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Fractal Analysis
Fractal Analysis Fractal geometry is a branch of mathematics that has attracted much attention in a variety of fields. The applications of this geometry found in real-world situations are based on the observation that few natural structures are actually composed of Euclidean lines and planes, but rather have a characteristic known as self-similarity. This means that there exists a hierarchy of detail at ever finer scales, so that viewing the structure at higher magnification simply reveals more detail and the overall appearance does not change. One of the commonly cited examples of fractal geometry is the observation that the coastlines of the world are not straight, and hence that determining their length is an ill-posed question. Measuring the length of a coastline produces a result that is a function of the length of the measuring tool. Walking along a coastline on a map with dividers set to 100 km produces a polygonal approximation that misses the smaller scale irregularities. Repeating the operation with a 10 km or 1 km stride length produces a longer result. Walking along the actual ground with a meter stick or crawling along it with a micrometer each produce even longer estimates. The interesting fact that connects these various measurements is that a plot of the log of the measured length vs. the length of the measurement tool produces a straight line (Richardson 1961). Furthermore, the slope of that line varies for different coastlines, being much steeper, for example, for the rugged coast of Norway than for the much smoother coast of Florida. Indeed, the human notion of smoothness vs. roughness is very well described by that slope. Figure 1 shows an example, with data from three of the Hawaiian islands. Mathematically, the fractal dimension of the coastline is calculated as (2km) where m is the slope of the line on the log–log plot ; this is a number between 1 and
Total distance (km)
300
200
100 0.1
Hawaii D =1.046 ± 0.004 Oahu D =1.122 ± 0.003 Kauai D =1.071± 0.002 1 Stride length (km)
Figure 1 Richardson plots for the coastlines of three Hawaiian islands.
10
1.9999 … The name fractal is a play on words introduced by Mandelbrot (1982), who is responsible for much of the development and popularization of the concept. It combines the words ‘‘ fracture ’’ (because, as we will see, fracture is one way to produce fractal dimensions) and ‘‘ fractional ’’. The idea of a fractional dimension is perhaps strange to those schooled in the traditional Euclidean geometry in which a line has a dimensionality of 1 (e.g., a length measured in centimeters), a plane of 2 (an area measured in cm#), and a volume of 3 (cm$). According to fractal geometry a coastline might have a dimensionality of 1.1863. It is convenient to think of the fractional part of the dimension as representing the tendency of the line to ‘‘ spread out ’’ into the two-dimensional area of the surface on which it lies. We will see that, similarly, a rough surface has a dimension that is between 2 and 2.9999 … and represents the tendency of the surface to spread out into the volume in which it resides. For a fractal surface, there is an interesting and useful relationship between the surface fractal dimension (greater than 2) and the fractal dimension of a coastline (greater than 1) constructed by intersecting that surface with a plane parallel to the nominal surface orientation. If the surface is isotropic, i.e., has no directionality or ‘‘ lay ’’ in the machining sense, the dimension of the line is exactly 1.0 less than that of the surface. In other words, if the coastline of Norway has a higher dimension than that of Florida, then the mountainous terrain is rougher by the same amount than the beaches. This relationship illustrated in Fig. 2.
1. Measurement Procedure 1 : Boundary Lines The relationship between the coastline dimension and the surface dimension can be exploited as a measurement tool for surfaces. In the so-called ‘‘ slit-island ’’ technique (Mandelbrot et al. 1984) the surface, for instance a fracture surface of a material, can be plated with a contrasting material and polished parallel to the nominal surface orientation to reveal islands where the mountains are intersected, and these can be measured to determine a dimension. Adding 1 then gives the surface dimension. Measuring the boundary dimension of the coastline can be accomplished in several ways. The technique of ‘‘ walking ’’ along the boundary with a series of different rulers is conceptually straightforward, but hard to implement in practice. In most cases, the measurement is performed in a computer. The digitized image of the islands (acquired using a digital camera for instance) has limited resolution and it is rarely possible to cover a sufficiently large range of stride lengths for good precision. In addition, the pixels are arrayed in a regular x,y grid that prevents constructing a polygonal approximation in which the strides are all the same length. 1
Fractal Analysis
Figure 2 Intersecting a fractal surface with a horizontal plane produces a fractal coastline.
There are several other approaches that are used. One method is to cover the image with a grid and count the number of squares in the grid through which the boundary passes. Repeating this process with grids of different size, and plotting the log of the number of squares vs. the log of the grid size, also produces a linear plot whose slope can be used to determine the dimension. Implementing a grid-counting method is straightforward using software, but again for a typical digital image with limited resolution it can be difficult to cover a large enough range of grid sizes to get adequate precision. Another method is to dilate the boundary line into a band and measure the area of the band. Plotting the log of the area (or more commonly the area divided by the radius of the dilation) vs. the log of that radius also produces a linear graph, and again the slope can be used to calculate the dimension. In practice, this is usually the preferred method for making the determination because there are very efficient tools for carrying it out and the precision is quite good. Figure 3 shows an example of the procedure. The image shows the slit islands produced by sectioning a fracture surface in a ceramic, with an image-processing function (the Euclidean distance map) used to assign a gray-scale value to each pixel in the islands according to the distance of that pixel from the nearest background point. The cumulative histogram of the number of pixels as a function of gray level (on logarithmic scales) shows the straight-line relationship, and from the slope of the line the fractal dimension is obtained. There are other methods available as well. One is plotting the area vs. the perimeter of each island in the image. Because of the finite resolution of the image, 2
the perimeter of the smaller islands is measured with more influence from the pixel size than that of larger islands, whereas the area is not so affected. The slope of this line (as usual on log axes) gives the dimension. Or simply plotting the number of islands as a function of size will provide a dimension ; for a very rough surface there will be a very rapid increase in the relative number of smaller islands as the many small peaks on the surface are intersected, and conversely for smooth surfaces the rate of increase is smaller. All of these techniques have been used in some applications, and one of the frustrations in comparing data from the literature is that they do not all give the same numerical results and indeed do not even measure quite the same thing from a mathematical point of view (Russ 1994). The numerical results are also influenced to some degree by the image resolution and size of the pixels used, and very strongly affected by the range of scales covered in the plots. The safest approach is to perform comparisons only between measurements made on different samples using identical preparation and analysis procedures, which in practice means that data from different laboratories cannot usually be compared. Finally, the methods are only really applicable to surfaces in which there is no directionality. This is unfortunately not the usual situation. It may apply to some chemically deposited or etched surfaces, or shotblasted machined surfaces (provided that the substrate crystallography does not impose structure), but is unlikely to be appropriate for most machined surfaces produced by grinding, turning, or polishing, or for most fracture surfaces, all of which have an inherent directionality associated with them.
Fractal Analysis
4.800 4.700
log (area)
4.600 4.500 4.400 4300 4.200 4.100 4.000 0.000
0.250
0.500 0.750 log (distance)
1.000
1.250
Fractal Dim = 1.468
Figure 3 Measurement of the fractal dimension of islands using the cumulative histogram of the Euclidian distance map.
2. Measurement Procedure 2 : Elevation Profiles Preparing slit-island images of rough surfaces is difficult and also destroys the surface. It would be much more convenient in many cases to work with elevation profiles. These may be prepared metallographically by sectioning the surface normally (i.e.,
parallel to a line normal to the nominal surface), or by using stylus profilometers that scan a line along the surface and measure the vertical deflection of the tip. These devices are in common use for measuring conventional surface roughness parameters such as Sa (the mean absolute deviation from a best-fit straight 3
Fractal Analysis line), which is used as a quality control index in machining. Many of the devices record the actual profile in a computer, so that fractal analysis may be performed. A common mistake in analyzing these profiles is to employ the same methods described above for the slitisland boundaries. The problem is that the vertical elevation profile is self-affine rather than self-similar (Mandelbrot 1985). This mathematical distinction is important : vertical elevation profiles must typically be enlarged by different scale factors in the lateral and vertical directions to display similar apparent roughness as they are magnified. Also, a scanned stylus cannot see undercuts in the surface profile. Some methods, such as the Richardson walk, will still produce straight line plots. But the slope does not give the desired fractal dimension. A grid count method would require that the grids be constructed of rectangles, not squares, and that the aspect ratio of the rectangles be varied as their size changed. The dilation technique can still be used, but the dilating element is no longer a circle but a horizontal line. No commercial profilometer equipment at this writing performs a correct fractal analysis of the profile. It is unfortunate that the apparent ease of obtaining vertical elevation profiles of surfaces and the misunderstanding of the appropriate measurement procedures has resulted in serious errors in a great deal of the published data. Any results based on vertical elevation profiles should be examined with suspicion and great care to see if correct interpretation and analysis procedures have been used. And, of course, the problem still remains of dealing with surfaces that are not inherently isotropic. An elevation profile is typically performed in a carefully selected orientation across (or more rarely along) the lay direction of the surface. Even if the profile itself is correctly analyzed, the resulting value may not provide a useful characterization of the surface.
3. Measurement Procedure 3 : Surface Scans Several kinds of modern instrumentation produce direct measurements of elevation for surface areas. The atomic force microscope (and its many offshoots) has lateral and vertical resolution of nanometers, but typically scans very small areas. Scanned stylus instruments have similar vertical resolution but their lateral resolution is limited to micrometers because of the size of the stylus. Interference microscopes have performances similar to the scanned stylus instruments but have difficulty with steep surfaces. They are also sensitive to surface films or oxides and do not produce consistent results with dielectric materials, but they are much faster than scanned stylus methods. All of these tools produce an image in which each pixel has a value representing the local elevation of the 4
uppermost point on the surface. These so-called range images are often displayed with gray scale or false color representing elevation, or may be rendered as surface visualizations using computer graphics. They can also be measured to determine the fractal dimension of the surface. As a word of caution, one of the most commonly used tools for examining rough surfaces is the scanning electron microscope (SEM). The images from this tool do not represent surface elevation and cannot be measured to determine a fractal dimension as discussed above. In some cases, the brightness of the SEM image may be a monotonic function of surface slope (e.g., in the absence of compositional variations). In this case the pixel brightness values from a fractal surface will be mathematically fractal (this is also the case for a visible light image with diffuse lighting). The dimension of this image can be measured and may be correlated with the dimension of the surface, but the correlation depends upon factors such as the electron voltage, surface composition, and detector position and may not be of much intrinsic use. All of the measurement techniques listed above for boundaries and profiles can be adapted to the measurement of surfaces. Some are more difficult to perform correctly than others because the surfaces are self-affine. Indeed, a very large fraction of published data has not been calculated appropriately and is therefore of little use except to reinforce the observation that many rough surfaces are indeed fractal. The most efficient procedure for measurement of the fractal dimension of surfaces, and one which allows characterization of anisotropic surfaces as well, seems to be through Fourier analysis. Plotting the amplitude of the Fourier power spectrum as a function of frequency (again, using the usual logarithmic axes) produces a straight line. The slope of the line can vary from 0 (white noise, with equal power at all frequencies) to 2 (for a Euclidean surface). The fractal dimension of the surface is calculated simply as 3k(m\2). Furthermore, the variation in the slope with direction can be determined as a measure of the anisotropy of the surface. Figure 4 shows an example of an isotropic surface in which the slope does not vary with direction, and an anisotropic surface in which it does. Notice that in both cases the plot of log(amplitude) vs. log(frequency) is linear, and the histogram of the phase information in the Fourier transform is random. The latter important requirement is often overlooked when performing fractal analysis. Anisotropy of fractal surfaces is a complex subject but includes at least two different types of directional variation. One possibility is simply an amplitude variation with direction, and the other is a dimension variation. In Fig. 4 the two directional or rose plots show the slope and intercept of the power spectrum slope, which represent these two different parameters.
Fractal Analysis (a)
Slope min =1.438 max =1.697
Mean slope = 1.560
log(mag) vs. log(freq)
Phase histogram
Icept min = 8.199i105 max = 1.179i106
As a thought experiment, consider a surface consisting of an extrusion in which the profile across the lay is fractal but in the extrusion direction the profiles are ideal straight lines. For this surface, the fractal dimension would be the same in all directions except exactly along the extrusion direction. In practice this would never be measured so the dimension would be isotropic. But the topothesy, a measure of the lateral scale of the fractal (technically, the lateral distance
over which the expected change in slope is one radian), would vary continuously with direction.
4. Typical Applications Comparatively little quantitative fractal analysis of surfaces (both natural or otherwise) has been performed. Most of the published results seem to be 5
Fractal Analysis (b)
Slope min = 0.325 max =1.024
Mean slope = 0.60408
log(mag) vs. log(freq)
Phase histogram
Icept min = 6.648i104 max = 9.759i106
Figure 4 Fourier analysis of two fractal surfaces : (a) isotropic and (b) anisotropic. Plots show the slope and intercept of the log(amplitude) vs. log(frequency) plot as a function of direction, the averaged plot, and a histogram of the phase from the Fourier transform.
content with demonstrating the existence of fractal geometry in a given instance, and have not gone very far in correlating the dimension with the formation of the surface or with its properties. 6
In at least three situations there are intriguing and potentially very useful relationships between surface fractal dimension and material properties or surface behavior. Attempts to understand the physics behind
Fractal Analysis these relationships are topics of research, and are not reviewed here. The situations are as follows. Brittle fracture produces surfaces that have a fractal geometry (ductile fracture generally does not). Generally speaking, the ‘‘ rougher ’’ the visual appearance of the fracture surface the more energy was required to produce the fracture, and this is confirmed by correlation between the fractal dimensions and the fracture toughness values for the materials (Passoja 1988, Mecholsky 1992). These correlations have been shown to apply even for metals, ceramics, and glasses, which have different types of atomic bonds and atomic arrangements (Fahmy et al. 1991). Besides experimental data, models of crack formation and propagation are used to investigate the physical basis for these relationships. Machined surfaces (including ground and polished surfaces) are never perfectly planar, and the roughness has important consequences for the behavior of those surfaces in their intended application (Russ 1997, Thomas 1999). Sliding contact involves either dry friction or the dispersal of lubricant across the surface, and surface geometry controls both. Thermal and electrical conductivity between surfaces is largely controlled by the (few) points of actual contact. By analogy to the slit-island measurement technique, it is clear that there will be a few large islands of contact and many smaller ones. Some of these will be elastically deformed and some may be plastically deformed. Mathematical analysis of these effects (Greenwood and Williamson 1966, Majumdar and Bhushan 1991) seems to support observational data but does not yet provide a complete understanding. Deposited surfaces are produced by a variety of chemical and ballistic procedures, and here at least the theoretical modeling of the surface geometry is well understood and in good agreement with experiment (Stanley and Ostrowsky 1986, Meakin 1983). In the limit of diffusion-limited aggregation, molecules or particles are more likely to adhere to the outer branches of a growing forest rather than penetrate to the ground. This produces a fractal surface. Altering the sticking probability of the particles, introducing initial velocities, temperature gradients, etc., alters the geometry in predictable ways. It seems likely that as more experimental and
theoretical work is performed, the importance of fractal geometry to the understanding of material surfaces will continue to grow. Since surfaces are one of the most important research and engineering areas in modern materials, this should provide a valuable tool for characterization that will be used to understand and better control the relationships between material and process variables, surface geometry, and service behavior. See also: Surface Roughening Bibliography Fahmy Y, Russ J C, Koch C C 1991 Application of fractal geometry measurements to the evaluation of fracture toughness of brittle intermetallics. J. Mater. Res. 6(9), 1856–61 Greenwood J A, Williamson J P B 1966 The contact of nominally flat surfaces. Proc. R. Soc. London A295, 300–19 Majumdar A, Bhushan B 1991 Fractal model of elastic–plastic contact between rough surfaces. ASME J. Tribol. 113, 1–11 Mandelbrot B B 1982 The Fractal Geometry of Nature. Freeman, New York Mandelbrot B B 1985 Self-affine fractals and fractal dimension. Phys. Scr. 32, 257–60 Mandelbrot B B, Passoja D E, Paullay A G 1984 Fractal character of fracture surfaces of metals. Nature 308, 721–2 Meakin P 1983 Diffusion-limited cluster formation in two, three and four dimensions. Phys. Re. A 27, 604 Mecholsky J J 1992 Application of fractal geometry to fracture in brittle materials. Scanning ’92. FAMS, Atlantic City, NJ Passoja D E 1988 Fundamental relationships between energy and geometry in fracture. In: Fractography of Glasses and Ceramics. American Ceramic Society, Westerville, OH, pp. 101–26 Richardson L F 1961 The problem of contiguity : an appendix of statistics of deadly quarrels. General Systems Yearbook 6, 139–87 Russ J C 1994 Fractal Surfaces. Plenum, New York Russ J C 1997 Fractal dimension measurement of engineering surfaces. In: Mainsah E et al. (eds.) Metrology and Properties of Engineering Surfaces. Chalmers University Press, Go$ teborg, Sweden, pp. 43–82 Stanley H E, Ostrowsky N (eds.) 1986 On Growth and Form . Martinus Nijhoff, Boston, MA Thomas T R 1999 Rough Surfaces. Imperial College Press, London
J. C. Russ
Copyright ' 2001 Elsevier Science Ltd. All rights reserved. No part of this publication may be reproduced, stored in any retrieval system or transmitted in any form or by any means : electronic, electrostatic, magnetic tape, mechanical, photocopying, recording or otherwise, without permission in writing from the publishers. Encyclopedia of Materials : Science and Technology ISBN: 0-08-0431526 pp. 3247–3254 7
Total distance (km)
300
200
100 0.1
Hawaii D =1.046 ± 0.004 Oahu D =1.122 ± 0.003 Kauai D =1.071± 0.002 1 Stride length (km)
Figure 1 Richardson plots for the coastlines of three Hawaiian islands.
10
1.9999 … The name fractal is a play on words introduced by Mandelbrot (1982), who is responsible for much of the development and popularization of the concept. It combines the words ‘‘ fracture ’’ (because, as we will see, fracture is one way to produce fractal dimensions) and ‘‘ fractional ’’. The idea of a fractional dimension is perhaps strange to those schooled in the traditional Euclidean geometry in which a line has a dimensionality of 1 (e.g., a length measured in centimeters), a plane of 2 (an area measured in cm#), and a volume of 3 (cm$). According to fractal geometry a coastline might have a dimensionality of 1.1863. It is convenient to think of the fractional part of the dimension as representing the tendency of the line to ‘‘ spread out ’’ into the two-dimensional area of the surface on which it lies. We will see that, similarly, a rough surface has a dimension that is between 2 and 2.9999 … and represents the tendency of the surface to spread out into the volume in which it resides. For a fractal surface, there is an interesting and useful relationship between the surface fractal dimension (greater than 2) and the fractal dimension of a coastline (greater than 1) constructed by intersecting that surface with a plane parallel to the nominal surface orientation. If the surface is isotropic, i.e., has no directionality or ‘‘ lay ’’ in the machining sense, the dimension of the line is exactly 1.0 less than that of the surface. In other words, if the coastline of Norway has a higher dimension than that of Florida, then the mountainous terrain is rougher by the same amount than the beaches. This relationship illustrated in Fig. 2.
1. Measurement Procedure 1 : Boundary Lines The relationship between the coastline dimension and the surface dimension can be exploited as a measurement tool for surfaces. In the so-called ‘‘ slit-island ’’ technique (Mandelbrot et al. 1984) the surface, for instance a fracture surface of a material, can be plated with a contrasting material and polished parallel to the nominal surface orientation to reveal islands where the mountains are intersected, and these can be measured to determine a dimension. Adding 1 then gives the surface dimension. Measuring the boundary dimension of the coastline can be accomplished in several ways. The technique of ‘‘ walking ’’ along the boundary with a series of different rulers is conceptually straightforward, but hard to implement in practice. In most cases, the measurement is performed in a computer. The digitized image of the islands (acquired using a digital camera for instance) has limited resolution and it is rarely possible to cover a sufficiently large range of stride lengths for good precision. In addition, the pixels are arrayed in a regular x,y grid that prevents constructing a polygonal approximation in which the strides are all the same length. 1
Fractal Analysis
Figure 2 Intersecting a fractal surface with a horizontal plane produces a fractal coastline.
There are several other approaches that are used. One method is to cover the image with a grid and count the number of squares in the grid through which the boundary passes. Repeating this process with grids of different size, and plotting the log of the number of squares vs. the log of the grid size, also produces a linear plot whose slope can be used to determine the dimension. Implementing a grid-counting method is straightforward using software, but again for a typical digital image with limited resolution it can be difficult to cover a large enough range of grid sizes to get adequate precision. Another method is to dilate the boundary line into a band and measure the area of the band. Plotting the log of the area (or more commonly the area divided by the radius of the dilation) vs. the log of that radius also produces a linear graph, and again the slope can be used to calculate the dimension. In practice, this is usually the preferred method for making the determination because there are very efficient tools for carrying it out and the precision is quite good. Figure 3 shows an example of the procedure. The image shows the slit islands produced by sectioning a fracture surface in a ceramic, with an image-processing function (the Euclidean distance map) used to assign a gray-scale value to each pixel in the islands according to the distance of that pixel from the nearest background point. The cumulative histogram of the number of pixels as a function of gray level (on logarithmic scales) shows the straight-line relationship, and from the slope of the line the fractal dimension is obtained. There are other methods available as well. One is plotting the area vs. the perimeter of each island in the image. Because of the finite resolution of the image, 2
the perimeter of the smaller islands is measured with more influence from the pixel size than that of larger islands, whereas the area is not so affected. The slope of this line (as usual on log axes) gives the dimension. Or simply plotting the number of islands as a function of size will provide a dimension ; for a very rough surface there will be a very rapid increase in the relative number of smaller islands as the many small peaks on the surface are intersected, and conversely for smooth surfaces the rate of increase is smaller. All of these techniques have been used in some applications, and one of the frustrations in comparing data from the literature is that they do not all give the same numerical results and indeed do not even measure quite the same thing from a mathematical point of view (Russ 1994). The numerical results are also influenced to some degree by the image resolution and size of the pixels used, and very strongly affected by the range of scales covered in the plots. The safest approach is to perform comparisons only between measurements made on different samples using identical preparation and analysis procedures, which in practice means that data from different laboratories cannot usually be compared. Finally, the methods are only really applicable to surfaces in which there is no directionality. This is unfortunately not the usual situation. It may apply to some chemically deposited or etched surfaces, or shotblasted machined surfaces (provided that the substrate crystallography does not impose structure), but is unlikely to be appropriate for most machined surfaces produced by grinding, turning, or polishing, or for most fracture surfaces, all of which have an inherent directionality associated with them.
Fractal Analysis
4.800 4.700
log (area)
4.600 4.500 4.400 4300 4.200 4.100 4.000 0.000
0.250
0.500 0.750 log (distance)
1.000
1.250
Fractal Dim = 1.468
Figure 3 Measurement of the fractal dimension of islands using the cumulative histogram of the Euclidian distance map.
2. Measurement Procedure 2 : Elevation Profiles Preparing slit-island images of rough surfaces is difficult and also destroys the surface. It would be much more convenient in many cases to work with elevation profiles. These may be prepared metallographically by sectioning the surface normally (i.e.,
parallel to a line normal to the nominal surface), or by using stylus profilometers that scan a line along the surface and measure the vertical deflection of the tip. These devices are in common use for measuring conventional surface roughness parameters such as Sa (the mean absolute deviation from a best-fit straight 3
Fractal Analysis line), which is used as a quality control index in machining. Many of the devices record the actual profile in a computer, so that fractal analysis may be performed. A common mistake in analyzing these profiles is to employ the same methods described above for the slitisland boundaries. The problem is that the vertical elevation profile is self-affine rather than self-similar (Mandelbrot 1985). This mathematical distinction is important : vertical elevation profiles must typically be enlarged by different scale factors in the lateral and vertical directions to display similar apparent roughness as they are magnified. Also, a scanned stylus cannot see undercuts in the surface profile. Some methods, such as the Richardson walk, will still produce straight line plots. But the slope does not give the desired fractal dimension. A grid count method would require that the grids be constructed of rectangles, not squares, and that the aspect ratio of the rectangles be varied as their size changed. The dilation technique can still be used, but the dilating element is no longer a circle but a horizontal line. No commercial profilometer equipment at this writing performs a correct fractal analysis of the profile. It is unfortunate that the apparent ease of obtaining vertical elevation profiles of surfaces and the misunderstanding of the appropriate measurement procedures has resulted in serious errors in a great deal of the published data. Any results based on vertical elevation profiles should be examined with suspicion and great care to see if correct interpretation and analysis procedures have been used. And, of course, the problem still remains of dealing with surfaces that are not inherently isotropic. An elevation profile is typically performed in a carefully selected orientation across (or more rarely along) the lay direction of the surface. Even if the profile itself is correctly analyzed, the resulting value may not provide a useful characterization of the surface.
3. Measurement Procedure 3 : Surface Scans Several kinds of modern instrumentation produce direct measurements of elevation for surface areas. The atomic force microscope (and its many offshoots) has lateral and vertical resolution of nanometers, but typically scans very small areas. Scanned stylus instruments have similar vertical resolution but their lateral resolution is limited to micrometers because of the size of the stylus. Interference microscopes have performances similar to the scanned stylus instruments but have difficulty with steep surfaces. They are also sensitive to surface films or oxides and do not produce consistent results with dielectric materials, but they are much faster than scanned stylus methods. All of these tools produce an image in which each pixel has a value representing the local elevation of the 4
uppermost point on the surface. These so-called range images are often displayed with gray scale or false color representing elevation, or may be rendered as surface visualizations using computer graphics. They can also be measured to determine the fractal dimension of the surface. As a word of caution, one of the most commonly used tools for examining rough surfaces is the scanning electron microscope (SEM). The images from this tool do not represent surface elevation and cannot be measured to determine a fractal dimension as discussed above. In some cases, the brightness of the SEM image may be a monotonic function of surface slope (e.g., in the absence of compositional variations). In this case the pixel brightness values from a fractal surface will be mathematically fractal (this is also the case for a visible light image with diffuse lighting). The dimension of this image can be measured and may be correlated with the dimension of the surface, but the correlation depends upon factors such as the electron voltage, surface composition, and detector position and may not be of much intrinsic use. All of the measurement techniques listed above for boundaries and profiles can be adapted to the measurement of surfaces. Some are more difficult to perform correctly than others because the surfaces are self-affine. Indeed, a very large fraction of published data has not been calculated appropriately and is therefore of little use except to reinforce the observation that many rough surfaces are indeed fractal. The most efficient procedure for measurement of the fractal dimension of surfaces, and one which allows characterization of anisotropic surfaces as well, seems to be through Fourier analysis. Plotting the amplitude of the Fourier power spectrum as a function of frequency (again, using the usual logarithmic axes) produces a straight line. The slope of the line can vary from 0 (white noise, with equal power at all frequencies) to 2 (for a Euclidean surface). The fractal dimension of the surface is calculated simply as 3k(m\2). Furthermore, the variation in the slope with direction can be determined as a measure of the anisotropy of the surface. Figure 4 shows an example of an isotropic surface in which the slope does not vary with direction, and an anisotropic surface in which it does. Notice that in both cases the plot of log(amplitude) vs. log(frequency) is linear, and the histogram of the phase information in the Fourier transform is random. The latter important requirement is often overlooked when performing fractal analysis. Anisotropy of fractal surfaces is a complex subject but includes at least two different types of directional variation. One possibility is simply an amplitude variation with direction, and the other is a dimension variation. In Fig. 4 the two directional or rose plots show the slope and intercept of the power spectrum slope, which represent these two different parameters.
Fractal Analysis (a)
Slope min =1.438 max =1.697
Mean slope = 1.560
log(mag) vs. log(freq)
Phase histogram
Icept min = 8.199i105 max = 1.179i106
As a thought experiment, consider a surface consisting of an extrusion in which the profile across the lay is fractal but in the extrusion direction the profiles are ideal straight lines. For this surface, the fractal dimension would be the same in all directions except exactly along the extrusion direction. In practice this would never be measured so the dimension would be isotropic. But the topothesy, a measure of the lateral scale of the fractal (technically, the lateral distance
over which the expected change in slope is one radian), would vary continuously with direction.
4. Typical Applications Comparatively little quantitative fractal analysis of surfaces (both natural or otherwise) has been performed. Most of the published results seem to be 5
Fractal Analysis (b)
Slope min = 0.325 max =1.024
Mean slope = 0.60408
log(mag) vs. log(freq)
Phase histogram
Icept min = 6.648i104 max = 9.759i106
Figure 4 Fourier analysis of two fractal surfaces : (a) isotropic and (b) anisotropic. Plots show the slope and intercept of the log(amplitude) vs. log(frequency) plot as a function of direction, the averaged plot, and a histogram of the phase from the Fourier transform.
content with demonstrating the existence of fractal geometry in a given instance, and have not gone very far in correlating the dimension with the formation of the surface or with its properties. 6
In at least three situations there are intriguing and potentially very useful relationships between surface fractal dimension and material properties or surface behavior. Attempts to understand the physics behind
Fractal Analysis these relationships are topics of research, and are not reviewed here. The situations are as follows. Brittle fracture produces surfaces that have a fractal geometry (ductile fracture generally does not). Generally speaking, the ‘‘ rougher ’’ the visual appearance of the fracture surface the more energy was required to produce the fracture, and this is confirmed by correlation between the fractal dimensions and the fracture toughness values for the materials (Passoja 1988, Mecholsky 1992). These correlations have been shown to apply even for metals, ceramics, and glasses, which have different types of atomic bonds and atomic arrangements (Fahmy et al. 1991). Besides experimental data, models of crack formation and propagation are used to investigate the physical basis for these relationships. Machined surfaces (including ground and polished surfaces) are never perfectly planar, and the roughness has important consequences for the behavior of those surfaces in their intended application (Russ 1997, Thomas 1999). Sliding contact involves either dry friction or the dispersal of lubricant across the surface, and surface geometry controls both. Thermal and electrical conductivity between surfaces is largely controlled by the (few) points of actual contact. By analogy to the slit-island measurement technique, it is clear that there will be a few large islands of contact and many smaller ones. Some of these will be elastically deformed and some may be plastically deformed. Mathematical analysis of these effects (Greenwood and Williamson 1966, Majumdar and Bhushan 1991) seems to support observational data but does not yet provide a complete understanding. Deposited surfaces are produced by a variety of chemical and ballistic procedures, and here at least the theoretical modeling of the surface geometry is well understood and in good agreement with experiment (Stanley and Ostrowsky 1986, Meakin 1983). In the limit of diffusion-limited aggregation, molecules or particles are more likely to adhere to the outer branches of a growing forest rather than penetrate to the ground. This produces a fractal surface. Altering the sticking probability of the particles, introducing initial velocities, temperature gradients, etc., alters the geometry in predictable ways. It seems likely that as more experimental and
theoretical work is performed, the importance of fractal geometry to the understanding of material surfaces will continue to grow. Since surfaces are one of the most important research and engineering areas in modern materials, this should provide a valuable tool for characterization that will be used to understand and better control the relationships between material and process variables, surface geometry, and service behavior. See also: Surface Roughening Bibliography Fahmy Y, Russ J C, Koch C C 1991 Application of fractal geometry measurements to the evaluation of fracture toughness of brittle intermetallics. J. Mater. Res. 6(9), 1856–61 Greenwood J A, Williamson J P B 1966 The contact of nominally flat surfaces. Proc. R. Soc. London A295, 300–19 Majumdar A, Bhushan B 1991 Fractal model of elastic–plastic contact between rough surfaces. ASME J. Tribol. 113, 1–11 Mandelbrot B B 1982 The Fractal Geometry of Nature. Freeman, New York Mandelbrot B B 1985 Self-affine fractals and fractal dimension. Phys. Scr. 32, 257–60 Mandelbrot B B, Passoja D E, Paullay A G 1984 Fractal character of fracture surfaces of metals. Nature 308, 721–2 Meakin P 1983 Diffusion-limited cluster formation in two, three and four dimensions. Phys. Re. A 27, 604 Mecholsky J J 1992 Application of fractal geometry to fracture in brittle materials. Scanning ’92. FAMS, Atlantic City, NJ Passoja D E 1988 Fundamental relationships between energy and geometry in fracture. In: Fractography of Glasses and Ceramics. American Ceramic Society, Westerville, OH, pp. 101–26 Richardson L F 1961 The problem of contiguity : an appendix of statistics of deadly quarrels. General Systems Yearbook 6, 139–87 Russ J C 1994 Fractal Surfaces. Plenum, New York Russ J C 1997 Fractal dimension measurement of engineering surfaces. In: Mainsah E et al. (eds.) Metrology and Properties of Engineering Surfaces. Chalmers University Press, Go$ teborg, Sweden, pp. 43–82 Stanley H E, Ostrowsky N (eds.) 1986 On Growth and Form . Martinus Nijhoff, Boston, MA Thomas T R 1999 Rough Surfaces. Imperial College Press, London
J. C. Russ
Copyright ' 2001 Elsevier Science Ltd. All rights reserved. No part of this publication may be reproduced, stored in any retrieval system or transmitted in any form or by any means : electronic, electrostatic, magnetic tape, mechanical, photocopying, recording or otherwise, without permission in writing from the publishers. Encyclopedia of Materials : Science and Technology ISBN: 0-08-0431526 pp. 3247–3254 7