# Correlation modeling between process condition of sandblasting and surface texture: A multi-scale approach

Hsin Shen Ho, Maxence Bigerelle, Renald Vincent, Raphaël Deltombe

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Hsin Shen Ho, Maxence Bigerelle, Renald Vincent, Raphaël Deltombe. Correlation modeling between process condition of sandblasting and surface texture: A multi-scale approach. Scanning, 2016, 38 (3), pp.191-201. 10.1002/sca.21254. hal-03448262

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# Correlation Modeling Between Process Condition of Sandblasting and Surface Texture: A Multi-Scale Approach

HSIN SHEN HO,<fnref n="1" />,<fnref n="2" /> MAXENCE BIGERELLE,<fnref n="3" /> RENALD VINCENT,<fnref n="2" /> AND RAPHAEL DELTOMB<fnref n="3" />

**Summary:** In the present study, the influence of sandblasting condition (working pressure) on surface texture is modeled, relying on a multi-scale approach and statistical analysis. To improve the correlation modeling between the process condition and surface texture, special effort is made to identification of an optimal parameter set, including 3D roughness parameters, cut-off lengths, filter types and model types. A power law relationship is identified between the pressure and $S_{dq}$ computed with a cut-off length of 120 µm using a low-pass filter. Experimental and theoretical arguments are provided for justification. SCANNING 38:191–201, 2016. © 2015 Wiley Periodicals, Inc.

**Key words:** surface finishing, surface topography, multi-scale analysis, statistical analysis

## Introduction

Sandblasting is one of the commonly used surface treatment methods. It is traditionally used for surface cleaning and rust removal, nowadays is widely applied to increase the surface roughness on the micro-scale level to facilitate cell adhesion (Anselme and Bigerelle, 2005), and improve surface bonding (Rodrigues et al., 2009) and surface wettability (Kim et al., 2012).

Surface characterization has received much attention for the purpose of modelling the cause-effect relation between the surface texture and its functional application.

Considering the complexity and irregularity of the surface texture (e.g., by sandblasting), one of the major difficulties in such work consists in identifying an appropriate scale that can reveal the features of the micro-complexity of the surface texture. The multi-scale analysis, defined using the fractal theory (Mandelbrot, '77), is therefore implemented in numerous work, such as Brown and Siegmann (2001), Vulliez et al. (2014) and Berglund et al. (2010a, 2010b). They reported correlations between surface roughness and functional properties, and these correlations vary with respect to scale.

As to the roughness parameters for surface characterization, the arithmetic mean deviation ($R_a$) is commonly used as the only candidate for surface texture analysis in academic and industrial fields. For instance, $R_a$ is exhibited to be a promising parameter to describe the textures of polished surfaces in optical appliance applications (Shiou and Asmare, 2015) and of machined surfaces in aerospace applications (Xu et al., 2015). However, this roughness parameter may appear insufficient to tackle the complexity of other engineering surface textures, hence additional roughness parameters of ISO are needed (Nowicki, '85; Bigerelle et al., 2013), even new roughness parameters are developed by users for their specific applications (Anselme and Bigerelle 2005; Vulliez et al., 2014). In spite of the proliferation of new roughness parameters, very often, not all of them are fully utilized for texture analysis, which is termed by Whitehouse ('94) as “parameter rash”. This ignorance seems to be a consequence of the fact that not many studies incline to question the pertinence of the roughness parameters which are used to describe surface textures, therefore effort is made to address such issue in the present work. Moreover the three dimensional (3D) roughness parameters are considered and analyzed, which are believed to provide richer information compared to ones of 2D (Nwaogu et al., 2013).

Surface characterization for the correlation modeling between the process condition and the resulted

Address for reprints: H. S. Ho, School of Mechanical Engineering, Zhengzhou University, 100 Science Road, 450001 Zhengzhou, China
E-mail: hsinshen.ho@zzu.edu.cn

Received 24 April 2015; Accepted with revision 21 July 2015

DOI: 10.1002/sca.21254

Published online 6 August 2015 in Wiley Online Library (wileyonlinelibrary.com).

<footnote n="1">¹School of Mechanical Engineering, Zhengzhou University, 100 Science Road, Zhengzhou, China</footnote>

<footnote n="2">²Pole EME, Centre Technique des Industries Mécaniques, 52 avenue Félix Louát, Senlis, France</footnote>

<footnote n="3">³Equipe Matériaux, Surfaces et Mise en forme, TEMPO, LAMIH, Université de Valenciennes, Campus Mont Houy, Valenciennes Cedex 9, France</footnote>
surface texture is the main concern of the present
work, with the ultimate aim to obtain the desirable
surface texture by controlling the process condition.
The sandblasting process parameters such as duration,
angle and size of abrasives are frequently used to alter
the texture of sandblasted surfaces. Qualitative assess-
ments have been performed to describe the effect of
those process conditions on surface texture (Bouzid
and Bouaouadja, 2000; Zhou et al., 2011; Wang et al.,
2014). For instance, the maximum height of the profile
(Rᵢ) is exhibited to increase linearly with impact angle
and blasting duration (Bouzid and Bouaouadja, 2000),
and R<sub>α</sub> is reported to increase with blasting duration
(Zhou et al., 2011) and size of abrasives (Wang et al.,
2014).

The contributions of the present study are as
follows: first, a multi-scale analysis is carried out in
order to reveal the inherent feature of the sandblasted
texture, considering two (low and high) Gaussian
filters together with a series of cut-off lengths; second,
the bootstrap method is implemented to quantify the
aleatory uncertainty of surface data, assuming them to
be isotropic; third, attempt is made to obtain a
quantitative correlation between the sandblasting
condition and resulted surface texture based on
regression analysis; fourth, considering all the 3D
roughness parameters of ISO25178-2 and EUR
15178N, a computational protocol is established to
identify a most relevant parameter set, including 3D
roughness parameters, scale, filter type and regression
model, for modeling the effect of the sandblasting
condition on the surface texture. The proposed
methodology is finally applied to the sandblasted
surface of stainless steel AISI 304L with the working
pressure as the process parameter.

The remainder of the paper is organized as
follows. Section Material and Measurement Method
presents the material and method used in this work.
Section Computational Protocol describes the com-
putational protocol dedicated to identifying the
optimal parameter set for textures analysis, includ-
ing 3D roughness parameter, evaluation scale, filter
type and regression model. Section Results and
Discussion is devoted to analyzing the surface
textures, the relevance of the obtained optimal
parameters set is discussed. Concluding remarks
are given in Section Conclusions.

**Material and Measurement Method**

**Material and Sandblasting Process**

In this study, the stainless steel AISI 304L is the
target material, which has the following chemical
composition (% in weight): C (0.022), Mn (1.58), Si
(0.35), S (0.024), P (0.039), Ni (8.08), Cr (18.60), Mo

(0.39), Cu (0.60), N (0.046) and Fe (base). Seven
substrates are prepared; each of them is a 5 mm thick
cylindrical plate with a diameter of 50 mm. These
substrates are automatically polished with pre-defined
experimental settings (load: 150 N, duration: 3 min), a
mirror surface is obtained using a diamond paste of
3 µm.

Substrates are blasted in a sandblasting machine CSF
70 V by silicon carbide (SiC) particles with diameters
ranging from 0.15 mm to 0.25 mm. Fixing the angle
(90°), the distance (5 cm) and the duration (60 s) of the
sandblasting process, the effect of the working pressure
is in particular investigated on the surface texture by
setting its value as: 1, 2, 3, 4, 5, 6, and 7 bar(s).

**Roughness Measurement and Estimation**

Roughness measurements are conducted with a
three-dimensional non-contact optical profilometer
(Zygo NewView™ 7300, Zygo Corp.). The white
light interferometer (20 × Mirau objective) is equipped
with a high speed camera (320 × 240 pixels). The
optical resolution is 710 nm in x, y directions, which is
related to the numerical aperture of the objective and
achieved through the Sparrow criterion. The sampling
interval is 1.09 µm. Setting the working distance as
4.70 mm, a region of 0.35 mm × 0.26 mm is measured.
A total of 16 individual measurements are stitched
together with an overlap of 20%, producing a stitched
region of 1.19 mm × 0.89 mm for reliable roughness
measurement. Twenty stitched regions are created per
substrate.

In order not to affect the micro-geometry of surfaces,
the geometrical form is removed by rectifying the
measured surfaces with a third degree polynomial fitting;
the reduction of the residual variance is insignificant
when increasing the polynomial degree from two to four
(Mezghani et al., 2009). To fully characterize the
measured surface texture, a total of 55 3D roughness
parameters are computed in agreement to ISO 25178-2
(2012) and EUR 15178N (1993). In addition, the edge
effect of measured surfaces is properly managed for the
reliability and accuracy of the roughness estimates. The
roughness parameters are listed in Table I, which are
discussed later for their relevance for the characterization
of the sandblasted surfaces.

**Computational Protocol**

The present study is conducted based on a proposed
computational protocol that relies on a multi-scale
technique integrated with computer-based bootstrap
method (see Fig. 1). It is elaborated in the following
sections.
<table><caption>TABLE I Surface roughness parameters used in the present study</caption><thead><tr><th></th><th></th><th></th></tr></thead><tbody><tr><td colspan="3">ISO 25178 (2012)</td></tr><tr><td colspan="3">Amplitude parameters</td></tr><tr><td>S<sub>a</sub></td><td>Arithmetic mean height</td><td>S<sub>a</sub> = <sup>1</sup>&frasl;<sub>A</sub> &iint;<sub>A</sub> |z(x, y)| dxdy<br>with A being the definition area</td></tr><tr><td>S<sub>q</sub></td><td>Root mean square height of the surface</td><td>S<sub>q</sub> = &radic;<span style="text-decoration: overline;"><sup>1</sup>&frasl;<sub>A</sub> z<sup>2</sup>(x, y)</span> dxdy<br>with A being the definition area</td></tr><tr><td colspan="3">Spatial parameters</td></tr><tr><td>S<sub>al</sub></td><td>Auto-correlation length</td><td>S<sub>al</sub> = min<sub>tx,ty&isin;R</sub> &radic;(tx<sup>2</sup> + ty<sup>2</sup>)<br>where R = {&(tx, ty) : ACF(tx, ty) &le; s, 0 &le; s &lt; 1}</td></tr><tr><td>S<sub>tr</sub></td><td>Texture-aspect ratio</td><td>S<sub>tr</sub> = <sup>min</sup>&frasl;<sub>tx,ty&isin;R</sub> &radic;(tx<sup>2</sup>+ty<sup>2</sup>)<br>where R = {&(tx, ty) : ACF(tx, ty) &le; s, 0 &le; s &lt; 1}<br>Q = {&(tx, ty) : ACF(tx, ty) &ge; s**}</td></tr><tr><td colspan="3">**denotes the property that the ACF&ge;s on the straight line connecting the point (tx,ty) to the origin</td></tr><tr><td colspan="3">Hybrid parameters</td></tr><tr><td>S<sub>dq</sub></td><td>Root mean square gradient</td><td>S<sub>dq</sub> = &radic;<span style="text-decoration: overline;"><sup>1</sup>&frasl;<sub>A</sub> &iint;<sub>A</sub> [((partial(x,y))/dx)<sup>2</sup> + ((partial(x,y))/dy)<sup>2</sup>] dxdy<br>with A being the definition area</td></tr><tr><td>S<sub>dr</sub></td><td>Developed interfacial area ratio</td><td>S<sub>dr</sub> = <sup>1</sup>&frasl;<sub>A</sub> &iint;<sub>A</sub> (&radic;[1 + ((partial(x,y))/dx)<sup>2</sup> + ((partial(x,y))/dy)<sup>2</sup>]) - 1 dxdy<br>with A being the definition area</td></tr><tr><td colspan="3">EUR 15178N (1993)</td></tr><tr><td colspan="3">Functional indices</td></tr><tr><td>S<sub>bi</sub></td><td>Surface bearing index</td><td>S<sub>bi</sub> = <sup>S<sub>q</sub></sup>&frasl;<sub>Trueheight(0.05)</sub></td></tr></tbody></table>

## Multi-Scale Technique

The multi-scale technique is performed with the use of two types (high-pass and low-pass) of Gaussian filters together with 23 cut-off lengths ranging from 5 µm to 1000 µm to separate surface

roughness from surface waviness at different wave-lengths. Figure 6 (A–B) illustrate the effects of coupling of filter type and cut-off length on measured surfaces shown in Figure 2, it is observed that filter type and cut-off length can affect significantly surface texture.

<figure><img src="image_2.png" /><figcaption>Fig 1. Framework of the methodology proposed for identifying the optimal parameter set. $S_i^*(f, \epsilon)$ denotes the roughness parameter $S_i$ estimated based on $k$th surface sample with the filter type (f) and cut-off length ($\epsilon$).</figcaption></figure>
<figure><img src="image_3.png" /><figcaption>Fig 2. Optical micrographs of surfaces blasted at 1 (A), 3 (B), 7 (C) bars.</figcaption></figure>

## Computer-Based Bootstrap Method

Due to the limited amount of available experimental data, the computer-based bootstrap method is adopted, which, relying on the resampling technique, consists in generating a high number of simulated bootstrap samples by randomly sampling with data replacement from the original experimental data set (Efron, '79).

Applying the bootstrap method to the 20 stitched surface regions, 10⁴ bootstrap samples are generated, based on which all the roughness parameters are computed with different settings of filter type and cut-off length. Subsequently, the correlation between each computed roughness parameter and the working pressure is assessed based on four types of regression models, which are linear, power, exponential and logarithm models. The coefficient of determination R², with a value∈[0;1], is invoked here to indicate the model quality as it is demonstrated to be highly capable of providing the goodness-of-fit summary statistic for a large variety of mathematical regression models (Cameron and Windmeijer, '97). It has been employed to find the best scale for characterizing relative area with adhesion (Brown and Siegmann, 2001) and for correlating curvature with fatigue limit (Vulliez et al., 2014).

## Optimal Parameter Set

For every scale, filter type and regression model, the probability density functions (PDFs) of each

roughness parameter and its related coefficient of determination are empirically constructed from the simulated bootstrap surface samples. Their descriptive statistics such as mean, 5th percentile and 95th percentile are also derived. Collect all the coefficients of determination computed with different parameter combinations and class them in terms of mean, 5th percentile and 95th percentile in a descending order, the pertinence of the parameter set is simply demonstrated by the classification order, which refers to the goodness-of-fit of the parameter set for the underlying law governing the sandblasting process. The acceptance of the most relevant parameter set is based on the highest mean value of the coefficient of determination.

## Results and Discussion

### Results

#### *Measured surface analysis*

Figure 2 and Figure 3 illustrate surfaces of stainless steel AISI 304L substrates after sand-blasting. It is shown that sandblasting results in the formation of ragged surfaces, i.e. the surface texture (c.f. Fig. 2) and roughness level (c.f. Fig. 3) of blasted substrates are greatly modified, implying that they are significantly dependent on impact pressure.

<figure><img src="image_4.png" /><figcaption>Fig 3. 3D representation of surfaces blasted at 1 (A), 3 (B), 7 (C) bars.</figcaption></figure>
### Optimal parameter set

Based on the multi-scale technique integrated with computer-based bootstrap method, the relevance of more than 10⁴ possible combinations, consisting of 55 roughness parameters, two filter types, 23 cut-off lengths and four regression models, is assessed to seek an optimal parameter set for the description of the functional correlation between the working pressure and surface roughness.

Figure 4(A–D) illustrates all the possible combinations of parameters that are sorted in a descending order according to their relevance indicated by the coefficient of determination ($R^2$). The mean of $R^2$ is up to 0.99, suggesting the presence of an optimal parameter set: the power law function describes best the interrelationship between the working pressure and the roughness parameter called root mean square surface slope ($S_{dq}$) computed at the cut-off length of 120 µm using a low-pass filter (refer to Fig. 5). This finding is found to be in accordance with Berglund et al. (2010a) where $S_{dq}$ varies with scale.

The evolution of $S_{dq}$ is given as a function of the pressure in Figure 5; $S_{dq}$ exhibits a tendency of monotonically increasing with the pressure. Indeed, when the pressure increases, the particle impact on the substrate’s surface becomes stronger, leading to the formation of wider craters and deeper indents, and consequently the local slope of each facet and its orientation become larger. The fact that the fitted curve goes through the confidence intervals implies that the found optimal parameter set is suitable for the description of the correlation between the surface roughness and working pressure.

### Result illustration

*Filter type.* The surface textures obtained with the use of a high-pass filter and low-pass filter together with

<figure><img src="image_5.png" /><figcaption>Fig 4. Classification order of the relevance of parameter sets for modeling the pressure effect on the surface texture. Modeling of $S_{dq}$ with the power law function (A) and logarithmic function (B) at a cut-off length of 120 µm and modeling of $S_{tr}$ (C) and $S_{bi}$ (D) with the linear function at a full horizontal scale, using a low-pass filter.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig 5. Modeling of $S_{dq}$ and working pressure with power law function for a cut-off length of 120 µm using a low-pass filter.</figcaption></figure>

three cut-off lengths (µm: 22, 120, 1000 - full horizontal scale) for three working pressures (bar(s): 1, 3, 7) are represented in Figure 6(A and B), respectively. The impact of abrasive particles on the substrate surfaces is observed to be isotropic regardless of filter types, implying that sandblasted surfaces have the same characteristics in all directions. Figure 6(A), representing surfaces obtained with a high-pass filter and 120 µm, shows that the size of impact craters increases with increasing working pressure. Figure 6(B), illustrating surfaces obtained with a low-pass filter, reveals the presence of macro-waviness at full horizontal scale and reflects the geometry of the abrasive particles used for sandblasting at 120 µm; the use of a low-pass filter neglects the phenomenon of localized deformation of a surface.

*Scale.* In Figure 7, the value of $R^2$ for the modeling of $S_{dq}$ with the power law model using a low-pass filter is computed as a function of the scale varying from 10 µm to 1000 µm; $R^2$ is exhibited to be a convex function of the scale. 120 µm is exhibited to be the optimal scale. And other scales, such as 10 µm and 1000 µm, describe the characteristics of sandblasted surfaces in a suboptimal sense.

*Roughness parameter.* Figure 8 shows the PDFs of $S_{dq}$ for the examined seven working pressures, computed at the relevant scale using pertinent filter type. The PDFs are perfectly disconnected for the seven pressures, implying that $S_{dq}$ is capable of describing the pressure effect on the surface texture.

*Regression model.* Figure 9(A-D) illustrate the modeling of $S_{dq}$ and the working pressure based on linear, logarithmic, exponential and power law functions. As indicated by $R^2$, the power law model in Figure 9(D) exhibits the best performance (with $R^2=0.995$) in comparison to the linear model (with $R^2=0.970$ in Fig. 9(A)), logarithmic model (with $R^2=0.935$ in Fig. 9(B)) and exponential model (with $R^2=0.925$ in Fig. 9(C)).
<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /><figcaption>Fig 6. Textures of surfaces blasted at the working pressures (bar(s): 1, 3, 7) with the use of a high-pass filter (A) and a low-pass filter (B) together with cut-off lengths equal to 22 µm, 120 µm and 1000 µm.</figcaption></figure>

To further illustrate the relevance of obtained results, another two roughness parameters calculated over the full horizontal scale ($\epsilon$ = 1000 µm) with a low-pass filter and fitted with the linear model function are assessed here – the surface anisotropy ($S_{tr}$) and the bearing index ($S_{bi}$). The former represents the uniformity of the texture aspect and the latter characterizes the quality of the bearing property. In Figure 9(E), the value of $S_{tr}$ shows significant variability in

response to the change of the working pressure, which varies between 0.82 and 0.90. As the value of $S_{tr}$ approaches 1, the properties of surfaces are thus isotropic. This finding is coherent in somehow with the observation that sandblasted surfaces do have the same characteristics in all directions (c.f. Fig. 6(A–B)). As for $S_{bi}$, it is observed to evolve with the pressure in a disordered manner (as seen in Fig. 9(F)), hence it is insensitive to the variation of the pressure. Under
<figure><img src="image_9.png" /><figcaption>Fig 7. Variation of R² as a function of the scale for S<sub>dq</sub> using a low-pass filter with power law model.</figcaption></figure>

the hypothesis of a linear model and with the use of a low-
pass filter at full horizontal scale, the parameters S<sub>tr</sub> and S<sub>bi</sub>
are not suitable for discriminating the pressure effect, where
R<sup>2</sup> equals to 0.21 and 0.02 for S<sub>tr</sub> and S<sub>bi</sub>, respectively.

Figure 10 represents the (non-normalized) PDF of R²
for the modeling of S<sub>dq</sub> and the pressure based on four
regression models. The PDFs are exhibited to be fairly in
shape and well-fitted by Gaussian bell-curves, i.e., the
estimated PDFs of R² are quite broad, suggesting that the
estimate of R² possesses a significant variability. This
shows the necessity of using the full PDF of R² to
discriminate the optimality of the parameter set. A
single value of R² can give a false decision on the

<figure><img src="image_10.png" /><figcaption>Fig 8. PDFs of S<sub>dq</sub> modeled with power law function for a cut-off length of 120 using a low-pass filter, under examined working pressures.</figcaption></figure>

pertinence of the investigated model when it is chosen
by chance in the overlapping part of the adjacent PDFs.
The power law function is found to have absolute
superiority compared to the others.

Discussion

Relevance of the selected roughness parameter

The parameter S<sub>dq</sub> is exhibited in Figure 5 to be a
monotonically increasing function of the pressure. The
modeling of arithmetic mean height of surfaces (S<sub>a</sub>) and
the shortest autocorrelation length (S<sub>al</sub>) with power law
function at 120 µm using a low-pass filter are also found

<figure><img src="image_11.png" /><figcaption>Fig 9. Modeling of S<sub>dq</sub> with linear (A), logarithmic (B), exponential (C) and power law (D) functions, using a low-pass filter at 120 µm; modeling of S<sub>tr</sub> (E) and S<sub>bi</sub> (F) with linear function using a low-pass filter at full horizontal scale.</figcaption></figure>
<figure><img src="image_12.png" /><figcaption>Fig 10. PDFs of R² for S<sub>dq</sub> calculated using a low-pass filter and a cut-off length equal to 120 µm.</figcaption></figure>

to possess similar behavior, as illustrated in Figure 11. It is worth noting that S<sub>a</sub> is a widely used roughness parameter due to the simplicity of calculation, the statistical robustness and the ease of interpretation while S<sub>al</sub>, which is one of the parameters that gives rise to the notion of distance between lumps and bumps, is well adapted to represent the notion of gaps between peaks and valleys in the case of an isotropic surface.

The PDFs of S<sub>dq</sub> is shown in Figure 8. The PDFs of S<sub>a</sub> and S<sub>al</sub> for the seven examined working pressures are illustrated respectively in Figure 12(A–B); these distributions are well fitted by Gaussian curves. Figure 12(A–B) show that S<sub>a</sub> and S<sub>al</sub> are inappropriate to discriminate pressure effects due to three reasons: I) the values of S<sub>a</sub> and S<sub>al</sub> are particularly large compared to the ones of S<sub>dq</sub>, II) the dispersion effect is especially pronounced for S<sub>a</sub> but notably S<sub>al</sub> and III) the

### Relevance of the selected scale

The scale of 120 µm, which is demonstrated to be relevant for the parameter S<sub>dq</sub> in Figure 7, underlines the robustness and accuracy of the methodology proposed in the present study. The scale of 120 µm is situated in the range of the pertinent spectrum with R² ≥ 0.90, covering a band from 15 µm to 1000 µm; the difference between the maximum and the minimum values of R² that is equal to 0.085 highlights the accuracy of the proposed methodology with respect to the scale. In this broad-band scale, S<sub>dq</sub> is dominantly pertinent with R² ≥ 0.90, verifying the robustness of the proposed methodology with respect to the roughness parameter.

### Relevance of the selected filter type

The repetitive hitting of abrasive particles on the substrate's surface forms a succession of peaks and valleys with sharp edges; the consideration of an appropriate scale for the measurement of the slopes of these peaks and valleys has been a major problem. Multiple methods are proposed in the literature, for instance, the slope calculation method defined in the standard ISO 25178-2 (2012) and the Lagrangian interpolation polynomial method with seven points in orthogonal directions (Whitehouse and Phillips '82). Although the former method is frequently used, it could not be implemented in the present study as it involves an integral within the definition area of a scale-limited surface. According to the latter method, the computation of the parameter S<sub>dq</sub> is as follows:

$$S_{dq} = \sqrt{\frac{1}{(M-6)(N-6)} \sum_{j=4}^{N-3} \sum_{i=4}^{M-3} \rho_{ij}^2} \quad (1)$$

with

$$\rho_{ij} = \left\{ \left\{ \frac{1}{60\Delta x} \left[ -\eta(x_{i-3}, y_j) + 9\eta(x_{i-2}, y_j) - 45\eta(x_{i-1}, y_j) + 45\eta(x_{i+1}, y_j) - 9\eta(x_{i+2}, y_j) + \eta(x_{i+3}, y_j) \right] \right\}^2 + \left\{ \frac{1}{60\Delta y} \left[ -\eta(x_i, y_{j-3}) + 9\eta(x_i, y_{j-2}) - 45\eta(x_i, y_{j-1}) + 45\eta(x_i, y_{j+1}) - 9\eta(x_i, y_{j+2}) + \eta(x_i, y_{j+3}) \right] \right\}^2 \right\}^{1/2}$$

distributions of S<sub>a</sub> and S<sub>al</sub> overlap each other severely specifically at higher pressures. As for the parameter S<sub>dq</sub>, the probability distribution of each examined pressure is distinct, indicating that S<sub>dq</sub> is able to discriminate the pressure effect. In fact, impact craters are formed when SiC particles hit the substrate's surface. The repetitive hitting may cause the formation of a plastically deformed layer and the creation of new craters on the previously formed craters, resulting then in the making of a succession of small peaks and valleys with sharp edges, where their slopes can be characterized by S<sub>dq</sub>. Indeed, S<sub>dq</sub> is a 3D hybrid parameter that is described by both the texture amplitude and spacing.

where η(x<sub>i+k</sub>, y<sub>i+k</sub>) is the height coordinate located in (x<sub>i+k</sub>, y<sub>i+k</sub>), M is the number of points in the profile, N is the number of profiles, and Δx and Δy are the sampling intervals.

Δx or Δy, which is the size of the pixel deducted directly from the camera resolution and the field of view, is indirectly pre-defined by the user. S<sub>dq</sub> is computed by averaging height coordinates at the seven nearest points with a maximum distance of 7Δx as an evaluation scale, hence only those peaks and valleys with the width in the order of Δx are concerned. At this scale, the high frequency noises generated by e.g. camera measurement noises are quite important, leading to a rather low signal-to-noise ratio. The estimated S<sub>dq</sub> at this scale is less
<figure><img src="image_13.png" /><figcaption>Fig 11. Modeling of S<sub>a</sub> and S<sub>al</sub> with power law function, for a cut-off length of 120 µm using a low-pass filter.</figcaption></figure>

robust as a result. It is therefore important to quantify the variability of the slopes over a wide range of scales using both high-pass and low-pass filters, as represented in Figure 13(A-B), respectively.

In Figure 13(A), the average slope seems to remain constant until they get close to sampling interval. Note that the high pass filter only allows passing the topographic details at finest scales (i.e. the highest spatial frequencies), and therefore, the slope is calculated at these scales in the measurement. Moreover, the calculated results of the slope are likely to contain significant uncertainty, indicating a low accuracy in the estimation of S<sub>dq</sub> in the presence of strong measurement noise. In Figure 13(B), the average slope is exhibited to be a monotonically increasing function of the scale. This scaling law is shown to be pressure-dependent, i.e. for a given evaluation scale, larger the craters and deeper the indent become when higher the pressure is. Yet, slope differences between the smallest and largest craters tend to become insignificant when the pressure increases. Indeed, the quasi-homogeneous distribution of crater size is obtained at higher pressure condition due to the formation of a hard- and plastically-deformed layer as a consequence of the repetitive hitting of abrasive particles, on which abrasive particles can hardly crush. It is important to note that these results would be the same for any surface that contains some fine scale irregularities.

Relevance of the selected regression model

In what follows, the efficiency of the power law model in describing the correlation between the parameter S<sub>dq</sub> and the pressure is justified with experimental and theoretical arguments based on the principles of contact mechanics. SiC particles are considered to be elastic spheres of radius D, indenting the elastic subtract to depth h, and thus creating a contact area (≈ craters) with a radius a, it holds that

$$a^2 = Dh \qquad (2)$$

Assume S<sub>a</sub> to vary in the same way as h, then

$$S_a \propto h \qquad (3)$$

The nozzle (working) pressure p can be proportional to the force F applied to the surface, it may then hold that

$$F \propto p \qquad (4)$$

According to Johnson ('70),

$$F \propto h^{3/2} \qquad (5)$$

By combining Eqs. (3)-(5), it is yielded that

$$S_a \propto p^{2/3} \qquad (6)$$

Similarly, it is admitted that S<sub>al</sub> is proportional to the contact semi-width a,

<figure><img src="image_14.png" /><figcaption>Fig 12. PDFs of S<sub>a</sub> (A) and S<sub>al</sub> (B) modeled with power law function for a cut-off length of 120 using a low-pass filter, under examined working pressures.</figcaption></figure>
<figure><img src="image_15.png" /><figcaption>Fig 13. Evolution of $S_{dq}$ versus scale using a high-pass filter (A) and a low-pass filter (B) in the log-log representation.</figcaption></figure>

$$S_{al} \propto a \qquad (7)$$

When Eq. (7) is related to Eqs. (2), (4) and (5), the relation between $S_{al}$ and the pressure is

$$S_{al} \propto p^{1/3} \qquad (8)$$

It is experimentally demonstrated in Figure 11(A–B) that power law function can be employed to describe the parameters $S_a$ and $S_{al}$ as the function of the pressure:

$$S_a = 0.224p^{0.828} \qquad (9)$$

$$S_{al} = 52.81p^{0.211} \qquad (10)$$

Assume $S_{dq}$ to be proportional to $S_a / S_{al}$ in an average sense, so

$$S_{dq} \approx 2S_a/S_{al} \qquad (11)$$

leading to

$$S_{dq} \propto p^{1/3} \qquad (12)$$

When experimental results in Eqs. (9) and (10) are applied to Eq. (11), $S_{dq}$ is written as

$$\begin{aligned} S_{dq} &= 2 \times 0.224p^{0.828}/52.81p^{0.211} \\ &= 0.0085p^{0.611} \approx 0.010p^{0.60} \end{aligned} \qquad (13)$$

Experimentally (c.f. Fig. 9(D)),

$$S_{dq} = 0.010p^{0.69} \qquad (14)$$

The experimental result in Eq. (14) is exhibited to be remarkably close to the theoretical one in Eq. (13), justifying the approximation made in Eq. (11). Yet, the theoretical function in Eq. (12) seems to be different from the ones in Eq. (13) and Eq. (14), which is mainly due to the following assumption and approximation: (i) the impact of the particles is assumed to be purely elastic, (ii) the superficial friction and particle sliding are neglected. In fact, the relationship defined in Eq. (12) can be refined by formulating the particle impact as a work-energy problem, which however remains challenging to be addressed. It can be concluded that the use of the power law model, suggested by the proposed methodology, to describe the

functional relationship between the parameter $S_{dq}$ and the pressure is justified theoretically and experimentally based on the principles of contact mechanics.

## Conclusions

The proposed methodology that relies on a multi-scale technique integrated with computer-based boot-strap method has been shown to enable to identify a relevant parameter set for the description of the pressure effect on the sandblasted surface texture. The effect of pressure variation on substrate’s texture is found to be best depicted by the roughness parameter called root mean square surface slope ($S_{dq}$) computed at the scale of 120 µm with a low-pass filter. The relationship between the working pressure and this roughness parameter is exhibited to be best modeled by the power law function with $R^2$ equal to 0.995; theoretical and experimental analyses based on the principles of contact mechanics are provided for justification.

## Acknowledgements

The authors are grateful to the reviewers for their constructive comments, which are valuable for improving the quality of the manuscript.

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