Determination of diffusion coefficients from semiintegrated d.c. and a.c. voltammetric data: Overcoming the edge effect at macrodisc electrodes

Accepted Manuscript Determination of diffusion coefficients from semiintegrated d.c. and a.c. voltammetric data: overcoming the edge effect at macrodisc electrodes Alexandr N. Simonov, Elena Mashkina, Peter J. Mahon, Keith B. Oldham, Alan M. Bond PII: DOI: Reference:

S1572-6657(15)00083-1 http://dx.doi.org/10.1016/j.jelechem.2015.02.020 JEAC 2015

To appear in:

Journal of Electroanalytical Chemistry

Received Date: Revised Date: Accepted Date:

8 December 2014 17 February 2015 17 February 2015

Please cite this article as: A.N. Simonov, E. Mashkina, P.J. Mahon, K.B. Oldham, A.M. Bond, Determination of diffusion coefficients from semiintegrated d.c. and a.c. voltammetric data: overcoming the edge effect at macrodisc electrodes, Journal of Electroanalytical Chemistry (2015), doi: http://dx.doi.org/10.1016/j.jelechem.2015.02.020

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Determination of diffusion coefficients from semiintegrated d.c. and a.c. voltammetric data: overcoming the edge effect at macrodisc electrodes Alexandr N. Simonov,† Elena Mashkina,† Peter J. Mahon,§ Keith B. Oldham,‡,* Alan M. Bond†,* † ‡

§

School of Chemistry, Monash University, Clayton, Victoria 3800, Australia Department of Chemistry, Trent University, Peterborough ON, Canada K9J 7B8 Faculty of Science, Engineering and Technology, Swinburne University of Technology,

Hawthorn, Victoria 3122, Australia

ABSTRACT. Unless the area of an inlaid disc electrode is sufficiently large, and/or the scan-rate fast enough, the ‘plateau’ of a semiintegrated d.c. voltammogram or aperiodic component of an a.c. voltammogram has a slope. This phenomenon, which has its origin in non-planar diffusion at the edge of the disc, interferes with an otherwise efficient method of determining diffusion coefficients. Methods of circumventing this difficulty are presented and tested with simulated and experimental data. *

Corresponding authors. E-mails: [email protected] (K.B.O.);

[email protected] (A.M.B.)

1

1. Introduction It is well known [1, 2] that semiintegration of an ideal d.c. voltammogram from a planar electrode generates a sigmoidally shaped response when the semiintegral M of the current I is plotted versus time or potential. This ideal scenario is illustrated by the black curves in Figure 1. A similar plateau is developed in a.c. voltammetry when the aperiodic component of the current [3] is semiintegrated. The plateau of the sigmoid has a height Mlim that is uninfluenced by ohmic polarisation, by the kinetics of the electron-transfer process, or by the coupling of the electron transfer to certain homogeneous chemical reactions. Mlim depends only on the number n of transferred electrons, Faraday’s constant F, the area A of the planar electrode, and two properties of the reactant: its bulk concentration cR, and diffusion coefficient DR. Therefore, measurement of the plateau height can be used to find any one of these five parameters, if the others are known. It is the diffusion coefficient that is most often in greatest doubt and that is customarily measured by this route. Here we address those electrode reactions that can be described by the stoichiometry

  P(soln)  ne , R(soln)  

(1:1)

where R is the reactant and the product, P, is initially absent. Throughout this document, the upper of the alternative ± signs relates to an oxidation of R and the lower to its reduction. The equation that describes the limiting semiintegral is

M lim  nFAcR DR

(1:2)

In the derivation of this equation it is assumed that R is transported solely by semiinfinite planar diffusion. However, the most common voltammetric electrodes are inlaid macrodiscs, having radii on the order of one millimetre, and inlaid electrodes of such sizes suffer from edge diffusion [4], in addition to planar diffusion. If A

DR tmax , where tmax is the duration of the

2

voltammetric scan (mainly determined by the scan-rate, v), then edge diffusion may legitimately be ignored and the limiting semiintegral displays a horizontal plateau, as in the black curve of Figure 1b.

Figure 1. a. D.c. cyclic voltammograms, simulated using the following parameters: n = 1, cR = 1 mM, DR = 10-9 m2 s-1, r = 25 mm (black), 1.5 (blue) or 0.5 mm (red), v = 0.1 V s-1, k0 =  (black, blue, and solid red) or 10-4 m s-1 (dashed red), T = 298 K. Despite the current density being non-uniform, the current is plotted normalised by the electrode area. b. The corresponding semiintegrals, normalised by division by Mlim, for the forward branches. In many experiments, however, the disc radius, r, is about 1.0 mm, the experiment often lasts approximately 10 s, and the diffusivity is close to 10-9 m2 s-1. In such circumstances, A is

3

certainly greater than DR tmax , but not overwhelmingly so. Accordingly edge current, which is not taken into account in equation 1:2, is significant, and its contribution causes the semiintegral to climb continuously, as in the red and blue curves of Figure 1b, instead of levelling off into a true plateau. This destroys our ability to make a straightforward measurement of DR from the plateau height. The voltammograms in Figure 1a are not experimental; they are simulated. The advantage of simulation here is that it enables the edge effects to be examined without danger of obfuscation by other voltammetric interferences such as uncompensated resistance, impurities, and capacitance. Simulations of voltammograms derived from a commercially available package [5] were carried out using a two-dimensional diffusion model that allows diffusion parallel to the electrode plane. Simulated voltammograms for reactions that are not reversible assume ButlerVolmer kinetics with α = 0.50 and an assigned standard rate constant k0. The same model was used for reversible processes, with k0 set to 100 m s-1, inasmuch as higher values lead to indistinguishable results. 2000 or 215 data points per volt were used for d.c. or a.c. voltammograms,

respectively.

The

aperiodic

components

were

extracted

from

a.c.

voltammograms via application of a Fourier transform – band filtering (rectangular window, bandwidth 5 Hz) –inverse Fourier transform sequence of operations [6]. Semiintegration was carried out using an established algorithm [7]. In Figure 1b, the corresponding semiintegral responses are reported, normalised by division by the ideal Mlim. In Figure 1b notice the pronounced change in the shape of the plateau brought about by a decrease in electrode area. Note also that the sloping plateau is not significantly affected by quasireversibility. This study aims to allow the calculation of DR from such plateaus, notwithstanding their slopes.

4

2. Background The obvious remedy for a sloping plateau is to restore preponderance to the value of A compared with DR tmax , either by increasing the size of the electrode or by increasing the scan rate. These measures are certainly effective, but there are circumstances in which such increases serve to undermine the purpose for which the voltammetry is being conducted. For example, a rather slow scan rate is essential in a.c. voltammetry to maintain adequate timescale separation between a.c. and d.c. phenomena. Likewise, large electrodes lead to excessively large currents, which may overburden instrumentation. Thus it would be valuable to tolerate the slope and find methods of extracting the diffusivity DR from a sloping experimental semiintegral ‘plateau’. In this study, two methodologies will be advanced by which the reactant’s diffusivity may be estimated from the sloping plateau. The first approach, dubbed the Convolution Method, is powerful but requires extensive mathematical processing. The second is a collection of empirical strategies, grouped under the title Extrapolation Methods. In pursuing either route, it is assumed that all parameters on the right-hand side of equation 1:2, other than DR itself, are known exactly. Though simulations have been performed (for example, [4]), no mathematical model yet exists that can predict edge currents at an inlaid disc electrode under conditions of potential-ramp voltammetry. There is, however, a well-established theory that incorporates edge current in a different electrochemical experiment, namely in potential-leap chronoamperometry. The methods elaborated in Sections 3 and 4 rely on results derived by modelling this alternative experiment.

5

In potential-leap chronoamperometry, a large potential is suddenly imposed on an inlaid disc electrode at which reaction 1:1 can occur, and the ensuing current is monitored. At times smaller than about A/2DR the current is accurately described by the equation [8, 9]  1 DR DR t smaller  I (t )   nFAcR DR      terms  A 2A  t

(2:1)

Each of the bracketed terms has a distinct geometrical significance [10]. The first term leads directly to Cottrell’s equation, showing it to arise from the planar surface. The second bracketed term contributes a time-independent current nFcR DR A that is proportional to the length of the disc’s perimeter. The current arising from the third bracketed term has its origin in the curvature of the disc’s edge. The negative terms are a consequence of competition for diffusant between different segments of the edge. Evidently, with the exception of the cottrellian contribution, all the terms in equation 2:1 arise – in one way or another – from the edge of the disc. That is, the edge current at short enough times in potential-leap chronoamperometry is  DR DR t smaller  I edge (t )   nFAcR DR     terms  2A  A

(2:2)

On semiintegration:  D t D t smaller  M edge (t )   nFAcR DR  2 R  R   terms  A 4A 

(2:3)

This is the edge contribution that adds to Mlim for the potential-leap experiment at times small in comparison to A/2DR. It is the corresponding formula for potential-scan voltammetry (or the forward scan of cyclic voltammetry) to which we would like to have access, but which is presently unknown.

6

The same study [9, 11] that led to equation 2:1 also reported the expression  D 8 16(2  9) A smaller  I (t )   nFAcR DR  4 R    , terms  5t 9 DR 11t 3  A

(2:4)

which describes the current in potential-leap chronoamperometry at times greater than about A/2DR. Taken together, the two equations describe the entire response.

3. Convolution Method Semiintegration is useful in electrochemistry because the semiintegral of the current is often linearly related to the concentrations of the electroactive species at the electrode surface. Primarily, semiintegration is applicable in voltammetry under conditions in which the electrochemistry is simple and semiinfinite planar diffusion prevails, as when large planar electrodes are used, together with excess supporting electrolyte. Despite these restrictions, a tribute to the utility of semiintegration is its incorporation into the menus of commercial instrumentation. Convolution voltammetry aims to extend the realm of applicability. Semiintegration may be regarded as a special case of the mathematical operation known as convolution. Thus viewed, the semiintegral of a current is the result of convolving I(t) with the function 1

t :

d 1/2 1 1 M (t )  1/2 I (t )   I (t )   I (t  ) d  dt t  0 t

(3:1)

The converse operation, semidifferentiation, may likewise be considered as a convolution: t

I (t ) 

d1/2 1 dM 1 dM M (t )   (t )   (t  ) d  1/2 dt t d t  d t 0

(3:2)

though it is often referred to as a ‘deconvolution’.

7

Convolution of I(t) with a function g(t) other than 1

t has been shown [9] to be capable of

providing information about surface concentrations under conditions in which the electrochemistry is not simple or when transport is not wholly by semiinfinite planar diffusion. The result of such a convolution is analogous to a semiintegral and has been termed an ‘extended semiintegral’ [11], here denoted by M x(t): t

M x (t )  g (t )  I (t )   g () I (t  ) d 

(3:3)

0

Again, there is a converse (deconvolution) operation t

I (t )  h(t ) 

dM x dM x (t )   h() (t  ) d  , dt dt 0

(3:4)

but the functions g(t) and h(t) are not now necessarily identical. Of present concern is an extended semiintegral designed to address transient voltammetric conditions at a disc electrode subject to both planar and edge diffusive transport. For the rather elaborate derivations of the appropriate g(t) and h(t) functions under reversible conditions, and for the algorithms used to implement the operations in equations 3:3 and 3:4, please see the literature [12-15]. Suffice it to note that these functions are derived from the series in 2:1 and 2:4. The functions g(t) and h(t) contain the diffusion coefficient which, of course, is unknown in the present problem. To overcome this lack, a crude value of DR may be guessed initially, and repeatedly refined by convolution until an unchanging value eventuates. It is this procedure of iterated convolution that was employed in the present study.

8

4. Extrapolation Methods Equation 2:3 describes the edge current in potential-leap chronoamperometry at a macrodisc electrode. Can this equation be adapted to apply to the ramp used in d.c. or a.c. voltammetry? The significance of t is clear for the leap: it is the elapsed time since the potential was stepped, but there is no clear-cut t = 0 for the ramp because Estart can take a range of values without affecting the voltammogram. However, it can be argued that, as far as the present problem is concerned, the best replacement for tleap = 0 is tramp = t1/2, the time at which the ramp potential reaches the half-wave potential. The rationale for this choice is that half of the semiintegral is accrued before that time, the other half arising thereafter. Therefore the ramp should produce a plateau that is not too different from that produced by a leap at the instant t1/2. The most important term on the right-hand side of equation 2:3 is 2nFcR DR At and so it may be posited that the approximation

M edge (t )  2nFcR DR A(t  t1/2 )  2nFcR DR

A ( E  E1/2 ) v

holds, E1/2 being the potential at which Mplanar(t) =

1 2

(4:1)

M lim . It should be stressed that this is a

crude approximation to the semiintegral of the voltammetric edge current, but one that might suffice in cases for which the edge effect leads to a mild slope in the plateau. In the plateau region of the semiintegrated voltammogram, it is the sum of equation 4:1, arising from edge diffusion, and the limiting semiintegral from 1:2, coming from planar diffusion, that will be manifest:    DR M (t )  M lim  M edge   nFAcR DR 1  2 ( E  E1/2 )  Av  

(4:2)

It is from this equation that all the methods of the present section derive.

9

Equation 4:2 predicts a linear relationship, in the plateau region, between the semiintegral and

( E  E1/2 ) . The intercept of that linear relationship at E = E1/2 can provide a value of the diffusion coefficient 2

 Intercept   M vs.   E  E    1/2   , DR     nFc A R    

(4:3)

as can the slope of the linear relationship:

DR 

  v Slope   M vs.   E  E1/2   . A 2nFcR

(4:4)

Evidently, performing a linear regression of experimental M values versus the corresponding

( E  E1/2 ) values will allow two determinations of DR to be made. In view of the crudity of our approximations we cannot expect (nor do we find) that the results of 4:3 and 4:4 coincide. A compromise, which has promise of providing a better value than that from either method, is to perform a constrained linear regression. By temporarily defining y  2M / (nFcR A3 ) ,

x  ( E  E1/2 ) / v and   2 DR / A , equation 4:2 contracts to

y    2 x

(4:5)

In a constrained linear regression of this equation, the best straight line through (x,y) points is selected, subject to the constraint that the slope must equal the square of the intercept: constrained y     2 x   linear regression

DR 

A 2 4

(4:6)

Of course, carrying out any of these regressions requires a knowledge of the half-wave potential. In the simulation studies presented in Section 5, this is not a problem because E1/2 is a known parameter, but in real experiments, such as those reported in Section 7, knowledge of the

10

exact location of the half-wave point may be tenuous. Estimates of E1/2 may perhaps be gleaned from other experiments, or from another aspect of the same experiment. In a.c. voltammetry, for example, E1/2 is accessible from nodes or extrema in the harmonic spectra. The point of inflection of the semiintegrated voltammogram also provides a good approximation to E1/2. If the worst comes to the worst, one may guess E1/2, determine DR on that basis, locate the ‘true’ plateau and reassess the half-wave’s location. Remember that it was only an approximation, in the first place, to assert that the edge current’s semiintegral in the later stages of a ramp experiment duplicates that from a leap at E1/2. There is, however, a regression capable of delivering DR that does not require knowledge of E1/2. Differentiation of equation 4:2 gives

dM A   nFcR DR dE v( E  E1/2 )

(4:7)

and this may be combined with equation 4:2 itself into 2(nFcR DR )2 A d E M   nFcR A DR  . v dM

(4:8)

Equation 4:8 predicts that a plot, or a linear regression, of M versus dE/dM (that is, versus the reciprocal of the local slope of the semiintegrated voltammogram curve) should give a straight line and that DR can be found from either the intercept or slope via the following formulae  dE    Intercept  M vs. d M     DR   nFcR A      

DR 

1 nFcR

2

v dE   Slope  M vs. 2A d M  

(4:9)

(4:10)

neither of which involves the half-wave potential.

11

It is to be expected that, whichever extrapolation method is employed, better results will be found the longer the plateau extends. Long ramps, however, increase the danger of some unwanted electrochemistry intervening at more extreme potentials. Appreciate, however, that both planar and edge diffusion continue unabated, once the plateau has been reached, even if the potential ceases to advance. Hence the ‘capped ramp’ waveform [16] may beneficially replace the pure ramp, allowing the sloping plateau to continue almost indefinitely. The various extrapolation methods remain valid, provided that ( E  E1/2 ) / v is replaced by t – t1/2. In semiintegrated cyclic voltammetry, the plateau region even extends into the early stages of the backward branch, though capacitive current may interfere unless careful corrections are made.

5. Simulation Results Inlaid discs of four radii were modelled, their areas being 1960, 19.6, 7.07 and 0.785 mm2; these values are to be compared with the DRtmax value of 0.01 mm2. The largest disc, for which the inequality A

DR tmax is abundantly satisfied, was included as a control; it delivered a

perfectly flat semiintegral plateau, as expected (Figure 1). The other three plateaus were sloping and they were analysed by the methods previously described. Other values used in the simulations were n = 1, T =298 K, cR = 1 mM, DR = DP = 10-9 m2 s-1, Estart = -0.25 V, E0 = 0 V, and v = 0.1 V s-1. A value of k0 that was effectively infinite was used in simulating the reversible semiintegrated voltammograms, whereas the quasireversible examples employed k0 = 10-4 m s-1. In performing the regression analyses, it was found that the plots of M versus

  E  E1/ 2 

were linear with coefficients of determination of at least R2 = 0.997, for potentials 0.170 V or more beyond E0 in the case of reversible voltammograms, or more than 0.235 V beyond E0 when quasireversible. The corresponding ranges for plots of M versus dE/dM started at potentials

12

0.300 or 0.400 V beyond E0. Examples of the application of extrapolation methods to these data are shown in Figure 2. Slopes and intercepts were found by linearly regressing such data.

Figure 2. Simulated semiintegral data plotted as the normalised semiintegral versus (a)

  E  E1/2  or (b) d E / d M together with the regression line, for the following:

r = 25 mm, reversible reaction (black squares and black line); r = 1.5 mm, reversible reaction (blue squares and blue line); r = 0.5 mm radius disc, reversible reaction (filled red squares and full red line); r = 0.5 mm, quasireversible reaction (open red squares and dashed red line). Other simulation parameters are as in Figure 1.

13

Data derived from these regressions, as well as from the convolution method, are assembled in Table 1. The tabulated results show remarkable success for the convolution method; less for the extrapolation methods. Methods that rely on the slope are seen to be inferior to those based on the intercept, which generally have errors of less than 1%. Use of constrained linear regression (Eq 4:6) is especially efficient. As expected, the results become progressively worse as the disc’s size diminishes due to the contributions from the smaller terms in equation 2:1. There is no noteworthy difference in success between the reversible and quasireversible results. Table 1. Results of applying several methods to recover the input value of DR (1.000∙10-9 m2 s-1) from simulated semiintegrated d.c. voltammograms. See the text for the simulation parameters. r / mm 25 2.5 1.5 0.5 a

k0 / m s-1 ∞

109 DR / m2 s-1 as determined by Convolution

Eq 4:3

Eq 4:4 a

Eq 4:6

Eq 4:9

Eq 4:10

n.a.

1.008

n.a.

0.997

1.008

n.a.

10

0.997

1.008

n.a.

n.a.

1.008

n.a.

∞

0.996

1.006

0.925

1.002

0.992

1.027

10

0.996

1.003

0.964

1.001

0.995

1.009

∞

0.996

1.004

0.979

1.002

0.993

1.031

10

0.997

1.000

1.011

1.001

0.997

1.017

∞

0.997

0.995

1.046

1.007

0.977

1.075

0.996

0.985

1.071

1.006

0.983

1.069

-4

-4

-4

-4

10

not applicable.

The results in Table 1 and Figures 1 and 2 relate to d.c. voltammetry, but we also modelled the semiintegrals of the aperiodic current in a.c. voltammetry at an inlaid disc electrode. Figure 3a shows examples of the aperiodic currents in a.c. cyclic voltammograms. The a.c. frequency and amplitude used in the simulations were 39.01 Hz and 0.100 V, with a scan rate of 0.07451 V s-1. Reversible a.c. voltammograms and two instances of quasireversibility (k0 = 10-3 m s-1 and 10-4 m s-1) were studied, other parameters being as in the d.c. simulations. Figure 3b shows the

14

result of semiintegrating the forward branches of the cyclic voltammograms portrayed in Figure 3a, while Figure 3c shows one of the extrapolation methods being exploited. Results obtained from applying convolution and extrapolation methods are recorded in Table 2. Table 2. Results of applying several methods to recover the input value of DR (1.000∙10-9 m2 s-1) from simulated semiintegral of the aperiodic component of a.c. voltammograms. See the text for the simulation parameters. 109 DR / m2 s-1 as determined by

r / mm

k0 / m s-1

Convolution

Eq 4:3

Eq 4:4

Eq 4:6

Eq 4:9

Eq 4:10

∞a

∞

n.a. b

1.002

n.a.

n.a.

1.001

n.a.

∞

0.996

0.974

1.083

1.004

0.974

1.254

10-3

0.996

0.974

1.084

1.004

0.976

1.251

10-4

0.996

0.972

1.088

1.004

0.970

1.261

0.5 a b

simulated using a planar diffusion model. not applicable.

The results in Table 2 show that, once again, the convolution method yields impeccable results. With the exception of those employing constrained linear regression, extrapolation results that rely on the slope give very poor results, while those based on the intercept are typically 2.5% low. The cause of the marginally poorer performance of the extrapolation methods in analysis of the aperiodic components of a.c. voltammograms compared with d.c. voltammetry is not understood completely, but may be associated with faradaic rectification current [3]. In principle, the assumptions invoked in Section 4 should apply to any mechanism that meets the requirements for applicability of equation 1:2. Indeed, success of application of the extrapolation methods to several multistage electrode mechanisms is demonstrated in Supporting Information (Figs. S1-3).

15

Figure 3. Simulated aperiodic components of cyclic a.c. voltammetry. a. Currents, normalised by division by the area, for the reversible and quasireversible cases. b. The results of semiintegrating the forward branches and dividing by Mlim. c. One of the extrapolation methods is applied and the regression lines are shown (the red line is obscured by being almost coincident with the blue line). Black lines and squares relate to the reversible case and r = ∞; blue lines and squares are for the reversible case and r = 0.5 mm; red lines and squares are for k0 = 10-4 m s-1 and r = 0.5 mm. For other parameters, see the text.

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6. Experimental Details Tetra-(n-butyl)ammonium hexafluorophosphate ((n-Bu)4NPF6; 98%, Wako) was recrystallised twice from ethanol (96%, Merck Emplura) and dried under vacuum at 40°C. Acetonitrile (Merck; HPLC grade), ferrocene (Fc; 98%, EGA Chemie), cobaltocenium hexafluorophosphate (CcPF6; Strem), hexaammineruthenium(III) chloride ([Ru(NH3)6]Cl3; 98%, Sigma-Aldrich), potassium ferricyanide (K3[Fe(CN)6]; >99.0%, Sigma-Aldrich), and potassium chloride (KCl, >99.5%, Merck Emsure) were used as received. For removal of oxygen from solutions, high purity nitrogen (99.999%, H2O < 3 ppm; O2 < 2 ppm), saturated with CH3CN or H2O vapour as appropriate, was employed. Electrochemical experiments were performed in a three-electrode cell at ambient temperature (24 ± 1 oC) using custom-made voltammetric instrumentation [17] in a ‘staircase’ regime, though with a minuscule potential step of 2 μV. A high surface area Pt wire separated from the working electrode compartment by a glass frit was used as the auxiliary electrode. A Pt wire placed inside a Luggin capillary positioned in the vicinity of the surface of the working electrode and filled with the appropriate solution was employed as a quasi-reference electrode in CH3CN. The potential of this quasi-reference electrode varied slightly giving rise to small variation in E0 values vs. Pt from experiment to experiment. An analogous arrangement, but using Ag|AgCl (1 M KCl) reference electrode was used in an aqueous media. A glassy carbon (GC) disk (nominally 3 mm diameter) (BAS) was employed as the working electrode. Procedures used for pre-treatment of the GC electrode and cleaning of the glassware are described elsewhere [18, 19]. The uncompensated resistance, Ru, was determined by electrochemical impedance spectroscopy at potentials devoid of faradaic current. Analysis for Ru was based on use of a simple RC circuit for the relevant conditions. Analysis of Ru and values so calculated were

17

included in the corresponding simulations. The surface area of the GC electrode (A = 6.7 mm2) was determined on the basis of the Randles-Ševčik formula by measuring d.c. voltammetric oxidation peak currents (scan rate, v = 0.100-1.00 V s-1) or from the Cottrell equation applied to chronoamperograms in the diffusion-controlled regime [20] for oxidation of 0.10-4.0 mM Fc in CH3CN (100 mM (n-Bu)4NPF6) (Ru ≤ 60 Ω) with n = 1 and DFc = 2.4∙10-9 m2 s-1 [21]. The validity of the A value was confirmed by comparison of background-corrected experimental d.c. voltammograms for oxidation of Fc with simulations based on a two-dimensional diffusion model using reversible electron-transfer kinetics. To achieve background correction, d.c. and a.c. voltammograms derived from the electrolyte alone were obtained under exactly the same conditions as in the presence of electroactive species and point-by-point subtraction of the current data was performed. To reconcile differences in the potential of the quasi-reference electrode used for experiments in acetonitrile (100 mM (n-Bu)4NPF6) in the presence and in the absence of Fc0, small concentration of CcPF6 was added to the electrolyte solutions and the reversible potential of the Cc+/0 process was used as a reference point. Equality of the (pseudo)capacitive background current in the absence and presence of the electroactive species in solution was assumed, which should be valid if neither R or P are adsorbed in significant amount on the electrode surface. Other experimental parameters include the following. Scan rates (for d.c. and a.c. voltammograms): 0.07078 V s-1 for reaction 6:1 when cR = 1.0 mM, but 0.11176 V s-1 when cR = 11 mM, 0.07451 V s-1 for reaction 6:2, and 0.08941 V s-1 for reaction 6:3. A.c. frequencies: 219.01 Hz for reaction 6:1, 72.01 Hz for reaction 6:2, and 9.015 for reaction 6:3. A.c. amplitude: 0.080 V universally. Uncompensated resistance: 5 Ω for reactions 6:1 and 6:3, 69 Ω for reaction 6:2 when cR = 1 mM, but 58 Ω when cR = 10 mM. The temperature was 297 K.

18

  Ru(NH3 )62 (aq)  e Ru(NH3 )36 (aq)  

with 1000 mM KCl

(6:1)

  Fc+ (acetonitrile) + e Fc(acetonitrile)  

with 100 mM (n-Bu)4NPF6

(6:2)

  Fe(CN)64- (aq)  e Fe(CN)3-6 (aq)  

with 1000 mM KCl

(6:3)

7. Experimental Results The various methods of extracting diffusivities from d.c. and a.c. cyclic voltammograms were applied to reactions 6:1-6:3, all of which are n =1 examples of reaction 1:1. In each case, DR was measured by several routes. These are varied processes inasmuch as the first two reactions behave reversibly, whereas the reduction of ferricyanide acts quasireversibly. Moreover, the uncompensated resistance encountered in the voltammetry of reaction 6:2 is significant. In contrast, ohmic interference is minimal in the voltammetry of reactions 6:1 and 6:3 because they take place in a highly conducting solution. In the interest of brevity, results are shown graphically in Figure 4 only for the voltammetry of reaction 6:2. We chose this reaction for detailed analysis because it is one of the few systems for which very careful measurements of diffusivity have been made [21]. Nevertheless, results for all three reactions are included in Table 3. In the d.c. voltammograms, data in the column of this table headed ‘E1/2’ is an estimate of the half-wave potential taken as the peak potential in semidifferentiated forward branch of voltammogram (that is, the point of inflection of the semiintegral curve). In the case of a.c. voltammograms, it is the midpoint potential for the central minima in the 4th harmonic component. In the main, the data in Table 3 reinforce the conclusions of our studies with simulated data. The convolution method performs excellently. As far as extrapolation methods are concerned, those based on intercepts yield more consistent results than those based on slopes, but the

19

constrained approach is more successful than either. Again, there is a difference between the results for aperiodic components in a.c. voltammetry compared with those for d.c. possibly due to faradaic rectification. Greater consistency is found for Fc0/+ in CH3CN than for [Ru(NH3)6]3+/2+ or [Fe(CN)6]3-/4- in water, perhaps due to uncertainties in the background correction procedure associated with contribution from functional groups on a GC surface in aqueous medium. Table 3. Estimation of DR by several methods based on semiintegration of background corrected d.c.

Process

Mode

voltammograms and aperiodic components of a.c. voltammograms for three electrode processes.

cR / mM

Fc0/+

d.c.

1.0

E1/2 / V

a

109 DR / m2 s-1 as determined by Convolution

Eq 4:3

Eq 4:4

Eq 4:6

0.576

2.4

2.4

2.4

2.4

10

0.589

2.4

2.4

2.6

2.4

1.0

0.573

2.4

2.5

2.0

2.4

10

0.575

2.5

2.4

2.7

2.5

1.0

-0.193

0.61

0.62

0.49

0.61

11

-0.197

0.65

0.62

0.93

0.65

1.0

-0.194

0.63

0.61

0.97

0.63

11

-0.197

0.66

0.64

0.84

0.66

d.c.

1.0

0.201

0.78

0.73

1.5

0.78

in H2O (1000 mM KCl) a.c.

1.0

0.203

0.75

0.70

1.5

0.75

in CH3CN (100 mM (n-Bu)4NPF6)

[Ru(NH3)6]3+/2+

a.c. d.c.

in H2O (1000 mM KCl)

a.c. [Fe(CN)6]3-/4a

0/+

vs. Pt quasi-reference electrode for the Fc process (E1/2 is unique for each experiment) and vs. Ag|AgCl (1 M KCl) for the [Ru(NH3)6]3+/2+ and [Fe(CN)6]3-/4- processes.

20

Figure 4. a. Background corrected d.c. voltammograms and aperiodic components from a.c. voltammograms for oxidation of Fc. b. The corresponding semiintegrals of the forward oxidative branches. c. Extrapolation of the semiintegrals and the regression lines. Blue (left ordinate axes) and red (right ordinate axes) data relate to cR = 1.0 mM and 10 mM, respectively. Solid lines and filled squares relate to d.c. voltammetry; dashed lines and open squares relate to aperiodic components of a.c. voltammograms. See Section 6 for experimental details.

21

The data reported as Table S1 and Fig. S4 (Supporting Information) show the use of equations 4:9 and 4:10 for the estimation of DR from the experimental data to be less successful than extrapolation methods based on equations 4:3-4:6 and convolution. Do the methods give the correct diffusion coefficients? Here we turn to the ferrocene results, where a value of 2.4∙10-9 ± 0.1∙10-9 m2 s-1 was recovered via the iterated convolution and constrained regression analysis in all cases. Importantly, the DR value of 2.4∙10-9 m2 s-1 taken from literature (Ref. [21] and references therein) was used to determine A (see Section 6), required for application of the extrapolation and convolution methods. The diffusion coefficients of [Fe(CN)6]3- and [Ru(NH3)6]3+ determined by convolution and equations 4:3 and 4:6 conform well to the literature data for 0.5-1.0 M KCl aqueous solutions [22-24]. Finally, excellent agreement is found between experimental background corrected data and theoretical curves modelled with the use of DR listed in Table 3 (as shown in Figs. S5 and S6).

8. Conclusions Provided that the electrode area A far exceeds the product of the diffusivity DR of the reactant and the forward scan duration tmax, the height of the plateau in semiintegrated d.c. or a.c. voltammetry can be used, via the formula  M lim  DR     nAFcR 

2

(8:1)

to calculate the diffusivity from an experiment conducted at an inlaid disc. However if the ratio A/DRtmax is less than about 100, the plateau has a noticeable slope, arising from edge diffusion, that impedes this approach to measuring diffusivities for both reversible and quasireversible processes.

22

Several methods have been explored whereby the diffusivity may be calculated despite the slope. Each method has been tested using simulated data and experimental results from three diverse electrode processes. The most successful method, which uses iterated convolutions, is mathematically complicated. Somewhat simpler to apply, and almost as effective, is the constrained linear regression method. Less successful, but simple, is a straightforward extrapolation of the linearised plateau back to the half-wave potential. Linearisation is achieved by plotting M versus

( E  E1/2 ) .

9. Supporting Information Figs. S1-S3 demonstrating application of extrapolation methods to analysis of semiintegrated data simulated for multistep mechanisms. Table S1 and Fig. S4 showing the results of analysis of experimental d.c. voltammograms with the use of equations 4:9 and 4:10. Figs. S5 and S6 comparing experimental and simulated voltammetric data.

10. Acknowledgments Financial support from the Australian Research Council is gratefully acknowledged. The authors deny any financial, or other, conflict of interest in disseminating this material.

11. References [1] K.B. Oldham, Tables of semiintegrals, Journal of Electroanalytical Chemistry 430 (1997) 114. [2] A.M. Bond, E.A. Mashkina, A.N. Simonov, A Critical Review of the Methods Available for Quantitative Evaluation of Electrode Kinetics at Stationary Macrodisk Electrodes, Developments in Electrochemistry, John Wiley & Sons, Ltd, 2014, pp. 21-47. [3] K.B. Oldham, J.C. Myland, A.M. Bond, E.A. Mashkina, A.N. Simonov, The aperiodic current, and its semiintegral, in reversible a.c. voltammetry: Theory and experiment, Journal of Electroanalytical Chemistry 719 (2014) 113-21. [4] J. Myland, K. Oldham, The excess current in cyclic voltammetry arising from the presence of an electrode edge, Journal of Solid State Electrochemistry (2014) 1-11.

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[5] Digielch-Professional 7.F; http://www.elchsoft.com/DigiElch/DigiElch7/Default.aspx. [6] E.A. Mashkina, A.N. Simonov, A.M. Bond, Optimisation of windowing for harmonic recovery in large-amplitude Fourier transformed a.c. voltammetry, Journal of Electroanalytical Chemistry 732 (2014) 86-92. [7] A.M. Bond, K.B. Oldham, G.A. Snook, Use of the Ferrocene Oxidation Process To Provide Both Reference Electrode Potential Calibration and a Simple Measurement (via Semiintegration) of the Uncompensated Resistance in Cyclic Voltammetric Studies in High-Resistance Organic Solvents, Analytical Chemistry 72 (2000) 3492-6. [8] P.J. Mahon, K.B. Oldham, The transient current at the disk electrode under diffusion control: a new determination by the Cope–Tallman method, Electrochimica Acta 49 (2004) 5041-8. [9] P.J. Mahon, K.B. Oldham, Diffusion-Controlled Chronoamperometry at a Disk Electrode, Analytical Chemistry 77 (2005) 6100-1. [10] K.B. Oldham, Edge effects in semiinfinite diffusion, Journal of Electroanalytical Chemistry and Interfacial Electrochemistry 122 (1981) 1-17. [11] P.J. Mahon, K.B. Oldham, Voltammetric modelling via extended semiintegrals, Journal of Electroanalytical Chemistry 445 (1998) 179-95. [12] K.B. Oldham, Convolution: a general electrochemical procedure implemented by a universal algorithm, Analytical Chemistry 58 (1986) 2296-300. [13] P.J. Mahon, K.B. Oldham, Convolutive modelling of electrochemical processes based on the relationship between the current and the surface concentration, Journal of Electroanalytical Chemistry 464 (1999) 1-13. [14] P.J. Mahon, K.B. Oldham, Convolutive modelling of the disk electrode geometry under reversible conditions, Electrochimica Acta 49 (2004) 5049-54. [15] C.L. Bentley, A.M. Bond, A.F. Hollenkamp, P.J. Mahon, J. Zhang, Applications of Convolution Voltammetry in Electroanalytical Chemistry, Analytical Chemistry 86 (2014) 207381. [16] M. Goto, K.B. Oldham, Semiintegral electroanalysis. Neopolarographic plateau, Analytical Chemistry 46 (1974) 1522-30. [17] A.M. Bond, N.W. Duffy, S.-X. Guo, J. Zhang, D. Elton, Changing the Look of Voltammetry, Analytical Chemistry 77 (2005) 186 A-95 A. [18] G.P. Morris, A.N. Simonov, E.A. Mashkina, R. Bordas, K. Gillow, R.E. Baker, D.J. Gavaghan, A.M. Bond, A Comparison of Fully Automated Methods of Data Analysis and Computer Assisted Heuristic Methods in an Electrode Kinetic Study of the Pathologically Variable [Fe(CN)6]3–/4– Process by AC Voltammetry, Analytical Chemistry 85 (2013) 117807. [19] A.N. Simonov, G.P. Morris, E.A. Mashkina, B. Bethwaite, K. Gillow, R.E. Baker, D.J. Gavaghan, A.M. Bond, Inappropriate Use of the Quasi-Reversible Electrode Kinetic Model in Simulation-Experiment Comparisons of Voltammetric Processes That Approach the Reversible Limit, Analytical Chemistry 86 (2014) 8408-17. [20] A.J. Bard, L.R. Faulkner, Electrochemical Methods: Fundamentals and Applications, 2nd ed., John Wiley & Sons, Inc., New York, USA 2001. [21] J. Janisch, A. Ruff, B. Speiser, C. Wolff, J. Zigelli, S. Benthin, V. Feldmann, H. Mayer, Consistent diffusion coefficients of ferrocene in some non-aqueous solvents: electrochemical simultaneous determination together with electrode sizes and comparison to pulse-gradient spinecho NMR results, Journal of Solid State Electrochemistry 15 (2011) 2083-94.

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[22] S.J. Konopka, B. McDuffie, Diffusion coefficients of ferri- and ferrocyanide ions in aqueous media, using twin-electrode thin-layer electrochemistry, Analytical Chemistry 42 (1970) 1741-6. [23] L. Bortels, B. Van den Bossche, J. Deconinck, S. Vandeputte, A. Hubin, Analytical solution for the steady-state diffusion and migration involving multiple reaction ions Application to the identification of Butler-Volmer kinetic parameters for the ferri-/ferrocyanide redox couple, Journal of Electroanalytical Chemistry 429 (1997) 139-55. [24] P. Sun, M.V. Mirkin, Kinetics of Electron-Transfer Reactions at Nanoelectrodes, Analytical Chemistry 78 (2006) 6526-34.

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D.c. and a.c. voltammetric edge current causes the semiintegral plateau to slope Diffusivity can be derived from the sloping semiintegral via several methods The most successful method based on iterated convolutions is complicated Simpler and almost as effective is the constrained linear regression method Less successful, but simple, are straightforward extrapolation methods