a computational and experimental study of the cyclic

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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

www.elsevier.com/locate/jelechem

Journal of

ElectroanalyticalChemistry

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http://mail%20to:%[email protected]/



[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.

194 T.J. Davies et al. / Journal of Electroanalytical Chemistry 566 (2004) 193–216



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].




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.

T.J. Davies et al. / Journal of Electroanalytical Chemistry 566 (2004) 193–216 197



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.




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).




‘‘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.




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.




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.




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.




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.




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.




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




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.




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.




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.




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).




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.




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




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.




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:




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.




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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a computational and experimental study of the cyclic

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