### Smoothing theory

It is very important to establish a reasonable and effective mathematical model of the smoothing effect in the computer-controlled polishing process. There have been some studies on the smoothing effect by using elastic tools mentioned above.

As shown in Fig. 1(a), the ripple errors on the surface cause uneven contact between the polishing pad and the workpiece, resulting in inhomogeneity in the pressure distribution. The peak of the ripple has an additional pressure difference *P*_{add} compared to that of the trough. According to the Preston equation, the material removed during the polishing process is proportional to the polishing pressure. Therefore, more material is removed at the peak of the ripple than at the trough. As a result, the workpiece will become smoother, thereby achieving the smoothing effect of the surface errors. The Preston equation shows that the material removal of workpiece satisfies:

$$ \Delta \varepsilon =K\cdot {P}_{add}\cdot v\cdot \Delta t $$

(1)

where Δ*ε is* the amount of change in the amplitude of the ripple errors after single smoothing, *K* is a constant parameter, *v* is the relative speed between the polishing pad and the workpiece, and Δ*t is* the time of single smoothing based on the existing surface errors.

For visco-elastic polishing tools, such as pitch pads or RC pads, a parametric smoothing model [11] has been established:

$$ SF=\frac{\Delta \varepsilon }{\Delta Z}=k\cdot \left({\varepsilon}_{ini}-{\varepsilon}_0\right) $$

(2)

where *SF* is the smoothing factor, defined as the ratio of Δ*ε* (the amount of change in the amplitude of the ripple errors after single smoothing) to Δ*Z* (the depth of material removal on the surface of the workpiece after single smoothing under the premise that the contact pressure at the peak and the trough of the ripple errors is the same). To a certain extent, Δ*Z* can be understood as the depth of material removal on the surface of the workpiece without ripple errors under the same processing conditions. *ε*_{ini} is the initial amplitude of the ripple errors before smoothing and *ε*_{0} is the final amplitude after multiple smoothing which indicates the limit of the smoothing process, that is, the amplitude does not change with the increase of smoothing time when the amplitude of ripple errors decreases to *ε*_{0}. The magnitude of *SF* characterizes the smoothing capacity to the ripple errors in the polishing process. It can be seen that the smoothing factor *SF* has a linear correlation with the surface roughness of the workpiece, and the proportional coefficient is *k*. In the parametric smoothing model:

$$ k=\frac{\kappa_{total}}{P} $$

(3)

$$ \frac{1}{\kappa_{total}}=\frac{1}{\kappa_{elastic}}+\frac{1}{\kappa_{others}} $$

(4)

where *P* is the pressure between the polishing pad and the workpiece, *κ*_{total} is the material coefficient of the polishing pad, which is related to the elastic material coefficient *κ*_{elastic} and the overall material coefficient *κ*_{others} of other structures.

The parametric smoothing model indicates that the smoothing factor is related to the material parameters and polishing pressure during the polishing process. However, it is difficult to reflect the relationship between the parameters of smoothing process and other process. It is not intuitive to infer the material removal rate of the actual polishing process because no factor is included in this model to describe the evolution of the surface error with time. Therefore, based on the parametric model, a time-dependent smoothing evolution model has also been proposed and applied [13].

Using the mathematical expression of the smoothing factor in the parametric smoothing model, the following equation can be obtained:

$$ \frac{d\varepsilon}{d Z}=\frac{d\left(\varepsilon -{\varepsilon}_0\right)}{d Z}=-k\cdot \left(\varepsilon -{\varepsilon}_0\right) $$

(5)

Then

$$ \varepsilon =\left({\varepsilon}_{ini}-{\varepsilon}_0\right)\cdot {e}^{-k\cdot Z}+{\varepsilon}_0 $$

(6)

where *ε* is the amplitude of the ripple errors on the workpiece surface after several times of smoothing process, *Z* is the total material removal depth of the workpiece surface without ripple errors under the same smoothing time *t* and processing conditions.

With *Z* developed by the Preston equation, similar as Eq.1, combined with Eq.3, the above equation yields:

$$ \varepsilon =\left({\varepsilon}_{ini}-{\varepsilon}_0\right)\cdot {e}^{-{\kappa}_{total}\cdot K\cdot v\cdot t}+{\varepsilon}_0 $$

(7)

The smoothing model represented by Eq.7 reveals that the surface ripple errors converge exponentially with time during the smoothing process. In the general application of the model, the data points of the polishing process are often fitted by an exponential function, and the smoothing efficiency is measured by the obtained fitting parameter values. However, how to calculate a more accurate predicted curve of smoothing process from the various process parameters of the model is still an urgent problem to be solved. The theoretical model established next is to find the relationship between the practical polishing parameters and the smoothing efficiency by analyzing the actual smoothing parameters in computer-controlled polishing.

### Predictable evolution smoothing model

Starting from Eq.6, the material removal depth *Z* under different processing conditions can be obtained by relatively accurately modeling and simulating of the theoretical tool influence function (*TIF*). In computer-controlled polishing, the polishing pad usually moves in a specific mode. The most common mode for pitch pad is double planetary motion, as shown in Fig. 2.

The polishing velocity *v* of any contact point (*r, α*) on the polishing pad surface varies with the locations, which can be expressed as:

$$ v\left(r,\alpha \right)={\omega}_1\sqrt{r^2{\left(1+n\right)}^2+{e}^2{n}^2-2 ren\left(1+n\right)\cos \alpha } $$

(8)

where

$$ n=\frac{\omega_2}{\omega_1} $$

(9)

Under a uniform pressure distribution of the polishing layer, the *TIF* of the double planetary motion pad (the average removal within a period *T*) satisfies:

$$ {\displaystyle \begin{array}{c} TIF=K\cdot P\cdot v\\ {}=K\cdot P\cdot \frac{\underset{-\theta }{\overset{\theta }{\int }}v\left(r,\alpha \right) d\alpha}{T}\\ {}=K\cdot P\cdot \frac{\omega_1}{2\pi}\underset{-\theta }{\overset{\theta }{\int }}v\left(r,\alpha \right) d\alpha \end{array}} $$

(10)

where the integration interval *θ* satisfies the following condition:

$$ \theta =\left\{\begin{array}{c}2\pi \\ {}2\operatorname{arccos}\left(\frac{r^2+{e}^2-{R}^2}{2 re}\right)\\ {}0\end{array}\right.{\displaystyle \begin{array}{c}\\ {}\\ {}\end{array}}{\displaystyle \begin{array}{c}r\le R-e\\ {}R-e<r\le R+e\\ {}r>R+e\end{array}} $$

(11)

The *TIF* is related to the position *r* as can be seen from the above equation. Meanwhile, in the polishing process, the area of a 2-D *TIF* image contains all the points at which the polishing pad can produce material removal. Therefore, an overall analysis of the *TIF* area is carried out to establish a comprehensive average effect of material removal. Then the total volume removal rate (*VRR*) in the area of the tool influence function satisfies:

$$ {\displaystyle \begin{array}{c} VRR=\iint TIF\left(x,y\right) dxdy\\ {}=K\cdot P\cdot \frac{\omega_1}{2\pi}\underset{0}{\overset{2\pi }{\int }} d\phi \underset{0}{\overset{R}{\int }} rV(r) dr\\ {}=K\cdot P\cdot {\omega}_1\underset{0}{\overset{R}{\int }} rV(r) dr\end{array}} $$

(12)

where

$$ V(r)=\frac{1}{\omega_1}\underset{-\theta }{\overset{\theta }{\int }}v\left(r,\alpha \right) d\alpha =\underset{-\theta }{\overset{\theta }{\int }}\sqrt{r^2{\left(1+n\right)}^2+{e}^2{n}^2-2 ren\left(1+n\right)\cos \alpha } d\alpha $$

(13)

During the actual polishing process, due to the non-uniformity of the velocity distribution generated by the polishing pad movement mode, the material removal depth at different positions in a specific dwell time is different. Hence, the average removal depth of each polishing point is taken as the total material removal depth *Z* of the workpiece to achieve an objective consideration of the smoothing effect. *Z* satisfies the following equation:

$$ Z=\frac{VRR}{\pi {R}^2}\cdot t=K\cdot P\cdot \frac{\omega_1}{\pi {R}^2}\underset{0}{\overset{R}{\int }} rV(r) dr\cdot t $$

(14)

Substituting Eq.3 and Eq.14 into Eq.6, a complete multi-parametric smoothing model can be obtained as follows:

$$ {\displaystyle \begin{array}{c}\varepsilon =\left({\varepsilon}_{ini}-{\varepsilon}_0\right){e}^{-{\kappa}_{total}\cdot \frac{VRR}{P\cdot \pi {R}^2}\cdot t}+{\varepsilon}_0\\ {}=\left({\varepsilon}_{ini}-{\varepsilon}_0\right){e}^{-{\kappa}_{total}\cdot K\cdot \frac{\omega_1}{\pi {R}^2}\underset{0}{\overset{R}{\int }} rV(r) dr\cdot t}+{\varepsilon}_0\end{array}} $$

(15)

An error decreasing factor (*EDF*) is defined to characterize the efficiency of the exponential convergence over time of the surface ripple errors of the workpiece during the smoothing process, and its equation satisfies:

$$ EDF={\kappa}_{total}\cdot \frac{VRR}{P\cdot \pi {R}^2}={\kappa}_{total}\cdot K\cdot \frac{\omega_1}{\pi {R}^2}\underset{0}{\overset{R}{\int }} rV(r) dr $$

(16)

In this way, the predictable smoothing evolution model Eq.15 with complete parameters of the entire polishing process is simplified as:

$$ \varepsilon =\left({\varepsilon}_{ini}-{\varepsilon}_0\right){e}^{- EDF\cdot t}+{\varepsilon}_0 $$

(17)

In this model, the surface ripple errors converge exponentially with a certain smoothing efficiency which depends on the magnitude of *EDF*: larger *EDF* implies higher efficiency. The convergence curve of the whole smoothing process and then the volume removal rate can be theoretically predicted with the given process parameters. However, due to the instability of the pitch layer and the inhomogeneity of the pressure distribution, the actual removal rate might deviate from the theoretic prediction. Therefore, it is necessary to bring in the volume removal efficiency from an actual polishing spot to calculate the *EDF* based on Eq.16.

### Correction of *EDF* solution process

According to the parameterized smoothing theory, *κ*_{total} is com posed of *κ*_{elastic} and *κ*_{others}. The elastic coefficient *κ*_{elastic} of the pitch layer is related to the spatial frequency *f* of the workpiece surface ripple errors [18], while *κ*_{others} is possibly affected by the geometry of the polishing tool itself, material, polishing slurry, and also the spatial frequency *f* of ripple errors. Therefore, a parameter *C*, called the slope correction factor, is used instead of *κ*_{others} [11].

$$ {\kappa}_{total}=\frac{1}{\frac{1}{\kappa_{elastic}(f)}+\frac{1}{C(f)}} $$

(18)

The influence of the spatial frequency *f* of the ripple errors on the factor *EDF* will be discussed in the experimental part.

In the parametrized smoothing model, according to Eq.2 and Eq.3, fitting a series of continuous experiment data of the smoothing factor *SF* and the surface ripple errors as shown in Fig. 3(a), a straight line with fitting slope *k* as shown in Fig. 3(b) can be obtained. Then *κ*_{total} can be calculated with *k* and the pressure *P* of the polishing pad from Eq.3. At the same time, the convergence curve of the ripple errors in the whole smoothing process can be inferred in accordance with the smoothing model. However, comparing the experimental results, there is a certain difference between the *κ*_{total} calculated by using the slope *k* and the *κ*_{total} obtained by the reverse calculation of the experimental results, which leads to the deviation of the predicted curve and the actual smoothing curve. Hence, it’s of vital importance to quantitatively analyze and modify the solution process of the *EDF* based on the parameterized model in combination with experimental phenomena.

Several short-time pre-processing is usually carried out to achieve the purpose of predicting the smoothing effect, and then the actual smoothing factor *SF* is calculated by combining these data. As shown in Fig. 3(a), during the pre-polishing process, the surface ripple errors converge exponentially. The actual experimental data is only a series of points on the curve at equal time intervals, denoted as *data 1*, *data 2*, *data 3*. The linear fit lines corresponding to these data points are shown in the Fig. 3(b). According to the definition of the smoothing factor *SF* from Eq.2 and Eq.17, for *data 1*, the following relationship equation between smoothing factor *SF* and the error decreasing factor *EDF* can be obtained:

$$ {SF}_1=\frac{\Delta \varepsilon }{\Delta Z}=\frac{\varepsilon_1-{\varepsilon}_2}{\frac{VRR}{\pi {R}^2}\left({t}_2-{t}_1\right)}=\frac{Ae^{- EDF\cdot {t}_1}-{Ae}^{- EDF\cdot {t}_2}}{\frac{VRR}{\pi {R}^2}\left({t}_2-{t}_1\right)}=\frac{Ae^{- EDF\cdot {t}_1}\left(1-{e}^{- EDF\cdot \Delta t}\right)}{\frac{VRR}{\pi {R}^2}\Delta t} $$

(19)

Similarly, at the point of *data 2*, *SF* satisfies:

$$ {SF}_2=\frac{Ae^{- EDF\cdot {t}_2}\left(1-{e}^{- EDF\cdot \Delta t}\right)}{\frac{VRR}{\pi {R}^2}\Delta t}=\frac{Ae^{- EDF\cdot {t}_1}{e}^{- EDF\cdot \Delta t}\left(1-{e}^{- EDF\cdot \Delta t}\right)}{\frac{VRR}{\pi {R}^2}\Delta t} $$

(20)

Then the slope of the line shown in Fig. 3 (b) satisfies:

$$ {k}_0=\frac{SF_1-{SF}_2}{\varepsilon_1-{\varepsilon}_2}=\frac{Ae^{- EDF\cdot {t}_1}{\left(1-{e}^{- EDF\cdot \Delta t}\right)}^2}{\frac{VRR}{\pi {R}^2}{Ae}^{- EDF\cdot {t}_1}\left(1-{e}^{- EDF\cdot \Delta t}\right)}=\frac{1-{e}^{- EDF\cdot \Delta t}}{\frac{VRR}{\pi {R}^2}\Delta t} $$

(21)

Substituting the expression of *EDF* in Eq.16 into the above equation gives:

$$ {k}_0=\frac{1-{e}^{-\frac{\kappa_{total}}{P}\cdot \frac{VRR}{\pi {R}^2}\Delta t}}{\frac{VRR}{\pi {R}^2}\Delta t} $$

(22)

It can be seen that under the premise of keeping the process parameters the same, the fitting slope *k*_{0} will change with the difference of the single pre-processing time Δ*t*. Affected by this, the *κ*_{total} obtained by Eq.3 combining the fitting slope *k*_{0} and the polishing pressure *P* also becomes a variable affected by the single pre-processing time Δ*t*, which does not conform to the essence of the processing process. *κ*_{total} is defined as the material coefficient of the polishing pad, which is usually constant without changing the structure or material of the polishing pad. Therefore, the abnormal experimental phenomena obtained above can be reasonably explained by Eq.22. During the actual polishing process, the relationship between the true value of *κ*_{total} and the linear slope *k*_{0} obtained by fitting the parametric smoothing model no longer satisfies Eq.3, which does not mean that Eq.3 is not suitable for the whole smoothing process。The slope *k* in Eq.3 is calculated by fitting multiple sets of data points obtained from the complete smoothing process (the ripple errors of the workpiece reach the smoothing limit from the initial amplitude through multiple smoothing processes). Several short-term smoothing during the pre-processing process, however, causes a certain degree of difference between the fitted slope *k*_{0} calculated by the fewer data points and the fitting slope *k* of the complete smoothing process. Under this inducement, the *κ*_{total} (expressed as *κ*_{total_(unfixed)} in the experimental part) and *EDF* (expressed as *EDF*_{_(unfixed)} in the experimental part) calculated by Eq.3 cannot match the actual values obtained from the smoothing process.

In fact, taking the limit value for the single pre-processing time Δ*t*, Eq.22 satisfies:

$$ \underset{\Delta t\to 0}{\lim}\frac{1-{e}^{-\frac{\kappa_{total}}{P}\cdot \frac{VRR}{\pi {R}^2}\Delta t}}{\frac{VRR}{\pi {R}^2}\Delta t}=\frac{\kappa_{total}}{P} $$

(23)

Obviously, since the sampling interval Δ*t* of the two data points is not zero in the actual polishing, the value of *κ*_{total} calculated based on the fitted slope *k* and Eq.3 is smaller than its actual value. In the case of multiple sampling and ensuring that the sampling interval Δ*t* is small enough, the above equation can be satisfied approximately, which is hard to implement in the pre-processing process. Calculating the value of *κ*_{total} through Eq. 22 can effectively avoid the influence of the sampling interval Δ*t* on the solution process of *κ*_{total} and obtain the true material coefficients normalized by sampling interval and the error decreasing factor of the complete smoothing process. Due to the differences in the structure and characteristics of different polishing pads, the material properties need to be tested after replacing the polishing pad for smooth processing, that is, several short-time pre-processing is required. The real material properties of the current polishing pad can be obtained by substituting the calculated smoothing factor *SF* and the fitting slope *k*_{0} into Eq.22, and then the convergence curve of ripple errors in the whole smoothing process can be simulated to realize the prediction and guidance of smoothing processing. At this point, a modified smoothing model has been established completely.