Asperity-peak heights
The asperity-peak heights increase with an increasing surface roughness for all the selected asperity-peak criteria, as shown in Figure 7. The differences between the selected asperity-peak criteria are almost negligible (within 10%) and are within the data scatter for all the surface roughnesses. Therefore, the asperity heights are not influenced by the changes in the asperity-peak identification criteria, though the highest asperity peaks are always identified with the 7PP criteria. The asperity-peak heights are around 0.018 µm for the smoothest surface and they increase to 0.47 µm for the roughest surface.
Figure 7: Asperity-peak heights in relation to the roughness parameter Ra for the 3PP, 5PP and 7PP criteria (Δx = 0.1875 µm).
3.2 Effect of the peak-threshold value Δz on the asperity-peak properties
Number of asperity peaks per profile
The variation in the number of asperity peaks with peak-threshold values is shown in Figure 8. The number of asperity peaks decreases with the increasing surface roughness, although it decreases with increasing peak-threshold values from 0% to 10% Rq. For the smoothest surface the number of asperity peaks is around 1400, and this varies only slightly for different peak-threshold values, all within the data scatter. However, as the surface roughness increases, the differences between the peak-threshold values become more pronounced. The difference in the number of asperity peaks between the smallest (0% Rq - 3PP) and the largest (10% Rq - M3PP-10) peak-threshold values is two-fold (500 asperity peaks) already for the second smoothest surface (Ra = 0.032 µm). The differences increase to as much as 650 asperity peaks for the roughest surface (Ra = 0.644 µm), where no asperity peaks are even identified for the 5% Rq and 10% Rq peak-threshold values. Accordingly, for the rough surfaces, the differences in the number of asperity peaks depending on the peak-threshold value used are enormous and become very unrealistic, while for very smooth, polished, surfaces, the influence of the peak-threshold value is negligible.
Figure 8: Number of asperity peaks per profile for different surface roughnesses according to the calculated peak-threshold values. The different columns represent the various peak-threshold values for the 3PP and M3PP criteria (Δx = 0.1875 µm).
Asperity-peak radii
In Figure 9 the asperity-peak radii for the different surface roughnesses according to the calculated peak-threshold values are presented. The asperity-peak radii decrease with increasing surface roughness as well as with increasing peak-threshold values. For the smoothest surface, the values of the asperity-peak radii were about 3.5 µm. Again, the differences between the different peak-threshold values are small, within the data scatter. But with increasing surface roughness, the differences in the asperity-peak radii between the different peak-threshold values become greater. This behaviour is the same as for the number of asperity peaks (see Figure 8). The difference in the asperity-peak radii between the 0% Rq and 10% Rq peak-threshold values for the second smoothest surface (Ra = 0.032 µm) is already 40% (a reduction of the asperity-peak radius from 2.6 µm to 1.5 µm), while for the second roughest surface (Ra = 0.190 µm) the difference is almost 80% (reduction of the asperity-peak radius from 2.4 µm to 0.5 µm). Again, some values of the asperity-peak radii for the roughest surface could not even be calculated, i.e., for the 5% Rq and 10% Rq peak-threshold values, since no asperity peaks were identified (Figure 8).
Figure 9: Radius of the asperity peaks for different surface roughnesses according to the calculated peak-threshold values. The different columns represent the various peak-threshold values for the 3PP and M3PP criteria.
Asperity-peak heights
Figure 10 shows the asperity-peak heights for the different surface roughnesses according to the variation in the peak-threshold values. The values of the asperity-peak height increase with the increasing surface roughness. The asperity-peak heights for the smooth surface are around 0.015 µm and increase to around 0.46 µm for the roughest surface. The effect of the peak-threshold value on the asperity-peak height is negligible, irrespective of the surface roughness, which is the opposite compared to the number and the radii of the asperity peaks. Namely, for every surface-roughness condition the asperity-peak heights for all the peak-threshold values are within the data scatter. Again, some asperity-peak heights could not be calculated for the 5% and 10% Rq at the roughest surface, since no asperity peaks were identified under those conditions (see Figure 8). It is clear that the data scatter is increasing with the increasing surface roughness. This is because with fewer asperity peaks being identified, the relative variation in the data becomes larger.
Figure 10: Height of the asperity peaks for different surface roughnesses according to the calculated peak-threshold values. The different columns represent the various peak-threshold values for the 3PP and M3PP criteria (Δx = 0.1875 µm).
3.3 Effect of the surface-profile resolution Δx on the asperity-peak properties
Number of asperity peaks per profile
Figure 11 shows the number of asperity peaks for different Δx distances between the surface-profile data points for the 3PP criterion. The number of asperity peaks for the Δx = 0.1875 µm, 0.375 µm and 0.75 µm Δx distances decreases with the increasing surface roughness and seem to level out for the rougher surfaces. In contrast, for the 1.125 µm and 1.875 µm Δx distances, the effect of levelling out is much less pronounced, as the values seem to be more constant over the whole surface-roughness range. The number of asperity peaks also decreases with the increasing Δx distance. The biggest absolute difference between the Δx distances is for the smoothest surface, where the change in the Δx distance from 0.1875 µm to 1.875 µm results in almost 1200 fewer asperity peaks being identified. The difference for the roughest surface is smaller, around 500 asperity peaks.
Figure 11: Number of asperity peaks in relation to the roughness parameter Ra for different Δx distances for the 3PP criterion.
Asperity-peak radii
The asperity-peak radii decrease with the increasing surface roughness for all the selected Δx distances, as shown in Figure 12. However, the asperity-peak radii increase with increasing Δx distances for all the surface roughnesses. The values of the asperity-peak radii tend to level out for the rougher surfaces. It is clear that the asperity-peak radii increase dramatically for the largest Δx distance compared to the smallest Δx distance, especially for smooth surfaces, where radii above even 130 µm are calculated. The difference between the smallest and largest Δx distances, i.e., the effect of the data resolution, is thus more than 35 times for the smoothest surface, but decreases to only 10 times for the roughest surface.
Figure 12: Asperity-peak radii in relation to the roughness parameter Ra for different Δx distances for the 3PP criteria.
Asperity-peak heights
The asperity-peak heights increase with the increasing surface roughness for all the selected Δx distances (see Figure 13). However, for any selected surface roughness, the asperity-peak heights are not influenced by changes in the Δx distances. For the two roughest surfaces there is a slight tendency for the asperity-peak height to increase with increasing Δx distances, but the asperity-peak heights for certain surface roughnesses are still within the data scatter and the variations are below 30 nm. The values of the asperity-peak heights are therefore almost the same as those presented in Figures 7 and 10.
Figure 13: Asperity-peak heights in relation to the roughness parameter Ra for different Δx distances for the 3PP criteria.
4. Discussion
Effect of the criteria of neighbouring points (3PP, 5PP and 7PP)
The number of asperity peaks for the 3PP criterion decreases with increasing surface roughness (from 1400 to 650 asperity peaks) and levels out for the rougher surfaces, while the values are almost constant for the 5PP (at 260 asperity peaks) and the 7PP (at 200 asperity peaks) criteria, as is clear from Figure 5. The results for the 3PP criterion thus have a much more trustworthy physical background than those for the 5PP and 7PP criteria and are in agreement with many theoretical and experimental observations for smooth and rough surfaces [9, 30]. Namely, the number of asperity peaks is reported to decrease with an increasing surface roughness, as is the case for the 3PP criterion.
It is interesting to note that the radii of the asperity peaks for the 3PP are in the range between 3.5 µm and 2.3 µm for the selected surface roughnesses, which is quite a small change compared to the great change in surface roughness. For the 5PP and 7PP criteria, the variation in the asperity-peak radii is greater, i.e., from 7 µm to 2.3 µm, which is still rather small compared to the effect of the surface-profile resolution Δx (see Figure 12).
Of course, the heights of the asperity peaks for the smooth surface are smaller than for the rougher surfaces, because smaller surface deviations are already considered as asperity peaks (see Figure 7). It is also interesting to see that the asperity-peak heights are almost the same (within the data scatter) for the 3PP, 5PP and 7PP, regardless of the surface roughness. Figure 14 shows the relationship between the actually measured parameter Ra and the asperity-peak heights for the 3PP criterion. It is clear from Figure 14 that a full linear correlation (R2 = 1) between the surface parameter Ra and the heights of the asperity peaks can be drawn for Ra values lower than 0.2 µm. However, the correlation is only slightly imperfect (R2 = 0.99) if the whole roughness range is taken into consideration.
Figure 14: Asperity-peak height in relation to the roughness parameter Ra for the 3PP criterion.
Accordingly, we can conclude that from these selected asperity-peak identification criteria, the 3PP criterion appears as the most appropriate for an asperity-peak identification for a broad roughness range of engineering surfaces. In addition, the changes in the asperity-peak radii are relatively small compared to the changes in the asperity-peak number and the asperity-peak heights for the 3PP criterion.
Effect of the peak-threshold value Δz (correction in the z-direction)
The number of identified asperity peaks decreases with increasing surface roughness as well as with increasing peak-threshold value (Figure 8). The differences become very apparent and greatly influence the number of identified asperity peaks (Figure 8). However, for rough surfaces, the use of peak-threshold values becomes unrealistic (no asperity peaks found, see Figure 8). Therefore, a constant peak-threshold value cannot be used throughout the whole surface-roughness range. Instead, the peak-threshold value should be a function of the surface roughness in order to obtain a more realistic number of asperity peaks. However, more profound analyses and a correlation between the peak-threshold values and the surface roughness exceed the scope of this paper.
The peak-threshold value also has an indicative effect on the asperity-peak radii. With an increasing peak-threshold value, the asperity-peak radii decrease for any given surface roughness (Figure 9). In addition, the asperity-peak radii also decrease with increasing surface roughness (Figure 9). Again, the influence of the peak-threshold value is minimal for the smoothest surface, but gradually increases as the surfaces get rougher.
Figure 15 shows typical asperity peaks with radii for the smooth and rough surfaces. For the smooth surfaces the asperity peaks are expected to be lower at a given asperity-peak width 2Δx (Figure 15a) compared to the asperity peaks on the rough surfaces, resulting in higher asperity-peak radii. For rough surfaces, the asperity peaks are higher at a given asperity-peak width 2Δx (Figure 15b) and thus have smaller radii compared to the smooth surfaces.
Figure 15: Asperity-peak radii for a) smooth surfaces and b) rough surfaces.
For smooth surfaces the absolute variation in the peak-threshold value is small (Table 3), and thus also the asperity-peak number and the radii differ by only a small value (Figures 8 and 9). In contrast, for rough surfaces, the absolute differences between the different peak-threshold values are larger and so also the asperity-peak number and the radii differentiate greatly (see Figures 8 and 9).
The asperity-peak heights, on the other hand, again increase with increasing surface roughness, regardless of the peak-threshold values (Figure 10). The asperity-peak heights slightly differ between the different asperity-peak threshold values, especially for rougher surfaces, but the calculated data is almost all within the scatter.
In our study we used real surfaces with five distinctively different surface-roughness values, in order to cover a broad range of relevant engineering-surface conditions and thus obtain more general relevance for these results. We can conclude based on these analyses that the peak-threshold value has much less effect on the asperity-peak number and the radii for smooth surfaces compared to rough surfaces. The peak-threshold value of 10% Rq, proposed [9] for the smooth surface (Bhushan and Poon criterion of Rq < 0.05 µm), has no effect on the number of asperity peaks and their radii for the smoothest surface, but has a critical influence on the number of asperity peaks and their radii already for the second smoothest surface, i.e., Ra = 0.032 µm and Rq = 0.041 µm, which is still considered as a smooth surface according to the Bhushan and Poon criterion (Rq < 0.05 µm). Both these surfaces should thus have a peak-threshold value of 10% Rq. However, for the three highest surface roughnesses, where the Rq values are above 0.05 µm, the proposed range of peak-threshold values completely dominates the analysis and certainly becomes inappropriate, since it even results in zero asperities, which is not realistic. It seems that the asperity-peak criterion with the peak-threshold value in its present form is not the most appropriate for asperity-peak identification and should be studied and developed substantially to become useful for real engineering surfaces across a broad range, especially for rough surfaces.
To illustrate this further, Figure 16 shows the ratio of the number of asperity peaks for different peak-threshold values divided by the number of asperity peaks found with the 3PP criterion, without the use of the peak-threshold value as a function of the surface roughness. It can be seen that for a small peak-threshold value (0.5% Rq), the ratio of the asperity peaks is almost constant for values of Ra below 0.1 µm, which means that the influence of the peak-threshold value is small. However, if the surface roughness is increased, then the influence of the peak-threshold value becomes more apparent. Larger peak-threshold values result in smaller ratios and show a significant influence of the peak-threshold value also for small surface roughnesses.
Figure 16: Ratio of the asperity peaks for different peak-threshold values in relation to the surface roughness.
Accordingly, it appears that the proposed peak-threshold values are too general for all possible engineering roughnesses since the results suggest too little effect for the smoothest and too much effect for the roughest surfaces. Based on our results, the use of the peak-threshold criterion is questionable and should be further studied as a function of the surface roughness for which it is implemented before it can be recommended for use.
Effect of the surface-profile resolution Δx
The number of asperity peaks is influenced by changes in the Δx distances and also by the surface roughness (Figure 11). For the three smallest Δx distances (0.1875 µm, 0.375 µm and 0.75 µm Δx distances) the number of asperity peaks seems to be levelling out with increasing surface roughness. On the other hand, for larger Δx distances (1.125 µm and 1.875 µm) the number of asperity peaks is little affected by the changes in surface roughness (Figure 11). It again seems that Δx distances above 1 µm are not the most appropriate when trying to identify asperity peaks. Namely, as explained in the literature [9, 30] and shown in our work, the number of asperity peaks should decrease with an increasing surface roughness, which is not the case for Δx distances above 1 µm (Figure 11).
Figure 12 shows the asperity-peak radii for different Δx distances. The asperity-peak radii increase dramatically for higher Δx distances, especially for smooth surfaces, where radii even above 130 µm were calculated. In the past, the reported range of asperity-peak radii in the literature was between 0.3 µm and 200 µm [6-8, 10, 31], but even some higher numbers were reported [4]. Some papers provided radii that are in good agreement with our findings for Δx = 0.1875 µm and the 3PP criterion, i.e., about 0.3 µm to 7 µm [4, 7, 10, 31], while some other papers reported much larger radii compared to our findings for Δx = 0.1875 µm, i.e., 20 µm to 500 µm [4, 6, 8, 9]. The latter, these very large asperity-peak radii, are reported when the surfaces were very smooth (Rq < 0.05 µm) and the Δx distances were above 1 µm, which is much higher than in our study. As was explained, by selecting different data-acquisition distances Δx, different asperity-peak radii can be calculated, even for the same surfaces [4, 9, 10, 19]. Therefore, any references to asperity-peak number and radii are strongly dependent on the selected Δx distances during the profile's data acquisition. However, according to our data, Δx distances below 1 µm should be used, and these will result in relatively smaller asperity-peak radii.
To summarize, for now it seems that the 3PP criteria should be used for asperity-peak identifications on rough surfaces. The use of asperity-peak criteria with peak threshold values seems too general and complicated. However, if the ''correct'' Δx distances were to be correlated with a different surface roughness, then the 3PP criterion would provide a good asperity-peak identification tool, even without the use of peak-threshold values.
5. Conclusions
On the basis of the experimental work in this study, the following conclusions can be drawn:
The 3PP criteria, unlike the 5PP and 7PP criteria, have a much more trustworthy physical background as the number of asperity peaks decreases with an increasing surface roughness;
The number of asperity peaks and their radii decreases with increasing surface roughness as well as with an increasing peak-height threshold value. The radii of the asperity peaks for the smoothest surface were found to be around 3.5 µm, but the values decrease below 1 µm for the rougher surfaces and high peak-threshold values.
Asperity peak heights increase from 0.015 µm for the smoothest surface and to around 0.46 µm for the roughest surface. The peak-threshold value has little effect on the asperity-peak heights;
The Δx distance (data resolution in the x-direction) has a large influence on the asperity-peak properties. Larger Δx distances result in a smaller number of asperity peaks and an increase in their radii, but do not affect the asperity-peak heights;
Δx distances above 1 µm seem to be less appropriate for asperity-peak identification. Use of better resolution in x-direction is thus suggested;
The proposed peak-threshold values (criteria for z-direction) are too general since the results suggest that the peak-threshold values are too small for the smoothest surfaces, but they are also too high for the rougher surfaces. Clear guidelines for their use is still missing;
If the ''correct'' Δx distances (x-direction data resolution) could be correlated with a different surface roughness, the 3PP criterion would provide a good asperity-peak identification tool, even without the use of peak-height threshold values.
Acknowledgement
The authors would like to thank European Social Fund for partial financial support.
