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A computational and experimental study of the cyclicvoltammetry response of partially blocked electrodes, part III:
interfacial liquid–liquid kinetics of aqueous vitamin B12s
with random arrays of femtolitre microdroplets of
dibromocyclohexane
Trevor J. Davies, Benjamin A. Brookes, Richard G. Compton *
Physical and Theoretical Chemistry Laboratory, Oxford University, South Parks Road, Grande-Bretagne, Oxford OX1 3QZ, UK
Received 16 September 2003; received in revised form 5 November 2003; accepted 7 November 2003
Abstract
The cyclic voltammetric response of electrodes modified with catalytically reactive microdroplets is modelled using finite dif-
ference simulations and a method is presented for the determination of kinetic parameters for the coupled heterogeneous reaction.
The method is first applied to investigate the liquid–liquid reaction between pure trans-1,2-dibromocyclohexane (DBCH) micro-
droplets, deposited on the surface of a basal plane pyrolytic graphite electrode, and vitamin B12s in aqueous solution. Second, cyclic
voltammetry on electrodes modified with microdroplets of DBCH diluted in dodecane is employed to determine the apparent bi-
molecular interfacial rate constant for the initial step in the DBCH(oil)jB12s (aq) reaction. The results are compared and contrasted
with a previous SECM/ITIES study of a similar reaction.
Ó 2003 Elsevier B.V. All rights reserved.
Keywords: Partially blocked electrodes; Finite difference methods; Modified electrodes; Microdroplets; Randomly arranged ensembles; Vitamin B12
1. Introduction
Charge transfer reactions at an electrode surface
partially covered with an inert material display charac-
teristic transient properties [1–3]. For example, in the
case of cyclic voltammetry with a quasi-reversible redox
couple, increasing the fractional coverage of inert ma-
terial, h, causes an increase in peak to peak separation
and (eventually) a decrease in peak height. Such systems
are not uncommon and it is relatively easy to modify or
intentionally ‘‘block’’ electrode surfaces [2–5]. Thus, a
good quantitative description of partially blocked elec-
trodes is potentially very useful to a large number of
research workers. However, in many experimental situ-
ations the blocking material is of microscopic propor-
tions, i.e., the inhomogeneities and/or the distances
between them are small compared to the thickness of the
diffusion layer. This results in non-linear diffusion in
regions close to the uncovered electrode (within the
diffusion layer), thus complicating the modelling pro-
cess. Amatore et al. [1] were the first to consider this case
successfully and developed a one-dimensional model for
an ensemble of microdisk electrodes dispersed under a
blocking film. By describing the observed voltammetric
trends, their work resulted in a strategy for estimating
the fractional coverage of the blocking film and the size
of the active sites. In Part I of this series, we developed a
two-dimensional numerical method to describe the
transient response of a symmetric array of inert disks on
an electrode surface (the exact inverse of the system
studied by Amatore) [2]. In a subsequent paper, this
model was successfully extended to arrays of randomly
distributed disks and, more importantly, microdroplets
* Corresponding author. Tel.: +44-1-865-275-413; fax: +44-1-865-
275-410.
E-mail address: [email protected] (R.G.
Compton).
0022-0728/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved.
doi:10.1016/j.jelechem.2003.11.026
Journal of Electroanalytical Chemistry 566 (2004) 193–216
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Journal of
ElectroanalyticalChemistry
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[3]. Indeed, Part II concluded with an electrochemical
method for determining the average radius of inert mi-
crodroplets deposited on an electrode surface.
A rather interesting situation arises when the block-
ing material is not inactive. Such systems are created
when electrodes are intentionally blocked with reactive
material that can lead to coupled heterogeneous chemical reactions. In the case of reactive microdroplets, e.g., oil
droplets on an electrode surface submerged in aqueous
media, the accompanying chemical reaction(s) takes
place at the liquidjliquid (oiljwater) interface. Such re-
actions are important to study due to their wide rele-
vance in a number of different research fields. For
example, catalytic electrochemical reactions in emulsi-
fied media, where the key reaction takes place at the
liquidjliquid interface, 1 have proved to be a viable al-
ternative to similar reactions in organic solvents [6,7].
This is important industrially – as well as providing a
cheaper alternative in which to carry out syntheses,
emulsions (especially sono-emulsions) are much
‘‘greener’’ media. Furthermore, liquid–liquid systems
are important in our understanding of some funda-
mental biological problems. For example, Senda and co-
workers [8] have used the nitrobenzenejwater interface
to model proton transfer across the biological mem-
branejsolution interface.
Previous studies of the rates of liquid–liquid reac-
tions have involved both ‘‘direct’’ and ‘‘indirect’’ ap-
proaches. The ‘‘classical’’ electrochemical experiment is
that of Samec and Maracek [9], where a four-electrode
configuration is used to measure interfacial charge
transfer. In the last decade, Bard and co-workers [10]have developed a method based on scanning electro-
chemical microscopy (SECM), where an UME is used
to probe charge transfer reactions directly at the in-
terface between two immiscible electrolyte solutions
(ITIES) [11]. A novel approach was that of Banks et al.
[12], who, when working with particular sono-emul-
sions were able to relate bulk measurements to the
heterogeneous rate constant for their specific liquid–
liquid reaction. In the following work we develop a
new approach based on the transient response of
electrodes modified with water-insoluble oil microdro-
plets in aqueous solutions. In particular we study the
case where the droplet surface reacts with an electro-
generated mediator in a catalytic pathway that leads to
the regeneration of the original electroactive species.
Using cyclic voltammetry we are able to probe such
systems and obtain kinetic data on the coupled liquid–
liquid reactions. The theoretical treatment is signifi-
cantly different from that for previous methods of
ITIES investigation by the fact that both electron
transfer and the coupled interfacial heterogeneous re-
action occur at the electrode surface.
The contrasts between coupled homogeneous and
heterogeneous reactions are interesting. In the latter,
electron transfer and the coupled heterogeneous chemi-cal reaction(s) occur exclusively at the electrode and
droplet surface, respectively. This contrasts the homo-
geneous analogues, where the coupled chemical reactions
occur within the diffusion layer. Mathematically, the
difference is even more apparent. For coupled homoge-
neous reactions, the kinetic terms are contained in the
diffusion equations whereas the corresponding hetero-
geneous kinetic terms appear only in the boundary
conditions. Thus, modifying a numerical method for
different mechanisms is relatively simple for the hetero-
geneous case when compared to the task involved in
modifying a coupled homogeneous simulation.
The electrocatalytic reaction of vitamin B12r (the
Co(II) form of vitamin B12) with trans-1,2-dibromocy-
clohexane (DBCH) in certain organic solvents is a well-
documented example of a coupled homogeneous
chemical reaction which regenerates the electroactive
material [13]. The reaction pathway is thought to pro-
ceed via one of two mechanisms where the rate deter-
mining step is either an SN2-type nucleophilic attack (A)
or an E2-type elimination (B) [18]:
CoðIIÞL þ eÀ¢CoðIÞL Electrode
Co
ðI
ÞL
þRBr2
RBr – Co
ðIII
ÞL
þBrÀ Solution
RBr – CoðIIIÞL RBrÅ þ CoðIIÞL Solution
CoðIÞL þ RBrÅ CoðIIÞL þ R0 þ BrÀ Solution
ðAÞ
CoðIIÞL þ eÀ¢CoðIÞL Electrode
CoðIÞL þ RBr2 Br – CoðIIIÞL þ BrÀ þ R0 Solution
Br – CoðIIIÞL þ CoðIÞL BrÀ þ 2CoðIIÞL Solution
ðBÞ
In the above pathways, RBr2
represents DBCH, R0
is
cyclohexene, Co(II)L is vitamin B12r and Co(I)L vi-
tamin B12s. Recently, we developed a heterogeneous
analogue of this system where microdroplets of
DBCH were deposited onto an electrode surface and
then immersed into an aqueous solution of vitamin
B12 [7]. The coupled chemical reaction between DBCH
and vitamin B12s occurs exclusively at the surface of
the microdroplet, and the reaction is heterogeneous.
In this paper we simulate the transient response at a
partially blocked electrode where the block regenerates
electroactive material via a separate heterogeneous
reaction:
1 Although true for sono-emulsions, in bicontinuous microemul-
sions, for example, the reaction occurs between the liquid-adsorbed
surfactantjliquid interface. The liquidjliquid interface is, therefore, a
model experimental system for surfactant based emulsions.
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axis of symmetry 0 < Z < 1; R ¼ 0;
oa
o R¼ 0;
ob
o R¼ 0;
unit cell edge 0 < Z < 1; R ¼ R0;
oa
o R ¼ 0;ob
o R ¼ 0;
diffusion layer Z 1; 06 R6 R0;
oa
oZ ¼ 0;
ob
oZ ¼ 0;
where a0 and b0 are the normalized concentrations of A
and B at Z ¼ 0. Note that the boundary condition for
the block is the only difference between pathway (D) and
the simple redox reaction previously discussed [2,3].
Following the procedure outlined in Part I [2], we nor-
malize R, Z and t with respect to the parameter R0:
r ¼ R R0
ð6Þ
z ¼ Z
R0
ð7Þ
s ¼ Dt
R20
: ð8Þ
As a result, we can define three dimensionless parame-
ters: mdl (dimensionless scan rate), k 0dl (dimensionless
electron transfer rate constant) and k 0het (dimensionless
heterogeneous rate constant)
mdl ¼ FR20
DRT m ð9Þ
k 0dl ¼ k 0 R0
Dð10Þ
k 0het ¼ k het R0
D: ð11Þ
Using the dimensionless parameters and assuming Da
¼ Db, we can make the substitution u ¼ a ¼ 1 À b into
Eqs. (4) and (5) to obtain just one equation that de-
scribes the mass transport:
ou
ot ¼ o2
o z 2
þ o2
or 2þ 1
r
o
or
u ð12Þ
with the boundary conditions:
electrode z ¼ 0; r b 6 r 6 1;ou
o z ¼ k 0f
À þ k 0bÁ
u0 À k 0b
block z ¼ 0; 06 r < r b;ou
o z ¼ u0ð À 1Þk 0het
axis of symmetry 0 < z < 1; r ¼ 0;ou
or ¼ 0
unit cell edge 0 < z < 1; r ¼ 1;ou
or ¼ 0
diffusion layer z 1; 06 r 6 1;ou
o z ¼ 0
where k 0f ¼ k 0dl exp½Àa F ð E ðt Þ À E 00Þ= RT , k 0b ¼ k 0dl exp½ð1ÀaÞ F ð E ðt Þ À E 00Þ= RT , r b is the dimensionless block
radius (r b ¼ Rb= R0Þ and u0 is the value of u at Z ¼ 0.
2.2. Finite difference formulation
We solve Eq. (12) using a fully implicit finite differ-
ence method with a geometrically expanding mesh
technique identical to that described in Part I [2]. The
only modification is in the implementation of the block
boundary condition which concerns mesh points
i ¼ 0; 1; . . . ; ðnb À 1Þ: 2
oui
o z ¼ U i;0ð À 1Þk 0het
% À l20U i;2 À l0 þ l1ð Þ2
U i;1 þ l1 l1 þ 2l0ð ÞU i;0
l0l1 l1 þ l0ð Þ
!
) U i;0 ¼/ l0 þ l1ð Þ2U i;1 À/l2
0U i;2 þ k 0het
/ l1 þ 2l0ð Þl1 þ k 0het
ð13Þ
where / ¼ 1=l0l1ðl0 ¼ l1Þ and the mesh numbering is
illustrated in Fig. 2. The modelling then follows the
same method as described in Part I [2], where the di-
mensionless current, w, is calculated using Eq. (14): 3
w ¼ 2 ffiffiffiffiffimdl
p Z
1
r b
oa
o z
z ¼0
r dr : ð14Þ
The relationship between the true current, I , and the
dimensionless current, w, is given by
I ¼ A½ bulk Aelec
ffiffiffiffiffiffiffiffiffiffiffi F 3 Dv
RT
r w ð15Þ
Fig. 2. Illustration of the mesh numbering system in the vicinity of the
electrode surface.
2 Misprint in [2]: The denominator in Eq. (13) contains an l1 term
which is missing in the corresponding equations in [2].3 Misprint in [2]: The factor of 2 was omitted in Eq. (14) in [2].
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where Aelec is the area of the electrode (or single diffusion
domain) and the number of electrons transferred is
equal to 1.
Optimum values for the grid expansion factors and
initial grid spacing, determined via convergence testing
over a range of k 0het, h; mdl and k 0
dl, were identical to those
used in Part I for the case of an inert, rather than areactive, block [2].
2.3. Simulation results
Initial simulations were carried out using parameters
relevant to the experimental problem discussed below.
In most simulated voltammograms, the dimensionless
current, w, is preferred over the true current, I , due to its
wider relevance. For example, when dealing with linear
sweep or cyclic voltammetry of a simple redox couple at
macro electrodes, it is well known that the largest di-
mensionless current is 0.4463 (where the mass transport
is linear diffusion only) [15]. In all simulations the tem-
perature was 293 K, although slight variations made a
negligible difference.
2.3.1. Inert vs. reactive
Fig. 3 illustrates the simulated cyclic voltammograms
for three different heterogeneous rate constants, k het ¼ 0
(no reaction), 0.01 and 10 cm sÀ1, at a diffusion domain
with block radius Rb ¼ 2:5 lm and coverage h ¼ 0:5where the scan rate is 50 mV sÀ1. The other parameters
are listed in the figure legend. The effect of the catalytic
regeneration of species A is immediately obvious by the
increased currents which increase with k het (for suitablylow values of k het where the heterogeneous reaction is
not diffusion controlled). The absence of a reverse peak
in the reactive block voltammograms is similar to that
observed for coupled catalytic homogeneous reactions
and is essentially due to the same factors [13]. In the
homogeneous analogue, any B species formed at the
electrode surface is oxidized back to A by the homoge-
neous mediator. Thus, upon scan reversal, there is no B
present in the vicinity of the electrode surface and noreverse peak is seen. Our heterogeneous analogue is
slightly different in that the re-oxidation of B to A oc-
curs only at the block surface. This implies that mass
transport of B from the uncovered electrode to the block
must be rapid or else we would see a reverse peak.
The voltammograms for the reactive blocks appear to
have a limiting current suggesting a constant flux of A is
being delivered to the uncovered electrode surface. On
closer inspection, there are actually gentle peaks at
E ¼ À0:32 V for k het ¼ 0:01 cm sÀ1 and E ¼ À0:61 V for
k het ¼ 10 cm sÀ1. However, the fact that the peak cur-
rents are almost maintained suggests that block regen-
eration causes some kind of quasi-steady state to be
achieved where solution depletion due to the one-elec-
tron reduction in the vicinity of the electrode surface is
compensated by the coupled heterogeneous reaction.
Finally, the ‘‘peak’’ potential, E p, appears to increase
with k het. Comparison with the homogeneous analogue
is complicated because of mediator depletion.
A similar situation would arise for a simple one-
electron redox couple and an array of collector–gener-
ator electrodes with infinitely thin insulating layers. In
this case the unit cell could be described as that in Fig. 1,
the only difference being that the block is now electrode
material held at a constant potential, E c. Varying thevalue of E c would then be equivalent to varying k het and
we would expect to see the same voltammetric trends.
2.3.2. Analysis of the concentration profiles
Fig. 4 illustrates concentration profiles taken at the
peak potentials for all three domains in Fig. 3. Note that
since h ¼ 0:5, r b ¼ 0:707 and the dark area around
r ¼ r b is due to the high density of mesh points required
to minimize the error caused by the singularity at r ¼ r band z ¼ 0. Again, the effect of the reactive block is im-
mediately obvious, especially in the case of the fast
heterogeneous reaction (k het
¼10 cms
À1), where we
observe an almost vertical drop in a when E ¼ À0:61 V.
Immediately over the block, B is converted rapidly into
A, by the coupled heterogeneous reaction, so that the
concentration of A is ½Abulk and a ¼ 1. However, at the
unblocked electrode surface, the potential is such that A
undergoes a rapid one-electron reduction to B causing
½A z ¼0 and a to equal zero. Hence, in the immediate vi-
cinity of the point r ¼ r b, we have a sharp drop in [A] as
we crossover from the blocked to the unblocked region.
As we move away from the electrode surface, this abrupt
change evolves into a smoother profile. For the slower
heterogeneous reaction, k het
¼0:01 cmsÀ1, the concen-
Fig. 3. Simulated dimensionless current voltammograms for diffusion
domains, where Rb ¼ 2:5 lm, k 0 ¼ 0:005 cm sÀ1, D ¼ 2:0 Â 10À6
cm2 sÀ1, m ¼ 0:05 VsÀ1, h ¼ 0:5 and k het ¼ 0; 0:01, and 10 cm sÀ1.
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tration of A over the block surface is less than the bulk
concentration and the gradients around the singularity
are much gentler.
In the case of inert blocks (k het ¼ 0 c m sÀ1), mass
transport to the electrode surface is supplied by diffusion
arising from concentration gradients caused by the in-
creasing voltage. When the block is reactive we have a
situation where the regeneration of A also leads to (quite
steep) concentration gradients. The concentration pro-
file for k het ¼ 10 cmsÀ1 might lead one to believe that
transport due to block regeneration dominates and
generally this is true for domains of high coverage and
not too excessive scan rates, vide infra. The sharp drop
in a at r ¼ r o results in a flux of A that greatly exceeds
that caused by the voltage increase and we observe a
greatly enhanced current. For slower heterogeneous re-
actions, the profiles are less abrupt and we observe a far
lesser increase in current. Note also that the concen-
tration profiles for species B are the inverse of those in
Fig. 3 (u
¼a
¼1
Àb). Thus, there is a large flux of B
from the uncovered electrode to the reactive block as
implied by the absence of a reverse peak.
As observed in the differences between Fig. 4(i) and
(ii) for the ‘‘reactive’’ domains, the flux of A from the
region of the block to the uncovered electrode is not
independent of potential . For example, it is not until the
concentration of A across the uncovered electrode sur-
face is zero that we observe a peak current. The reason
for this can be attributed to two factors. First, looking at
the profiles for the k het ¼ 10 cmsÀ1 domain, the large
gradient at the singularity results in a massive flux of A
to the uncovered electrode surface which has a major
influence on the overall current. This is illustrated in
Fig. 5 where we have plotted ðoa=o z Þ z ¼0 (a variable that
is proportional to the current density) taken at the peak
potential across the uncovered electrode surface for the
(a) k het ¼ 0 cm sÀ1 and (b) k het ¼ 10 cm sÀ1 domains. In
the case of the inert block, the current density varies
little across the electrode surface. This contrasts the re-
active block where we see a large decrease in
ðoa=o z
Þ z
¼0
Fig. 4. Dimensionless concentration profiles for the diffusion domains in Fig. 3, taken at E ¼ (i) )0.05 V, (ii) )0.32 V and (iii) )0.61 V, where
k het ¼ (a) 0 cmsÀ1, (b) 0.01 cm sÀ1 and (c) 10 cm sÀ1.
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as we move away from the singularity. For fast coupled
heterogeneous reactions where the value of a at the
blocked surface is always 1, the maximum flux and,
therefore, maximum current will occur at a potential
when [A] at the uncovered electrode surface is zero.
Second, because of the solution replenishment in the
vicinity of the electrode surface, bulk solution depletion
has a lesser influence on the voltammograms. For the
case of the inert block, we have an example of the well
documented ‘‘increasing rate of electron transfer vs.
depletion of bulk solution’’ [16]. As the potential in-
creases, ½A z ¼0 decreases (which should lead to steeper
concentration gradients and therefore higher flux) but at
the same time the concentration of A in the vicinity of
the electrode surface is depleted (leading to shallower
concentration gradients). Thus, we have a trade off be-
tween two opposing factors and observe a peak poten-
tial where ½A z ¼0 is greater than zero (remember that our
redox couple is quasi reversible, k 0 ¼ 0:005 cmsÀ1). In
the case of the reactive blocks, reactant depletion in the
vicinity of the electrode surface is lessened – not only
does the reactive block supply a large flux of A, it does so
at distances close to the electrode surface. Therefore, the
steepest concentration gradients occur when ½A z ¼0 ¼ 0,after which we observe a more or less constant current
(‘‘steady’’ state). This also explains why we observe an
increase in E p with k het (until we reach the diffusion
limited regime, vide infra). Faster coupled heteroge-
neous reactions can supply more A to the electrode
surface, meaning that we need to reach higher potentials
where our electrode kinetics are fast enough to con-
vert all the excess reactant at the uncovered electrode
surface.
On closer inspection of Fig. 4 two things become
apparent. First, in all three profiles, (a)–(c), the diffusion
layer thickness is approximately constant at each of the
potentials. Thus, it appears that a non-zero k het has little
influence on the diffusion layer thickness. Second, a
Fig. 5. Plots of ðoa=o z Þ z ¼0 (taken at the peak potential, E pÞ against r
over the uncovered electrode surface for the k het ¼ 0 c m sÀ1 (a) and
k het ¼ 10 cmsÀ1 (b) domains in Fig. 3. The inset in (b) is a close-up of
the region near the r -axis. Note that r b ¼ 0.707.
Fig. 6. Dimensionless concentration profiles in the immediate vicinity
of the electrode surface for those profiles illustrated in (a) Fig. 4(a)(iii)
and (b) Fig. 4(c)(iii).
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‘‘kink’’ just above the surface of the electrode is visible
in the concentration profiles for the reactive blocks.
Fig. 6 illustrates a close-up of this region for Fig. 4(a)(iii)
(k het ¼ 0 c m sÀ1, E ¼ À0:61 V) and (c)(iii) (k het ¼ 10
cm sÀ1, E ¼ À0:61 V). The magnitude of the concen-
tration gradient in Fig. 6(a) is approximately equal to
that of the section labelled ‘‘ X ’’ in Fig. 6(b) and we canassociate this with the depletion of A at the electrode
surface leading to diffusion from the bulk solution. The
section labelled ‘‘Y ’’ in Fig. 6(b) is clearly steeper and
must arise from block regeneration. Thus, we have an
idea of the dominance of block regeneration over scan
rate in contributing to mass transport for this particular
domain. Also, it is clear to see that the effect of the block
is confined to a region close to the electrode surface. The
depth of region Y gives us an idea of how far regener-
ated reactant has diffused in the time taken to reach the
stated potential, which in this case is about 1.5 lm.
2.3.3. The effect of scan rate
So far we have considered only one scan rate, 50
mV sÀ1. What happens if we increase the rate of the
potential sweep? Fig. 7 illustrates linear sweep voltam-
mograms for the same domain in Fig. 4(c) where the
scan rates are 0.1, 0.5 and 2.0 V sÀ1. As observed for a
simple quasi-reversible redox reaction, the peak dimen-
sionless current, wmax, decreases as the scan rate in-
creases, although the relative decrease appears quite
large. In contrast to the inertly blocked case, the peak
potential, E p is independent of scan rate ( E p ¼ À0:61 V
for all three scans). Even though the scan rate increases,
the potential at which the electrode kinetics are fastenough to achieve ½A z ¼0 remains the same. Therefore,
according to the reasons discussed above we would ex-
pect E p to be independent of scan rate. The corre-
sponding concentration profiles (taken at E ¼ À0:61 V)
are displayed in Fig. 8. As expected, the diffusion layer
thickness decreases with increasing scan rates, but this is
the only useful information we can obtain from the
Fig. 7. Dimensionless current linear sweep voltammograms for a dif-
fusion domain, where Rb ¼ 2:5 lm, h ¼ 0:5, k 0 ¼ 0:005 cm sÀ1,
D ¼ 2:0 Â 10À6 cm2 sÀ1, k het ¼ 10 cm sÀ1 and m ¼ 0:1, 0.5 and 2.0 V sÀ1.
Fig. 8. Dimensionless concentration profiles, taken at E ¼ E p ¼ À0:61
V, for the simulated voltammograms in Fig. 7, where m¼ (a) 0.1 V sÀ1,
(b) 0.5 V sÀ1 and (c) 2.0 V sÀ1.
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larger profiles. On observing the close-ups in Fig. 9,
where we have focussed on the characteristic kinks, two
things become apparent. First, in the region after the
kink the concentration gradient increases with scan rate
which is what we would expect for a gradient induced by
voltage increase – as the scan rate increases, the time for
bulk solution depletion decreases, resulting in steeper
gradients. Second, in the region between the kink and
the electrode surface, the profiles appear to be identical.
As mentioned above, the profile in this region can be
attributed solely to block regeneration. Unlike scan rate,
we cannot vary block regeneration so we would expect a
constant effect. More importantly, combining Eqs. (14)and (15) to eliminate w, we can write the current as:
I ¼ 2p R0 A½ bulk FD
Z 1
r b
oa
o z
z ¼0
r dr ð16Þ
in which we have used Aelec ¼ p R20, the area of the dif-
fusion domain. The concentration profiles in Fig. 9 are
plots of a against r and z at a potential corresponding to
the peak current. Close to the electrode surface, all three
profiles are identical which suggests that the integral in
Eq. (16) is constant. Thus, according to Eq. (16), the
peak current will be independent of scan rate. This phe-
nomenon is illustrated in Fig. 10 where we have re-
plotted Fig. 7 replacing the y -axis with I =½Abulk instead
of dimensionless current (the current values appear low
because the voltammograms are for a single diffusion
domain of radius R0 ¼ Rb= ffiffiffih
p ¼ 3:5 lm). The increase
in current as we increase the scan rate from 0.1 to 2.0
V sÀ1 is tiny and illustrates what little effect scan rate has
on the voltammograms. The reason for this indepen-
dence of peak current with scan rate has already been
touched upon. Because block regeneration occurs ex-
clusively at the electrode surface, very large concentra-
tion gradients are developed such that the catalytic
reaction dominates mass transport to the whole of the
electrode surface. Scan rate has no influence on thecoupled heterogeneous reaction, and therefore, no in-
fluence on the concentration profiles in the immediate
Fig. 9. Dimensionless concentration profiles as illustrated in Fig. 8
where we have ‘‘zoomed in’’ on the region close to the electrode sur-
face, well within the diffusion layer.
Fig. 10. The corresponding real current linear sweep voltammograms
for those illustrated in Fig. 7. Bold: m ¼ 2:0 V sÀ1; dashed: m ¼ 0:5V sÀ1; solid: m ¼ 0:1 V sÀ1.
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vicinity of the electrode surface. Thus, the integral in Eq.
(16) is dominated by block regeneration and we observe
a peak current (‘‘limiting’’ current) which is independent
of scan rate.
Is this phenomenon true for all cases? The question
can be answered if we consider situations where block
regeneration might not dominate the mass transport toall of the unblocked electrode surface. In other words,
when will increasing the scan rate cause an increase in
the peak current? Judging from the concentration pro-
files, scan rate will never influence mass transport near
the blockedjunblocked electrode boundary – the gradi-
ent due to block regeneration is simply too high. How-
ever, the outskirts of the diffusion domain are the least
affected by block regeneration, so scan rate could have
an influence on mass transport to this region of the
uncovered electrode surface. Therefore, we should ex-
pect to see an increase in peak current with scan rate at
domains of low coverage, h, where the outskirts of the
domain are far from the block. As we increase the scan
rate, the time for regenerated material to diffuse from
the block to the outer regions of the domain decreases.
At the same time, the diffusion layer thickness decreases
and mass transport to the outer region of the electrode
should no longer be dominated by block regeneration.
This is illustrated in Fig. 11 whereR 1
r bðoa=o z Þ z ¼0r dr at
the peak potential (which according to Eq. (16) is pro-
portional to the peak current, I pÞ is plotted against
coverage for a range of scan rates at a domain where
k 0 ¼ 0:005 cm sÀ1, D ¼ 2:0 Â 10À6 cm sÀ1, k het ¼ 10
cm sÀ1 and the block radius is (a) 2:5 lm or (b) 10.0 lm.
Overlaid on both (a) and (b) are the corresponding plotsfor the inert domains (dashed lines). The figure is
slightly complicated by the increasing domain radius,
R0, for the decreasing h values but the trends are as
follows. At domains of high coverage, h > 0:5, block
regeneration dominates mass transport – not only do we
observe greatly increased currents but I p is independent
of scan rate. For 0:2 < h < 0:5 the outer regions of the
domains are too far from the block for it to have any
effect and we enter an intermediate region where diffu-
sion is also influenced by the increasing voltage. Thus,
we observe a significant increase in peak current with
increasing scan rate. Finally, for h < 0:1 the coverage is
so low that most of the uncovered electrode is ‘‘out of
range’’ from the influence of block regeneration and we
enter a region where diffusion is dominated by the in-
creasing voltage. Hence, we observe identical peak cur-
rents to the inert counter part. The change on going
from high to low coverage is more defined for the
Rb ¼ 10 lm situation because we are dealing with larger
domains. Note that at low scan rates, m < 0:1 V sÀ1, for
some domains (e.g., Rb ¼ 2:5 lm) block regeneration
has a major influence on mass transport over the full
range of coverage. Fig. 12(a) illustrates the concentra-
tion profile, taken at E
¼ À0:6 V (a potential where
reduction of A to B is rapid), where R0 ¼ 20 lm,
h ¼ 0:1, k het ¼ 10 cmsÀ1, k 0 ¼ 0:005 cmsÀ1, D ¼ 2:0 Â10
À6cm s
À1 and m
¼2:0 V s
À1, i.e., a domain where mass
transport is not dominated by block regeneration. No-
tice that there is no characteristic kink in the profile and
the concentration gradient out the outskirts of the do-
main is what we would expect from an inert domain
with the same parameters. Fig. 12(b) illustrates the
corresponding simulated cyclic voltammogram, along
with that for the inert case. As observed, the effect of the
reactive block is minimal. Although the block supplies a
large flux of A at distances close to the electrode surface,
it only does so for areas of the uncovered electrode surface
that are close to the blocked junblocked boundary. For
instance, in Fig. 6 we approximated the ‘‘range’’ of
(a)
(b)
Fig. 11. Plot of I p=2p R0½Abulk FD against h for diffusion domains where
D ¼ 2.0 Â 10À6 cm2 sÀ1, k 0 ¼ 0:005 cmsÀ1, m ¼ 0:05, 0.1, 0.5 and 2.0
V sÀ1, and Rb ¼ (a) 2.5 lm, and (b) 10 lm. The solid curves corre-
spond to k het ¼ 10 cm sÀ1 and the dashed curves correspond to k het ¼ 0
cm sÀ1.
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block regeneration as 1.5 lm (for the specified param-
eters). Therefore, when dealing with large domains of
low coverage, the area of the uncovered electrode that
block regeneration will influence is small compared to
(1 À hÞp R20 (the total uncovered area). Additionally,
because diffusion to and from the domain outskirts is
approximately the same as that for the inertly blocked
domain, we observe a reverse peak. The voltammogram
for the reactive block in Fig. 12 also illustrates the ab-
sence of a ‘‘limiting current’’ which is observed for do-
mains of high coverage. This is to be expected for
domains where block regeneration has little influence on
the mass transport to the electrode surface. Fig. 13 il-
lustrates a similar plot to Fig. 11(a); the only difference
is that k het ¼ 0:01 cmsÀ1. As seen, a slower catalytic
reaction results in lower peak currents and a smaller
region over which peak current is independent of scan
rate.
2.3.4. The effect of k 0
In the case of a given coverage with an inert block,
the peak dimensionless current, wmax, varies between
two limiting cases (reversible and irreversible electrode
kinetics) depending on the scan rate, m, and magnitude
of the electron transfer rate constant, k 0
[2]. The linearsweep voltammograms in Fig. 14(a) demonstrate the
effect of varying k 0 for a diffusion domain, where
Rb ¼ 2:5 lm, h ¼ 0:5, D ¼ 2:0 Â 10À6cm sÀ1, k het ¼ 10
cm sÀ1, m ¼ 0:1 V sÀ1 and k 0 ¼ 101 and 10À5 cm sÀ1. As
we move from fast (k 0 ¼ 10 cm sÀ1) to slow
(k 0 ¼ 0:00001 cm sÀ1) electrode kinetics we observe an
expected change in the voltammogram shape, i.e., a shift
in the peak potential, but only a very slight decrease in
the peak dimensionless current: 7:330 À 7:314 ¼ 0:016,
corresponding to a 0.2% decrease. For the same domain
where the blocks are inert, Fig. 14(b), the decrease is
much more significant: 0:414À
0:312¼
0:102, corre-
sponding to a 25% decrease. Similar trends are observed
for domains of high coverage (h > 0:4), providing the
coupled heterogeneous reaction is fast enough (k het > 1
cm sÀ1).
The reason for these observations has already been
discussed – we observe a peak current when the poten-
tial is such that the electrode kinetics are fast enough to
achieve ½A z ¼0 ¼ 0. The larger the electron transfer rate
constant, the lower is the required potential, hence as we
increase k 0 we observe a decrease in E p. The slight de-
crease in current is due to the increased depletion of A
that is not fully compensated by block regeneration.
Fig. 12. (a) Dimensionless concentration profile, taken at E ¼ E p and
(b) the corresponding linear sweep voltammogram (solid curve) for a
domain where Rb ¼ 20 lm, h ¼ 0:1, k het ¼ 10 cm sÀ1, k 0 ¼ 0:005
cm sÀ1, D ¼ 2:0 Â10À6 cm2 sÀ1 and m ¼ 2:0 V sÀ1. Overlaid as a dashed
curve in (b) is the corresponding linear sweep voltammogram for the
inert domain, i.e., k het ¼ 0 cmsÀ1.
Fig. 13. Plot of I p=2p R0½Abulk FD against h for diffusion domains where
Rb
¼2:5 lm, D
¼2:0
Â10À6 cm2 sÀ1, k 0
¼0:005 cm sÀ1, and m
¼0:05,
0.1, 0.5 and 2.0 VsÀ1. The solid curves correspond to k het ¼ 0:01cm sÀ1 and the dashed lines correspond to k het ¼ 0 cmsÀ1.
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An interesting situation occurs when we vary k 0 in
domains of low coverage (h < 0:05). Figs. 15(a)–(d) il-
lustrate the simulated linear sweep voltammograms for a
domain where h ¼ 0:02, m ¼ 0:0 2 V sÀ1, Rb ¼ 3:0 lm
and k 0 ¼ 101, 10À1, 10À3 and 10À5 cm sÀ1. In all four
figures we have plotted the voltammograms for both the
reactive (k het ¼ 100 cmsÀ1) and inert (k het ¼ 0 c m sÀ1)
block. The domain is relatively large ð R0 ¼ Rb=
ffiffiffih
p ¼ 17 lm) so mass transport will be influenced by
the increasing voltage. Hence, we do not observe a
‘‘limiting current’’ and as we decrease k 0 there is a de-
crease in wmax. More interestingly, there appears to be a
pre-peak that becomes more defined as the electrode
kinetics become slower. The origin of this peak can be
understood by considering the voltammograms for the
same domain but with an inert block, which are overlaid
in Fig. 15. As seen, the pre-peak is almost identical to
the peak for the inertly blocked domain. In the con-
centration profiles displayed previously a severe con-
centration gradient at the blockedjunblocked electrode
boundary was labelled as the major source of the in-
creased currents. These profiles were all taken at po-tentials where the reduction of A to B at the electrode
surface was rapid and able to ‘‘compete’’ with the cou-
pled chemical reaction. In the case of the pre-peak,
observed in Fig. 15, the potential is such that reduction
of A to B at the unblocked electrode surface is simply
not fast enough to generate a large concentration gra-
dient similar to those seen in previous profiles. There-
fore, we observe a peak where the mass transport is
dominated by the effect of the increasing voltage (scan
rate) with block regeneration playing a minor role. It is
not until we reach higher potentials that block regen-
eration plays a major role in mass transport and we
observe the larger peak currents. In the case of the
fastest electrode kinetics (k 0 ¼ 10 cmsÀ1) a pre-peak is
not observed because rapid reduction is achieved at
much lower potentials so block regeneration plays its
role much earlier on in the scan. If we increase the scan
rate and/or increase the coverage then the pre-peak will
be shifted to more negative potentials but the main peak
(due to block regeneration) will stay at approximately
the same potential. Thus at higher values of m and h we
do not observe such anomalies.
2.3.5. The effect of k het
Fig. 16 illustrates linear sweep voltammograms for adomain where Rb ¼ 3:0 lm, h ¼ 0:5 and k het is varied
between 10À1 and 105 cm sÀ1 (the other parameters are
given in the figure legend). Similar plots are obtained for
domains where block regeneration dominates the mass
transport to the electrode surface (i.e., h > 0:4). There
are two major features. First, the system appears to
become diffusion controlled for k het > 103 cm sÀ1. Ini-
tially, this value may seem large in size; all electro-
chemists know that heterogeneous electron transfer
reactions are diffusion controlled when k 0 > 1–10
cm sÀ1, depending on the system parameters. However,
a comparison with Butler–Volmer kinetics is invalid. A
more detailed discussion and mathematical treatment
can be found in Appendix A, but the general point is as
follows. In cyclic voltammetry, for example, diffusion
controlled electron transfer redox reactions are electro-
chemically reversible – an equilibrium is maintained at
the electrode surface at all times. The coupled hetero-
geneous reaction studied in this work is chemically ir-
reversible – there is no equilibrium.
Second, as we increase k het, we observe a corre-
sponding increase in E p until we reach the diffusion
controlled regime. The reason for this has already been
touched upon. For domains where mass transport is
Fig. 14. Dimensionless current linear sweep voltammograms for a
single diffusion domain where k het ¼ (a) 10 cm sÀ1 and (b) 0 cmsÀ1. In
both (a) and (b) D ¼ 2:0 Â 10À6 cm2 sÀ1, Rb ¼ 2:5 lm, h ¼ 0:5, m ¼ 0:1V sÀ1, and k 0 ¼ 101 and 10À5 cm sÀ1.
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dominated by block regeneration we observe a peak
current when ½A z ¼0 ¼ 0. Faster coupled reactions sup-
ply more A to the electrode surface meaning we need to
reach higher potentials where the electron transfer re-
action is fast enough to reduce all of the regenerated A.
2.3.6. Summary of the factors discussed
From the preceding discussion we can categorize the
voltammetry of catalytically reactive partially blocked
electrodes into three main areas, as given in Table 1.
2.4. Relationship with experimental system
The experimental system to be compared with the
theory described above is the catalytic regeneration of
aqueous vitamin B12r (Co(II)L) by microdroplets of 1,2-
trans-dibromocyclohexane (DBCH) immobilized on the
electrode surface. By comparison with the homogeneous
pathway for this reaction (pathway (A)), we can write
the heterogeneous analogue as:
Co
ðII
ÞL
ðaq
Þ þeÀ¢
k f
k b
Co
ðI
ÞL
ðaq
ÞUnblocked electrode
RBr2ðoilÞ þ CoðIÞLðaqÞk 1BrÀðaqÞ þ CoðIIÞLðaqÞ
þ RBrÅ Drop surface ðEÞ
RBrÅ þ CoðIÞLðaqÞ k 2BrÀðaqÞ þ CoðIIÞLðaqÞ
þ R0ðoilÞ Drop surface or solution
where the first chemical step (k 1Þ is the reaction with
DBCH, the second step is the reduction of the RBrÅ
radical (k 2Þ and R0
is cyclohexene. A similar scheme can
be written for the E2 mechanism (pathway B). However,
Fig. 15. Simulated voltammograms for a diffusion domain, where Rb ¼ 3:0 mm, h ¼ 0:02, D ¼ 2:0 Â 10À6 cm2 sÀ1 and m ¼ 0:02 V sÀ1. In each part the
two voltammograms are for a reactive (k het ¼ 10 cmsÀ1) and inert (k het ¼ 0 cmsÀ1) block. Parts (a)–(d) correspond to different electron transfer rate
constants: k 0 ¼ (a) 101, (b) 10À1, (c) 10À3 and (d) 10À5 cm sÀ1.
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since the two reaction pathways (A) and (B) are kinet-
ically indistinguishable [18], the results in this section are
also applicable to the liquidjliquid equivalent of path-
way (B). The overall flux of Co(I)L at the droplet sur-
face can be written as:
À Dd CoðIÞL½
dZ Z ¼0
¼ flux due to first step
þ flux due to second step ð17Þwhere we could write the flux due to the two individual
steps as:
À Dd CoðIÞL½
dZ
Z ¼0
first step
¼ k 1 CoðIÞL½ Z ¼0 ð18Þ
À Dd CoðIÞL½
dZ
Z ¼0
second step
¼ k 2 CoðIÞL½ Z ¼0 ð19Þ
However, the rate of the second step depends entirely on
the rate of the first and because it is the first step which is
rate determining we express Eq. (17) as:
À Dd CoðIÞL½
dZ
Z ¼0
¼ 2k 1 CoðIÞL½ Z ¼0: ð20Þ
Note that this is equivalent to saying RBrÅ is at a steady
state. Hence, the relationship between k het (theory) and
k 1 (experiment) is given by:
k het ¼ 2k 1: ð21ÞA major difference between our scheme and the homo-
geneous analogue involves the organic mediator,
DBCH. In the homogeneous mathematical treatment,
the depletion of the mediator is accounted for in the
mass transport equations. However, in our model we
have assumed that depletion of DBCH at the droplet
surface has a negligible effect on the rate of the coupled
heterogeneous reaction. The validity of this assumption
will be discussed later.
2.5. Accounting for the random distribution of microdro-
plets
The voltammetric response of a modified electrode is
the sum response of all the diffusion domains present (in
the experimental systems involved in this work there are
over 100,000 domains on the electrode surface). Because
microdroplet distribution is random, we require a pro-
tocol that allows us to predict the voltammetric re-
sponse of an ensemble of randomly distributed diffusion
domains.
When working with inert microdroplets in a previous
paper [3], we used the peak dimensionless current, wmax,
as a variable for characterising the modified electrode.In the work presented here we will use a similar method
to determine a value of k 1 for the reaction in pathway
(E). That is, for given values of D, k het; k 0 and m, we will
focus primarily on the relationship between the volume
of blocking material (DBCH microdroplets) and the
peak dimensionless current, wmax.
The following method is a modification of Model B1
described in [3]. We assume that all droplets on the
electrode surface are hemispherical with a constant ra-
dius, Rb. The validity of these two assumptions, partic-
ularly the monodisperse distribution, is discussed in Part
II [3]. The number of droplets on the electrode surface,
N block, can be calculated using
Fig. 16. Dimensionless current linear sweep voltammograms for a
domain, where Rb ¼ 3:0 lm, h ¼ 0:5, k 0 ¼ 0:005 cm sÀ1, D ¼2:0 Â 10À6 cm2 sÀ1, m ¼ 0:02 V sÀ1, and the heterogeneous rate con-
stant for the reactive block, k het, varies between 100 and 105 cm sÀ1.
Table 1
Categorization of catalytically reactive partially blocked electrodes
Block regeneration dominates
mass transport
Both block regeneration and scan rate influence mass
transport to the electrode surface
Mass transport dominated by the
increasing voltage
Observe greatly enhanced
‘‘limiting’’ currents
I p increases with m and k 0 Small increase in I p and no ‘‘limiting’’
current observed
‘‘ I p’’ (almost) independent
of m and k 0A pre-peak is observed at low h and m Voltammetry (almost) identical to
k het ¼ 0 domains
E p depends on when [A z ¼0 ¼ 0 Back peak observed in CV
No back peak observed in CV
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N block ¼ 3V block
2 Rb Aelec
ð22Þ
where V block is the volume of blocking material (a known
experimental quantity) and Rb, the DBCH droplet ra-
dius, can be determined via experiments with an inert
redox couple (vide infra). The number of domains with
radius R0 þ d R0 on the electrode surface, N R0, is given by
the total number of domains, N block, multiplied by the
probability of the domainÕs occurrence:
N R0¼ N block P R0ð Þd R0 ð23Þ
where P ð R0Þ is given by
P R0ð Þ ¼ 2p R0 N block
Aelec
eÀp R2
0 N block
Aelec : ð24Þ
We then vary R0 between Rb and 3h R0i. For R0 < Rb, the
domain is completely blocked so wmax ¼ 0, and the
probability that domains of radius R0 > 3 < R0 > occur
is negligible. Given values of k 0
, D, m and k het, we canconvert to dimensionless variables and simulate a peak
dimensionless current for each domain, wmaxð R0Þ, in the
R0 range:
wmax R0ð Þ ¼ f h; mdl; k 0dl; k 0het
À Á: ð25Þ
The peak dimensionless current due to the whole elec-
trode is then given by the following integral:
wmax ¼ p N block
Aelec
Z 3 R0h i
0
R20wmax R0ð Þ P R0ð Þd R0 ð26Þ
in which the ðp N block= AelecÞ R20 P ð R0Þ term is used to weight
each simulated wmax value according to the size and
quantity of each domain.
Fig. 17 illustrates a plot of simulated ðp N block= AelecÞ R2
0 P ð R0Þwmaxð R0Þ against R0, where V block ¼4:1 Â 10À5 cm3, k 0 ¼ 0:005 cmsÀ1, D ¼ 2:0 Â 10À6 cm2
sÀ1, k het ¼ 10 cmsÀ1, Rb ¼ 2:5 lm, m ¼ 0:1 V sÀ1 and
Aelec ¼ 0:2011 cm2. From Eq. (26) it follows that the
theoretically predicted current for the whole electrode is
given by the area under the curve. When working with aparticular redox couple of known k 0 and D, for differ-
ent values of k het;V block and m, we can simulate such
curves and thus generate a table of theoretical data
which can be compared with those obtained experi-
mentally. Determining k hetðor 2k 1Þ is then a matter of
interpolation.
3. Experimental
3.1. Chemical reagents
All reagents were of the highest grade available
commercially and were used without further purifica-
tion. These were dodecane (Aldrich, 99+%), trans-1,
2-dibromocyclohexane (Aldrich, GC), potassium hexa-
cyanoferrate(II) trihydrate (Lancaster, 99%), hydroxo-
cobalamin hydrochloride (vitamin B12a, Sigma, 98%),
potassium chloride (Fluka, AnalR), acetonitrile (Fisher,
dried and distilled) and ethyl acetate (Fisher, HPLC).
Water, with a resistivity of not less than 18 MX cm, used
to make the electrolyte and buffer solutions was taken
from an Elgastat system (USF, Bucks., UK). All solu-
tions were out gassed with oxygen-free nitrogen (BOCGases, Guilford, Surrey, UK) for at least 20 min prior to
experimentation. All experiments were conducted at
22 Æ 3 °C. Experiments involving vitamin B12a were
conducted in a pH 2.5 phosphate buffer prepared with
an ionic strength of 0.2 M (0.1 M sodium dihydrogen-
phosphate and 0.1 M phosphoric acid (85 wt% solution
in water) both from Aldrich). At this pH, the Co(II)
complex (vitamin B12rÞ exists in its ‘‘base off’’ form,
which is easier to reduce than the ‘‘base on’’ form [17].
Vitamin B12a has a cobalt(III) centre and undergoes
two consecutive one-electron reductions to B12r (Co(II))
and B12s (Co(I)). In this work we were interested only in
the B12r=B12s couple and so to avoid the complication
caused by the presence of B12a ,a bulk electrolysis of B12a
to B12r was performed before each experiment as un-
dertaken by Rusling and co-workers [18]. The electrol-
ysis was stopped after about 4 h during which ca. 1.5
equivalents of the required charge had passed. After the
pre-electrolysis the solution had a brown colour, indi-
cating the presence of vitamin B12r [19] and linear sweep
voltammetry with a 3.0-mm diameter glassy carbon
electrode showed the absence of the B12a – B12r reduction
peak. A constant flow of nitrogen was kept over the B12r
solution at all times.
Fig. 17. Simulated curve of ðp N block= AelecÞ R20 P ð R0Þwmaxð R0Þ vs. R0 for
an electrode modified with ‘‘reactive’’ blocks of surface radius
Rb ¼ 2:5 lm and k het ¼ 10 cmsÀ1. The other parameters are m ¼ 0:1V sÀ1, D ¼ 2:0 Â 10À6 cm2 sÀ1, k 0 ¼ 0:005 cm sÀ1, Aelec ¼ 0:20 cm2 and
V block ¼ 4:1 Â 10À5 cm3. The area under the curve is the predicted peak
dimensionless current, wmax, for a modified electrode with the stated
parameters.
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3.2. Instrumentation
Electrochemical experiments were under taken in a
conventional three-electrode cell, employing a 5.1 mm
diameter basal plane pyrolytic graphite electrode (bppg,
Le Carbone Ltd., Sussex, UK), a platinum wire counter
electrode and a saturated calomel reference electrode(Radiometer, Copenhagen, Denmark). Electrochemical
data were recorded using a commercial computer con-
trolled potentiostat (AUTOLAB PGSTAT30, Eco-
Chemie, Utrecht, The Netherlands).
The bppg working electrode was modified with mi-
crodroplets of DBCH by solvent evaporation of a 2–10
lL aliquot of DBCH + acetonitrile stock solution. Sim-
ilarly, modification with microdroplets of a mixture of
DD and DBCH were obtained via evaporation of a 4 lL
aliquot of DD + DBCH + ethyl acetate solution – it was
found that volumes in excess of 4 lL did not stay con-
fined to the bppg surface. The electrode was cleaned
immediately prior to experimentation by rinsing with
acetone and water and the surface was renewed by
polishing on carborundum paper (P1000 grade, Acton
and Bormans, Stevanage, UK).
3.3. Computation
The voltammetric response of reactive diffusion do-
mains was simulated using the same protocols described
in Part I [2]. Concentration profiles were obtained via
Matlab 5.1.0.
4. Results and discussion
Experiments with DBCH and vitamin B12r were split
into two sections. First, the bppg electrode was modified
with varying amounts of pure DBCH in order to obtain
a rate constant for the DBCHjB12s liquid–liquid reac-
tion. Second, to mimic the experiments performed by
Rusling and co-workers [18], the bppg electrode was
modified with various ratios of DBCH and dodecane
(an inert oil).
4.1. Electrodes modified with pure DBCH
4.1.1. Determination of DBCH droplet radius
As mentioned in Section 2.5, before performing any
‘‘reactive’’ experiments with DBCH-modified electrodes,
for the purpose of data analysis it is necessary to obtain
a DBCH droplet radius, Rb. This was achieved by re-
cording the voltammetry of a basal plane electrode
modified with microdroplets of DBCH in a solution of
ferrocyanide (a medium in which DBCH is inert).
Fig. 18 illustrates cyclic voltammograms at a DBCH-
modified bppg electrode in 2.04 mM ferrocyanide + 0.1
M KCl. The microdroplets appear not to react with ei-
ther species in the redox couple, the voltammetry dis-
playing all the characteristics of inert partially blocked
electrodes [1–3]. Analysis of voltammograms obtained
from the naked electrode with DigiSimÓ gave a diffusion
coefficient, D ¼ 5:6 Æ 0.3 Â 10À6 cm2 sÀ1, and electron
transfer rate constant, k 0 ¼ 0:01 Æ 0.002 cm sÀ1, for the
Fe(CN4À6 )/Fe(CN3À
6 ) redox couple, both values in
Fig. 18. Cyclic voltammograms for the oxidation of 2.04 mM potas-
sium hexacyanoferrate(II) trihydrate aqueous solutions containing 0.1
M potassium chloride at a 5.1 mm diameter basal plane pyrolyticgraphite electrode modified with 0, 120, 300 and 600 nmol of DBCH.
The scan rate was 0.2 V sÀ1.
Fig. 19. Simulated relationship (according to Model B1Þ between wmax
and Rb for a 5.1 mm basal plane electrode modified with 4.1 Â 10À5
cm3 of inert blocking material at four different scan rates (0.1, 0.2, 0.5
and 1.0 VsÀ1), where D ¼ 5:6 Â 10À6 cm2 sÀ1 and k 0 ¼ 0:01 cmsÀ1.
The dashed lines represent the experimental data for DBCH blocking
in ferrocyanide solution and the dots are the best agreement between
theory and experiment.
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agreement for that determined previously on carbon
electrodes [20]. With values for D, k 0 and I p for different
scan rates and DBCH blocking volumes, we were able to
simulate the electrode response according to Model B1
[3]. Fig. 19 illustrates the theoretically predicted varia-
tion of peak dimensionless current with droplet radius at
four different scan rates where V block ¼ 4:1 Â 10À5
cm3
, Aelec ¼ 0:2011 cm2, D ¼ 5:6 Â 10À6 cm2 sÀ1 and
k 0 ¼ 0:01 cmsÀ1. Overlaid as dashed lines are the ex-
perimental results – the peak currents for each scan rate
were converted into dimensionless form using Eq. (15).
The points where the dashed and solid lines cross are
highlighted in the figure and correspond to the best
agreement between theory and experiment. From a
number of plots like Fig. 19, we were able to determine
the (model B1Þ droplet radius as Rb ¼ 2:5 Æ 0.5 lm.
4.1.2. Reaction with vitamin B 12s
Initial experiments with DBCH modified electrodes
gave poor reproducibility at low block volumes, i.e.,
V block < 2 Â 10À5 cm3 corresponding to around 150
nmol. For this reason, subsequent experiments were
performed on electrodes modified with V block > 3 Â 10À5
cm3 DBCH. To obtain values of D and k 0 for the
B12r=B12s redox couple necessary for the simulations,
cyclic voltammograms with a naked electrode were re-
corded over a range of scan rates. Subsequent analysis
with DigiSimÓ gave D ¼ 2:0 Æ 0:1 Â 10À6 cm2 sÀ1 and
k 0 ¼ 0:005 Æ 0:002 cm sÀ1, which agrees well with values
previously determined [18].
Fig. 20(a) illustrates cyclic voltammograms recorded
at 0.05 VsÀ1
in 1.18 mM vitamin B12r, where the elec-trode is naked and modified with 300 nmol of DBCH
(corresponding to V block ¼ 4:1 Â 10À5 cm3). The reduc-
tion of vitamin B12r to B12s in pH 2.5 phosphate buffer at
a bppg electrode occurs quite close to solvent break-
down leading to scans where peaks are observed just
before the potential becomes too negative. However, the
differences between the two scans are clear and very
much like those observed for the simulated voltammo-
grams in Fig. 3. Not only do we observe the absence of a
back peak, but also the current at the modified electrode
is greatly enhanced with what appears to be a limiting
plateau before we enter solvent breakdown. All this
suggests that we are dealing with a fast coupled heter-
ogeneous reaction where mass transport is dominated by
block regeneration. The catalytic reaction consumes
DBCH and, as expected, repetitive scans show decreased
peak currents, the decrease being more significant for
the lower scan rates. The influence of this droplet de-
pletion is discussed later.
To confirm that the catalytic increase in current was
solely due to DBCH, voltammograms were recorded
with similar block volumes of pure dodecane, an ex-
ample of which is illustrated in Fig. 20(b). These gave a
decrease in current, when compared to the naked results,
consistent with the predicted response for a droplet ra-
dius of approximately 4–5 lm. This suggests that the
presence of any oxygen dissolved in the microdroplets
had a negligible effect on the observed voltammetry.
Also, scans recorded with a DBCH-modified electrode
in phosphate buffer (i.e., no B12rÞ showed no appreciable
direct reduction of DBCH within the potential window.
Fig. 21 illustrates voltammograms recorded at four
different scan rates in 1.18 mM vitamin B12r, where the
Fig. 20. (a) Cyclic voltammograms for the reduction of 1.18 mM B12r
solution in aqueous pH 2.5 phosphate buffer at a 5.1 mm diameter
bppg electrode modified with 0 (naked) and 300 nmol of DBCH. The
scan rate was 0.05 V sÀ1. (b) Cyclic voltammograms for the reduction
of 1.21 mM B12r solution in aqueous pH 2.5 phosphate buffer at a 5.1
mm diameter bppg electrode modified with 0 (naked) and 6:4
Â10À5
cm3 dodecane. The scan rate was 0.2 V sÀ1.
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bppg electrode is modified with (a) 300 nmol and (b) 600
nmol DBCH. Only the forward scans are plotted (to
avoid over crowding) and the m ¼ 0:0 5 V sÀ1 scan is
printed in bold to give a clearer representation. As we
increase V block
, the quality of the scans decreases but a
peak/limiting current is still visible. At the highest cov-
erage, increasing the scan rate appears to have little ef-
fect on the recorded voltammograms. Similarly, in
Fig. 21(a) the relative increase in peak/limiting current
on going from 0.05 to 1.0 V sÀ1 is extremely small
compared to what one would expect from a m1=2 de-
pendence, for example.
With knowledge of Rb, D, k 0; Aelec and V block, the
theoretically predicted peak dimensionless current, wmax,
can be calculated for chosen values of k het and m via the
approach described in Section 2.5. Fig. 22(a) illustrates
the simulated relationship between wmax and m for the
modified electrodes in Fig. 21, where k het ¼ 10, 100 and
1000 cm sÀ1. The corresponding relationships between I pand m are displayed in Fig. 22(b). In both (a) and (b) the
experimental results are overlaid as circles. As observed
in Fig. 22(b), the simulated results suggest I p should be
(more or less) independent of m at all three values of k het
considered. This is a similar situation to that discussedin Section 2.3.3 but the cause is slightly more compli-
cated. For example, Fig. 10 in which we observed a peak
current–scan rate independence concerned a single dif-
fusion domain where mass transport was dominated by
block regeneration. Our modified electrode surface is an
ensemble of domains, not all of which show such be-
haviour. To consider the effect of having a range of
domains (each with Rb ¼ 2:5 lm), we need to take into
account their relative numbers. This is illustrated in
Fig. 23 where we have plotted P ð R0Þ against h for the
modified bppg electrode in Fig. 21(a) (V block ¼ 4:1 Â10À5 cm3). Notice that domains with h > 1:0 do have a
probability of occurring. This is the basis of how our
model takes into account droplet overlap and is dis-
cussed in more detail in Part II [3] (in an attempt to
focus on the 0 < h < 1 region we have omitted the rest
of the distribution function which tends to zero for
higher values of hÞ. Referring to Fig. 11(a) in Sec-
tion 2.3.3, we can separate Fig. 23 into three regions:
wmax ¼ 0 corresponding to hP 1; 0:4 < h < 1 where
block regeneration dominates mass transport to the
electrode surface; and h < 0:4 where the voltammetry is
influenced by scan rate. The block volume is such that
domains in the h < 0:4 region contribute a small pro-
portion to the total number of domains on the electrodesurface. Hence, the voltammetric response will be
dominated by domains in the region 0:4 < h < 1 (re-
member that domains in the region h ! 1 have no vol-
tammetric response), each of which has a peak current
which is independent of scan rate. If the major contri-
bution to the total current comes from a collection of
domains which posses an I p – m independence, the ob-
served peak current for the whole electrode should be
independent of scan rate. In addition, the recorded
voltammograms should show little variance with m, as
seen in Fig. 10. Increasing the block volume results in an
even smaller contribution from the h < 0:4 region so we
should expect a similar response from the electrode
blocked with 600 nmol DBCH.
However, the predicted I p – m independence is observed
only at the higher coverage (ignoring the 0.2 V sÀ1 re-
sult) and high scan rates, suggesting that the peak cur-
rents for the low scan rate data are too low. This
discrepancy between theory and experiment probably
results from our initial assumption that the reactivity of
the block is unaffected by DBCH depletion. Table 2 lists
the approximate percentage depletion of DBCH for
both sets of (forward) scans in Fig. 21. The DBCH de-
pletion is calculated as follows. Since the electrode is
Fig. 21. Linear sweep voltammograms for the reduction of 1.18 mM
B12r solution in aqueous, pH 2.5, phosphate buffer at a 5.1 mm di-
ameter bppg electrode modified with (a) 300 and (b) 600 nmol of
DBCH. The scan rates are labelled on the figures (0.05, 0.1, 0.5 and 1.0
V sÀ1).
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heavily blocked and the catalytic currents are greatly
enhanced, we can approximate the current as being en-
tirely due to the reduction of regenerated B12r. In
pathway (A), the reaction of one DBCH molecule re-
sults in the generation of two B12r
species, so the moles
of DBCH involved in the reaction is equal to half the
moles of electrons transferred. This allows us to calcu-
late the loss of DBCH (we know how much is deposited
before the scan) which leads to a percentage depletion.
The numbers in Table 2 suggest that the percentage
depletion is not significantly high. However, we need to
remember that the reaction occurs only at the droplet
surface (the liquidjliquid interface) – it is heterogeneous.
Therefore, a diffusion layer, similar to that produced in
a simple potential step experiment, will be formed as the
scan progresses – the longer the scan (i.e., the lower the
scan rate) the larger this diffusion layer becomes. Hence,
at low scan rates we observe a lower I p than expected
due to the currents being limited by DBCH diffusion
within the droplet, a factor unaccounted for in our
modelling. Upon increasing the scan rate, the depletion
is less and the DBCH diffusion layer has a shorter time
in which to grow. Therefore, the effect of DBCH de-
pletion becomes negligible and we observe the predicted
I p – m independence. At higher blocking volumes the area
of uncovered electrode is smaller, leading to lower cur-
rents, and the total microdroplet surface area is in-
creased. Both of these factors lessen the effect of DBCH
depletion and we obtain results that show better agree-
ment with theory over a larger range of scan rates.
In the light of these findings, further experiments
were conducted at scan rates in the range 0 :56 m6 4:0V sÀ1 with similar blocking amounts. Analysis of these
results also suggested a value of k het to be 150 < k het <
Fig. 22. Simulated variation of (a) wmax with scan rate and (b) I p with scan rate for the modified electrodes in Fig. 21 where k het ¼ 101, 102 and 103
cm sÀ1. Part (i) corresponds to the electrode modified with 300 nmol of DBCH and (ii) corresponds to the 600 nmol modification. The experimental
results are plotted as circles.
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250 cmsÀ1. According to Eq. (21), this suggests that the
rate constant for the liquid–liquid reaction between
DBCH and B12s is 75 < k 1 < 125 cm sÀ1.
4.1.3. The double peak anomaly
The quasi-reversible kinetics of the B12r=B12s redox
couple suggest that a double peak, as discussed in
Section 2.3.4, should be observed for domains of low
coverage at low scan rates. However, attempts to ob-serve this anomaly with modified electrodes of a suit-
able coverage were unsuccessful, primarily due to two
reasons. First, a modified electrode is an ensemble of
diffusion domains, which in our case vary in size. Even
at low block volumes, there will still be domains of
higher coverage (i.e., domains for which no double
peak is observed) that contribute to the overall vol-
tammetry. Because the observed current is a sum of
that for all the domains on the electrode surface, the
double peak signal, due to the small range of domains
we are interested in, could be lost. Second, at low
block volumes and low scan rates the effect of droplet
depletion is large.
4.2. Electrodes modified with a mixture of DBCH and
dodecane
This section examines the voltammetry of electrodes
modified with diluted DBCH oil droplets, the major
constituent of the drop being dodecane. Therefore,
comparisons of experimental results with simulationslead to the determination of an effective heterogeneous
rate constant, k eff 1 , for the liquid–liquid reaction. We
are interested in the variation of k eff 1 with the DBCH
concentration.
4.2.1. Determination of DBCH/dodecane droplet radius
Following the same method described in Section
4.1.1, the inert blocking of dodecane and DBCH was
investigated with solutions of 50–70 mM dodecane and
1.5–4.1 mM DBCH in ethyl acetate, corresponding to
microdroplet DBCH concentrations of 0.096 6
[DBCH] 6 0.32 M. In all cases the droplet radius was
determined to be Rb ¼ 4:5 Æ 0:5 lm.
4.2.2. Reaction with vitamin B 12s
Fig. 24 illustrates voltammograms recorded at a bppg
electrode modified with 4 ll of 70 mM dodecane and 3.1
mM DBCH solution (corresponding to [DBCH] ¼ 0.19
M and V block ¼ 6:5 Â 10À5 cm3) in a 1.15 mM solution of
pH 2.5 vitamin B12r. The difference between the vol-
tammograms recorded at ‘‘high’’ (1.0, 2.0 and 3.0 V sÀ1)
and ‘‘low’’ (0.1 V sÀ1) scan rates agrees with that dis-
cussed earlier. Although our reactant has been diluted,
about 5% of the volume applied is DBCH. Therefore, as
an initial estimate, we would not expect the effectiveheterogeneous rate constant to decrease by much more
than an order of magnitude, a value still high enough for
block regeneration to dominate mass transport to the
electrode surface. Also, the block volume we are work-
ing with is large enough for domains that do show an
I p – m dependence to have a negligible contribution. At
the higher scan rates we observe this expected behaviour
– the voltammetry is almost independent of the rate of
the potential sweep. At the low scan rates, the effect of
DBCH depletion is even more apparent for these mod-
ified electrodes. Not only is the current smaller than
expected, a clear forward peak and slight back peak are
also observed. On closer inspection of the 0.1 V sÀ1
Fig. 23. The distribution of diffusion domains on a 5.1 mm diameter
bppg electrode surface modified with 4.1Â 10À5 cm3 of blocking ma-terial, where the individual block radius is 2.5 lm.
Table 2
Approximate percentage depletion of DBCH (up to )1.1 V vs. SCE on the forward scans) on the modified electrodes in Fig. 21
m/V sÀ1 300 nmol 600 nmol
DBCH depletion/nmol Depletion/% DBCH depletion/nmol Depletion/%
0.05 11 3.7 8.1 1.3
0.1 6.1 2.0 4.0 0.67
0.5 1.4 0.47 0.83 0.14
1.0 0.73 0.24 0.41 0.07
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voltammogram, a slight shoulder can be seen before the
reductive peak. Similar features have been observed in
homogeneous media and are associated with reactant
(DBCH) depletion [24]. Thus, when the electrode is
modified with ‘‘diluted’’ reactant, the limiting effect of
diffusion within the droplet is enhanced.
Fig. 25 illustrates the simulated variation between k eff het
and I p for the modified electrode in Fig. 24 at 1.0, 2.0
and 3.0 VsÀ1. The high block volume results in vol-
tammetry dominated by domains of high coverage and
we observe only a slight increase in I p with m over the
range shown. Using the high scan rate experimental data
and these simulated curves, the k eff het (and therefore k eff
1 Þvalues were determined via interpolation, as illustrated
in Fig. 25 for the m ¼ 2:0 V sÀ1 scan, for a range of
DBCH concentrations. The results are illustrated in
Fig. 26 and suggest a linear relationship between k eff 1 and
[DBCH], the gradient of which is 21 MÀ1 cm sÀ1 and can
be equated to the apparent bimolecular rate constant(k ¼ k eff
1 /[DBCH]). In a similar experiment, Rusling and
co-workers [18] investigated the liquid–liquid reaction
between aqueous B12r and DBCH diluted in benzonit-
rile. Using the SECM/ITIES method described by Bard
and co-workers [11], they determined effective hetero-
geneous rate constants for different concentrations of
DBCH in benzonitrile. Their results also indicated a
linear relationship with a value of 3.0 MÀ1 cm sÀ1 for the
apparent bimolecular rate constant. The difference (al-
most an order of magnitude) between the two apparent
bimolecular rate constants could be due to a number of
reasons. First, benzonitrile and dodecane are signifi-
cantly different organic solvents: benzonitrile is polar
whereas dodecane is not. Thus, in the two liquids, the
solvation of the reactants and transition state will vary,
leading to different Gibbs energy of activation values for
the two systems. Second, our work has clearly shown
that droplet depletion can have a limiting effect on the
observed current. If it is neglected to model diffusion on
both sides of the ITIES one must work with system pa-
rameters where the effect of reactant depletion is negligi-
ble. Since our approach is based on transient
voltammetry we were able to do this easily by operating
at high scan rates. Once we had observed the predicted
Fig. 24. Cyclic voltammograms for the reduction of 1.15 mM B12r
solution in aqueous, pH 2.5, phosphate buffer at a 5.1 mm diameter
bppg electrode modified with 6.5Â 10À5 cm3 of 0.19 M DBCH in
dodecane. The illustrated voltammograms correspond to scan rates of
0.1, 1.0, 2.0 and 3.0 V sÀ1.
Fig. 25. Simulated variation of the peak current with k eff 1 for the
modified electrodes in Fig. 24, where the scan rates are 1.0, 2.0 and 3.0
V sÀ1. The dashed line is an example of the interpolation procedure by
which we determine k eff 1 .
Fig. 26. Dependence of the effective heterogeneous rate constant, k eff 1 ,
on the concentration of DBCH in dodecane.
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I p – m independence we could be certain our currents were
not limited by diffusion within the microdroplets.
However, the SECM/ITIES technique used by Bard and
co-workers [11] and Rusling and co-workers [18] is
based on steady-state measurements. Currents are re-
corded at times when there will be well-established dif-
fusion layers on both sides of the ITIES. Indeed, Unwinand co-workers [21,22] have investigated the conse-
quence of reactant depletion in the second phase on such
steady state measurements at the ITIES. Their work [22]
suggests that the concentrations used by Rusling and co-
workers [18] falls into the region where the constant
composition assumption is invalid, leading to an un-
derestimate of the actual bimolecular rate constant. In-
terestingly, the concentrations of DBCH required for
the constant composition assumption to be valid would
result in SECM currents limited by diffusion of B12r
rather than the rate of the liquid–liquid reaction [18,22].
5. Conclusions
In this paper we have seen how the principles devel-
oped in Parts I and II for the simulation of uncompli-
cated electron transfer reactions at an electrode partially
covered with inert blocking material can be extended
further to model irreversible liquid–liquid reactions at
the surface of microdroplets dispersed over an electrode
surface. The transient technique developed in this work
has a number of advantages over current probes for
reactions occurring at the interface between two im-
miscible electrolyte solutions (ITIES). Not only are theexperimental requirements relatively simple, compared
to scanning electrochemical microscopy (SECM) for
example, but in theory we can measure very fast liquid–
liquid reactions before being limited by diffusion. Fur-
thermore, when studying fast liquid–liquid reactions
(k het > 1 cmsÀ1) at high block volumes, we can easily
identify results where the current is limited by diffusion
in the second phase, a factor unaccounted for in our
modelling. When determining heterogeneous (bimolec-
ular) rate constants, this enables us to use the constant
composition assumption with some degree of confi-
dence, unlike in the SECM/ITIES method where reac-
tant depletion in the second phase can often be
problematic [21,22].
Acknowledgements
The authors thank the EPSRC for studentships for
T.J.D. and B.A.B. We further thank Russell Evans, Jay
Wadhawan, Michael Hyde, Robert Jacobs and Phillip
Greene for their assistance regarding experimental pro-
cedures, and Prof. J.F. Rusling for initiating our interest
in vitamin B12s chemistry.
Appendix A. Can we compare values of khet and k0
corresponding to diffusion limitation?
Consider the following coupled heterogeneous elec-
trode reaction where electron transfer at a planar macro
electrode is preceded by an irreversible heterogeneous
reaction:
Ak hetB
B þ eÀ¢
k f
k b
C:ðFÞ
After t ¼ 0, the potential is such that the reduction of B
to C is rapid and limited by the irreversible heteroge-
neous reaction. Therefore, the flux of B to the electrode
surface will be equal to the flux of A and we can write:
I
FAelec
¼ DB
o B½ o x
x¼0
¼ k het A½ x¼0 ðA:1Þ
where DB is the diffusion coefficient of B and the othersymbols have already been defined. With B and C ini-
tially absent, the equations that describe such a system
are:
diffusion equation:
o A½ ot
¼ Do
2 A½ o x2
ðA:2Þ
boundary conditions:
t ¼ 0 x A½ ¼ A½ bulk ðA:3Þt > 0 x 1 A½ ¼ A½ bulk ðA:4Þ
t > 0 x ¼ 0 Do A½ o x
x¼0
¼ k het A½ x¼0 ðA:5Þ
where D is the diffusion coefficient of species A, [Abulk is
the bulk concentration of A, and [A x¼0 is the concen-
tration of A at the electrode surface. A Laplace trans-
formation of Eq. (A.2) and the first two boundary
conditions leads to:
A½ ¼ A½ bulk
sþ f sð ÞeÀð s= DÞ1=2 x ðA:6Þ
where f ð sÞ is a factor to be determined and we have
transformed from t to s space. The Laplace transform of
the final boundary condition (Eq. (A.5)) allows the de-
termination of f ð sÞ:
f ð sÞ ¼ À H i A½ bulk
s H i þ s1=2ð Þ ðA:7Þ
where H i ¼ k het= D1=2. Substituting for f ð sÞ in Eq. (A.6)
gives:
A½ ¼ A½ bulk
sÀ H i A½ bulkeÀð s= DÞ1=2 x
s H i þ s1=2ð Þ ðA:8Þ
Now, we evaluate the Laplace transform of Eq. (A.1) to
obtain an expression for the transformed current:
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I ¼ FAelec Do A½ o x
! x¼0
¼ FAeleck het A½ bulk
s1=2 H i þ s1=2ð Þ ðA:9Þ
and the current is given by the inverse transform:
I
¼ FAeleck het A
½ bulke H 2
it erfc H it 1=2À Á: ð
A:10
ÞThe diffusion limited current for this system, I d;i, is given
by the Cottrell equation [15]:
I d;i ¼ FAelec D1=2 A½ bulk
p1=2t 1=2: ðA:11Þ
We can compare the above irreversible heterogeneous
kinetics with electron transfer kinetics by considering a
simple potential step experiment on a one-electron redox
couple, where the potential applied is not high enough
to cause rapid reduction and B is initially absent:
A þ eÀ¢
k f
k b
B: ðGÞ
The current for this arrangement is given by
I ¼ FAelec k f A½ x¼0
À À k b B½ x¼0
Á ðA:12Þwhere k f and k b are defined in Eqs. (2) and (3), respec-
tively. Following a similar procedure to that described
above, we can express the current as [15]:
I ¼ FAeleck f A½ bulke H 2r t erfc H rt 1=2À Á
; ðA:13Þwhere H r ¼ ðk f þ k bÞ= D, assuming the diffusion coeffi-
cients of the two species are equal. The diffusion con-
trolled current for the potential step experiment, I d;r is
given by [15]:
I d;r ¼ FAelec D1=2 A½ bulk
p1=2t 1=2 1 þ hð Þ ðA:14Þ
Fig. 27. Simulated voltammetric response for the ‘‘potential step’’ experiments described in the text over (a) 10ls and (b) 1 s, where the kinetic rate
constants vary between 10À3 and 101 cm sÀ1. Plots labelled (i) correspond to the irreversible heterogeneous reaction. Plots labelled (ii) correspond to
the electron transfer reaction where E ¼ À0:005 V vs. E 00. In both simulations T ¼ 298 K and D ¼ 1:0 Â 10À5 cm2 sÀ1. The diffusion limited response
for each situation is plotted as a dashed line.
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where
h ¼ e F E À E 00ð Þ
RT : ðA:15ÞFig. 27 illustrates the voltammetric response for both
systems described in the above text over (a) short
(t
¼10 ls) and (b) long (t
¼1 s) time scales. The plots
labelled ‘‘(i)’’ correspond to the irreversible heteroge-neous reaction, whereas ‘‘(ii)’’ corresponds to the elec-
tron transfer reaction. The diffusion limited response is
overlaid as a dashed curve. In both parts (a) and (b) it is
clearly observed that the electron transfer reaction be-
comes diffusion controlled significantly earlier than the
irreversible heterogeneous reaction, for the same k 0 and
k het values. It could be argued that the figures differ
because they correspond to two slightly different ex-
perimental systems. However, that is the main point – a
direct comparison between values of diffusion limited k 0
and k het is inappropriate because of the way diffusion
limited is ‘‘defined’’ for the two types of reaction. In thecase of Butler–Volmer electron transfer, the reaction
becomes diffusion controlled once equilibrium is
achieved at the electrode surface. This is certainly not
the case for irreversible heterogeneous kinetics where the
reaction becomes diffusion limited when mass transport
can no longer supply enough reactant. This is high-
lighted in the difference between the derivations of Eqs.
(A.11) and (A.14). In the Cottrell equation (A.11) we use
the boundary condition t > 0, ½A x¼0 ¼ 0, whereas the
Nernst boundary condition is employed for reversible
electron transfer reactions. Therefore, as well as having
different equations that describe the voltammetric re-
sponse ((A.10) and (A.13)), we also have different
equations that describe the diffusion limited response
((A.11) and (A.14)).
Fig. 27 demonstrates that a diffusion limited k 0 value
corresponds to a diffusion limited k het value which is
within an order of magnitude larger. For example, in
Fig. 27(a) the k 0 ¼ 10 cmsÀ1 response appears diffusion
limited from the very start, whereas that for k het ¼ 10
cm sÀ1 does not. To get the same response from the ir-
reversible kinetics, the value of k het needs to be increased
to 20 cm sÀ1. This method of comparison is for a system
where mass transport is described by linear diffusion. In
the reactive partially blocked electrode experiment de-scribed within this paper, mass transport is described by
convergent diffusion. Upon increasing the mass-transfer
coefficient, not only do the diffusion limited regimes
occur at higher values of k 0 and k het, but it would also be
reasonable to expect an increase in the difference be-
tween diffusion limited k het and diffusion limited k 0 val-
ues. Indeed, in a theoretical treatment for an electron
transfer preceded by adsorption of the electroactive
species at a rotating disc electrode, Harland and
Compton [23] demonstrated that the chronoampero-
metric response became diffusion limited around
k het ¼ 1000 cm sÀ1. They went on to investigate the ki-
netics of bromine adsorption onto poly-crystalline
platinum and determined the irreversible heterogeneous
rate constant to be ca. 220 cm sÀ1
[23]. Therefore, thefact that our partially blocked electrode system also
becomes diffusion controlled just after k het ¼ 1000
cm sÀ1 is not unreasonable.
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