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Electrochemical determination of the fractal dimension of fractured surfaces
Acta metall, mater. Vol. 40, No. 8, pp. 1819 1826, 1992 Printed in Great Britain. All rights reserved
0956-715192 $5.00 + 0.00 Copyright @ 1992 Pergamon Press Ltd
E L E C T R O C H E M I C A L D E T E R M I N A T I O N OF THE F R A C T A L D I M E N S I O N OF F R A C T U R E D S U R F A C E S A. IMRE, T. PAJKOSSY and L. NYIKOS Department of Chemistry, Central Research Institute for Physics, P.O.B. 49, H-1525 Budapest, Hungary (Received 7 October 1991; in revised forrn 7 February 1992)
Abstract--An electrochemical method, based on the analysis of the time dependence of the diffusional flux of molecular species to a surface, for accurately (~< _+0.02) determining the fractal dimension of rough surfaces (2 ~< D r < 2.4) in the 1 100/~m size range is described. We discuss how this method can be used for the measurement of the fractal dimension of fractured steel surfaces prepared by the Charpy impact test. R~sum~-On pr6sente une m6thode qui permet de d6terminer pr6cis6ment (~< + 0,02) la dimension fractale des surfaces rudes (2 ~< Df < 2,4) dans la gamme de largeur de 1 100/zm. Cette m~thode electrochimique repose sur l'analyse de diffusion des esp6ces moleculaires vers la surface, en fonction du temps. On discute la possibilit6 d'utiliser ladite m&hode pour la d6termination de dimension fractale des surfaces en aciers fractur~es, corrige6s par le test Charpy. Zusammenfassung Eine elektrochemische Methode, die auf der Analysis der Zeitabhfingigkeit des Diffusionsflusses yon Molekfilspecies zu einer Oberflfiche beruht, ist beschriebenen ffir die genauc Bestimmung (~< _+0.02) der Fraktaldimension yon groben Oberfl/ichen (2~
1. INTRODUCTION The characterization of the surface geometry of fractured surfaces has been attracting growing interest. The fractal dimension, Dr, of fractured surfaces, obtained for example by the Charpy impact test, was measured by several groups, either by analysing some cross-section profile [2, 3, 5] or by using a method called "slit-island" of ground surfaces [1, 2, 4-6], and correlation functions of bidimensional cuts [7]. Unfortunately, the accurracy of these methods is quite modest, the published error limits of Df (if specified at all) are in the order of 0.1. In addition, the determination of D~, by analyzing a profile or a cut seems to be controversial from a theoretical point of view [5, 7, 8]. In this paper, we describe an alternative, more accurate electrochemical method, based on the measurement of the time dependence of diffusion controlled currents and suitable for determining the fractal dimension of fractured surfaces of metals in the 1 100 ~ m size range. To demonstrate the feasibility of the proposed method, measurements were carried out employing fractured steel samples of various tempering temperatures. Correlations of Df and tempering temperature or impact energy were found and are reported. Nevertheless, the paper is
methodological in nature, and our aim is to introduce and suggest the use of this electrochemical method to metallurgists who seek correlations of Df and metallurgical parameters. The structure of this paper is the following: First, the theoretical background is summarized in Section 2. Section 3 deals with experimental details. In Section 4 we demonstrate the applicability of the method by a set of measurements made using gold replicas of fractured surfaces of various steel samples. In Section 5 we discuss the advantages and disadvantages of the new method and compare it with existing ones. Finally, in Section 6 we summarize the main findings and conclusions.
2. THEORY Diffusion controlled f l u x o f molecular or ionic species to an interface
Let us consider the following system: Assume an electrode (typically a metal) immersed into an electrolyte (typically an aqueous solution) containing some inert salt of high concentration and an electroactive (redox) substance of low concentration. The former ensures the high conductivity of the solution whereas the latter provides the possibility
1819
1820
IMRE et al.: THE FRACTAL DIMENSION OF FRACTURED SURFACES
of charge transfer across the electrode-electrolyte interface. Here we have to distinguish between Faradaic current and charging current. While the former is due to the charge transfer across the interface the latter causes charge accumulation in the double layer. Let the solution be homogeneous before "time zero" (t = 0) and let the electrode potential have such a value that no reaction proceeds at the interface and thus no net current flows across the cell. At t = 0 a potential jump is applied and therefore the electroactive substance of the solution becomes reactive. We assume that the following conditions hold: (i) The charge transfer rate is infinitely high following the potential jump. Thus, the redox substance gets depleted close to the surface where its concentration drops to zero and, consequently, the overall rate of the electrode reaction is controlled by the diffusion of the electroactive substance from the bulk of the solution to the interface. (ii) The solution resistance is zero, thus the voltage drop in the solution can be neglected. (iii) Electroactive molecules in the solution are transported only by diffusion; migration and convection effects are neglected. (iv) The interface is planar, thus diffusion proceeds along one spatial coordinate.
A
~ t 4 solution
~
~
ts
2 tl Z
B
-I-
2
-4 ~
-3
-2
._L.
. . _ ~ L ~
-1
O
If all the above conditions hold, then the time dependence of the current, L is described by the Cottrell equation [9] I = AnFcb~/tn
1
I o 9 za(t/s)
(1)
where A is the diffusion front area (in the planar case it is equal to the electrode surface area), nF is the charge needed for the electrochemical reaction of one mole of the electroactive substance, and cb and D denote the bulk concentration and the diffusion coefficient, respectively. This transient diffusion limited current will hereafter be referred to as the Cottrell current. Qualitatively, the decay of the current can be explained by the fact that the solution layer adjacent to the surface becomes depleted and this layer--called the diffusion layer--gets thicker and thicker and thus the concentration gradient, being the driving force of diffusion, decreases. In general, the diffusion layer width at time t is approximately 2,,/2D~. The surface area determination of a planar electrode is simple: by measuring the I(t) function and by using equation 1, A can be calculated provided n, cb and D are known. If the electrode surface is not planar, then the diffusion layer is not uniform along the surface [Fig. I(A)], except in the limits of very short and very long times [q and t4 in Fig. I(A), respectively]. In these time ranges the diffusion layer thickness is much smaller and larger than the smallest and largest irregularities, respectively, and thus, just like in the case of the planar diffusion, equation (1)
O.O -
\
C
dimension= 2.4
try o "...t
-1.0
-
-4
,
I
-3
,
I
-2 1 °al,~ o ( E / c m )
Fig. 1. (A) Concentration profiles as a function of time (t I < t2 < t~ < t4) in the vicinity of a rough (fractal) working electrode following a potential jump. The lines represent the border of the depleted zones. (B) Time dependence of the diffusion controlled current of a fractal electrode (solid line). The classical t -m behaviour prevailing in the short and long time limits are indicated by the dashed lines. (C) The theoretical area-yardstick length dependence of a fractal surface (solid line). Left, middle and right portions correspond to microscopic, fractal and macroscopic areas, respectively.
IMRE et al.: THE FRACTAL DIMENSION OF FRACTURED SURFACES holds. The diffusion front area, .4, to be used in equation (1) is the microscopic and macroscopic area, respectively. In the intermediate time range [t2 and t3 in Fig. I(A)], as time goes on, the diffusion layer smears out and A continuously changes. Then the diffusion layer width plays the role of a time-dependent yardstick length, e(t), and the Cottrell equation [equation (1)] can be generalized to the form I = A (e)nFc b x/~-/t~z.
(2)
Because the yardstick length depends on time, the I oct -~/2 decay is modified. As it can be shown [10-13] in the cases of fractal interfaces the overall time dependence changes to I o c t - ' , with = (Df-- 1)/2
(3)
where Of is the fractal dimension of the surface. By determining the exponent a from the log(/) vs log(t) plot, we can calculate the fractal dimension from equation (3). There is another way of analysing the I ( t ) dependence. By this, we express A(e) from equation (2), and get A (e) = I x x / ~ / ( n F C b ) .
(4)
This means that the transient current can be transformed to an apparent area and, taking e = ~ , we can determine surface area as function of yardstick length. The fractal dimension, Df, can be determined from the slope of log[A(e)] vs log[e] in accordance with the following equation [14] log[`4 (e)] = const - (2 - Df) × log(e).
(5)
Thus, if we measure the Cottrell current transient [Fig. I(B)], and transform it to log[A(e)] vs log(e) [Fig. I(C)] in accordance with equation (3) and by employing e = 2x/~-t, and using the simple approximation that this dependence is linear, the fractal dimension can be determined from the slope, utilizing equation (5). The above line of thought is fairly general: because no assumption on the actual geometry has been made, the result holds for all types of fractals. Real systems can be regarded to be fractals only in a certain size range limited by two cut-offs eminand emax. Beyond these limits the surface is smooth (two dimensional) and, accordingly, the time dependence of the Cottrell current before a certain shorter and after a certain longer time limit is of the t -1/2 form [equation (1)], and within this time range the dependence shows the t -" form, in accordance with equation (5) [Fig. I(B)]. The fractal description is an approximation of rough surfaces. If the dependence of log(area) vs log(yardstick length) is approximately linear in a certain size range, then the surface can be regarded as fractal in this range, otherwise the fractal approximation is not justified.
1821
Using the electrochemical method of area measurement one has to be aware of the following limitations: (a) According to assumption (i), the rate of chargc transfer on the metal surface should be infinite. In reality, if the rate is not sufficiently high, then the current at short times is smaller than the theoretical value predicted by equation (2). To minimize this effect, one should use a fast redox couple and should keep the electrode surface clean. (b) Another condition is that the solution resistance should be zero. In practice the finite solution conductivity puts an upper limit on the current. To minimize this effect, high inert salt concentration and small interelectrode distances are preferable. Conditions (a) and (b) both cause negative deviations from the theoretical Cottrell current at short times. (c) A positive deviation appears due to the charging current which is nearly independent of the concentrations. This, effect can be suppressed to some extent by increasing the ratio of the Faradaic current to the charging current by increasing the concentration of the redox substance. (d) Spontaneous convections always appear causing an extra flux of species to the interface thus increasing the current. This effect is pronounced at long times and limits the duration of the measurements to about 10 s. One can decreasse the disturbances caused by the convections by avoiding vibrations and by mounting the working electrode horizontally. 3. E X P E R I M E N T A L
Sample preparation Carbon steel specimens (DIN C45) were employed. These were quenched and tempered at various temperatures in the range from 200 to 600"~C for 30 rain, and broken by Charpy impact test. The impact energy varied with the different samples (see Table 1). As the scanning electrode microscope (SEM) pictures of Fig. 2 show, the fractured surfaces are not smooth. The Cottrell experiment requires samples which are inert electrochemically. Since pure gold is ideal for such purposes the first step was to prepare gold replicas of the fractured steel surfaces. High quality replicas could be obtained in the following way: the fractured faces were pressed into a 6ram x 6ram × 1 mm gold pellet by a hydraulic press with a typical pressure of 5 MPa. Such a Table 1. The temperingtemperature(T), impact energy (E) and fractal dimension (D0 of samples Sample T (C) E (J) Dr 1 -12 2.15+0.02 2 200 20 2.19_+0.02 3 300 17 2.20+_0.03 4 450 63 2.14 ± 0.(12 5 600 123 2.11 +0.02
1822
IMRE et al.: THE FRACTAL DIMENSION OF FRACTURED SURFACES
pressing makes the gold fill up perfectly also the voids of several micron size, as it is shown in Fig. 3. After the pressing procedure the gold was separated from the steel by anodically dissolving the latter in dilute (1 N) H2SO 4. The residual iron and its oxides were removed by boiling the gold sample in conc. HC1 for several minutes. To get a well defined macroscopic electrode surface area, discs of 3.5 mm diameter were cut from the middle part of the gold replicas by using a punch. In what follows we shall refer to these replicas as rough gold electrodes. Before the electrochemical measurements the following cleaning procedure was used: Organic contamination (e.g. fingerprints) was removed by organic solvents (such as CC14) and by keeping the samples in a furnace at 600°C for 15 min. Prior to each measurement the electrodes were soaked for 5 min in conc. HNO 3 then rinsed with and kept under distilled water. Cell design and solution composition
These electrochemical measurements require solutions containing only a small amount if any organic contaminant. Because of this, high purity distilled water rather than deionized water should be used. The inert salt acting as supporting electrolyte must
Fig. 2. SEM pictures a fractured steel surface (A) and of its gold replica (B).
Fig. 3. SEM pictures of tetragonal pits made on a steel surface by using the diamond pyramid of a Vickers microscope (A and B) and of the gold replica (C and D).
IMRE et al.:
THE FRACTAL DIMENSION OF FRACTURED SURFACES
electrode:
~~aa~oUr2t~d lectrode
1823
n
lttened nd
Insula,
O'7"ml
rking electrode: .5ram disc the gold replica
Plastic syringe
I Fig. 4. Electrochemical cell.
also be of high purity, preferably recrystallized, without electroactive contamination. We optimized the solution composition on the basis of the consideration of Section 2, points (i) and (ii). The solution contained 15 mM K4[Fe(CN6)] as the electroactive substance and 3 M NaNO3 as inert salt. The 15 mM is a concentration large enough to have a Cottrell current which is considerably larger than the charging current. The rate of the ferro-ferricyanide transformation is also high enough and the resistance of the solution--with the cell construction described below--low enough to obtain good measurements between 1 ms and 10 s. The electrochemical measurements were carried out in a special three electrode cell (illustrated in Fig. 4). The distance between the working electrode and the counter electrode could be set to be smaller than 1 mm. In this case the resistance of the solution is sufficiently small (around 1 f2), but still satisfies the condition of semi-infinite diffusion in the timeframe of the measurement. The block diagram of the measurement system is shown in Fig. 5. The cell was connected to a potentiostat (type Electroflex, Hungarian make). A digitalto-analog converter (DAC) provided the voltage jump for the potentiostat. The setup contained two voltmeters: a Keithley 194A high speed voltmeter with 16 bits resolution was used as a current meter by measuring the voltage drop across a Rm = 5.0 f~ resistor; another instrument, a Keithley 192 digital multimeter was used for calibration and offset correction purposes. Both voltmeters and the DAC were controlled by a Hewlett-Packard 86A desktop computer via its GPIB interface. In the measurements, the potential was switched from 0 mV to 650 mV vs saturated calomel electrode and after 10 s back to 0 inV. The first jump triggered the high speed voltmeter which measured the current and stored the digitized values in its internal 64 K
AM40,8(?
buffer. The data were read out by the controller after the end of the measurement. The Keithley 192 voltmeter measured the output voltage of the current-tovoltage converter of the potentiostat before the jump, thus the offset of the Keithley 194A could be corrected for. The current transient raw data points (their number being 32738) were compressed (averaged) in such a way that the points become equidistant on a logarithmic timescale. 4. RESULTS 1. We calibrated the setup by using smooth, polished gold samples. A typical measured current vs time plot is shown in Fig. 6(A). The solid line shows the current predicted by equation 1, assuming a diffusion coefficient of 5 x 10 6cmZ/s. In the following parts we use this value. Note that any error of this value has virtualy no influence on the determination of fractal dimension, because the latter is related to the slope of the line. Although the points of Fig. 6(A) fit almost perfectly to the line of slope - 1 / 2 , a transformation according to equation (3) [displayed in Fig. 6(B)] shows
GPB I Potentiost at
CELL ~
@
.L--
Wor king
GPIB Controller
Fig. 5. Block diagram of measurement setup. Abbreviations: DAC: digital-to-analog converter, DVM: Digital voltmeter, GPIB: General Purpose Interface Bus.
1824
IMRE
et al.:
THE FRACTAL DIMENSION OF FRACTURED SURFACES
that the area has a weak dependence on the yardstick length. From the slope, the fractal dimension is determined to be 2.02. As the results of a number of such measurements made on rough gold electrodes show, the reproducibility of this value is about -I-0.02 in the 5-100/~m size range (Fig. 7). Beyond these limits the reproducibility is somewhat poorer, because the measured current vs time curves are rather sensitive to contaminations and to vibrations. 2. We measured the time dependence of the current on the gold replicas. As Fig. 6(C) shows, the dependence is steeper than the /,-1/2 o n e indicated by a dotted line for comparison. The slope of full line is - 0 . 5 9 from which, in accordance with equation (5), Df = 2.19 is found. Accordingly, on the log(area) vs log(t) plot of Fig. 6(D), the slope of the solid line is - 0 . 1 9 , corresponding to D f = 2.19. Note that the dependence can be well approximated as linear, there-
A
"",...., -s-3
u~
mension = 2.02
)
-2 " ~
2
fore this particular fractured surface can indeed be regarded to be fractal between 5 and 100 #m. 3. Similar linear log(A) vs log(e) plots were obtained on other rough gold samples. The fractal dimensions are listed in Table 1. 4. As we found in the preliminary measurements, the different regions of fractured surfaces may have different character of roughness and, consequently, different fractal dimension. This might be one of the reasons why the results found in the literature are sometimes at variance [1, 5, 7]. Since we did not aim to study the lateral dependence of roughness, we prepared the samples by cutting out and testing only the middle part of the replicas. 5. In order to facilitate comparison between our fractal dimension data and those published by others, we plot the fractal dimension as a function of impact energy and of tempering temperatures. We observed
-z
-1
i 2
o
-4
-2
-3
1Ogjo(E/cra)
1o 9 jo(t/s;
C
dimension
= 2.1g
2
-4
-%3
-1.0
-2
-1
0 J o 9 ~o(t/s~
-
1
.
2
-3
~
-2
1 o91o(-C/cra)
Fig. 6. Diffusion controlled current on smooth (A) and rough (C) samples and the area-yardstick length dependences for smooth (B) and rough (D) cases. The rough electrode is sample No. 2.
IMRE et al.: THE FRACTAL DIMENSION OF FRACTURED SURFACES a weak decrease of fractal dimension with the increase of impact energy [Fig. 8(A)]. N o t e that Mandelbrot et al. [1] obtained a stronger dependence, Pande et al. [5] did not obtain any dependence, and Bouchard et al. [6] published a fractal dimension of 2.2 + 0.1 which was reported to be independent of impact energy. Our results are also in this region: a weighted least mean squares fit of the points of Fig. 8(A) gives a slope which does not significantly differ from zero. The weighted average of the points gives an average of 2.17 _ 0.04. Similarly, the fractal dimension shows weak if any dependence on the tempering temperature [Fig. 8(B)].
A 2.2
I
5. DISCUSSION
2.0
Advantages and disadz~antages o f the method In order to compare this method of determination of the fractal dimension with other ones we emphasize that the diffusion layer width acts as a time-dependent yardstick length, and thus it complies with the definition of the Hausdorff dimension [12]. This is a great advantage over the methods based on the analysis of the distribution of some length of some cross-section. For example, one can determine the fractal dimension from the slope of the power spectrum of a cross-sectional profile [1, 2, 5, 10, 12]. However, this method must be used with caution because the slope of a power spectrum does not depend on the magnitude of the peaks of the profile whereas the fractal dimension obviously does. Similar and other problems appear if the size distribution of horizontal cuts (the "slit islands") are analysed [4, 5].
\
L~
, I
I I
20
J
I
40
,
J
60
,
I
80
,
I
,
]00
I
120
Impoct energy
[J]
B
o C 2.2
2. E
1825
O0
200
400
600
Temperature [~C] Fig. 8. Fractal dimension as a function of impact energy (A) and as a function of tempering temperature (B).
o.T - . 7 o
"-'---t
-.9
-1.1
4
-3
-2
i oglo(£/cm) Fig. 7. Reproducibility of area-yardstick length dependences for a rough electrode. Curves 1-4 are from measurements before which the electrode was cleaned as described in Section 3. Note the parallelism of curves 1 to 4. The importance of the proper cleaning is illustrated by curve 5: it is the result of a measurement when fingerprints were intentionally left on the electrode surface.
The method requires a gold replica of the rough surface. In cases of fractured surfaces of hard metals it is straightforward to press the metal into a gold pellet, however, other methods can be found: for example, for less hard substances one can attempt to prepare plastic replicas and to cover them with a gold layer by using vacuum evaporation. The fact that a gold replica must be prepared sets an upper limit on emin, because one cannot press gold into the very small pores and voids. We estimate that the upper limit of the fractal dimension is around 2.3 or 2.4. Even if a perfect replica of an electrode of highly ramified structure were available, the effect of the nonzero solution resistance would greatly influence the Cottrell current in the ms range, i.e. in the # m region.
1826
IMRE et al.: THE FRACTAL DIMENSION OF FRACTURED SURFACES 6. SUMMARY AND CONCLUSIONS
A new experimental method has been developed for determining the fractual dimension of rough surfaces (2 ~< D f < 2.4) in the 1-100 # m size range. This method can be applied for fractured steel surfaces by using a replica preparation technique. Its applicability was demonstrated by measuring the fractal dimension of fractured carbon steel surfaces prepared by the Charpy impact test. The measured fractal dimensions are in the range of 2.1 and 2.2. We found that the results depend on which part of the broken surface is analysed, because the fractal dimension of these samples is apparently not uniform laterally. The measured fractal dimensions showed weak if any dependence on tempering temperature and impact energy. However, we feel that the question whether the fractal dimension of fractured surfaces does depend on these and on other metallurgical and solid state parameters or not is still an open one. To give a definite answer, further systematic studies on various metals and numerous samples employing precise methods of surface area measurements like the one described in this paper are required. Acknowledgements--The authors are indebted to Dr E. Czoboly for the steel samples, to Dr Z. V6rtesy for the
micrographs, and to Dr R. Schiller for his continuous interest during the work. Support for this work by the Hungarian research foundation OTKA under Contract No. 1840 is gratefully acknowledged.
REFERENCES
I. B. B. Mandelbrot, D. E. Passoja and A. J. Paullay, Nature 308, 721 (1984). 2. E. E. Underwood and K. Banerji, Mater. Sci. Engng 80, 1 (1986). 3. Z. G. Wang, D. I. Chen, X. X. Jiang, S. H. Ai and C. H. Shih, Scripta metall. 22, 827 (1988). 4. C. W. Lung and S. Z. Zhang, Physica D 38, 242 (1989). 5. C. S. Pande, L. E. Richards, N. Louat, B. D. Dempsey and A. J. Schwoeble, Acta metall. 35, 1633 (1987). 6. C. W. Lung and Z. Q. Mu, Phys. Rev. B 38, 11781 (1988). 7. E. Bouchaud, G. Lapasset and J. Planes, Europhys. Lett. 13, 73 (1990). 8. L. V. Meisel, J. Phys. D. Appl. Phys. 24, 942 (1991). 9. A. J. Bard and L. R. Faulkner, Electrochemical Methods, Chap. 5. Wiley, New York (1980). 10. L. Nyikos and T. Pajkossy, Electrochim. Acta 31, 1347 (1985). ll. T. Pajkossy and L. Nyikos, ibid. 34, 171 (1989). 12. A. P. Borosy, L. Nyikos and T. Pajkossy, Electrochim. Acta 36, 163 (1991). 13. T. Pajkossy, J. Electroanal. Chem. 300, 1 (1991). 14. B. B. Mandelbrot, The Fractal Geometry of Nature. Freeman, San Francisco (1982).
0956-715192 $5.00 + 0.00 Copyright @ 1992 Pergamon Press Ltd
E L E C T R O C H E M I C A L D E T E R M I N A T I O N OF THE F R A C T A L D I M E N S I O N OF F R A C T U R E D S U R F A C E S A. IMRE, T. PAJKOSSY and L. NYIKOS Department of Chemistry, Central Research Institute for Physics, P.O.B. 49, H-1525 Budapest, Hungary (Received 7 October 1991; in revised forrn 7 February 1992)
Abstract--An electrochemical method, based on the analysis of the time dependence of the diffusional flux of molecular species to a surface, for accurately (~< _+0.02) determining the fractal dimension of rough surfaces (2 ~< D r < 2.4) in the 1 100/~m size range is described. We discuss how this method can be used for the measurement of the fractal dimension of fractured steel surfaces prepared by the Charpy impact test. R~sum~-On pr6sente une m6thode qui permet de d6terminer pr6cis6ment (~< + 0,02) la dimension fractale des surfaces rudes (2 ~< Df < 2,4) dans la gamme de largeur de 1 100/zm. Cette m~thode electrochimique repose sur l'analyse de diffusion des esp6ces moleculaires vers la surface, en fonction du temps. On discute la possibilit6 d'utiliser ladite m&hode pour la d6termination de dimension fractale des surfaces en aciers fractur~es, corrige6s par le test Charpy. Zusammenfassung Eine elektrochemische Methode, die auf der Analysis der Zeitabhfingigkeit des Diffusionsflusses yon Molekfilspecies zu einer Oberflfiche beruht, ist beschriebenen ffir die genauc Bestimmung (~< _+0.02) der Fraktaldimension yon groben Oberfl/ichen (2~
1. INTRODUCTION The characterization of the surface geometry of fractured surfaces has been attracting growing interest. The fractal dimension, Dr, of fractured surfaces, obtained for example by the Charpy impact test, was measured by several groups, either by analysing some cross-section profile [2, 3, 5] or by using a method called "slit-island" of ground surfaces [1, 2, 4-6], and correlation functions of bidimensional cuts [7]. Unfortunately, the accurracy of these methods is quite modest, the published error limits of Df (if specified at all) are in the order of 0.1. In addition, the determination of D~, by analyzing a profile or a cut seems to be controversial from a theoretical point of view [5, 7, 8]. In this paper, we describe an alternative, more accurate electrochemical method, based on the measurement of the time dependence of diffusion controlled currents and suitable for determining the fractal dimension of fractured surfaces of metals in the 1 100 ~ m size range. To demonstrate the feasibility of the proposed method, measurements were carried out employing fractured steel samples of various tempering temperatures. Correlations of Df and tempering temperature or impact energy were found and are reported. Nevertheless, the paper is
methodological in nature, and our aim is to introduce and suggest the use of this electrochemical method to metallurgists who seek correlations of Df and metallurgical parameters. The structure of this paper is the following: First, the theoretical background is summarized in Section 2. Section 3 deals with experimental details. In Section 4 we demonstrate the applicability of the method by a set of measurements made using gold replicas of fractured surfaces of various steel samples. In Section 5 we discuss the advantages and disadvantages of the new method and compare it with existing ones. Finally, in Section 6 we summarize the main findings and conclusions.
2. THEORY Diffusion controlled f l u x o f molecular or ionic species to an interface
Let us consider the following system: Assume an electrode (typically a metal) immersed into an electrolyte (typically an aqueous solution) containing some inert salt of high concentration and an electroactive (redox) substance of low concentration. The former ensures the high conductivity of the solution whereas the latter provides the possibility
1819
1820
IMRE et al.: THE FRACTAL DIMENSION OF FRACTURED SURFACES
of charge transfer across the electrode-electrolyte interface. Here we have to distinguish between Faradaic current and charging current. While the former is due to the charge transfer across the interface the latter causes charge accumulation in the double layer. Let the solution be homogeneous before "time zero" (t = 0) and let the electrode potential have such a value that no reaction proceeds at the interface and thus no net current flows across the cell. At t = 0 a potential jump is applied and therefore the electroactive substance of the solution becomes reactive. We assume that the following conditions hold: (i) The charge transfer rate is infinitely high following the potential jump. Thus, the redox substance gets depleted close to the surface where its concentration drops to zero and, consequently, the overall rate of the electrode reaction is controlled by the diffusion of the electroactive substance from the bulk of the solution to the interface. (ii) The solution resistance is zero, thus the voltage drop in the solution can be neglected. (iii) Electroactive molecules in the solution are transported only by diffusion; migration and convection effects are neglected. (iv) The interface is planar, thus diffusion proceeds along one spatial coordinate.
A
~ t 4 solution
~
~
ts
2 tl Z
B
-I-
2
-4 ~
-3
-2
._L.
. . _ ~ L ~
-1
O
If all the above conditions hold, then the time dependence of the current, L is described by the Cottrell equation [9] I = AnFcb~/tn
1
I o 9 za(t/s)
(1)
where A is the diffusion front area (in the planar case it is equal to the electrode surface area), nF is the charge needed for the electrochemical reaction of one mole of the electroactive substance, and cb and D denote the bulk concentration and the diffusion coefficient, respectively. This transient diffusion limited current will hereafter be referred to as the Cottrell current. Qualitatively, the decay of the current can be explained by the fact that the solution layer adjacent to the surface becomes depleted and this layer--called the diffusion layer--gets thicker and thicker and thus the concentration gradient, being the driving force of diffusion, decreases. In general, the diffusion layer width at time t is approximately 2,,/2D~. The surface area determination of a planar electrode is simple: by measuring the I(t) function and by using equation 1, A can be calculated provided n, cb and D are known. If the electrode surface is not planar, then the diffusion layer is not uniform along the surface [Fig. I(A)], except in the limits of very short and very long times [q and t4 in Fig. I(A), respectively]. In these time ranges the diffusion layer thickness is much smaller and larger than the smallest and largest irregularities, respectively, and thus, just like in the case of the planar diffusion, equation (1)
O.O -
\
C
dimension= 2.4
try o "...t
-1.0
-
-4
,
I
-3
,
I
-2 1 °al,~ o ( E / c m )
Fig. 1. (A) Concentration profiles as a function of time (t I < t2 < t~ < t4) in the vicinity of a rough (fractal) working electrode following a potential jump. The lines represent the border of the depleted zones. (B) Time dependence of the diffusion controlled current of a fractal electrode (solid line). The classical t -m behaviour prevailing in the short and long time limits are indicated by the dashed lines. (C) The theoretical area-yardstick length dependence of a fractal surface (solid line). Left, middle and right portions correspond to microscopic, fractal and macroscopic areas, respectively.
IMRE et al.: THE FRACTAL DIMENSION OF FRACTURED SURFACES holds. The diffusion front area, .4, to be used in equation (1) is the microscopic and macroscopic area, respectively. In the intermediate time range [t2 and t3 in Fig. I(A)], as time goes on, the diffusion layer smears out and A continuously changes. Then the diffusion layer width plays the role of a time-dependent yardstick length, e(t), and the Cottrell equation [equation (1)] can be generalized to the form I = A (e)nFc b x/~-/t~z.
(2)
Because the yardstick length depends on time, the I oct -~/2 decay is modified. As it can be shown [10-13] in the cases of fractal interfaces the overall time dependence changes to I o c t - ' , with = (Df-- 1)/2
(3)
where Of is the fractal dimension of the surface. By determining the exponent a from the log(/) vs log(t) plot, we can calculate the fractal dimension from equation (3). There is another way of analysing the I ( t ) dependence. By this, we express A(e) from equation (2), and get A (e) = I x x / ~ / ( n F C b ) .
(4)
This means that the transient current can be transformed to an apparent area and, taking e = ~ , we can determine surface area as function of yardstick length. The fractal dimension, Df, can be determined from the slope of log[A(e)] vs log[e] in accordance with the following equation [14] log[`4 (e)] = const - (2 - Df) × log(e).
(5)
Thus, if we measure the Cottrell current transient [Fig. I(B)], and transform it to log[A(e)] vs log(e) [Fig. I(C)] in accordance with equation (3) and by employing e = 2x/~-t, and using the simple approximation that this dependence is linear, the fractal dimension can be determined from the slope, utilizing equation (5). The above line of thought is fairly general: because no assumption on the actual geometry has been made, the result holds for all types of fractals. Real systems can be regarded to be fractals only in a certain size range limited by two cut-offs eminand emax. Beyond these limits the surface is smooth (two dimensional) and, accordingly, the time dependence of the Cottrell current before a certain shorter and after a certain longer time limit is of the t -1/2 form [equation (1)], and within this time range the dependence shows the t -" form, in accordance with equation (5) [Fig. I(B)]. The fractal description is an approximation of rough surfaces. If the dependence of log(area) vs log(yardstick length) is approximately linear in a certain size range, then the surface can be regarded as fractal in this range, otherwise the fractal approximation is not justified.
1821
Using the electrochemical method of area measurement one has to be aware of the following limitations: (a) According to assumption (i), the rate of chargc transfer on the metal surface should be infinite. In reality, if the rate is not sufficiently high, then the current at short times is smaller than the theoretical value predicted by equation (2). To minimize this effect, one should use a fast redox couple and should keep the electrode surface clean. (b) Another condition is that the solution resistance should be zero. In practice the finite solution conductivity puts an upper limit on the current. To minimize this effect, high inert salt concentration and small interelectrode distances are preferable. Conditions (a) and (b) both cause negative deviations from the theoretical Cottrell current at short times. (c) A positive deviation appears due to the charging current which is nearly independent of the concentrations. This, effect can be suppressed to some extent by increasing the ratio of the Faradaic current to the charging current by increasing the concentration of the redox substance. (d) Spontaneous convections always appear causing an extra flux of species to the interface thus increasing the current. This effect is pronounced at long times and limits the duration of the measurements to about 10 s. One can decreasse the disturbances caused by the convections by avoiding vibrations and by mounting the working electrode horizontally. 3. E X P E R I M E N T A L
Sample preparation Carbon steel specimens (DIN C45) were employed. These were quenched and tempered at various temperatures in the range from 200 to 600"~C for 30 rain, and broken by Charpy impact test. The impact energy varied with the different samples (see Table 1). As the scanning electrode microscope (SEM) pictures of Fig. 2 show, the fractured surfaces are not smooth. The Cottrell experiment requires samples which are inert electrochemically. Since pure gold is ideal for such purposes the first step was to prepare gold replicas of the fractured steel surfaces. High quality replicas could be obtained in the following way: the fractured faces were pressed into a 6ram x 6ram × 1 mm gold pellet by a hydraulic press with a typical pressure of 5 MPa. Such a Table 1. The temperingtemperature(T), impact energy (E) and fractal dimension (D0 of samples Sample T (C) E (J) Dr 1 -12 2.15+0.02 2 200 20 2.19_+0.02 3 300 17 2.20+_0.03 4 450 63 2.14 ± 0.(12 5 600 123 2.11 +0.02
1822
IMRE et al.: THE FRACTAL DIMENSION OF FRACTURED SURFACES
pressing makes the gold fill up perfectly also the voids of several micron size, as it is shown in Fig. 3. After the pressing procedure the gold was separated from the steel by anodically dissolving the latter in dilute (1 N) H2SO 4. The residual iron and its oxides were removed by boiling the gold sample in conc. HC1 for several minutes. To get a well defined macroscopic electrode surface area, discs of 3.5 mm diameter were cut from the middle part of the gold replicas by using a punch. In what follows we shall refer to these replicas as rough gold electrodes. Before the electrochemical measurements the following cleaning procedure was used: Organic contamination (e.g. fingerprints) was removed by organic solvents (such as CC14) and by keeping the samples in a furnace at 600°C for 15 min. Prior to each measurement the electrodes were soaked for 5 min in conc. HNO 3 then rinsed with and kept under distilled water. Cell design and solution composition
These electrochemical measurements require solutions containing only a small amount if any organic contaminant. Because of this, high purity distilled water rather than deionized water should be used. The inert salt acting as supporting electrolyte must
Fig. 2. SEM pictures a fractured steel surface (A) and of its gold replica (B).
Fig. 3. SEM pictures of tetragonal pits made on a steel surface by using the diamond pyramid of a Vickers microscope (A and B) and of the gold replica (C and D).
IMRE et al.:
THE FRACTAL DIMENSION OF FRACTURED SURFACES
electrode:
~~aa~oUr2t~d lectrode
1823
n
lttened nd
Insula,
O'7"ml
rking electrode: .5ram disc the gold replica
Plastic syringe
I Fig. 4. Electrochemical cell.
also be of high purity, preferably recrystallized, without electroactive contamination. We optimized the solution composition on the basis of the consideration of Section 2, points (i) and (ii). The solution contained 15 mM K4[Fe(CN6)] as the electroactive substance and 3 M NaNO3 as inert salt. The 15 mM is a concentration large enough to have a Cottrell current which is considerably larger than the charging current. The rate of the ferro-ferricyanide transformation is also high enough and the resistance of the solution--with the cell construction described below--low enough to obtain good measurements between 1 ms and 10 s. The electrochemical measurements were carried out in a special three electrode cell (illustrated in Fig. 4). The distance between the working electrode and the counter electrode could be set to be smaller than 1 mm. In this case the resistance of the solution is sufficiently small (around 1 f2), but still satisfies the condition of semi-infinite diffusion in the timeframe of the measurement. The block diagram of the measurement system is shown in Fig. 5. The cell was connected to a potentiostat (type Electroflex, Hungarian make). A digitalto-analog converter (DAC) provided the voltage jump for the potentiostat. The setup contained two voltmeters: a Keithley 194A high speed voltmeter with 16 bits resolution was used as a current meter by measuring the voltage drop across a Rm = 5.0 f~ resistor; another instrument, a Keithley 192 digital multimeter was used for calibration and offset correction purposes. Both voltmeters and the DAC were controlled by a Hewlett-Packard 86A desktop computer via its GPIB interface. In the measurements, the potential was switched from 0 mV to 650 mV vs saturated calomel electrode and after 10 s back to 0 inV. The first jump triggered the high speed voltmeter which measured the current and stored the digitized values in its internal 64 K
AM40,8(?
buffer. The data were read out by the controller after the end of the measurement. The Keithley 192 voltmeter measured the output voltage of the current-tovoltage converter of the potentiostat before the jump, thus the offset of the Keithley 194A could be corrected for. The current transient raw data points (their number being 32738) were compressed (averaged) in such a way that the points become equidistant on a logarithmic timescale. 4. RESULTS 1. We calibrated the setup by using smooth, polished gold samples. A typical measured current vs time plot is shown in Fig. 6(A). The solid line shows the current predicted by equation 1, assuming a diffusion coefficient of 5 x 10 6cmZ/s. In the following parts we use this value. Note that any error of this value has virtualy no influence on the determination of fractal dimension, because the latter is related to the slope of the line. Although the points of Fig. 6(A) fit almost perfectly to the line of slope - 1 / 2 , a transformation according to equation (3) [displayed in Fig. 6(B)] shows
GPB I Potentiost at
CELL ~
@
.L--
Wor king
GPIB Controller
Fig. 5. Block diagram of measurement setup. Abbreviations: DAC: digital-to-analog converter, DVM: Digital voltmeter, GPIB: General Purpose Interface Bus.
1824
IMRE
et al.:
THE FRACTAL DIMENSION OF FRACTURED SURFACES
that the area has a weak dependence on the yardstick length. From the slope, the fractal dimension is determined to be 2.02. As the results of a number of such measurements made on rough gold electrodes show, the reproducibility of this value is about -I-0.02 in the 5-100/~m size range (Fig. 7). Beyond these limits the reproducibility is somewhat poorer, because the measured current vs time curves are rather sensitive to contaminations and to vibrations. 2. We measured the time dependence of the current on the gold replicas. As Fig. 6(C) shows, the dependence is steeper than the /,-1/2 o n e indicated by a dotted line for comparison. The slope of full line is - 0 . 5 9 from which, in accordance with equation (5), Df = 2.19 is found. Accordingly, on the log(area) vs log(t) plot of Fig. 6(D), the slope of the solid line is - 0 . 1 9 , corresponding to D f = 2.19. Note that the dependence can be well approximated as linear, there-
A
"",...., -s-3
u~
mension = 2.02
)
-2 " ~
2
fore this particular fractured surface can indeed be regarded to be fractal between 5 and 100 #m. 3. Similar linear log(A) vs log(e) plots were obtained on other rough gold samples. The fractal dimensions are listed in Table 1. 4. As we found in the preliminary measurements, the different regions of fractured surfaces may have different character of roughness and, consequently, different fractal dimension. This might be one of the reasons why the results found in the literature are sometimes at variance [1, 5, 7]. Since we did not aim to study the lateral dependence of roughness, we prepared the samples by cutting out and testing only the middle part of the replicas. 5. In order to facilitate comparison between our fractal dimension data and those published by others, we plot the fractal dimension as a function of impact energy and of tempering temperatures. We observed
-z
-1
i 2
o
-4
-2
-3
1Ogjo(E/cra)
1o 9 jo(t/s;
C
dimension
= 2.1g
2
-4
-%3
-1.0
-2
-1
0 J o 9 ~o(t/s~
-
1
.
2
-3
~
-2
1 o91o(-C/cra)
Fig. 6. Diffusion controlled current on smooth (A) and rough (C) samples and the area-yardstick length dependences for smooth (B) and rough (D) cases. The rough electrode is sample No. 2.
IMRE et al.: THE FRACTAL DIMENSION OF FRACTURED SURFACES a weak decrease of fractal dimension with the increase of impact energy [Fig. 8(A)]. N o t e that Mandelbrot et al. [1] obtained a stronger dependence, Pande et al. [5] did not obtain any dependence, and Bouchard et al. [6] published a fractal dimension of 2.2 + 0.1 which was reported to be independent of impact energy. Our results are also in this region: a weighted least mean squares fit of the points of Fig. 8(A) gives a slope which does not significantly differ from zero. The weighted average of the points gives an average of 2.17 _ 0.04. Similarly, the fractal dimension shows weak if any dependence on the tempering temperature [Fig. 8(B)].
A 2.2
I
5. DISCUSSION
2.0
Advantages and disadz~antages o f the method In order to compare this method of determination of the fractal dimension with other ones we emphasize that the diffusion layer width acts as a time-dependent yardstick length, and thus it complies with the definition of the Hausdorff dimension [12]. This is a great advantage over the methods based on the analysis of the distribution of some length of some cross-section. For example, one can determine the fractal dimension from the slope of the power spectrum of a cross-sectional profile [1, 2, 5, 10, 12]. However, this method must be used with caution because the slope of a power spectrum does not depend on the magnitude of the peaks of the profile whereas the fractal dimension obviously does. Similar and other problems appear if the size distribution of horizontal cuts (the "slit islands") are analysed [4, 5].
\
L~
, I
I I
20
J
I
40
,
J
60
,
I
80
,
I
,
]00
I
120
Impoct energy
[J]
B
o C 2.2
2. E
1825
O0
200
400
600
Temperature [~C] Fig. 8. Fractal dimension as a function of impact energy (A) and as a function of tempering temperature (B).
o.T - . 7 o
"-'---t
-.9
-1.1
4
-3
-2
i oglo(£/cm) Fig. 7. Reproducibility of area-yardstick length dependences for a rough electrode. Curves 1-4 are from measurements before which the electrode was cleaned as described in Section 3. Note the parallelism of curves 1 to 4. The importance of the proper cleaning is illustrated by curve 5: it is the result of a measurement when fingerprints were intentionally left on the electrode surface.
The method requires a gold replica of the rough surface. In cases of fractured surfaces of hard metals it is straightforward to press the metal into a gold pellet, however, other methods can be found: for example, for less hard substances one can attempt to prepare plastic replicas and to cover them with a gold layer by using vacuum evaporation. The fact that a gold replica must be prepared sets an upper limit on emin, because one cannot press gold into the very small pores and voids. We estimate that the upper limit of the fractal dimension is around 2.3 or 2.4. Even if a perfect replica of an electrode of highly ramified structure were available, the effect of the nonzero solution resistance would greatly influence the Cottrell current in the ms range, i.e. in the # m region.
1826
IMRE et al.: THE FRACTAL DIMENSION OF FRACTURED SURFACES 6. SUMMARY AND CONCLUSIONS
A new experimental method has been developed for determining the fractual dimension of rough surfaces (2 ~< D f < 2.4) in the 1-100 # m size range. This method can be applied for fractured steel surfaces by using a replica preparation technique. Its applicability was demonstrated by measuring the fractal dimension of fractured carbon steel surfaces prepared by the Charpy impact test. The measured fractal dimensions are in the range of 2.1 and 2.2. We found that the results depend on which part of the broken surface is analysed, because the fractal dimension of these samples is apparently not uniform laterally. The measured fractal dimensions showed weak if any dependence on tempering temperature and impact energy. However, we feel that the question whether the fractal dimension of fractured surfaces does depend on these and on other metallurgical and solid state parameters or not is still an open one. To give a definite answer, further systematic studies on various metals and numerous samples employing precise methods of surface area measurements like the one described in this paper are required. Acknowledgements--The authors are indebted to Dr E. Czoboly for the steel samples, to Dr Z. V6rtesy for the
micrographs, and to Dr R. Schiller for his continuous interest during the work. Support for this work by the Hungarian research foundation OTKA under Contract No. 1840 is gratefully acknowledged.
REFERENCES
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