# Determine The Number Of Asperity Peaks Biology Essay

**Published:** **Last Edited:**

This essay has been submitted by a student. This is not an example of the work written by our professional essay writers.

At present there are no generally accepted and experimentally confirmed, 2D or 3D, deterministic, asperity-deformation models to evaluate the real contact area in tribological applications. One of the key obstacles is that there are no clear and experimentally verified criteria about how to define and consequently determine the "actual" load-carrying asperity peaks. As a result, this work attempts to clarify how different, arbitrarily selected, asperity-peak identification criteria affect the calculated asperity-peak properties, i.e., the number, radii and heights. Such an analysis is still missing from the literature on 2D and 3D, asperity-peak analyses and is required for a better understanding of the physical meaning and engineering feasibility, and thus more realistic assumptions about these criteria.

Different criteria that take into account the number of required neighbouring points (i.e., 3, 5 and 7 points), the peak-threshold value (z-direction) and the effect of the data resolution in the x-direction were applied in this study. Five different real surface roughnesses in the broad engineering range from RaÂ =Â 0.003Â Âµm to RaÂ =Â 0.70Â Âµm were evaluated. The results show the huge influence of these pre-selected criteria for which no verified guidelines exist. Although contact-deformation conditions based on experimental evidence are still required, several obvious and relevant conclusions can be drawn: (i) the 3-point asperity-peak criteria are more trustworthy than the 5 or 7 point criteria; (ii) an x-direction data resolution Î”x below 1Â Âµm should be used to limit the important effect on the calculated number of asperity peaks; (iii) the peak threshold value (z-direction criteria) depends to a large extent on the surface roughness and lacks guidelines for use in its current form.

Keywords: surface topography, roughness, asperity peak, real contact area, identification criteria

## 1. Introduction

When trying to characterise the real contact area in tribological contacts, the topography, roughness, load and material properties are the main influencing parameters. However, these parameters are difficult to determine reliably due to them changing continuously at the micro-asperity level. It is probably for this reason that in the majority of tribological publications a nominal contact area is used to calculate the contact pressures and the temperatures. However, this simplification may greatly overestimate the size of the real contact area, and this underestimates the severity of the real contact conditions that occur between two rough surfaces [1]. More realistic, higher contact pressures and temperatures can cause different behaviour of the materials and lubricants in tribological contacts, which can have a significant influence on the tribological properties [1]. It is thus very important to estimate the real contact areas as accurately as possible in order to better understand the real contact conditions.

Surface topography is characterised by variations in the form, waviness and roughness [2, 3]. Because of the nature of surface topography, the real contact area in a tribological contact is a momentary sum of the "micro-contacts" Ai and is only a fraction of the nominal contact area [4-6], as shown in Figure 1.

Figure 1: Real contact area of flat/flat contact.

However, the definition of a micro-contact is very general in its nature, because there exist surface irregularities as well as surface roughnesses down to the atomic scale. Thus, even the micro-contacts Ai consist of other, smaller micro-contacts, such as Ai,i shown in Figure 1, if we scale down the lengths of interest. 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.

At present there are no generally accepted and experimentally confirmed, 2D or 3D, deterministic, asperity-deformation models to evaluate the real contact area in tribological contacts. One of the key obstacles is that there are no general criteria about how to define and determine the "actual" load-carrying asperity peaks. As a result, this work attempts to clarify how different, arbitrarily selected, asperity-peak identification criteria affect the calculated asperity-peak properties, i.e., the number, radii and heights. Although there have been attempts in the past, there is no consensus on their relevance due to the lack of any experimental verification and complex mathematical analyses with many assumptions, limiting their use for a highly specialised audience rather than establishing a ready-to-use engineering basis.

For the purpose of this research, steel specimens with five distinctively different surface roughnesses were prepared and then measured using a stylus-tip profiler and the 2D data 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 - 3, 5 and 7 points) as well as the corrections in the z-direction and the resolution of the profile measurement in the x-direction were evaluated for a broad range of engineering surface roughnesses (Ra between 0.003Â Âµm and 0.644Â Âµm).

It is known from the literature that the asperity-peak properties, calculated or measured from 2D surface profiles, can be different compared to those that are calculated or measured from the 3D surface topography [7-11]. Although we realize that 2D surface profiles are less accurate than 3D surface topography, 2D profilometry is still widely used in both industry and academia, and so a 2D analysis remains important and relevant - especially for a comparison with scientific past results and industrial best practices and values (for tribological function), which are mostly 2D. Furthermore, a 2D analysis could provide us with an important basic understanding of asperity-peak identification criteria, which can easily be expanded to a 3D surface topography, and this is planned for future work. However, it should be noted that 3D surface analyses can - due to their complexity - also cause large errors if they are not properly used, as well as lack of standardised 3D parameters still limits their wider use in more profound and comprehensive analyses.

## 1.1 Models for real-contact-area calculations

One of the first real-contact-area models was proposed by Archard in 1957 [3]. He suggested that the asperities should be modelled on different scales. An asperity should have upon it a collection of smaller asperities, each of which supports a collection of even smaller asperities. For a rough, flat surface in contact with a smooth, rigid flat, he discovered that if multiple scale roughness is assumed, the correlation between the real contact area and the load is linear. In 1966 Greenwood and Williamson presented their contact model (GW model) to describe the conditions between a rough surface and an ideally flat, rigid plane, by considering the statistical nature of the surfaces [6]. The deformation theory behind the model is based on Hertz contact theory [12]. This model assumes that a rough surface consists of asperity peaks that have ideal spherical tips of the same radius. The asperity-peak height distribution (a statistical function), however, has to be defined in advance; Greenwood and Williamson assumed a Gaussian distribution of the asperity-peak heights as well as an exponential distribution. For the exponential distribution, the real contact area is directly proportional to the load and almost the same conclusion was drawn for the Gaussian distribution [6].

Several modifications to their model were proposed later [13-16]. Greenwood and Trip expanded the model in 1970 to the contact of two rough surfaces and concluded that the contact between two rough surfaces is not significantly different from the contact between a rough surface and a flat plane, although some modifications to the calculations are needed [13]. In 1975 Bush et al. upgraded the GW contact model (i.e., the BGT model). They still considered a statistical distribution of the asperity-peak heights, but assumed that the asperities are elliptical in shape [15]. In 2006 Greenwood additionally simplified the BGT contact model but obtained very similar results [14].

The above-mentioned real-contact-area models [6, 14-16] are considered to be ''statistical'' and they require the asperity-peak properties (the asperity-peak density, the average asperity-peak radii, etc.) as an input parameter for the calculation. These properties are calculated from surface profiles with the use of statistical analyses of the surface profiles or topography. However, such analyses can never replicate the exact behaviour of the surface asperities; instead they just provide an average ''picture'' of the surface, on which we base many assumptions and thus develop uncertainties and errors.

In 1971 Nayak introduced a random process model of rough surfaces [17]. He suggested that the power spectral density of an isotropic, Gaussian surface contains all the data required for a calculation of the real contact area. With the help of spectral moments, which can be calculated from the surface profiles, the density and radii of the surface asperity peaks can be estimated [9, 18, 19]. The main problem with this model is that the spectral moments are calculated from the measured surface profiles. Namely, as several authors reported [4, 9, 10, 19], the parameters calculated from the surface profiles or the surface topographies depend greatly on the surface-measuring technique, the instrument, its resolution and the use of filters. As a result, the asperity-peak properties and thus the real-contact-area calculations can vary significantly depending on the measuring instrumentation, the procedures and the post analyses.

On the other hand, the problem of surface-parameter dependence on the measuring instrument and the measuring procedures can be eliminated by the use of a fractal analysis, where the 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]. However, these concepts were not adopted for deterministic contact models.

In recent years, with advancing computational power, numerical models with computer simulations are often used to calculate real contact areas [4, 24-27]. With these ''deterministic'' models, the statistical functions for the asperity peaks on the surface are replaced with simple, but real, measured geometries. In this way the calculation does not depend on the statistical characterization and the typical ''averaging'' of the surfaces. Nevertheless, these models still require input data about the real surface, which needs to be measured with surface-measuring instruments in 2D or 3D, and the level to which the measured surface data are considered as (relevant) micro-asperities must also be determined by using certain arbitrary criteria.

## 1.2 Asperity-peak identification for deterministic contact models

When using deterministic, real-contact-area models instead of, e.g., statistical models, each asperity on the surface can be identified and its height and radius can be calculated. However, the key issue is, how do we identify the ''relevant'' asperities on the surface that actually influence the deformation, temperature and load-carrying properties? Namely, if an ''irrelevant'' asperity is mistakenly identified, it becomes equal to a relevant one and the contact morphology changes significantly. So it is very important that only relevant asperity peaks are identified on the surface for a realistic evaluation. A few criteria on how to identify the relevant asperities on the surface exist in the literature [7-9]. These are described in more detail below.

## The 3-point peak criterion (3PP criterion)

In 1984 Greenwood suggested using a 3-point peak (3PP) criterion on a 2D surface profile [8]. An asperity peak is defined as a point that is higher than its two closest neighbours, as schematically shown in Figure 2. However, only asperity peaks above a profile mean line were taken into consideration. This is due to the fact that contacts between two rough surfaces are expected to occur on the highest asperity peaks, certainly above the profile mean line [6]. The valleys and the peaks above and below the profile mean line will thus have no effect on the real contact area, at least when the loads are in a realistic engineering range.

Figure 2: The 3-point peak (3PP) criterion with the presented height differences Î”z on a 2D profile.

The 3-point peak on a 2D profile must therefore satisfy the following criteria:

ziÂ >Â zi-1, zi+1,

with the additional condition

ziÂ >Â m.

The asperity-peak radius Î² can be calculated as the radius of a circumcircle through the peak point and its two closest neighbours, shown in Figure 2. Each individual asperity peak i, found according to the 3PP criterion, is thus characterized with an asperity-peak height zi and a radius Î²i.

Obviously, in these analyses, discrete points are used, which are separated from each other by a certain distance Î”x (see Figure 2). This is a distance corresponding to the profile measuring length L divided by the number of acquired discrete (x,z) data points in the profile, typically defined by the software of the measuring machine.

## The 5-point peak criterion (5PP criterion)

A 5-point peak is defined as a point that is higher than its four closest neighbour points (Figure 3b). Basically, it is the same as a 3PP, just that each asperity peak must have two lower neighbour points on each side. This method was rarely used in the literature [7] and its effects on an asperity-peak determination and consequently on the contact properties are thus even less clear than for the 3PP criterion.

## The 7-point peak criterion (7PP criterion)

A 7-point peak is defined as a point that is higher than its six closest neighbour points (Figure 3c). It is a variation of 3PP and 5PP, but with even more restrictive criteria. No such asperity-peak definition was found in the literature, but we introduce it to obtain a trend of the effect of the criterion that is restricted by the number of neighbouring points.

Figure 3 shows the difference between the 3PP, 5PP and 7PP criteria. It is clear that with an increasing number of neighbouring points for the asperity-peak definition, the asperity peaks become wider. These three criteria (3PP, 5PP and 7PP) basically determine the width of an asperity peak at the root of a so-identified asperity peak.

Figure 3: a) 3PP, b) 5PP and c) 7PP criteria on a 2D profile.

## The modified 3-point peak criterion with a peak threshold value (M3PP criterion)

In 1995 Bhushan and Poon [9] proposed a modified 3-point peak (M3PP) criterion. They defined an asperity peak as a point higher than its two neighbour points (the same as in [8] for the 3PP), but with the additional criteria that the height difference Î”zi between the two neighbouring points and the peak point must be greater than a pre-defined peak-threshold value (see Figure 2). They proposed a peak-threshold value of 10%Â Rq forÂ smooth surfaces (RqÂ <Â 0.05Â Âµm), and threshold values below 10%Â Rq forÂ rougher surfaces (RqÂ >Â 0.05Â Âµm) [4].

A 3-point peak with peak-threshold value criteria (M3PP) on a 2D profile must therefore satisfy the following criteria:

ziÂ >Â zi-1, zi+1,

with the additional conditions

ziÂ >Â m

minÂ (Î”z1,i, Î”z2,i)Â >Â peak-threshold value.

The asperity-peak radius Î² is again calculated in the same way as for the 3PP, but with satisfactory additional peak-threshold-value criteria, as described.

## 2. Experimental details

## 2.1 Specimen geometry and the surface roughness

Stainless-steel 100Cr6 cylindrical discs with five different roughnesses were prepared for the purpose of this research. The dimensions of the cylindrical discs were Î¦Â 24Â mmÂ xÂ 8Â mm. The specimens were initially cut from a steel rod and then machined to the desired geometry. The surface hardness of the steel specimens was 63Â Â±Â 1Â HRC, measured with a microhardness tester (Leitz Miniload, Wild Leitz GmbH, Wetzlar, Germany).

The different roughnesses were obtained with a sequence of grinding and polishing steps to achieve values of Ra in the range between 0.003Â Âµm and 0.70Â Âµm using a surface-grinding machine (RotoPol-21with RotoForce-3 module, Struers, Denmark).

The surface-roughness parameters were measured using a stylus-tip profiler (T8000, Hommelwerke GmbH, Schwenningen, Germany) according to the DIN 4768 standard. Eight measurements in different directions were performed on each specimen and the surface parameters Ra and Rq were determined. 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 is clear that measurements of the roughness, even in different directions, had relatively low standard deviations, i.e., below 5%, which is negligible compared to the distinctive differences among the five selected roughness ranges.

Table 1: Values of Ra and Rq for different surface roughnesses.

Surface condition

Ra, Âµm

Rq, Âµm

## Roughness 1

0.003Â Â±Â 0.001

0.004Â Â±Â 0.001

## Roughness 2

0.032Â Â±Â 0.001

0.041Â Â±Â 0.001

## Roughness 3

0.073Â Â±Â 0.003

0.094Â Â±Â 0.008

## Roughness 4

0.190Â Â±Â 0.006

0.249Â Â±Â 0.011

## Roughness 5

0.644Â Â±Â 0.016

0.843Â Â±Â 0.020

## 2.2 Measurements of the surface profiles for an analysis of the asperity-peak properties

For the analysis of the asperity-peak properties, additional surface profiles needed to be measured, using the same stylus profiler. Namely, the data from profiles used for the surface-roughness measurements could not be used for the asperity-peak analysis due to the different measuring lengths that were used; these are required for measurements of the surface roughness according to DIN 4768. This would lead to different lateral resolutions Î”x, since the same number of data points are always recorded on different measuring lengths. However, for the purpose of this study, the same Î”x is needed for all the profiles; otherwise different lateral resolutions would lead to different asperity-peak parameter properties [4, 9, 10, 19].

A lateral resolution Î”xÂ =Â 0.1875Â Âµm was selected for our measurements and can be achieved with most contact and optical profilers [28] and also with AFMs [29]. Table 2 presents the selected measuring conditions for a determination of the asperity-peak properties, valid for all the surface-roughness conditions used.

Table 2: Profile measuring conditions for all the surface roughnesses.

## Measuring length

mm

## Sample

## points

## Î”x resolution

Âµm

## Measuring speed

mm/s

1.5

8000

0.1875

0.15

Samples of each surface roughness were measured six times in different directions to obtain the representative 2D profiles for an analysis of the asperity-peak properties. Figure 4 shows a representative profile for every surface roughness.

Figure 4: Surface profiles for the asperity-peak analysis of the average roughness values a) RaÂ =Â 0.003Â Âµm, b) RaÂ =Â 0.032Â Âµm, c) RaÂ =Â 0.073Â Âµm, d) RaÂ =Â 0.190Â Âµm and e) RaÂ =Â 0.644Â Âµm. Different scales are used for the z axis in order to obtain a clearer presentation of the profiles' topography characteristics.

Prior to the surface-profile analysis, each profile was filtered using a Gaussian filter in order to eliminate the effect of profile tilt.

In addition to the 3PP, 5PP and 7PP criteria, a modified criterion with a variation of the peak-threshold value in the z-direction was also used in this work. However, this was only applied to the 3PP criterion, i.e., to form the M3PP criterion. Accordingly, different peak-threshold values were used for the asperity-peak identification. These were selected in the range proposed by Bhushan and Poon [9] in a sequence of five different Rq values, as presented in Table 3. The 3PP criterion is thus actually the M3PP with a 0% Rq peak-threshold value (Table 3).

Table 3: Peak-threshold values for the 3PP and M3PP criteria for different surface-roughness conditions and peak-threshold values.

## CRITERIA

## 3PP

## M3PP-0.5

## M3PP-1

## M3PP-2

## M3PP-5

## M3PP-10

## Peak threshold value

0% Rq

Âµm

0.5% Rq

Âµm

1% Rq

Âµm

2% Rq

Âµm

5% Rq

Âµm

10% Rq

Âµm

Roughness 1

RqÂ =Â 0.004Â Âµm

0

2.0Â·10-5

4.1Â·10-5

8.2Â·10-5

2.0Â·10-4

4.1Â·10-4

Roughness 2

RqÂ =Â 0.041Â Âµm

0

2.0Â·10-4

4.1Â·10-4

8.2Â·10-4

2.0Â·10-3

4.1Â·10-3

Roughness 3

RqÂ =Â 0.094Â Âµm

0

4.7Â·10-4

9.4Â·10-4

1.9Â·10-3

4.7Â·10-3

9.4Â·10-3

Roughness 4

RqÂ =Â 0.249Â Âµm

0

1.2Â·10-3

2.4Â·10-3

5.0Â·10-3

1.2Â·10-2

2.4Â·10-2

Roughness 5

RqÂ =Â 0.843Â Âµm

0

4.2Â·10-3

8.4Â·10-3

1.7Â·10-2

4.2Â·10-2

8.4Â·10-2

After the elimination of the profile tilt and the calculations of the peak-threshold values, the profiles were further analysed with in-house-developed software. The surface profiles for a selected surface roughness are imported into the software and the asperity peaks are identified according to all the different asperity-peak criteria (Table 3). The asperity-peak radii and heights were also calculated for these profiles. After all the profiles were analysed, the average number of asperity peaks, the average height and the average radii were calculated for a selected surface roughness. The procedure is then repeated for all the selected surface roughnesses.

In order to also introduce the effect of the Î”x data resolution on the asperity-peak properties, the surface profiles from our measurements were modified in such a way that the different Î”x distances were considered for the existing profiles. Thus a different data resolution was obtained by using only every 2nd, 4th, 6th or 10th profile data point. These modified profiles were taken into consideration and used in the asperity-peak-properties analysis - in exactly the same way as explained above. The complete set of Î”x resolutions analysed is presented in Table 4. This variation was employed only for the 3PP criterion.

Table 4: Variation of Î”x resolutions.

## ''Original'' Î”x

resolution, Âµm

Variation of Î”x resolution, Âµm

Î”x

2Â·Î”x

4Â·Î”x

6Â·Î”x

10Â·Î”x

0.1875

0.375

0.750

1.125

1.875

## 3. Results

## 3.1 Effect of neighbouring points (asperity-peak width) on the asperity-peak properties

## Number of asperity peaks per profile

Figure 5 shows the number of asperity peaks for the 3PP, 5PP and 7PP criteria. For the 3PP criteria, the number of asperity peaks decreases with the increasing surface roughness, from 1400 asperity peaks for the smoothest surface to 650 asperity peaks for the roughest surface. It seems that the number of asperity peaks levels out for the rougher surfaces. However, the 5PP and 7PP criteria have almost no effect on the number of asperity peaks. The number of asperity peaks for the 5PP is almost constant throughout the whole roughness range at a value of 260 asperity peaks, while the values are around 200 for the 7PP criteria, but again almost constant for all roughnesses (except for the smoothest surface, where the number of asperity peaks is even lower).

Figure 5: Number of asperity peaks in relation to the roughness parameter Ra for 3PP, 5PP and 7PP criteria (Î”xÂ =Â 0.1875Â Âµm).

## Asperity-peak radii

The asperity-peak radii decrease with the increasing surface roughness for all three criteria (see Figure 6). The values are around 3.6 Âµm for the 3PP at the smoothest surface and decrease to a value of 2.3Â Âµm for the roughest surface. For the 5PP and 7PP, the radii for the smoothest surface are about 6.9Â Âµm, and they again decrease towards the highest surface roughness to values similar to the 3PP, i.e., about 2.3Â Âµm. There is only a small difference in the radii calculated with the 5PP and 7PP, but the 5PP always has the higher radii values. The difference between the 3PP and the other two criteria is almost two-fold for the smoothest surface, but this decreases with an increasing surface roughness.

Figure 6: Asperity-peak radii in relation to the roughness parameter Ra for the 3PP, 5PP and 7PP criteria (Î”xÂ =Â 0.1875Â Âµm).

## 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 0.018Â Âµm for the smoothest surface and they increase to 0.47Â Âµm for the roughest surface.

Figure 7: Asperity-peak heights in relation to the roughness parameter Ra for the 3PP, 5PP and 7PP criteria (Î”xÂ =Â 0.1875Â Âµm).

## 3.2 Effect of the peak-threshold value Î”z 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%Â Rq - 3PP) and the largest (10%Â Rq - M3PP-10) peak-threshold values is two-fold (500 asperity peaks) already for the second smoothest surface (RaÂ =Â 0.032Â Âµm). The differences increase to as much as 650 asperity peaks for the roughest surface (RaÂ =Â 0.644Â Âµm), where no asperity peaks are even identified for the 5%Â Rq and 10%Â 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 (Î”xÂ =Â 0.1875Â Âµm).

## 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 3.5Â Âµm. 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%Â Rq and 10%Â Rq peak-threshold values for the second smoothest surface (RaÂ =Â 0.032Â Âµm) is already 40% (a reduction of the asperity-peak radius from 2.6Â Âµm to 1.5Â Âµm), while for the second roughest surface (RaÂ =Â 0.190Â Âµm) the difference is almost 80% (reduction of the asperity-peak radius from 2.4Â Âµm to 0.5Â Âµm). Again, some values of the asperity-peak radii for the roughest surface could not even be calculated, i.e., for the 5%Â Rq and 10%Â 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 0.015Â Âµm and increase to around 0.46Â Âµm 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%Â 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 (Î”xÂ =Â 0.1875Â Âµm).

## 3.3 Effect of the surface-profile resolution Î”x on the asperity-peak properties

## Number of asperity peaks per profile

Figure 11 shows the number of asperity peaks for different Î”x distances between the surface-profile data points for the 3PP criterion. The number of asperity peaks for the Î”xÂ =Â 0.1875Â Âµm, 0.375Â Âµm and 0.75Â Âµm Î”x distances decreases with the increasing surface roughness and seem to level out for the rougher surfaces. In contrast, for the 1.125Â Âµm and 1.875Â Âµm Î”x 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 Î”x distance. The biggest absolute difference between the Î”x distances is for the smoothest surface, where the change in the Î”x distance from 0.1875Â Âµm to 1.875Â Âµm 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 Î”x distances for the 3PP criterion.

## Asperity-peak radii

The asperity-peak radii decrease with the increasing surface roughness for all the selected Î”x distances, as shown in Figure 12. However, the asperity-peak radii increase with increasing Î”x 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 Î”x distance compared to the smallest Î”x distance, especially for smooth surfaces, where radii above even 130Â Âµm are calculated. The difference between the smallest and largest Î”x 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 Î”x distances for the 3PP criteria.

## Asperity-peak heights

The asperity-peak heights increase with the increasing surface roughness for all the selected Î”x distances (see Figure 13). However, for any selected surface roughness, the asperity-peak heights are not influenced by changes in the Î”x distances. For the two roughest surfaces there is a slight tendency for the asperity-peak height to increase with increasing Î”x distances, but the asperity-peak heights for certain surface roughnesses are still within the data scatter and the variations are below 30Â 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 Î”x 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 3.5Â Âµm and 2.3Â Âµm 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 7Â Âµm to 2.3Â Âµm, which is still rather small compared to the effect of the surface-profile resolution Î”x (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 (R2Â =Â 1) between the surface parameter Ra and the heights of the asperity peaks can be drawn for Ra values lower than 0.2Â Âµm. However, the correlation is only slightly imperfect (R2Â =Â 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 Î”z (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 2Î”x (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 2Î”x (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%Â Rq, proposed [9] for the smooth surface (Bhushan and Poon criterion of RqÂ < 0.05Â Âµm), 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., RaÂ =Â 0.032Â Âµm and RqÂ =Â 0.041Â Âµm, which is still considered as a smooth surface according to the Bhushan and Poon criterion (RqÂ <Â 0.05Â Âµm). Both these surfaces should thus have a peak-threshold value of 10%Â Rq. However, for the three highest surface roughnesses, where the Rq values are aboveÂ 0.05Â Âµm, 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 0.1Â Âµm, 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 Î”x

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 (0.1875Â Âµm, 0.375Â Âµm and 0.75Â Âµm 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 (1.125Â Âµm and 1.875Â Âµm) the number of asperity peaks is little affected by the changes in surface roughness (Figure 11). It again seems that Î”x distances above 1Â Âµm 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 1Â Âµm (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 130Â Âµm were calculated. In the past, the reported range of asperity-peak radii in the literature was between 0.3Â Âµm and 200Â Âµm [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 Î”xÂ =Â 0.1875Â Âµm and the 3PP criterion, i.e., about 0.3Â Âµm to 7Â Âµm [4, 7, 10, 31], while some other papers reported much larger radii compared to our findings for Î”xÂ =Â 0.1875Â Âµm, i.e., 20Â Âµm to 500 Âµm [4, 6, 8, 9]. The latter, these very large asperity-peak radii, are reported when the surfaces were very smooth (RqÂ <Â 0.05Â Âµm) and the Î”x distances were above 1Â Âµm, 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 1Â Âµm 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.

## 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 3.5Â Âµm, but the values decrease below 1Â Âµm for the rougher surfaces and high peak-threshold values.

Asperity peak heights increase from 0.015Â Âµm for the smoothest surface and to around 0.46Â Âµm 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 1Â Âµm 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.

## Acknowledgement

The authors would like to thank European Social Fund for partial financial support.