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Electrochemical technique for the determination of fractal dimension of dental surfaces
Colloids and Surfaces B: Biointerfaces 32 (2003) 375 /381 www.elsevier.com/locate/colsurfb
Electrochemical technique for the determination of fractal dimension of dental surfaces Ali Eftekhari * Electrochemical Research Center, P.O. Box 19395-5139, Tehran, Iran Received 26 May 2003; received in revised form 1 July 2003; accepted 6 August 2003
Abstract Fractal dimension of a carious tooth surface was determined using an electrochemical method. The method was based on time-dependent diffusion towards electrode surfaces, which is one of the most useful and reliable methods for the determination of fractal dimension of electrode surfaces. For this purpose, the tooth was covered with a gold layer, which acted as an electrode in electrochemical experiments. It is suggested that the fractal dimension can be used as a quantitative measure of the state of dental surfaces. The method presented demonstrates the power of electrochemical techniques for the determination of fractal dimension of surface of non-conducting objects. # 2003 Elsevier B.V. All rights reserved. Keywords: Fractal surface; Electrodeposition; Non-conducting substrate; Gold-coated electrode; Tooth surfaces; Dental applications
1. Introduction Since the revolutionary discovery of fractal geometry by Mandelbrot [1], it has been used for analysis of different objects in various branches of science and technology [2 /7]. It has also been successfully proposed in electrochemical science [8 /29], due to the essential role of electrode surfaces and interfacial problems in electrochemistry. Several methods have been proposed for the determination of fractal dimension of electrode * Corresponding author. Tel.: /98-21-204-2549; fax: /9821-205-7621. E-mail address: [email protected] (A. Eftekhari).
surfaces by means of electrochemical techniques. Fractal dimension is a quantitative parameter for analysis of fractal objects, which is widely used for comparative investigations of fractal objects. It is also a powerful quantitative factor for analysis of rough surface structures. In the present work, it is aimed to report fractality of dental surfaces for the first time and application of fractal dimension as a measure for the investigation of dental surfaces. The methodology is of practical interest, as surface analysis of teeth can be used in dental science. It is also a useful approach for studies of dental materials to find biomaterials similar to natural teeth (from a surface structure point of view).
0927-7765/03/$ - see front matter # 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.colsurfb.2003.08.004
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A. Eftekhari / Colloids and Surfaces B: Biointerfaces 32 (2003) 375 /381
2. Experimental Deposition of a smooth gold layer on the electrode surface was used for gold masking purpose, as this method provides uniform gold layer on every side of the electrode surface [26]. Indeed, this approach has been introduced as an efficient approach for fractal analysis of surfaces. It has been described that even for the electrode surfaces, which have a fast redox systems (e.g., electroactive films participating in reaction with diffusing ions from the electrolyte solution), gold covering of the electrode is more reliable method for fractal analysis [28]. This approach has also been successfully used for fractal studies of different surfaces [25 /29]. The gold deposition process was carried out according to the available method for the deposition of metals on non-conducting substrates [30]. The ideal thickness for the investigation of structure of the tooth interface is monolayer Au film, as the fractal dimension of the Au film depends on the surface coverage. However, deposition of only a monolayer Au film is difficult to achieve. Thus, an Au film with just a few layers thickness was deposited in this research. This is suitable for our purpose as the fractal dimension at this range of thickness (approximately 4 /10 layers) is independent of the film thickness. This is the range that a complete Au surface is formed but there is not still an opportunity for the formation of complex patterns of the electrodeposit, and possible errors are ignorable. On the other hand, for a comparative study (such this research), the surface structure of the Au layer deposited can be considered as an excellent factor of the dental surfaces. Then, the other sides of the fabricated electrode was coated with an insulating cover except its cross section, which is the active part of the electrode in the electrochemical measurements. A schematic structure of the gold-coated tooth electrode is presented in Fig. 1. The only available active area of the electrode, which can participate in the electrochemical reactions, is the central part of the cross section of the electrode (the area specified in the vertical view in Fig. 1). The most important section of teeth surfaces is the part covered with enamel, which is available in buccal environment
Fig. 1. A schematic of the gold-coated tooth electrode.
(the deep gray section illustrated in Fig. 1). However, as the most important section of a tooth in dental decay is its cross section, the lateral sides of the tooth were also covered by the insulating cover. The fractal dimension of the electrode surface was determined using an electrochemical method based on time-dependency of the diffusion-limited current after a potential step. The most important requirement for this method is the existence of a very fast electrochemical redox couple on the electrode surface, which is available for gold electrode. Electrochemical reaction of ferricyanide (a wellcharacterized anion), on the gold-coated tooth electrode, was used as a redox-probe to determine the fractal dimension of the tooth electrode surface. The electrolyte was an aqueous solution of 3 M NaCl and 15 mM K4Fe(CN)6 and 15 mM K3(CN)6. The initial potential was 0.6 V where no electrochemical reduction of Fe(CN)3 occurs. By 6 stepping the potential to low value of ca. 0 V,
A. Eftekhari / Colloids and Surfaces B: Biointerfaces 32 (2003) 375 /381
377
1 2
(the value of fractal
essentially all ferricyanide ions are reduced to Fe(CN)4 6 . The electrochemical experiments were performed using a low-noise homemade potentiostat connected to a computer running CorrView software. The experiments were carried out at room temperature using a conventional three-electrode electrochemical cell containing the working electrode. The working electrodes were prepared by gold masking of a tooth. A platinum rod and a saturated calomel electrode (SCE) were used as counter and reference electrode, respectively. A Luggin capillary was placed between the working electrode and reference electrode to minimize the ohmic drop. It is known that log-log scale curves of chronoamperograms is accompanied by a set of dispersed data at long experiment times, which is due to tiny differences in the current as a result of small perturbations. For better understanding the methodology and to show typical curves, such dispersed data were omitted. It should be emphasized that this action does not change the curve slope employed for the determination of the fractal dimension.
3. Results and discussion The most of available methods for the determination of the fractal dimension of surfaces using both electrochemical and non-electrochemical techniques are based on estimating the value of fractal parameter. The fractal parameter (a ) can be easily transformed to the fractal dimension (Df) according to the following equation [15]: a (Df 1)=2
Eq. (1), the power of
parameter) corresponds to the integer dimension of two from Euclidean geometry point of view. It has been described that for the rough (fractal) electrodes, the Cottrell equation is transformed to an extended form as [11]: I(t)sF ta
(2)
where sF is a proportionality factor. This indicates that in a certain range of time, the current is a power-law function of time, and the fractal dimension is included in the exponent, a /(Df / 1)/2. Therefore, the fractal dimension can be determined from slope of the current /time relationship plotted in a log /log scale. The validity of this method has been examined by numerical calculations including Monte Carlo simulations of random walk [11] and by experiments using artificial fractal electrodes and real ones [11,20]. Fig. 2 shows the log I /log t curve of a goldcoated carious tooth electrode in ferricyanide solution. According to Pajkossy and Nyikos [11], the slope of the curve equals the fractal parameter (a ). Here, we found a /0.660 and, according to Eq. (1): Df /2.320. For validity test of the results obtained from fractal analysis of the tooth electrode, fractal dimension of the electrode surface was also determined with another method. Cyclic voltam-
(1)
Although this relation was first proposed by de Gennes [31] in a somehow different context, it has also been successfully used for the electrochemical methods. To determine the fractal dimension of the tooth electrodes, chronoamperometric technique was employed. This is one of the oldest methods for this purpose, which is based on extended form of Cottrell equation. In the conventional electrodes, the diffusion controlled current shows the well-known inverse square root time dependence, as I 8/t 1/2 [32]. According to
Fig. 2. A typical log I /log t curve of the system under investigation (the gold-coated carious tooth electrode in ferricyanide solution).
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A. Eftekhari / Colloids and Surfaces B: Biointerfaces 32 (2003) 375 /381
metry is another useful electrochemical method for the determination of fractal dimension of electrode surfaces. This method is based on scan rate dependency of the cathodic peak current, which its concepts and proposition has been elaborated by Strømme et al. [18,19]. They have reported that there is an exponential dependence between the cathodic peak current and potential scan rate as the following one: Ipc8na or Ipc s?F na
(3)
Thus, the fractal parameter is equal to the slope of the cathodic peak current versus the scan rate plotted in a log /log scale. This approach can be used by collecting the values of the cathodic peak currents of cyclic voltammograms recorded at different potential scan rates. Similar to the former method, fractal dimension can be estimated from the value of fractal parameter in accordance with Eq. (1). A typical series of data are presented in Fig. 3 for the values of the cathodic peak currents obtained from cyclic voltammetric measurements at different scan rates. As the curve is in the log /
log scale, slope of the curve indicates the value of 0.656 for the fractal parameter a . Therefore, the value of 2.313 can be obtained for the fractal dimension of the tooth electrode surface. Comparison of the results obtained from two different techniques employed (chronoamperometry and cyclic voltammetry) indicates that there is a good agreement between the values of the fractal dimension of the tooth electrode (Table 1). It should be noted that the values estimated for the fractal dimension of dental surfaces are useful for comparison purpose, and we cannot claim that the surface structures can be exactly defined by the four-digit numbers calculated as the fractal dimensions. To shed light on the applied significance of fractal analysis of dental surfaces, the fractal dimension of a healthy tooth surface was also estimated. To this aim, a carious tooth with lateral dental decay (in the interface of two neighbor teeth), which is known in dentistry as a Class II dental decay, was employed. In this case, the upper side of the tooth is remained safe, and for our
Fig. 3. Relationship of the cathodic peak current (Ipc) to potential scan rate (n ) plotted in the log /log scale (log Ipc /log n ) for the carious tooth electrode (m), healthy tooth electrode (k) and artificial enamel electrode (j). The data were obtained from cyclic voltammetric studies of the system at different scan rates.
A. Eftekhari / Colloids and Surfaces B: Biointerfaces 32 (2003) 375 /381
379
Table 1 Comparison of the results obtained from two different techniques Method
Fractal parameter a
Experiment range
Correlation coefficient
RSDa (%)
Estimated Df
Chronoamperometry Cyclic voltammetry
0.6600 0.6566
0.01 /10 (t /s) 20 /800 (n /mV s 1)
0.98843 0.99633
5.4 3.2
2.320 2.313
a
Relative standard deviation, estimated for five successive measurements.
investigation can be considered as a healthy tooth. Similarly, fractal dimension of a smooth surface made from an artificial enamel (a kind of ceramiclike material using in clinical dentistry to restore carious enamel) was also determined based on the approach proposed. The results obtained from cyclic voltammetric measurements are illustrated in Fig. 3. It can be concluded from different curves that dental decay significantly increases the fractal dimension of tooth surface (comparison of carious and healthy teeth). It is indicative of the fact that the existence of natural tooth in buccal environment is accompanied by surface roughening. Another important parameter for understanding fractality of an object is cut-offs. Indeed, inner and outer cut-offs indicates the scale of fractality, as a real fractal object does not show fractal behavior at length zero to infinity [1]. To find that a dental surface has fractality in what scale, it is needed to estimate corresponding cut-offs. To this aim, the length scales traversed on the dental surface under minimum and maximum scan rates were calculated. In this condition, the length scale can be estimated from the diffusion layer, which acts as a yardstick length. The diffusion layer thickness can be obtained as: DX zFADCbulk =ipeak
(4)
where z is the number of electrons transferred per electroactive species in the redox reaction, F , A and D are the Faraday constant, the surface area and the diffusion coefficient, respectively. The value of D can be calculated using Randles/ Sevcik equation [32]. Therefore, we are able to estimate the inner and outer cutoffs based on the experimental results. The results are illustrated in Table 2. The distinction between inner and outer cutoffs can be used to investigate the fractal morphology of a surface.
The results suggest that dental decay causes a decrease in the fractality scale, which might be attributed to the formation of tiny porosities in the dental caries. According to the data presented, the relatively large value of the outer cutoffs suggests that the bumpiness of the film along the surface is sensed to be different from the perpendicular direction, which is indicative of a self-affine structure. Dental decay causes decrease of the outer cut-off of the dental surface and increase of the fractal dimension. In this case, the inner cut-off scales in two different directions, viz. perpendicular and parallel to the surface, are approximately the same, which suggests a self-similar structure. Surprisingly, the results suggest that dental decay is accompanied by the formation of self-similar fractal pattern. One may think about possibility and reliability of the gold masking approach for the determination of fractal dimension of the substrate surface, as the generated surface may have a considerable roughness due to the Au-deposits. Surely, there is no claim that the approach applied here determines the exact value of fractal dimension of the tooth surfaces. Indeed, the results are satisfactory for a comparative study, which is the aim of the present research. On the other hand, the possible error made by the Au-deposit own structure is not very significant. To investigate the suitability of the deposition experiments to deposit uniform gold mask, a thin film of gold was deposited under the same condition on a smooth Au electrode surface (with a roughness factor lesser than 2), which can be referred to as a completely smooth surface with an integer dimension of 2. The fabricated electrode has a slope of 0.509 in the log ip /log n curve, which is very close to ideal case of Randels/Sevick equation.
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Table 2 Comparison of fractal studies of different dental surfaces using cyclic voltammetric measurements Sample
Carious enamel Healthy enamel Artificial enamel
Cut-offs/nm Inner
Outer
141 175 201
238 279 312
Fractal parameter a
Fractal dimension Df
Correlation coefficient
0.6566 0.5904 0.5555
2.313 2.180 2.111
0.99633 0.99764 0.99976
The present work is of great interest from methodology point of view. In fact, the approach proposed provides a new opportunity for fractal studies of biosurfaces using electrochemical techniques. Most of biosurfaces are non-conducting (similar to dental surfaces) and even in the case of conducting biosurfaces, no fast redox system is available to use phenomenon of ‘‘diffusion towards surface’’ to use electrochemical measurements for fractal study. Interestingly, the present approach can be employed for fractal analysis of various biosurfaces.
4. Conclusions It was demonstrated that fractal structure (generally surface characterization) of non-conducting objects such as teeth could be investigated using electrochemical methods. The results are indicative of the dependence of the fractal dimension of tooth surface on the dental decay, which has a clinical interest. As the teeth surfaces are very important in dental affairs, fractal dimension of them can be used as a quantitative factor for their investigations. On the other hand, it is shown that electrochemical techniques can be used for fractal analysis of non-conducting objects including biosurfaces. In fact, this provides a new opportunity, e.g., although electrochemical methods have been widely used for the investigation of dental materials; but for the cases of natural teeth, it seems that there is no relationship between electrochemistry and dentistry, as natural teeth are non-conducting materials. The present paper was a short communication to report a new approach for the determination of the fractal dimension of non-conducting materials
using electrochemical methods. Determination of the fractal dimension of teeth surfaces, as a quantitative factor, can be used for applied investigations in dental research and even for clinical purposes. Truly, the approach proposed cannot be used as an in vivo method; however, the results are of interest to use fractal geometry in dentistry.
References [1] B.B. Mandelbrot, Fractal Geometry of Nature, Freeman, San Francisco, 1983. [2] M.F. Barnsley, Fractals Everywhere, 2nd ed, Academic Press, New York, 1993. [3] H.O. Peitgen, D. Saupe (Eds.), The Science of Fractal Image, New York, Springer, 1988. [4] F. Moon, Chaotic and Fractal Dynamics: An Introduction for Applied Scientists and Engineers, Wiley, New York, 1992. [5] K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, Wiley, New York, 1990. [6] W.G. Rothschild, Fractals in Chemistry, Wiley, New York, 1998. [7] D. Avnir (Ed.), The Fractal Approach to Heterogeneous Chemistry, Wiley, Chichester, UK, 1989. [8] A. Le Mehaute, G. Crepy, Solid State Ionics 8&9 (1983) 17. [9] L. Nyikos, T. Pajkossy, Electrochim. Acta 30 (1985) 1533. [10] L. Nyikos, T. Pajkossy, Electrochim. Acta 31 (1986) 1347. [11] T. Pajkossy, L. Nyikos, Electrochim. Acta 34 (1989) 171. [12] T. Pajkossy, L. Nyikos, Electrochim. Acta 34 (1989) 181. [13] L. Nyikos, T. Pajkossy, A.P. Borosy, S.A. Martemyanov, Electrochim. Acta 35 (1990) 1423. [14] A.P. Borosy, L. Nyikos, T. Pajkossy, Electrochim. Acta 36 (1991) 163. [15] T. Pajkossy, J. Electroanal. Chem. 300 (1991) 1. [16] A. Imre, T. Pajkossy, L. Nyikos, Acta Metall. Mater. 40 (1992) 1819. [17] T. Pajkossy, A.P. Borosy, A. Imre, S.A. Martemyanov, G. Nagy, R. Schiller, L. Nyikos, J. Electroanal. Chem. 366 (1994) 69.
A. Eftekhari / Colloids and Surfaces B: Biointerfaces 32 (2003) 375 /381 [18] M. Strømme, G.A. Niklasson, C.G. Granqvist, Solid State Commun. 96 (1995) 151. [19] M. Strømme, G.A. Niklasson, C.G. Granqvist, Phys. Rev. B 52 (1995) 14192. [20] Y. Dassas, P. Duby, J. Electrochem. Soc. 142 (1995) 4175. [21] M. Strømme Mattsson, G.A. Niklasson, C.G. Granqvist, Phys. Rev. B 54 (1996) 17884. [22] S. Ye, A.K. Vijh, L.H. Dao, J. Electrochem. Soc. 144 (1997) 1734. [23] K. Aoki, Y.A. Teragishi, J. Electroanal. Chem. 441 (1998) 25. [24] P. Tomczyk, Y. Ono, Y. Hisamitsu, M. Mohamedi, I. Uchida, Electrochem. Solid-State Lett. 2 (1999) 431.
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[25] A.R. Imre, L. Balazs, Fractals 4 (2000) 349. [26] A. Eftekhari, Mendeleev Commun. 12 (2002) 122. [27] A. Eftekhari, Electrochim. Acta 47 (2002) 4347 (Erratum 48 (2002) 290). [28] A. Eftekhari, Electrochim. Acta 48 (2003) 2831. [29] A. Eftekhari, Appl. Surf. Sci., in press; doi: 10.1016/S01694332(03)00842-0 [30] S. Ono, T. Osaka, K. Naitoh, Y. Nakagishi, J. Electrochem. Soc. 146 (1999) 160. [31] P.-G. de Gennes, C.R. Acad. Sci. Paris Series II 295 (1982) 1061. [32] A.J. Bard, L.R. Faulkner, Electrochemical Methods, Wiley, New York, 1980.
Electrochemical technique for the determination of fractal dimension of dental surfaces Ali Eftekhari * Electrochemical Research Center, P.O. Box 19395-5139, Tehran, Iran Received 26 May 2003; received in revised form 1 July 2003; accepted 6 August 2003
Abstract Fractal dimension of a carious tooth surface was determined using an electrochemical method. The method was based on time-dependent diffusion towards electrode surfaces, which is one of the most useful and reliable methods for the determination of fractal dimension of electrode surfaces. For this purpose, the tooth was covered with a gold layer, which acted as an electrode in electrochemical experiments. It is suggested that the fractal dimension can be used as a quantitative measure of the state of dental surfaces. The method presented demonstrates the power of electrochemical techniques for the determination of fractal dimension of surface of non-conducting objects. # 2003 Elsevier B.V. All rights reserved. Keywords: Fractal surface; Electrodeposition; Non-conducting substrate; Gold-coated electrode; Tooth surfaces; Dental applications
1. Introduction Since the revolutionary discovery of fractal geometry by Mandelbrot [1], it has been used for analysis of different objects in various branches of science and technology [2 /7]. It has also been successfully proposed in electrochemical science [8 /29], due to the essential role of electrode surfaces and interfacial problems in electrochemistry. Several methods have been proposed for the determination of fractal dimension of electrode * Corresponding author. Tel.: /98-21-204-2549; fax: /9821-205-7621. E-mail address: [email protected] (A. Eftekhari).
surfaces by means of electrochemical techniques. Fractal dimension is a quantitative parameter for analysis of fractal objects, which is widely used for comparative investigations of fractal objects. It is also a powerful quantitative factor for analysis of rough surface structures. In the present work, it is aimed to report fractality of dental surfaces for the first time and application of fractal dimension as a measure for the investigation of dental surfaces. The methodology is of practical interest, as surface analysis of teeth can be used in dental science. It is also a useful approach for studies of dental materials to find biomaterials similar to natural teeth (from a surface structure point of view).
0927-7765/03/$ - see front matter # 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.colsurfb.2003.08.004
376
A. Eftekhari / Colloids and Surfaces B: Biointerfaces 32 (2003) 375 /381
2. Experimental Deposition of a smooth gold layer on the electrode surface was used for gold masking purpose, as this method provides uniform gold layer on every side of the electrode surface [26]. Indeed, this approach has been introduced as an efficient approach for fractal analysis of surfaces. It has been described that even for the electrode surfaces, which have a fast redox systems (e.g., electroactive films participating in reaction with diffusing ions from the electrolyte solution), gold covering of the electrode is more reliable method for fractal analysis [28]. This approach has also been successfully used for fractal studies of different surfaces [25 /29]. The gold deposition process was carried out according to the available method for the deposition of metals on non-conducting substrates [30]. The ideal thickness for the investigation of structure of the tooth interface is monolayer Au film, as the fractal dimension of the Au film depends on the surface coverage. However, deposition of only a monolayer Au film is difficult to achieve. Thus, an Au film with just a few layers thickness was deposited in this research. This is suitable for our purpose as the fractal dimension at this range of thickness (approximately 4 /10 layers) is independent of the film thickness. This is the range that a complete Au surface is formed but there is not still an opportunity for the formation of complex patterns of the electrodeposit, and possible errors are ignorable. On the other hand, for a comparative study (such this research), the surface structure of the Au layer deposited can be considered as an excellent factor of the dental surfaces. Then, the other sides of the fabricated electrode was coated with an insulating cover except its cross section, which is the active part of the electrode in the electrochemical measurements. A schematic structure of the gold-coated tooth electrode is presented in Fig. 1. The only available active area of the electrode, which can participate in the electrochemical reactions, is the central part of the cross section of the electrode (the area specified in the vertical view in Fig. 1). The most important section of teeth surfaces is the part covered with enamel, which is available in buccal environment
Fig. 1. A schematic of the gold-coated tooth electrode.
(the deep gray section illustrated in Fig. 1). However, as the most important section of a tooth in dental decay is its cross section, the lateral sides of the tooth were also covered by the insulating cover. The fractal dimension of the electrode surface was determined using an electrochemical method based on time-dependency of the diffusion-limited current after a potential step. The most important requirement for this method is the existence of a very fast electrochemical redox couple on the electrode surface, which is available for gold electrode. Electrochemical reaction of ferricyanide (a wellcharacterized anion), on the gold-coated tooth electrode, was used as a redox-probe to determine the fractal dimension of the tooth electrode surface. The electrolyte was an aqueous solution of 3 M NaCl and 15 mM K4Fe(CN)6 and 15 mM K3(CN)6. The initial potential was 0.6 V where no electrochemical reduction of Fe(CN)3 occurs. By 6 stepping the potential to low value of ca. 0 V,
A. Eftekhari / Colloids and Surfaces B: Biointerfaces 32 (2003) 375 /381
377
1 2
(the value of fractal
essentially all ferricyanide ions are reduced to Fe(CN)4 6 . The electrochemical experiments were performed using a low-noise homemade potentiostat connected to a computer running CorrView software. The experiments were carried out at room temperature using a conventional three-electrode electrochemical cell containing the working electrode. The working electrodes were prepared by gold masking of a tooth. A platinum rod and a saturated calomel electrode (SCE) were used as counter and reference electrode, respectively. A Luggin capillary was placed between the working electrode and reference electrode to minimize the ohmic drop. It is known that log-log scale curves of chronoamperograms is accompanied by a set of dispersed data at long experiment times, which is due to tiny differences in the current as a result of small perturbations. For better understanding the methodology and to show typical curves, such dispersed data were omitted. It should be emphasized that this action does not change the curve slope employed for the determination of the fractal dimension.
3. Results and discussion The most of available methods for the determination of the fractal dimension of surfaces using both electrochemical and non-electrochemical techniques are based on estimating the value of fractal parameter. The fractal parameter (a ) can be easily transformed to the fractal dimension (Df) according to the following equation [15]: a (Df 1)=2
Eq. (1), the power of
parameter) corresponds to the integer dimension of two from Euclidean geometry point of view. It has been described that for the rough (fractal) electrodes, the Cottrell equation is transformed to an extended form as [11]: I(t)sF ta
(2)
where sF is a proportionality factor. This indicates that in a certain range of time, the current is a power-law function of time, and the fractal dimension is included in the exponent, a /(Df / 1)/2. Therefore, the fractal dimension can be determined from slope of the current /time relationship plotted in a log /log scale. The validity of this method has been examined by numerical calculations including Monte Carlo simulations of random walk [11] and by experiments using artificial fractal electrodes and real ones [11,20]. Fig. 2 shows the log I /log t curve of a goldcoated carious tooth electrode in ferricyanide solution. According to Pajkossy and Nyikos [11], the slope of the curve equals the fractal parameter (a ). Here, we found a /0.660 and, according to Eq. (1): Df /2.320. For validity test of the results obtained from fractal analysis of the tooth electrode, fractal dimension of the electrode surface was also determined with another method. Cyclic voltam-
(1)
Although this relation was first proposed by de Gennes [31] in a somehow different context, it has also been successfully used for the electrochemical methods. To determine the fractal dimension of the tooth electrodes, chronoamperometric technique was employed. This is one of the oldest methods for this purpose, which is based on extended form of Cottrell equation. In the conventional electrodes, the diffusion controlled current shows the well-known inverse square root time dependence, as I 8/t 1/2 [32]. According to
Fig. 2. A typical log I /log t curve of the system under investigation (the gold-coated carious tooth electrode in ferricyanide solution).
378
A. Eftekhari / Colloids and Surfaces B: Biointerfaces 32 (2003) 375 /381
metry is another useful electrochemical method for the determination of fractal dimension of electrode surfaces. This method is based on scan rate dependency of the cathodic peak current, which its concepts and proposition has been elaborated by Strømme et al. [18,19]. They have reported that there is an exponential dependence between the cathodic peak current and potential scan rate as the following one: Ipc8na or Ipc s?F na
(3)
Thus, the fractal parameter is equal to the slope of the cathodic peak current versus the scan rate plotted in a log /log scale. This approach can be used by collecting the values of the cathodic peak currents of cyclic voltammograms recorded at different potential scan rates. Similar to the former method, fractal dimension can be estimated from the value of fractal parameter in accordance with Eq. (1). A typical series of data are presented in Fig. 3 for the values of the cathodic peak currents obtained from cyclic voltammetric measurements at different scan rates. As the curve is in the log /
log scale, slope of the curve indicates the value of 0.656 for the fractal parameter a . Therefore, the value of 2.313 can be obtained for the fractal dimension of the tooth electrode surface. Comparison of the results obtained from two different techniques employed (chronoamperometry and cyclic voltammetry) indicates that there is a good agreement between the values of the fractal dimension of the tooth electrode (Table 1). It should be noted that the values estimated for the fractal dimension of dental surfaces are useful for comparison purpose, and we cannot claim that the surface structures can be exactly defined by the four-digit numbers calculated as the fractal dimensions. To shed light on the applied significance of fractal analysis of dental surfaces, the fractal dimension of a healthy tooth surface was also estimated. To this aim, a carious tooth with lateral dental decay (in the interface of two neighbor teeth), which is known in dentistry as a Class II dental decay, was employed. In this case, the upper side of the tooth is remained safe, and for our
Fig. 3. Relationship of the cathodic peak current (Ipc) to potential scan rate (n ) plotted in the log /log scale (log Ipc /log n ) for the carious tooth electrode (m), healthy tooth electrode (k) and artificial enamel electrode (j). The data were obtained from cyclic voltammetric studies of the system at different scan rates.
A. Eftekhari / Colloids and Surfaces B: Biointerfaces 32 (2003) 375 /381
379
Table 1 Comparison of the results obtained from two different techniques Method
Fractal parameter a
Experiment range
Correlation coefficient
RSDa (%)
Estimated Df
Chronoamperometry Cyclic voltammetry
0.6600 0.6566
0.01 /10 (t /s) 20 /800 (n /mV s 1)
0.98843 0.99633
5.4 3.2
2.320 2.313
a
Relative standard deviation, estimated for five successive measurements.
investigation can be considered as a healthy tooth. Similarly, fractal dimension of a smooth surface made from an artificial enamel (a kind of ceramiclike material using in clinical dentistry to restore carious enamel) was also determined based on the approach proposed. The results obtained from cyclic voltammetric measurements are illustrated in Fig. 3. It can be concluded from different curves that dental decay significantly increases the fractal dimension of tooth surface (comparison of carious and healthy teeth). It is indicative of the fact that the existence of natural tooth in buccal environment is accompanied by surface roughening. Another important parameter for understanding fractality of an object is cut-offs. Indeed, inner and outer cut-offs indicates the scale of fractality, as a real fractal object does not show fractal behavior at length zero to infinity [1]. To find that a dental surface has fractality in what scale, it is needed to estimate corresponding cut-offs. To this aim, the length scales traversed on the dental surface under minimum and maximum scan rates were calculated. In this condition, the length scale can be estimated from the diffusion layer, which acts as a yardstick length. The diffusion layer thickness can be obtained as: DX zFADCbulk =ipeak
(4)
where z is the number of electrons transferred per electroactive species in the redox reaction, F , A and D are the Faraday constant, the surface area and the diffusion coefficient, respectively. The value of D can be calculated using Randles/ Sevcik equation [32]. Therefore, we are able to estimate the inner and outer cutoffs based on the experimental results. The results are illustrated in Table 2. The distinction between inner and outer cutoffs can be used to investigate the fractal morphology of a surface.
The results suggest that dental decay causes a decrease in the fractality scale, which might be attributed to the formation of tiny porosities in the dental caries. According to the data presented, the relatively large value of the outer cutoffs suggests that the bumpiness of the film along the surface is sensed to be different from the perpendicular direction, which is indicative of a self-affine structure. Dental decay causes decrease of the outer cut-off of the dental surface and increase of the fractal dimension. In this case, the inner cut-off scales in two different directions, viz. perpendicular and parallel to the surface, are approximately the same, which suggests a self-similar structure. Surprisingly, the results suggest that dental decay is accompanied by the formation of self-similar fractal pattern. One may think about possibility and reliability of the gold masking approach for the determination of fractal dimension of the substrate surface, as the generated surface may have a considerable roughness due to the Au-deposits. Surely, there is no claim that the approach applied here determines the exact value of fractal dimension of the tooth surfaces. Indeed, the results are satisfactory for a comparative study, which is the aim of the present research. On the other hand, the possible error made by the Au-deposit own structure is not very significant. To investigate the suitability of the deposition experiments to deposit uniform gold mask, a thin film of gold was deposited under the same condition on a smooth Au electrode surface (with a roughness factor lesser than 2), which can be referred to as a completely smooth surface with an integer dimension of 2. The fabricated electrode has a slope of 0.509 in the log ip /log n curve, which is very close to ideal case of Randels/Sevick equation.
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Table 2 Comparison of fractal studies of different dental surfaces using cyclic voltammetric measurements Sample
Carious enamel Healthy enamel Artificial enamel
Cut-offs/nm Inner
Outer
141 175 201
238 279 312
Fractal parameter a
Fractal dimension Df
Correlation coefficient
0.6566 0.5904 0.5555
2.313 2.180 2.111
0.99633 0.99764 0.99976
The present work is of great interest from methodology point of view. In fact, the approach proposed provides a new opportunity for fractal studies of biosurfaces using electrochemical techniques. Most of biosurfaces are non-conducting (similar to dental surfaces) and even in the case of conducting biosurfaces, no fast redox system is available to use phenomenon of ‘‘diffusion towards surface’’ to use electrochemical measurements for fractal study. Interestingly, the present approach can be employed for fractal analysis of various biosurfaces.
4. Conclusions It was demonstrated that fractal structure (generally surface characterization) of non-conducting objects such as teeth could be investigated using electrochemical methods. The results are indicative of the dependence of the fractal dimension of tooth surface on the dental decay, which has a clinical interest. As the teeth surfaces are very important in dental affairs, fractal dimension of them can be used as a quantitative factor for their investigations. On the other hand, it is shown that electrochemical techniques can be used for fractal analysis of non-conducting objects including biosurfaces. In fact, this provides a new opportunity, e.g., although electrochemical methods have been widely used for the investigation of dental materials; but for the cases of natural teeth, it seems that there is no relationship between electrochemistry and dentistry, as natural teeth are non-conducting materials. The present paper was a short communication to report a new approach for the determination of the fractal dimension of non-conducting materials
using electrochemical methods. Determination of the fractal dimension of teeth surfaces, as a quantitative factor, can be used for applied investigations in dental research and even for clinical purposes. Truly, the approach proposed cannot be used as an in vivo method; however, the results are of interest to use fractal geometry in dentistry.
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