# Multiscale Characterizations of Surface Anisotropies

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## Abstract

**:**

## 1. Introduction

^{2}of 0.96 [21], clearly discriminated progressions of edge rounding by mass finishing [22], and found variations in stitching of profiles in measurements of aspheric lenses [23].

## 2. Materials and Methods

#### 2.1. Surfaces and Measurements

#### 2.2. Multiscale Curvature Tensor Calculations

**k**and

_{1}**k**are indicative of anisotropy. A principal direction vector can be decomposed into direction cosines, and then to three directional angles: α, β, and γ, between

_{2}**k**and the global coordinate system in which the measured topographies are described:

_{1}**e**

_{x},

**e**

_{y}, and

**e**

_{z}(Figure 1). Distributions of α, β, and γ, over all triangular patches, can be plotted for each scale in spherical coordinates. For anisotropic surfaces, an evident peak or peaks in the distribution is expected to appear, indicating the dominant directions. Whereas, for more isotropic surfaces, more uniform distributions are expected. These results can also be visualized in a horizontal coordinate system, also known as topocentric, bfigurey expressing the orientation of

**k**in angular coordinates: altitude, or elevation, and azimuth [36]. The reason why a spherical coordinate system is not used is because the polar angle, in that case, is measured from a fixed zenith direction (z-axis), whereas in the horizontal coordinate system, elevation is measured from a reference plane, associated with a horizon, which is more intuitive. When principal directions of maximal curvature are coplanar with datum, the elevation is zero, whereas the polar angle would be 90 degrees.

_{1}#### 2.3. Bandpass Filtering for Multiscale Analyses

^{®}7.4 software (DigitalSurf, Besançon, France). Non-measured heights in the measurements were filled-in with using smart shape interpolation.

#### Step 1: Defining spatial frequency bands (scales) for bandpass filters and filtering.

#### Step 2: Calculating conventional topographic characterization parameters.

#### Step 3: Creating polar plots from conventional topographic characterization parameters.

## 3. Results

#### 3.1. Visual Impressions of Anisotropy

#### 3.2. Multiscale Characterizations of Surface Anisotropies by Bandpass Filtering

#### 3.3. Multiscale Characterizations of Surface Anisotropie by the Direction of Maximum Curvature

**k**, directions of maximum curvatures, with arrows on height maps surfaces at two scales, five and twenty times their original sampling intervals. Other scales are included in Supplementary Materials. Arrows are plotted for a limited number of regions to improve perceptions. Regions indicating ridges and grooves, that are clearly anisotropic, are visible for MilledC in all scales. This is evident in the similar orientation of

_{1}**k**for every location. In contrast, for µEDMed, orientations of principal direction of maximum principal curvature vary with region as expected for isotropic surface. For MilledF finer scale orientations of

_{1}**k**are mostly aligned in x, corresponding to anisotropic features created by cutting tool rotation and interactions between the tool edge and workpiece material. For larger scales, arrows are indicative of the bi-directional pattern created by the feed. Visualization of

_{1}**k**for L-PBFed topography indicate different types of directional features. At the finest scale, wrinkle-like features created by solidifying melt pools can be seen, as well as vertical grooves on the pillar like structures, and valleys between them.

_{1}**k**, is expressed in the HCS and associated certain bin. More

_{1}**k**vectors in a direction indicates more anisotropy. Some are shown in Figure 7, other scales are in the Supplementary Materials. An azimuth angle of 90 or 270 degrees corresponds to the alignment with x-axis, and 0 or 180 degrees to y-axis.

_{1}**k**is aligned with 90 and 270 degree azimuths for all scales, while elevation is mostly below 5 degrees. Some fine scale features appear to be inclined at slightly steeper angles, although this is only visible for scales less than 7.846 µm (10× original sampling interval). These features are located around the cylindrical form, which was intentionally not removed to test this effect. Regardless of those fine scale features, the nature of MilledC anisotropy is generally two-dimensional.

_{1}**k**for µEDMed examples is evidently more dispersed than MilledC. The dominant azimuths of 0 and 180 degrees, as well as 90 and 270 degrees, relate to the scanning direction of the electrodes during electric discharge machining. Elevation is always less than 5 degrees and is related to general flatness of the topography with some slightly inclined slopes of the discharge craters. This confirms the isotropic nature of µEDMed topographies.

_{1}**k**in HCS for MilledF show how its anisotropy depends on scale. Between 2 and 18 µm, principal directions of maximum curvatures are aligned with 90 and 270 degree azimuths. Between 20 and 50 µm a second dominant direction is present for azimuth between 0 and 180 degrees. At 60 µm, the first dominant direction is not visible, whereas at 80 µm, the second is not evident. For larger scales, both dominant directions are present. The elevation for all scales analyzed is always close to 0 degrees. This is due to the fact that the form (or the general shape) was removed prior to the analysis.

_{1}**k**are 0 and 180 degrees, as well as 90 and 270 degrees, although dispersion is visibly greater than for MilledC and MilledF. The elevation is found to be up to 90 degrees for the finest scales, when distribution is presented in a logarithmic scale for magnitude (Figure 8), and it is related to the curvature of pillar like features. Clearly the anisotropy of this surface topography is 3D and scale-dependent.

_{1}#### 3.4. Conventional Approach Based on Fourier Transform in Polar Coordinates

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Relations between curvature principal directions

**k**and

_{1}**k**and normal vector and visualization of geometrical direction angles for principal direction

_{2}**k**in spherical coordinates.

_{1}**Figure 2.**Renderings of measured topographies of (

**a**) MilledC, (

**b**) µEDMed, (

**c**) L-PBFed, and (

**d**) MilledF. Please note that black arrows indicate visual impression of apparent anisotropy.

**Figure 3.**Polar plots showing dominant directions of anisotropy with texture direction (Std) in degrees, magnitude with the complement of the texture aspect ratios (1 − Str), and scales vertically with scale number corresponding to the band numbers from Table 2, with 1 the smallest scale as the highest, calculated for: (

**a**) MilledC, (

**b**) µEDMed, (

**c**) L-PBFed, and (

**d**) MilledF.

**Figure 4.**Directions of maximum principal curvatures calculated for MilledC, µEDMed, and MilledF at two scales: 5× and 30× the original sampling interval, plotted together with color-coded height maps. Please note that red arrows indicate a direction of maximum curvature at a given location.

**Figure 5.**Directions of maximum principal curvature calculated for L-PBFed at the original sampling interval. Please note that red arrows indicate a direction of maximum curvature at a given location.

**Figure 6.**2D histograms created from direction cosines of maximum curvature calculated for (

**a**) MilledC, (

**b**) µEDMed, (

**c**) L-PBFed, and (

**d**) MilledF at the indicated scales.

**Figure 7.**Linear histograms of maximum principal curvature directions in horizontal coordinate system (HCS), i.e., topocentric coordinates at three different scales.

**Figure 8.**Logarithmic histograms of maximum principal curvature directions in horizontal coordinate system (HCS), i.e., topocentric coordinates at two different scales for the L-PBFed surface.

**Figure 9.**Rosette plots created with a conventional, non-multiscale method using Fourier spectra in polar coordinates, for (

**a**) MilledC, (

**b**) µEDMed, (

**c**) SLMed, and (

**d**) MilledF.

Surface | MilledC | MilledF | µEDMed | L-PBFed |
---|---|---|---|---|

Original sampling interval [µm] | 0.790 | 2.000 | 0.125 | 0.260 |

5× original sampling interval [µm] | 3.950 | 10.000 | 0.625 | 1.300 |

20× original sampling interval [µm] | 15.800 | 40.000 | 2.500 | 5.200 |

25× original sampling interval [µm] | 19.750 | 50.000 | 3.125 | 6.500 |

40× original sampling interval [µm] | 31.600 | 80.000 | 5.000 | 10.400 |

**Table 2.**Wavelengths of the nesting indices, center, low-pass, and high-pass for multiscale (also known as sliding) bandpass filtering, all in [µm].

MilledC | µEDMed | L-PBFed | MilledF | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

No. | Center | Low | High | Center | Low | High | Center | Low | High | Center | Low | High |

1 | 3.0 | - | 4.0 | 0.422 | - | 0.563 | 1.1 | - | 1.5 | 6 | - | 8 |

2 | 4.5 | 3.0 | 6.0 | 0.563 | 0.375 | 0.750 | 1.5 | 1.0 | 2.0 | 9 | 6 | 12 |

3 | 6.0 | 4.0 | 8.0 | 0.844 | 0.563 | 1.125 | 2.3 | 1.5 | 3.0 | 12 | 8 | 16 |

4 | 9.0 | 6.0 | 12.0 | 1.125 | 0.750 | 1.500 | 3.0 | 2.0 | 4.0 | 18 | 12 | 24 |

5 | 12.0 | 8.0 | 16.0 | 1.688 | 1.125 | 2.250 | 4.5 | 3.0 | 6.0 | 24 | 16 | 32 |

6 | 18.0 | 12.0 | 24.0 | 2.250 | 1.500 | 3.000 | 6.0 | 4.0 | 8.0 | 36 | 24 | 48 |

7 | 24.0 | 16.0 | 32.0 | 3.375 | 2.250 | 4.500 | 9.0 | 6.0 | 12.0 | 48 | 32 | 64 |

8 | 36.0 | 24.0 | 48.0 | 4.500 | 3.000 | 6.000 | 12.0 | 8.0 | 16.0 | 72 | 48 | 96 |

9 | 48.0 | 32.0 | 64.0 | 6.750 | 4.500 | 9.000 | 18.0 | 12.0 | 24.0 | 96 | 64 | 128 |

10 | 72.0 | 48.0 | 96.0 | 9.000 | 6.000 | 12.000 | 24.0 | 16.0 | 32.0 | 144 | 96 | 192 |

11 | 96.0 | 64.0 | 128.0 | 13.500 | 9.000 | 18.000 | 36.0 | 24.0 | 48.0 | 192 | 128 | 256 |

12 | 144.0 | 96.0 | 192.0 | 18.000 | 12.000 | 24.000 | 48.0 | 32.0 | 64.0 | 270 | 192 | 348 |

13 | 192.0 | 128.0 | 256.0 | 27.000 | 18.000 | 36.000 | 72.0 | 48.0 | 96.0 | 384 | 256 | 512 |

14 | 270.0 | 192.0 | 348.0 | 36.000 | 24.000 | 48.000 | 96.0 | 64.0 | 128.0 | 522 | 348 | 696 |

15 | 384.0 | 256.0 | - | 48.000 | 32.000 | - | 144.0 | 96.0 | 192.0 | 768 | 512 | - |

16 | N/A | N/A | N/A | N/A | N/A | N/A | 192.0 | 128.0 | 256.0 | N/A | N/A | N/A |

17 | N/A | N/A | N/A | N/A | N/A | N/A | 288.0 | 192.0 | 384.0 | N/A | N/A | N/A |

18 | N/A | N/A | N/A | N/A | N/A | N/A | 384.0 | 256.0 | 512.0 | N/A | N/A | N/A |

19 | N/A | N/A | N/A | N/A | N/A | N/A | 576.0 | 384.0 | - | N/A | N/A | N/A |

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**MDPI and ACS Style**

Bartkowiak, T.; Berglund, J.; Brown, C.A.
Multiscale Characterizations of Surface Anisotropies. *Materials* **2020**, *13*, 3028.
https://doi.org/10.3390/ma13133028

**AMA Style**

Bartkowiak T, Berglund J, Brown CA.
Multiscale Characterizations of Surface Anisotropies. *Materials*. 2020; 13(13):3028.
https://doi.org/10.3390/ma13133028

**Chicago/Turabian Style**

Bartkowiak, Tomasz, Johan Berglund, and Christopher A. Brown.
2020. "Multiscale Characterizations of Surface Anisotropies" *Materials* 13, no. 13: 3028.
https://doi.org/10.3390/ma13133028