A COMPUTER AIDED FRACTAL CALCULATION IN TRIBOLOGY RESEARCH Jian LI and Chengqing YUAN Wuhan Research Institute of Materials Protection, 430030, Wuhan, CHINA; e-mail: lijianwuhan@163.net Weimin LIU State Key Laboratory of Solid Lubrication, Lanzhou, 730000, CHINA; e-mail: wmliu@ns.lzb.ac.cn Meng HUA MEEM Dept., City University of Hong Kong, 83 Tat Chee Avenue, HONG KONG; e-mail: memnghua@cityu.edu.hk SUMMARY Using fractal in studying tribology is becoming more important. Although digitizing and processing the surface topography image of a friction surface and wear particle are the means for calculating of the respective fractal dimensions, they are extremely troublesome. To establish a method for calculating and analyzing the fractal phenomena of a surface, the use of computer software and hardware is rather essential. A computer program was developed for auto-calculating: (i) the profile of fractal dimension using the existing methods of yardstick, power spectrum, and structure function; (ii) surface fractal dimension using the methods of slit island and box counting. The program also used W-M standard fractal function curve to improve the accuracy of prediction. Values predicted by all these methods show a power law trance, and demonstrate the individual advantages and limitations on practical calculation. Predicted profile of fractal dimension suggested that the yardstick method gave large error when some pivotal points were abandoned. Generally, structure function method predicted relatively more precisely for surface profile of high fractal dimension. Power spectrum method gave relatively good precision in predicting surface profile of low fractal dimension. The accuracy of calculating the surface fractal dimension is also analyzed by auto-relation coefficient. Furthermore, examples of fractal analysis on TiN surface and/or wear particles were performed by the various methods.

Keywords: Tribology, Fractal, Surface, Topography, Wear particle

1 INTRODUCTION Surface is normally composed of mutually super-imposed roughnesses with a large number of length-scales that can be characterized by the standard deviation of surface peaks and troughs [1-2]. The multi-scale nature of surface makes the variances and derivatives of surface peaks and troughs, and other roughness parameters varying with the resolution and the filter of a measuring instrument. Hence, the values of rough surfaces obtained are generally not unique and should be characterized in retaining their structural information at all scales. As a result, the accurate quantification of the multiscale nature of surface roughness is essential.

Nanoscale measurement of rough surfaces magnifies the near real profile of the surfaces and thus provides unique property of the data. Since the topography of a surface profile is statistically self-affinity, its similarity under different magnifications can thus be statistically characterized by fractal geometry [3-5]. Fig. 1 demonstrates a typical sketch map of fractal surface topography from different length scales and resolutions. A specific feature of fractal approach is its ability to characterize surface roughness by scale-independent parameters. Generally, information of fractal analysis of the roughness features at all length scales always exhibits fractal behavior. Wear normally generates a great deal of wear particles, which can be measured by ferrography and observed by microscope. The analysis of wear types may usually vary depending on the

experience of a researcher, and thus leads to large subjective deviation and quantitative divergence. However, fractal dimension can be used to quantify its character of the edge profile and texture surface of wear particles using available fractal theories [6-7].

In recent years, many researchers use fractal theory in the area of tribological investigations, and subsequently derived several fractal mathematical models. Generally, the fractal of friction surface and wear particle involves with digitizing and processing of the surface topography image. Therefore, calculating of fractal dimensions for friction surface and wear particles is rather complicated. Computer software and hardware for calculating and analyzing the fractal information thus become essentially important.

Fig. 1: The frictional surface with fractal character [3]

2 COMPUTER FRACTAL DATA AND CALCULATION

Profile instrument like AFM (atomic force microscope) or FFM (friction force microscope) etc., can be used to measure 2D or 3D surface topography whose data can hen be used to calculate fractal dimension. Generally, the data of 2D or 3D topography images of a surface can be obtained by analog signals from profile instrument or AFM etc. Since the bulk of data so obtained is always large, methods to process properly these signals properly and effectively by computer must therefore be sought.

2.1 Fractal calculation of profile

Aiming at relating the generic surface topography scanned by ferrography image, SEM image, and optical microscope image, etc., to the fractal dimension of surface profile and/or particle shape, conversion of surface topography into digital colorized image, then into gray image and subsequently into binary image, must be performed. To facilitate computer recognition of boundary, coloring differently the individual profiles and backgrounds allows the rapid identification of edges of surface profiles.

Through tracking the coordinates of points on boundary by proper computer programming, it is anticipated that the fractal dimension of profile can be estimated using a computer program with adequate fractal calculation methods. By doing so, the bulk of heavy workloads can thus be reduced if manual calculation I performed instead.

Profile

Yard-stick method Power spectrum

method

Profile Fractal

Image Processing

Data

Fig. 2: Sequence of Fractal Dimension Calculation of a Surface Profile

Yard-stick method, power spectrum method, structure function method, box counting method and variation method are the typical methods available in the literature for calculating profile fractal dimension [6-8]. A computer program for auto-calculating of fractal dimension using the first three aforementioned fractal calculation methods is studied. To ensure that the program is useful in calculating the fractal phenomena, theoretical curves generated by W-M function [10] as stated in Eq. (1) below is used as a standard fractal function and its D as an ideal fractal dimension for comparison.

,)sin()( )( 2112 <<>= ∑+∞

−∞=

− Dttfk

kkD λλλ (1)

Three W-M fractal function curves with λ = 1.55 (Fig. 3) for D = 1.2, 1.5 and 1.8, respectively, were generated by a computer. Fractal dimensions calculated by the yard-stick method, the power spectrum method and the structure function method using the data from the W-M curves were compared with the ideal fractal dimension D. The percentage of deviation for the individual methods and D values was also estimated (Table 1).

(a) W-M curve of fractal dimension D = 1.2

(b) W-M curve of fractal dimension D = 1.5

(c) W-M curve of fractal dimension D = 1.8 Fig. 3: Standard W-M fractal function curve generated

by computer

Ideal fractal dimension 1.2 1.5 1.8 Yard-stick method 1.2895 1.5681 1.8613 Power spectrum method 1.1999 1.4354 1.7283 Structure function method 1.3164 1.5450 1.8088 Relative error of yard-stick method

7.4% 4.5% 3.4%

Relative error of power spectrum method

0.01% 4.3% 3.9%

Relative error of structure function method

9.7% 3.0% 0.49%

Table 1: comparison of computed fractal dimension with Ideal fractal dimension

Results show that values of fractal dimension estimated by the three fractal-calculation methods used in the computer program gives accuracy ranging from 0.01 % to 9.7 %. Also, the data from all three methods seem to fellow the trace of a power law. Individually, they show accurate in calculating certain fractal values whilst having relatively higher deviation in estimating the others. When using yard-stick method, the abandonment of some pivotal points would create relatively larger error. When all points are used in the fractal calculation by the power spectrum method, it gives approximation in the conversion of those discrete peaks to frequency. Generally, the structure function method gives relatively smaller level of approximation in calculating fractal information using the peak information directly.

However, the use of subtracting approach for those very close points does bring computation error by computer. Data in Table 1 show that the structure function method generally gives relatively more precise calculation than the other two methods for surface profile of high fractal dimension, whilst the power spectrum method provides the best precision for surface profile of low fractal dimension. 2.2 Fractal calculation of surface

Fractal dimension of profile is comparatively simple to calculate. It is sometimes not enough to describe a surface that needs to be described by fractal dimension of surface, normally with 2 < D < 3. Calculation of sur-face fractal dimension usually deals with even larger bulk of data, thus higher amount of workloads. For better analysis of fractal characteristics, calculation of the fractal dimension of surface by computer program is even more significant than that of profile.

Existing methods for calculating surface fractal dimension mainly include box counting method, slit island method and project method etc. The computer program developed by the authors, in its present stage, uses: (i) box counting method [11-12] to calculate the surface fractal dimension based on image information, and (ii) slit island method [13] to work out surface fractal dimension based on 3D data. Fig.4 illustrates the sequence of calculating the surface fractal dimension in the developed program.

AFM

3-D Data Files

Surface Fractal Dimension

Surface Fitting

Surface Fitting

Fitting Equation

Fitting Equation

2-D Data Files

Image

)1()1(

1 1

)()(),( −−

= =−−== ∑ ∑ ji

p

i

q

jij yyxxyxfz a

Fig. 4: Sequence for fractal Calculation of Surface

Method Fractal dimension Object

Based on 3D data

Based on image

Slit island method 2.41 2.63 Auto-relation confidence of slit island method 0.991 0.982

Box counting method — 2.59 Auto-relation confidence of slit island method — 0.979

Table 2: The surface fractal dimension of the methods

used for Fig.4

A 3D surface topography from 3D data and a 3D surface topography from optical image are showed in Fig.4. The calculated surface fractal dimension for Fig.4 using the aforementioned two methods are tabulated in Table 2. All the estimated confidence levels of the two methods were above 97 %, showing good reliability of the calculation.

3 APPLICATION OF THE PROGRAM TO PREDICT FRACTAL DIMENSIONS OF TIN COATING SURFACE AND WEAR PARTICLES IN LUBRICANT OIL

The study of TiN coating surface in sliding contacts of the rough surface was traditionally based on the two-dimensional surface roughness. Researchers have shown that the fractal analysis is a better approach in representing the true contact and the variance in a tribological surface pairs [14-17] and in characterizing surfaces. Study in [18] suggested that surface coated with TiN also tends to exhibit fractal character.

(a)

(b)

Fig. 5: TiN coating surface profile of scratch abrasion and normal wear

Fig. 6: The wear surface topography of TiN coating surface

Figs. 5(a) and 5(b) are their respective profile, and Fig. 6 shows the texture images of scratch abrasion and nor-mal wear of a TiN coating surface, Fig. 7 illustrates the corresponding 3-D surface topography of scratch abra-sion and normal wear. These images, after proper

transformation, can be used to work out the fractal dimension by the above analyzing methods. The respec-tive fractal dimension D at different levels of abrasion on TiN coating surfaces by the various methods is clearly tabulated in Table 3, which demonstrates that the levels of abrasion of TiN surface may be described by fractal dimension. Generally, when the degree of scratch abrasion TiN coating surface increases, the fractal dimension of its profile increases whilst the fractal dimension of its surface decreases. Because the texture of TiN coating surface is much more complex than its surface profile, the study of surface fractal dimension is thus more essential to tribology research.

Fig. 7: 3D surface topography of Scratch abrasion and normal wear

Object

Fractal dimension Method

Scratch abrasion

Normal abrasion

Yard-stick method 1.75 1.44 Power spectrum

method 1.63 1.47

Structure function method 1.71 1.42

Slit island method 2.63 2.80 Box counting

method 2.59 2.67

Table 3: Fractal dimension at various abrasion degrees on TiN coating surfaces

Fig. 8: Ferrography digital image of wear particles

Shape profile of wear image (Fig. 8) can be fully characterized by edge curve. Fractal dimension not only describes wear particle profile but also wear particle surface texture. For examples, the fractal dimensions for the shape of three wear particles: (a) a sphericity wear particle of fatigue wear (Fig. 9(a)), (b) a cutting wear particle of abrade wear (Fig. 9(b)), and (c) an oxide pieces cumulate wear particle (Fig. 9(c)), are tabulated in Table 4.

(a) (c)

(b)

Fig. 9: Shape of the three wear particles in Fig.8

Object Fractal

dimension Method

Fig.9 (a) Fig.9 (b) Fig.9(c)

Yard-stick method 1.74 1.86 1.79

Power spectrum method 1.52 1.91 1.87

Structure function method 1.56 1.82 1.77

Box counting method 2.57 2.35 2.41

Table 4: Fractal dimension of wear particles calculated by the different methods

Values of the fractal dimension in Table 4 can be regarded as a shape parameter distinguishing the wear types of different wear particles. Although fractal dimensions shown in Tables 3 and 4 varied with the use of different mathematical methods, their trend is approximately similar. The meaningful application of fractal analysis in tribology research generally requires the comparison of the fractal dimensions calculated by a same method. Furthermore, different methods have different sensitivity in analyzing the TiN coating surface and wear particles. This should be adequately cons-idered when any meaningful investigation is initiated.

4 CONCLUSION Computer program to auto-calculate fractal dimension of both profile and surface was initiated. Fractal dimension may be used to characterize surface topography and wear particle.

Calculation by the program developed suggests that:

(1) Method of fractal calculation influences the studying of fractal. The automatically calculating of fractal dimension using fractal models with image processing technology is viable. However, the improve-ment of the veracity of fractal dimension evaluation is still increasingly commanding a great deal of interests.

(2) TiN coating surface has fractal character. The TiN coating surface with different levels of abrasion can be characterized by fractal dimension.

(3) The fractal dimension, D, of TiN coating surface are scale-independent and related to fractal calculation methods. Fractal dimension is a parameter for distinguishing the wear types of different wear shape of wears particles.

(4) Fractal theory application on tribology is worth for studying the intrinsic law of surface topography and wear particle. 5 ACKNOWLEDGEMENT The work described in this paper was partially supported by a grant from CityU (Project No. 710037). 6 REFERENCES [1] S. Wilson; A. T. Alpas: TiN coating wear mecha-nisms in dry sliding contact against high speed steel. Surface and Coatings Technology 108-109(1998) 369-376 [2] J. Takadoum; H. Houmid Bennani: Influence of substrate roughness and coating thickness on adhesion. friction and wear of TiN films, Surface and Coatings Technology 96(1997) 272-282 [3] Paul S Addison: Fractals And Chaos An Illustrated course. Institute of Physics Publishing, London 1997 [4] T.R. Thomas; B.G. Rosen; N. Amini: Fractal characterization of anisotropy of rough surface. Wear, 1999 (232): 41-50 [5] Claude Tricot; Pierre Ferland; George Baran: Fractal analysis of worn surfaces. Wear, 1994(172): 127-133 [6] P. Podsiadlo; G.W. Stachowiak: Evaluations of boundary fractal methods for the characterization of wear particles. Wear, 1998(217): 24-34 [7] T.B. Kirk; G.W. Stachowiak; A.W. Batchelor: Fractal parameters and computer image analysis applied to wear particles isolated by ferrography. Wear, 1991(145): 347-365 [8] S. Ge; S. Suo: The Computation Methods for Fractal Dimension of Surface Profiles. Transaction of Tribology, Vol.17, No.4, 1997, Page 354-362

[9] A. Majumdar and B. Bhushan: Fractal Model of Elastic-Plastic Contact Between Rough Surfaces. Trans-actions of the ASME Journal of Tribology Vol. 113 Jan. 1991 [10] Kenneth Falconer: Fractal geometry – mathemati-cal foundations and applications. 1990 [11] Pentland A.P.: Fractal based description of natural scenes. IEEE Trans. Pattern Aral.&Machine Intell.1984.6(6):661~674 [12] Sarker N. et al.: An efficient approach to estimate fractal dimension of texture image. Pattern Recognition, 1992,25(9): 1035~1041 [13] Malinverno A.: A simple method for estimate the fractal dimension of self-affine series. Geophs Res Lett, 1990,17(6): 1953~1956

[14] S. Ganti and B. Bhushan: Generalized fractal analysis and its applications to engineering surfaces. Wear, 180(1995) 17-34 [15] N. K. Myshkin; S. A. Chizhik and V. V. Gorbunov: Nanoscale topography and tribological problems. Tribology International Vol. 28. No. 1. 1995 39-43 [16] L. He; J. Zhu: The fractal character of processed metal surfaces. Wear 208(1997) 17-24 [17] A. Majumdar and B. Bhushan: Role of fractal geometry in roughness characterization and contact mechanics of surface, Transactions of the ASME Journal of Tribology 112(1990) 205-216 [18] J. Li; M. Hua; C.Q.Yuan and H.Y. Ma: Fractal characteristic of the surface topography of TiN coating. Proceeding of 3rd International Symposium on Tribo-Fatigue, Beijing, Oct. 22-26, 2000, 309-3314.