Cyclic voltammetry in weakly supported media: The reduction of the cobaltocenium cation in acetonitrile – Comparison between theory and experiment

Journal of Electroanalytical Chemistry 650 (2010) 135–142

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Cyclic voltammetry in weakly supported media: The reduction of the cobaltocenium cation in acetonitrile – Comparison between theory and experiment Juan G. Limon-Petersen, Edmund J.F. Dickinson, Stephen R. Belding, Neil V. Rees, Richard G. Compton ⇑ Department of Chemistry, Physical and Theoretical Chemistry Laboratory, Oxford University, South Parks Road, Oxford OX1 3QZ, United Kingdom

a r t i c l e

i n f o

Article history: Received 12 July 2010 Received in revised form 17 August 2010 Accepted 19 August 2010 Available online 6 September 2010 Keywords: Ohmic drop Migration Nernst–Planck–Poisson equations Cobaltocenium Weakly supported voltammetry Supporting electrolyte

a b s t r a c t Experimental cyclic voltammetry at a hemispherical mercury microelectrode in acetonitrile solution, containing 3 mM cobaltocenium hexafluorophosphate and different concentrations of supporting electrolyte, is compared with theoretical simulations using the Nernst–Planck–Poisson system of equations, without the assumption of electroneutrality, and is found in to be in good agreement. Deviations from diffusion-only theory are analyzed in terms of migration and potential drop in the solution as a function of the concentration of supporting electrolyte. We are unaware of previous reports in which non-steadystate cyclic voltammetry without supporting electrolyte has been quantitatively and fully simulated, so this work opens up a new area for voltammetry. Ó 2010 Elsevier B.V. All rights reserved.

1. Introduction The power of voltammetry for the measurement of kinetic and thermodynamic redox data and for understanding the mechanisms of reactions involving electron transfer is widely recognized [1] and utilized. That said, experiments are conventionally restricted to solvents containing large amounts of electrolyte, either naturally or else through the deliberate addition of ‘supporting’ or ‘background’ electrolyte. The purpose of this is to ensure that the potential drop between the electrode and the solution occurs over a sufficiently short distance to be compatible with electron transfer via tunneling. Consequently the electric field associated with the potential drop is confined to a narrow interfacial region and is very close to zero in bulk solution. Hence, transporting ions to or from the electrode occurs only by diffusion; migration does not participate. We have considered elsewhere the concentration of supporting electrolyte required for a cyclic voltammetry experiment to be quantitatively diffusional [2]. It is the case that the addition of supporting electrolyte can introduce problems to an experiment. In particular, the altered ionic strength can change the values of the kinetic and thermodynamic properties sought. Also, complexation of the redox species of interest with the added electrolyte can occur, while for the case of many non-polar solvents it is impossible to dissolve the required ⇑ Corresponding author. Tel.: +44 (0) 1865 275413; fax: +44 (0) 1865 275410. E-mail address: [email protected] (R.G. Compton). 1572-6657/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jelechem.2010.08.011

levels of supporting electrolyte needed for rigorous elimination of migrational effects. For these reasons we have recently developed theoretical and simulation procedures aimed at permitting the quantitative prediction of voltammetry at micro and macroelectrodes (but not nanoelectrodes) under weakly supported conditions [3–8]. The model developed [2,3] uses the Nernst–Planck–Poisson system of equations but avoids the traditional approach based on the assumption of electroneutrality which has been most commonly employed under steady-state conditions [9–17] but has also been applied to dynamic simulations [18–20]. Rather, the assumption of a negligibly small electrical double layer is adopted (see below) which offers computational accuracy and simplicity as well as physical insight. In previous papers we have shown that this approach can be used quantitatively to simulated chronoamperometry for a variety of systems [4–7]. However, most electrochemists use cyclic voltammetry rather than chronoamperometry and so, in this paper, we demonstrate that cyclic voltammetry of the reduction of the cobaltocenium cation in acet onitrile solution:

CoCpþ2 þ e ¢ CoCp2

ð1Þ

can be precisely simulated with excellent agreement between theory and experiment. A microelectrode is used to avoid confusion with possible convective effects noted in some weakly supported systems at macroelectrodes [19,20]; the electrode is nonetheless large enough to exhibit non-steady-state behaviour. Systems involving the formation of a neutral species, as in Eq. (1), were

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explicitly excluded from an early treatment [18]; this partly dictates our choice of chemical system. This work seeks to demonstrate that the rigorous analysis of cyclic voltammetric data under weakly supported conditions is now viable rather generally at both micro and macroelectrodes.

A range of conventional normalisations are introduced [2,8] to simplify these equations and reduce the number of independent variables in the system such that the most general possible conclusions may be drawn:

Ci C A F/ h¼ RT r R¼ re D s ¼ 2A t re ci ¼

2. Theoretical model 2.1. Establishment of the model We have a solution containing a monovalent cationic electroactive species A+, which is capable of undergoing electron transfer to form a neutral species B:

Aþ þ e ¢ B0

ð2Þ

The solution is additionally supported by a concentration Csup of a monovalent inert salt MX, which is presumed to be completely dissociated. An extra equivalent of X is initially present as the counter-ion of the electroactive species. The continuum Nernst–Planck equation is used to describe the flux of any species i at any point in solution. Assuming convection to be negligible, this equation has a contribution from diffusion and from migration:

  zi F C i r/ Ji ¼ Di rC i þ RT

ð3Þ

where for species i, Ji is the flux vector, Di is the diffusion coefficient, Ci is concentration, zi is the species charge, / is potential, F is the Faraday constant, R is the gas constant and T is temperature. Conservation of mass then requires that the space-time evolution of the concentration of i obeys:

F

X

s 0

i

zi C i

ð5Þ

ð11Þ

where C A is the bulk concentration of species A and re is the electrode radius. The resulting set of dimensionless Nernst–Planck–Poisson (NPP) equations are then:

@ci Di @ 2 ci 2 @ci @ci @h @2h 2 @h ¼ þ þ c i 2 þ ci þ zi 2 R @R R @R @ s DA @R @R @R @R 2 X @ h 2 @h ¼ R2e þ zi c i @R2 R @R i

!! ð12Þ ð13Þ

where the dimensionless variable Re is defined:

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F 2 C A Re ¼ r e RT s 0

ð14Þ

and may be considered to represent the scale of the electrode relative to the solution Debye length [2].

2.2. Dimensionless coordinates Having established the Poisson equation as most appropriate, we may hence write, in a hemispherically symmetric space surrounding the Hg electrode and extending into bulk solution, with spatial coordinate r, for i = A+, B0, M+, X:

!! @ 2 C i 2 @C i zi F @C i @/ @2/ 2 @/ þ Ci 2 þ Ci þ þ r @r RT @r @r @r r @r @r2

@ / 2 @/ F ¼ þ @r 2 r @r s 0

X i

The NPP equations are solved subject to ten boundary conditions: one for each species, plus potential, at each of the two boundaries R = 1 and R ? 1. At R = 1, Butler–Volmer kinetics are applied, relating the flux of A normal to the electrode to the surface concentrations, ci,0, of A and B in the form:

  @cA  ¼ K 0 expðaðhapp  hðR ¼ 1ÞÞÞ cA;0 @R R¼1

 expðð1  aÞðhapp  hðR ¼ 1ÞÞÞ cB;0

where s is the dielectric constant of the solvent medium, and 0 is the permittivity of free space. The electroneutrality approximation has offered a popular P alternative, however: it is assumed that i zi ci ¼ 0 for all species i, thus introducing the necessary further condition to solve the Nernst–Planck equations. In our experience, however, the formulation of the electroneutrality approximation with unequal diffusion coefficients does not significantly aid computational runtime or convergence, as it would for Amatore’s reformulation [10], and so it offers no real computational advantage as well as introducing an additional unnecessary approximation.

2

ð10Þ

ð4Þ

The Nernst–Planck equations are not directly solvable for a set of species, since they introduce the electric field as an additional unknown. A further condition is therefore required to solve a Nernst–Planck system. In our theoretical model we invoke the Poisson equation to relate potential and charge in accordance with Maxwell’s equations:

@C i ¼ Di @t

ð9Þ

2.3. Boundary conditions

@C i ¼ r  Ji @t

r2 / ¼ 

ð8Þ

zi C i

ð6Þ ð7Þ



ð15Þ

The dimensionless heterogeneous rate constant is defined: 0

K0 ¼

k re DA

ð16Þ

with k0 being the dimensional heterogeneous rate constant. a is the transfer coefficient, and happ is the dimensionless applied potential:

happ ¼

F ðE  E/f Þ RT

ð17Þ

where E/f is the formal potential of the A+/B couple. For a cyclic voltammetry experiment happ is varied linearly in s through a triangular waveform, at a dimensionless scan rate r:

  2 @happ   ¼ F re @ s  RT D

r ¼ 

v

ð18Þ

where v is the dimensional scan rate, j@E/@tj. From conservation of mass:

  DB @cB  @cA  ¼  DA @R R¼1 @R R¼1

ð19Þ

and assuming M and X to be inert and insoluble in the electrode:

  @cM  @cX  ¼ ¼0 @R R¼1 @R R¼1

ð20Þ

J.G. Limon-Petersen et al. / Journal of Electroanalytical Chemistry 650 (2010) 135–142

The boundary condition for potential at the electrode surface assumes that the double layer is negligible in extent compared to the diffusion layer, such that the charge on the electrode is fully compensated at R  1. Then the enclosed charge is vanishing at this point, and so, from Gauss’s law, the electric field is zero:

 @h  ¼0 @RR¼1

ð21Þ

This zero-field approximation has shown wide applicability [3,5,6] and remains valid where the diffusion layer extends beyond a few nanometres. The R ? 1 boundary is represented by limiting the simulation space to a finite value which greatly exceeds the thickness of the diffusion layer for a diffusion-only system, being Rmax ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6 ðDmax =DA Þsmax where Dmax is the largest diffusion coefficient in the system. The concentrations of A, B, M and X are set to their bulk values at this boundary:

cA ¼ 1 cB ¼ 0

ð22Þ ð23Þ

cM ¼ csup

ð24Þ

cX ¼ 1 þ csup

ð25Þ

Finally, the potential is constrained such that a reference potential of h = 0 occurs at R ? 1, which on the assumption of electroneutrality outside the simulation space yields the expression:

hðR ¼ Rmax Þ þ Rmax

 @h  ¼0 @RR¼Rmax

ð26Þ

Surface flux at the hemisphere, j, is recorded for simulated voltammetric output as:

 @cA  j¼ @R R¼1

ð27Þ

such that the corresponding dimensional current, i, is:

i ¼ 2pFC A DA r e j

ð28Þ

2.4. Numerical methods The NPP equations are solved using a finite difference method. An expanding grid is used corresponding to previous work [2], with a region of extremely dense regular grid spacing close to the electrode that expands proportionally to R further from the electrode. This provides an efficient and accurate simulation. A regular time grid was employed. Optimal parameters for these space and time grids have been previously established and correspond to those used in past works [2]. The equations are discretised by a fully implicit method and solved by the iterative Newton-Raphson method, exactly as in previous work [2]. All simulations were programmed in C++ and run on a desktop computer (Intel Core2 Quad 2.4 GHz, 2 GB RAM), with running times of 10 min per voltammogram being typical. 3. Experimental 3.1. Chemicals and equipment All solutions were made with acetonitrile solvent (MeCN, HPLC grade, Fisher Scientific). Cobaltocenium hexafluorophosphate (>98%, Strem Chemicals) was used as the electroactive species, with tetra-n-butylammonium perchlorate (TBAP, P99%, Fluka) as supporting electrolyte. A three electrode cell was used: a 25 lm radius platinum disk or a 25 lm radius mercury hemispherical electrode was used as a working electrode. The use of a hemispherical electrode is consistent with the choice of simulation in a perfectly hemispherical geometry. The mercury microelectrode was pre-

137

pared by electrodepositing mercury on a platinum disk, and the size was controlled by passing the necessary amount of current to form a hemisphere, using the procedure stated in Bard et al. [21] and Limon-Petersen et al. [22]. A commercial ‘No leak’ reference electrode comprising Ag/Ag+ in a PEEK barrel fitted with a membrane junction (66-EE009, Cypress electrodes) was used in order to minimize contamination of the solution. Note that voltammograms were recorded using a PG-STAT12 potentiostat (Autolab, Utrecht, Netherlands) without the ‘ohmic drop correction’ used or theorised in some early studies of weak support [18,20], so as to permit a clear comparison of theory and experiment. All solutions were thoroughly degassed with N2 before each experiment, and an inert atmosphere was maintained during the experiments. 3.2. Experimental procedure Solutions containing 3 mM of cobaltocenium hexafluorophosphate and different concentrations of TBAP (150, 30 and 0 mM) were studied. The cobatocenium-cobaltocene couple was selected for its fast electron kinetics [23]. The solution containing 150 mM of TBAP was used to estimate the diffusion coefficient of the cobaltocenium cation and the corresponding neutral cobaltocene, by performing double potential step chronoamperometry at a 50 lm diameter platinum microdisk electrode. The experimental results were analyzed using the theoretical methodology described by Klymenko et al. [24] giving DCoCpþ ¼ 1:8  105 cm2 s1 and 2 DCoCp2 ¼ 2:3  105 cm2 s1 . The former is in good agreement with previous experimental data [23,25]; we are unaware of any literature values for the latter. Using the microhemispherical mercury electrode, cyclic voltammograms were recorded for the different supporting electrolyte concentrations, at scan rates of 0.05, 0.1, 0.2, 0.5 and 1 V s1, in a potential window between 0.6 and 1.2 V vs. Ag/Ag+. A cleaning regime was applied before each recorded cyclic voltammogram. The cleaning procedure consisted of running three cyclic voltammograms in predegassed pure acetonitrile from 0.6 to 3.5 V vs. Ag wire in order to remove any contamination. 4. Results and discussion 4.1. Theoretical results Cyclic voltammograms were simulated using the NPP simulation method described above and the parameters r ¼ 10; Re ¼ 3  103 ; DA ¼ DB ¼ DMþ ¼ DX ; K 0 ¼ 104 and a = 0.5, for different concentrations of supporting electrolyte: csup = 100, 10, 1, 0.1, 0.01 and 0. The results are presented in Fig. 1A where the voltammetry can be seen to change dramatically from fully-supported (where csup = 100) to self-supported, with the magnitudes of the peak currents, the peak-to-peak separation, and the mass transport-limited currents all increasing with decreased support. As with any electrolytic process, the transfer of electrons to or from the electrode perturbs the electric field immediately adjacent to the electrode. The migrational transport of ions will act to reestablish local electroneutrality, with concomitant changes in the concentration profiles of the various species in this region. For the reduction of a cation as in the voltammetry of cobaltocenium, negative charge is added to solution, so cations are attracted and anions are repelled from the vicinity of the electrode. In the case of high levels of supporting electrolyte, there are high concentrations of ions available to respond to the perturbed electric field and move accordingly (via migration); the electroactive species, which is detected at the electrode, has such a low

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csup

100

10

1

0.1

0.01

0

0

j

θPET−θref

0

-2

-2

-4

A

-4 -20

-10

0

10

B 20

-20

-10

0

10

θM-θref

2.0

20

1.5

10

θM-θPET

Ω

θM -θref

1.0

C 0.5

0

D -10

0.0

-20 -20

-10

0

10

20

0

2

θM-θref

4

6

τ

Fig. 1. Simulations at different concentrations of supporting electrolyte using the parameters r ¼ 10; Re ¼ 103 ; DA ¼ DB ¼ DMþ ¼ DX ; K 0 ¼ 104 and a = 0.5. (A) cyclic voltammetry, (B) potential drop in solution vs. applied potential, (C) inferred resistance of solution vs. applied potential, and (D) potential at the plane of electron transfer vs. dimensionless time.

concentration relative to the inert electrolyte that it carries an insignificant proportion of the migrational current and so is observed to show purely diffusional behaviour. As the levels of supporting electrolyte fall, so the proportion of migrational current carried by the electroactive species increases until its migration becomes observable in the voltammetry. The currents (peak and limiting) are noted to increase since the cobaltocenium cation experiences a Coulombic attraction in response to its own reduction, and so migration augments mass transport in this case. Peak separation is also seen to increase, due to the lowering of the potential driving force experienced by the electroactive species at the plane of electron transfer (PET, see Fig. 1B) caused by the perturbed electric field (commonly termed ‘ohmic drop’), and the consequently increased overpotential required to achieve a transportlimited current. Note that in the model used the PET corresponds to the layer of solution immediately adjacent to the electrode. Fig. 1C illustrates the inferred resistance of the solutions with different concentrations of supporting electrolyte, calculated by ref Ohm’s Law (X ¼ hPET h : the potential at the plane of electron transj fer, hPET, minus the potential at the reference electrode, href, is divided by the flux, j). For the highest concentrations, a linear relationship is found as traditionally expected, but it is clear that at lower support levels, the complex dependences of both fluxes and potential drop in solution cause Ohm’s Law to no longer be obeyed in terms of a constant resistance. Hence, in general, the ‘electrical resistance’ of a solution is constant in neither time nor space for all weakly-supported solutions. For this reason, we prefer the terminology ‘potential drop’ rather than ohmic drop. Fig. 1D shows how the lowering of the effective potential difference at the PET for weakly-supported solutions has an effect on the

effective voltage scan rate there, such that in the case of zero added supporting electrolyte, there is a clear slowing of the effective scan rate close to the peak potentials at the PET, shown in greater detail in Fig. 2 where the dimensionless effective scan rate at the plane of electron transfer (rPET) is plotted vs. dimensionless time, where rPET is calculated as;

  @ðhapp  hPET Þ   @s

rPET ¼ 

ð29Þ

The perceived overpotential acting on the reactant at the surface no longer follows a triangular waveform. The effects of supporting electrolyte concentration on the peak current and potential were also investigated in detail, and the trends identified in Fig. 3A found to be replicated across the full range of voltage scan rates and supporting electrolyte concentrations. Note that the increase in the reverse (i.e. oxidative) peak (Fig. 4B) is not as large as for the forward (reductive) peak (Fig. 4A). This is due to the species being oxidised being neutral cobaltocene, which is therefore unaffected by electric field gradients and not subject to migrational transport. The data presented in Figs. 3 and 4 are available as Supplementary Data. For high support ratios the familiar steady-state (low r) and Randles-Ševcˇík (high r) limits are recovered. At low support, and low scan rate, an increase over the fully-supported limiting current is noted, with a factor of 1.98 which is in good agreement with the value 1.94 derived by Oldham [26]. The corresponding steady-state potential drop is 3.63, with 3.45 predicted [26]. The small discrepancy may be attributable to the approximation of electroneutrality by Oldham, which is not used in our theory.

139

11.5 11.0 10.5 10.0 9.5 9.0 8.5 8.0 7.5 7.0 6.5 6.0 5.5 5.0 4.5

hexafluorophosphate and 150 mM of TBAP at scan rates of 1000, 500, 200, 100 and 50 mV s1. The data are presented in Fig. 5 (solid lines) for 1000–50 mV s1 and the currents can be seen to increase with scan rate. For a planar electrode a square root dependence of peak current with scan rates is expected [1]. However, for hemispherical electrodes of the size studied the response is intermediate between convergent microelectrode diffusion and the planar electrode response. This feature also explains the decrease of the peak-to-peak separation with scan rate [27]. The cyclic voltammograms of the solution containing 150 mM TBAP were simulated with the program DigiSimTM, assuming diffusion-only transport. DigiSimTM has an option for hemispherical diffusion, which was selected. The parameters used for the simulation are stated in Table 1 and the simulations are shown in Fig. 5 (open circles). The simulations are found to be in good agreement with the experimental data. In particular the good fit between the experiment and the simulations allows us to infer that the amount of supporting electrolyte was high enough to ensure diffusion-only flux, and that negligible migration of electroactive species occurs. The recorded experimental data were also compared with the NPP simulation model, for the solution containing the highest concentration of supporting electrolyte. Good mutual agreement was obtained between DigiSim, the NPP simulations, and the experimental results. In Fig. 5, the two theoretical curves (DigiSim and NPP simulations) overlap exactly at the highest support ratio;

0 0.01 0.1 1 10 100

0

2

4

6

8

τ Fig. 2. Effective scan rate at the plane of electron transfer vs. dimensionless time at different concentrations of supporting electrolyte, using the parameters from Fig. 1.

4.2. Experimental results Cyclic voltammograms were first run using the hemispherical mercury electrode in a solution containing 3 mM cobaltocenium

A

B

0

0

-3

-2

-6

-4

j PF

-9

-6

θ PF

-12

-8

-15 -18 -2

2 1 0 -1 -2

-10 -1

0

-1

log ( σ 1 )

-2

2 3 -3

l

og

2

1

0

-12

)

(c

su

-14

p

-2

-1

0

log ( σ

1

2

log ( csup)

σPET

J.G. Limon-Petersen et al. / Journal of Electroanalytical Chemistry 650 (2010) 135–142

3 -3

)

Fig. 3. Simulations using the parameters Re ¼ 103 ; DA ¼ DB ¼ DMþ ¼ DX ; K 0 ¼ 104 and a = 0.5 to obtain forward (reductive) (A) peak height (jPF) and (B) peak position (hPF) for the reduction of a singly positively charged species as functions of scan rate and concentration of supporting electrolyte.

B

12

9

11

8

10

7

9 8 7 6 5 4 3 2 1

θ PB

5 4 3

-2

-1

log ( c

sup

0

)

-1 1

2 -2

) sup

)

2 3 2 1 0

log ( c

-3

6

log ( σ

j PB

A

1 -3 -2 -1 0 1 2 -2

-1

0

1

2

3

) log ( σ

Fig. 4. Simulations using the parameters stated in Fig. 3 to obtain backward (oxidative) (A) peak height (jPB) and (B) peak position (hPB) as functions of scan rate and concentration of supporting electrolyte.

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J.G. Limon-Petersen et al. / Journal of Electroanalytical Chemistry 650 (2010) 135–142 -1

50 mV s

100 mV s

-1

0.0

0.0

-8

I/A

I/A

-5.0x10

-7

-1.0x10

-1.5x10

-1.0x10

-7

-2.0x10

-7

-7

-1.2

-1.0

-0.8

-0.6

-1.2

-1.0

-1

-1

500 mV s

0.0

0.0

I/A

I/A

-0.6

V vs Ag/Ag

200 mV s

-7

-1.0x10

-2.0x10

-0.8 +

+

V vs Ag/Ag

-1.0x10

-7

-2.0x10

-7

-7

-1.2

-1.0

-0.8

-1.2

1Vs

-1.0

-0.8

-0.6

+

+

V vs Ag/Ag

V vs Ag/Ag -1

-7

1.0x10

I/A

0.0

Experimental

-7

-1.0x10

Theoretical

csup=50 csup=10

-7

-2.0x10

Traces

-7

-3.0x10

-1.2

-1.0

-0.8

-0.6

+

V vs Ag/Ag

Fig. 5. Comparison between theory and experiment at different concentrations of supporting electrolyte, for scan rates of 50, 100, 200, 500 and 1000 mV s1 and different concentrations of supporting electrolyte csup = 50, 10 and traces of ionic impurities (no intentionally added supporting electrolyte).

therefore, for the sake of clarity, just the NPP simulations are presented. Next, cyclic voltammograms were recorded in a solution containing 3 mM cobaltocenium hexafluorophosphate and 30 mM TBAP. The experimental data are presented in Fig. 5 by dashed lines. It is observed that a broadening in the peak-to-peak separation occurs, compared to that in the case of higher support concentrations, along with an increase in current. Comparison between simulation and experiment was made using the parameters in Table 1, except for the diffusion coefficient of the cobaltocenium cation, which was estimated by means of best fitting to be DCoCpþ ¼ 1:92  105 cm2 s1 , which is close to literature val2 ues [28]. The small difference in diffusion coefficients of cobalt-

ocenium at different concentrations of supporting electrolyte has been previously studied in detail by Amatore et al. [10]. Fig. 5 shows theory (squares) and experiment (dashed lines) to be in good agreement. Finally, cyclic voltammograms were recorded in a solution containing only 3 mM cobaltocenium hexafluorophospate and no deliberately added supporting electrolyte, although the complete absence of the latter is experimentally impossible to achieve exactly because of trace impurities in the solution. The experimental results obtained are presented in Fig. 5 (dotted line), and show a larger increase in the peak-to-peak separation compared to the full or partial support cases. The faradaic current is much larger than those observed in the previous cases.

J.G. Limon-Petersen et al. / Journal of Electroanalytical Chemistry 650 (2010) 135–142

in the present case such a change was not inferred, and the kinetic parameters remained fast enough not to influence the shape of the cyclic voltammogram. Interestingly we note that an early treatment of reversible cyclic voltammetry, but under the assumption of electroneutrality, did ‘‘not consider” the simulation of processes involving the formation of uncharged species (such as CoCp2 in the present case) since ‘‘the concentration of ions” in the depletion layer approaches zero at the electrode surface, ‘‘producing very high local resistance” [18].

Table 1 Values used for the simulations. Parameter

Value

CA k0a DCoCpþ

3 mM 10 cm s1 1.8 ± 0.1  105 cm2 s1*

DCoCp2 DClO4 b DTBAPþ b re

2.3 ± 0.1  105 cm2 s1 3  105 cm2 s1 2.1  105 cm2 s1 25  104 cm

2

a The heterogeneous rate constant was set high enough to maintain electrochemically ‘reversible’ conditions. b Calculated by the Wilke–Chang Method [30]. * A value of DCoCpþ ¼ 1:92  105 cm2 s1 was used in 2 the simulations with lower supporting electrolyte (see Section 4.2).

5. Conclusions

To compare to the experiments, the simulation used the same parameters for diffusion coefficients as presented in the previous case with 30 mM supporting electrolyte. The simulated voltammogram had a slightly larger peak current than the experimental result (Fig. 6). Therefore, trace inert ionic impurities were considered in the simulation. The impurities were modeled in terms of an elevated csup, assuming that the impurities in solution have the same charge and diffusion coefficients as the support species. A calibration plot was made varying the concentrations of trace electrolyte (Fig. 6); it was found that traces of 60 lM of inert electrolyte gave the best agreement with theory, and this is consistent with the 98% purity stated by the supplier for cobaltocenium hexafluorophosphate, if the remaining 2% is ionic. The E/f was modified to reach the best agreement between theory and experiment. The shift can be explained by the activity coefficients of cobaltocenium and cobaltocene depending on the ionic strength of the solution. Since the ionic strength dependence of these activity coefficients is not independently known, the reduction potential was treated as an empirical variable for each of the three ionic strengths studied. It is interesting to note that whilst previous experimental studies [29] suggest that a variation in the heterogeneous rate constant (k0) with the concentration of supporting electrolyte is observable,

-7

-2.0x10

-7

I/A

-2.5x10

150μM -7

-3.0x10

120μM 90μM 60μM 30μM 0

-7

-3.5x10

-1.2

-1.1

141

-1.0 +

E / V vs Ag/Ag

Fig. 6. Simulation of the self-supported reduction of the 3 mM cobaltocenium hexafluorophosphate at v = 1 V s1 with different trace levels of monovalent electrolyte MX in the range of 0–150 lM; the experimental voltammetry is shown as circles.

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