Принципы нанометрологии (1027623), страница 57
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However, mostsurfaces of engineering interest are smooth if viewed at a sufficiently largescale, and the fractal dimension can change with respect to scale.Two approaches have been used to adapt fractal analysis to engineeringprofiles and surfaces. One approach is to treat the profiles as self-affine,meaning that they have a scaling exponent that varies with scale [47]. Theother approach is to divide the scales into regions.
For example, most surfacesare rough at fine scales, and smooth at larger scales, and a smooth–roughcrossover (SRC) scale can be used to define the boundary between rough(described by fractal geometry) and smooth (described by Euclidean geometry). In the rough region the fractal dimension can be used to characterizeroughness; however, the relative lengths and relative areas at particularscales, which are used to determine the fractal dimension, may be moreuseful.
The SRC is determined as the scale at which the relative lengths orareas exceed a certain threshold. There may be other crossover scales,separating scale regions where different surface creation mechanisms havecreated geometries with different complexities.8.4.1 Linear fractal methodsThe fractal dimension and the length-scale fractal complexity are determinedfrom the slope of a log-log plot of relative lengths against scale [48]. Therelative lengths are the calculated lengths, determined from a series of virtualtiling exercises, divided by the nominal length (see Figure 8.23).
The nominallength is the straight line length, or the length of the profile used in the lengthcalculation projected onto the datum. In a virtual tiling exercise the length ofthe profile at a certain scale is calculated by stepping along the measuredprofile with a line segment whose length is that scale. The exercise isFractal methodsFIGURE 8.23 Line segment tiling on a profile.repeated in the series by using progressively different lengths and plotting thelogarithm of the relative lengths against the logarithm of the correspondingscale for determining that relative length. Linear interpolation is usedbetween measured heights to maintain consistency in the length of the linesegments.The slope of the graph in Figure 8.23 is determined over some appropriaterange of scales where the plot is approximately linear.
The scale region isindicated with the slope. Slope multiplied by minus 1000 is the linear fractalcomplexity parameter,Rlfc ¼ 1000 ðslopeÞ:(8.38)One minus the slope of the length-scale plot, whose value is a negativenumber, is the fractal dimension,Dls ¼ 1 ðslopeÞ:(8.39)While the slope of the length-scale plot is generally negative or zero whenthere are periodic structures on the surface, aliasing can result in small-scaleregions with positive slopes.
In these cases local minima in relative lengthscan be found at integer multiples of the wavelength of the periodic structures[49]. The finest linear scale in this analysis that has meaning is the samplinginterval, and the largest is the length of the measured profile.Length-scale fractal analysis has found fewer applications than area-scalefractal analysis. Some examples of its use include determining anisotropy fordiscriminating different kinds of dental microwear [50], discrimination oftool usage and there is some indication that length-scale fractal analysis maybe useful in understanding the skin effect in high-frequency electrical253254C H A P T ER 8 : Surface topography characterizationtransmissions.
The relative lengths as a function of scale have also been usedto compare instruments [51].The relative length at a particular scale is related to the inclination on thesurface, f, at that scale. Inclinations on a surface vary as a function of scale(see Figure 8.24). The relative length parameter is given byRrel ¼X 1 pi Lcosfi(8.40)where L is the total nominal length of the profile and pi is the nominal, orprojected length of the ith segment.The relative length can give an indication of the amount of the surfacethat is available for interaction.
The relative area, calculated from an arealmeasurement, however, gives a better indication, because it contains moretopographic information.When the analysed profile is sufficiently long, an SRC can be observed. Atthe largest scales the relative lengths will tend towards a minimum weightedaverage of the reciprocal of the cosine of the average inclination of the analysed profile. If the profile is levelled, this will be one, the minimum relativelength. In any case, the slope at the largest scales will be zero so that thefractal dimension will be one, the minimum for a profile. At larger scales therelative lengths deviate significantly from one and the SRC has been reached.A threshold in relative length can be used to determine the crossover in scale.There may be other crossover scales dividing regions of scale that havedifferent slopes on the relative length-scale plot.
This possibility of multipleFIGURE 8.24 Inclination on a profile.Fractal methodsslopes on the length-scale plot is a characteristic of a scale-sensitive fractalanalysis.8.4.2 Areal fractal analysisThe areal fractal methods are in many ways similar to the linear methodsdiscussed in section 8.4.1. As with the profile analyses there are manymethods that can be used to estimate the fractal dimension of a rough arealsurface. Two areal methods can be found in ISO 25178 part 2 [25], volumescale and area-scale methods.8.4.2.1 Volume-scale analysisVolume-scale analysis, also known as the variation method, estimates thevolume between morphological opening and closing envelopes abouta surface.
The volume is estimated using nominally square, structuringelements. The size of the structuring elements is varied and the change ofvolume (Svs) is noted. The logarithm of the volume is plotted against thescale of the elements, i.e. the length of the sides of the square structuringelements. As the scale increases so does the volume. The fractal dimension isthe slope of the plot, d, plus two.
As with the length-scale analysis of engineering surfaces, volume-scale analysis can produce a plot with several slopesin different scale regions with corresponding crossover scales, making thisa scale-sensitive type of fractal analysis.8.4.2.2 Area-scale analysisArea-scale analysis estimates the area of a rough surface as a function ofscale. Area-scale analysis uses repeated virtual tiling exercises of themeasured surface with triangles whose area represents the scale of theanalysis. For each tiling exercise the triangles are all the same size. The tilingexercises are repeated with different-sized triangles until the desired range ofscales is covered (see Figure 8.25).The maximum range of areal scales that is potentially meaningful inarea-scale analysis of a measured surface is from the finest areal scales,which would be half the square of the sampling interval, to the largest,which would be half of the region measured at the large scales.
This is fora measurement that is approximately square with equal sampling intervalsin each direction.Linear interpolation is used between measured heights to maintainconsistency in the area of the triangles. The relative area (Srel) is the calculated area divided by the nominal or projected area. Therefore, the minimumrelative area is one. As with the relative length, the relative area is an indication of the inclinations on the surface.255256C H A P T ER 8 : Surface topography characterizationFIGURE 8.25Tiling exercises for areascale analysis.The logarithm of the relative area is plotted against the logarithm of thescale to create an area-scale plot.
The slope on this graph is related to thearea-scale fractal complexity, Safc,Safc ¼ 1000ðslopeÞ:(8.41)The scale range over which the slope has been determined can also be usefulin discriminating surfaces, and in understanding surface texture formation andits influence on surface behaviour. The fractal dimension is given byDas ¼ 2 2 ðslopeÞ:(8.42)The slopes of the area-scale plots used in these calculations are negative.The calculated fractal dimensions are greater than or equal to two and lessthan three.The above methods are scale-sensitive fractal analyses, recognising thatactual surfaces cannot be well characterized by a single fractal dimension.When the analysed region is sufficiently large there is an SRC. At the largerscales the relative areas tend towards the weighted average of the reciprocal ofthe cosine of the slopes of the unlevelled surface, as shown in equation (8.40).Srel will be one at the large scales if the measured surface is sufficiently largeand properly levelled.
In any event the slope of the relative area-scale graphwill be generally zero, at sufficiently large scales if a sufficiently large region isanalysed. Therefore, the fractal dimension tends towards two, or theEuclidean dimension, at large scales.Comparison of profile and areal characterizationArea-scale analysis has a clear physical interpretation for many applications. Many interactions with surfaces are related to the area available tointeract and with the inclinations on the surface. The relative area can serveto characterize surfaces in a manner directly related to the functioning forthese kinds of interactions. For example, equations for heat, mass and chargeexchange contain area terms or density terms implying area.
Because the areaof a rough surface depends on the scale of observation, or calculation, to usea calculated or measured area for a rough surface in heat, mass or chargeexchange calculations, the appropriate scale for the exchange interactionmust be known.Adhesion is an area where area-scale analysis has found application, forexample thermal spray coating adhesion [52], bacterial adhesion [53] andcarburizing [54], which depends on mass exchange. Area-scale analysis alsoappears useful in electrochemical impedance [55], gloss [56] and scattering[57]. The relative area at a particular scale can be used as a parameter fordiscrimination testing over a range of scales.
This kind of scale-baseddiscrimination has been successful on pharmaceuticals [58], microwear onteeth [50] and ground polyethylene ski bases [59]. Area-scale analysis canalso be used to show the effects of filtering by comparing the relative areas ofmeasurements with different filtering at different scales.8.5 Comparison of profile and areal characterizationWith the long history and usage of profile parameters, knowledge has beenbuilt up and familiarity with profile methods has developed. It will, therefore,often be necessary to compare profile and areal parameters.
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