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Asperity-peak criteria are necessary in order to identify the relevant, load-carrying asperity peaks in the contacts between rough surfaces. Once these load-carrying asperity peaks are identified, the real contact area can be calculated using deterministic contact models. This work focuses on the effect that different asperity-peak identification criteria have on the identified asperity-peak properties (number, radii and heights) for rough surfaces. Different criteria, which take into account the number of required neighbouring points (i.e., 3, 5 and 7 points), and also the peak-threshold value (z-direction), were applied in this study and analysed for five different real surface roughnesses in the broad engineering range from Ra = 0.003 µm to Ra = 0.70 µm. In addition, the effect of the data resolution in the x-direction on the asperity-peak properties is also evaluated.
The results show that criteria with three neighbouring points result in much more trustworthy asperity-peak properties than those with five and seven neighbouring-points criteria. The results also show that the x-direction data resolution has an important influence on the number of asperity peaks and their radii, but has little effect on the peak heights; the x-direction data resolution values of Î”x below 1 µm should be used. Peak threshold value (z-direction criteria) has very diverse effect for rough and smooth surfaces and lacks clear guidelines for its use. It seems from the results that the criteria with only three neighbouring points (3PP) should be used for asperity-peak identification, especially if the data resolution in the x-direction could be correlated with different surface roughnesses.
Keywords: surface topography; identification; asperity-peak; radii; real contact area.
Real contact area is an important parameter when evaluating tribological contacts. Using real contact area for the calculations of contact temperatures and contact pressure results in much higher values compared to calculations with nominal contact area [Kalin]. Such increase in contact temperatures and contact pressure can thus cause different mechanical and tribological behaviour of materials in the tribological contacts. This is especially important for materials, that are very sensitive to changes in temperature, i.e. for polymers [pogaÄnik]. So in order to better understand the behaviour of tribological contacts, it is very important to estimate the real contact area as accurately as possible.
There are several different models available for the calculation of real contact area, which can be grouped into three main categories: statistical, fractal and ''deterministic'' models. The most known and widely used statistical model is Greenwood-Williamson model, which can be used for the contact of rough surface against ideally flat, rigid plane . This model assumes that surfaces consist of sphere-shaped asperities with Gaussian heights distribution. The results show the direct proportionality of real contact area and normal load. Due to same drawbacks of Greenwood-Williamson contact model, several modifications were later proposed [13-16]. Several authors reported that properties of surface asperities depend greatly on the surface-measuring technique, the instrument, its resolution and the use of filters [4, 9, 10, 19, pogaÄnik]. In order to avoid surface-parameter dependence on measuring instrument, fractal surface analyses were introduced. With the use of fractal models, surface roughness becomes scale-independent and thus provides surface-roughness information regardless of the resolution and length scale. Such a model was presented by Majumdar and Bhushan [20, 21], and several other authors [22, 23].
With the advancing computational power, ''deterministic'' contact models are becoming more popular [4, 24-27]. The statistical functions for the asperity peaks on the surface are replaced with simple, but real, measured geometries with measurable number, radii and heights. In this way the calculation does not depend on the statistical characterization and the typical ''averaging'' of the surfaces.
However, the problem of these models is the identification of relevant asperity-peaks, that transfer the load in the tribological contact. So a level, to which the measured surface data are considered as (relevant) micro-asperities must be determined by using certain arbitrary criteria. The level of what is considered as a relevant ''micro-asperity", or micro-contact, therefore depends on our ability to identify and quantify them, as well as our ability to determine their influence in bearing loads, heat transfer, etc. Accordingly, it is crucial to determine which asperity peaks do have an influence on the contact conditions and are able to resist external loads.
A determination of the load-carrying asperity peaks is always needed when using deterministic contact models for real contact-area calculations. However, to do this, in accordance with the above discussion, we first need to identify the asperity peaks. Methods for determining the asperity peaks are seldom described in the literature and are not well established, and this may also be one of the important obstacles to their use for the real contact area in tribological models.
This work focuses on a review of the existing asperity-peak identification criteria for 3D rough surfaces. For the purpose of this research, steel specimens with five distinctively different surface roughnesses were prepared and then measured using an optical interferometer and surface topographies were analysed to calculate the number of asperity peaks, their radii and their heights. The effects of the different asperity-peak identification criteria (the number of neighbouring points that define an asperity-peak) as well as the corrections in the z-direction and the resolution of the profile measurement in x- and y-direction were evaluated for a broad range of engineering surface roughnesses (Ra between 0.005 µm and 0.529 µm).
1.2 Asperity-peak identification for deterministic contact models in 3D
Surface topographies can be measured with different machines, for example with stylus profilers, optical interferometers or AFMs. 3D topography, measured with stylus profiler, consist of several parallel profile measurements, which can be later combined to 3D image of the surface. As for the measurements with optical interferometers or AFMs, 3D topographies can be obtained directly from a single measurement.
Figure 3 presents discrete points of surface, measured either by stylus profiler, optical interferometer or AFM. Asperity-peak can be defined as a point higher than its closest neighbor points, as shown in Figure 3. Red colored point is asperity-peak, blue colored dots are neighboring points. The most widely used asperity-peak identification criteria in the literature are 5- and 9-point rectangular definitions (Figure 3a and b) [GW2, T1, E39, E40, E41, E50 ].
Figure : Asperity-peaks on 3D surfaces a) 5-point rectangular asperity-peak definition, b) 9-point rectangular summit definition, c) 4-point triangular asperity-peak definition and d) 7-point hexagonal asperity-peak definition [P2, P3].
Some authors [P2, P3] also proposed different summit definitions in 3D. They proposed a triangular asperity-peak definition (Figure 3c) and a hexagon asperity-peak definition (Figure 4d). But these definitions require different distances between parallel surface measurements and different measuring starting point, in order to get equal spacing between summit point and its neighbour points. However, such measurements are hard to perform and are rarely used in praxis.
Figure 4 shows discrete points of surface measurement. Lines on Figure are contour lines and represent the height of the points. Greenwood [GW2] noted that with a 5-point asperity-peak definition in 3D (Figure 3a), there is a possibility of finding false asperity-peaks, as shown in Figure 4. Figure 4a shows a saddle point (A) that is wrongly identified as an asperity-peak, if 5-point peak definition is used. On Figure 4b a ridge point (B) is presented which is also falsely interpreted as an asperity-peak with 5-point peak definition. Greenwood suggested using 9-point peak for asperity-peak identification in 3D. Increasing the number of points reduces the risk of missing an asperity-peak, but a finite possibility of missing asperity-peaks always exist [T17].
Figure : False asperity-peak identification with 5-point rectangular definition; a) saddle point and b) ridge point. Reproduced after [GW2].
Another way of identifying asperity-peaks is with surface pattern recognition. The principle was first introduced by Maxwell in 1870 [M4]. He suggested dividing a landscape (or surface) into regions consisting of hills (peaks) and regions consisting of dales (valleys). However, using Maxwell analysis results in over-segmentation of surface into tiny, shallow peaks/valleys, instead of identifying important peaks or valleys [M3]. Due to several drawbacks of Maxwell's proposal, different pattern recognition procedures were later introduced to improve the identification asperity-peaks or valleys [M5, M6, M7, M8, M9].
For the purpose of this research, 5 different asperity-peak identification criteria were used.
The 5-point peak criterion in 3D (5PP-3D criterion)
Several authors suggested using a 5-point peak (5PP-3D) criterion for identifications of asperity-peaks for 3D topographies [M1, M2, GW2, M10]. An asperity-peak is defined as a point that is higher than its four closest neighbours, as schematically shown in Figure 2. Figure 2 also shows locations of of different optical interferometer measurements in x and y direction (lines i and j).
Due to the fact, that contacts between rough surfaces is expected to occur only on seldom number of highest asperity-peaks [K6], only peaks above profile mean surface are taken into consideration.
Figure 2: The 5-point peak criterion (5PP-3D) with presented x and y grid.
The 5-point peak on a 3D topography can be mathematically defined as:
zi,j > zi-1,j; zi+1,j; zi,j-1; zi,j+1
with the additional condition
zi,j > m.
The radius Î² of identified asperity-peak is calculated as a radius of a sphere, fitted to the 5 points of the asperity-peak. The fitting was done according to the least squares method. Identified asperity-peak i is thus characterised with height zi,j and radius Î²i.
The modified 5-point peak criterion in 3D (M5PP-3D criterion)
In 1995 Bhushan and Poon [K9] proposed a modification to the 3-point peak asperity-peak identification criterion. The details are presented in [M11]. Their theory is valid for 2D asperity-peaks, but can be applied to 5PP-3D criterion, i.e. to get a modified 5-point peak criterion called M5PP-3D. Additional condition is that 4 neighbouring points has to be lower than the peak point for a certain value, named peak threshold value. Additional conditions can be mathematically written as
Î”z1,i = zi,j - zi-1,j
Î”z2,i = zi,j - zi+1,j
Î”z3,i = zi,j - zi,j-1
Î”z4,i = zi,j - zi,j+1
min (Î”z1,i, Î”z2,i, Î”z3,i, Î”z4,i) > peak-threshold value.
Bhushan and Poon proposed a peak-threshold value of 10% Rq for smooth surfaces (Rq < 0.05 µm), and threshold values below 10% Rq for rougher surfaces (Rq > 0.05 µm) [K4]. The same sphere fitting is applied as for the 5PP-3D criterion.
The 9-point peak criterion in 3D (9PP-3D criterion)
A 9-point peak criterion in 3D (9PP-3D) is defined as a point higher than its eight closest neighbour points (Figure 3b). It is similar to the 5PP-3D criterion, but with additional conditions, that can be mathematically written as
zi,j > zi-1,j-1; zi-1,j+1; zi+1,j-1; zi+1,j+1.
The asperity-peak radius is again calculated as a radius of a sphere, least-square fitted to all 9 points of the identified asperity-peak.
The modified 9-point peak criterion in 3D (M9PP-3D criterion)
A modified 9-point peas in 3D (M9PP-3D) is defined in the same way as 5PP-3D criterion, but with even more restrictive criteria. The four most distant points from the center of the asperity-peak must be smaller than its two closest neighbours. These additional conditions can be mathematically written as
zi-1,j-1 < zi,j-1; zi-1,j
zi-1,j+1 < zi,j+1; zi-1,j
zi+1,j-1 < zi,j-1; zi+1,j
zi+1,j+1 < zi,j+1; zi+1,j.
No such asperity-peak identification criterion was found in the literature, but we introduce it to analyse the effects of different criteria on properties of asperity-peaks.
2. Experimental details
2.1 Specimen geometry and the surface roughness
For the purpose of this research, stainless steel (100Cr6) samples were prepared with 5 different surface roughnesses. Samples were cylindrical in shape (24 mm diameter, 8 mm height). Sample surface roughness was 63 ± 1 HRC, measured with Leitz Miniload microhardness tester (Leitz Miniload, Wild Leitz GmbH, Wetzlar, Germany).
5 different surface roughnesses, ranging from Sa = 0.005 µm to Sa = 0.529 µm, were achieved using different sequence of abrasive papers on surface-grinding machine (RotoPol-21with RotoForce-3 module, Struers, Denmark).
Surface roughness parameters Sa and Sq were measured using a 3D optical microscope (ContourGT-K0, Bruker, Arizona, USA). 20x magnification lens was used for the measurements, which resulted in Î”x = Î”y =0.187 µm lateral resolution in x and y direction respectively. The total scanned area was 0.24 mm x 0.18 mm. In order to minimize the effect of noise, surface averaging function was used.
For each sample, 5 measurements on different locations were taken in order to calculate surface roughness parameters. The average values of the roughness parameters together with their standard deviations were calculated for every surface-roughness condition. The results are presented in Table 1. It can be seen from the results, that relatively low standard deviations were calculated, i.e. below 10%, which is negligible, especially compared to distinctive differences among the five selected surface roughnesses.
Table 1: Values of Sa and Sq for different surface roughnesses.
0.005 ± 0.001
0.007 ± 0.001
0.057 ± 0.003
0.091 ± 0.007
0.116 ± 0.005
0.161 ± 0.007
0.218 ± 0.014
0.289 ± 0.018
0.529 ± 0.030
0.660 ± 0.025
2.2 Measurements of the surface topographies for the analysis of the asperity-peak properties
For the purpose of asperity-peak analysis, the measurements from surface roughness calculations were used. As mentioned above, each sample was measured 5 times on different location in order to get a representative topography of different surfaces.
Prior to the surface-topography analysis, each measurement was filtered to eliminate the effect of surface tilt. No additional filtering was applied.
In addition to the 5PP-3D, 9PP-3D and M9PP-3D criteria, a modified criterion with a variation of the peak-threshold value in the z-direction was also used in the analyses. Such modification was only used with 5PP-3D criterion, to form the M5PP-3D criterion. Different peak-threshold values were selected in the range proposed by Bhushan and Poon [K9] in a sequence of five different Rq values, as presented in Table 3. The 5PP criterion is thus actually the M5PP-3D with a 0% Rq peak-threshold value (Table 3).
Table 3: Peak-threshold values for the M5PP-3D criterion for different surface-roughnesses and peak-threshold values.
Peak threshold value
Rq = 0.007 µm
Rq = 0.091 µm
Rq = 0.161 µm
Rq = 0.289 µm
Rq = 0.660 µm
Levelled surface topographies were then analysed using specific developed software. For selected surface roughness, data (x, y and z coordinates) is imported into the software and the asperity peaks are identified according to different asperity-peak criteria (5PP-3D, M5PP-3D, 9PP-3D and M9PP-3D). In addition, asperity-peak radii and heights were also calculated for these topographies. After all 5 measurements for one surface roughness were analysed, the average number, height and radii were calculated for a selected surface roughness. The procedure is then repeated also for other surface roughnesses.
To analyse the effect of data resolution Î”x and Î”y on the asperity-peak properties, surface topographies from our measurements were modified in such a way that different data resolution were obtained. Different data resolution was obtained by using only every 2nd, 4th, 6th or 10th row and column from measurement data. These modified topographies were taken into consideration and used in the asperity-peak-properties analysis - in exactly the same way as explained above. Different lateral resolutions used in analyses are presented in Table 4. Such data variation was only employed with 5PP-3D criterion.
Table 4: Variation of Î”x and Î”y resolutions.
''Original'' Î”x and Î”y
Variation of Î”x and Î”y resolutions, µm
It can be seen that modified data resolutions in x and y directions are the same. In order to simplify the presentation of the results, differences in data resolutions will only be referred to as changes in Î”x distances.
3.1 Effect of different asperity-peak criteria on the asperity-peak properties
Number of asperity-peaks per area
Number of asperity peaks per scanned area for three different asperity-peak criteria are presented on Figure 5. The number of asperity peaks decreases with the increasing surface roughness both for 5PP-3D and 9PP-3D criteria. The number of asperity-peaks also decrease with increasing surface roughness for M9PP-3D criterion. However, the decrease is much smaller compared to the other two criteria. The asperity-peak number for 5PP-3D decreases from 1.1E5 (smooth surface) to 2.4E4 (rough surface), which is the factor of 4. It seems that the number of asperity-peaks starts levelling out with increasing surface roughness, except for M9PP-3D, were the number of asperity-peaks decreases logarithmically.
Figure 5: Number of identified asperity-peaks in relation to the roughness parameter Ra for 5PP-3D, 9PP-3D and M9PP-3D criteria (Î”x = Î”y = 0.180 µm).
Figure 6 shown the asperity-peak radii at different surface roughnesses for three selected identification criteria. Asperity-peak radii decrease with increasing surface roughness for selected identification criteria. The radii for 5PP-3D and 9PP-3D criteria are very similar at all roughnesses. The values decrease from 4.2 µm at the smoothest surface to 0.8 µm for the roughest surface. There is only a small difference in the radii calculated with the 5PP-3D and 9PP-3D, but the 5PP-3D always results in higher radii values. For the M9PP-3D identification criteria, the radii values decrease from 5.6 µm to 1.3 µm with increasing surface roughness. The values are in average 35 % higher compared to 5PP-3D and 9PP-3D criteria.
Figure 6: Asperity-peak radii in relation to the roughness parameter Ra for 5PP-3D, 9PP-3D and M9PP-3D criteria (Î”x = 0.180 µm).
The asperity-peak heights for selected identification criteria are presented on Figure 7. The heights increase with increasing surface roughness for 5PP-3D, 9PP-3D and M9PP-3D criteria. The height differences between 5PP-3D and 9PP-3D are small and are within data scatter. As for M9PP-3D criterion, the heights are in average 20 % lower compared to the other two criteria. The asperity-peak heights are around 0.01 µm for the smoothest surface and increase to 0.71 µm for the roughest surface.
Figure 7: Asperity-peak heights in relation to the roughness parameter Ra for 5PP-3D, 9PP-3D and M9PP-3D criteria (Î”x = 0.180 µm).
3.2 Effect of the peak-threshold value on the asperity-peak properties
Number of asperity peaks per area
The number of asperity peaks per area decreases with the increasing surface roughness (Figure 8). The number also decreases with increasing peak-threshold values from 0 % to 10 % Rq. The number of asperity-peaks for the smoothest surface decreases form 1.1E5 to 5.7E4 when threshold values increase from 0 % to 10 % Rq. This is for about 80 %. For the other surface roughnesses, the relative differences of identified asperity-peaks within one surface roughness are much bigger - can be up to 4.5 times. The values are all below 2.5E4. It can also be noted, that apart from the smoothest surface roughness, the decrease of number of asperity-peaks with increasing peak threshold values is linear. It can also be concluded, that the influence of peak threshold value is smaller for smoother surfaces compared to rough surfaces.
Figure 8: Number of asperity-peaks per area for different surface roughnesses according to different peak-threshold values. The different columns represent the various peak-threshold values for the 5PP and M5PP criteria (Î”x = 0.180 µm). DODAJ OKVIRÄŒKE
The asperity-peak radii for the different surface roughnesses according to peak-threshold values are presented on Figure 9. The asperity-peak radii decrease both with increasing surface roughness and increasing peak threshold values. The values of asperity peak radii for the smoothest surface are between 4.3 µm and 3.8 µm. For the other surface roughnesses, the radii decrease 3 times and are in the range between 1.6 µm and 0.3 µm. For the smoothest surface, the differences between different peak threshold values are relatively small. However, the differences increase slightly for the other surface roughnesses. The difference in the asperity-peak radii between the 0% Rq and 10% Rq peak-threshold values for the second smoothest surface (Ra = 0.057 µm) is already 80% (a reduction of the asperity-peak radius from 1.6 µm to 0.6 µm), while for the other rougher surfaces, the differences are between 55 % and 65 %.
Figure 9: Asperity-peak radii for different surface roughnesses according to the calculated peak-threshold values. The different columns represent the various peak-threshold values for the 5PP and M5PP criteria (Î”x = 0.180 µm). DODAJ OKVIRÄŒKE
Figure 10 shows the asperity-peak heights for the different surface roughnesses and different peak-threshold values. The values of the asperity-peak height for all surfae roughnesses (except for the smoothest surface) increase with the increasing surface roughness as well as with increasing peak threshold values. The asperity-peak heights for the smooth surface are around 0.009 µm and are constant for all peak threshold values. Except for the smoothest surface, peak threshold value has big effect on asperity-peak heights. The increase in heights is almost linear with increasing peak threshold values. The differences within same surface roughnesses can be as much as 2x but these differences decrease with increasing surface roughness. The maximum heights at rough surface can be as much as 0.9 µm.
Figure 10: Asperity-peak heights for different surface roughnesses according to the calculated peak-threshold values. The different columns represent the various peak-threshold values for the 5PP and M5PP criteria (Î”x = 0.180 µm). DODAJ OKVIRÄŒKE
3.3 Effect of the surface-topography Î”x resolution on the properties of asperity-peaks for 5PP-3D citerion
Number of asperity peaks per area
Figure 11 shows the number of asperity peaks per area for different data resolutions Î”x and Î”y for 5PP-3D criterion. The number of asperity-peaks decreases with increasing surface roughness and also decreases with increasing Î”x and Î”y distances between raw data. The values for the Î”x and 2Î”x seems to be levelling out for the higher surface roughnesses; for the other data resolutions, the values are much lower and seems to decrease with a linear tendency. The differences in identified asperity-peaks for the smoothest surface is factor of 50 between Î”x and 10 Î”x resolutions. However, with increasing surface roughness, the differences between different resolutions become smaller. For the roughest surface with 10 Î”x, as little as 1000 asperity-peaks were identified.
Figure 11: Number of asperity-peaks per area in relation to the roughness parameter Ra for different Î”x distances (5PP-3D).
Figure 12 shows asperity-peak radii for different Î”x and Î”y resolutions at selected surface roughnesses. The values of radii increase with increasing data resolutions for all surface roughnesses. However, with increasing surface roughnesses, the asperity-peak radii decrease. Again it seems that the values of the asperity-peak radii level out for the rougher surfaces. It can be seen from the results that asperity-peak radii dramatically increase for the larger data resolutions, especially at smooth surfaces. The maximum radius calculated was 270 µm. The difference between the smallest and largest Î”x distances, i.e., the effect of the data resolution, is thus more than 60 times for the smoothest surface, but decreases to only 7 times for the roughest surface.
Figure 12: Asperity-peak radii in relation to the roughness parameter Ra for different Î”x distances (5PP-3D).
Figure 13 presents heights of the identified asperity-peaks for different data resolutions at selected surface roughnesses. The heights of asperity-peaks increase with increasing surface roughnesses for all data resolutions. It seems that data resolution has little effect on the heights of asperity peaks. For all individual surface roughnesses, the heights data seems to be within scatter. For the roughest surface there is a slight tendency for the asperity-peak height to increase with increasing Î”x distances. The heights of asperity-peaks are between 0.009 µm for the smoothest surface and increase above 0.8 µm for the roughest.
Figure 13: Asperity-peak heights in relation to the roughness parameter Ra for different Î”x distances (5PP-3D criterion).
Effect of different asperity-peak identification criteria on asperity-peak properties
The number of asperity-peaks per area decreases with increasing surface roughness both for 5PP-3D and 9PP-3D identification criteria and seems to be levelling out for the rougher surfaces (at 22000 asperity-peaks for 5PP-3D and 13000 for 9PP-3D criterion (Figure x). The results for the 5PP-3D and 9PP-3D criteria thus have a much more trustworthy physical background than those for the M9PP-3D criterion and are in agreement with many theoretical and experimental observations [9, 30]. It is reported that the number of asperity-peaks decreases with increasing surface roughness. This is also true for M9PP-3D criterion, but the values on number of asperity peaks ate too low compared to the literature findings . We can conclude thet the M9PP-3D criterion is not the most appropriate for the identification of asperity-peaks on the rough surfaces.
Asperity peak radii for 5PP-3D and 9PP-3D criteria are in the range between 4.1 µm and 0.9 µm for the selected surface roughnesses, which is quite a small change compared to the great change in surface roughness. The range is almost the same also for the M9PP-crietrion, however, the values of asperity-peaks are a bit higher (between 5.5 µm and 1.3 µm). The effects are still rather small, especially when compared to the results where the effect of Î”x was taken into consideration (see Figure 12).
It is obvious that the heights of the asperity-peaks are smaller for smoother surfaces compared to rougher ones, because surface deviations are already considered as asperity peaks. Figure 14 shows the relationship between surface roughness parameter Ra and the asperity-peak heights for the 5PP-3D criterion. Almost a perfect linear correlation (R2 = 0.97) is found between peak heights and surface roughness for surface roughness below 0.25 µm. However, the correlation is only slightly imperfect (R2 = 0.94) if the whole roughness range is taken into consideration. The slope of the curve is much higher compared to other observations from the literature [Tomanik].
Figure 14: Asperity-peak height in relation to the roughness parameter Ra for the 5PP-3D criterion.
It seems from the analysis of different asperity-peak identification criteria that 5PP-3D for 9PP-3D criteria appear as the most appropriate for the identification of asperity-peaks for real 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 both criteria.
Effect of the peak-threshold value on the properties of identified asperity-peaks
It can be seen from Figure 8 that the number of identified asperity-peaks decrease with increasing surface roughness. In addition, the number also decreases wit increasing peak-threshold values. 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.
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.
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  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.
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.
napiÅ¡i da so razlike med posameznimi hrapavostmi z enakimi PTV relativno majhne, razen primerjavi z najbolj gladko povrÅ¡ino
It seems from the results that peak-threshold value criteria could be an effective tool for the identification of the relevant asperity-peaks on the surfaces. However, the current recommendations for the peak threshold values are to general. Additional research is needed in order to provide correlation between surface roughness and peak-threshold values, but this exceeds the scope of this paper.
Effect of the surface data resolution Î”x on the properties of asperity-peaks
The number of asperity peaks is influenced by changes in the Î”x distances and also by the surface roughness (Figure 11). For the three smallest Î”x distances (0.1875 µm, 0.375 µm and 0.75 µm Î”x distances) the number of asperity peaks seems to be levelling out with increasing surface roughness. On the other hand, for larger Î”x 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 Î”x 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 Î”x distances above 1 µm (Figure 11).
Figure 12 shows the asperity-peak radii for different Î”x distances. The asperity-peak radii increase dramatically for higher Î”x 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 . Some papers provided radii that are in good agreement with our findings for Î”x = 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 Î”x = 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 Î”x distances were above 1 µm, which is much higher than in our study. As was explained, by selecting different data-acquisition distances Î”x, 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 Î”x distances during the profile's data acquisition. However, according to our data, Î”x 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'' Î”x 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.
On the basis of the experimental work in this study, the following conclusions can be drawn:
The 5PP-3D and the 9PP-3D criteria seems the most appropriate methods to identify the relevant asperity-peaks on the real engineering surfaces;
The number of asperity peaks and their radii decreases with increasing surface roughness as well as with an increasing peak 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 Î”x distance (data resolution in the x-direction) has a large influence on the asperity-peak properties. Larger Î”x distances result in a smaller number of asperity peaks and an increase in their radii, but do not affect the asperity-peak heights;
Î”x 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'' Î”x 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.
The authors would like to thank European Social Fund for partial financial support.