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The hanging meniscus rotating disk (HMRD) Part 2. Application to simple charge transfer reaction kinetics
.io~fllo.t o1~
ELSEVIER
Journal of Electroanalytical Chemistry 385 (1995) 39-44
The hanging meniscus rotating disk (HMRD) Part 2. Application to simple charge transfer reaction kinetics H.M. Villullas *, M. Lopez Teijelo INFIQC, Depto. de Fisico Qu[mica, Facultadde Ciencias Qu[micas, Universidad Nacional de Cdrdoba, Sue.If, C.C.61, 5016 Cdrdoba, Argentina Received 9 August 1994; in revised form 21 September 1994
Abstract The hanging meniscus rotating disk (HMRD) is a configuration in which the electrode material is studied without first being mounted in an insulating mantle. It was recently shown that the hydrodynamic behavior of this system is similar to that of the conventional rotating-disk electrode. In this paper the applicability of this configuration to the study of simple charge transfer reaction kinetics is analyzed and some experimental data are presented. The kinetic parameters for simple charge transfer reactions can be obtained using the HMRD as well as on conventional rotating-disk electrodes.
1. ingroducfion
The rotating-disk electrode (RDE) [1] has been widely used in research work in electrochemistry. One of its most important applications has been to the study of charge transfer reaction kinetics [2-4]. The RDE is quite simple to construct and commonly used forms involve an electrode material embedded into Teflon, epoxy resin or another plastic. However, for soft materials and metal single cystals, the usual procedure of press-fitting the sample into the insulating mantle can damage the crystal structure severely. To overcome the difficulties of mounting single crystals as rotating-disk electrodes, the hanging meniscus rotating disk (HMRD) has recently been developed [5] as a combination of the pendant meniscus method [6] and the conventional RDE. The HMRD uses a cylinder with no external shroud which is simply lowered until contact with the electrolyte is made and then raised to develop a hanging meniscus [5,7]. Measurements for the ferro/ferricyanide couple on Au have demonstrated that the hydrodynamic behavior of the HMRD is similar to that of a conventional R D E [5,7,8]. Linear plots of limiting current (I L) vs. tot/2, where to is the rotation rate, have been obtained and show the
* Corresponding author. 0022-0728/95/$09.50 © 1995 Elsevier Science S.A. All rights reserved SSDI 0022-0728(94)03763-9
same slope as that expected for a conventional RDE and a small negative intercept [7]. However, the omission of the insulating shroud and the presence of the electrolyte meniscus introduce different hydrodynamic boundaries. Once the liquid in the hydrodynamic layer reaches the disk edge, it changes direction and flows down along the meniscus surface. This reversal of flow direction at the edge results in a reduction of effective electrode radius for the convective diffusion current. According to Cahan [9], this perturbed layer should have a thickness similar to the thickness 8o of the hydrodynamic boundary layer. Thus the effective radius is given by Rerf = R 0 K(v/to) !/2, where R0 is the geometric radius and K is a proportionality constant. We have recently studied the hydrodynamic behavior of the HMRD by varying experimental parameters such as electrode material, electrode diameter and meniscus height [8]. A correction term was introduced into Levich equation to account for the reduction of the effective electrode radius promoted by the change in flow direction [9]. From this [8], IL(HMRD) =
0.620nFD2o/3 v - i/t,c~ tot/2 ¢rR02 x[l-
2KRotCvlto) '/2]
(1)
where R 0 is the geometric radius, K is a proportionality constant and all the other symbols have their usual meanings [10,11]. The first term in Eq. (1) is exactly the
H.M. V'dlulla& M. Lopez Teijelo/Journal of Electroanalytical Chemistry 385 (1995) 39-44
Levich equation and the second term, which ~not depend on rotation speed, accounts for the negative intercept and is given by
intercept ffi-(O.620nFD2/3v-l/6c~a)2KRoiv 1/2 (2) where the quantity in parentheses is exactly the slope of the standard Levich equation. The validity of Eq. (I) has been demonstrated [8] and allowed us to explain the value of the slope and the negative intercept. The thickness of the flow reverml r e , o n was shown to be a function of rotation speed, meniscus height and electrode radius. The value of the proportionality constant K depends on meniscus height h m as follows:
K~k(hm-ho)
(3)
where h0 is the critical meniscus height above which h m should be set to prevent meniscus climb on the cylinder side [8] (lateral wetting occurs at and below h 0 and Eq, (1) is no longer valid). The values of K and h0 were found to depend on specific properties of the electrode Imeniscus interface (contact angle) and the constant k appears to be a function of disk diameter
[8]. The usual procedure for extracting quantitative kinetic information for charge transfer reactions taking place on a RDE is based on the method of Frumkin and Tedoradse [12], i.e. for first-order kinetics by plotting 1/1 vs. ~o-n/2. However, these plots are not expected to be linear, for the HMRD. In t.his paper we discuss the applicability of this configuration to studies of the kinetics of simple charge transfer reactions under mixed control.
2.Experimental Experiments were carried out on Au, Pt and glassy carbon (GC) electrodes. These materials were used as cylinders mounted in an HMRD configuration as described elsewhere [5]. Conventional RDEs (Pine Instruments) were also used for comparison. Experiments were run using ferro/ferricyanide and Fe2+/Fe 3+ redox couples as test reactions. Solutions were prepared from AR reagents and ultrapure water produced by a MilliRo-MilliQ system. A glass cell for rotating electrodes (Pine Instruments) was employed. A gold sheet separated by a glass frit served as the counter-electrode. All potentials were measured against a HgIH82SO4 I1 M SO~- reference electrode. All experiments were performed using an AFSR Pine rotator. Experiments were carried out at different rotation rates ranging from 400 to 8100 rev min-n. The meniscus height was measured carefully (+ 0.001 cm) using a dial gauge as described elsewhere [5].
3. Remits and discussion
3.1. Current-rotation rate relations It has been shown that, for any meniscus height for which lateral wetting does not occur, the IL--tO1/2 plots for the HMRD have a nonzero intercept [7,8] as shown in Fig. 1 for ferrocyanide oxidation on gold. The corresponding data for the RDE are included in all the figures for comparison. Data obtained for this system were treated by the method of Frumkin and Tedoradse [12], i.e. by plotting I / I vs. ¢o-1/2 (for first-order kinetics) as shown in Fig. 2. The nonlinear behavior due to the independent term in the IL-¢On/z relationship for the HMRD (Eq. (1)) is clearly seen. Since the effective radius tends towards the true radius when the rotation rate is increased [8], the curve for the HMRD approaches the linear behavior shown by the conventional RDE in the region of high oJ. Although the presence of an independent term in the IL-¢O~/2 equation for the HMRD seems to introduce a limitation on the use of this configuration in kinetic studies, it is rather simple to show that it can be overcome easily. The current for a simple charge transfer reaction taking place at any given potential E can be written as
1 = nFak( E)c~ (4) where k(E) is the potential-dependent rate constant, c s is the surface concentration of reacting species, p is the order of the reaction and the other symbols have their usual meanings [10,11]. On a RDE and for nonlimiting-current conditions, the current will be given by
/ROE 0.620nFAD~/3v- 1/%1/2( Ct~ _ Cs) =
(5)
Defining the current lm~, in the absence of any mass transfer effects as
lm~ = nFAk( E)( c~ )"
(6)
1,6 RDE
J---~
,,/.r"
.~/'Jm ~J'HMRD
/ 0.06 . . . .
!
J 2'0 ",,~ "4b" ,',,
"6b . . . .
80
Fig. 1. Plot o f ! L vs. 03I/2 for ferrocyanide oxidation on Au: [] R D E ; m H M R D at a meniscus height of 0.185 cm.
H.M. Villullas, M. Lopez Teijelo/Journal of Electroanalytical Chemistry 385 (1995) 39-44
2.0
and combining these equations for first-order kinetics, we obtain the very well-known equation where B is the Levich constant and Bo~~/2 is the limiting current. According to Eqs. (2) and (3), the limiting current on a H M R D can be rewritten as
where
z=2gvt/2Rol
(9)
If the Levich constant B is replaced by B'= B/ct; for nonlimidng-current conditions, the current on a HMRD wi~,l be given by
(10)
I/2 - B'Z)( cq; -c,.)
The limiting current measured oil tile HMRD will be smaller than that measured on a RDE because of the presence of a region at the edge of the disk where reversal of flow direction occurs [8,9]. However, the available area for the charge transfer reaction should be equal to the total electrode area. Thus Eqs. (4) and (6) should also hold for the HMRD at any meniscus height as long as lateral wetting does not occur, i.e. at any h m > ho. Combining Eqs. (4), (6) and (10) for first-order kinetics yields 1 / ( BoJ il2 + BZ)
(11)
It is clear from a comparison of Eqs. (7) and (11) that the general equation for first-order kinetics 111 = 111..,. + l I I L
(12)
is also valid for the HMRD. Thus, for first-order kinetics, a plot of 1/1 vs. 1/1L should be linear with a slope of unity. Moreover, if the whole electrode area of the HMRD is available for the charge transfer reaction, plots of 1/1 vs. 1/1L for the
3.0 s,
J
7
.r"'~'/~ RDE L,~.5
0'~.~0
E1 E2
\ \
(8)
IttHMRD) = B t ° t l 2 - B Z
1/1HMRD = 1/lma x +
b
(7)
1 / / R D E = l lima x + 1/B(o l/2
i.MR.) = (
41
'0.02 . . . .
0.04
'
'
'0.06
w-v~/rpm-V ~ Fig. 2. Plot of 1/I vs. t o - i / 2 for ferrocyanide oxidation on Au: ra RDE; [] H M R D at a meniscus height of 0.185 cm. The current was measured at constant overpotential (0.080 V).
0"40. 4
1.'2
( WZ,) / m,4'
2.0
Fig. 3. Plot of 1 / i vs. 1/! L for ferroeyanide oxidation on Au at t w o different overpotentials ( E I = 0.080 V and E 2 = 0.130 V): 13 RDE; I H M R D at a meniscus height of 0.185 cm.
HMRD and for the RDE should be coincident, yielding exactly tile same extrapolation at 1/1L = 0. In fact, this was observed and is shown in Fig. 3 for the oxidation of ferrocyanide on gold at two different overpotentials. It is worth noting that data for both the RDE and the HMRD lie exactly on the same straight line for each potential and that the slopes are equal to unity as expected. It should also be noted that data points corresponding to the HMRD are displaced with respect to the corresponding points measured on the RDE at identical rotation rate. A pair of data points obtained at the same rotation speed are indicated as a and b in Fig. 3 to illustrate this displacement, which is due to the fact that smaller currents are measured on the HMRD configuration. Although the calculation of rate constants, in itself, is not of particular interest in the present work, it is the aim in most studies of reaction kinetics. It is worth pointing out that data for the HMRD lie exactly on the same straight lines as those measured on the RDE and consequently yield the same value of Ima x a t any given potential. This obviously implies that the values of the Tafel slope and the reaction order can be obtained in the same way as for data measured using the conventional RDE [2-4]. It is well known that the Levich constant B is a function of potential when the system is studied close to equilibrium [11]. Thus, 1/1 vs. to -I/2 plots for a first-order charge transfer reaction on a RDE will show a variation in the slope in this potential region. Such a variation should also affect the 1/I vs. 1/I L plots, and the expected increase in slope for decreasing overpotentials was observed. Nevertheless, the values of lm, x obtained by extrapolation t o 1/1 L = 0 are still valid.
3.2. Influence of meniscus height It is interesting to note that, for increasing meniscus height, the curvature observed in 1/1 vs. to -~/2 plots
H.M. V'dhdlas,M. Lop~ Teijelo/Journal of Eiectroanalytical Chemistry 385 (1995) 39-44
42¸¸¸¸
T
0.8
hm //era
0.240 .u 0.222 ,.,~ 0.204
RDE
3.3. Shape of current-potential curves in the nonlimiting current region
0.2 0.~,
meniscus height above the point of lateral wetting, i.e. This restriction is based on the conditions for which the equations given above are valid and it is also extremely important in systems where lateral wetting may introduce undesirable contributions from the cylinder side, as in the case of single-crystal electrodes.
h m > h o [8].
0
0.20
Flg, 4, Plot of 1ti vs. ~= t/2 for the reductionof Fe3+ on RDE and HMRD GC electrodes. The meniscus heights for the HMRD are indicated on the figure, The current was measured at constant overpotential (0.40 V). becomes more pronounced, as shown for the reduction of Fe 3+ on GC electrodes in Fig. 4. In contrast, the 1/I vs. I/IL plots (Eq. (12))should coincide for all values of meniscus height, provided that lateral wetting is avoided by setting h m above ho. This behavior is illustrated in Fig. 5. It is clearly seen that all data lie on the same straight line of unit slope. Since the current values obtained on the H M R D are smaller than those obtained on the RDE and decrease for higher meniscii, it is clearly seen that data points are shifted with respect to those corresponding to the RDE at the same rotation rate. Again, since the effective radius approaches the geometric radius as the rotation rate is increased [8], differences become smaller at the highest values of limiting current. it should be emphasized that, although it seems clear that measurements taken at different meniscus heights will be equivalent and will yield the same kinetic parameters, the HMRD must be used at a 0,8
It seems clear from the data presented above that the current /max in the absence of any mass transfer effect is the same for both the RDE and the H M R D configurations. This behavior demonstrates that the whole area of the disk is available for the charge transfer reaction in the H M R D despite the reduction of effective radius that characterizes its hydrodynamic behavior [8]. For convective diffusion, a region of reversal flow direction operates at the edge of the disk on the HMRD, yielding limiting currents corresponding to the effective area which is smaller than the geometric area [8,9]. As shown in Fig. 1, the limiting current on the HMRD is smaller than that measured on the R D E (provided that the HMRD is correctly used, i.e. h m > h0). In the potential region of mixed control, the activation current Im~ and the mass-transport-controlled current I L combine to yield the total current (Eq. (12)). Since the mass-transport-controlled current It~HMRD) on the HMRD is smaller, the total current in the nonlimiting-current potential region will also be smaller. From the equations given above, the relation between these currents is 1/IHMRD = IlIRD E + Z~-i/211L(HMRD)
To illustrate this behavior, curves for a first-order charge transfer process were computer simulated. The
hm / c m
T
.),0.240
~ 0,6
o,o
RDE/JV~,O0222
/,' t" 1,/"
oo5
-,,,
f
/,' /,,\
~0,4
(13)
~?,"
.
.
.
.
0,2 0.00
O -'°
.
6 ~,,p6oi/z ,
°%o
"
o:2
"
/
"
, :, o .
)
Fig. 5. Plot of 1/i vs. 1 / i L for the reduction of Ve 3+ on GC electrodes. Experimental conditions as in Fig. 4.
i
80
i oo
t
Fig. 6. Calculated 1-¢oW2 curves at 0.10 V overpotential for the R D E and H M R D configurations: k o = l × | 0 -3 cm s-Z; a = 0 . 5 ; T = 298 K; c~ -- 1 x 10 -3 M; v -- 0.01 cm 2 s - i; Do __ 1 × 10 -5 cm 2
s-I; Ro--0.30 cm. For the HMRD, K--2.50. The corresponding limitingcurrents (/L) are also included.
ll.M. Viihdlas, M. Lopez Teijelo/Journal of Electroanalytical Chemistry 385 (1995) 39-44
43
0.12
E/v
\
,'21 i
~
0.08
0.95 0.55 0.50
,,,,,,,, 6 t
\
0.45
I
0.04 .
.
.
.
RDE HMRD
0.40
j
ooo0
0.12
0 14
nrr
Ely
0.35
0~ ' 0.~
o.'8
1 3 -o
Fig. 7. Simulated I - E curves for simple charge transfer reactions (kt=l×10-3; k z = l × 1 0 -5 cm s -=) for the R D E and H M R D configurations. Calculations performed for ~ = 400 rev rain- !. Other parameters as for Fig. 6.
l-co =/2 curves calculated for an overpotential in the
nonlimiting=current region are shown in Fig. 6 for both configurations (limiting currents are also included). At high rotation rates the current difference #Rrm - IHMRD tends towards zero, as expected from the variation of effective radius with rotation rate [8]. However, for decreasing ~o this current difference approaches a constunt value given by the difference in the limiting currents, which corresponds to the negative intercept observed in the Levich plot for the HMRD. The current difference IRo E -IHMRD is also potential dependent. This implies that current-potential curves taken on the HMRD should show a slight change in shape compared with those measured on a conventional RDE. To illustrate these changes, calculated I - E curves for two different values of rate constant are shown in Fig. 7 for both the HMRD and the RDE. It is clear that in the potential region where the reaction rate is under kinetic control the curves cointide. In the limiting-current region, the difference is constant and equals the value of the negative intercept, while in the nonlimiting current region the rate at which the current approaches the limiting value will be 0.5
0.4 ~
w/~rn ....
RDE HMRD
,,,~'~~
0.3
~'~'0.2
~
.
6400
.-.'-~_..'=-.-.-." 3600
. . . . . . .
1600
0.1
0"0C
0.4
E/v
0.8
1.2
Fig. 8. Simulated I - E curves for the RDE and H M R D configurations at different rotation speeds. Calculations performed for k o = I x 10 -5 cm s- 1. Other parameters as for Fig. 6.
o
m
•
m
•
m
,, o o
Fig. 9. Current difference /RDE- /HMRD VS. ~1/2 curves for different overpotentiais. Calculations performed for ko = 1 × 10 -5 cm s-~. Other parameters as for Fig. 6.
a function of rotation rate. This is shown in Fig. 8, where it can be seen that the separation between the curves in the nonlimiting current region is more pronounced for the lowest value of ~o as consequence of the decrease in effective radius [8] that affects the mass transport process. The combined effect of rotation rate and potential on IRDE --IHMRO iS shown in Fig. 9 where this difference is plotted against ¢o1/2 for different overpotentials. In addition, it should be noted that when the shapes of the I - E profiles obtained on the RDE and HMRD are compared, the differences will also depend on meniscus height and other experimental variables. The smaller currents measured on the HMRD configuration in the nonlimiting and limiting current potential regions is a direct consequence of the reduction of effective radius produced by reversal of flow direction at the edge of the disk. It has already been shown that the effective radius is a function of meniscus height, electrode diameter and other properties such as the contact angle [8].
4. Conclusions Despite the differences in the hydrodynamic boundaries introduced in the HMRD because of the absence of an insulating mantle and the presence of an electrolyte meniscus, the hydrodynamic behavior of this configuration is similar to that of conventional rotating disks and can also be used to study the kinetics of simple charge transfer reactions. Although the reduction of the effective electrode radius caused by the reversal of flow direction at the edge of the disk affects the convective diffusion, it is clear that the whole electrode area is available for the charge transfer reaction on the HMRD. From the data and discussion presented above it seems quite clear that, for a simple charge transfer reaction, the activation current lma~ obtained on the HMRD at any given
44
H.M. V'dlullas,M. Lopez Teijelo/Journal of Electroanalyticai Chemistry 385 0995) 39-44
potential will be identical with the one obtained using a conventional RDE. This also applies to the potential region close ~o equil~rium. Tafel slopes and reaction orders can t,e calculated using the same procedure applied for th~ RDE. Measureme~,'s can be taken at any meniscus height provided~at no lateral wetting occurs. As long as h m is set above h0, the kinetic parameters calculated are independent of meniscus height. It has been demonstrated that the kinetic parameters for first-order charge transfer reactions can be obtained using the HMRD as well as on conventional rotating.disk electrodes. The equivalent equations for reaction orders other than unity should also hold. Additionally, the absence of an insulating shroud implies potential advantage for use in systems where making a seal at the electrode Jshroud interface is difficult. It also allows a wide variety of treatments to be applied to the electrode surface. This makes the HMRD a valuable tool for studying the kinetics of simple charge transfer reactions on single-crystal electrodes and other systems.
Acknowledsements This research has been supported by CONICET, CONICOR and Fundaci6n Antorchas. We also thank
valuable suggestions made by Professor E. Gileadi, Tel Aviv University.
References [I] V.G. Levich, Physicochemical Hydrodynamics. Prentice Hail, Englewood Cliffs,NJ, 1962. [2] A.C. Riddiford,Advances in Electrochemistryand Electrochem. ical Engineering,Vol. 4, Interscience,New York, 1966. [3] R.N. Adams, Electrochemistry at Solid Electrodes, Dekker, New York, 1969. [4] F. Opekar And P. Beran, J. Electroanal. Chem., 69 (1976) 1. [5] B.D. Cahan, H.M. Villullas and E.B. Yeager, Extended Abstracts, ECS Meeting, Atlanta, 1988, Abstract 532; B.D. Cahan, H.M. Villullas and E.B. Yeager, ,I. Electroanal. Chem., 306 (1991) 213. [6] D. Dickertmann, F.D. Koppitz and J.W. Schultze, Electrochim. Acta, 21 (1976) 967. [7] B.D. Cahan and H,M. Villullas, Extended Abstracts, ECS Meet. in8, Chicago, 1988, Abstract 707; B.D. Cahan and H.M. Villul. las, J. Electmanal. Chem., 307 (1991) 263. [8] H.M. Villullas and M. L6pez Teijelo, 43rd ISE Meeting, C6rdoba, Argentina, 1992, Abstract 6-014; H.M. Villullas and M. L6pez Teijelo, J. Electroanal. Chem., 384 (1995) 25. [9l B.D. Cahan, Prec. Syrup. on Modelling of Batteries and Fuel Cells, ECS Meeting, Phoenix, AZ, 1991. [10l A.J. Bard and L.R. Faulkner, Electrochemical Methods, Wiley, New York, 1980. [! 1] E. Gileadi, Electrode Kinetics, VCH, New York, 1993. [12] A. Frumkin and G. Tedoradse, Z. Elektrochem., 62 (1958) 251.
ELSEVIER
Journal of Electroanalytical Chemistry 385 (1995) 39-44
The hanging meniscus rotating disk (HMRD) Part 2. Application to simple charge transfer reaction kinetics H.M. Villullas *, M. Lopez Teijelo INFIQC, Depto. de Fisico Qu[mica, Facultadde Ciencias Qu[micas, Universidad Nacional de Cdrdoba, Sue.If, C.C.61, 5016 Cdrdoba, Argentina Received 9 August 1994; in revised form 21 September 1994
Abstract The hanging meniscus rotating disk (HMRD) is a configuration in which the electrode material is studied without first being mounted in an insulating mantle. It was recently shown that the hydrodynamic behavior of this system is similar to that of the conventional rotating-disk electrode. In this paper the applicability of this configuration to the study of simple charge transfer reaction kinetics is analyzed and some experimental data are presented. The kinetic parameters for simple charge transfer reactions can be obtained using the HMRD as well as on conventional rotating-disk electrodes.
1. ingroducfion
The rotating-disk electrode (RDE) [1] has been widely used in research work in electrochemistry. One of its most important applications has been to the study of charge transfer reaction kinetics [2-4]. The RDE is quite simple to construct and commonly used forms involve an electrode material embedded into Teflon, epoxy resin or another plastic. However, for soft materials and metal single cystals, the usual procedure of press-fitting the sample into the insulating mantle can damage the crystal structure severely. To overcome the difficulties of mounting single crystals as rotating-disk electrodes, the hanging meniscus rotating disk (HMRD) has recently been developed [5] as a combination of the pendant meniscus method [6] and the conventional RDE. The HMRD uses a cylinder with no external shroud which is simply lowered until contact with the electrolyte is made and then raised to develop a hanging meniscus [5,7]. Measurements for the ferro/ferricyanide couple on Au have demonstrated that the hydrodynamic behavior of the HMRD is similar to that of a conventional R D E [5,7,8]. Linear plots of limiting current (I L) vs. tot/2, where to is the rotation rate, have been obtained and show the
* Corresponding author. 0022-0728/95/$09.50 © 1995 Elsevier Science S.A. All rights reserved SSDI 0022-0728(94)03763-9
same slope as that expected for a conventional RDE and a small negative intercept [7]. However, the omission of the insulating shroud and the presence of the electrolyte meniscus introduce different hydrodynamic boundaries. Once the liquid in the hydrodynamic layer reaches the disk edge, it changes direction and flows down along the meniscus surface. This reversal of flow direction at the edge results in a reduction of effective electrode radius for the convective diffusion current. According to Cahan [9], this perturbed layer should have a thickness similar to the thickness 8o of the hydrodynamic boundary layer. Thus the effective radius is given by Rerf = R 0 K(v/to) !/2, where R0 is the geometric radius and K is a proportionality constant. We have recently studied the hydrodynamic behavior of the HMRD by varying experimental parameters such as electrode material, electrode diameter and meniscus height [8]. A correction term was introduced into Levich equation to account for the reduction of the effective electrode radius promoted by the change in flow direction [9]. From this [8], IL(HMRD) =
0.620nFD2o/3 v - i/t,c~ tot/2 ¢rR02 x[l-
2KRotCvlto) '/2]
(1)
where R 0 is the geometric radius, K is a proportionality constant and all the other symbols have their usual meanings [10,11]. The first term in Eq. (1) is exactly the
H.M. V'dlulla& M. Lopez Teijelo/Journal of Electroanalytical Chemistry 385 (1995) 39-44
Levich equation and the second term, which ~not depend on rotation speed, accounts for the negative intercept and is given by
intercept ffi-(O.620nFD2/3v-l/6c~a)2KRoiv 1/2 (2) where the quantity in parentheses is exactly the slope of the standard Levich equation. The validity of Eq. (I) has been demonstrated [8] and allowed us to explain the value of the slope and the negative intercept. The thickness of the flow reverml r e , o n was shown to be a function of rotation speed, meniscus height and electrode radius. The value of the proportionality constant K depends on meniscus height h m as follows:
K~k(hm-ho)
(3)
where h0 is the critical meniscus height above which h m should be set to prevent meniscus climb on the cylinder side [8] (lateral wetting occurs at and below h 0 and Eq, (1) is no longer valid). The values of K and h0 were found to depend on specific properties of the electrode Imeniscus interface (contact angle) and the constant k appears to be a function of disk diameter
[8]. The usual procedure for extracting quantitative kinetic information for charge transfer reactions taking place on a RDE is based on the method of Frumkin and Tedoradse [12], i.e. for first-order kinetics by plotting 1/1 vs. ~o-n/2. However, these plots are not expected to be linear, for the HMRD. In t.his paper we discuss the applicability of this configuration to studies of the kinetics of simple charge transfer reactions under mixed control.
2.Experimental Experiments were carried out on Au, Pt and glassy carbon (GC) electrodes. These materials were used as cylinders mounted in an HMRD configuration as described elsewhere [5]. Conventional RDEs (Pine Instruments) were also used for comparison. Experiments were run using ferro/ferricyanide and Fe2+/Fe 3+ redox couples as test reactions. Solutions were prepared from AR reagents and ultrapure water produced by a MilliRo-MilliQ system. A glass cell for rotating electrodes (Pine Instruments) was employed. A gold sheet separated by a glass frit served as the counter-electrode. All potentials were measured against a HgIH82SO4 I1 M SO~- reference electrode. All experiments were performed using an AFSR Pine rotator. Experiments were carried out at different rotation rates ranging from 400 to 8100 rev min-n. The meniscus height was measured carefully (+ 0.001 cm) using a dial gauge as described elsewhere [5].
3. Remits and discussion
3.1. Current-rotation rate relations It has been shown that, for any meniscus height for which lateral wetting does not occur, the IL--tO1/2 plots for the HMRD have a nonzero intercept [7,8] as shown in Fig. 1 for ferrocyanide oxidation on gold. The corresponding data for the RDE are included in all the figures for comparison. Data obtained for this system were treated by the method of Frumkin and Tedoradse [12], i.e. by plotting I / I vs. ¢o-1/2 (for first-order kinetics) as shown in Fig. 2. The nonlinear behavior due to the independent term in the IL-¢On/z relationship for the HMRD (Eq. (1)) is clearly seen. Since the effective radius tends towards the true radius when the rotation rate is increased [8], the curve for the HMRD approaches the linear behavior shown by the conventional RDE in the region of high oJ. Although the presence of an independent term in the IL-¢O~/2 equation for the HMRD seems to introduce a limitation on the use of this configuration in kinetic studies, it is rather simple to show that it can be overcome easily. The current for a simple charge transfer reaction taking place at any given potential E can be written as
1 = nFak( E)c~ (4) where k(E) is the potential-dependent rate constant, c s is the surface concentration of reacting species, p is the order of the reaction and the other symbols have their usual meanings [10,11]. On a RDE and for nonlimiting-current conditions, the current will be given by
/ROE 0.620nFAD~/3v- 1/%1/2( Ct~ _ Cs) =
(5)
Defining the current lm~, in the absence of any mass transfer effects as
lm~ = nFAk( E)( c~ )"
(6)
1,6 RDE
J---~
,,/.r"
.~/'Jm ~J'HMRD
/ 0.06 . . . .
!
J 2'0 ",,~ "4b" ,',,
"6b . . . .
80
Fig. 1. Plot o f ! L vs. 03I/2 for ferrocyanide oxidation on Au: [] R D E ; m H M R D at a meniscus height of 0.185 cm.
H.M. Villullas, M. Lopez Teijelo/Journal of Electroanalytical Chemistry 385 (1995) 39-44
2.0
and combining these equations for first-order kinetics, we obtain the very well-known equation where B is the Levich constant and Bo~~/2 is the limiting current. According to Eqs. (2) and (3), the limiting current on a H M R D can be rewritten as
where
z=2gvt/2Rol
(9)
If the Levich constant B is replaced by B'= B/ct; for nonlimidng-current conditions, the current on a HMRD wi~,l be given by
(10)
I/2 - B'Z)( cq; -c,.)
The limiting current measured oil tile HMRD will be smaller than that measured on a RDE because of the presence of a region at the edge of the disk where reversal of flow direction occurs [8,9]. However, the available area for the charge transfer reaction should be equal to the total electrode area. Thus Eqs. (4) and (6) should also hold for the HMRD at any meniscus height as long as lateral wetting does not occur, i.e. at any h m > ho. Combining Eqs. (4), (6) and (10) for first-order kinetics yields 1 / ( BoJ il2 + BZ)
(11)
It is clear from a comparison of Eqs. (7) and (11) that the general equation for first-order kinetics 111 = 111..,. + l I I L
(12)
is also valid for the HMRD. Thus, for first-order kinetics, a plot of 1/1 vs. 1/1L should be linear with a slope of unity. Moreover, if the whole electrode area of the HMRD is available for the charge transfer reaction, plots of 1/1 vs. 1/1L for the
3.0 s,
J
7
.r"'~'/~ RDE L,~.5
0'~.~0
E1 E2
\ \
(8)
IttHMRD) = B t ° t l 2 - B Z
1/1HMRD = 1/lma x +
b
(7)
1 / / R D E = l lima x + 1/B(o l/2
i.MR.) = (
41
'0.02 . . . .
0.04
'
'
'0.06
w-v~/rpm-V ~ Fig. 2. Plot of 1/I vs. t o - i / 2 for ferrocyanide oxidation on Au: ra RDE; [] H M R D at a meniscus height of 0.185 cm. The current was measured at constant overpotential (0.080 V).
0"40. 4
1.'2
( WZ,) / m,4'
2.0
Fig. 3. Plot of 1 / i vs. 1/! L for ferroeyanide oxidation on Au at t w o different overpotentials ( E I = 0.080 V and E 2 = 0.130 V): 13 RDE; I H M R D at a meniscus height of 0.185 cm.
HMRD and for the RDE should be coincident, yielding exactly tile same extrapolation at 1/1L = 0. In fact, this was observed and is shown in Fig. 3 for the oxidation of ferrocyanide on gold at two different overpotentials. It is worth noting that data for both the RDE and the HMRD lie exactly on the same straight line for each potential and that the slopes are equal to unity as expected. It should also be noted that data points corresponding to the HMRD are displaced with respect to the corresponding points measured on the RDE at identical rotation rate. A pair of data points obtained at the same rotation speed are indicated as a and b in Fig. 3 to illustrate this displacement, which is due to the fact that smaller currents are measured on the HMRD configuration. Although the calculation of rate constants, in itself, is not of particular interest in the present work, it is the aim in most studies of reaction kinetics. It is worth pointing out that data for the HMRD lie exactly on the same straight lines as those measured on the RDE and consequently yield the same value of Ima x a t any given potential. This obviously implies that the values of the Tafel slope and the reaction order can be obtained in the same way as for data measured using the conventional RDE [2-4]. It is well known that the Levich constant B is a function of potential when the system is studied close to equilibrium [11]. Thus, 1/1 vs. to -I/2 plots for a first-order charge transfer reaction on a RDE will show a variation in the slope in this potential region. Such a variation should also affect the 1/I vs. 1/I L plots, and the expected increase in slope for decreasing overpotentials was observed. Nevertheless, the values of lm, x obtained by extrapolation t o 1/1 L = 0 are still valid.
3.2. Influence of meniscus height It is interesting to note that, for increasing meniscus height, the curvature observed in 1/1 vs. to -~/2 plots
H.M. V'dhdlas,M. Lop~ Teijelo/Journal of Eiectroanalytical Chemistry 385 (1995) 39-44
42¸¸¸¸
T
0.8
hm //era
0.240 .u 0.222 ,.,~ 0.204
RDE
3.3. Shape of current-potential curves in the nonlimiting current region
0.2 0.~,
meniscus height above the point of lateral wetting, i.e. This restriction is based on the conditions for which the equations given above are valid and it is also extremely important in systems where lateral wetting may introduce undesirable contributions from the cylinder side, as in the case of single-crystal electrodes.
h m > h o [8].
0
0.20
Flg, 4, Plot of 1ti vs. ~= t/2 for the reductionof Fe3+ on RDE and HMRD GC electrodes. The meniscus heights for the HMRD are indicated on the figure, The current was measured at constant overpotential (0.40 V). becomes more pronounced, as shown for the reduction of Fe 3+ on GC electrodes in Fig. 4. In contrast, the 1/I vs. I/IL plots (Eq. (12))should coincide for all values of meniscus height, provided that lateral wetting is avoided by setting h m above ho. This behavior is illustrated in Fig. 5. It is clearly seen that all data lie on the same straight line of unit slope. Since the current values obtained on the H M R D are smaller than those obtained on the RDE and decrease for higher meniscii, it is clearly seen that data points are shifted with respect to those corresponding to the RDE at the same rotation rate. Again, since the effective radius approaches the geometric radius as the rotation rate is increased [8], differences become smaller at the highest values of limiting current. it should be emphasized that, although it seems clear that measurements taken at different meniscus heights will be equivalent and will yield the same kinetic parameters, the HMRD must be used at a 0,8
It seems clear from the data presented above that the current /max in the absence of any mass transfer effect is the same for both the RDE and the H M R D configurations. This behavior demonstrates that the whole area of the disk is available for the charge transfer reaction in the H M R D despite the reduction of effective radius that characterizes its hydrodynamic behavior [8]. For convective diffusion, a region of reversal flow direction operates at the edge of the disk on the HMRD, yielding limiting currents corresponding to the effective area which is smaller than the geometric area [8,9]. As shown in Fig. 1, the limiting current on the HMRD is smaller than that measured on the R D E (provided that the HMRD is correctly used, i.e. h m > h0). In the potential region of mixed control, the activation current Im~ and the mass-transport-controlled current I L combine to yield the total current (Eq. (12)). Since the mass-transport-controlled current It~HMRD) on the HMRD is smaller, the total current in the nonlimiting-current potential region will also be smaller. From the equations given above, the relation between these currents is 1/IHMRD = IlIRD E + Z~-i/211L(HMRD)
To illustrate this behavior, curves for a first-order charge transfer process were computer simulated. The
hm / c m
T
.),0.240
~ 0,6
o,o
RDE/JV~,O0222
/,' t" 1,/"
oo5
-,,,
f
/,' /,,\
~0,4
(13)
~?,"
.
.
.
.
0,2 0.00
O -'°
.
6 ~,,p6oi/z ,
°%o
"
o:2
"
/
"
, :, o .
)
Fig. 5. Plot of 1/i vs. 1 / i L for the reduction of Ve 3+ on GC electrodes. Experimental conditions as in Fig. 4.
i
80
i oo
t
Fig. 6. Calculated 1-¢oW2 curves at 0.10 V overpotential for the R D E and H M R D configurations: k o = l × | 0 -3 cm s-Z; a = 0 . 5 ; T = 298 K; c~ -- 1 x 10 -3 M; v -- 0.01 cm 2 s - i; Do __ 1 × 10 -5 cm 2
s-I; Ro--0.30 cm. For the HMRD, K--2.50. The corresponding limitingcurrents (/L) are also included.
ll.M. Viihdlas, M. Lopez Teijelo/Journal of Electroanalytical Chemistry 385 (1995) 39-44
43
0.12
E/v
\
,'21 i
~
0.08
0.95 0.55 0.50
,,,,,,,, 6 t
\
0.45
I
0.04 .
.
.
.
RDE HMRD
0.40
j
ooo0
0.12
0 14
nrr
Ely
0.35
0~ ' 0.~
o.'8
1 3 -o
Fig. 7. Simulated I - E curves for simple charge transfer reactions (kt=l×10-3; k z = l × 1 0 -5 cm s -=) for the R D E and H M R D configurations. Calculations performed for ~ = 400 rev rain- !. Other parameters as for Fig. 6.
l-co =/2 curves calculated for an overpotential in the
nonlimiting=current region are shown in Fig. 6 for both configurations (limiting currents are also included). At high rotation rates the current difference #Rrm - IHMRD tends towards zero, as expected from the variation of effective radius with rotation rate [8]. However, for decreasing ~o this current difference approaches a constunt value given by the difference in the limiting currents, which corresponds to the negative intercept observed in the Levich plot for the HMRD. The current difference IRo E -IHMRD is also potential dependent. This implies that current-potential curves taken on the HMRD should show a slight change in shape compared with those measured on a conventional RDE. To illustrate these changes, calculated I - E curves for two different values of rate constant are shown in Fig. 7 for both the HMRD and the RDE. It is clear that in the potential region where the reaction rate is under kinetic control the curves cointide. In the limiting-current region, the difference is constant and equals the value of the negative intercept, while in the nonlimiting current region the rate at which the current approaches the limiting value will be 0.5
0.4 ~
w/~rn ....
RDE HMRD
,,,~'~~
0.3
~'~'0.2
~
.
6400
.-.'-~_..'=-.-.-." 3600
. . . . . . .
1600
0.1
0"0C
0.4
E/v
0.8
1.2
Fig. 8. Simulated I - E curves for the RDE and H M R D configurations at different rotation speeds. Calculations performed for k o = I x 10 -5 cm s- 1. Other parameters as for Fig. 6.
o
m
•
m
•
m
,, o o
Fig. 9. Current difference /RDE- /HMRD VS. ~1/2 curves for different overpotentiais. Calculations performed for ko = 1 × 10 -5 cm s-~. Other parameters as for Fig. 6.
a function of rotation rate. This is shown in Fig. 8, where it can be seen that the separation between the curves in the nonlimiting current region is more pronounced for the lowest value of ~o as consequence of the decrease in effective radius [8] that affects the mass transport process. The combined effect of rotation rate and potential on IRDE --IHMRO iS shown in Fig. 9 where this difference is plotted against ¢o1/2 for different overpotentials. In addition, it should be noted that when the shapes of the I - E profiles obtained on the RDE and HMRD are compared, the differences will also depend on meniscus height and other experimental variables. The smaller currents measured on the HMRD configuration in the nonlimiting and limiting current potential regions is a direct consequence of the reduction of effective radius produced by reversal of flow direction at the edge of the disk. It has already been shown that the effective radius is a function of meniscus height, electrode diameter and other properties such as the contact angle [8].
4. Conclusions Despite the differences in the hydrodynamic boundaries introduced in the HMRD because of the absence of an insulating mantle and the presence of an electrolyte meniscus, the hydrodynamic behavior of this configuration is similar to that of conventional rotating disks and can also be used to study the kinetics of simple charge transfer reactions. Although the reduction of the effective electrode radius caused by the reversal of flow direction at the edge of the disk affects the convective diffusion, it is clear that the whole electrode area is available for the charge transfer reaction on the HMRD. From the data and discussion presented above it seems quite clear that, for a simple charge transfer reaction, the activation current lma~ obtained on the HMRD at any given
44
H.M. V'dlullas,M. Lopez Teijelo/Journal of Electroanalyticai Chemistry 385 0995) 39-44
potential will be identical with the one obtained using a conventional RDE. This also applies to the potential region close ~o equil~rium. Tafel slopes and reaction orders can t,e calculated using the same procedure applied for th~ RDE. Measureme~,'s can be taken at any meniscus height provided~at no lateral wetting occurs. As long as h m is set above h0, the kinetic parameters calculated are independent of meniscus height. It has been demonstrated that the kinetic parameters for first-order charge transfer reactions can be obtained using the HMRD as well as on conventional rotating.disk electrodes. The equivalent equations for reaction orders other than unity should also hold. Additionally, the absence of an insulating shroud implies potential advantage for use in systems where making a seal at the electrode Jshroud interface is difficult. It also allows a wide variety of treatments to be applied to the electrode surface. This makes the HMRD a valuable tool for studying the kinetics of simple charge transfer reactions on single-crystal electrodes and other systems.
Acknowledsements This research has been supported by CONICET, CONICOR and Fundaci6n Antorchas. We also thank
valuable suggestions made by Professor E. Gileadi, Tel Aviv University.
References [I] V.G. Levich, Physicochemical Hydrodynamics. Prentice Hail, Englewood Cliffs,NJ, 1962. [2] A.C. Riddiford,Advances in Electrochemistryand Electrochem. ical Engineering,Vol. 4, Interscience,New York, 1966. [3] R.N. Adams, Electrochemistry at Solid Electrodes, Dekker, New York, 1969. [4] F. Opekar And P. Beran, J. Electroanal. Chem., 69 (1976) 1. [5] B.D. Cahan, H.M. Villullas and E.B. Yeager, Extended Abstracts, ECS Meeting, Atlanta, 1988, Abstract 532; B.D. Cahan, H.M. Villullas and E.B. Yeager, ,I. Electroanal. Chem., 306 (1991) 213. [6] D. Dickertmann, F.D. Koppitz and J.W. Schultze, Electrochim. Acta, 21 (1976) 967. [7] B.D. Cahan and H,M. Villullas, Extended Abstracts, ECS Meet. in8, Chicago, 1988, Abstract 707; B.D. Cahan and H.M. Villul. las, J. Electmanal. Chem., 307 (1991) 263. [8] H.M. Villullas and M. L6pez Teijelo, 43rd ISE Meeting, C6rdoba, Argentina, 1992, Abstract 6-014; H.M. Villullas and M. L6pez Teijelo, J. Electroanal. Chem., 384 (1995) 25. [9l B.D. Cahan, Prec. Syrup. on Modelling of Batteries and Fuel Cells, ECS Meeting, Phoenix, AZ, 1991. [10l A.J. Bard and L.R. Faulkner, Electrochemical Methods, Wiley, New York, 1980. [! 1] E. Gileadi, Electrode Kinetics, VCH, New York, 1993. [12] A. Frumkin and G. Tedoradse, Z. Elektrochem., 62 (1958) 251.