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The ECE-DISP1 problem: general resolution via double channel electrode collection efficiency measurements
103
J. Electroanal. Chem., 297 (1991) 103-124 Elsevier Sequoia S.A., Lausanne
problem: general resolution via double The ECE-DISPl channel electrode collection efficiency measurements Patrick R. Unwin * Physical Chemistty (Received
Laboratory,
Oxford University, South Parks Roari, Oxford OXI 3QZ (UK)
25 May 1990; in revised form 7 August
1990)
AMTaCt T’he use of collection efficiency measurements, at the double channel electrode, is considered in the resolution of the “classical” ECE-DISPl problem. Particular attention is given to the case where the upstream (generator) electrode is held at a potential in the transport-limited current region, with the downstream (collector) electrode at a potential such that the initial heterogeneous electron transfer is reversed but sufficient over-potential is available to drive the second electron transfer, in the original sense, should it cccur. The theory is developed for the problem, relating the collection efficiency (the ratio of transport-limited currents at the collector and generator electrodes) to the geometries of the double electrode and channel cell, the solution velocity, the homogeneous rate constant, and the nature of the mechanism operating. It is shown that under the defined conditions the two mechanisms may be readily separated, and the kinetic domain accessible to the technique is identified. The theoretical predictions are tested on the reduction of fluorescein dianion in aqueous solution (pH 9.5-10.0). Collection efficiency measurements readily point to a DISPl assignment for this process, in agreement with previous work.
INTRODUCTION
The problem of identifying the origin of the second electron transfer heterogeneous (step (iii)) or homogeneous (step (iv))? - within the “classical” ECE-DISP mechanistic scheme [1,2] (considered below in term of a reduction process): A+e-+B
(Ei?
6)
B+C
69 (E,O >> EiO)
C+e-+D
(iii)
B+C+A+D
* Present
address:
0022-0728/91/$03.50
(iv)
Department
of Chemistry,
University
0 1991 - Elsevier Sequoia
S.A.
of Texas at Austin,
Austin,
TX 78712, USA.
104 TABLE
1
Definition of the ECE-DISP Mechanism
limiting mechanisms
step
63
(iii)
ECE
rds a
J
DISPl
rds
X
DISPZ
J
X
6)
(iv)
rds
a rds = rate determining step.
has received much attention in recent years [l-21]. Of the three limiting mechanisms (defined in Table 1) arising from the above scheme [2], only the DISP2 process is readily identified by virtue of it being second-order [2]. Resolving the ECE-DISPl competition has proved to be extremely difficult with many of the conventional electrochemical techniques [2,5]. For example, neither steady state transport-limited current measurements at rotating disc [6-S], channel [9] and ultramicro- [lo] electrodes nor half-wave potential-rotation speed characteristics at the rotating disc electrode [ll] in themselves distinguish between the two mechanisms within the bounds of typical experimental accuracy, although discrimination may be achieved if an independent value of the rate constant for step (ii) is available [9-111. In the case of linear sweep voltammetry, the shift in peak potential with scan rate is identical for the two processes when the wave is “two-electron” [2,12], whilst the peak current-scan rate characteristics are so close as to make discrimination unrealistic [13]. Likewise for single potential step chronoamperometry it is thought that the degree of experimental precision required prohibits general resolution [2,14-161, although this point has been disputed [17]. By turning to double potential step chronoamperometry (DPSC), with the first step to a potential in the transport-limited current region and the final step to a potential such that reaction (i) is reversed, but sufficient overpotential is still available to drive step (iii) in the original sense, should it occur, Sadant and co-workers have demonstrated that the ECE-DISPl problem can be resolved unambiguously [2,18]. This approach has been applied at both stationary [2,15,18-201 and rotating disc [5,21] electrodes. Bewick et al. [16] have also shown that the two competing pathways may be distinguished by monitoring the absorbance-time characteristics of species D following the application of a potential step and subsequent circuit opening. For completeness, mention should also be made of the extensive work of Saveant and co-workers [22-341 on indirect electrochemical methods, in which the ECE and DISP pathways can be characterised by providing a third reaction pathway which competes with the heterogeneous or homogeneous electron transfer to the intermediate C. “Double electrodes”, such as the rotating ring-disc electrode (RRDE) [35,36], double channel electrode (DChE) [37,38], paired microband electrode [39-411 and interdigitated array electrode [40,42] (amongst others), operated under steady-state
105
conditions in the “generator-collector” mode, may be used to provide analogous information to double step experiments at single electrodes. Surprisingly, however, collection measurements at double electrodes have not been applied explicitly to the ECE-DISPl problem, although the ECE process at the RRDE has been treated [43]. It is therefore the aim of this paper to consider, both theoretically and experimentally, the use of collection experiments in the general resolution of the ECE-DISPl problem. By setting the potentials at the (upstream) generator and (downstream) collector electrodes (Es_, and Em, respectively) relative to Eio and EiE such that Egen e Eio -=z E,, -=z E,O, those of steps (i)-(iv) which are relevant would be driven at the generator electrode and the following reactions would occur at the collector electrode (assuming the kinetics of the A/B and C/D couples to be sufficiently reversible): B - e- + A ( ECE/DISP~)
c + e-+
D (ECE)
The collection efficiency - the ratio of the transport-limited and generator electrodes N = IL,,, (collector) /Iti (generator)
c-i) (iii) currents at the collector
(1)
may then, under certain conditions, be quite different for the two mechanisms, thereby affording a route to the resolution of the problem. Until recently, the principal advantage of steady-state experiments over transient methods lay in the fact that the latter could be prone to complications when the period of interest (dictated by the kinetics of the reaction) was of the same timescale as the double layer charging process. However, the recent advent of the ultramicroelectrode [44-471 has considerably shortened the time over which double layer charging is significant [44-481, opening up the possibility of studying very fast reaction kinetics (on the microsecond timescale) via DPSC [48]. To compete with this, hydrodynamic electrodes must be able to deliver both a wide range of, and high, mass transport rates. Thus for the present work the DChE, as opposed to the hitherto more popular RRDE, has been selected, since it is anticipated that this criterion will be more readily satisfied using this electrode geometry [38]. In order to cover the widest possible range of mass transport rates, the theory for the problem is tackled numerically via the backwards implicit finite difference method (BIFDM) [49-511. This approach allows the solution of channel electrode problems using the full parabolic velocity profile to describe flow through the channel [49-511, whereas alternative analytical methods usually require the approximation of the parabolic profile by a linear one (the L&Que approximation [38,52]), thereby placing restrictions on the electrode geometries and solution velocities to which the theory may be applied [38,53]. The theoretical predictions are tested with experiments on the model DISPl system of the reduction of fluorescein dianion in aqueous solution (pH 9.5-10.0) [9,11,21].
106 THEORETICAL
FORMULATION
OF THE PROBLEM
ECE mechanism The steady-state
mass transport equations for species A, B and C within a ChE cell, when (a) the flow is fully developed and Poiseiulle in nature and (b) lateral and side edge diffusion can be neglected, are [9,38] D 8’[A]/+*
- uX8[A]/k
= 0
(2)
D 8*[B]/gy*
- u, a[B]/ax
- k,[B] = 0
(3)
D a*[C]/ay*
- v, a[C]/ax
+ kii[B] = 0
(4) The co-ordinate system for the DChE cell is defined in Fig. 1. In eqns. (2)-(4): kii is the rate constant for step (ii), D is a diffusion coefficient common to A, B and C and u, is the solution velocity in the x direction, given by [38] v,=q,{l-(y-h)*/h*}
(5)
where v, is the velocity in the centre of the channel. For the purpose of collection experiments, the upstream (generator) electrode is held at a potential in the transport-limited current region for the reduction process (i.e. the electron transfers denoted by steps (i) and (iii) are driven). The potential at the downstream (collector) electrode is such that the reduction of C is still driven, but B is oxidised to A (see above). The boundary conditions pertaining to this situation are: x=0,
ally:
y=o,
o
x1
[A]=[A]*,
[C]=O
(6)
a[B]/ay= -a[A]/ay, = ap]/ay = a[c]/ay = 0
[A] =o,
a[A]/ay [B] =O, ali
[B]=O,
X:
[cl =o
(7) (8)
[C] =0
(9)
a[A]/ay=a[B]/ay=a[C]/ay=O
(IO) where [A]* denotes the concentration of A in bulk solution. The collection efficiency (defined by eqn. 1) for the ECE mechanism is given by N=
x3(a[c]/ay),,C0
dx - /X3(a[Bl/8y)Y=0 x2
dx
dx}
01)
Fig. 1. Coordinate system for the double channel electrode cell
107
DISPl mechanism The mass transport equations for this case are: D i3*[A]/ay2 - u, a[A]/ax
+ kii[B] = 0
02)
D a*[B]/ay*
- 2k,,[B]
03)
- uX a[B]/ax
= 0
Again the upstream and downstream electrodes are held at potentials so that, respectively, reactions (i) and (-i) are driven. Thus the boundary conditions take the form: X=O,
all JY
[A] =[A]*
y=O,
O
[A]=O,
xi
a[~]/ay=
x*
[B] = 0
y= 2h,
ali
[B]=o a[B]/ay=
a[B]/ay=o
(14) -a[A]/ay
(15)
06) 07)
X: a[A]/ay=a[B]/ay=o
08)
The collection efficiency is now given by
dx
N=-
(19)
Method of solution Collection efficiencies for each of the mechanisms were calculated, ,for particular electrode and cell geometries and rate constants, by solving the specified mass transport equations with the appropriate boundary conditions using the backwards implicit finite difference method (BIFDM) [49-511. Details on the general implementation of the method can be found elsewhere [49-5154,551. New information, required for the present problem, is given in Appendix A. The calculations were carried out via a Fortran program, which was executed on the Oxford University Vax Cluster. A copy of the program is available from the author upon request. THEORETICAL
RESULTS
The collection efficiency, N, depends upon the DChE geometry (xi, x2, x3), the solution flow rate and the rate constant for the conversion of B to C in solution. The aim of this section is to illustrate the general effect of each of these parameters on N for the two mechanisms of interest, and to indicate the kinetic domain in which collection efficiency measurements can resolve the ECE-DISPl problem effectively. Effect of k, and V, An increase in k, or decrease in V, promotes the homogeneous decomposition of B both in the region of the generator and in its transit to the detector electrode. For the DISPl mechanism this is expected to lead to a reduction in the magnitude of N. This effect is born out in Fig. 2, which shows the dependence of Non k, and V, for
108
Fig. 2. A contour plot showing the dependence of Non kii and V, for the DISPl mechanism at a DChE defined by x1 = 0.10 cm, x2 = 0.15 cm, xg = 0.30 cm, d = 0.60 cm and h = 0.02 cm. A value of 10m5 cm’ s-l was assumed for D.
a DChE defined by x1 = 0.1 cm, xz = 0.15 cm, x3 = 0.3 cm, d = 0.6 cm and h = 0.02 cm. A value of N = - 0.360 is predicted for this geometry in the absence of kinetic complications (in the fast flow rate limit) and the data are seen to tend to
Fig. 3. A contour plot showing defined in Fig. 2).
the dependence
of Non
kii and Vf for the ECE mechanism
at a DChE (as
109
I .o.
0.5.
/-----
IogK
OISPI -0.1
-0.z
~ Fig. 4. Working curves of N vs. log,, K for the ECE and DISPl mechanisms at a DChE (as defined in Fig. 2).
this value under these conditions, but as kii is increased and V, decreased, N tends towards zero rapidly. The effect of k, and V, on N for the ECE mechanism is shown in Fig. 3 for the same DChE geometry. Overall the relationship is quite different from the DISPl case. Although N takes on negative values for small k, and/or large V, (as for DISPl), N becomes positive as these parameters are, respectively, increased and decreased. In the limit of very large k, and small V,, N tends towards zero. This is because under these conditions the ECE reaction is essentially “two-electron”, with neither of the intermediates B or C escaping from the generator electrode. As would be expected, the behaviour is similar to that found in double potential step chronoamperometry (DPSC) [2,18], for which the mass transport “parameter” relates to the times at which the currents on the forward and reverse transients are measured and the time for the potential step reversal. As with DPSC, it is clear that the large difference in the collection efficiency characteristics for the two mechanisms should allow the ECE-DISPl problem to be resolved unambiguously. For the range of parameters used to generate Figs. 2 and 3, Ddx,/Vrh -=K1. Under these conditions the LMque approximation holds [38], and so N is a unique function of the normal&d rate constant, K = k,( h2x;/4v;D)1'3
(20)
(a.i)
(a,ii)
(b,ii)
(c,i)
I Fig. 5. Concentration profiles for species (a) B and (b) C across l/10 of the channel at a DChE (of the geometry defined in Fig. 4) when an ECE mechanism operates. The data relate to K = (i) 0.191, (ii) 0.410 and (iii) 1.91. The contours are equally separated concentrations normalised with respect to the bulk solution concentration of A (i.e. [X]/[A] *, where X is B or C).
for each mechanism, as evidenced by Figs. 2 and 3. The numerical method can therefore be used to generate working curves for a particular DChE geometry. The result of this exercise, for the geometry defined above, is shown in Fig. 4 which serves to illustrate the large difference in the N-K characteristics for the two mechanisms. At small K (i.e. fast flow rate and/or small kii), N values are similar for the two processes, because under these conditions there is little decomposition of B (and consequent formation of C) on the timescale of the transit of B to the collector electrode. This point is illustrated by Fig. 5 (a,;) and (a$) for the ECE mechanism at the previously defined DChE with K = 0.191 (i.e. log,,(V,/cm3 s-l) = -0.5 and k, = 3 s-l, with the remaining parameters as defined above). The dramatic difference between the collection efficiency characteristics for the two mechanisms arises as K is increased. This leads to the build up of C and depletion of B over both electrodes, as depicted in Fig. 5 (b,i)-(c,ii) for the ECE mechanism. The fact that C is electroactive in the cathodic sense at the detector electrode (boundary condition 9) when the ECE mechanism operates means that N can take on positive values. This is obviously not the case for the DISPl mechanism.
111
-16
01
03 (x3-xJlcm
Fig. 6. The effect of gap and detector electrode lengths on N for the DISPl mechanism. The data relate to x1 = 0.1 cm, Vf = 0.1 cm3 s-l and kii = 2.0 s-l, with the other parameters as defined previously (i.e. K = 0.274).
Effect of gap and collector electrode lengths
Increasing the length of the gap between the generator and collector electrodes increases the transit time of B to the collector electrode, thereby promoting the decay of this species in solution. For the DISPl mechanism this leads to a diminution in the current at the collector electrode, as evidenced by Fig. 6. In the case of the ECE mechanism, Fig. 7 shows that N shifts from negative to positive
-03 .Ok
-12
0'1
03 (x3-x,llcm
0.5
Fig. 7. The effect of gap and detector electrode lengths on N for the ECE mechanism. The parameters are as defined in Fig. 6.
112
values with increasing gap length, since this serves to produce a large amount of C, which is subsequently collected at the downstream electrode. The length of the collector electrode, in relation to the gap and generator lengths, determines the quantities, and the distribution, of the various intermediates that are collected following their escape from the generator electrode. For the DISPl mechanism, increasing the length of the detector electrode (for a fixed gap length) leads to an increase in the amount of B collected, as evidenced by Fig. 6 which shows that N becomes more negative. Notice, however, that there is a limit to the collector length beyond which there is little, and eventually no, increase in N. This is expected given the instability of B in solution. For the ECE mechanism, the collector electrode length influences significantly the distribution of B and C in its vicinity and therefore both the magnitude and the sign of the current. For the case considered in Fig. 7, increasing the collector electrode length (when the gap is small) leads initially to N becoming more negative (collection of B is favoured). However, as the electrode is made longer the decay of B to C is promoted and so N shifts in the positive sense. For large gap lengths, B will have largely decayed to C in its transit to the collector electrode. N is therefore positive and increasing the collector length simply increases the amount of C collected (leading to an increase in N). Range of measurable rate constants The upper limit on the range of rate constants which may be measured via the technique is governed principally by (i) the highest rate of mass transport which may be attained at the generator electrode (in order to promote the escape of intermediates from this electrode for detection downstream) and (ii) the transit time between the generator and collector electrodes. In the consideration of the first point, the advantage of the non-uniformly accessible channel electrode over uniformly accessible electrodes (such as the rotating disc electrode) is that the rate of mass transport depends on both the solution velocity and the electrode length [38] (the shorter the electrode the higher the average rate of mass transport). In principle this allows the rate of mass transport to be varied over a very wide range, although this has not yet been fully realised practically. For the purpose of the following discussion, the parameters considered below have all been reported in various electrochemical experiments, but have yet to be employed together in a channel electrode experiment. The range of flow rates which may be implemented is limited at the upper end by the fact that flow must be laminar, i.e. the Reynolds number, Re = IJ,,h/v
(21)
where v is the kinematic viscosity of the solution, should not exceed 2000 [56]. It follows from eqn. (21) that the upper limit on u,, can be extended by shrinking the height of the channel. Taking a minimum value of h = 5 X 1O-3 cm (although shallower channels could be employed), a typical value of v = 0.01 cd s-i, and putting an upper limit of 200 on Re (to ensure that flow is well within the laminar regime), solution velocities up to u0 = 400 cm s-l could be employed. For a
113
conventional channel cell of width d = 0.6 cm, this corresponds to V, = 1.6 cm3 s-l, a value well below the maximum achieved practically of 66 cm3 s-l [57]. Given the above value of u,, the minimum electrode length for which diffusional edge effects contribute negligibly (< 1%) to the transport-limited current is ca. 10T3 cm [58], assuming a typical value of D = low5 cm2 s-l. Micro-band electrodes of this dimension are readily fabricated [39-421. Indeed, the technology exists to make much shorter electrodes [59]. As for the second point, the generator to collector transit time is minimised by using fast flow rates and a small gap. It follows from Figs. 6 and 7 that, in order to achieve discrimination between the two mechanisms in the fast kinetic limit, a long collector electrode is required, to ensure that the two mechanisms yield N values of opposite sign. A general DChE experiment involves measuring N over a wide range of flow rates. However, in order to assign an upper limit to the range of rate constants accessible in the case of the DISPl mechanism, the smallest value of N which can be measured at the fastest flow rate with reasonable accuracy (i.e. distinguished from zero current) is considered. Taking this value to be -0.02 (as considered previously for the EC reaction at the DChE [60]), together with the values for the parameters defined above, and a DChE characterised by x1 = 0.0010 cm, x2 = 0.0012 cm and x3 = 0.0112 cm, the maximum value of kii which could be determined for the DISPl reaction is calculated to be 7 X lo3 s-l. The question now arises as to whether discrimination from the ECE process can be achieved under these conditions. The value of k, corresponds to an effective number of electrons transferred at the upstream electrode n e.f = I,,&,,(
k, = 0)
(22)
of 1.331. Assigning a typical error of f0.02 to this neff value leads to k = 6.13 x lo3 (+ 0.61 x 103, -0.53 x 103) s-l if an ECE mechanism operates, and a resulting value of N = 0.167 ( + 0.008, - 0.010) for the defined DChE geometry. Thus, Iii,,- V, and N-V, analyses together. facilitate the ready resolution of the problem in the fast kinetic limit. If positive confirmation of an ECE mechanism alone is required, the kinetic domain accessible to the technique is greatly extended. For example, assuming N = 0.05 can be measured with an accuracy of f 0.01 at a DChE defined by x1 = 0.0010 cm, x2 = 0.0012 cm and x3 = 0.0212 cm, a process with k, = 2.0 x lo5 (+0.5 x 105, -0.3 X 105) s-l could be characterised. As for the measurement of slow kinetics, it is well established that hydrodynamic double electrodes provide a means of studying processes which are too slow for DPSC [61]. The investigation of slow processes requires the use of small mass transport rates. For the DChE, this dictates the use of small solution velocities and long electrodes [38]. The practical limits on these parameters are imposed by the point at which either lateral diffusion or natural convection become significant relative to forced lateral convection [38]. In experimental practice, no problems are encountered using a volume flow rate as low as low4 cm3 s-l and a generator electrode length of 0.4 cm (with h = 0.02 cm) [38]. If ILim- V, and N-V, analyses are again used together, then, taking neff = 1.1 as the minimum value which can be
114
measured accurately, it should be possible to characterise processes (ECE or DISPl) with k, = 6 X 10e3 s-l. Separation of the two mechanisms could then be achieved by employing a long gap and collector electrode (see above). For example, with x2 = 0.80 cm, x3 = 1.4 cm and the parameters defined above, values of N for the ECE and DISPl mechanisms of, respectively, +0.151 and -0.097 result. To summarise, DChE collection efficiency measurements are potentially extremely useful for the resolution of the ECE-DISPl problem, given the wide kinetic range (defined above) under which the two mechanisms can effectively be separated. EXPERIMENTAL
The apparatus for DChE voltammetry has been described previously [54] and so only additional details will be given here. The channel employed in this work had a cross-sectional dimension of 0.04 cm x 0.60 cm. Silver double electrodes were produced by casting two strips of silver foil (99.95% Goodfellow, Cambridge, UK) flush with one face of a resin block (Araldite: hardener HY219, resin CY219, Ciba-Geigy, Duxford, UK) which formed the coverplate to the channel [54,55]. Two DChE geometries were used: (A) x1 = 0.177 cm, x2 = 0.275 cm, x3 = 0.533 cm; (B) xi = 0.183 cm, x2 = 0.252 cm, x3 = 0.615 cm. An additional strip of silver foil located on the coverplate, upstream of the double electrode, functioned as a pseudo-reference electrode. Aqueous solutions of sodium fluorescein (ca. 3 mM) of pH 9.5-10-O and pH 13 were made up as described previously [9,21]. Collection efficiencies were determined at each flow rate by recording a currentvoltage curve for the reduction of the fluorescein dianion at the generator electrode. The resulting change in the current at the collector electrode, held at a potential corresponding to the foot of the generator electrode wave, was measured directly. All measurements were made under thermostatted conditions at 25 _+0.5 o C. EXPERIMENTAL
RESULTS
AND
DISCUSSION
As a standard DISPl system the reduction of fluorescein dianion (F2-) in aqueous solution @H 9.5-10.0) was investigated at a silver DChE. The reaction scheme takes the form [9,11,21]: generator electrode :
F2-+
solution :
F-3-+
solution : collector electrode :
e--s Fe3H+
FHe2-+ F-3-_
5
FH’Z-
Fe3-+ e-_,
F2-+
FH3-
F2-
Both transport-limited current ( Iti,,,) - flow rate (V,) characteristics at the generator electrode, and collection efficiency measurements, were of interest. The former measurements in isolation have been considered previously [9] and do not allow the ready resolution of the ECE-DISPl problem. Typical characteristics, obtained at pH 9.50 with the upstream electrode of DChE (A), are shown in Fig. 8. In the
115
150T
Fig. 8. Transport-limited current-flow rate characteristics for the reduction of F*- (solution pH = 9.50) at the upstream electrode of DChE (A).
absence of homogeneous kinetic complications a linear relationship between llim and V;‘/3 is predicted under the experimental conditions employed [62]: I,,
= 0.925nFw[F2-]*
( V,D2~;/h2d)1’3
(23)
where n is the number of electrons transferred per redox event. The behaviour for n = 1 was deduced by changing the solution pH to ca. 13 (through the addition of potassium hydroxide) where Fe3- is effectively stable on the timescale of channel electrode voltammetric measurements [9]. At pH 9.50 a smooth transition from twoto one-electron behaviour was observed as the flow rate was increased, as expected of an ECE-DISP system [9]. Under the experimental conditions utilised, the effective number of electrons transferred, nefr , is a unique function of the normal&d rate constant, K, for both the ECE and DISPl reactions [9]. Figure 9 shows the analysis of the Iti,,,-V, data in Fig. 8 using n,- K working curves for these mechanisms published elsewhere [9]. The data - plotted as K vs. qm2’3 (as suggested by eqn. 20) - analyse satisfactorily in terms of both mechanisms, confirming that this method alone does not permit mechanistic resolution. However, different values are yielded for k: 2.18 s-l (ECE) and 2.76 s-l (DISPl). Data obtained at pH 9.77 (using the upstream electrode of DChE (B) gave values for k of 1.30 s-l (ECE) and 1.42 s-l (DISPl). Collection efficiency data, obtained at the two pH defined above, are shown in Figs. 10a and b. The values of k, deduced from the neff - V, analyses can be used to simulate the behaviour expected for the two mechanisms. The results of this exercise are shown alongside the experimental data in Fig. 10. It is clear that the two mechanisms give rise to quite different collection efficiency characteristics, so that
116
Fig. 9. Analysis
of the data in Fig. 8 in terms of ECE (0) and DISPl
(0) mechanisms
[9].
resolution of the ECE-DISPl competition for this case is readily achieved. Comparison of the experimental data with the predicted behaviour for the two mechanisms points to a DISPl assignment for the reduction of F*-, in agreement with previous work [9,11,21].
----. +o .,_.
,*-----___
+0.1-
(01
-.
..
‘\
N
\
lb)
‘\
‘\
‘\
N '\ ECE '\ (V‘hn3S'P3
0-o:*
,---_
0.6
0%.O-$D)., cl
-Ol-
.. 0“.
O
--;--__ 0
'\ -o,‘\ *.
O-2
-Ol--
PJ~lcdi'l"~
I 0.6
04 '\ o-'-.Ry 0
:c\
‘,ECE \ '\ '\ '\
'. 0 '\
'\\ '1 '\ '\ 'b,, '\ '.
'\ .\ 0'
\
\\
'\\
'\
\
'\O
-02-
Fig. 10. Collection efficiency-flow rate characteristics for the reduction of F2- at (a) pH 9.50 using DChE (A), and (b) pH 9.77 using DChE (B). Also shown is the behaviour predicted theoretically for the ECE and DISPl mechanisms using the rate constants (defined in the text) deduced from limiting current measurements at the generator electrode.
117
As a check on the measurements, collection efficiencies were also determined for the reduction of F2- at pH 13 (for the reasons mentioned above). These data were found to be in good agreement (&- 2%) with theoretical predictions for the simple one-electron reduction of F2-, thereby confirming the validity of collection efficiency measurements with the DChE described. The above exercise serves to illustrate that Z,i,-V, and N- Vr measurements, employed together, provide a route to resolving the ECE-DISPl competition. The question naturally arises as to whether collection efficiency measurements in isolation allow the same objective to be realised. To answer this, N-Vr data obtained in the Ltv&que limit were converted to corresponding values of K using N-K working curves calculated for the two DChE geometries utilised. The resulting K-Vfp2/3 plots for each of the mechanisms at the two pHs are shown in Figs. lla and b. In contrast to Zlim - V, characteristics, a linear relationship between K and V,-2/3 results only for the DISPl, and not the ECE, mechanism. The former mechanism yields kii = (a) 1.75 s-’ and (b) 1.27 s-l - values which are in good agreement with those deduced from neft measurements. The results in Figs. lla and b suggest that analysis of collection efficiency data alone should be sufficient for resolving the ECE-DISPl problem. Of course, generator electrode limiting current-flow rate data are obtained when such measurements are made! These can also be analysed to provide a check on the mechanistic assignment, as illustrated above. The observation of a DISPl, rather than an ECE, mechanism for the reduction of fluorescein under the defined conditions is in agreement with the Amatore-Saveant rule that “ECE mechanisms do not occur in conditions where they could be characterised by electrochemical kinetic techniques” [63]. This rule, deduced with special regard to transient methods, was based on the fact that for the ECE mechanism to prevail the ECE-DISPl kinetic zone diagram dictates that log[ kii[A] */(k;i20’/2)] < -0.72 (where 0 is the measurement time in potential step experiments [14,18,64] and 0 = RT/Fu (u being the potential sweep rate) in linear sweep and cyclic voltammetry [13]). Simultaneous restrictions on the values of kii and 0 arise from the fact that nelf < 1.9. For aromatic or heteroaromatic substrates the disproportionation rate constant, ki,, is estimated to be diffusioncontrolled once the difference Eiz - Eio is greater than a few hundred millivolts [63]. With all these conditions and a substrate concentration, [A] *, at the millimolar level Amatore and Saveant illustrated that the ECE mechanism would operate only when measurement times were below one tenth of a microsecond [63] - inaccessible practically and hence the basis for the rule. A similar situation is likely to hold for other techniques [63]. The zone diagram for the ECE-DISPl competition at the channel electrode (see Appendix B) in Fig. 12 indicates that the larger the product ki,[A]*, the greater must be the value of kii for the ECE mechanism to dominate, and consequently the higher the rate of mass transport necessary to ensure that the upstream electrode reaction is not two-electron (i.e. sufficient intermediate escapes for collection downstream). For the single (and most favourable) case of nelf = 1.9 (which will yield a measurable collection efficiency), it follows from Fig. 12 that log K,, < 1.80. For ki, = 10” mol-’ dm3 s-’
118 0.P
(0) 0
K
0
0
/
0
04..
I__
0 0
0
0
O
0.
0
0
0
5
lo
(V,/ds-'12"
(b)
Fig. 11. Analysis (using eqn. 20) of the data in Fig. 10 in terms of the ECE (0) and DISPl (0) schemes, following conversion of N values to their kinetic equivalents for each mechanism via numerically calculated N-K working curves for the DChE geometries. (a) pH 9.50, DChE (A); (b) pH 9.77, DChE (B).
and a 1 mM substrate concentration must satisfy the following condition log( 4h4x$P/9V,%)
“3 < - 5.20
this implies
that the mass transport
parameters
(24)
Even if the transport parameters defined earlier, in the discussion of the highest rate of mass transport attainable at the channel electrode, are considered (together with D = 1O-5 cm’ s-l), the above quantity amounts to only - 3.7, which is well above that required for the observation of ECE characteristics. This therefore confirms
119
lY
a O-
-l.-
lelectron
I
0
-1
2 log K,
Fig. 12. Kinetic zone diagram for the ECE-DISPl conditions.
that the Amatore-SavCant transient, methods. APPENDIX
A. BACKWARD
competition
at the channel electrode under steady-state
rule applies to steady-state channel electrode, as well as
IMPLICIT
FINITE
DIFFERENCE
METHOD
The calculations of collection efficiencies using the BIFDM involves covering the three regions of interest in the cell (0 < x < xi, x, -Cx < x2 and x2 -Cx < x3) with a two-dimensional finite difference grid with increments in the x and y directions defined by: x,(k)=kAx,,
Ax,=x,/K,
k=O,l,...K,; j=O,l,...J;
y(j)=jAy,
Ay=2h/J
(Al) 642)
where x* is either xi, (x2-x1) or (xg -x2) and K* is the corresponding total number of x divisions for the particular zone. Typical values of J = 500-800 and K* = 1000 x*/x3 were employed in the calculations reported in this paper. Upon approximating the derivatives in the mass transport equations by their BIFD equivalents [49-511, invoking an explicit treatment of the kinetic terms [50], and applying the boundary conditions, the collection efficiency problem condenses to the solution of the following set of simultaneous equations: d, = b,u, + c*u* d, = ajuj_l d,_,
643)
+ bjuj + cjuj+ 1,
= aJ-Iu,_2
j=2
,... J-2
+ bJ--luJ--l
for each of the species requiring consideration.
(A41 645) Values for d,, dj (j = 2,. . . J - 1)
120 TABLE Al Values for d, mechanism
species
4
ECE
A B B C
E& g& d’,c &
A B B
$,
DISPl
$1
wne o-x,
-[‘?g!~kii(A.~)~/L’l+ ‘fgtk+l -N&%i(W2/4 +M&kiW~2/~l
o-x, x1 - x3 o-x3
+[?&kii(Ay)2/4 * t’&ii@v)2/W+ $:);&ky(Ay)2,~]
o-x,
o-x,
V&+l
Xl -
x3
TABLE A2 Values for dj (j=2,
3,...5-1)
mechanism
species
dj
zone
ECE
A B C
F+ giC,c-[~f+d‘W2/‘~l gj,k +[A;gj,kkti(AY)‘/oI
o-x,
A B
g;k
+[~f~~k&i(b’)2/~l
o-x,
g,,k
-[2+fk“ii(Av)2/Dl
o-x,
DISPl
o-x, o-x,
TABLE A3 Values for b, mechanism
species
4
zone
ECE, DISPl ECE, DISPl ECE, DISPl ECE ECE
A B B C C
1+2q 1+#4: 1+2h7 1+2x* 1+X:
o-x, o-x, x2-x3
o-x,, Xl -
x2-x) x2
and b, are given in Tables Al, A2 and A3. The expressions for aj (j = 2,. . . J - l), J - 1) and cj (j = 1,. . . J - 2) for all species and all zones are bj (j=2,... aj=
-hf
(A6)
bj = 1 + 2h;
(A71
cj=
(A81
-x;
In Tables Al, A2 and A3 and eqns. (A6)-(A8) “7 =
8D( Ax)h3d W;j(Ay)3(2h
-j
AY)
(A91
121
and gR = [B]/[A]
*
(AlO)
The computational strategy involves calculating the vector { uj ( j = 1, . . . J - l)}k from known {dj.(j= l,... J - l)}k by use of the Thomas algorithm [65,66]. The boundary condtttons (6) and (14) provide values respectively for the ECE and DISPl mechanisms, for the vectors {d }k=,, (for each of the species), from which { u }k co are calculated. These, in turn, are related to the new vectors { d }k= 1. The calculation thus proceeds down the cell through each of the three zones of interest. For the purpose of collection efficiency measurements, concentration profiles are evolved for A, B and C at the upstream electrode for the ECE mechanism, but only B and C need to be considered over the gap and the collector electrode. In the case of the DISPl process the A profile is generated at the upstream electrode only, and the B profile is calculated throughout all zones. The collection efficiencies are calculated from:
-A(x,
N =
- xZ)&,
i
1
)I(
AX+&)
(DISPI)
(AI2)
When the distance over which concentration changes in the direction normal to the DChE is small compared to the charmel height (i.e. for very fast flow rates and/or short DChEs, carrying out the finite difference calculations across the full channel becomes very inefficient. Under these conditions the region of interest in the Y-direction is therefore confined to a distance Ymax= 36, = (3/0.67)(
hx,D/~)“~
where 8, is the diffusion layer thickness at the trailing edge of the collector electrode (if the whole region 0 < x < x3 was a single electrode). The increments in the finite-difference grid in the y-direction are then given by AY
=y,m/J
This approach requires a change in boundary conditions (10) and (18) to Y +YIn,,
all x,
[A] + [A]*,
[B] +O,
[C] -0
(AI3)
which modifies the following matrix elements bJ_, = 2x;_, d,A_,.k = g_k,,, &,,=&i,~+%--l(I
+ 1 + A*,-,
(for A, B and C)
(AI4)
(ECE)
(AI51
+ [&&i(~Y)*,‘~])
(DISPl)
(AI6)
When compared to the original approach, the latter gives very accurate results, with a considerable saving in computer time.
122 APPENDIX B. KINETIC ZONE DIAGRAM FOR THE ECE-DISPI CHANNEL ELECTRODE
COMPETITION AT THE
When both the ECE and DISPl pathways contribute to the overall channel electrode reaction mechanism, the mass transport equations requiring consideration are: D a2[A]/ay2-u,
a[A]/ax+ki,[B][C]=O
(Bl)
D a2[B]/8y2
- U, a[B]/ax
- kii[B] - ki,[B][C]
= 0
(B2)
D a2[C]/8y2
- U, a[C]/ax
+ k,[B]
= 0
(B3)
- ki,IB][C]
In order to deduce the general kinetic zone diagram, conditions under which the LhiZque approximation [38], u, = 2 u, y/h, is valid are considered. Equations (Bl)(B3) may then be written in dimensionless form: a2a/i3t2 - 5 au/ax
+ K,bc = 0
a2b/at2-t ab/aX- Kb a2qa.p- ( at/ax + Kb -
(B4)
K,bc = 0
(B5)
K,bc
036)
where x = X/Xi
(B7)
5 = (2u,/hD~,)“~y
(B8)
Kd = ki,[A]*
(B9)
q=
( Iz~x;/~u,~D)~‘~
WIAI *
(q=a,
b or c)
(BlO)
and K is given by eqn. (20). It follows that the ECE-DISPl competition is governed by the two kinetic parameters K andKd; ECE being promoted by Kd + 0 and DISPl by K,, --, co for a given value of K. Equations (B4)-(B6) were solved with boundary conditions (6), (7) and (10) using the BIFDM (Appendix A). The following matrix elements change from those given in the preceding appendix for the pure ECE limiting mechanism (at the upstream electrode with semi-infinite boundary conditions in the y-direction): dpk = g;k +
[Xrg/IfkgF/ckiv CAY )‘[A] */D]
d~-*.l,=g~-l.k+h:-l(l+
(j=1,2,...&2)
[g,“-,,,g,‘_,,,ki,(A~)2[AI*/D])
d,B=g~k+hT([g~~+l] - [gPlkg~kkiv(AY)‘[AI*/D] - [gT,kii(hy)2/D]) diB=g~~-(XTg~k(Ay)‘/D){ g,S_*kiv[AI* +kti} dF=gJf,+
(j=2,3,...J-1)
(X:g~,(Ay)2/D){kii-g~kkiv[AI*}(j=1,2,...J-1)
To ensure that LCv&ue conditions applied, the calculation employed the following
123
parameters: V, = 0.1 cm3 s-l, h = 0.02 cm, xi = 0.1 cm, d = 0.6 cm and [A] * = 10e3 mol dmp3, with kii and k, being varied. The results obtained may be interpreted in terms of a kinetic zone diagram by considering the DISPl situation to apply when the current for the reduction of C is less than 5% of the total current and ECE behaviour corresponding to the reduction current of C being greater than, or equal to, 95% of that predicted for the pure ECE limiting mechanism [63]. The zone diagram deduced on this basis is shown in Fig. 12. The additional zones showing one- and two-electron behaviour correspond to n eff -c 1.05 and neff > 1.90, respectively. The behaviour observed is qualitatively similar to that found for transient techniques [13,14]; an increase in K displaces the system from one-electron behaviour into the ECE or DISPl zones, whilst the shift from ECE to DISPl occurs by decreasing K and/or increasing &. The zone diagram provides a means of deducing the conditions required for the operation of an ECE or DISPl mechanism. REFERENCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29
C.P. Andrieux and J.M. Saveant, J. Electroanal. Chem., 267 (1989) 15. C. Amatore, M. Gareil and J.M. Saveant, J. Electroanal. Chem., 147 (1983) 1, and refs. therein. M.D. Hawley and S.W. Feldberg, J. Phys. Chem., 70 (1966) 3459. R.N. Adams, M.D. Hawley and S.W. Feldberg, J. Phys. Chem., 71 (1967) 851. R.G. Compton, M.E. Laing, D. Mason, R.J. Northing and P.R. Unwin, Proc. R. Sot. Lond., A418 (1988) 113, and refs. therein. P.A. Malachesky, L.S. Marcoux and R.N. Adams, J. Phys. Chem., 70 (1966) 4068. S. Karp, J. Phys. Chem., 72 (1968) 1082. L.S. Marcoux, R.N. Adams and SW. Feldberg, J. Phys. Chem., 73 (1969) 2611. R.G. Compton, P.J. Daly, P.R. Unwin and A.M. Waller, J. Electroanal. Chem., 191 (1985) 15. M. Fleischmann, F. Lasserre and J. Robinson, J. Electroanal. Chem., 177 (1984) 115. R.G. Compton, R.G. Harland, P.R. Unwin and A.M. Waller, J. Chem. Sot., Faraday Trans. 1, 83 (1987) 1261. L. Nadjo and J.M. Saveant, J. Electroanal. Chem., 33 (1971) 419. C. Amatore and J.M. Saveant, J. Electroanaf. Chem., 85 (1977) 27. C. Amatore and J.M. Sav&nt, J. Electroanal. Chem., 102 (1979) 21. E.A. Ticianelli, L.A. Avaca and E.R. Gonzalez, J. Electroanal. Chem., 258 (1989) 379. A. Bewick, J.M. Mellor and B.S. Pons, Electrochim. Acta, 23 (1978) 77. P.J. Kinlen, R.D. Goodin and F.E. Woodard, J. Electroanal. Chem., 190 (1985) 185. C. Amatore and J.M. Saveant, J. Electroanal. Chem., 107 (1980) 353. C.P. Andrieux, P. Hapiot and J.M. Saveant, J. Am. Chem. Sot., 109 (1987) 3768. P. Hapiot, J. Moiroux and J.M. Saveant, J. Am. Chem. Sot., 112 (1990) 1337. R.G. Compton, D. Mason and P.R. Unwin, J. Chem. Sot., Faraday Trans. 1, 84 (1988) 2057. F. M’Halla, J. Pinson and J.M. Savtant, J. Electroanal. Chem., 89 (1978) 327. C. Amatore, J.M. Saveant and A. Thitbault, J. Electroanal. Chem., 103 (1979) 303. C. Amatore, J. Chaussard, J. Pinson, J.M. Saveant and A. Thiebault, J. Am. Chem. Sot., 101 (1979) 6012. J.M. Savtant, Act. Chem. Res., 13 (1980) 323. C. Amatore, J. Pinson, J.M. Saveant and A. Thiebault, J. Electroanal. Chem., 107 (1980) 59. C. Amatore, J. Pinson, J.M. Saveant and A. Thiebault, J. Electroanal. Chem., 107 (1980) 75. F. M’Halla, J. Pinson and J.M. Savtant, J. Am. Chem. Sot., 102 (1980) 4120. C. Amatore and J.M. Saveant, J. Electroanal. Chem., 123 (1981) 189.
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C. Amatore and J.M. Savt!ant, J. EIectroanal. Chem., 123 (1981) 203. C. Amatore, F.M’Halla and J.M. Savtant, J. Electroanai. Chem., 123 (1981) 219. C. Amatore, J. Pinson and J.M. Saveant, J. Electroanal. Chem., 123 (1981) 231. C. Amatore, J. Pinson, J.M. Saveant and A. Thitbault, J. Am. Chem. Sot., 103 (1981) 6930. C. Amatore, J. Pinson, J.M. Saveant and A. Thiebault, J. Am. Chem. Sot., 104 (1982) 817. W.J. Albery and M.L. Hitchman, Ring-Disc Electrodes, Clarendon Press, Oxford, 1971, and refs. therein. S. Bruckenstein and B. Miller, Act. Chem. Res., 10 (1977) 54, and refs therein. H. Gerischer, I. Mattes and R. Braun, J. Efectroanal. Chem., 10 (1965) 553. P.R. Unwin and R.G. Compton, in R.G. Compton and A. Hamnett (Eds.), Comp~bensive Chemical Kinetics, Vol. 29, Elsevier, Amsterdam, 1989, pp. 173-296, and refs. therein. D.G. Sanderson and L.B. Anderson, Anal. Chem., 57 (1985) 2388. A.J. Bard, J.A. Crayston, G.P. Kittlesen, T. Varco Shea and M.S. Wrighton, Anal. Chem., 58 (1986) 2321, and refs. therein. T. Varco Shea and A.J. Bard, Anal. Chem., 59 (1987) 2101. 0. Niwa, M. Morita and H. Tabei, Anal. Chem., 62 (1990) 447, and refs. therein. K.B. Prater and A.J. Bard, J. Electrochem. Sot., 117 (1970) 1517. M. Fleischmann, S. Pons, D. Rohson and P.P. Schmidt, Ultramicroelectrodes, Datatech Systems, 1987, and refs. therein. S. Pons and M. Fleischmann, AnaI. Chem., 59 (1987) 1391A, and refs. therein. R.M. Wightman, Science, 240 (1988) 415, and refs. therein. R.M. Wightman and D.O. Wipf in A.J. Bard (Ed.), Electroanalytical Chemistry, Vol. 15, Marcel Dekker, New York, 1989, p. 267, and refs. therein. C.P. Andrieux, P. Hapiot and J.M. Saveant, J. Phys. Chem., 92 (1988) 5992. J.L. Anderson and S. Moldoveanu, J. Eiectroanal. Chem., 179 (1984) 107. T.Y. Ou, S. Moldoveanu and J.L. Anderson, J. Electroanal. Chem., 247 (1988) 1. R.G. Compton, M.B.G. Pilkington and GM. Steam, J. Chem. Sot., Faraday Trans. I,84 (1988) 2155. M.A. Le+que, Ann. Mines Mem., Ser. 12, 13 (1928) 201. R.G. Compton and P.R. Unwin, J. Electroanal. Chem., 264 (1989) 27. R.G. Compton, G.M. Steam, P.R. Unwin and A.J. Barwise, J. Appl. Electrochem., 18 (1988) 657. R.G. Compton and P.R. Unwin, Philos. Tram R. Sot. Lond., A330 (1980) 1. R.B. Bird, W.E. Stewart and E.N. Lightfoot, Transport Phenomena, Wiley, New York, 1960, p. 47. F. Barr, C. Bernstein and W. Vielstich, in H. Gerischer (Ed.), Advances in Electrochemistry and Electrochemical Engineering, Vol. 13, Interscience, New York, 1984, pp. 261-353. K. Aoki, K. Tokuda and H. Matsuda, J. E&ztroanal. Chem., 217 (1987) 33. J.D. Seibold, E.R. Scott and H.S. White, J. Electroanal. Chem., 264 (1989) 281. K. Aoki, K. Tokuda and H. Matsuda, J. Eiectroanal. Chem., 79 (1977) 49. C.P. Andrieux and J. M. Sav&mt, in C.F. Bernasconi (Ed.), Investigation of Rates and Mechanisms of Reactions. Part II (Techniques of Chemistry), Vol. 6, Wiley, New York, 1986, pp. 305-391. V.G. Levich, Physicochemical Hydrodynamics, Prentice-Hall, Englewood Cliffs, NJ, 1962, pp. 112116. C. Amatore and J.M. Saveant, J. Electroanal. Chem., 86 (1978) 227. J.M. Saveant, Bull. Sot. Cbim. Fr., (1988) 225. L.H. Thomas, Watson Sci. Comput. Lab. Rept., Columbia University, New York, 1949. L. Lapidus and G.F. Pinder, Numerical Solution of Partial Differential Equations in Science and Engineering, Wiley, New York, 1982.
J. Electroanal. Chem., 297 (1991) 103-124 Elsevier Sequoia S.A., Lausanne
problem: general resolution via double The ECE-DISPl channel electrode collection efficiency measurements Patrick R. Unwin * Physical Chemistty (Received
Laboratory,
Oxford University, South Parks Roari, Oxford OXI 3QZ (UK)
25 May 1990; in revised form 7 August
1990)
AMTaCt T’he use of collection efficiency measurements, at the double channel electrode, is considered in the resolution of the “classical” ECE-DISPl problem. Particular attention is given to the case where the upstream (generator) electrode is held at a potential in the transport-limited current region, with the downstream (collector) electrode at a potential such that the initial heterogeneous electron transfer is reversed but sufficient over-potential is available to drive the second electron transfer, in the original sense, should it cccur. The theory is developed for the problem, relating the collection efficiency (the ratio of transport-limited currents at the collector and generator electrodes) to the geometries of the double electrode and channel cell, the solution velocity, the homogeneous rate constant, and the nature of the mechanism operating. It is shown that under the defined conditions the two mechanisms may be readily separated, and the kinetic domain accessible to the technique is identified. The theoretical predictions are tested on the reduction of fluorescein dianion in aqueous solution (pH 9.5-10.0). Collection efficiency measurements readily point to a DISPl assignment for this process, in agreement with previous work.
INTRODUCTION
The problem of identifying the origin of the second electron transfer heterogeneous (step (iii)) or homogeneous (step (iv))? - within the “classical” ECE-DISP mechanistic scheme [1,2] (considered below in term of a reduction process): A+e-+B
(Ei?
6)
B+C
69 (E,O >> EiO)
C+e-+D
(iii)
B+C+A+D
* Present
address:
0022-0728/91/$03.50
(iv)
Department
of Chemistry,
University
0 1991 - Elsevier Sequoia
S.A.
of Texas at Austin,
Austin,
TX 78712, USA.
104 TABLE
1
Definition of the ECE-DISP Mechanism
limiting mechanisms
step
63
(iii)
ECE
rds a
J
DISPl
rds
X
DISPZ
J
X
6)
(iv)
rds
a rds = rate determining step.
has received much attention in recent years [l-21]. Of the three limiting mechanisms (defined in Table 1) arising from the above scheme [2], only the DISP2 process is readily identified by virtue of it being second-order [2]. Resolving the ECE-DISPl competition has proved to be extremely difficult with many of the conventional electrochemical techniques [2,5]. For example, neither steady state transport-limited current measurements at rotating disc [6-S], channel [9] and ultramicro- [lo] electrodes nor half-wave potential-rotation speed characteristics at the rotating disc electrode [ll] in themselves distinguish between the two mechanisms within the bounds of typical experimental accuracy, although discrimination may be achieved if an independent value of the rate constant for step (ii) is available [9-111. In the case of linear sweep voltammetry, the shift in peak potential with scan rate is identical for the two processes when the wave is “two-electron” [2,12], whilst the peak current-scan rate characteristics are so close as to make discrimination unrealistic [13]. Likewise for single potential step chronoamperometry it is thought that the degree of experimental precision required prohibits general resolution [2,14-161, although this point has been disputed [17]. By turning to double potential step chronoamperometry (DPSC), with the first step to a potential in the transport-limited current region and the final step to a potential such that reaction (i) is reversed, but sufficient overpotential is still available to drive step (iii) in the original sense, should it occur, Sadant and co-workers have demonstrated that the ECE-DISPl problem can be resolved unambiguously [2,18]. This approach has been applied at both stationary [2,15,18-201 and rotating disc [5,21] electrodes. Bewick et al. [16] have also shown that the two competing pathways may be distinguished by monitoring the absorbance-time characteristics of species D following the application of a potential step and subsequent circuit opening. For completeness, mention should also be made of the extensive work of Saveant and co-workers [22-341 on indirect electrochemical methods, in which the ECE and DISP pathways can be characterised by providing a third reaction pathway which competes with the heterogeneous or homogeneous electron transfer to the intermediate C. “Double electrodes”, such as the rotating ring-disc electrode (RRDE) [35,36], double channel electrode (DChE) [37,38], paired microband electrode [39-411 and interdigitated array electrode [40,42] (amongst others), operated under steady-state
105
conditions in the “generator-collector” mode, may be used to provide analogous information to double step experiments at single electrodes. Surprisingly, however, collection measurements at double electrodes have not been applied explicitly to the ECE-DISPl problem, although the ECE process at the RRDE has been treated [43]. It is therefore the aim of this paper to consider, both theoretically and experimentally, the use of collection experiments in the general resolution of the ECE-DISPl problem. By setting the potentials at the (upstream) generator and (downstream) collector electrodes (Es_, and Em, respectively) relative to Eio and EiE such that Egen e Eio -=z E,, -=z E,O, those of steps (i)-(iv) which are relevant would be driven at the generator electrode and the following reactions would occur at the collector electrode (assuming the kinetics of the A/B and C/D couples to be sufficiently reversible): B - e- + A ( ECE/DISP~)
c + e-+
D (ECE)
The collection efficiency - the ratio of the transport-limited and generator electrodes N = IL,,, (collector) /Iti (generator)
c-i) (iii) currents at the collector
(1)
may then, under certain conditions, be quite different for the two mechanisms, thereby affording a route to the resolution of the problem. Until recently, the principal advantage of steady-state experiments over transient methods lay in the fact that the latter could be prone to complications when the period of interest (dictated by the kinetics of the reaction) was of the same timescale as the double layer charging process. However, the recent advent of the ultramicroelectrode [44-471 has considerably shortened the time over which double layer charging is significant [44-481, opening up the possibility of studying very fast reaction kinetics (on the microsecond timescale) via DPSC [48]. To compete with this, hydrodynamic electrodes must be able to deliver both a wide range of, and high, mass transport rates. Thus for the present work the DChE, as opposed to the hitherto more popular RRDE, has been selected, since it is anticipated that this criterion will be more readily satisfied using this electrode geometry [38]. In order to cover the widest possible range of mass transport rates, the theory for the problem is tackled numerically via the backwards implicit finite difference method (BIFDM) [49-511. This approach allows the solution of channel electrode problems using the full parabolic velocity profile to describe flow through the channel [49-511, whereas alternative analytical methods usually require the approximation of the parabolic profile by a linear one (the L&Que approximation [38,52]), thereby placing restrictions on the electrode geometries and solution velocities to which the theory may be applied [38,53]. The theoretical predictions are tested with experiments on the model DISPl system of the reduction of fluorescein dianion in aqueous solution (pH 9.5-10.0) [9,11,21].
106 THEORETICAL
FORMULATION
OF THE PROBLEM
ECE mechanism The steady-state
mass transport equations for species A, B and C within a ChE cell, when (a) the flow is fully developed and Poiseiulle in nature and (b) lateral and side edge diffusion can be neglected, are [9,38] D 8’[A]/+*
- uX8[A]/k
= 0
(2)
D 8*[B]/gy*
- u, a[B]/ax
- k,[B] = 0
(3)
D a*[C]/ay*
- v, a[C]/ax
+ kii[B] = 0
(4) The co-ordinate system for the DChE cell is defined in Fig. 1. In eqns. (2)-(4): kii is the rate constant for step (ii), D is a diffusion coefficient common to A, B and C and u, is the solution velocity in the x direction, given by [38] v,=q,{l-(y-h)*/h*}
(5)
where v, is the velocity in the centre of the channel. For the purpose of collection experiments, the upstream (generator) electrode is held at a potential in the transport-limited current region for the reduction process (i.e. the electron transfers denoted by steps (i) and (iii) are driven). The potential at the downstream (collector) electrode is such that the reduction of C is still driven, but B is oxidised to A (see above). The boundary conditions pertaining to this situation are: x=0,
ally:
y=o,
o
x1
[A]=[A]*,
[C]=O
(6)
a[B]/ay= -a[A]/ay, = ap]/ay = a[c]/ay = 0
[A] =o,
a[A]/ay [B] =O, ali
[B]=O,
X:
[cl =o
(7) (8)
[C] =0
(9)
a[A]/ay=a[B]/ay=a[C]/ay=O
(IO) where [A]* denotes the concentration of A in bulk solution. The collection efficiency (defined by eqn. 1) for the ECE mechanism is given by N=
x3(a[c]/ay),,C0
dx - /X3(a[Bl/8y)Y=0 x2
dx
dx}
01)
Fig. 1. Coordinate system for the double channel electrode cell
107
DISPl mechanism The mass transport equations for this case are: D i3*[A]/ay2 - u, a[A]/ax
+ kii[B] = 0
02)
D a*[B]/ay*
- 2k,,[B]
03)
- uX a[B]/ax
= 0
Again the upstream and downstream electrodes are held at potentials so that, respectively, reactions (i) and (-i) are driven. Thus the boundary conditions take the form: X=O,
all JY
[A] =[A]*
y=O,
O
[A]=O,
xi
a[~]/ay=
x*
[B] = 0
y= 2h,
ali
[B]=o a[B]/ay=
a[B]/ay=o
(14) -a[A]/ay
(15)
06) 07)
X: a[A]/ay=a[B]/ay=o
08)
The collection efficiency is now given by
dx
N=-
(19)
Method of solution Collection efficiencies for each of the mechanisms were calculated, ,for particular electrode and cell geometries and rate constants, by solving the specified mass transport equations with the appropriate boundary conditions using the backwards implicit finite difference method (BIFDM) [49-511. Details on the general implementation of the method can be found elsewhere [49-5154,551. New information, required for the present problem, is given in Appendix A. The calculations were carried out via a Fortran program, which was executed on the Oxford University Vax Cluster. A copy of the program is available from the author upon request. THEORETICAL
RESULTS
The collection efficiency, N, depends upon the DChE geometry (xi, x2, x3), the solution flow rate and the rate constant for the conversion of B to C in solution. The aim of this section is to illustrate the general effect of each of these parameters on N for the two mechanisms of interest, and to indicate the kinetic domain in which collection efficiency measurements can resolve the ECE-DISPl problem effectively. Effect of k, and V, An increase in k, or decrease in V, promotes the homogeneous decomposition of B both in the region of the generator and in its transit to the detector electrode. For the DISPl mechanism this is expected to lead to a reduction in the magnitude of N. This effect is born out in Fig. 2, which shows the dependence of Non k, and V, for
108
Fig. 2. A contour plot showing the dependence of Non kii and V, for the DISPl mechanism at a DChE defined by x1 = 0.10 cm, x2 = 0.15 cm, xg = 0.30 cm, d = 0.60 cm and h = 0.02 cm. A value of 10m5 cm’ s-l was assumed for D.
a DChE defined by x1 = 0.1 cm, xz = 0.15 cm, x3 = 0.3 cm, d = 0.6 cm and h = 0.02 cm. A value of N = - 0.360 is predicted for this geometry in the absence of kinetic complications (in the fast flow rate limit) and the data are seen to tend to
Fig. 3. A contour plot showing defined in Fig. 2).
the dependence
of Non
kii and Vf for the ECE mechanism
at a DChE (as
109
I .o.
0.5.
/-----
IogK
OISPI -0.1
-0.z
~ Fig. 4. Working curves of N vs. log,, K for the ECE and DISPl mechanisms at a DChE (as defined in Fig. 2).
this value under these conditions, but as kii is increased and V, decreased, N tends towards zero rapidly. The effect of k, and V, on N for the ECE mechanism is shown in Fig. 3 for the same DChE geometry. Overall the relationship is quite different from the DISPl case. Although N takes on negative values for small k, and/or large V, (as for DISPl), N becomes positive as these parameters are, respectively, increased and decreased. In the limit of very large k, and small V,, N tends towards zero. This is because under these conditions the ECE reaction is essentially “two-electron”, with neither of the intermediates B or C escaping from the generator electrode. As would be expected, the behaviour is similar to that found in double potential step chronoamperometry (DPSC) [2,18], for which the mass transport “parameter” relates to the times at which the currents on the forward and reverse transients are measured and the time for the potential step reversal. As with DPSC, it is clear that the large difference in the collection efficiency characteristics for the two mechanisms should allow the ECE-DISPl problem to be resolved unambiguously. For the range of parameters used to generate Figs. 2 and 3, Ddx,/Vrh -=K1. Under these conditions the LMque approximation holds [38], and so N is a unique function of the normal&d rate constant, K = k,( h2x;/4v;D)1'3
(20)
(a.i)
(a,ii)
(b,ii)
(c,i)
I Fig. 5. Concentration profiles for species (a) B and (b) C across l/10 of the channel at a DChE (of the geometry defined in Fig. 4) when an ECE mechanism operates. The data relate to K = (i) 0.191, (ii) 0.410 and (iii) 1.91. The contours are equally separated concentrations normalised with respect to the bulk solution concentration of A (i.e. [X]/[A] *, where X is B or C).
for each mechanism, as evidenced by Figs. 2 and 3. The numerical method can therefore be used to generate working curves for a particular DChE geometry. The result of this exercise, for the geometry defined above, is shown in Fig. 4 which serves to illustrate the large difference in the N-K characteristics for the two mechanisms. At small K (i.e. fast flow rate and/or small kii), N values are similar for the two processes, because under these conditions there is little decomposition of B (and consequent formation of C) on the timescale of the transit of B to the collector electrode. This point is illustrated by Fig. 5 (a,;) and (a$) for the ECE mechanism at the previously defined DChE with K = 0.191 (i.e. log,,(V,/cm3 s-l) = -0.5 and k, = 3 s-l, with the remaining parameters as defined above). The dramatic difference between the collection efficiency characteristics for the two mechanisms arises as K is increased. This leads to the build up of C and depletion of B over both electrodes, as depicted in Fig. 5 (b,i)-(c,ii) for the ECE mechanism. The fact that C is electroactive in the cathodic sense at the detector electrode (boundary condition 9) when the ECE mechanism operates means that N can take on positive values. This is obviously not the case for the DISPl mechanism.
111
-16
01
03 (x3-xJlcm
Fig. 6. The effect of gap and detector electrode lengths on N for the DISPl mechanism. The data relate to x1 = 0.1 cm, Vf = 0.1 cm3 s-l and kii = 2.0 s-l, with the other parameters as defined previously (i.e. K = 0.274).
Effect of gap and collector electrode lengths
Increasing the length of the gap between the generator and collector electrodes increases the transit time of B to the collector electrode, thereby promoting the decay of this species in solution. For the DISPl mechanism this leads to a diminution in the current at the collector electrode, as evidenced by Fig. 6. In the case of the ECE mechanism, Fig. 7 shows that N shifts from negative to positive
-03 .Ok
-12
0'1
03 (x3-x,llcm
0.5
Fig. 7. The effect of gap and detector electrode lengths on N for the ECE mechanism. The parameters are as defined in Fig. 6.
112
values with increasing gap length, since this serves to produce a large amount of C, which is subsequently collected at the downstream electrode. The length of the collector electrode, in relation to the gap and generator lengths, determines the quantities, and the distribution, of the various intermediates that are collected following their escape from the generator electrode. For the DISPl mechanism, increasing the length of the detector electrode (for a fixed gap length) leads to an increase in the amount of B collected, as evidenced by Fig. 6 which shows that N becomes more negative. Notice, however, that there is a limit to the collector length beyond which there is little, and eventually no, increase in N. This is expected given the instability of B in solution. For the ECE mechanism, the collector electrode length influences significantly the distribution of B and C in its vicinity and therefore both the magnitude and the sign of the current. For the case considered in Fig. 7, increasing the collector electrode length (when the gap is small) leads initially to N becoming more negative (collection of B is favoured). However, as the electrode is made longer the decay of B to C is promoted and so N shifts in the positive sense. For large gap lengths, B will have largely decayed to C in its transit to the collector electrode. N is therefore positive and increasing the collector length simply increases the amount of C collected (leading to an increase in N). Range of measurable rate constants The upper limit on the range of rate constants which may be measured via the technique is governed principally by (i) the highest rate of mass transport which may be attained at the generator electrode (in order to promote the escape of intermediates from this electrode for detection downstream) and (ii) the transit time between the generator and collector electrodes. In the consideration of the first point, the advantage of the non-uniformly accessible channel electrode over uniformly accessible electrodes (such as the rotating disc electrode) is that the rate of mass transport depends on both the solution velocity and the electrode length [38] (the shorter the electrode the higher the average rate of mass transport). In principle this allows the rate of mass transport to be varied over a very wide range, although this has not yet been fully realised practically. For the purpose of the following discussion, the parameters considered below have all been reported in various electrochemical experiments, but have yet to be employed together in a channel electrode experiment. The range of flow rates which may be implemented is limited at the upper end by the fact that flow must be laminar, i.e. the Reynolds number, Re = IJ,,h/v
(21)
where v is the kinematic viscosity of the solution, should not exceed 2000 [56]. It follows from eqn. (21) that the upper limit on u,, can be extended by shrinking the height of the channel. Taking a minimum value of h = 5 X 1O-3 cm (although shallower channels could be employed), a typical value of v = 0.01 cd s-i, and putting an upper limit of 200 on Re (to ensure that flow is well within the laminar regime), solution velocities up to u0 = 400 cm s-l could be employed. For a
113
conventional channel cell of width d = 0.6 cm, this corresponds to V, = 1.6 cm3 s-l, a value well below the maximum achieved practically of 66 cm3 s-l [57]. Given the above value of u,, the minimum electrode length for which diffusional edge effects contribute negligibly (< 1%) to the transport-limited current is ca. 10T3 cm [58], assuming a typical value of D = low5 cm2 s-l. Micro-band electrodes of this dimension are readily fabricated [39-421. Indeed, the technology exists to make much shorter electrodes [59]. As for the second point, the generator to collector transit time is minimised by using fast flow rates and a small gap. It follows from Figs. 6 and 7 that, in order to achieve discrimination between the two mechanisms in the fast kinetic limit, a long collector electrode is required, to ensure that the two mechanisms yield N values of opposite sign. A general DChE experiment involves measuring N over a wide range of flow rates. However, in order to assign an upper limit to the range of rate constants accessible in the case of the DISPl mechanism, the smallest value of N which can be measured at the fastest flow rate with reasonable accuracy (i.e. distinguished from zero current) is considered. Taking this value to be -0.02 (as considered previously for the EC reaction at the DChE [60]), together with the values for the parameters defined above, and a DChE characterised by x1 = 0.0010 cm, x2 = 0.0012 cm and x3 = 0.0112 cm, the maximum value of kii which could be determined for the DISPl reaction is calculated to be 7 X lo3 s-l. The question now arises as to whether discrimination from the ECE process can be achieved under these conditions. The value of k, corresponds to an effective number of electrons transferred at the upstream electrode n e.f = I,,&,,(
k, = 0)
(22)
of 1.331. Assigning a typical error of f0.02 to this neff value leads to k = 6.13 x lo3 (+ 0.61 x 103, -0.53 x 103) s-l if an ECE mechanism operates, and a resulting value of N = 0.167 ( + 0.008, - 0.010) for the defined DChE geometry. Thus, Iii,,- V, and N-V, analyses together. facilitate the ready resolution of the problem in the fast kinetic limit. If positive confirmation of an ECE mechanism alone is required, the kinetic domain accessible to the technique is greatly extended. For example, assuming N = 0.05 can be measured with an accuracy of f 0.01 at a DChE defined by x1 = 0.0010 cm, x2 = 0.0012 cm and x3 = 0.0212 cm, a process with k, = 2.0 x lo5 (+0.5 x 105, -0.3 X 105) s-l could be characterised. As for the measurement of slow kinetics, it is well established that hydrodynamic double electrodes provide a means of studying processes which are too slow for DPSC [61]. The investigation of slow processes requires the use of small mass transport rates. For the DChE, this dictates the use of small solution velocities and long electrodes [38]. The practical limits on these parameters are imposed by the point at which either lateral diffusion or natural convection become significant relative to forced lateral convection [38]. In experimental practice, no problems are encountered using a volume flow rate as low as low4 cm3 s-l and a generator electrode length of 0.4 cm (with h = 0.02 cm) [38]. If ILim- V, and N-V, analyses are again used together, then, taking neff = 1.1 as the minimum value which can be
114
measured accurately, it should be possible to characterise processes (ECE or DISPl) with k, = 6 X 10e3 s-l. Separation of the two mechanisms could then be achieved by employing a long gap and collector electrode (see above). For example, with x2 = 0.80 cm, x3 = 1.4 cm and the parameters defined above, values of N for the ECE and DISPl mechanisms of, respectively, +0.151 and -0.097 result. To summarise, DChE collection efficiency measurements are potentially extremely useful for the resolution of the ECE-DISPl problem, given the wide kinetic range (defined above) under which the two mechanisms can effectively be separated. EXPERIMENTAL
The apparatus for DChE voltammetry has been described previously [54] and so only additional details will be given here. The channel employed in this work had a cross-sectional dimension of 0.04 cm x 0.60 cm. Silver double electrodes were produced by casting two strips of silver foil (99.95% Goodfellow, Cambridge, UK) flush with one face of a resin block (Araldite: hardener HY219, resin CY219, Ciba-Geigy, Duxford, UK) which formed the coverplate to the channel [54,55]. Two DChE geometries were used: (A) x1 = 0.177 cm, x2 = 0.275 cm, x3 = 0.533 cm; (B) xi = 0.183 cm, x2 = 0.252 cm, x3 = 0.615 cm. An additional strip of silver foil located on the coverplate, upstream of the double electrode, functioned as a pseudo-reference electrode. Aqueous solutions of sodium fluorescein (ca. 3 mM) of pH 9.5-10-O and pH 13 were made up as described previously [9,21]. Collection efficiencies were determined at each flow rate by recording a currentvoltage curve for the reduction of the fluorescein dianion at the generator electrode. The resulting change in the current at the collector electrode, held at a potential corresponding to the foot of the generator electrode wave, was measured directly. All measurements were made under thermostatted conditions at 25 _+0.5 o C. EXPERIMENTAL
RESULTS
AND
DISCUSSION
As a standard DISPl system the reduction of fluorescein dianion (F2-) in aqueous solution @H 9.5-10.0) was investigated at a silver DChE. The reaction scheme takes the form [9,11,21]: generator electrode :
F2-+
solution :
F-3-+
solution : collector electrode :
e--s Fe3H+
FHe2-+ F-3-_
5
FH’Z-
Fe3-+ e-_,
F2-+
FH3-
F2-
Both transport-limited current ( Iti,,,) - flow rate (V,) characteristics at the generator electrode, and collection efficiency measurements, were of interest. The former measurements in isolation have been considered previously [9] and do not allow the ready resolution of the ECE-DISPl problem. Typical characteristics, obtained at pH 9.50 with the upstream electrode of DChE (A), are shown in Fig. 8. In the
115
150T
Fig. 8. Transport-limited current-flow rate characteristics for the reduction of F*- (solution pH = 9.50) at the upstream electrode of DChE (A).
absence of homogeneous kinetic complications a linear relationship between llim and V;‘/3 is predicted under the experimental conditions employed [62]: I,,
= 0.925nFw[F2-]*
( V,D2~;/h2d)1’3
(23)
where n is the number of electrons transferred per redox event. The behaviour for n = 1 was deduced by changing the solution pH to ca. 13 (through the addition of potassium hydroxide) where Fe3- is effectively stable on the timescale of channel electrode voltammetric measurements [9]. At pH 9.50 a smooth transition from twoto one-electron behaviour was observed as the flow rate was increased, as expected of an ECE-DISP system [9]. Under the experimental conditions utilised, the effective number of electrons transferred, nefr , is a unique function of the normal&d rate constant, K, for both the ECE and DISPl reactions [9]. Figure 9 shows the analysis of the Iti,,,-V, data in Fig. 8 using n,- K working curves for these mechanisms published elsewhere [9]. The data - plotted as K vs. qm2’3 (as suggested by eqn. 20) - analyse satisfactorily in terms of both mechanisms, confirming that this method alone does not permit mechanistic resolution. However, different values are yielded for k: 2.18 s-l (ECE) and 2.76 s-l (DISPl). Data obtained at pH 9.77 (using the upstream electrode of DChE (B) gave values for k of 1.30 s-l (ECE) and 1.42 s-l (DISPl). Collection efficiency data, obtained at the two pH defined above, are shown in Figs. 10a and b. The values of k, deduced from the neff - V, analyses can be used to simulate the behaviour expected for the two mechanisms. The results of this exercise are shown alongside the experimental data in Fig. 10. It is clear that the two mechanisms give rise to quite different collection efficiency characteristics, so that
116
Fig. 9. Analysis
of the data in Fig. 8 in terms of ECE (0) and DISPl
(0) mechanisms
[9].
resolution of the ECE-DISPl competition for this case is readily achieved. Comparison of the experimental data with the predicted behaviour for the two mechanisms points to a DISPl assignment for the reduction of F*-, in agreement with previous work [9,11,21].
----. +o .,_.
,*-----___
+0.1-
(01
-.
..
‘\
N
\
lb)
‘\
‘\
‘\
N '\ ECE '\ (V‘hn3S'P3
0-o:*
,---_
0.6
0%.O-$D)., cl
-Ol-
.. 0“.
O
--;--__ 0
'\ -o,‘\ *.
O-2
-Ol--
PJ~lcdi'l"~
I 0.6
04 '\ o-'-.Ry 0
:c\
‘,ECE \ '\ '\ '\
'. 0 '\
'\\ '1 '\ '\ 'b,, '\ '.
'\ .\ 0'
\
\\
'\\
'\
\
'\O
-02-
Fig. 10. Collection efficiency-flow rate characteristics for the reduction of F2- at (a) pH 9.50 using DChE (A), and (b) pH 9.77 using DChE (B). Also shown is the behaviour predicted theoretically for the ECE and DISPl mechanisms using the rate constants (defined in the text) deduced from limiting current measurements at the generator electrode.
117
As a check on the measurements, collection efficiencies were also determined for the reduction of F2- at pH 13 (for the reasons mentioned above). These data were found to be in good agreement (&- 2%) with theoretical predictions for the simple one-electron reduction of F2-, thereby confirming the validity of collection efficiency measurements with the DChE described. The above exercise serves to illustrate that Z,i,-V, and N- Vr measurements, employed together, provide a route to resolving the ECE-DISPl competition. The question naturally arises as to whether collection efficiency measurements in isolation allow the same objective to be realised. To answer this, N-Vr data obtained in the Ltv&que limit were converted to corresponding values of K using N-K working curves calculated for the two DChE geometries utilised. The resulting K-Vfp2/3 plots for each of the mechanisms at the two pHs are shown in Figs. lla and b. In contrast to Zlim - V, characteristics, a linear relationship between K and V,-2/3 results only for the DISPl, and not the ECE, mechanism. The former mechanism yields kii = (a) 1.75 s-’ and (b) 1.27 s-l - values which are in good agreement with those deduced from neft measurements. The results in Figs. lla and b suggest that analysis of collection efficiency data alone should be sufficient for resolving the ECE-DISPl problem. Of course, generator electrode limiting current-flow rate data are obtained when such measurements are made! These can also be analysed to provide a check on the mechanistic assignment, as illustrated above. The observation of a DISPl, rather than an ECE, mechanism for the reduction of fluorescein under the defined conditions is in agreement with the Amatore-Saveant rule that “ECE mechanisms do not occur in conditions where they could be characterised by electrochemical kinetic techniques” [63]. This rule, deduced with special regard to transient methods, was based on the fact that for the ECE mechanism to prevail the ECE-DISPl kinetic zone diagram dictates that log[ kii[A] */(k;i20’/2)] < -0.72 (where 0 is the measurement time in potential step experiments [14,18,64] and 0 = RT/Fu (u being the potential sweep rate) in linear sweep and cyclic voltammetry [13]). Simultaneous restrictions on the values of kii and 0 arise from the fact that nelf < 1.9. For aromatic or heteroaromatic substrates the disproportionation rate constant, ki,, is estimated to be diffusioncontrolled once the difference Eiz - Eio is greater than a few hundred millivolts [63]. With all these conditions and a substrate concentration, [A] *, at the millimolar level Amatore and Saveant illustrated that the ECE mechanism would operate only when measurement times were below one tenth of a microsecond [63] - inaccessible practically and hence the basis for the rule. A similar situation is likely to hold for other techniques [63]. The zone diagram for the ECE-DISPl competition at the channel electrode (see Appendix B) in Fig. 12 indicates that the larger the product ki,[A]*, the greater must be the value of kii for the ECE mechanism to dominate, and consequently the higher the rate of mass transport necessary to ensure that the upstream electrode reaction is not two-electron (i.e. sufficient intermediate escapes for collection downstream). For the single (and most favourable) case of nelf = 1.9 (which will yield a measurable collection efficiency), it follows from Fig. 12 that log K,, < 1.80. For ki, = 10” mol-’ dm3 s-’
118 0.P
(0) 0
K
0
0
/
0
04..
I__
0 0
0
0
O
0.
0
0
0
5
lo
(V,/ds-'12"
(b)
Fig. 11. Analysis (using eqn. 20) of the data in Fig. 10 in terms of the ECE (0) and DISPl (0) schemes, following conversion of N values to their kinetic equivalents for each mechanism via numerically calculated N-K working curves for the DChE geometries. (a) pH 9.50, DChE (A); (b) pH 9.77, DChE (B).
and a 1 mM substrate concentration must satisfy the following condition log( 4h4x$P/9V,%)
“3 < - 5.20
this implies
that the mass transport
parameters
(24)
Even if the transport parameters defined earlier, in the discussion of the highest rate of mass transport attainable at the channel electrode, are considered (together with D = 1O-5 cm’ s-l), the above quantity amounts to only - 3.7, which is well above that required for the observation of ECE characteristics. This therefore confirms
119
lY
a O-
-l.-
lelectron
I
0
-1
2 log K,
Fig. 12. Kinetic zone diagram for the ECE-DISPl conditions.
that the Amatore-SavCant transient, methods. APPENDIX
A. BACKWARD
competition
at the channel electrode under steady-state
rule applies to steady-state channel electrode, as well as
IMPLICIT
FINITE
DIFFERENCE
METHOD
The calculations of collection efficiencies using the BIFDM involves covering the three regions of interest in the cell (0 < x < xi, x, -Cx < x2 and x2 -Cx < x3) with a two-dimensional finite difference grid with increments in the x and y directions defined by: x,(k)=kAx,,
Ax,=x,/K,
k=O,l,...K,; j=O,l,...J;
y(j)=jAy,
Ay=2h/J
(Al) 642)
where x* is either xi, (x2-x1) or (xg -x2) and K* is the corresponding total number of x divisions for the particular zone. Typical values of J = 500-800 and K* = 1000 x*/x3 were employed in the calculations reported in this paper. Upon approximating the derivatives in the mass transport equations by their BIFD equivalents [49-511, invoking an explicit treatment of the kinetic terms [50], and applying the boundary conditions, the collection efficiency problem condenses to the solution of the following set of simultaneous equations: d, = b,u, + c*u* d, = ajuj_l d,_,
643)
+ bjuj + cjuj+ 1,
= aJ-Iu,_2
j=2
,... J-2
+ bJ--luJ--l
for each of the species requiring consideration.
(A41 645) Values for d,, dj (j = 2,. . . J - 1)
120 TABLE Al Values for d, mechanism
species
4
ECE
A B B C
E& g& d’,c &
A B B
$,
DISPl
$1
wne o-x,
-[‘?g!~kii(A.~)~/L’l+ ‘fgtk+l -N&%i(W2/4 +M&kiW~2/~l
o-x, x1 - x3 o-x3
+[?&kii(Ay)2/4 * t’&ii@v)2/W+ $:);&ky(Ay)2,~]
o-x,
o-x,
V&+l
Xl -
x3
TABLE A2 Values for dj (j=2,
3,...5-1)
mechanism
species
dj
zone
ECE
A B C
F+ giC,c-[~f+d‘W2/‘~l gj,k +[A;gj,kkti(AY)‘/oI
o-x,
A B
g;k
+[~f~~k&i(b’)2/~l
o-x,
g,,k
-[2+fk“ii(Av)2/Dl
o-x,
DISPl
o-x, o-x,
TABLE A3 Values for b, mechanism
species
4
zone
ECE, DISPl ECE, DISPl ECE, DISPl ECE ECE
A B B C C
1+2q 1+#4: 1+2h7 1+2x* 1+X:
o-x, o-x, x2-x3
o-x,, Xl -
x2-x) x2
and b, are given in Tables Al, A2 and A3. The expressions for aj (j = 2,. . . J - l), J - 1) and cj (j = 1,. . . J - 2) for all species and all zones are bj (j=2,... aj=
-hf
(A6)
bj = 1 + 2h;
(A71
cj=
(A81
-x;
In Tables Al, A2 and A3 and eqns. (A6)-(A8) “7 =
8D( Ax)h3d W;j(Ay)3(2h
-j
AY)
(A91
121
and gR = [B]/[A]
*
(AlO)
The computational strategy involves calculating the vector { uj ( j = 1, . . . J - l)}k from known {dj.(j= l,... J - l)}k by use of the Thomas algorithm [65,66]. The boundary condtttons (6) and (14) provide values respectively for the ECE and DISPl mechanisms, for the vectors {d }k=,, (for each of the species), from which { u }k co are calculated. These, in turn, are related to the new vectors { d }k= 1. The calculation thus proceeds down the cell through each of the three zones of interest. For the purpose of collection efficiency measurements, concentration profiles are evolved for A, B and C at the upstream electrode for the ECE mechanism, but only B and C need to be considered over the gap and the collector electrode. In the case of the DISPl process the A profile is generated at the upstream electrode only, and the B profile is calculated throughout all zones. The collection efficiencies are calculated from:
-A(x,
N =
- xZ)&,
i
1
)I(
AX+&)
(DISPI)
(AI2)
When the distance over which concentration changes in the direction normal to the DChE is small compared to the charmel height (i.e. for very fast flow rates and/or short DChEs, carrying out the finite difference calculations across the full channel becomes very inefficient. Under these conditions the region of interest in the Y-direction is therefore confined to a distance Ymax= 36, = (3/0.67)(
hx,D/~)“~
where 8, is the diffusion layer thickness at the trailing edge of the collector electrode (if the whole region 0 < x < x3 was a single electrode). The increments in the finite-difference grid in the y-direction are then given by AY
=y,m/J
This approach requires a change in boundary conditions (10) and (18) to Y +YIn,,
all x,
[A] + [A]*,
[B] +O,
[C] -0
(AI3)
which modifies the following matrix elements bJ_, = 2x;_, d,A_,.k = g_k,,, &,,=&i,~+%--l(I
+ 1 + A*,-,
(for A, B and C)
(AI4)
(ECE)
(AI51
+ [&&i(~Y)*,‘~])
(DISPl)
(AI6)
When compared to the original approach, the latter gives very accurate results, with a considerable saving in computer time.
122 APPENDIX B. KINETIC ZONE DIAGRAM FOR THE ECE-DISPI CHANNEL ELECTRODE
COMPETITION AT THE
When both the ECE and DISPl pathways contribute to the overall channel electrode reaction mechanism, the mass transport equations requiring consideration are: D a2[A]/ay2-u,
a[A]/ax+ki,[B][C]=O
(Bl)
D a2[B]/8y2
- U, a[B]/ax
- kii[B] - ki,[B][C]
= 0
(B2)
D a2[C]/8y2
- U, a[C]/ax
+ k,[B]
= 0
(B3)
- ki,IB][C]
In order to deduce the general kinetic zone diagram, conditions under which the LhiZque approximation [38], u, = 2 u, y/h, is valid are considered. Equations (Bl)(B3) may then be written in dimensionless form: a2a/i3t2 - 5 au/ax
+ K,bc = 0
a2b/at2-t ab/aX- Kb a2qa.p- ( at/ax + Kb -
(B4)
K,bc = 0
(B5)
K,bc
036)
where x = X/Xi
(B7)
5 = (2u,/hD~,)“~y
(B8)
Kd = ki,[A]*
(B9)
q=
( Iz~x;/~u,~D)~‘~
WIAI *
(q=a,
b or c)
(BlO)
and K is given by eqn. (20). It follows that the ECE-DISPl competition is governed by the two kinetic parameters K andKd; ECE being promoted by Kd + 0 and DISPl by K,, --, co for a given value of K. Equations (B4)-(B6) were solved with boundary conditions (6), (7) and (10) using the BIFDM (Appendix A). The following matrix elements change from those given in the preceding appendix for the pure ECE limiting mechanism (at the upstream electrode with semi-infinite boundary conditions in the y-direction): dpk = g;k +
[Xrg/IfkgF/ckiv CAY )‘[A] */D]
d~-*.l,=g~-l.k+h:-l(l+
(j=1,2,...&2)
[g,“-,,,g,‘_,,,ki,(A~)2[AI*/D])
d,B=g~k+hT([g~~+l] - [gPlkg~kkiv(AY)‘[AI*/D] - [gT,kii(hy)2/D]) diB=g~~-(XTg~k(Ay)‘/D){ g,S_*kiv[AI* +kti} dF=gJf,+
(j=2,3,...J-1)
(X:g~,(Ay)2/D){kii-g~kkiv[AI*}(j=1,2,...J-1)
To ensure that LCv&ue conditions applied, the calculation employed the following
123
parameters: V, = 0.1 cm3 s-l, h = 0.02 cm, xi = 0.1 cm, d = 0.6 cm and [A] * = 10e3 mol dmp3, with kii and k, being varied. The results obtained may be interpreted in terms of a kinetic zone diagram by considering the DISPl situation to apply when the current for the reduction of C is less than 5% of the total current and ECE behaviour corresponding to the reduction current of C being greater than, or equal to, 95% of that predicted for the pure ECE limiting mechanism [63]. The zone diagram deduced on this basis is shown in Fig. 12. The additional zones showing one- and two-electron behaviour correspond to n eff -c 1.05 and neff > 1.90, respectively. The behaviour observed is qualitatively similar to that found for transient techniques [13,14]; an increase in K displaces the system from one-electron behaviour into the ECE or DISPl zones, whilst the shift from ECE to DISPl occurs by decreasing K and/or increasing &. The zone diagram provides a means of deducing the conditions required for the operation of an ECE or DISPl mechanism. REFERENCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29
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