Modelling cyclic voltammetry without digital simulation

doi:10.1016/j.electacta.2011.05.044

Electrochimica Acta

Volume 56, Issue 28, 1 December 2011, Pages 10612-10625

https://doi.org/10.1016/j.electacta.2011.05.044 Get rights and content

Abstract

A semi-analytical method of modelling cyclic voltammetry is described, derived, exemplified, verified, and advocated. A listing of the mechanistic schemes that can be addressed by the procedure includes E, EE, CE, EC and ECE, all with various degrees of reversibility. The approach has advantages and disadvantages when compared with digital simulation.

Highlights

► “Black box” digital simulation is not the only way to model cyclic voltammetry. ► A single equation describes a cyclic voltammogram in our semi-analytical approach. ► Our method can treat the following mechanisms: E, EE, CE, EC, ECE, C′E, square, etc.

Introduction

Cyclic voltammetry has become the default investigatory tool for electroanalytical chemists. This technique is customarily performed under the following conditions:

(a)
A three-electrode cell is served by a potentiostat. The uncompensated resistance is small and ignored.
(b)
Supporting electrolyte is present at a concentration large enough that the migration of the species of interest can be ignored. The solution is quiescent. These two features ensure that transport, to and from the working electrode, is purely diffusive.
(c)
The working electrode is, or behaves as if it were, a planar macroelectrode, edge effects being negligible. This, together with the provision of an unobstructed zone in front of the electrode, allows diffusion to be treated as planar and semiinfinite.
(d)
Generally, only one pertinent species is present initially. If this species is not itself electroactive, it converts to an electroreactant by a “preceding” homogenous chemical reaction. The electroproduct may, or may not, undergo a “following” chemical reaction and/or a second electrode reaction.
(e)
At the commencement of the experiment, the potential of the working electrode is chosen, whenever feasible, to have a value at which the faradaic current is either negligible or small.
(f)
The potential of the working electrode is linearly scanned at a constant rate from a starting potential, to a reversal potential, and back to its starting value.
(g)
The formal potential(s) of the electrode reaction(s) under study generally lie(s) well within the doubly scanned potential range.
(h)
The current is measured as a function of time. However, cyclic voltammograms are mostly reported as dual curves of the forward and backward current branches versus potential.
(i)
The nonfaradaic current arising from the charging of the double layer at the working electrode is either ignored or crudely compensated by adding or subtracting the product of the scan rate and the capacitance, assumed constant.

Throughout what follows, it is assumed that all the above standard conditions hold.

Cyclic voltammetry is executed in two varieties according as the initial direction of the potential scan is positive-going or negative-going. This polarity choice usually corresponds to whether an oxidation or a reduction is the prime subject of the investigation. To avoid the need to duplicate all our equations to cater to the two varieties, we adopt a notation in this article that provides polarity inclusivity. Thus, in depicting the electron-transfer reaction that the first scan elicits by the equation $R \mp e^{-} \to P$ we use R to denote the electroreactant (not “reduced form”!) and P to denote the electroproduct irrespective of whether the process is an oxidation (upper sign) or a reduction (lower sign). The alternative $\underset{lower}{upper}$ signs in process (1:1) also occur in many of the mathematical equations that follow, for example in Eqs. (2:2), (2:4). Do not make the easy mistake of imagining that the signs change on reversal; they do not!

Section snippets

Methods of modelling cyclic voltammetry

Thankfully, the days when cyclic voltammograms were analyzed solely on the basis of the location of their peaks are almost past. Mostly, nowadays, the entire cyclic voltammogram is modelled and concordance between the experimental curve and a synthetic curve provides the basis on which conclusions are drawn.

There are three broad routes by which cyclic (and other) voltammograms may be modelled: $voltammetric modelling \{\begin{array}{c} mathematical analysis \\ semi-analytical methods \\ digital simulation \end{array}$ although the

Relations obeyed by diffusing solutes

Intrinsic to the models that will be developed later in this article are relationships that link the diffusive flux of a solute species at the electrode surface to its concentration there, and to its bulk concentration, if any. Here we shall investigate two scenarios: in the first, there is a single diffusant; in the second there are two interconverting diffusants.

The goal of this section is to derive Eqs. (3:8), (3:28), none of which incorporates the x spatial coordinate. It is by the use of

Linking to the electrical variables

The derivations of the previous section have effectively removed the spatial coordinate, x, from the problem. We now address the need to relate the surface concentrations and surface flux densities of species involved in the electrode reaction to the electrical variables – the electrode potential E(t) and the faradaic current I(t). Ultimately it is the time-dependent relation between I and E that is sought in modelling cyclic voltammetry.

When the electron transfer reaction is no more

Semiintegration algorithm

The operation of semiintegration with respect to time, signified here by the operator d^−1/2/dt^−1/2 is the $v = - 1 / 2$ instance of a generalized “differintegration” operator [13] applied to some function f of the variable t. It can be defined [14] by the limiting operation $\frac{d^{v}}{{dt}^{v}} f (t) = \lim_{δ \to 0} \{\frac{δ^{- v}}{Γ {- v}} \sum_{n = 0,1}^{(t / δ) - 1} \frac{Γ {n - v}}{Γ {n + 1}} f (t - n δ)\}$ in which Γ {} denotes the gamma function [15] and n is a summation index. In electrochemical applications, t is invariably time, while f may be a current, a flux density, or a

Numerical modelling of cyclic voltammograms

The principles embodied in this article apply to any form of voltammetry in which the potential is caused to change with time. Our treatment is well adapted to experiments in which the potential is changed gradually (though not necessarily linearly), rather than in steps; it is not restricted to cyclic voltammetry. Nevertheless, in view of the ubiquity of cyclic voltammetry, we shall give examples only of that popular form of potentiodynamic voltammetry.

The only aspect that makes cyclic

Scheme E_r: Reversible electron transfer without homogeneous chemistry

This is the classic case that yields a cyclic voltammogram of the familiar “hybrid” shape. The diagram in Fig. 3 illustrates the simple mechanism.

In this simplest of cases, the algorithm is derived by applying Eq. (3:8) to each species, the electroreactant and the electroproduct, yielding $c_{R}^{s} (t) = c_{R}^{b} + \frac{1}{\sqrt{D_{R}}} \frac{d^{- 1 / 2}}{d t^{- 1 / 2}} j_{R}^{s} (t)$ and $c_{P}^{s} (t) = \frac{1}{\sqrt{D_{P}}} \frac{d^{- 1 / 2}}{d t^{- 1 / 2}} j_{P}^{s} (t)$ and then replacing the surface flux densities by the current through use of Eqs. (4:1), (4:2) $c_{R}^{s} (t) = c_{R}^{b} \mp \frac{1}{F A \sqrt{D_{R}}} \frac{d^{- 1 / 2}}{d t^{- 1 / 2}} I (t)$ and $c_{P}^{s} (t) = \frac{\pm 1}{F A \sqrt{D_{P}}} \frac{d^{- 1 / 2}}{{dt}^{- 1 /}}$

Scheme E_q: Quasireversible electron transfer without homogeneous chemistry

The mechanism addressed in this section, illustrated schematically in Fig. 4, differs from that in Section 7 only in that the electron transfer is not now necessarily reversible. As before, solutes R and P are transported to and from the electrode by diffusion processes uncomplicated by any concurrent homogeneous kinetics, but only species R is present in the bulk.

Equations (7:1), (7:2), (7:3) apply unchanged, but instead of combining (7:3), (7:4) through the Nernst equation, the full

Scheme CE_r: reversible electron transfer preceded by homogeneous kinetics

Here we consider a scheme in which the original electropassive substrate S isomerizes to produce the electroactive species R, which undergoes an electron transfer to give a product P, as depicted in Fig. 6. The nominal bulk concentration of the substrate is c^b, but the establishment of the equilibrium detailed in (2:5) ensures that the actual bulk concentrations are $c_{S}^{b} = \frac{c^{b}}{1 + K}$ and $c_{R}^{b} = \frac{K c^{b}}{1 + K}$ Because the electroproduct P diffuses straightforwardly away from, and later towards, the electrode and the

Scheme E_qC: Quasireversible electron transfer followed by homogeneous kinetics

Here we consider the scheme in which the product P of an electron-transfer reaction undergoes a first (or pseudo-first) order chemical reaction, as outlined in Fig. 8. For maximal generality, the chemical step is treated as bidirectional.

The diffusion of the electroreactant R is uncomplicated and it therefore obeys Eq. (3:8), which in this case takes the form of the first equality in the expression $c_{R}^{s} (t) = c_{R}^{b} + \frac{1}{\sqrt{D_{R}}} \frac{d^{- 1 / 2}}{d t^{- 1 / 2}} j_{R}^{s} (t) = c_{R}^{b} \mp \frac{1}{F A \sqrt{D_{R}}} \frac{d^{- 1 / 2}}{d t^{- 1 / 2}} I (t)$ the second step being a consequence of

Scheme C′E: Homogeneous regeneration of the electroreactant

In this scheme, an electropassive species X, present in large concentration, homogeneously converts the electroproduct P of an electron transfer reaction back to the electroreactant R as outlined in Fig. 10. This is referred to as a “catalytic” scheme because the electrode reaction serves to catalyze the conversion of X to some other electropassive product Y. The large excess of X serves to make the homogeneous $X + P \overset{\vec{k}}{⟶} R + Y$ reaction pseudo-first order in P, with a rate constant $\vec{k}$ that

Scheme E_qE_q: Two quasireversible electron transfers without homogeneous kinetics

Here we consider the case of two successive electron transfers, linked via an intermediate species I. Each electrode reaction has its own set of electrode kinetic constants, as described in item (vi) of Section 2, that we distinguish by use of 1 or 2 subscripts. The original reactant R is present at concentration $c_{R}^{b}$ in the bulk, where neither the intermediate I nor the final product P exists. The proportionation reactions P + R ⇄ 2I do not occur. All three species freely diffuse to and from the

Scheme E_rCE_r: Two reversible electron transfers coupled by homogeneous kinetics

Here the product, species I, of one electron transfer is electropassive, but it converts homogeneously to an electroactive isomer, species J, which undergoes a second electron transfer. Only the first electroreactant, species R, is present in the bulk. Fig. 14 summarizes the events. As in the preceding section, species I and J are prohibited from taking part in any conproportionation or disproportionation reaction.

Species R and P undergo straightforward diffusion and each therefore obeys Eq.

Square scheme: Alternative electron transfers with homogeneous interconversions

Fig. 16 depicts what is commonly called a “square scheme”. Two reactants, R₁ and R₂, interconvert homogeneously through first (or pseudo-first) order processes. Both are electroactive, yielding products P₁ and P₂ that are absent initially; subsequently these products also interconvert by first-order chemical reactions. Thermodynamics requires a relationship between the two solution-phase equilibrium constants $R_{1} ⇄_{{\overset{\leftarrow}{k}}_{R}}^{{\vec{k}}_{R}} R_{2} \frac{{\vec{k}}_{R}}{{\overset{\leftarrow}{k}}_{R}} = K_{R} and P_{1} ⇄_{{\overset{\leftarrow}{k}}_{P}}^{{\vec{k}}_{P}} P_{2} \frac{{\vec{k}}_{P}}{{\overset{\leftarrow}{k}}_{P}} = K_{P}$ on the one hand, and the formal

Conclusions

The final equation in each of Sections 7.11, and the sum of a pair of equations in Sections 12 Scheme E, 13 Scheme E, are algorithms whereby cyclic voltammetry may be conveniently modelled. Similar treatments can address mechanisms not considered here. These equations are semi-analytical formulas because they incorporate summations. The summations arise from replacing a semiintegral by a formulation that is exact in the limit of a vanishingly small time-step δ Prior to the replacement of the

Acknowledgement

We are grateful for the financial support of the Natural Sciences and Engineering Research Council of Canada.

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Electrochimica Acta

Modelling cyclic voltammetry without digital simulation

Abstract

Highlights

Introduction

Section snippets

Methods of modelling cyclic voltammetry

Relations obeyed by diffusing solutes

Linking to the electrical variables

Semiintegration algorithm

Numerical modelling of cyclic voltammograms

Scheme Er: Reversible electron transfer without homogeneous chemistry

Scheme Eq: Quasireversible electron transfer without homogeneous chemistry

Scheme CEr: reversible electron transfer preceded by homogeneous kinetics

Scheme EqC: Quasireversible electron transfer followed by homogeneous kinetics

Scheme C′E: Homogeneous regeneration of the electroreactant

Scheme EqEq: Two quasireversible electron transfers without homogeneous kinetics

Scheme ErCEr: Two reversible electron transfers coupled by homogeneous kinetics