# Conditions of the Presence of Bimodal Amplitude Distribution of Two-Process Surfaces

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Considerations

## 3. The Analysis of Generated Surfaces

## 4. The Analysis of Measured Surfaces

^{2}contained 1024 × 1024 data points. Forms were removed by polynomials of the second degree. A digital filtration was not used.

## 5. Conclusions

- Limiting conditions of bimodal height distribution of two-process surface topography were developed. They depend on the ratio of the standard deviations of the valley and plateau parts Svq/Spq and on the material ratio at the transition between plateau and valley portions Smq. Based on these conditions, bimodal and unimodal height probability distributions were correctly discriminated for modeled and measured surfaces.
- The bimodal ratio increased when the Svq/Spq ratio increased. Typically, the upper peak is the major mode. However, for low values of the Smq parameter and for low Svq/Spq ratio, the lower peak, which corresponds to the material ratio of 50%, can be the major mode.
- When the Smq parameter is not lower than 50%, unimodal amplitude distribution exists. The mode and the smallest slope of the material ratio curve appear at the material ratio of 50%.
- For unimodal height distribution and the value of the Spq parameter smaller than 50%, the mode corresponds to the Smq material ratio.
- The results are functionally important because of the high tribological significance of the material ratio curve. In particular, the position of its smallest slope deserves attention.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**Probability plots of material ratio curves for different two-process surfaces; the Smq parameter is smaller than 50% (

**a**) and the Smq parameter is higher than 50% (

**b**) with plateau depth Pd and the vertical distance between two modes DIS.

**Figure 3.**Example of generation of two-process surface topography: the plateau surface: Sq = 0.3 µm, CL = 20 µm (

**a**), the valley surface: Sq = 1.7 µm, CL = 150 µm (

**b**), two-process surface: Spq = 0.3 µm, Svq = 1.7 µm, Smq = 72.7% (

**c**), Sq is the rms. height, CL is the correlation length.

**Figure 4.**Contour plots (

**a**,

**d**) surface probability plot (

**b**,

**e**), material ratio curve and probability distribution (

**c**,

**f**) of two-process isotropic modeled surfaces of the Smq parameter of 10% (

**a**–

**c**) and 20% (

**d**–

**f**), the other parameters of both surfaces are Spq = 0.12 µm, Svq = 0.5 µm, both surfaces have bimodal ordinate distribution.

**Figure 5.**Contour plots (

**a**,

**d**) surface probability plot (

**b**,

**e**), material ratio curve and probability distribution (

**c**,

**f**) of two-process isotropic modeled surfaces of the Smq parameter of 10% (

**a**–

**c**) and 20% (

**d**–

**f**), the other parameters of both surfaces are Spq = 0.12 µm, Svq = 0.7 µm, both surfaces have bimodal ordinate distributions.

**Figure 6.**Contour plots (

**a**,

**d**,

**g**) surface probability plot (

**b**,

**e**,

**h**), material ratio curve and probability distribution (

**c**,

**f**,

**i**) of two-process isotropic modeled surfaces of the Smq parameter of 10% (

**a**–

**c**), 20% (

**d**–

**f**) and 30% (

**g**–

**i**), the other parameters of both surfaces are Spq = 0.12 µm, Svq = 1.0 µm, all surfaces have bimodal ordinate distributions.

**Figure 7.**Conditions of bimodal height distributions of two-process surfaces; bimodal distribution appears for the Smq parameter within the shaded area.

**Figure 8.**Contour plots (

**a**,

**d**) surface probability plot (

**b**,

**e**), material ratio curve and probability distribution (

**c**,

**f**) of two-process isotropic modeled surfaces of the Smq parameter of 40% (

**a**–

**c**) and 80% (

**d**–

**f**), the other parameters of both surfaces are Spq = 0.12 µm, Svq = 0.7 µm, both surfaces have unimodal ordinate distributions.

**Figure 9.**Contour plots (

**a**,

**d**,

**g**) surface probability plot (

**b**,

**e**,

**h**), material ratio curve and probability distribution (

**c**,

**f**,

**i**) of two-process measured surfaces characterized by the following parameters: Spq = 0.31 µm, Svq = 1.61 µm, Smq = 24% (

**a**–

**c**), Spq = 0.49 µm, Svq = 2.89 µm, Smq = 15% (

**d**–

**f**) and Spq = 0.43 µm, Svq = 1.69 µm, Smq = 20% (

**g**–

**i**), three surfaces have bimodal ordinate distributions.

**Figure 10.**Contour plots (

**a**,

**d**) surface probability plot (

**b**,

**e**), material ratio curve and probability distribution (

**c**,

**f**) of two-process measured surfaces characterized by the following parameters: Spq = 0.61 µm, Svq = 2.04 µm, Smq = 38% (

**a**–

**c**) and Spq = 0.43 µm, Svq = 2.51 µm, Smq = 84%, both surfaces have unimodal ordinate distributions.

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**MDPI and ACS Style**

Pawlus, P.; Reizer, R.; Wieczorowski, M.
Conditions of the Presence of Bimodal Amplitude Distribution of Two-Process Surfaces. *Materials* **2020**, *13*, 4037.
https://doi.org/10.3390/ma13184037

**AMA Style**

Pawlus P, Reizer R, Wieczorowski M.
Conditions of the Presence of Bimodal Amplitude Distribution of Two-Process Surfaces. *Materials*. 2020; 13(18):4037.
https://doi.org/10.3390/ma13184037

**Chicago/Turabian Style**

Pawlus, Pawel, Rafal Reizer, and Michal Wieczorowski.
2020. "Conditions of the Presence of Bimodal Amplitude Distribution of Two-Process Surfaces" *Materials* 13, no. 18: 4037.
https://doi.org/10.3390/ma13184037