Journal of Mining Science, Vol. 44, No. 3, 2008

FRACTAL ANALYSIS OF THE HIERARCHIC STRUCTURE OF FOSSIL COAL SURFACE

A. D. Alekseev, T. A. Vasilenko, and A. K. Kirillov

UDC 622.537.8

The fractal analysis is described as method of studying images of surface of fossil coal, one of the natural sorbent, with the aim of determining its structural surface heterogeneity. The deformation effect as a reduction in the dimensions of heterogeneity boundaries is considered. It is shown that the theory of nonequilibrium dynamic systems permits to assess a formation level of heterogeneities involved into a sorbent composition by means of the Hurst factor. Atomic-powered microscope, fractal dimension, natural sorbents

INTRODUCTION

The structure of porous media is usually studied with application of an effective method based on the theory of fractals [1, 2]. A power pore density by radii f (r ) A r  B and the divergence of a pore space surface when r o 0 are considered as indicating factors for the pore space surface being fractal. The pore volume being finite, power ȼ is related to the Hausdorff fractal dimension d f as B d f  1 .

The pore space surface can be considered as a fractal exclusively when 3  B  4 , viz. different porous materials have a fractal surface dimension within 2  d f  3 [2]. Here, methods of small-angle X-ray and neutron scattering [3], adsorption [4] and mercury porosimetry [5] were used. The method proposed in this paper for evaluating the fractal dimension of surface structure of a sorbent (coal) is based on investigations into the surface relief by scanning with an atomic-powered microscope. The Hausdorff dimension was determined for boundaries of structures heightened in a digitized surface image in a luminance range I of 256 levels. HarFa explorer [6] makes it possible to obtain additional qualitative data on structures seen in images. For instance, Fourier analysis and wavelet-analysis provide insight into spectral characteristics of 2D images and their 1D cross-sections along coordinate axes, and the fractal analysis yields distribution of structural units of an object image with provision for luminance range and color gradation. THEORY

Let us introduce basic concepts to set forth with the subject of the present paper. Statistically self-affine curve is a set Ys (t ) of points such that h S 1 ˜ Ys (ht ) is statistically equivalent to Ys (t ) for any real h [7]. Parameter h specifies the scale transformation of curve Ys (t ) with analogy parameter 0 d S d 1 . Curve Ys (t ) has a self-affine structure in all scales defined by t considered continuous. Institute of Physics of Rock Formation Processes, National Academy of Sciences, E-mail: [email protected], Donetsk, Ukraine. Translated from Fiziko-Tekhnicheskie Problemy Razrabotki Poleznykh Iskopaemykh, No. 3, pp. 14-24, May-June, 2008. Original article submitted November 20, 2008. 1062-7391/08/4403-0235 ”2008 Springer Science + Business Media, Inc. 235

Approximation of set Ys (t ) of points by a histogram with an augment H reveals that as H o 0 the curve length

¦i Y (ti  H )  Y (ti )

ramifies like H S [8]. Then Hausdorff dimension of Ys (t ) equals to:

G f [YS ] 1  S . For smooth curves, S = 0 and G f

(1)

1 . For an arbitrary continuous self-affine curve Ys (t ) with

0 d S d 1 , the Hausdorff dimension G f is restricted to the range from 1 to 2 and is called the Hausdorff

dynamic dimension. As a self-affine curve, a time series of a process, describing behavior of a dynamic system is usually considered. The fractal dimension for geometric objects is obtaoned by computation of number of squares, covering an object boundary or the whole object, including its boundary. In the first case, we deal with the curve the fractal dimension of which is determined similarly to the method considered above. The obtained fractal dimension is called the Hausdorff structural (spatial) dimension d f . It is essential to note a connection between d f and dynamic dimension G f [7]. Herewith it is assumed that a certain fractal set F is reflected onto the self-affine time series Ys (t ) . For Euclidean space dimension n 3d f and G f are related as G f  d f

3 . The latter equality permits to switch from geometric objects to

representations of fractal properties of time series, developed in the turbulence theory [8, 9]. The conception of a process power spectrum appears useful. Its presentation as G( f ) ~ f D , where f is frequency, makes it possible to move from the time-series formula by Berry ( D 5  2G f ) [10] to a formula that relates spatial fractal dimension d f and spectrum factor D : D 2G f  1 . What is the Hausdorff fractal dimension of a closed curve is readily understood from a relation of the curve length L and area S q inscribed into the said curve. In fact, according to [11]:

L ~ Sq

df /2

,

(2)

d f is found from a slope of a straight line for (2) plotted in logarithmic coordinates:

2L d f log Sq  a0 . For a circle, d f

(3)

1 by (2); for a strongly irregular curve, the Hausdorff dimension tends to 2.

PROCEDURE

Images of a few grades of coal were studied in order to derive fractal dimensions of scanned objects. Measurements were made at V. E. Lashkarev Institute of Semiconductor Physics, NAS, Kiev, by silicon probes RTESP14, Veeco, Co., with the nominal tip radius of 10 nm in air, in tapping mode be means of a serial atomic-power microscope (APM) NanoScope IIIa (Veɟco, Co.). The scanning method under consideration makes it possible to obtain not only an amplitude but an oscillation phase of the probe. 3D images of the coal surface relief and 2D images of the surface layer relief and phase state, being different in luminance, are provided for each coal grade. Images transformed from color versions into black-and-white ones were studied in 256 level luminance range I. In this case, different-scale structural formations can be analyzed with evaluation of fractal dimension D for a number of object types: 1) boundaries of different scale formations, black-and-white (BW); 2) black + blackand-white (B + BW); 3) white + black-and-white (W + BW). The calculation method for fractal dimension is described in [6] and is based on the number of image elements (N) indicated above and falling at square boxes under reduction of their sizes. A fractal dimension is determined from the slope 236

of function ln(N ) , where logarithm of square side ln(r ) is taken as an argument on the abscissa axis. The analysis of the curve slope included in HarFa, permits to calculate image scales, meeting the preset confidence probability for an obtained average value of a fractal dimension. The linear analysis (Box Counting Method) enables to calculate fractal dimensions DBW , DB WB , DW  BW (hereinafter symbols are taken after [6]). According to [4], fractal dimensions can have values within 0 – 2, however, for studying topology of a 2D object boundary, a physically based fractal dimension of the object boundaries is considered as between 1 and 2. Of interest is fractal dimension DBW characterizing boundaries of structural formations of surfaces in 2D images. If a fractal dimension of structural objects preserved its value within the entire interval of the square size variations, it could be reckoned a fractal. But real objects exhibit a multi-fractal property when the scale variations result in the fractal dimension changes. Therefore, there arises a necessity to study the entire spectrum of fractal dimensions, which are built by means of the fractal analysis method Range [6], when the fractal spectrum is calculated for the whole range of image luminance. Building the fractal spectrum involves situations when the fractal dimension value remains constant within a rather large range of luminance, and in these cases, we can say that we deal with a “monofractal”. If the dimension varies within the entire luminance range I (namely, within linear scales r), the study object is a multifractal. The validity of conclusions relative to a fractal dimension value depends on the pixel dimension of a digital image. For the fractal analysis by HarFa program, we took images of the surface relief and phase state that were obtained for the following grades of coals from the Donetsk Coal Basin: KZh (Krasnoarmeiskaya Mine), D (Trudovskaya Mine) and A (Mine 2-2bis PO Shakhtersk-Antratsit). As dimensions of the microscope-scanned fields were 3u3 ȝm, the complete images of 3u3 ȝm and cut-from-initial-images fields of 1u1 ȝm were analyzed. The data on KZh coal preliminarily subjected to uniaxial loading up to 2GPa in a high-pressure chamber were studied as well. This paper presents the data on processing 1u1 ȝm images. The surface relief image obtained by APM characterizes the height of a given point relative to a reference point, in nanometers: more intensive background corresponds to a higher point of the relief (Fig. 1).

Fig. 1. 3D image of the surface relief for KZh coal, obtained by the atomic-power microscope

237

TABLE 1. Fractal Dimensions, Spectral Characteristics and Hurst Factors for Structure of Coal Surface Relief Images Coal grade

D KZh A

DBW

Box 1.2486 r 0.0422 1.3322 r 0.0457 1.3025 r 0.0473

Range 1.256 1.470 1.393

Box 1.4972 1.6644 1.6050

H

Gf

D Range 1.512 1.940 1.786

Box 1.7514 1.6678 1.6975

Range 1.744 1.530 1.607

Box 0.2486 0.3322 0.3025

Range 0.256 0.470 0.393

N ote: Box stands for the Box Counting Method (linear method for fractal analysis); Range means the fractal spectrum method

Similarly, the luminance distribution in an image of the APM probe oscillation provides insight into a phase composition of a coal substance in view of its petrographic composition. Conversion of an initial image into black-and-white color followed by the luminance range, allows distinguishing objects of the preset luminance and analyzing them in terms of the fractal theory. The presented data include images of both the relief (amplitude) and the probe oscillation phase. RESULTS

Fractal Dimension

In Table 1 the fractal dimensions DBW for boundaries are cited. The dimensions, confidence intervals obtained from a straight-line equation for the confidence probability Ɋ = 0.95 are given. This condition is fulfilled practically for all the versions within the scale range ln(r )  2.082 y  0.693 in terms of relative units (from 29 to 150 nm). An example of a relief image and a fractal spectrum, obtained by the Range Method, is demonstrated in Fig. 2. Values of DBW vary within the luminance range, but the maximum values are observed at I 40  140 (Fig. 2b). In luminance range I 70  105 , maximum DBW 1.47 is for KZh coal, DBW 1.393 is for anthracite and DBW 1.256 is for D coal. Comparing single-type images reveals differences in fractal dimensions obtained by the linear analysis for different coal grades; nevertheless, the differences can be assumed negligible, considering the root-mean-square errors. The differences in structure observed in 2D spectral analysis are more pronounced.

Fig. 2. (a) Image 1u1 ȝm of KZh coal surface relief and (b) its fractal spectrum; within I = 90 – 120, maximum DBW = 1.47 238

When using the Range Method to build fractal spectra, revealed differences are greater: for KZh coal in the range I 80  105 , DBW t 1.846 , the maximum DBW 1.863 is when I 92 . Close values for maximum fractal dimensions: 1.692 and 1.658, respectively, are obtained for D and A coals. Determination of Structural Image Area

HarFa software is useful to evaluate area of structure images within a wide range of luminance. It suffices to build the fractal spectrum within a prescribed luminance range. Once an initial image being converted into a black-and-white, the luminance range in terms of a relative scale from 0 to 255 levels is applied to single out black and white structures. The method of the fractal dimension evaluation implies covering an image with boxes, different in dimensions: the number of boxes, N, is actually equal to the number of boxes covering black images with boundary (B+BW) and white images with boundary (W+BW). Once this method produces the number of boxes with the reduction in their sizes, we have a good approximation is obtained for areas of figures with different luminances; thus, we have:

ln N d f ln(r )  a0 ,

(4)

where coefficient d f defines a straight line slope to the abscissa axis, free member a0 nominally equals the number of minimal size boxes covering either color. In our case, different variants of boundary luminance intervals of black-and-white color are considered. The minimal dimension of a box side was taken equal to two pixels (4 nm), the maximal was to 115 pixels (230 nm). During calculations by (4), in the image luminance range from 0 to 255 relative units, variants when the lower boundary was assumed from 80 to 120 and the upper boundary from 150 to 200 were studied. In this case, the structure identifying the brightest sections of an image and corresponding to a phase component, aliphatics, was presented in black color (Fig. 3) (aliphatics is a disordered structure of coal substance, mainly contained in periphery-flank matrix; aromatics is an ordered carbon structure, composing crystallites). The other brighter sections of the image are shown in white color in Fig. 3, and they correspond to aromatics. When selecting variants withat the lower boundary of 30 – 45 and upper boundary of 80 – 100, we have the opposite relation between colors determining different phase components. The specific choice of boundaries resulted in the image in black-and-white color approximately corresponding to the relation of intensity in a color image, which was accepted for structures related to two phases: aliphatics and aromatics.

Fig. 3. Black-and-white image of phase composition of KZh coal surface, white color corresponds to an aromatic component of coal substance 239

The aliphatics to aromatics relation obtained by HarFa for KZh coal complies relatively well with the data reported by other authors who used conventional methods [12]. The aromatic component for KZh coal was 0.78 r 0.09. Data for Deformed Specimens

The image fractal analysis was also used for two KZh coal specimens subjected to uniaxial loading up to 2 GPa. The field of image was 1u1 ȝm. The analysis results obtained by the same scheme are given below in Table 2, Figs. 4 and 5. The linear fractal analysis shows that the structure of the surface image after the surface deformation has substantially less dimension as compared to an unloaded specimen (Table 2). The maximal fractal dimension obtained by the Range Method for the surface relief of specimen No. 1 is DBW 1.128 and DBW 1.573 for the phase (Figs. 4 and 5). DBW 1.222 and DBW = 1.641 are for specimen No. 2. It is important to note that two specimens of the same coal GRADE, loaded uniaxially, exhibit good match of fractal dimension values for the surface relief and different values for the phase composition, the shape of fractal spectra confirming this statement. The spectrum for the phase component of specimen No. 1 can be characterized as monofraɫtal, but it is soundly multifractal for specimen No. 2 and has the maximum at I | 95 . TABLE 2. Fractal Dimension, Spectrum Characteristics and Hurst Factors for KZh Coal Surface Structure after Uniaxial Loading No. of specimen

1 2

1 2

Component of APM measurements

Amplitude Phase Amplitude Phase Amplitude Phase Amplitude Phase

Box Method DBW Gf D 1.193 1.9035 1.0965 r 0.0326 2.329 1.3355 1.6645 r 0.0510 1.244 1.878 1.1220 r 0.036 0.978 2.012 0.9880 r 0.0293 Range Method for Fractal Spectrum 1.128 1.256 1.872 1.573 2.146 1.427 1.222 1.444 1.778 1.641 2.282 1.359

H 0.0965 0.6645 0.122 0.0

0.128 0.573 0.222 0.641

Note: For fractal dimension DBW, the confidence intervals are reported for 95 % probability

Fig 4. (a) Image of KZh coal surface relief after uniaxial loading up to 2 GPa and (b) its fractal spectrum; within I = 50 – 150, maximum DBW = 1.128 240

Fig 5. (a) Image of phase composition of KZh coal surface after uniaxial loading up to 2 GPa and (b) its fractal spectrum; within I = 75 – 150, maximum DBW = 1.573

Hurst Factor

Let us analyze the fractal dimension of the surface images for different coal grades from the standpoint of the fractal theory and theory of nonequilibrium dynamic systems. By [8] the fractal dimensions obtained for geometric objects ( d f ) and the dimensions determined by analyzing time series of different processes in nature, technics and biology ( G f ) are interrelated. At being so, the Hurst factor, which allows evaluation of stochasticity of a process and irregularity of the curve [13], can serve an essential characteristic of the nonequilibrium quasi-stationary state of a system [8]. Meaning of Hurst factor ɇ follows from the expression for the root-mean-square deviation of xt particle coordinates in molecular movement, that being a component of the expression for diffusion velocity: ( xt  xt  't ) 2

1/ 2

v 't H .

(5)

Ordinary diffusion has H 1/ 2 . The movement is jump-wise (irregular curve) if 0  H  1 and is smooth when H 1 for small time intervals 't . Value H 1/ 2 in (5) corresponds to random fluctuations in the system, probabilities of which are distributed by Gauss. For the Gauss time series, the Hausdorff dimension is G f 2  1/ 2 3 / 2 [8]. Considering the power spectrum for time series as power dependence Sp ~ 1/ f D , relation D 1 2H is valid [8]. Thereto, Hausdorff dynamic dimension G f coincides with the Graf dimension Dg introduced by Mandelbrott [11]. The Graf dimension for fractal molecular movement (5) is

determined by expression Dg

2  H . Then we have Dg

(5  D ) / 2 within 1  D  3 for the process

with the spectrum shape 1/ f D . It is essential to note that Dg o 1 for D t 3 , namely, the curve gets smooth. Dg

3 / 2 for the random Gauss process when H

0.5 , D 2 , so it is imperative to take these

relations into account when calculating the Hurst factor from Hausdorff’s structural dimensions. From the above we calculated Hurst factors ɇ, power spectrum degrees D , dynamic fractal dimension G f for coal surface images. Based on these parameters, a conclusion can be made on the orderliness degree for the hierarchic structure of coal substance and stochasticity of the equivalent time series (Tables 1 and 2). 241

DISCUSSION

Fractal dimensions obtained by analyzing the coal surface images by APM makes it possible to understand the change in the structure of coal substance in the metamorphism line and under highpressure loading. Actually, D, KZh fossil coals and anthracite are located in the metamorphism line in the direction of the rise of the aromatic component in the coal substance. In nature, coal structure forms under different conditions under variable high pressure and environment temperature factors. Therefore, the orderliness of the structure, mainly consisting of carbon atoms, improves during transition from brown coals to anthracite. The coal formation conditions should be taken into account in theoretical parameters, evaluating the fractal characteristics of the study specimens. The authors have analyzed the images with APM scanning area of 1u1 ȝm. This approach allows comparing fractal characteristics of images of the same pixels. The data obtained by the linear analysis (Box Method) and fractal spectra (Range Method) were used [6]. The tabulated maxima of Hausdorff’s structural dimensions for boundaries of surface structures, DBW , corresponding to the values of the image luminance I | 100 in a respective scale, were presented for the second method. The data in Tables 1 and 2 were calculated from DBW according to the expressions mentioned in the Hurst factor calculation procedure. Spectral index D determines the slope of the oscillation spectrum for the time series, plotted in the double logarithmic coordinates ln G  ln( f ) , where f is an oscillation frequency, G( f ) is a function of spectrum density. At D 2 a stochastic process is called the Gauss process. At D 1 , a stochastic process is known as a flicker noise. As the white noise corresponds to D 0 , the flicker noise is called “rose” noise, and Gauss’ noise is “brown”. There are cases when D ! 2 . Then a stochastic process is referred to a “dark-brown” or even “black” process [8]. So, value ɇ = 0.5 will comply with a stochastic boundary of the coal surface image structure having the Gauss distribution. In its turn, the tendency of parameter D to one becomes consistent with the non-equilibrium, a more stressed state of the structural organization of a substance. Comparison of The Surface Relief Data For Three Coal Specimens (Table 1). Both methods of evaluation of the fractal dimension show the conformity with the relative characteristics of coal specimens. As the Hurst factor for KZh coal is close to 0.5, the relief of its surface can be considered as the most irregular in stochastic terms and can be expressed as a spectral index value close to two (Table 1). Under similar consideration, the surface relief structure for D coal specimen could be estimated as the most ordered. Anthracite ranks an intermediate position. These conclusions refer to the scale range for heterogeneities, corresponding to maximum values of DBW . Comparison of Data on Coal Specimens Uniaxially Loaded up to 2 GPa (Table 2). The data obtained for two KZh coal specimens differ in a certain way. The linear analysis, corresponding to integral characteristics within the dimension interval ln(r )  2.082 y  0.693 , exhibited a sharp reduction in the Hausdorff structural fractal dimension down to one, thus, resulting in the approximation of the spectral index to one and the Hurst factor to zero. This suggests that the coal deformation changed the equilibrium structure of the coal medium as compared to the initial specimen and transferred it into an appreciably stressed state. The latter conclusion concerns the surface relief. As the fractal dimension tends to increase due to the fragmentation of the initial specimen into smaller objects under deformation of metal specimens [14], in our case, the reduction in the fractal dimension can be explained by agglomeration of coal powder under the uniaxial load up to 2 GPa in a high-pressure chamber. Actually, a larger-scale structure can be seen in images of a deformed specimen as compared 242

to an unstrained one (Fig. 4). Approximately 5-fold increase is observed for the characteristic scale of the structure image. Yet, the fractal dimension increased up to DBW 1.664 for the phase composition of specimen No. 1, moreover, the Hurst factor was close to 0.5, and the spectral index exceeded 2 (Table 2). This means that the structural components of the coal substance have accepted a less-ordered organization, and the specific dimensions of heterogeneities diminished. APM scanning of the specimen surface along spatial coordinate x permits recovering the efficient potential responsible for power characteristics of a probe, interacting with the surface. The numerical modeling assumes its scale (power) dependence in the space of wave numbers q 1/ 'x with factor E : c(q) q  E [15], being rather universal for real surfaces, when the factor ranges within 0.7  E  1.0 . Index q is taken for the wave vector that is related to spatial coordinate x of surface potential U (x) through its Fourier image: q2

U ( x) U 0 ³ c(q) cos( qx  M q )dq , q1

where M q is an oscillation phase. In a general case, potential U (x) is stochastic. The scaling concept of the Fourier image potential enables to simplify its reconstruction relative to the test surface relief. The section of 256u256 pixels was cut out to find factor E in the image of scanned coal specimen surface 1u1 ȝm in size, and the Fourier image of luminance distribution was calculated by lines along each cut, spectra being averaged. Factor E was evaluated by a slope of the straight line in graph logU (q)  log(q / qmax ) plotted in logarithmic coordinates. For three specimens of KZh, D, and A coals, the assumed values of E ranged from 0.827 r 0.105 to 0.947 r 0.091, namely, the surface potential, hence, force [9], the microscope probe interacts with the KZh coal surface, is evaluated by a wider interval of scales for a wave number as compared to anthracite ( E 0.947 ). Because of the reverse dependence between a wave number and linear dimensions, it can be taken that interaction of a probe with anthracite surface is distinguished for the characteristic scale of heterogeneities, being substantially less than those for KZh coal. The similar relationship of the characteristic scales for surface heterogeneities of the test coal specimens is confirmed by the data on the rough treatment of relief images. In fact, the mean square values for heterogeneities along z-coordinate are 4.61; 12.61, and 6.34 ȝm for KZh, D, and A coals, respectively. CONCLUSIONS

There are grounds to consider the undertaken analysis of APM images as an efficient approach to investigate the structure formation for a natural sorbent, fossil coal. Evaluating the fractal dimension for surface heterogeneities provides additional information of the hierarchic structure orderliness for the coal substance and makes it possible to reveal specific features of its organization for different specimens. The Hausdorff dimension, being an objective parameter, is obtained for the boundaries of different-scale heterogeneities and permits to estimate a level of the specimen structure organization within a large-scale range. It is important to emphasize that application of this method promotes the recovery of the surface potential, being the fundamental base of the scanning microscopy process. The variations of the fractal dimension of the surface structure under deformation implies on segregation or clustering of substance heterogeneities. The increase in the fractal dimension of the pore space surface has been obtained earlier from the data on small-angle X-ray dissemination for 243

T coal specimen [17]. The exponent for pore distribution by size equaled 3.95 for the initial specimen, and the exponent growth up to 4.56 was gained under the uniaxial compression up to 6 GPa due to decrease in the number of pores. Once Hausdorff dimension is related to factor B 2(d f  1) , it means that the initial value of d f

2.475 increased up to 2.780 after the deformation. Since the relief images

obtained by APM reveal the surface heterogeneities rather than the pore system of specimens, there is no contradiction between the variation of the fractal surface dimension for pores and heterogeneities that can be seen in 2D images, hence, we are dealing with different-scale objects. HarFa — Harmonic and Fractal Image Analysis, version 5.1 was applied to obtain data on processing of the specimen relief images. The authors are grateful to A. Filippov and A. Zavdoveev for the fruitful discussion of the research data, and to O. Litwin for the granted opportunity to undertake specimen scanning at V. E. Loshkarev’s Institute of Semiconductor Physics, National Academy of Sciences of Ukraine, Kiev.

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