{array}{ll}E\left(\eta, P\left(x,y,{R}_b,d,\omega, Y,\nu, \varphi, {\eta}_1,{\eta}_2\right),V\left(x,y,S,\varphi, {R}_b,d\right),{H}_w,\beta \right)& ={\eta \delta}_p^2\tan \beta V\left(x,y,S,\varphi, {R}_b,d\right)\\ {}& \qquad =\frac{2\eta {W}_pV\left(x,y,S,\varphi, {R}_b,d\right)}{\pi {H}_w\tan \beta }=\frac{2\eta {R}_p^2P\left(x,y,{R}_b,d,\omega, Y,\nu, \varphi, {\eta}_1,{\eta}_2\right)V\left(x,y,S,\varphi, {R}_b,d\right)}{H_w\tan \beta}\end{array}} $$

(4)

According to the kinematics theory, the relative velocity between the polishing pad and the target surface in the polishing area can be expressed as

$$ V\left(x,y,S,\varphi, {R}_b,d\right)=\frac{\pi S}{30}\sqrt{{\left(y\cot \varphi -\left({R}_b-d\right)\right)}^2{\left(\sin \varphi \right)}^2+{x}^2{\left(\cos \varphi \right)}^2} $$

(5)

where x² + y² ≤ (R_b)² − (R_b − d)², S is angular velocity in rpm, φ is the inclination angle, d is the polishing depth in mm, and R_b is the radius of the bonnet in mm.

The pressure distribution between the polishing pad and the target surface is very complex and still not well understood as resulting from multiple influence factors including the elastic response of the polishing pad, the hydrodynamic forces due to fluid flow at the interface and the viscoelastic relaxation of the polishing bonnet, etc. In this study, the viscoelastic properties of the polyurethane pad is considered, and the polishing bonnet in contact with the flat workpiece surface was assumed to be a viscous sphere on a hard plane regardless of the contribution of slurry hydrodynamic pressure, pad asperities, contact-surface instability, and pad-abrasive-workpiece contact. According to Brilliantov and Poschel (1998), the total stress P(x, y, R_b, d, ω, Y, ν, φ, η₁, η₂) is a sum of the elastic part of the stress tensor ${\sigma}_{zz}^{el}$ and the dissipative part of the stress tensor ${\sigma}_{zz}^{dis}$. ${\sigma}_{zz}^{el}$ is the known solution for the Hertz contact problem (Landau and Lifshitz 1959) as expressed by Eq. 6 as follows:

$$ {\sigma}_{zz}^{el}={E}_1\frac{\partial {u}_z}{\partial z}+\left({E}_2-\frac{E_1}{3}\right)\left(\frac{\partial {u}_x}{\partial x}+\frac{\partial {u}_y}{\partial y}+\frac{\partial {u}_z}{\partial z}\right)={p}_0{\left(1-\frac{x^2}{a^2}-\frac{y^2}{a^2}\right)}^{1/2} $$

(6)

where u_x, u_y, and u_z denote the x-, y-, and z-direction displacement field of the classic Hertz contact problem; E₁ = Y/(1 + ν) and E₂ = Y/3(1 − 2ν) denote the elastic material constants with Y and ν being the Young modulus and the Poisson ratio of the polishing pad, respectively; $a=\sqrt{dR_b}$ denotes the radius of contact area; and p₀ = 3F_N/(2πa²) denotes the maximum contact pressure; F_N is the total elastic force, acting by the surface (in normal direction) on the polishing pad as expressed in Eq. 7:

$$ {F}_N=\frac{2}{3}\frac{Y}{\left(1-{v}^2\right)}{R_b}^{1/2}{d}^{3/2} $$

(7)

According to the kinematics theory and contact mechanics (Brilliantov and Poschel 1998; Zheng et al. 2011; Johnson 1987), ${\sigma}_{zz}^{dis}$ can be expressed as

$$ {\displaystyle \begin{array}{l}{\sigma}_{zz}^{dis}={\eta}_1\frac{\partial {\dot{u}}_z}{\partial z}+\left({\eta}_2-\frac{\eta_1}{3}\right)\left(\frac{\partial {\dot{u}}_x}{\partial x}+\frac{\partial {\dot{u}}_y}{\partial y}+\frac{\partial {\dot{u}}_z}{\partial z}\right)\\ {}=\frac{\left(1-2v\right)\left(1+v\right)}{Y}\cdot \frac{\omega \cos \varphi \left(2{\eta}_2+{\eta}_1/3\right){p}_0x\left({R}_b-d\right)}{a^2}\\ {}\qquad \cdot {\left(1-\frac{x^2}{a^2}-\frac{y^2}{a^2}\right)}^{-1/2}+\frac{{\left(1-v\right)}^2}{Y}\cdot \frac{\omega \cos \varphi \left(2{\eta}_2+{\eta}_1/3\right)\pi {p}_0x}{2a}\end{array}} $$

(8)

where ${\dot{u}}_x$, ${\dot{u}}_y$, and ${\dot{u}}_z$ denote the time derivative of the displacement field in x-, y-, and z-direction, respectively; η₁ and η₂ are the coefficients of viscosity, related to shear and bulk deformation, respectively; ω is the angular velocity; and φ is the inclined angle.

As a result, the pressure distribution at the polishing contact area can be expressed by

$$ {\displaystyle \begin{array}{l}P\left(x,y,{R}_b,d,\omega, Y,\nu, \varphi, {\eta}_1,{\eta}_2\right)={\sigma}_{zz}^{el}+{\sigma}_{zz}^{dis}\\ {}={p}_0{\left(1-\frac{x^2}{a^2}-\frac{y^2}{a^2}\right)}^{1/2}+\frac{\left(1-2v\right)\left(1+v\right)}{Y}\cdot \frac{\omega \cos \varphi \left(2{\eta}_2+{\eta}_1/3\right){p}_0x\left({R}_b-d\right)}{a^2}\cdot \\ {}{\left(1-\frac{x^2}{a^2}-\frac{y^2}{a^2}\right)}^{-1/2}+\frac{{\left(1-v\right)}^2}{Y}\cdot \frac{\omega \cos \varphi \left(2{\eta}_2+{\eta}_1/3\right)\pi {p}_0x}{2a}\end{array}} $$

(9)

4 Generation of Microstructures by Bonnet Polishing

4.1 Polishing Mechanism by Multi-axis Ultra-precision Polishing

As shown in Fig. 5, CCUP is implemented by using a seven-axis freeform polishing machine, Zeeko IRP200 from Zeeko^TM Ltd. of the UK, in which four axes control the workpiece motion and the other three axes control the polishing head. There are mainly two types of polishing methods, one is mechanical polishing, and the other is fluid jet polishing. The bonnet polishing is the mechanical polishing (see Fig. 6), which makes use of a plastic bonnet covered with a polishing cloth for performing the polishing process.

Fig. 5

Ultra-precision freeform polishing machine (Zeeko IRP200 from the UK): (a) polishing machine; (b) seven-axis motion of the machine

Fig. 6

Cutting mechanics for CCUP based on mechanical polishing: (a) polishing tools; (b) polishing geometry

4.2 Surface Generation Mechanisms

The CCUP is a complex process and is quite different from other ultra-precision machining processes such as single-point diamond turning and raster milling; the surface generation by CCUP is dominated by the polishing tool influence function (TIF) instead of the pure geometry of cutting tool. The influence function is affected by a number of factors such as workpiece material, polishing cloth, machining parameters, geometry of workpiece, polishing strategies, and especially the slurry. This provides an important means to generate structured surfaces. One of the key issues about CCUP is the investigation of TIF.

Any workpiece surfaces may be approximated locally by a convex, concave, or flat geometry. The influence function models are different for these three shapes, since the contact conditions between the workpiece and the polishing tool (e.g., bonnet) are different leading to varying deformations of the bonnet. As a result, the pressure distribution in the contact region and the magnitude of relative velocity between polishing tool and workpiece are different and hence the material removal rate (MRR). Moreover, the material removal characteristics of the polishing tool, the slurry, and the polishing cloth also play an important role in surface generation and have a significant impact on TIF. When establishing an integrated model to simulate and predict the surface generation in ultra-precision polishing process, all these factors need to be taken into account.

4.2.1 Theoretical Modeling of Influence Function for Bonnet on a Fat Surface

One of the most important issues in the simulation of surface generation in CCUP is the determination of TIF, which indicates the tool material removal characteristics. The theoretical base of the surface topography simulation model is focused on the modeling of the TIF. Based on Preston’s Law (Preston 1927), the TIF is proportional to the polishing tool pressure on the surface of the workpiece and the relative velocity between the tool and the workpiece. Some previous research work (Walker et al. 2000, 2002, 2003) shows that the distribution of pressure of the tool appears to be in the shape of a Gaussian function. In the surface topography simulation model, a modified Gaussian function (MGF) is employed to construct the TIF in CCUP.

As shown in Fig. 7a, a Gaussian function is expressed as

$$ f(x)=a\exp \left(-\frac{1}{2}{\left|\frac{x-c}{b}\right|}^2\right) $$

(10)

Fig. 7

Graphical illustrations for parameters in modified Gaussian function (MGF)

Based on Eq. 10, the modified Gaussian function is expressed

$$ f(x)=a\exp \left(-\frac{1}{2}{\left|\frac{x-c}{b}\right|}^{\lambda}\right) $$

(11)

where a is the height of the curve’s peak and b controls the width of the curve, and this can be derived as the half width at the point with the curve height of a a exp (−0.5). The parameter λ can be derived as

$$ \lambda =\frac{\ln \left(\ln \left({k}_1\right)/\ln \left({k}_2\right)\right)}{\ln \left({w}_1/{w}_2\right)} $$

(12)

where k₁ is a positive value near 0, w₁ is the width of the curve at the height of k₁a, k₂ is a positive value near 1, and w₂ is the width of the curve at the height of k₂a.

Fig. 8 shows the graphical illustration of polishing mechanism by a bonnet on a flat surface. d is the polishing depth in mm, R is the radius of the bonnet, ϕ is the diameter of spot size, and φ is the inclination angle. In Fig. 8b, O_b is the center of the bonnet, O_p is the center of the polishing area, P is a polishing point within the polishing area, L is the axis of rotation of the bonnet, Q is the intersection of line L with the polished surface area, O_r is the swing center of point P, r_P is the swing radius, PK ⊥ O_PQ, PN ∥ O_bO_P, and N is in the spherical surface with radius R and center O_b. Angular velocity expressed in vector form is (norm{direction}).

$$ \omega =2\pi S\left\{0,-\sin \left(\varphi \right),\cos \left(\varphi \right)\right\} $$

(13)

where S is the angular velocity.

Fig. 8

Graphical illustration of polishing mechanism by bonnet on flat: (a) polishing tool and workpiece; (b) detailed geometry in the polishing area

The point P(x_p, y_p, z_p) is a point in the contact area and is determined by angle αand θ, as shown in Fig. 8b. The vector of $\overrightarrow{O_bP}$ is represented as

$$ \overrightarrow{O_bP}=D\sec \left(\alpha \right)\left\{\tan \left(\alpha \right)\sin \left(\theta \right),\tan \left(\alpha \right)\cos \left(\theta \right),-1\right\} $$

(14)

$$ \left| KQ\right|=\left|{O}_PQ\right|-\left|{O}_PK\right|=D\tan \left(\varphi \right)-{y}_p $$

(15)

Therefore,

$$ {r}_P=\sqrt{{\left|{O}_rK\right|}^2+{\left| PK\right|}^2}=\sqrt{{\left(\left(D\tan \left(\varphi \right)-{y}_p\right)\cos \left(\varphi \right)\right)}^2+{x}_p^2} $$

(16)

$$ \left|{\mathbf{v}}_{\mathbf{r}}\right|={r}_P\left|\boldsymbol{\upomega} \right|=\frac{\pi }{30}S\sqrt{{\left(D\tan \left(\varphi \right)-{y}_p\right)}^2+{x}_p^2} $$

(17)

The direction of the velocity v_r at point P can be obtained by using vector operation (cross product):

$$ {\mathbf{v}}_{\mathbf{r}}=\omega \times \overrightarrow{O_bP}=\left|{\mathbf{v}}_{\mathbf{r}}\right|\left\{{v}_{rx},{v}_{ry},{v}_{rz}\right\} $$

(18)

The feed speed can be expressed as

$$ {\mathbf{v}}_{\mathbf{f}}=\left|{\mathbf{v}}_{\mathbf{f}}\right|\left\{{v}_{fx},{v}_{fy},{v}_{fz}\right\} $$

(19)

For polishing a flat surface, the relative velocity at the point P(x_p, y_p, z_p) is

$$ {v}_p=\sqrt{{\left(\left|{\mathbf{v}}_{\mathbf{r}}\right|{v}_{rx}+\left|{\mathbf{v}}_{\mathbf{f}}\right|{v}_{fx}\right)}^2+{\left(\left|{\mathbf{v}}_{\mathbf{r}}\right|{v}_{ry}+\left|{\mathbf{v}}_{\mathbf{f}}\right|{v}_{fy}\right)}^2} $$

(20)

The distribution of the pressure of the bonnet on the workpiece is taken as a Gaussian curve shape, which is expressed as a modified Gaussian function (MGF)

$$ P(r)=a\exp \left(-\frac{1}{2}{\left|\frac{r-c}{b}\right|}^{\lambda}\right) $$

(21)

where r = |PO_b| = D tan (α), c = 0; P_m is the maximum pressure at the center of the imprint. Therefore,

$$ P\left(\alpha \right)={P}_m\exp \left(-\frac{1}{2}{\left(\frac{D\tan \left(\alpha \right)}{b}\right)}^{\lambda}\right) $$

(22)

The parameter b controls the width of the curve, and λ determines the shape of the curve. The two parameters are determined by air pressure, polishing cloth material, etc.

According to Preston’s Law, the integrated material removal rate (IMRR) is

$$ {M}_p=k\cdot {P}_P\cdot {V}_P\cdot {t}_d $$

(23)

where M_P is the material removal at point P, k is Preston coefficient, P_P is polishing pressure, V_P is the magnitude of relative velocity at point P, and t_d is dwell time. As a result, MRR at point P(α, θ) is

$$ {M}_p\left(\alpha, \theta \right)=k\cdot {P}_m\cdot \exp \left(-\frac{1}{2}\left(\frac{D\tan \left(\alpha \right)}{b}\right)\right)\cdot {\left({\left(\left|{\mathbf{v}}_{\mathbf{r}}\right|{v}_{rx}+\left|{\mathbf{v}}_{\mathbf{f}}\right|{v}_{fx}\right)}^2+{\left(\left|{\mathbf{v}}_{\mathbf{r}}\right|{v}_{ry}+\left|{\mathbf{v}}_{\mathbf{f}}\right|{v}_{fy}\right)}^2\right)}^{1/2}\cdot {t}_d $$

(24)

The above equation is the polishing tool material removal characteristics, or influence function. A series of preliminary simulation and experimental studies based on the above theoretical model have been undertaken to generate various structured surfaces, which will be presented in the coming section.

4.2.2 Simulation Experiments for Generating Different Surface Patterns

Table 3 tabulates the design of the simulation experiments. Three groups of experiments are conducted under different spacing distance, different polishing depth, and different variable process angles, respectively. Figures 9, 10, and 11 show the different surface patterns generated by the designed three groups of the experiments. It is interesting to note that different structured surfaces can be generated by bonnet polishing with changing the polishing parameters and strategies, which provides some new approaches and solutions for producing structured surfaces, especially for the hard machining materials.

Table 3 Design of simulation experiments

Fig. 9

Structured surface generation by different spacing in mm (horizontal spacing, spx; vertical spacing, spy)

Fig. 10

Structured surface generation by different polishing depths in mm

Fig. 11

Structured surface generation by different inclination angle in degrees

4.2.3 Real Polishing Experiments for Generating Different Surface Patterns

A series of practical polishing experiments have also been conducted in the present study to further validate the proposed model. The polishing machine used in the experiments is a seven-axis freeform polishing machine IRP200 manufactured by Zeeko Ltd., UK. Table 4 shows the polishing parameters, workpiece materials, slurries, and polishing tools employed in the polishing experiments.

Table 4 Polishing parameters used in the experimental studies

To determine the tool influence function (TIF) in the polishing process, the test of TIF was firstly undertaken under the same polishing conditions but with a longer dwell time for the convenience of the data measurement. In the present study, the dwelling time for TIF test is 300 s. The polished surface for TIF test was measured by Talysurf PGI 1240 (Taylor-Hobson Ltd., UK), and the measured area data are shown in Fig. 12a. The measured data are processed in two steps. Firstly, the measured TIF data (X = Y = 0) are centered by x–y shift, and the data center is determined by an inscribed circle inside the rectangular formatted data (X–Y plane). Secondly, the data outside the inscribed circle (X–Y plane) are trimmed. Hence, the data is fitted to find the coefficients in the proposed model. Figure 12b shows the processed data which indicate TIF in the contact area of the polishing bonnet on the workpiece surface. As shown in Fig. 11, it is found that the TIF is asymmetric and the reason for the asymmetry is due to the inclination angle of the bonnet on workpiece surface which causes the asymmetric relative velocity between the polishing tool and the workpiece surface, and hence the MRR is asymmetrical as referred to Eqs. 23 and 24. To find the coefficients in Eq. 24, nonlinear regression method is used based on the measured data for TIF test. The coefficients were found to be k = 3.566 × 10⁻⁸ (Preston coefficient), b = 0.7446, and c = 3.0432.

Fig. 12

(a) Measured TIF test data and (b) data after processing

Figure 13 shows the mathematical extrapolation of tool influence function obtained in the polishing process. Figure 13a depicts the 3D topography, while Fig. 13b and c is the TIF profiles in XZ and YZ planes, respectively. It is interesting to note that the two profiles in XZ and YZ planes are different in shape. This makes the polishing process more flexible to generate different surface structures.

Fig. 13

Mathematical extrapolation of tool influence function (TIF) in the polishing process: (a) 3D topography; (b) in XZ plane; (c) in YZ plane

The polishing experiments are designed by using different polishing strategies with different spacing distances as show in Table 5. The topography of the polished workpiece surface was measured by a white-light interferometer Wyko NT 8000 from Veeco Instruments Inc., USA.

Table 5 Group A: experimental design for the raster polishing tests

Table 6 illustrates the comparisons between the measured and the simulated results of the workpiece surface polished under various conditions, i.e., conditions A1–A5. It is interesting to note that the patterns of the predicted structured surfaces exhibit a good agreement with the measured structured surfaces. This further validates the proposed model. This not only provides an important means for generating different surface structures or patterns by ultra-precision polishing approach but also makes the polishing process more controllable and predictable.

Table 6 Comparison between the measured and simulated results for the structured surface generation

For further verification, some surface parameters (root-mean-squared value, Sq; peak-to-valley value, St) are determined for the measured surface data and the predicted ones. Figure 14 shows a graphic illustration of the comparison of the predicted and measured results. As shown in Fig. 14, the parametric values are of the same order of magnitude. It is also interesting to note that the predicted values of Sq are all larger than the measured values (except data no. A5), while the predicted values of St are all smaller than the measured ones. This can be explained as follows. The model proposed in the present study only considers the geometry of the TIF. In other words, the surface generation is predicted based on the shape of TIF.

Fig. 14

A comparison between the measured and predicted results for the surface parameters: (a) root-mean-squared value (Sq) comparisons; (b) peak-to-valley value (St) comparisons

However, the polishing process also involves other factors such as interference between two concessive polishing contacts, the influence of polishing fluid, and interaction between the workpiece material and abrasive particles in the polishing fluid. As a result, the predicted St is less, and the predicted Sq is larger than the measured values, since in the real polishing process, there are additional factors which remove high points on the surface. Moreover, there is a good agreement of the trend between the measured and predicted results. There exist some deviations between the measured and predicted results which are due to the fact that ultra-precision polishing is a complex machining process. The surface generation is affected by a lot of materials and process factors. Some other factors would need to be considered such as the effect of slurry, machine characteristics, dynamic factors, etc.

5 Microstructured Surface Generation by Bonnet Polishing with Different Precess

In bonnet polishing, there are different precess polishing, including single precess, continuous precess, and swing precess, as shown in Fig. 15. Figure 15a is the “single precess,” and Fig. 15b is “continuous precess” polishing regime (Beaucamp and Namba 2013), in which the precess angle is constant. Figure 15c and d shows the two types of swing precess bonnet polishing processes which include climb and vertical swing precess bonnet polishing processes. In the vertical swing precess bonnet polishing process, the swing plane of polishing spindle is perpendicular to the feed direction of the polishing tool, while the swing plane of polishing spindle consists of the normal direction of the target surface and the feed direction of the polishing tool in climb swing precess bonnet polishing. The generation of complex 3D structured surfaces is affected by many factors which include point spacing, track spacing, swing speed, swing angle, head speed, tool pressure, tool radius, feed rate, polishing depth, polishing cloth, polishing strategies, polishing slurry, etc.

Fig. 15

(a) Single precess, (b) continuous precess, (c) climb swing precess, and (d) vertical swing precess bonnet polishing across the raster tool path

5.1 Microstructured Surface Generation by Single Precess Polishing

In practice, polishing is a multistep process conducted by repeatedly running particular designed polishing cycles until the expected surface finish and form error are obtained. Within each cycle, the polishing tool sweeps through the polished surface under adopted polishing tool path and desired dwell time map. Hence, an important part of the surface generation model is dwell time and tool path planning (Tam et al. 1999) as considering time efficiency and surface quality improvement. Based on the discussion in previous sections, bonnet polishing appears to be a relative and cumulative polishing process for various materials, and the surface generation, ΔZ_i, of bonnet polishing can be assumed to be a linearly cumulative effect of dwell time, T₀, together with the material removal rate, MRR_ij, under the same polishing condition as follows:

$$ \Delta {Z}_i=\sum \limits_{j=1}^N{MRR}_{ij}\cdot {T}_0,\qquad \left(i=1,2,3\cdots M\right) $$

(25)

where M is the sample point number, N is the trajectory point number along the polishing path, and MRR_ij is the material removal of jth sample point when the ith trajectory point along the polishing path is polished by the bonnet. The value of MRR_ij depends on the material removal rate and the contract area of the bonnet. It will be non-zero if the jth sample point locates within the contract area of the ith trajectory point and will be zero if the sample point locates outside the contract area.

In the present study, raster polishing was used to understand the surface generation mechanisms by single precess bonnet polishing and verify the effectiveness of the surface generation model. The polishing conditions in all the paths are assumed to be constant, and hence the tool influence function is stable and constant, and the polishing paths are also assumed to be evenly spaced straight lines on a flat surface as can be seen in Fig. 16. When the predicted influence function follows straight path lines with a constant surface feed rate, the removal profile is constant along each and all the path lines. Hence, the material removal distribution is constant along the path line direction and waves in the orthogonal direction arising from the overlapping of the removal profiles affiliated with adjacent polishing path lines for the raster polishing. In this case, the surface generation may be simplified to be a 2D problem which can be solved numerically for given polishing path spacing and path number. The polishing path spacing represents the translation distance of removal profile along the orthogonal direction, and the feed rate determines the dwell time of each polished spot and hence the surface height of removal profile along the path line direction.

Fig. 16

A schematic diagram of the surface generation model by single precess bonnet polishing in the raster polishing

Experimental studies for single precess bonnet polishing by aster polishing were carried out. The polishing strategies used for generating the designed surface pattern are shown in Table 7, while the surface generation model has been purposely built and programmed by using MATLAB software package. Figure 17 shows a comparison between the measured and simulated results in the pattern test. It is found that the simulated surface pattern by the surface generation model agree well with that for the experimental results. The peak-to-valley (PV) value and the root-mean-squared (RMS) value are determined for the measured surface data and the predicted ones. The PV value of the predicted data is 0.803 μm, and the RMS value is 229 nm, while the PV value and RMS value of the measured surface data are 0.876 μm and 110 nm, respectively. It is interesting to note that the parametric values are of the same order of magnitude. This further help to explain and predict the relative and cumulative polishing process and the linearly cumulative effect of dwell time together with the constant material removal rate under the identical polishing condition in surface generation of bonnet polishing.

Table 7 Parameters for generating surface pattern

Fig. 17

Comparison between the measured and simulated results of the pattern test

To further verify the effectiveness of the surface generation model, a least-square-based surface matching method is used to evaluate the deviation of the measured surface from the corresponding simulated surface (Ren et al. 2012). Due to the misalignment between the coordinate frames of the coordinate frames of the measured surface and the simulated surface, the surface matching is required to search for an optimal Euclidean motion for the measured surface so that it is well aligned with the simulated surface as close as possible. After that, the deviation of the simulated surface and the measured surface is considered to be the prediction error of the proposed model as shown in Fig. 18. It is turned out that the peak-to-valley (PV) value of the prediction error is 0.4231 μm and the root-mean-squared (RMS) value is 64.8 nm. The result reveals that the surface generation model can be successfully used for predicting and better explaining the surface generation in bonnet polishing. This further validates the technical feasibility and effectiveness of the surface generation model in bonnet polishing.

Fig. 18

Evaluated prediction error of the testing pattern for bonnet polish

5.2 Microstructured Surface Generation by Swing Precess Bonnet Polishing

5.2.1 Principle of Swing Precess Bonnet Polishing

It is well known that surface generation by bonnet polishing is dominated by the influence function of the polishing tool instead of the pure geometry of the cutting tool that is quite different from other ultra-precision machining processes such as single-point diamond turning and ultra-precision raster milling. Taking the advantages of high polishing efficiency and flexibly controlled influence function in the bonnet polishing process, the swing precess bonnet polishing (SPBP) method for generating microstructured surfaces is realized by the relative and cumulative process of varied tool influence function through the combination of specific polishing tool orientation and tool path. As shown in Fig. 19, the SPBP method is a sub-aperture finishing process in which the polishing spindle is swung around the normal direction of the target surface within the scope of swing angle while moving around the center of the bonnet.

Fig. 19

Schematic diagram of swing precess bonnet polishing

5.2.2 Tool Path Generator for Swing Precess Bonnet Polishing

To realize swing precess bonnet polishing, a tool path generator (TPG) has been purposely built which is used for generating the CNC files for polishing as shown in Fig. 20. The TPG is composed of three modules which are input module, processing module, and output module. The input module is used to acquire the polishing parameters (e.g., head speed, spacing, tool pressure, bonnet radius, feed rate, etc.) and the polishing strategies (e.g., spacing and swing speed). Hence, the processing module determines the spot size, track coordinates, polishing gesture, and control parameters. The necessary CNC file of the SPBP tool path is generated by the output module. In this study, the TPG has been developed by the MATLAB programming software package. Hence, SPBP can be undertaken by a seven-axis polishing machine through two kinds of motions which include feed motion (X-, Y-, Z-, C-axis) and swing motion (A-, B-, H-axis), respectively.

Fig. 20

Schematic diagram of tool path generator for SPBP

5.2.3 A Multi-scale Material Removal Model for SPBP

Since the produced 3D structured surfaces are effected by various polishing parameters, the prediction model needs to be developed for effective selection of the polishing conditions and better understanding of the SPBP process. The surface generation of the polishing process can be regarded as the convolution of the influence function and the dwell time map along the pre-specific tool path (Tam and Cheng 2010). To simulate the generation of 3D structured surfaces by SPBP, a multi-scale material removal model has been developed based on the prior work (Cao and Cheung 2016). The model is used for predicting the tool influence function in SPBP based on the study of contact mechanics, kinematics theory, and abrasive wear mechanism. The material removal of the SPBP can be expressed by

$$ MRR\left(x,y,t\right)=\frac{2\eta {K}_{ac}{V}_c{tV}_r\left(x,y\right)P\left(x,y\right)}{H_w\tan \alpha }{\left(\frac{R_a}{\sigma_z}\right)}^{\frac{1}{2}} $$

(26)

where η is the volume fraction of a wear groove removed as wear debris, K_ac is the coefficient related to the particle size distribution and the hydrodynamics condition, V_c is the volume fraction of the polishing slurry, t is the polishing time, H_w is the hardness of polished workpiece, α is the semi-angle of the cone particle, R_a is the radius of the pad asperities, σ_z is the standard deviation of asperity heights, V_r(x, y) is the relative velocity distribution between the polishing pad and the target surface in the polishing area, and P(x, y) is the pressure distribution at the polishing contact area.

$$ V\left(x,y,S,\varphi, {R}_b,d\right)=\frac{\pi S}{30}\sqrt{{\left(y\cot \varphi -\left({R}_b-d\right)\right)}^2{\left(\sin \varphi \right)}^2+{x}^2{\left(\cos \varphi \right)}^2} $$

(27)

where x² + y² ≤ (R_b)² − (R_b − d)², S is angular velocity in rpm, φ is the swing angle, d is the polishing depth in mm, and R_b is the radius of the bonnet in mm.

$$ {\displaystyle \begin{array}{l}P\left(x,y,{R}_b,d,\omega, Y,\nu, \varphi, {\eta}_1,{\eta}_2\right)={p}_0{\left(1-\frac{x^2}{a^2}-\frac{y^2}{a^2}\right)}^{1/2}+\\ {}\frac{\left(1-2v\right)\left(1+v\right)}{Y}\cdot \frac{\omega \cos \varphi \left(2{\eta}_2+{\eta}_1/3\right){p}_0x\left({R}_b-d\right)}{a^2}\cdot {\left(1-\frac{x^2}{a^2}-\frac{y^2}{a^2}\right)}^{-1/2}+\\ {}\frac{{\left(1-v\right)}^2}{Y}\cdot \frac{\omega \cos \varphi \left(2{\eta}_2+{\eta}_1/3\right)\pi {p}_0x}{2a}\end{array}} $$

(28)

where $a=\sqrt{dR_b}$ denotes the radius of contact area and Y and ν are the Young modulus and the Poisson ratio, respectively, while η₁ and η₂ are the coefficients of viscosity related to shear and bulk deformation, respectively. ω is the angular velocity, and p₀ = 3F_N/(2πa²) denotes the maximum contact pressure, and F_N is the total elastic force, acting on the surface (in normal direction) on the polishing pad:

$$ {F}_N=\frac{2}{3}\frac{Y}{\left(1-{v}^2\right)}{R_b}^{1/2}{d}^{3/2} $$

(29)

5.2.4 Surface Generation Model of Swing Precess Bonnet Polishing

A surface generation model has been developed for swing precess bonnet polishing based on the surface generation mechanism of the relative and cumulative removal process in the polishing process, together with the predicted tool influence function by the multi-scale material removal model as described in previous section. As shown in Fig. 21, the track coordinates of the polishing bonnet, along with the polishing path, are determined in terms of polishing parameters and strategies. The tool influence function of every polishing point is then computed by the multi-scale material removal model with respect to the corresponding polishing gesture and track coordinates. Hence, the surface generation by SPBP is simulated by the aggregation of the amount of material removal of the superposed tool influence function at every sampling point, together with a Gaussian Process model (Williams and Rasmussen 2006).

Fig. 21

Schematic diagram of the surface generation mechanisms of swing precess bonnet polishing across the raster tool path

6 Experimental Studies for Bonnet Polishing of Microstructured Surface

To realize the technological merits as well as the performance of the multi-scale material removal model and the surface generation model in this chapter, a series of experiments were conducted on a Zeeko IRP 200 ultra-precision freeform polishing machine, as shown in Fig. 22a. Cylindrical nickel copper samples with a diameter of 25.4 mm were machined by single-point diamond turning to ensure their consistent initial surface finish. The parameters used in single-point diamond turning include spindle speed, 1500 rpm; feed rate, 8 mm per min; depth of cut, 4 μm; and tool radius, 2.47315 mm. The surface roughness and 3D surface topography of the samples were measured by a Zygo Nexview 3D optical surface profiler. As shown in Fig. 22b, the arithmetic roughness, R_a, of the samples before polishing is found to be 103 nm.

Fig. 22

(a) Zeeko IRP200 Ultra-precision freeform polishing machine and (b) measured surface topography of the samples before polishing

Two groups of experiments are designed, said Group A and Group B. In Group A of experiments, line tests of swing precess bonnet polishing were conducted with different polishing parameters in order to study the effect of polishing parameters on the generation of structured surfaces. Experiments conducted in Part B are pattern tests which aim to verify the feasibility of the proposed swing precess bonnet polishing method under various polishing conditions and evaluate the performance of the multi-scale material removal model and surface generation model for the swing precess bonnet polishing.

6.1 Group A: Effect of Parameters on the Generation of Structured Surfaces

6.1.1 Experimental Setup

This group of experiments aims to investigate how single process parameter affects the generation of 3D structured surface with swing precess bonnet polishing. The polishing parameters are selected based on the work range of typical parameters in bonnet polishing system (Zeng and Blunt 2014). Green silicon carbide (GC) grit #4000 and LP-26 polishing cloth were used. The tool radius, the tool pressure, and the spindle speed are 20 mm, 1.2 bar, and 1500 rpm, respectively. Other polishing settings are shown in Table 8. As shown in Fig. 23, the 3D structured surfaces generated on the polished area were characterized by the peak-to-valley (PV) height, the width, and the length of a rectangle that can cover a cycle of the structure.

Table 8 Polishing parameters used in line tests of swing precess bonnet polishing

Fig. 23

Characterization of the surface feature parameters of the generated 3D structured structures

6.1.2 Results and Discussion

Figures 24, 25, 26, 27, 28, and 29 show the results of the characterization of the 3D structures generated by only a single one-way tool path. It is interesting to note that the shape of each cycle of the structures is varied under different polishing conditions. It is believed that this may affect the pattern of the 3D structures formed on a surface. Moreover, precess angles and feed rate significantly affect the density of the 3D structures, as shown in Figs. 24 and 25. The smaller the precess angle and the slower the feed rate, the higher the density of the 3D structures. Figure 26 shows that the density of the 3D structures increases with the increasing swing speed. This can be explained by

$$ N=\frac{L{\omega}_s}{4{F}_rA} $$

(30)

Fig. 24

Effect of precess angle on the surface generation in line test: (a) 2D cross-sectional profile with PV values and (b) width and length of the 3D structures

Fig. 25

Effects of feed rate on the surface generation in line test: (a) 2D cross-sectional profile with PV values and (b) width and length of the 3D structures

Fig. 26

Effects of swing speed on the surface generation in line test: (a) 2D cross-sectional profile with PV values and (b) width and length of the 3D structures

Fig. 27

Effect of tool offset on the surface generation in line test: (a) 2D cross-sectional profile with PV values and (b) width and length of the 3D structures

Fig. 28

Effects of point spacing on the surface generation in line test: (a) 2D cross-sectional profile with PV values and (b) width and length of the 3D structures

Fig. 29

Effects of swing direction on the surface generation in line test: (a) 2D cross-sectional profile with PV values and (b) width and length of the 3D structures

where N is the number of swing cycle, L is the length of the polishing area in mm, ω_s is the swing speed in degree per min, F_r is the feed rate in mm/min, and A is the precess angle in degree. The smaller the precess angle and the higher the swing speed mean the smaller the swing cycle, while the slower the feed rate means the longer polishing time. Hence, both of them cause the higher density of the structures.

Figures 24a and 25a show that the PV values increase as the precess angle and feed rate are increased. This can be explained by

$$ {T}_c=\frac{4A}{\omega_s} $$

(31)

where T_c is time of each swing cycle. A larger precess angle means longer time of each swing cycle and hence causes a larger PV value. The smaller the feed rate means the higher the density of the structures, and this may cause the smaller PV values. Figure 24b shows that the width and the length of the 3D structures increase with increasing precess angle. Figure 25b shows that the width of the 3D structures increases with increasing feed rate, while the length of the 3D structures shows no obvious variation trend. Precess angle 15^o, feed rate 75 mm per min, and swing speed 200° per minute give similar 3D structures, which have a larger width-to-length ratio.

Figures 27, 28, and 29 show that the tool offset, point spacing, and swing direction have no significant influence on the density of the 3D structures. Figure 27a shows that the PV values of tool offset of 0.14 mm is obviously smaller than that of 0.28 and 0.42 mm, while Fig. 27b shows that the width and the length of the 3D structures increase with increasing the tool offset. This can be explained by a larger tool offset presenting a larger contact area and higher contact pressure and hence a higher material removal rate. Tool offsets of 0.28 and 0.42 mm give similar PV values, and this infers that there may exist a critical tool offset which if exceeded would lead to the breaking of the increasement of PV values with the increasing tool offset.

Figure 28 shows that the effect of point spacing on the peak-to-valley (PV) height, the width, and the length of the 3D structures is negligible. Figure 29 shows that the swing directions mainly affects the shape of the 3D structure but shows no significant influence on the PV values. Although experimental study was conducted on studying effect of the single process parameter on the generation of 3D structured surfaces, the significance of process parameters and the interactions among them should be further investigated in the further study.

6.2 5.2 Group B: Results of Experimental and Theoretical Investigation of Pattern Tests

6.2.1 Experimental Setup

In Group B, four sets of parameters were selected to generate 3D structured surfaces, as shown in Table 9. The tool pressure and the spindle speed were set at 1.2 bar and 1500 rpm, respectively. Twelve samples were polished with three samples for each set.

Table 9 Polishing parameters used in pattern tests of swing precess bonnet polishing

6.2.2 The Repeatability Under Various Polishing Conditions

Table 10 shows the results of pattern tests measured by a Zygo Nexview 3D optical surface profiler, and Table 11 shows the photos of the patterns generated by swing precess bonnet polishing. It is interesting to note that complex 3D structured patterns are formed on the surfaces. The results show that different kinds of the complex 3D structured surfaces can be generated by using different polishing parameters. It is also found that the 3D topographies of the 3D structured patterns generated on the surfaces have a good consistency in surface shape when using the same polishing parameters. Since there are some noise points for the original measured data without filtering, especially for the case of set 3(b), the direct comparison of measured data for each set is not acceptable.

Table 10 Measured topography of structured pattern generated by swing precess bonnet polishing

Table 11 Photos of structured pattern generated by swing precess bonnet polishing

To further evaluate the feasibility and stability of the proposed SPBP method, the deviation of measured results with same polishing conditions was evaluated by the repeatability test. For every set of experiment, three measurement data were obtained, named set a, set b, and set c. The measurement data set a were used as the reference data. Then measurement data set b and measurement data set c were compared to the reference data. The error maps were then obtained, and the root-mean-square (RMS) value of the error maps were calculated and shown in Table 12. As shown in Table 13, the result shows that the RMS values of the error maps vary from 0.2 to 0.5 μm which demonstrate that the repeatability of the experiment is reasonable. This implies that swing precess bonnet polishing can be used for generating different kinds of 3D structured patterns in deterministic and stable way, especially for difficult-to-machine and ferrous materials. However, the deformation of polishing bonnet and the abrasion wear of the polishing cloth could affect the stability of the produced pattern, and hence both of them should be critically controlled. The polishing conditions should also be further optimized for obtaining the structured patterns with high surface finish.

Table 12 The error maps of experimental results for pattern tests

Table 13 The root-mean-squared (RMS) value of the error maps of experimental results for pattern tests

6.2.3 Experimental Verification of Surface Generation Model for SPBP

To evaluate the performance of the multi-scale material removal model and surface generation model for SPBP, the simulation experiments were conducted using the same polishing parameters with practical polishing experiments as shown in Table 9. Figure 30 shows the simulated 3D structured surface generated by the theoretical model. It is interesting to note that the surface topography of the simulated results shows a reasonably good agreement with that of the experimental data. This means that the theoretical model can be successfully used for the selection of polishing parameters for designed structured pattern and better understanding of the SPBP process.

Fig. 30

Simulated topography of structured pattern by the developed theoretical model

To compare the simulation results and measured results quantitatively, the measured results have to be registered to the simulated results by a freeform surface characterization method which has been purposely built based on an iterative closest point (ICP) method (Besl and Mckay 1992). To improve the accuracy and robustness of the registration process, the simulated surface was first filtered with a Gaussian zero-order regression filter (Brinkmann et al. 2001). Hence, the measured surface was registered to the filtered simulated surface with the ICP method, and the transformation matrix was obtained. Based on the calculated transformation matrix, the error map showing the deviation of the simulation results and the measured results was determined by registering the measured results with the original simulated results. Figure 31 shows the surface matching and error evaluation process for case parameter set 4. As shown in Fig. 31a and b, the simulated topography and measured topography of the complex 3D structured surface generated by SPBP are found to agree well. To further verify the performance of the models quantitatively, the deviation of the measured surface from the corresponding simulated surface was determined by the freeform surface characterization method. Figure 31c shows the results after registration of the measured surface and the simulated surface, while Fig. 31d shows the deviation between the simulated and measured results.

Fig. 31

The surface matching and error evaluation process for case set 4

Table 14 shows the comparison results between the simulated and measured results of swing precess bonnet polishing. The simulated results were found to agree reasonably well with the experimental results with the root-mean-squared (RMS) value of the prediction error from 0.1 to 0.5 μm. Since the surface generation of SPBP is the relative and cumulative process of varied tool influence function, the prediction error of surface generation model is the accumulation of the error of material removal model under different polishing gestures. The comparison results show that the theoretical model can be successfully used for the prediction and better understanding of the SPBP process.

Table 14 The root-mean-squared (RMS) value of the error maps between the simulated and experimental results

7 Summary

Microstructured surfaces are surfaces possessing specially designed functional textures which are widely used in the development of a wide range of products such as structured surfaces for improving adhesion property of the mold and die and fluid films sealing. However, they are difficult to be fabricated to high precision and high accuracy by traditional machining technology. As one of the computer-controlled ultra-precision polishings (CCUP), bonnet polishing is an enabling machining technology which is capable of fabricating ultra-precision freeform surfaces made of difficult-to-machine materials with sub-micrometer form accuracy and surface roughness in the nanometer range. It addresses the limitations of materials that can be machined by single-point diamond turning or ultra-precision raster milling processes. However, most of the previous research work has focused on the control of surface finish and the control of form errors of the surfaces. Research on the generation of microstructured surface by bonnet polishing has received relatively little attention.

In this chapter, a theoretical and experimental investigation of microstructured surface generation by using bonnet polishing was presented. The surface generation by bonnet polishing has been studied which include the theoretical and experimental analysis of material removal rate (MRR) or tool influence function (TIF) which significantly affects the surface generation in bonnet polishing. A surface topography simulation model has been built, and hence a model-based simulation system has been established for the modeling and simulation of microstructured surface generation by using bonnet polishing. A series of simulation and practical polishing experiments have been undertaken to verify the surface topography simulation model and to evaluate the performance of the model-based simulation system. The experimental results provide an important means to demonstrate the capability of the model-based simulation system in the prediction of surface generation in terms of the form error and various patterns of the 3D microstructures generated by using bonnet polishing.