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Diffusion to fractal surfaces—IV. The case of the rotating disc electrode of fractal surface
Elecrrochimica Acra, Vol. 35, No. 9, pp. 1423-1424, 1990 Printed in Great Britain.
0013-4686/90 $3.00 + 0.00 0 1990. Perpmon Press pk.
DIFFUSION TO FRACTAL SURFACES-IV. THE CASE OF THE ROTATING DISC ELECTRODE OF FRACTAL SURFACE L. NYIKOS,* T. PAJKOSSY,* A. P. J~OROSY*and S. A. MARTEMYANOV~ *Central Research Institute for Physics, H-1525 Budapest, P.O. Box 49, Hungary ?A. N. Frumkin Institute of Electrochemistry,
117071 Moscow V-71, Leninski prospect, 31, U.S.S.R.
(Received 13 June 1989; in revised form 5 December
1989)
Abstract-The limiting current of a fast electrode reaction on a rotating, inactive disc with an electrochemically active fractal subset was theoretically predicted to he proportional to w’, with o being the rotation frequency, and the exponent a was determined by the fractal dimension, D,, of the active area as a = (Df - 1)/2. The experimental verification is presented. Key words: Levich equation, partially active surfaces, hydrodynamics.
EXPERIMENTAL
INTRODUCTTON The Cottrell response, ie the decay of diffusioncontrolled current of particles diffusing from an initially homogeneous medium to a completely absorbing fractal boundary was previously shown and verified[l, 21 to exhibit a t -” time dependence
rather than the conventional f-r’* decay, with the exponent CI being determined solely by the fractal dimension, 0,[3], of the interface as a = (Dr- 1)/2. Since the diffusion length-a time-dependent “yardstick”-measures the fractal dimension proper, this result is quite general As a consequence of this modified time dependence, the shape of the linear sweep and cyclic voltammograms also change[4] and it is the same CI value which enters the modified expressions describing the voltammograms measured at fractal interfaces. In this note we deal with another consequence of the generalized diffusion kinetics which might serve as a tool to describe the Levich response of the partially active or blocked but otherwise smooth rotating disc electrode (rde) provided the active area can be modelled as fractal. On an rde, the current is under the control of both convection and diffusion. The Levich expression[5] describing the current, j, as a function of the angular velocity, w, of this electrode in an electrolyte containing redox species of fast electrode kinetics predicts a jawri2 relation. As we have recently shown[6], the Levich equation can be generalized for the rde with fractal (Or < 2) surface and j(o) depends on w as: j(w)ccw’, (1) with a = (D, - 1)/2-again with the same exponent as in the cases above. In what follows we describe the experiment performed to verify this predicted power-law behaviour.
Fractal rdes were constructed by using platinum electrodes of well-defined fractal geometry with D,= log(3)/log(2) = 1.585 and having the same pattern as that used in Ref.[Z]. The method of manufacturing was very similar to that described in Ref.[Z] but an approx. 1 pm thick S&N, layer was deposited through a fractal mask for the insulation of the major part of the electrode instead of photoresist to enhance chemical stability. Several rdes were manufactured by using cylindrical teflon holders. Silicon rubber or polypropylene was used for the insulation of the silver epoxy contacts and for forming the planar face of the rde s. For control purposes, conventional (Or = 2) platinum rdes were also made by using exactly the same methods with the only difference that the fractal mask was absent. The electrodes were rotated by a Tacussel ED1 rotator driven by a programmable dc voltage source. The rotation frequency was measured by means of a lamp-mirror-photodiode-amplifiercounter arrangement. The current was measured potentiostatically at -0.2 V potential us see in an electrolyte containing approx. 5 mM of KJFe(CN),J, 5 mM Kr[Fe(CN)J, 10 mM KCN and 0.5 M NaNO,. A standard electrochemical cell and a potentiostat type EF-430 (Hungarian make) was used. The measurement was controlled and data were processed by a Hewlett-Packard HP 86A microcomputer. The instruments were hooked up to the computer uiu an HP-IB interface. The rotation speed was set by the computer program. To check system stability, the speed was scanned up and down. The counter (Keithley 775A) and the multimeter digitizing the current (Keithley 192) were read out by the microcomputer. In order to normalize currents to unit microscopic area, the active area of the electrodes was determined as follows. Limiting currents were measured in the
1423
L. NYIKOSet al.
1424
Cottrell response of the fractal electrodes (before the conventional-to-fractal crossover, cjI Ref.[Z]) and the
I
diffusion coefficient obtained with the Tacussel rde were used to estimate the active, microscopic area of the fractal N
I
'E, a .;
%
electrodes.
RESULTS The measured current us angular frequency data pairs are shown in Fig. 1. The results of the control experiments (lower traces) closely follow the Levich equation. For the fractal rdes, the measured data points fall on straight lines drawn with the slope predicted by equation (l), thus verifying the theoretical result of our previous work[6].
$
log f.lJ/s-’ Fig. 1. Rotating disc electrode current (normalized to unit microscopic area) as a function of the angular velocity. Data points on lines l-3 were measured on fractal electrodes whereas those around lines 4-6 were measured on conventional r&s. The lines are drawn with the slopes predicted by equation (1) and the classical Levich equation, respectively. same solution by using the conventional, maskless electrodes and a Tacussel platinum rde with a wellknown area, and the ratio of the respective currents yielded the area ratios. The short-time portion of the
REFERENCES 1. L. Nyikos and T. Pajkossy, Electrochim. Acta 31, 1347 (1986). 2. T. Pajkossy and L. Nyikos, Electrochim. Acta 34, 171 (1989). 3. B. B. Mandelbrot, The Fractal Geometry of Nature, Freeman, San Francisco (1982). 4. T. Pajkossy and L. Nyikos, Electrochim. Acta 34, 181 (1989). 5. V. G. Levich, Physicockemical Hydrodynamics, Prentice
Hall, Englewood Cliffs (1962). 6. L. Nyikos, T. Pajkossy and S. A. Martemyanov, Elektrokhimiya 25, 1543 (1989).
0013-4686/90 $3.00 + 0.00 0 1990. Perpmon Press pk.
DIFFUSION TO FRACTAL SURFACES-IV. THE CASE OF THE ROTATING DISC ELECTRODE OF FRACTAL SURFACE L. NYIKOS,* T. PAJKOSSY,* A. P. J~OROSY*and S. A. MARTEMYANOV~ *Central Research Institute for Physics, H-1525 Budapest, P.O. Box 49, Hungary ?A. N. Frumkin Institute of Electrochemistry,
117071 Moscow V-71, Leninski prospect, 31, U.S.S.R.
(Received 13 June 1989; in revised form 5 December
1989)
Abstract-The limiting current of a fast electrode reaction on a rotating, inactive disc with an electrochemically active fractal subset was theoretically predicted to he proportional to w’, with o being the rotation frequency, and the exponent a was determined by the fractal dimension, D,, of the active area as a = (Df - 1)/2. The experimental verification is presented. Key words: Levich equation, partially active surfaces, hydrodynamics.
EXPERIMENTAL
INTRODUCTTON The Cottrell response, ie the decay of diffusioncontrolled current of particles diffusing from an initially homogeneous medium to a completely absorbing fractal boundary was previously shown and verified[l, 21 to exhibit a t -” time dependence
rather than the conventional f-r’* decay, with the exponent CI being determined solely by the fractal dimension, 0,[3], of the interface as a = (Dr- 1)/2. Since the diffusion length-a time-dependent “yardstick”-measures the fractal dimension proper, this result is quite general As a consequence of this modified time dependence, the shape of the linear sweep and cyclic voltammograms also change[4] and it is the same CI value which enters the modified expressions describing the voltammograms measured at fractal interfaces. In this note we deal with another consequence of the generalized diffusion kinetics which might serve as a tool to describe the Levich response of the partially active or blocked but otherwise smooth rotating disc electrode (rde) provided the active area can be modelled as fractal. On an rde, the current is under the control of both convection and diffusion. The Levich expression[5] describing the current, j, as a function of the angular velocity, w, of this electrode in an electrolyte containing redox species of fast electrode kinetics predicts a jawri2 relation. As we have recently shown[6], the Levich equation can be generalized for the rde with fractal (Or < 2) surface and j(o) depends on w as: j(w)ccw’, (1) with a = (D, - 1)/2-again with the same exponent as in the cases above. In what follows we describe the experiment performed to verify this predicted power-law behaviour.
Fractal rdes were constructed by using platinum electrodes of well-defined fractal geometry with D,= log(3)/log(2) = 1.585 and having the same pattern as that used in Ref.[Z]. The method of manufacturing was very similar to that described in Ref.[Z] but an approx. 1 pm thick S&N, layer was deposited through a fractal mask for the insulation of the major part of the electrode instead of photoresist to enhance chemical stability. Several rdes were manufactured by using cylindrical teflon holders. Silicon rubber or polypropylene was used for the insulation of the silver epoxy contacts and for forming the planar face of the rde s. For control purposes, conventional (Or = 2) platinum rdes were also made by using exactly the same methods with the only difference that the fractal mask was absent. The electrodes were rotated by a Tacussel ED1 rotator driven by a programmable dc voltage source. The rotation frequency was measured by means of a lamp-mirror-photodiode-amplifiercounter arrangement. The current was measured potentiostatically at -0.2 V potential us see in an electrolyte containing approx. 5 mM of KJFe(CN),J, 5 mM Kr[Fe(CN)J, 10 mM KCN and 0.5 M NaNO,. A standard electrochemical cell and a potentiostat type EF-430 (Hungarian make) was used. The measurement was controlled and data were processed by a Hewlett-Packard HP 86A microcomputer. The instruments were hooked up to the computer uiu an HP-IB interface. The rotation speed was set by the computer program. To check system stability, the speed was scanned up and down. The counter (Keithley 775A) and the multimeter digitizing the current (Keithley 192) were read out by the microcomputer. In order to normalize currents to unit microscopic area, the active area of the electrodes was determined as follows. Limiting currents were measured in the
1423
L. NYIKOSet al.
1424
Cottrell response of the fractal electrodes (before the conventional-to-fractal crossover, cjI Ref.[Z]) and the
I
diffusion coefficient obtained with the Tacussel rde were used to estimate the active, microscopic area of the fractal N
I
'E, a .;
%
electrodes.
RESULTS The measured current us angular frequency data pairs are shown in Fig. 1. The results of the control experiments (lower traces) closely follow the Levich equation. For the fractal rdes, the measured data points fall on straight lines drawn with the slope predicted by equation (l), thus verifying the theoretical result of our previous work[6].
$
log f.lJ/s-’ Fig. 1. Rotating disc electrode current (normalized to unit microscopic area) as a function of the angular velocity. Data points on lines l-3 were measured on fractal electrodes whereas those around lines 4-6 were measured on conventional r&s. The lines are drawn with the slopes predicted by equation (1) and the classical Levich equation, respectively. same solution by using the conventional, maskless electrodes and a Tacussel platinum rde with a wellknown area, and the ratio of the respective currents yielded the area ratios. The short-time portion of the
REFERENCES 1. L. Nyikos and T. Pajkossy, Electrochim. Acta 31, 1347 (1986). 2. T. Pajkossy and L. Nyikos, Electrochim. Acta 34, 171 (1989). 3. B. B. Mandelbrot, The Fractal Geometry of Nature, Freeman, San Francisco (1982). 4. T. Pajkossy and L. Nyikos, Electrochim. Acta 34, 181 (1989). 5. V. G. Levich, Physicockemical Hydrodynamics, Prentice
Hall, Englewood Cliffs (1962). 6. L. Nyikos, T. Pajkossy and S. A. Martemyanov, Elektrokhimiya 25, 1543 (1989).