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Percolation of cadmium across a mercury film
Materials Chemistry and Physics 78 (2003) 796–799
Percolation of cadmium across a mercury film K. Malek a,∗ , F. Gobal b a
Delft Department of Chemical Technology, Delft University of Technology, Julianalaan 136, 2628 BL Delft, The Netherlands b Department of Chemistry, Sharif University of Technology, P.O. Box 11365-9516, Tehran, Iran Received 23 April 2002; received in revised form 31 July 2002; accepted 12 August 2002
Abstract Electrodeposition/dissolution of cadmium onto a film of mercury shows some deviations from the natural liquidity of mercury caused by the reduction of Cd onto it. Percolation and fractal analyzes were done on the surface and the bulk of the mercury film during diffusion of Cd species (atoms). These show that the fractal dimensions of the Cd-inserted mercury film are about 2.11 and 2.54 near the surface of the mercury film and at deeper points inside the film, respectively. The insertion process has a negligible effect on the surface morphology of the mercury film and there is a phase transition in the bulk, as well as a geometrical transition during the Cd-insertion (de-insertion) process. This corresponds to a percolation threshold of about 0.2 mol l−1 Cd content. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Thin films; Coatings; Electrochemical techniques; Transport properties
1. Introduction In recent years there has been a growing interest in the study of the diffusion of electrochemically generated species in the electrode matrix due to its effect on the morphology [1] and physico-chemical properties [2] of electrochemical deposited films, employed in the construction of sensors, optical devices, batteries, memories [3] etc. An understanding of the relevant diffusion processes is necessary if optimum results are to be achieved by diffusion anneals. On the other hand, atom diffusion is an important part of the science and technology of a mixed-metal mercury electrode [4] (e.g. mercury cadmium telluride, MCT). A mercury electrode is basically an ideally smooth electrode without any roughness inside it and in the boundary of solution. This property appears in the impedance response of a mercury drop electrode in an inert solution as a straight line parallel to the imaginary axis in a Nyquist plot [5]. Structural changes in the morphology during diffusion of small species have an important effect on the type and amount of diffusivity. In general, two distinct diffusion regimes can be identified: (a) diffusion occurs without any net chemical fluxes; i.e. the material is chemically homogeneous and (b) the material is chemically inhomogeneous so that diffusion gives rise to net chemical fluxes. Regime (a) represents diffusion at chemical equilibrium and gives rise to self-diffusion coefficients. For ∗ Corresponding author. Tel.: +31-15-27-84685; fax: +31-15-27-88267. E-mail address: [email protected] (K. Malek).
the interpretation and rationalization of the diffusion properties, the concepts of fractal dimensions of the matrix of the doped films [6] and percolation of dopants (intercalates) in the film have been used [7]. In our previous studies [8–10] we have applied the above concepts to describe the properties of doped species and their behavior during diffusion through the film. The purpose of the present work is to investigate the electrodeposition/dissolution of cadmium into mercury films, focusing on possible deviations from the natural liquidity of mercury caused by the reduction of Cd. This can be done by characterization of the percolative and fractal structure of the surface and the bulk of the resultant film during diffusion of Cd species (atoms). For this purpose, cyclic voltammetry and electrochemical impedance spectroscopy methods have been used. 2. Experimental Films of mercury were prepared on copper rods, previously carefully cleaned by dichloromethane and dilute nitric acid followed by rinsing with distilled water. A two-electrode cell, copper in contact with a drop of mercury forming the cathode and a platinum plate forming the anode, in a 1 M KCl solution is used. Increasing the potential slowly forms a thin layer of fairly stable mercury on the copper surface. Although copper is partially soluble in mercury, the bulk of the film is pure mercury except for
0254-0584/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 2 5 4 - 0 5 8 4 ( 0 2 ) 0 0 3 8 6 - 3
K. Malek, F. Gobal / Materials Chemistry and Physics 78 (2003) 796–799
the copper/mercury interface, which is a copper amalgam. The thickness of the film measured gravimetrically was 6 ± 1 m. Electrochemical deposition/dissolution of Cd onto this electrode was studied in a conventional three-electrode cell where the mercury film was the working electrode with its potential monitored against a standard calomel electrode and a platinum plate formed the counter electrode. The electrolyte was 0.01 M CdCl2 + 0.5 M KCl solution of Analar grade of Hopkins and Wiliams origin. Cyclic voltammetry (CV) and electrochemical impedance spectroscopy (EIS) studies were done using an EG&G electrochemical system comprising of a 273A potentiostat/galvanostat and a 1250 Lock-in-amplifier run by an IBM ps/value point computer with M270 and M398 software packages.
3. Results and discussion Fig. 1 represents the cyclic voltammograms of Cd deposition/dissolution onto a Hg film recorded at various potential sweep rates. As we are particularly interested in the diffusion of Cd in the film, delays of 180 s at the end of the cathodic half cycles have been exercised prior to the anodic runs. During the anodic half cycle, at high potential sweep rates there is a large peak possessing a shoulder, which is clearly observed as a distinct peak at lower potential sweep rates. The slopes of the potential sweep rate dependency on the anodic peaks currents in the log–log scale provides α which is related [11] to the fractal dimension, Df , of the surface (and the bulk) through the relation: Df − 1 α= (1) 2 Applying this expression to the results presented in Fig. 1 gives the fractal dimensions of the surface and bulk of the film to be around 2.11 and 2.54, respectively. The α values have been derived from a log–log plot of the main peak and the shoulder versus sweep rate (Fig. 2), within a reasonable order of magnitude to establish a fractal relation. Furthermore, attempts have been made to correct for the effect of
Fig. 1. Cyclic voltammograms signifying the process of cadmium deposition/dissolution onto mercury film at various potential sweep rates.
797
Fig. 2. Log–log plots of the peak current vs. sweep rate.
background capacitance current and internal resistance (IR) drops. The main anodic peak is due to the dissolution of Cd species concentrated either on or near the surface of the mercury film while the second peak arises from the dissolution of Cd atoms, originating from deeper points inside the mercury film. Thus, the fractal dimensions correspond to the surface and the bulk topography of the film as shown in Fig. 3. Applying a modified Sevcik equation for fractal electrodes [12] to the results of the bulk peak, the diffusion coefficient, D, of Cd in the Hg film is obtained as 2.2 × 10−12 m2 s−1 . Using Eq. (2): x =
zFADCb ip
(2)
and assuming Cb = 0.32 M for Cd on the basis of the volume of the film and the time of electrolysis (cathodic delay), the width of the diffusion layer, x, for the fastest and the slowest potential sweep rates have been obtained: 1.6 × 10−7 and 5.32 × 10−6 m, respectively. In Eq. (2), F is the Faraday constant, A is the area of the electrode (∼ =2 cm2 )
Fig. 3. Illustration of the surface and bulk topography of the mercury film during Cd deposition/dissolution.
798
K. Malek, F. Gobal / Materials Chemistry and Physics 78 (2003) 796–799
Fig. 5. Variation of the conductivity of the mercury film with cadmium concentration in the bulk of the film.
nearly 2.53 is expected for a three-dimensional percolative system [13,14]. It has been shown in a percolation system. β Df = D − (3) ϑ
Fig. 4. Impedance responses of the cadmium loaded in Hg/solution interface in the form of the Nyquist plots recorded at various direct-current offset potentials:(a) −0.9, −0.7; (b) −0.5 V vs. SCE. Inset shows the equivalent circuit used to analyze the impedance data.
and ip is the peak current. Interestingly enough, the diffusion lengths are in accordance with the independently measured film thickness. Fig. 4 represents the Nyquist diagrams of the cadmium loaded mercury/solution interface at various direct-current offset potentials. In all cases the loadings were performed at −1.4 volt versus standard calomel electrode (V/SCE) and the EIS studies were performed after constant anodic currents were reached. At −0.5 V/SCE, the Hg film is almost free of Cd and a single time constant characterizing the interface has been observed. A single rather squashed semi-circle appears in the Nyquist plot. At more negative potentials where the loading of Cd is higher, a semicircle terminating to a segment of a straight line at the low frequency end of the spectrum has been observed. The equivalent circuit has been shown in Fig. 4a, which is compatible with the results. In both cases the first squashed semicircle is associated with the mercury film capacitance–resistance characteristics while the semicircle in the medium-frequency range signifies the charge transfer associated with the dissolution of Cd. The straight line at the low-frequency end of the spectrum shows the dominance (activation) of mass transfer across the Hg film, which can be considered as a Warburg impedance. Fig. 5 represents the variation of the film’s conductivity with the bulk concentration of Cd. A percolation phenomenon, with the threshold at nearly 0.2 mol l−1 Cd content, has been observed. At both sides of this concentration the conductivity of the Hg/Cd system rises, with the steepest rise at the Cd increasing side. A fractal dimension of
where D = 3, and β and ϑ are the critical exponents (currently accepted values of β = 0.41 and ϑ = 0.88) which control the percolation probability and the correlation length in a percolation system, respectively. This ratio is virtually constant in a three-dimensional system with a Df value of 2.53. In the system under study, the Hg and Cd atoms can interdiffuse by an exchange process in the bulk [15] and a phase transition can occur at a percolation threshold potential of around −0.5 V/SCE [16]. This is also shown in the Nyquist diagram (Fig. 4) with a blocking interface in −0.5 V/SCE and a partially non-blocking interface in more negative (more Cd content) potentials with diffusing Cd atoms. Similar to most percolation phenomena, a phase and a geometrical transition can be occurred in threshold contents of Cd. In addition to the percolation concentration p, another parameter, the ratio r, of the rate of dissolution to the diffusion rate is introduced. For r 1, diffusion is the limiting step (as in our case for potentials higher than −0.5 V/SCE) and for low concentration of Cd atoms, clusters with a diffusion-limited aggregation structure [17] are obtained. In the opposite limit, when r goes to zero and when the concentration of Cd is high (close to the percolation threshold and in the potential more negative than −0.5 V/SCE) percolation-like patterns are formed. Chemical dissolution of one component may simply reveal the structure of the other. So, at large scales (m) the pattern cannot be fractal since the average ratio between the two components is fixed and at small scales (mm) the Cd-inserted Hg film can be a fractal as shown in this study. 4. Conclusions To summarize, fractal dimensions of Cd-inserted mercury films were calculated to be about 2.11 and 2.54 near the surface of the mercury film and at deeper points inside
K. Malek, F. Gobal / Materials Chemistry and Physics 78 (2003) 796–799
the film, respectively. The insertion process does not significantly affect the surface morphology of mercury and it appears smooth. It is shown that there is a phase transition phenomena during Cd-insertion (de-insertion) process into the Hg film. This corresponds to a percolation threshold of about 0.2 mol l−1 Cd content. In this case, the bulk of the film is a percolation system and has a dimension of 2.54 near the threshold. The percolation structure and fractal dimension of this binary-mixed-metal predicts an amalgam structure instead an alloy. References [1] M.D. Levi, E. Levi, D. Aurbach, M. Schmidt, R. Oesten, U. Heider, J. Power Sources 97/98 (2001) 525. [2] A. Van der Ven, M.K. Aydinol, G. Ceder, G. Kresse, J. Hafner, Phys. Rev. B 58 (1998) 2975. [3] P.J. Gellings, H.J.M. Bouwmeester (Eds.), The CRC Handbook of Solid State Electrochemistry, CRC Press, Boca Raton, FL (1997).
799
[4] D. Shaw, J. Electron. Mater. 24 (1995) 587. [5] J.R. MacDonald, Impedance Spectroscopy Emphasizing Solid Materials and Systems, Wiley, New York, 1987. [6] M. Stromme, J. Isidorson, G.A. Niklasson, C.G. Granqvist, J. Appl. Phys. 80 (1996) 233. [7] K. Malek, Thin Solid Films 408 (2002) 73. [8] K. Malek, F. Gobal, Synth. Met. 113 (2000) 169. [9] F. Gobal, K. Malek, M.G. Mahjani, M. Jafarian, Synth. Met. 108 (2000) 15. [10] K. Malek, M.-O. Coppens, Phys. Rev. Lett. 87 (2001) 125505. [11] T. Pajkossy, L. Nyikos, Electrochim. Acta 34 (1989) 171. [12] M. Stromme, G.A. Niklasson, C.G. Granqvist, Solid State Commun. 96 (1995) 151. [13] D. Stauffer, A. Aharony, Introduction to Percolation Theory, Francis & Taylor, London, 1985, p. 69. [14] M. Sahimi, Application of Percolation Theory, Francis & Taylor, London, 1994 (Chapter 12) pp. 193–224. [15] D. Avnir (Ed.), The Fractal Approach to Heterogeneous Chemistry, Wiley, New York, 1989. [16] K. Aoki, Y. Teragishi, J. Electroanal. Chem. 441 (1998) 25. [17] A. Bunde, S. Havlin (Eds.), Fractals and Disordered Systems, 2nd ed., Springer, Berlin, 1996, pp. 66–69.
Percolation of cadmium across a mercury film K. Malek a,∗ , F. Gobal b a
Delft Department of Chemical Technology, Delft University of Technology, Julianalaan 136, 2628 BL Delft, The Netherlands b Department of Chemistry, Sharif University of Technology, P.O. Box 11365-9516, Tehran, Iran Received 23 April 2002; received in revised form 31 July 2002; accepted 12 August 2002
Abstract Electrodeposition/dissolution of cadmium onto a film of mercury shows some deviations from the natural liquidity of mercury caused by the reduction of Cd onto it. Percolation and fractal analyzes were done on the surface and the bulk of the mercury film during diffusion of Cd species (atoms). These show that the fractal dimensions of the Cd-inserted mercury film are about 2.11 and 2.54 near the surface of the mercury film and at deeper points inside the film, respectively. The insertion process has a negligible effect on the surface morphology of the mercury film and there is a phase transition in the bulk, as well as a geometrical transition during the Cd-insertion (de-insertion) process. This corresponds to a percolation threshold of about 0.2 mol l−1 Cd content. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Thin films; Coatings; Electrochemical techniques; Transport properties
1. Introduction In recent years there has been a growing interest in the study of the diffusion of electrochemically generated species in the electrode matrix due to its effect on the morphology [1] and physico-chemical properties [2] of electrochemical deposited films, employed in the construction of sensors, optical devices, batteries, memories [3] etc. An understanding of the relevant diffusion processes is necessary if optimum results are to be achieved by diffusion anneals. On the other hand, atom diffusion is an important part of the science and technology of a mixed-metal mercury electrode [4] (e.g. mercury cadmium telluride, MCT). A mercury electrode is basically an ideally smooth electrode without any roughness inside it and in the boundary of solution. This property appears in the impedance response of a mercury drop electrode in an inert solution as a straight line parallel to the imaginary axis in a Nyquist plot [5]. Structural changes in the morphology during diffusion of small species have an important effect on the type and amount of diffusivity. In general, two distinct diffusion regimes can be identified: (a) diffusion occurs without any net chemical fluxes; i.e. the material is chemically homogeneous and (b) the material is chemically inhomogeneous so that diffusion gives rise to net chemical fluxes. Regime (a) represents diffusion at chemical equilibrium and gives rise to self-diffusion coefficients. For ∗ Corresponding author. Tel.: +31-15-27-84685; fax: +31-15-27-88267. E-mail address: [email protected] (K. Malek).
the interpretation and rationalization of the diffusion properties, the concepts of fractal dimensions of the matrix of the doped films [6] and percolation of dopants (intercalates) in the film have been used [7]. In our previous studies [8–10] we have applied the above concepts to describe the properties of doped species and their behavior during diffusion through the film. The purpose of the present work is to investigate the electrodeposition/dissolution of cadmium into mercury films, focusing on possible deviations from the natural liquidity of mercury caused by the reduction of Cd. This can be done by characterization of the percolative and fractal structure of the surface and the bulk of the resultant film during diffusion of Cd species (atoms). For this purpose, cyclic voltammetry and electrochemical impedance spectroscopy methods have been used. 2. Experimental Films of mercury were prepared on copper rods, previously carefully cleaned by dichloromethane and dilute nitric acid followed by rinsing with distilled water. A two-electrode cell, copper in contact with a drop of mercury forming the cathode and a platinum plate forming the anode, in a 1 M KCl solution is used. Increasing the potential slowly forms a thin layer of fairly stable mercury on the copper surface. Although copper is partially soluble in mercury, the bulk of the film is pure mercury except for
0254-0584/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 2 5 4 - 0 5 8 4 ( 0 2 ) 0 0 3 8 6 - 3
K. Malek, F. Gobal / Materials Chemistry and Physics 78 (2003) 796–799
the copper/mercury interface, which is a copper amalgam. The thickness of the film measured gravimetrically was 6 ± 1 m. Electrochemical deposition/dissolution of Cd onto this electrode was studied in a conventional three-electrode cell where the mercury film was the working electrode with its potential monitored against a standard calomel electrode and a platinum plate formed the counter electrode. The electrolyte was 0.01 M CdCl2 + 0.5 M KCl solution of Analar grade of Hopkins and Wiliams origin. Cyclic voltammetry (CV) and electrochemical impedance spectroscopy (EIS) studies were done using an EG&G electrochemical system comprising of a 273A potentiostat/galvanostat and a 1250 Lock-in-amplifier run by an IBM ps/value point computer with M270 and M398 software packages.
3. Results and discussion Fig. 1 represents the cyclic voltammograms of Cd deposition/dissolution onto a Hg film recorded at various potential sweep rates. As we are particularly interested in the diffusion of Cd in the film, delays of 180 s at the end of the cathodic half cycles have been exercised prior to the anodic runs. During the anodic half cycle, at high potential sweep rates there is a large peak possessing a shoulder, which is clearly observed as a distinct peak at lower potential sweep rates. The slopes of the potential sweep rate dependency on the anodic peaks currents in the log–log scale provides α which is related [11] to the fractal dimension, Df , of the surface (and the bulk) through the relation: Df − 1 α= (1) 2 Applying this expression to the results presented in Fig. 1 gives the fractal dimensions of the surface and bulk of the film to be around 2.11 and 2.54, respectively. The α values have been derived from a log–log plot of the main peak and the shoulder versus sweep rate (Fig. 2), within a reasonable order of magnitude to establish a fractal relation. Furthermore, attempts have been made to correct for the effect of
Fig. 1. Cyclic voltammograms signifying the process of cadmium deposition/dissolution onto mercury film at various potential sweep rates.
797
Fig. 2. Log–log plots of the peak current vs. sweep rate.
background capacitance current and internal resistance (IR) drops. The main anodic peak is due to the dissolution of Cd species concentrated either on or near the surface of the mercury film while the second peak arises from the dissolution of Cd atoms, originating from deeper points inside the mercury film. Thus, the fractal dimensions correspond to the surface and the bulk topography of the film as shown in Fig. 3. Applying a modified Sevcik equation for fractal electrodes [12] to the results of the bulk peak, the diffusion coefficient, D, of Cd in the Hg film is obtained as 2.2 × 10−12 m2 s−1 . Using Eq. (2): x =
zFADCb ip
(2)
and assuming Cb = 0.32 M for Cd on the basis of the volume of the film and the time of electrolysis (cathodic delay), the width of the diffusion layer, x, for the fastest and the slowest potential sweep rates have been obtained: 1.6 × 10−7 and 5.32 × 10−6 m, respectively. In Eq. (2), F is the Faraday constant, A is the area of the electrode (∼ =2 cm2 )
Fig. 3. Illustration of the surface and bulk topography of the mercury film during Cd deposition/dissolution.
798
K. Malek, F. Gobal / Materials Chemistry and Physics 78 (2003) 796–799
Fig. 5. Variation of the conductivity of the mercury film with cadmium concentration in the bulk of the film.
nearly 2.53 is expected for a three-dimensional percolative system [13,14]. It has been shown in a percolation system. β Df = D − (3) ϑ
Fig. 4. Impedance responses of the cadmium loaded in Hg/solution interface in the form of the Nyquist plots recorded at various direct-current offset potentials:(a) −0.9, −0.7; (b) −0.5 V vs. SCE. Inset shows the equivalent circuit used to analyze the impedance data.
and ip is the peak current. Interestingly enough, the diffusion lengths are in accordance with the independently measured film thickness. Fig. 4 represents the Nyquist diagrams of the cadmium loaded mercury/solution interface at various direct-current offset potentials. In all cases the loadings were performed at −1.4 volt versus standard calomel electrode (V/SCE) and the EIS studies were performed after constant anodic currents were reached. At −0.5 V/SCE, the Hg film is almost free of Cd and a single time constant characterizing the interface has been observed. A single rather squashed semi-circle appears in the Nyquist plot. At more negative potentials where the loading of Cd is higher, a semicircle terminating to a segment of a straight line at the low frequency end of the spectrum has been observed. The equivalent circuit has been shown in Fig. 4a, which is compatible with the results. In both cases the first squashed semicircle is associated with the mercury film capacitance–resistance characteristics while the semicircle in the medium-frequency range signifies the charge transfer associated with the dissolution of Cd. The straight line at the low-frequency end of the spectrum shows the dominance (activation) of mass transfer across the Hg film, which can be considered as a Warburg impedance. Fig. 5 represents the variation of the film’s conductivity with the bulk concentration of Cd. A percolation phenomenon, with the threshold at nearly 0.2 mol l−1 Cd content, has been observed. At both sides of this concentration the conductivity of the Hg/Cd system rises, with the steepest rise at the Cd increasing side. A fractal dimension of
where D = 3, and β and ϑ are the critical exponents (currently accepted values of β = 0.41 and ϑ = 0.88) which control the percolation probability and the correlation length in a percolation system, respectively. This ratio is virtually constant in a three-dimensional system with a Df value of 2.53. In the system under study, the Hg and Cd atoms can interdiffuse by an exchange process in the bulk [15] and a phase transition can occur at a percolation threshold potential of around −0.5 V/SCE [16]. This is also shown in the Nyquist diagram (Fig. 4) with a blocking interface in −0.5 V/SCE and a partially non-blocking interface in more negative (more Cd content) potentials with diffusing Cd atoms. Similar to most percolation phenomena, a phase and a geometrical transition can be occurred in threshold contents of Cd. In addition to the percolation concentration p, another parameter, the ratio r, of the rate of dissolution to the diffusion rate is introduced. For r 1, diffusion is the limiting step (as in our case for potentials higher than −0.5 V/SCE) and for low concentration of Cd atoms, clusters with a diffusion-limited aggregation structure [17] are obtained. In the opposite limit, when r goes to zero and when the concentration of Cd is high (close to the percolation threshold and in the potential more negative than −0.5 V/SCE) percolation-like patterns are formed. Chemical dissolution of one component may simply reveal the structure of the other. So, at large scales (m) the pattern cannot be fractal since the average ratio between the two components is fixed and at small scales (mm) the Cd-inserted Hg film can be a fractal as shown in this study. 4. Conclusions To summarize, fractal dimensions of Cd-inserted mercury films were calculated to be about 2.11 and 2.54 near the surface of the mercury film and at deeper points inside
K. Malek, F. Gobal / Materials Chemistry and Physics 78 (2003) 796–799
the film, respectively. The insertion process does not significantly affect the surface morphology of mercury and it appears smooth. It is shown that there is a phase transition phenomena during Cd-insertion (de-insertion) process into the Hg film. This corresponds to a percolation threshold of about 0.2 mol l−1 Cd content. In this case, the bulk of the film is a percolation system and has a dimension of 2.54 near the threshold. The percolation structure and fractal dimension of this binary-mixed-metal predicts an amalgam structure instead an alloy. References [1] M.D. Levi, E. Levi, D. Aurbach, M. Schmidt, R. Oesten, U. Heider, J. Power Sources 97/98 (2001) 525. [2] A. Van der Ven, M.K. Aydinol, G. Ceder, G. Kresse, J. Hafner, Phys. Rev. B 58 (1998) 2975. [3] P.J. Gellings, H.J.M. Bouwmeester (Eds.), The CRC Handbook of Solid State Electrochemistry, CRC Press, Boca Raton, FL (1997).
799
[4] D. Shaw, J. Electron. Mater. 24 (1995) 587. [5] J.R. MacDonald, Impedance Spectroscopy Emphasizing Solid Materials and Systems, Wiley, New York, 1987. [6] M. Stromme, J. Isidorson, G.A. Niklasson, C.G. Granqvist, J. Appl. Phys. 80 (1996) 233. [7] K. Malek, Thin Solid Films 408 (2002) 73. [8] K. Malek, F. Gobal, Synth. Met. 113 (2000) 169. [9] F. Gobal, K. Malek, M.G. Mahjani, M. Jafarian, Synth. Met. 108 (2000) 15. [10] K. Malek, M.-O. Coppens, Phys. Rev. Lett. 87 (2001) 125505. [11] T. Pajkossy, L. Nyikos, Electrochim. Acta 34 (1989) 171. [12] M. Stromme, G.A. Niklasson, C.G. Granqvist, Solid State Commun. 96 (1995) 151. [13] D. Stauffer, A. Aharony, Introduction to Percolation Theory, Francis & Taylor, London, 1985, p. 69. [14] M. Sahimi, Application of Percolation Theory, Francis & Taylor, London, 1994 (Chapter 12) pp. 193–224. [15] D. Avnir (Ed.), The Fractal Approach to Heterogeneous Chemistry, Wiley, New York, 1989. [16] K. Aoki, Y. Teragishi, J. Electroanal. Chem. 441 (1998) 25. [17] A. Bunde, S. Havlin (Eds.), Fractals and Disordered Systems, 2nd ed., Springer, Berlin, 1996, pp. 66–69.