Cyclic voltammetry: simulation and analysis of reaction mechanisms

Preface

-

Cyclic voltammetry (CV) has been in the Forefront of the study of electron t s conseyuences. With the cyclic voltarnmetric method one can simultaneously sctivale mc.lecules by electron transfer and probe subsequent ical reactions. The cyclic voltarnmetric response curve thus provides ativn about elwtron transfer kinetics and thermodynamics as well as the consequences of ekct con transfer. Ths book jntroduces cyclic vol tarnmetry and its application to the analysis of electrochemical reaction mechanisms. It also provides the experimentalist with a siruulation-based approach for the analysis of cyclic voltammograrns. Chapter I presents a brief summary of electrochemical principles. Emphasis is on a conceptud approach to the reduction potential and electrode kinetics. Chapter 2 introduces cxperimental and conceptual aspects of cyclic voltammetry. The relationship hetween electrode kinetics, chemical kinetics, and lored, find the imporpant concept of electrochemical reversibility Chapter 3 is a survey of the use of CV for the study of reaction mechanisms in anometallic, it~orgaruc.and pharmacological chemistry. introduces the method of simulation by explicit finite differences, only employed numerical method for CV analysis. Chapter 5 describes CVSIM, a general program for the simuIation of cyclic voltarnmetric experiments. A general program fcr the simulation of double experiments, DSTEP, is also provided. describes CVFIT, a program that combines CVSlM with n A

. ., ,

. .

.

.

,..-

-..

..

.

.

.

viii

PREFACE

simplex driver to find the least-squares best fit between experimental and simulated cyclic voltarnmograms. CVSIM, CVFIT. and DSTEP are included on the diskette with this book. They require the use of a PC-compatible (80 x 86) computer. A math coprocessor is not required, but it is recommended. I welcome questions or comments from those utilizing the simulation software (E mail : Gosser @ sci.ccny.cuny.edu).

Contents

Acknowledgments Many thanks go to Philip Rieger, James Rusling, and Brenda Shaw, for their encouragement of my interest in simulation. Acknowledgment is due to Qindong H n a n p and Feng Zhang, my co-workers at City College, for their many contributions to the work represented here. Helpful discussions with Ron Birke, my colleague at City College, are gratefully acknowledged. I would like to thank the researchers who have used the CVSIM program, especially Dwight Sweigart, Jay Kochi, Carlo Nervi, and Christian Amatore, whose comments have been particularly helpful. Finally, I wouId like to thank Edmund Immergut, of VCH, for his valuable advice and assistance.

Useful Equations xi 1. The Reduction Potential and Electrode Kinetics

David K. Gosser, Jr. May 1993

1.1 The Reduction Potential 1 1.2 Electrode Kinetics 12 References 24

2. The Cyclic Voltammetric Experiment

I

I

5

1'I $

27

2.1 An Overview 28 2.2 The Electrochemical Cell 30 2.3 Electrochemical Mechanisms: E&C Notation 35 2 4 Distortions of the Faradaic Response 56 2 5 Microelectrodes and Fast Scan Voltanlnletry 59 2.6 Potential Step Methods and Cyciic Voltamriletry 60 2.7 Construction of a Fast Potentiostat 64 2.8 Determination of the Number of Electrons 68 References 68

3. A Survey of Electrochemical Mechanisms 3.1 The CE Mechanism 72 3.2 Multielectron Transfer 75 3.3 Protonations at Equilibrium 77

71

1

CONTENTS

x

3.4 3.5 3.6 3.7 3.8

Catalytic Mwhanisms SO The Reduction of Nilrobenzoic Acid 89 Reduction of the Nit'rosonium Cation and Its Complexes 92 Reactivity of 17-, 18-, and 19-Electron Tungsten Complexes 95 Mechanisms Involving Adsorption 97 References 102 .

4. The Simulation of Electrochemical Experiments

Useful Equations

105

4.1 The Discretized Diffusion Equation 105 4.2 Evaluation of the Boundary Conditions 108 4.3 Dimensionless Units 109 4.4 Solution Chemical Kinetics 110 4.5 A Sample Simulation Program 110 References 114

5. CVSIM: A General Program for the Simulation

of Cyclic Voltammetry Experiments 5.1 5.2 5.3 5.4

115

An Overview of CVSIM 115

Extensions of the Simulation Method 117 Accuracy of Simulations 120 Installation and Use of CVSIM and CVGRAF 123 5.5 Examples 125 References 135

6 . CWIT: Simplex Data

Analysis with CVSIM

AGO

-

FE' = AG

,,.,, + SAG,, + constant

Nernst Equation 137

6.1 CVFIT: Simplex Data Analysis with CVSIM 137 6.2 Instructions for the use of CVFIT 138 6.3 A Prototype Analysis: The EC Mechanism 139 6.4 Some Final Comments on Simulation Analysis 145 References 148

Butler- Volmer Equation

k0

Appendix: Summary of Instructions for CVSJM, CVGRAF, CVFIT,andDSTEP 149 Index

=

=

stattdard heterogeneous rate constant (cm/s)

!

rr = cat hadic transfer coefficient 1 - a = anodic transfer coefficient, fl

:'

Cyclic Voltammetry (CV) E, (reversible one-electron transfer)

153

C

CYCLIC VOLTAMMETRY

THE REDUCTION POTENTIAL AND ELECTRODE KINETICS

...=> e-vac

Ox

Red ,

Figure 1-2 Thermodynamic cycle for reduction in solution.

standard free energy and has the usual relationship to the standard reduction potential (also referred to absolute scale), AG,",,= -nFE,',,. The existence of the oxidized and reduced forms of a rrdon couple thus determine a kind of Fermi 1eveI in solution, [see Pleskov and GurevichZfor a statistical mechanical derivation]. The Nernst equation for the half-reaction, referenced to vacuum, is then given by*:

Figure 1-1 The conduction band model. Conduction is possibfe when parrially fillcd valance electrons cnn be raised in energy by external fields.

potential of the electron in the conducting material. 111 thermodynamics, the electrochemical potential is defined as the change in free energy per unit species with everything else kept constant:

Eaba= E L

p,,,(e, conductor) = -

The units of the Fermi level free energy, usually expressed in electron volts, can be interconverted with other units that are more useful in other contexts (1 eV x N, = 96.485 kJ/mol = 23.06 kcal/mol = 8066 cm- '). Mow let us turn to the process in solution, the ferricyanidejferrocyanide transition. The free energy change associated with the transfer of an electron to an oxidized species in solution can be viewed in the context of a thermodynamic cycle (Figure 1-2). The total free energy is seen to be composed of: 1. The free energy associated with transfer of the oxidized species from solution to vacuum. This is the negative of the solvation energy, -AG,,,,,,o,,. 2. The free energy associated with the transfer of an electron to the oxidized species jn vacuum (electron attachment), AG,,,,,. This is approxinlalely the electron affinity (EAI of the oxidized species or the negative of the ionization potential (IP) of the reduced species. 3. The free energy associated with transfer of the reduced species from vacuum to solution, AG

'

; "

'

, ,.

We have:

+ AGsolv(red) - AGsolurax) (I) The subscript abs denotes that the referenw level for energy is established by the electron in vacuo (the "absolute" scale). If the oxidized and reduced species are at unit activity (at T = 298.15"K) then the free energy of reduction is a AGabs = AGel.at.

t

-

RT [Red] -In nF

[Ox]

A Le Chatelier "co~icentration" cffect raises or lowers the solution Fermi level by the rclative concentrations of the reduced and oxidized forms. When an electrode is placed in a solution containing the oxidized and reduced species, the Femi levels tend to equalize. Ths is achievsd through electron transfer between the electrode and either of the redox species. Electron transfer from the reduced species to the electrode will raise the Fermi level of the electrode and lower that of the sulution. Etectmn transfer from the electrode to the oxidized species will lower the Fermi level of the electrode and raise that of the solution. The total Fermi level energy is usually separated into a chemical part (the innate Femi level of the metal) and an electrical part (energy as a result of charging). Thus we can write FL,ota, = FL.in*,t, - Fd where #I is the electrical potential OF the electrode. At equiiihrium, the total free energy change must be zero for electron transfer: the Fermi levels are equal. In general, both Fenni levels might be expected to move to reach equilibrium. In practice, however, the electrode potential can be controlled (as in the C V experiment) and the solution must adjust (Figure 1-3). Alternatively. iF the electrode potential is not controlled, the solution species can determine the electrode Fermi level. I t i s useful to emphasize the distinction between the electrical potential 4. and rhe Fernli level or the electrode potential, E. The electrode potential cvns~stsof both the chemical and electrical energy associated with bringing an electron to The concen1raIion ltrms in Equation (2) should actually be written as activities

CYCLIC VOLTAMMETRY

THE REDUCTION PO'ITNTIAL AND ELECTRODE KINETICS I

,

Figure 14 Hypothetical determination of E. Note that a difference in electrode

'

potentials is determined. Since the Fermi levels (FL) of the metals in contact must equilibrate, the difference in elmtrode potentials is transmitted exactly to a difference in lead electrical potentials, if the leads are of the same material.

(FL- Fd)mo,- (FL - F4)ptU, = (FL - F ~ ) c , I , , - ( F L - F # ~ c ~= ( ~F(& I - 42)

Figure 1 3 Equilibration of the Fermi levels of the electrode and solution. The electrical potential 4, of the electrode, is due to charging of the electrode. It is this process that changes the Fermi level of the electrode from its innate value.

the electrode, whereas the electrical potential is that part of the energy due solely to electrostatic effects.

1.1.2 The Hydrogen Reference and the Absolute Potential The preceding discussion used the vacuum level as the reference, to emphasize the relationshp between gas phase properties such as TP and EA. Electrochemical measurements, however, must utilize a solution-based reference. The primary reference is based on the hydrogen ion reduction in aqueous solution: H&,uniiaciiutg

+ e-

+ ~ ~ ( g atm) , l

E0

= 0 volts

for which the potential is taken as E* = 0 volts. The potential of any other halfreaction can in theory be obtained by constructing a complete cell with the hydrogen reaction as the anodic (oxidation) reaction and another half-reaction as the cathodic (reduction) reaction (Figure 1-4). AH redox reactions are thus described by their tendency to undergo reduction relative to the hydrogen ion. For instance, the standard potentiaI for the reduction of Fe(CN):- to F~(CN):is E0 = 0.356, and the reaction:

What is the relationship between the vacuum (or absolute) potential reference and a reference based on the hydrogen ion reduction? This is the tendency tie., the free energy change) for an electron to move from vacuurn to reduce the hydrogen ion in aqueous solution. This quantity is very important in relating electrochemical measurements and gas phase measurements or spectroscopic measurements, and it has been measured to be about 4.42V;however, values : around 4.8 V have aIso been r e p ~ r t e d-. ~ Some physical insight into the origin of the absolute potential is obtained with a thermochemical cycle for hydrogen reduction (Figure 1-5). The individual steps are (1) desolvation of H+, (2) electron attachment in vacuo to give the hydrogen atom, and (3) formation of molecular hydrogen in the standard state. The system of standard reduction i potentials is described with respect to aqueous solutions. Much electrochemical

!: work involves nonaqueous solutions, and special consideration must be given to

measurements in nonaqueous media.4 In cyclic voltammetric studies it is

1 common to report the redudon potential of a chosen standard, such as the

B I '

is favorable.

ferrocinium ion, under the conditions of the experiment. The reduction potential o f such a large ion is not expected to change appreciably with solvent. 1.1.3 Practical Reference Half-Celb In p r a c t i ~ electrode , reactions other than hydrogen ion reduction are used to construct practical reference systems. Potentials determined using these hdfcells can be related back to the hydrogen reference or absolute potential scale if

THE REDUCTION POTENTIAL AND ELECTRODE KINETICS

CYCLtC VOLTAMMETRY

7

For lzrricyanidelferrocyanide versus the silver chloride reference, this is:

1

Note that for a fixed reference system, the cell potential depends on the cathodic reaction components. Again, we can note that nvnaqueous systems pose a particular problem. For instance, if the solvent used in the reference electrode differs greatly from the cell solution, a potential is created ljurlction potential).

- AG roc

1.1.4 Formal Potentials Figure 1-5

The E0 for a half-reaction is the potential of that renction versus the standard hydrogen electrode, with all species at unit activity. Most reduction potentials are not determined under such conhtions, so it is expedient to define a "formal" reduction potential. This i s a reduction potential measured under conditions where the reaction quotient in the Nernst equation is one and other "nonstandard" conditions are described: solvent, electrolyle, pN, and so on. Formal reduction potentials are represented by Eo'. Reduction potentials determined by cyclic voltanmetry are usually formal potentials. The difference between standard and formal potentials is not expected to be great. Other definitions of the formal potential are offered.'

Thermodynamic cycle for hydrogen ion reduction.

desired. The saturated Ag/AgC1 and Hg/Hg,CI, (calomel electrode) are commonly used secondary standards. The use of saturated solutions keeps the chloride concentrations constant and thus fixes the potential.

+

+

= 0.199 V

AgCI(s) e - = Ads) C1-(iaq, sat KCI)

E"

Hg,Cl,(s)

ED= 0.244 V

+2e-

=

2HgO) + 2C1-(aq, sat KCl)

The potential of a complete cell is the difference in potential between the cell h alf-reactions:

1.1.5 The Chemical Interpretation of the Reduction Potential

cathohc and anodic Ecell = E c

- Ea

,

RT -1n-nF

[Red] [Ox]

RT E , = E: ---InnF

[Red] [Ox]

E,

.

= E:

-

where the concentration ratios indicate general reaction quotient terms. , The relationship between the concentrations and the potential is given by the Nernst equation for the complete cell:

t<

t

,

,

where Q is the reaction quotient: aA + bB*cC

+ dD

i:

'

The process of reduction in aqueous solution has been represented by the thermodynamic cycle in Figure 1-2, and the resulting equation.

~ F E O=

- AG ,,.

-

dAG,,,

-

4.43F + constant

(5)

Eo is the standard reduction potential based on the hydrogen reference. The hydrogen reduction potential versus the vacuum is taken as 4.42 V. The constant term accounts for the use of other references, such as the standard calomel electrode (SCE) or the Ag/AgCI electrode. (A small contributioi~to the constant can also w m e from junction potentials due to varying solvents. etc.1 Again, we can note that the free energy of electron association is approximately equal to the electron affinity and to the negative of the ionization potential. Let us conside~the application of Equation (5) to the one-electron reduction of p-benzoquinone and to the oxidation of nitroxide spin label Tempo. Kebarle and Chowdhury have utilized gas phase equilibrium electron transfer measurements to determine the thermodynamics of gas phase electron attachment to many organic compounds, including a series of quinonm. The free .

., ,

.. ,

,,

,

.

.-.

. "".

. a -

CYCLIC VOLTAMMETRY

I

d

THE REDUCTION POTENTIAL AND ELECTRODE KINETICS

l

TEMPO' (g)

0

TEMW (O)

Figure 1 4 The reduction of benzoquinone reduction (top) and the oxidation of Tempo (bottom).

energy of electron association for benzoquinone is - 1.9 eV. This number reflects the energy of the lowest unoccupied molecular orbital (LUMO) of benzoquinone. The redox potential for benzoquinone in dirnethylfomarnide (DMF) is 0.0 V versus the SCE. The SAG,,,, can be estimated through Equation (5). Since the benzoquinone [BQ) is neutral, the solvation energy change can be identified with that of the anion. The constant term in Equation (5) is -0.241 V, to account for the use of the SCE. The free energy of solvation that is released upon going from the neutral to the BQ anion can then be estimated as 2.26eV.' The energetics of the reduction of BQ are summarized in Figure I-7(top). The energetics of solvation play an important role in the thermodynamics of electron transfer in solution. The thermodynamics of electron transfer can be envisioned as electron transfer from one Fermi level to another, with the Fermi level of solution determind by the Nernst equation, and the E0 determined by both LUMO energy and solvation changes. Often, valence bond structures, molecular orbital diagrams, and electronegativity considerations can be useful in understanding the physical and chemical origin of the reduction potential. Electron-withdrawing groups favor reduction (a more positive EO)by reducing the energy of the LUMO (much in the same way that the electrode potential is lowered when electrons are withdrawn), and electrondonating groups will tend to have the opposite effect, making reduction more difficult (a more negative EO). A plot of E0 versus AG,,, for benzoquinone (Figure 1-8) illustrates the combined effect of structural changes on the molecular orbital structure as well

* The estimates here differ from those of Reference 6 because of a different choioe for the absolute potential of the hydrogen electrode.

TEMPO+ taeO

!

-

TEMPO [a@ Figure 1-7 Thermodynamic cycles for the reduction of benzoquinone and dhe oxida-

; tion of Tempo. >I

!

as on the solvation.' The LUMO for the quinones is a nY-typeorbital centered around the highly electronegative oxygens. Ths accounts for the large positive electron affinity of the quinones compared to other similar compounds like pquinohmethane. The electronegative oxygens lower the LUMO energy. Kebarle and Chowdhury also studied the effect of methyl substituents on the EA and found that for each methyl group, the EA was decreased by 1.7 kcallmol. It can be seen from Figure 1-8 that there is no significant difference in d A G , , for the methyl-substituted quinone series. The electron-donating methyl groups act as an electron source that raises the LUMO of the quinones. The chloroquinone series is quite different. The electron-withdrawing effect lowers the LUMO, increasing the EA. It can also be seen from Figure 1-8 that dAG,,,, is not constant for the chloroquinones. This variability is explained by the increased ' charge delocalization caused by the electron-withdrawing effect of the chloro 1, - , substituents. From the Born model of solvation, it is expected that a larger radius will decrease solvation energy. Finally, we can look at the following

j

I

* For spectroscopic measurements, the equality between the ionization potential and the energy of the highest occupied molecular orbital (HOMO) or the electron affin~tyand the LUMO energy follows from Koopman's "theorem,"which asserts they are equal and opposite (for IP/HOMO) or $ equal (EAILUMO).This relation requires that no electronic relaxailon take place after removal of an electron, which is not generally true. Electrochemical measurements of &%wil thus also include any similar electronic relaxation.

10

THE REDUCTION POTENTIAL AND ELECTRODE KINETICS

CYCLIC VOLTAMMETRY

where N A = Avogadro's number (6.022 x e = elementary charge (1.602 x 10- l 9 C) E~ = permittivity of a vacuum (8.854 x 10-l2 F/mZ)

r

= radius

of ion (m) constant of solvent

E = &electric

The Born equation is not expected to give very accurate results (often calculated solvation energes are too high). However, it is useful for estimates and qualitative predictions. Dielectric constants for some common electrochemical solvents are given in the next chapter (Table 2-1). In estimating chemical properties, it is sometimes very usefuI to combine reduction potentials with other physical quantities. For instance, solution bond dissociation energies for organo acids have been estimated by combining reduction potentials with pK,

1.1.6 The Temperature Dependence of the Reduction Potential The entropy of reduction can be obtained by measuring the reduction potential as a function of temperature.

Figure 1-8 The variation of electron affinity (EA) with reduction potential. (Adapted from Reference 6.)

In such experiments the temperature dependence of the reference cell potential must be taken into ~ C C O U I I ~The . ~ most convenie~ltmethod of doing this entails the use of a "nonisothermal" cell, in w h c h the reference is kept at a . constant temperature whle the temperature of the remaining halF-cell reaction is i changed. ,

r'

series: benzoquinone, naphthoquinone, and anthraquinone. There is apparently no change in solvation energies here, indicating that the charge remains highly localized on the oxygens throughout the series. However, the LUMOs do change, as predicted by MO theory. The energetics of the oxidation of the nitroxide spin label Tempo (in DMF) have also been s t u d ~ dand , ~ are shown in Figure 1-7(bottom).The ionization potential (IP)of Tempo is 7.31 eV and the oxidation potential is 0.271 V versus the standard hydrogen electrode. The solvation energy change (cation to neutral) is then calculated as - 2.6 eV. Both the Tempo cation and the BQ anion have localized charge, and similar size. Barring specific solvent effects (e.g., solvent coordination), we expect from the Born model (Equation 61, which treats the solvent as a dielectric continuum, that the solvation energies will be comparable:

/

L

1.1.7 Innuence of Coupled Chemical Reactions on the Reduction Potential

h

It is common for a chemical reaction, such as a protooation, to be part of the overall reduction process. For instance, in aqueous solution. p-benzoquinone is reduced by a two-electronltwo-proton (?e-/2H +) process over a wide range of pH values. The reduction potential is:

ip

Another example is ascorbic acid (H,A). s1uch aIsu undergoes a 2e-/2H process to dehydroascorbatz (Dl:

;

H2A-2H+ - 2 e - d l 3

In this case a formal potential can be written which also takes into account ,. .

....-I ----A

CYCLIC: VOLTAMMETRY

12

the presence in en01 groups of dissociable proton^,'^ with pK,, = 4.10 and pK,, = 11.79. The resulting expression For the formal potential is:

where

F,,, = m2A]

+ [HA-] + [A2-]

1.2 Electrode Kinetics Electron transfer reactions at the electrode may not be rapid enough to maintain equilibrium concentrations of the redox couple species near the electrode surface. It is therefore necessary to consider the kinetics of the electron transfer process. The rate equation for heterogeneous electron transfer (Equation 8) expresses the flux of electrons at the electrode surface (Figure 1-9):

where A is the area of the electrode (cm). With a centimeter-basedsystem, the heterogeneous rate constant is expressed in centimeters per sewnd, and concentrations are given in moles per inilliliter (M x 10-9).

1.2.1 Transition State Theory for Electron Transfer

THE REDUCTION POTENTIAL

AND ELECTRODE K N T I C S

13

the electrode. We have seen that the free energy of the electron is equivalent to the electrode potential. Thus the reactant side of the transition state diagram (TSD) reflects the potential of the electrode. The product side of the TSD is the free energy of the reduced species in solution. The transition state for electron transfer must depend in some way on the physical and chemical changes that occur as a consequence of the electron transfer. We have already seen that solvation energy changes are a significant part of the process of electron transfer. The transition state is a nonequilibriurn solvated state somewhere between the oxidized and reduced forms. Often chemical or structural changes in the molecule itself occur as a consequence of electron transfer. In such cases, we might expect these changes to be reflected to some degree in the transition state as well. Transition state diagrams are sketched in Figure 1-10 for different electrode potentials. The effeqt of changing the potential is reflected partially in the energy ofthe transition state. The curves in the TSD series of Figure 1-10are related to the free energy relationships that have often been observed by physical organic ; chemists for a homologous series of chemical reactions (Figure 1-11). Such homologous reaction series often follow linear free energy relationships (LFER), wherein in the following type of relationship ktween rate constants ' and equilibrium constants is observed

lnIIhlkr~rcnl

(9a)

a 1nCktq/keq(rtfJ

where the rate and equilibrium constants of a series of homologous reactions are I compared to a reference reaction of the series. From the constants k , = Ilk, ! and k,/h= k , it can be easily shown that fi = (I - a).

The main task of electrochemical rate theory is to promote an understanhug of the physical basis of the heterogeneous rate constants k, and k Heterogeneous electron transfer can be envisioned with the aid of transition state diagrams. First consider the forward reaction, the transfer of an electron from the electrode to the oxidized species in solution adjacent to the electrode. A transition state diagram will in the first place reflect the reactant and the product free energy. The reactants are the oxidized species in solution and the electron in

,.

Ox

T

I cm

1

Red

Figure 1-9 The flux of electrons at an electrode surface is dacribed as moles of electrons per second through a surface area of one square centimeter.

e

0,

+ Ox

Red

Figure 1-10 Transition state diagrams for reduction at difirent electrode potentials -L-A.

.

CYCLIC VOLTAMMETRY

14

THE REDUCTION POTENTIAL AND ELECTRODE KINETICS

The fundamental equation of transition state theory is:

Together with the ratelequilibrium relationships this leads to linear free energy relationships for the forward and reverse reactions:

In an LFER, the transition state is interpreted as resembling to some degree the reactants and products. The degree of resemblance is given by the coefficients a and p. The transition state resembles to s o m e degree both the reactants and the

products. The degree of resemblance is given by the coefficient in the LFER (Equation 9). Changes in the free energy of the reactants or the products are reflected as fractional changes in the transition state energy. In the series of

organic reactions, the free energy is changed by substituent effects.

1.2.2

The Butler-Volmer Equation

Figure 1-1 I Linear free energy relationship obtain4 for the addition of semicarbazide to subs tiluted benzaldehydes. The slope, analogous to thc transfer coeficient in electrwhemistry, is 0.47. (Adapted from Reference 11.)

The situation in electrochemistry is unique. The free energy of the reactants is changed without any change in the overall reaction (without any chemical change in the reactants or products). It is not surprising that electrochemical kinetics have long been descrikd by an LFER that is called the Butler- Volmer equation. If a reference rate is chosen as the rate when E = E", the forward and reverse rates are equal (k, = A, = A,), and we have: AGf,,,,,,

= AGL,

A G,, ~,,,, = AG;,,

+ a F [ E - EO] -

[I

- u]F[E E q -

Sometimes u is referred to as the cathodic or forward transfer coefficient and fi = (1 - u) as the anodic or reverse transfer coeficient. From transition state theory the rate is exponentially related to the free' energy of activation, so that

whch is the Butler-Volmer equation. The standard rate constant k , is a measure of the intrinsic energy harrier. The transfer coefficient a is generally believed to reflect the nature of the transition state, in the same sense that is used in physical organic chemistry. Small values (or > 0.5) are indicative of a productlike transition state, and large values (a > 0.5) are indicative of a reactant-like transition state. Figure 1-12 shows several eIectron transfer reactions with corresponding d u e s of a. Anthracene12 has a = 0.55: the barrier for electron transfer consists largely of solvation reorganization energy, and the rate of

~6 Figure 1-12

+iNO,

a = 0.5

Electron transfer reactions and transkr coefficients.

THE REDUCTION POTENTIAL AND ELECTRODE KINETICS

CYCLIC VOLTAMMETRY

16

electron transfer is fast. Both tetraalkyltjn oxidatiot~@ = C).29)13 and alkyl halide reduction (a = 0.3)14 occur with concurretlt s t r u ~ t ~ r change al (from tetrahedral to pyramidal for the alkyltins and bond breaking for the alkyl halides); these structural changes contribute to the barrier far electron transfer. Cyclooctaletraene (a = 0.4) goes from a tub shape to n partially Battened state upon redu~tion.'~.'~ An interesting pharmaceutical, Arternisinin, has a value of a = 0.35: presumably the oxygen-oxygen bond in this endoperoxide is either lengthened or broken upon electron transfer. Methylcobalarnin has a large transfer coefficient (a= 0.78). The reason for this is not as easily explained, but it can be noted that the electron transfer and following chemical reaction occur in two separate activatsd steps. It hns been suggested that in this case the transfer coefficient will k large.I4 Whereas NO: has a relatively facile reduction, with a = 0.5,'~ calcuIations based on whent structural changes indicate that the electron transfer should be rather slow. This is indicative of a more complicated, "inner sphere'' mechanism, in which the donor (the electrode) and acceptor (NO:) are coordinated in such a way as to reduce the barrier to electron transfer.

1.2.3 Outer and Inner Sphere Interactions Electron transfer. heterogeneous or homogeneous, can be classified as outer sphere or inner sphere, according to the extent of interaction between the electron donor and the electron acceptor (Figure 1-1 3). Electron transfer reactions that take place with weak interactions between reactants are outer sphere and those with strong interactions are inner sphere. Outer sphere reactions are characterized by the relative absence of steric effects on the rate of electron transfer, while inner sphere reactions are very senslt~veto

17

ic effects, which prevent the close encounter needed for full interaction. This observation has been suggested as the major experimental criterion for distinguishing inner from outer sphere electron transfer.

'

2.4 Marcus Theory '

',

The theory that has found the most application for characterizing the nature of outer sphere electron transfers is that developed by R. Marcus, who was awarded the Nobel Prize in 1992 for these contributions. As it is employed in t stuhes, the Marcus theory provides a method by which one can relate the rate of electron transfer to solvation reorganization changes, structural changes, and the overall free energy change. Focusing on the electron transfer act, theory for electron transfer presumes a preequiiibrium factor far complzn formation, and the following electron

D

-

+ A-[D,Aj

Kc,

ID, A1 CD+, A-1 ke, It is the rate of the electron transfer step, kc,, that is discussed below. The "Marcusian" transition state diagram presents a reaction coordinate that is composed of solvent reorganization as one moves from reactant to product. Transition state diagrams of this sort can be applied to electrode reactions or homogeneous electron transfer (Figure 1-14). The free energy along the , coordinate varies as a square of the deviation from the equilibrium position. The ',physicaImeaning is that small changes in the solvent medium around the ion

T 1 G

(D

Figure 1-13 Schematic representation of outer and inner sphere elcciron transfer

reactions.

1 . A1

D+ , A-1

Figure 1-14 Transition state diagram for electron transfer acoording to Marcus theory. .

.

.

-

.

A

CYCLIC VOLTAMMETRY

18

can give rise to large changes in the electronic energy of the molecule. The intersection of the curves gives the transition state energy. When the driving force AG = 0, the intersection of the two curves occurs at an energy of A G= ~ 144. This is called the intrinsic free energy barrier, where I is related lo the optical and static dielectric constants.

-

THE REDUCTION POTENTlA L AND ELECTRODE KINETICS

19

activation energy increases with driving force, is dificult to observe with electrochemical experiments* but has been observed in other contexts." The theory is similar for homogeneous electron transfer, except that because ions involved, the value of , i is expected to be larger. Also, the work rent as a result of the differencein work of bringing reactants and products together in solution compared to bringing a single ion to electrode. A term is sometimes included in reorganization energy to accommodate changes in vibrational energies for the molecule/ion transition.

In polar solvents E, << c,,,,, and the optical dielectric term dominates. The general expression for the activation energy is given by

where w is the work required to bring the Ox and Red t o the electrode. Often the work term is neglected. Figure 1-15 shows the variation of the activation energy with driving force for a typical value of I, 1.00eV. Inspection of Equation (12) (after expansion) shows that the free energy relationship, neglecting the quadratic term. i s equivalent to the Butler-Volmer equation, with the transfer coefficient set to 0.5. This i s shown in Figure 1-1 5 to be true for smaIl deviations from the reduction potential. We can also note that for the highly endoergic or exoergic regions, the transfer coefficient (the change in activation energy per change in driving force) is given by:

Several cyclic voltammetric studies have shown the expected change in the transfer coefficient with ~otentia1.l~ The highly exoergic region, where the

0

where f is the force constant and Sd is a bond length change. These are sometimes called "inner sphere" contributions to the reorganization energy: they are not to be confused with an "inner sphert" interactions, as described

1.2.5 Kochi's Free Energy Relationship for Electron Transfer portance of inner sphere interactions to electron transfer was first recognized by H . Taube, who received the Nobel Prize for work in this area. The compIexity of the problem, however, has prevented the development of a quantitative theory comparable to Marcus theory for outer sphere reactions. In practical terms, inner sphere interactions generally lower the activation barrier for electron transfer compared to outer sphere electron transfers. As a con. sequence, they may play important roles in electron transfer in biochsmistry and atalysis of electrode reactions. A characteristic inner sphere reaction ted by Kochi is the electron transfer of iridium hexachloridz with 1tin.l3 A five-coordinate transition state around the tin is indicated, bherein the positive attraction in the ion pair stabilizes the transition state ,(Figure 1-16). In the case of inner sphere electron transfer, steric congestion prevents the close encounter needed to stabilize the TS, and lower rates of homogeneous electron transier are observed. In contrast to outer sphere <electrontransfer, there i s no obvious relationship between homogeneous and ,heterogeneousrates, as the interactions between D/A complexes in solution are generally divergent from electrode-molecule interactions. A new method to characterize inner sphere electron transfer .reactions has been developed by KochiL3 and designated FERET (free energy

HutlerYoirrler

+ Figure 1-15 Variation of activation energy with potential according to Marcus theory. The straight ltne corresponds to a linearized version of Marcus thcory, equivalent to the Butler-Volmer equation.

I/ - I f -

'I

+

p{- ir.] a;

Figure 1-16 Typiiral inner sphere complex. This is k + u x the continuum of energy levels in the electrhde (see rcfewn~u:13a).

THE REDUCTION POTENTIAL AND ELECTRODE W T [ C S

relationship for electron transfer). The sequence of events for electron transfer between a general donor and acceptor can be written:

[D ',A-]

+ products

(fast)

The products can be either separated D' and A - or a collapsed ion pair. For endergonic driving force,the observed rate constant is given by:

In the endergonic region, a cycle can relate the thertnodynamics and kinetics of the electron transfer process (Figure 1-17): Figure 1-18 Thermochmical cycle for optical electron transfer.

The work terns w, and w, are associated with bringmg the reactants or products together. For neutral reactants, w, can be n e g l ~ t e d . Another cycle involves spectroscvpic measurement oft he optical analogue of electron transfer, the charge transfer transition energy, hvc, (Figure 1-18). The work term for inner sphere interaction of the ion-pair can be evaluated as follows: [D,A]% [D', A-]* (15 ) w,*= hv, - IP, + EA, + w,

'.

If a common acceptor is chosen, and a reference donor, then a comparative work term for the ion-pair can be defined:

-

Aw,* w,*- w:'

= Ahv,

- AIP,

(16) Finally, utilizing a combination cycle for thermal electron transfer and a optical charge transfer (Figure 1-19), the relative free energy of activation for electron transfer can be exprsssed as follows:

\

/i

This equation, a generalized free energy relationship for electron transfer IFERET). has the advantage aleonsisting entirely of terms that can be evaluated from experiment. Ths approach has been utiIized for a comparative study of the inner sphere reactions of nitronium ions at an electrode and with aromatic substrates in solution, providing insight into the activation process.lg

c Figure 1-17 Therrnochemical cycle for thermal electron trader.

D+

A-

Is*

Figure 1-19 Overall thermochernical cycle that establishes a r e energy relationship for electron transfer.

CYCLIC VOLTAMMETRY

22

REDUCTION POTENTIAL AND ELECTRODE KINETICS

1.2.6 Comparing Homogeneous and Heterogeneous

Electron Transfer Rates The rates of heterogeneous and homogeneous reactions can be compared in the context of transition state theory:

where Z is a preexponential factor that includes a preequilibrium term for formation of the donor acceptor complex, and a frequency factor. For heterogeneous reactions Z,, is on the order of lo4-'cm/s, and for homogeneous reactions Z,, is on the order of 10" M-l s-'.''

1.2.7 T h e Double Layer and Electrode Kinetics In most electrochemical experiments, an inert salt is added (about 0.1-1 M) to increase the solution conductivity and to minimize mass transport by migration. The structure of the electrode-solution interface under such conditions has an effect on the kinetics of electrode processes. In general, an electrode has net charge that is related to its potential. Every electrode material has a potential of zero charge (pzc): the potentiai at which the net charge is zero. At potentials negative to the pzc, the electrode is negatively charged, and at potentials positive to the pzc, the electrode has a net positive charge. Consider a negatively charged electrode in contact with an electrolyte solution: positive ions will be attracted to the electrode and negative ions repelled. Some ions may be immobilized at the surface. Themal motion is balanoed by electrostatic forces so that there will be an excess of positive ions near the electrode relative to the bulk solution (Figure 1-20, top). The details of the structure will depend on solvent properties (dielectric constant) and electrolyte charge and concentration. This is called the Gouy-Chapman-Stern model for the double layer (charged electrode and charged solution) at the electrode-solution interface. As a consequence of the structure of the double layer, the potential changes decays in a gradual manner as distance from the electrode (Figure 1-20,bottom). The electrical field (V/cm) can be very large across the electrode-solution interface. The potential decay will be steeper as the concentration of electrolyte is large and as the polarization is increased (the deviation of the potential from the potential of zero charge). The effect of the double layer on electrode kinetics is illustrated in Figure 1-21. A charged particle approaching the electrode requires work to move through the potential drop to reach a point near enough to the electrode to transfer an electron. This work term either increases or decreases the activation energy compared to the ideal case, where the potential drop is infinitely steep and there is no work associated with movement of the charged particle. Values of k, and a or 0 determined without taking into account double-layer effects are termed "uncorrected values." The task of double-layer theory, insofar as

X

1-20 Sketch 01 the distribution of ions near a chargcd electrode ( :electrode layer) and the corresponding potcntial drop.

MmcefrPmelecMwke

-21 Origin of the double-layer correction. An ion must move across a potcn-

CYCLIC VOLTAMMETRY

24

electrode kinetics are concerned,is to determine the value of the potential drop from the OHP (closest approach) to the solution. The Butler-Volmer equation taking this correction (SE) into consideration is written:

THE REDUCTION POTENTIAL AND ELECTRODE KINETICS 13. Kochi,

25

J. K. Angew. Chem., 1.1. Ed. Engl. Im, 27, 1227.

13a. Marcus, Rudoiph A. An~ewandteChemie International Edition in English 1993, 32 1 1 11. 14. Saveant, J. hn. Single E/t~rrrunTrumfer a d Nut-ieophilic Subsrirwion. Advanoes in Organic Chemistry, Vol. 26. Academ~cPress: London, 1%. 15. Allendorfer, R. D.; Rieger.

P H.J. .4m.C b n SDC. 1965.87,2336.

16. Fry, A. J.; Britfon, W.E, EdF. Top~rsIn Orgnrriz E l e c t r ~ ~ b s m i s f1st r ~ ed.. , Plenum Eess: New

York, 1986.

For charged species, the concentration terms in the Butler-Volmer equation must also be modified as follows:

17. Lee, K. Y.; Kuchynka, D. J.; Krxlhi, J. K.Inorg. Chem. 1990,29,4196. 18. Chem Rev. 1992, 92 (3).

19. Kochi, J. K. Acc. Chem. R P S 1992, 25, 39.

20. Brunschwig, B. S.; Logan, J.; Newton, M. D.; Sutin, N.J . Am. C h ~ mSVC,1980,102, 5798 21. Bard, A. J., Faulkuer. L. R. Electmrkmical Methods. John Wiley & Sons: New York, 1980

The parameters necessary to make a correct model of the double layer are often not easily obtained. However, the uncorrected values of electrode kinetic parameters may retain significance under many circumstances. For instance, for a series of compounds studied under similar caudition~kOand a values can be compared. Double-layer effectsare also minimized by large electrolyte concentrations (0.1-5 M).The 6 E term is smallest near the potential af zero charge (pzc), and largest at extreme potential values, far from the pzc of the electrodes. A good overview of double-layer theory is given in Bard and Faulkner's text2' and by Rieger."

References I . Gurney, R. W . Ions in Solution. Dover: New York, 1962.

2. Pleskov, Y.V.. Gurrvich, Y . Y . Smicodtlctor Ph~toeIectrockmistry.Consultants Bureau: New York, 1986.

3. Antropov, L. I. Tkerw~~ical Electrochemistry. Mir: Moscow, 1972. 4. K o r y i ~3.: Dvorak, I. Principles o j Electrochemistry, 1st ed. John Wiiey & Sons: New York, 1987. 5. Goodisman, J . Eltctrochemistry: Themeticul FouPrdationq 1st e d John Wiley M

Sons: New

York, 19B7. 6. Kebarle, P.;Chowdhury, 5. C k m . Rev. 1987, 87,514. 7. Summerman, W.: Deffner, U. Tetrahedron 1975. 31. 593.

8. Bordwell, F. G.;Cheng, .I.-P.; Harrelson, J. A,, Jr. I . Am. Chkmm.Soc. 1988, 110, 1129. 9. Hupp, J. H.; Weaver, M. J. I A O T CJz~am. ~ 1984, 23, 3639. 10. Harris, D. Quantitative C h i c 0 1 Anl~igsis.2nd ed. W. H. Fmman: New York, 1987. 11. Lemer, J. E.; Grunwalk, E. Rates mi Equilihrio ojOrgotur Rcuctic~ns.Dover: New York, 1963.

12. Kojima, H.; Bard, A. J. J , Am. C h t . Soc. 1975, 97,6317.

22 Rieger, P. H- EIc~trochemislry.Rentice-Hall: Englewood Cliffs, NJ,1987.

CHAPTER

The Cyclic mmetric Experiment

In Chapter I we explored the fundamental relationship between the electrode ;potential and a redor couple in solution. It was also pointed out that if the ,( potential of an electrode is controlled externally, the solution can be made to ; "adjust" by electron transfer to approhch equilibrium with the electrode ;'potential. In many electrochemical experiments, the solution initially has only $:oneform of a redox couple present, and the electrode is initially set at a potential ?: .:such that this form does not undergo electron transfer. This ensures that the jpperiment begins at zero farddaic current. The electrode potential is then
THE CYCLIC VOLTAMMETRIC EXPERIMENT

CYCLIC VOLTAMMETRY

a.

Time

b. W

C.

Time

d.

Figure 2-2

CV waveforms,

Inexpensive analog instruments as well as microprocessor-controlled units are commercially available. The equipment can include useful features such as IR compensation, data smoothing, background subtraction, and automated display capabilities. Commercial instruments offer scan rates of up to several hundred volts per second. However, units that have very high scan rate capability (to 106V/s) can be constructed at relatively low cost. The clectrochemical reaction of interest takes place at the working electrode (WE). Electrical current at the WE due to electron transfer is termed faraduic. current. An auxiliary, or "counter" electrode (AE) is driven by the potentiostntic circuit to balance the faradaic process at the WE with an electron t r ~ n u f e rof opposite direction (e.gif reduction takes place at the WE, oxidation takes plam 'at the AE). The process at the A E is typically not of interest, and in most -experiments the small currents observed mean that the electrolytic products at the AE have no influence on the processes at the WE. The faradaic current at the WE is transduced to a potential output at a selected sensitivity, expressed in amperes per volt, and recorded in a digital or alog form. The CV response is plotted as current versus potential. Figure 2-3a ows the shape of a C V current response for a typical reduction process. uring the forward sweep the-oxidized form is r e d u ~ d while , on the reverse weep the reduced form near the electrode is reoxidized. Chemical reaction ,

'

2.1 An Overview A potentiostat system sets the control parameters of the experiment. Its purpose is to jmposs on an electrode (the working electrode) a cyclic linear potentid rwerp and to output the raulting mrrent-potential curve. Tbs sweep is described in general by its initial (Ei), switching (E,), final (Ed potentials, sweep (or wan) rate (r, in V/s). The potential as a function of time is:

E = E~ + v t (forward sweep) E = Es - v t (reverse sweep) sweeps are possible, such as the option of a second More pontential. ~ ~ l t icycles ~ l care sometimes used, but in many instances these linear sweep voitammetry not be more idormative than a single cycle. The ( L S ~ is) ulcd for a h d f ~ d CV. e Figure 2-2 illustrates various possible .

.

.

.-

I

n -

.

.A,.-.

_.d_A...

-2

.

. .,

.

. --,.l. .. " -

. .

Y-YY

. . ..

. .

,

.

CYCLIC VOLTAMMETRY

TIIE CYCLIC VOLTAMMETRIC EXPERIMENT

Audllary

Elccbodc

worhg E l m m d e . RHRmce Elmuode

Figure 24 (a) Typical electrochemical cell. (b) Lugin capillary reference.

-1 -1.3

-1.4

-1.5

-1.8

PotentinI/V

Figure 2-3 CVs for (a) ferricyanide reduction and (b) methylcobalamin reduction.

coupled to the electrode reaction can drastically affect the shape of the CV response. Compare the current response just described with that of the reduction of methylcobalamin (Figure 2-3b}, where no reverse peak is observed. The absence of the reverse peak indicates that a following chemical reaction has removed the reduced species.

contains solvent. electrolyte, one or more principal elcctroactive species, and possibly added reagents that will undergo reactions with the electrolytic .products. Before the experiment, it is necessary to remove dissolved oxygen, which has a cathohc sip~ialtllat can interfere with observed current response, !Thisis normally done by purging the solution with an inert gas such as N, or Ar. >Removal of dssolved oxygen CWI also be effected with freeze-pump-thaw acuum line. Electrochemical experiments cat1 be performed in glove oxes, or in vacuum for electroactive species, that are sensitive to air

,

2.2 The Electrochemical Cell 2.2.1 Cell Design A typical cell design for a cyclic voltammetric experiment is shown in Figure 24a. The simplest approach is merely to have the three electrodes immersed in the solution in close proximity. A Luggin capillary (Figure 2-4b) further isolates the reference solution from the cell solution. At the outset of the experiment the d l

2.2.2 Solvents

The choice of solvent is determined by several factors, including conductance, .,solubilityof electrolyte and eleclroactive substance, and reactivity with electrolytic products. The solvent can also have important properties such as decreasing usually unwanted effects (e.g., adsorption of the electroactive species ode). Because of the importance of the solvent in electrochemicd is sometimes desirabIe to consider the physical and chemical e solvent in some detail. Solvent properties relevant to electroriments are listed in Table 2-1. The melting and boiling points 1 temperature range for most solvents (with some variation due P

THE CYCLIC VOLTAMMETRIC EXPERIMENT

Often, to minimize evaporation of the cell solvent (and consequent changes in concentrations), the purge gas (N,,, Ar) is passed through the same solvent used in the electrochemical cell. This step is particularly necessaxy for solvents with low vapor pressures. Mention should also be made of the increasing use of unconventional media electrochemical experiments. Experiments in nlicellar solutions and microemulsions, for instance, can solubilize or concentrate reactants in rni~elles.~ Cyclic voItammetric results have been obtained below the freezing point of the solvent: for instance, in frozen DMSO4hS and in perchloric

Table 2-1 Properties at Common Electrochemical Solvents Solvent ( m ~bp, , "c)

Dielmtric conslant

p p

H,0(071w Propylene carbonate

E: (Solvatochromatic) -

64.96

1.O 0.491

46.95

0.444

78.3

Acceptor number

Donor number

--54.8

(- 54.5,24 1.7)

Dimethyl sulfoxide

f 9.3

0.77

33

(18.5,189)

Dimethyl-

2.2.3 Electrolytes

lormamide ( - 60.4,153)

Acetonitrile

35.94

0.460

0.36

18.9

34.78

0.3?4

0.21

14.8

32.66

0.790

24.55

0.654

20.56

0.355

(-43.8,81)

Nitrobenzene

,

(5.8,210)

Methanol (-97.7,64.5) Ethanol

( - 114.5,78.3) Acetone ( - 94.7-56.1)

: i

0.44

37.9 12.5

'

'.

-

The electrolyte, added to enhance conductivity and ti-] minimize double-layer and migration current effects, is chosen on the basis of solubility in a gven solvent as well as inertness toward the eIectroactive substance and its electrolyproducts. There are of course many choices of electrolyte for use in aqueous solution. The tetraalkylammonium salts are the most commonly used noneous electrolytes. TetrabutyIammonium tetrafluoroborate (T3ATFB) and tetrabutyiammoninm hexafluorophosphate (TBAHFP) are recommended by Fry and Britton, who note that TBAHFPin acetonitrile has a particularly large useful potential hnge of 3.4 to - 2 9 V (vs. SCE).

+

Swrrrc: Reference 1.

Electrodes Working Electrode [WE) .2.4

to colligative effects).The dielectric constant is primarily an indicator of solvent polarity and solubilizing power. Other solvent properties, such as acceptor and donor numbers, indicate the ability of the solvent to participate in electron-pair donor-acceptor interactions. The spectral charge transfer energy (solvochromatic shift) of pyridinium-N-phenoxide betaine ,dye has also been used to characterize solvent^.^ Solvent mixtures sometimes are used to average properties. An example is the nonaqueous electrochemistry of rnethylcobalamin where mixtures of methanol and DMF achieve a balance ketwsen solubility and conductivity. Low temperature studies can be very useful in electrochemical investigations. Reactive species can be stabilized and reversible electrochemistry obtained. Solvent properties can change drastjcally with temperature. For example, DMF dimerizes at below -40°C and has a large potential window as well as a lower dielectric constant. In Fry and Britton's handy review of solvenrs and electrolytes? acetonitrile, ethanol, methanol, and methylene chloride are recommended as good oxidative (anodic) electrochemical solvents, while acetonitrile, DMF,and dimethyl sulfoxide (DMSO) are suggested for reductive (cathodic) electrochemistry. Acetonitrile is suggested as the best overall nonaqueous solvent on the basis of its electrochemical properties and its relative nontoxicity. The review of Fry and Bntton also is a good place to start when looking for purification methods.

disk electrodes are most commonly employed in CV experiments (Figure 2-5). Platinum, glassy carbon, gold, silver, or amalgams* are often used. The use :of carbon electrodes of various types has been reviewed.? The mercury drop electrode is useful for electrochemistry in aqueous solution at large negative

Disk working zlectrode.

Figure 2-5

Recip lor s silver amalgam electrode: polish the Ag wire electrode (ca. 1 mm diameter) with h e alumonurn oxide powder. Reduce at - I.OV(vs. SCE) for 3 minutes ih s el1 containing nitric acid (pH1.5) and a drop of mercury. While still holding the potential, touch the eledrode surfaw lo !he Hg drop. Rime xveral times with deionid water; immerse in deionized water for 50 miuutes. Polish with aluminum oxide p w d e r and rinse again. The electrode can be usedbr months. like an?.

r

Y

1

Y

.

U

"

. .^.,

,:..,

.

. . ..

.

.--

THE CYCLIC VOLTAIv'IMETRIC EXPERIMENT

potentials because of the sluggishness of the hydrogen reduction compared to other eIectrodes. However, for nonaqueous work and for electrochemistry at positive potentials, the solid electrodes are often preferred. The WE should also have a facile electron transfer with the electroactjve species. The factors that facilitate heterogeneous transfer are not always well understood, and careful pretreatment by polishing sonicating. or holding or cycling of the WE potential of electrodes can markedly improve performance. Electrode fouling occurs quite easily, and frequent polishing is sometimes necessary. Deleterious effects such as I R drop and capacitive charging time are greatly reduced as the electrode radius is made smaller. For cyclic voltammetry, the result is that much higher sweep rates can be achieved with ultramicroelectrodes (radius < 100pm).Most preliminary studies at moderate scan rates utilize electrodes having a radius of approximately 0.2 crn.

Reference Electrode IRE) The most common reference elwtrode systems used in aqueous solutions are Ag/AgCl and the calomel electrode. If aqueous-based references are used in nonaqueous solution, however, large liquid junction is produced; and often more serious, aqueous contamination of the nonaqueous cell occurs. Thus this combination is not recommended. The use of an AgJAg' non-aqueous-based reference is suggested for nonaqueous electrochemistry. To avoid large junction potentials the RE solvent should be as close in nature as possible to the cell solvent system. Often potentials are calibrated with a standard, such as ierracene or cobaltocene. Suggested standards are listed in Table 2-2, along with reduction potentials and other properties. Construction of an Ag/Agf reference for nonaqueous use is shown in Figure 2-6. Reference electrodes can drift with time and must be carefully maintained.

Table 2-2 Reference Conlpounds for Cyclic Voltammetry ---Substan&

--

FdCNjG -

"-

RwH,)f

+f2

E"'(YI 0.253vs. AgJAgCI, 1 M KC1 0.17 vs. SCE

+' -

Ferroaene 0.37vs. SCE F~(CP),/F~(CP);* Ternpo,Tempo' ** 0.271 vs. Ado. 1 M AeNO,/acetonitrile

--

D x 10'(cm2/s) Conditionsb - _ ------ --D,, = 0.76 pH 3.0 D,, = 0.63 0.1 M KCI a,, = 0.55 pH 7.0 Phosvhate buffer 0.6M TEAP in D,,, = 2.0 D,,, = 0.77

2.2.5

-

Glass frlt or vycor plug

Figure 26

Nonaqueous reference electrode.

Potential Window and Background Subtraction

f a solvent-electrolyte-WE system have a n can be collect& the region, solvent or electrolyte electrolysis reaches a level that obscures the signal. The potential window is greatly affected lyres. Distillatiotl aid drying can extend the it is wise to collect "background" spectralvent and electrofyte present. The potential window of the solvent -electroly te system can be established, and any obvious impurities can be identified before they cause havoc in the interpretation of results with the electroactjve species present. l o quantitative studies, the background can be subtracted from equivalent experiments with the electroactive substanor present. A particularly effective met hod proposed by Wightman and Wipf utilizes a flow injection cell to obtain background and experimental

ance in the 0.5-1DmM mprove detection limits ignal averaging.' Such ts dramatically, down to the nanomoiar

acetonltrile cl l MTBAP in acetoni[nle

.3 Electrochemical Mechanisms: E & C Notation A

Cp, cyclopentadienyl. TEAP, tetraethylammonium perchlorate: TBAP, telrahulylarnmonrum perchloralc.

Swrce:

* Baur. J. E. and Wightman. R. M.,J . ElectroaneL Chem.. 1991- 3W. 75** Summerman, W.and M n e r . U., T&rah~drofl.1975, J I . 593

this section we examine the CV response for the simplest electrochemical r i k d by the E&C notation: E represents , while C refers to a solution chemical

i

CYCLIC VOLTAMMETRY

36

s Red

Red -r Products

37

f i e voltage sweep and current response for this experiment a n shown in - - -Figure 2-7. The: data can be represented as a current-time curve, or, since the . - . potentld is linearly related to time for each half-cycle, as a current-potential curve (the usual representation). Note that reduction current is laken as positive. 1 . .- eep goes from left to right. For any CV, the direction of the initial ep snould be indicated or at least apparent (the initial sweep potential is always set, if possible, at a potential where zero faradaic current *rllrch

reaction that is coupled to an electrode reaction. Fur instance, the EC mechanism refers to an electrode reaction followed by a chemical reaction. Ox + e -

THE CYCLIC VOLTAMMETRIC EXPERWENT

-

ED',kO,rw

-

kchem

Subscripts are used to provide additional information. The subscript "T stands for reversible (meaning that both forward aud reverse processes are fast enough to maintain equilibrium or "Nernstian" conditions a t the surface), and "i" represents irreversible (only the forward reaction is significantk "iw and "r" are limiting cases of "q," or quasi-reversible (meaning that both the forward and reverse processes take place hut are not fast enough to be considered at equilibrium). Thus, in an E,Cimechanism the electrode reaction is fast and reversible and the chemical reaction is irreversible. After "ECE" the E&C nomenclature breaks down: ECE can mean several different things (and one cannot be sure what an "ECECCE" mechanism reiers to !). At this point the mechanism needs to be further defined- for instance, by a reaction sequence or schematic drawing that indicates all possible chemical and electrochemical events considered. We note that every reductive mechanism has an oxidative analogue. For instance, the oxidative EC is:

..

I

I

4

! s.!c

0

Time

Thus, all the theoretical results quoted for a reductive mechanism can be immediately applied to the analogous oxidative process. The only dlaerence is in the sign of the current and the direction of the potential sweep.

2.3.1 The "E," Mechanism The simplest possible mechanism is the one-electron oxidation or reduction of a solution chemical species at the WE. Consider a typical experiment in which 1 m M of an electroactive species in an electrolyte solution undergoes a facile, reversible one-electron reduction with the4following parameters:

1

I

Experhenfa1 Settings E, = 0.0,E, = -0.5, E, = 0.0, V

I =

~V/S

Area of electrode A = 0.01 cm2 Cell temperature T = 298 K

1 1

Ox

+ e-

= Red,

Eo' = -0.25 V (i.e.. vs. a reference)

C,, = 1.00mM, Cred= 0.0111M

I 1

t

Potentla1

(\I)

Figure 2 7.. (a) Potential waveform. (b) Current-time

and icl

rnlrr~nt-nn*--*:-l

38

CYCLIC VOLTAMMETRY

THE CYCLIC VOLTAMMETRIC EXPERIMENT

Consider what happens near the working electrode in the foregoing example (refer to Figure 2-8, which shows selected points along the CV with the corresponding concentration profiles of Ox and Red). As the potential is swept past the reduction potential, the oxidized form is converted to the reduced form in proportions consistent with the Nernst equation. Because the potential sweep is fast (relative to the rate of diffusion of material to the electrode), the oxidized species is depleted near the electrode. As a consequence of this depletion, cyclic voltammograms have a peak shape, in contrast to the familiar polarographic sigmoidal wave. Sigmoidal CVs can be indicative of a catalytic regeneration of reactant near the electrode. The peak resulting from the reduction process is called the cathodic peak current; by convention, it is positive, and it occurs 28.5 mV negative to the EO(at 25OC). At the switching potential, the direction of the potential sweep is reversed. A negative anodic peak current is observed as the potential is swept past the Eo (the electron transfer is in the opposite direction). The anodic peak occurs about 28.5 mV positive of the Eo (the exact value depends very slightly on the switching potential). Note that the major perturbation on the initial concentrations of Ox and Red lies within the mean square diffusion length X, = (2Q x time)''', which is about 4Spm for this experiment-large with respect to the double-layer thickness.

Figure 2-8 The current response and corresponding concentration profiles (A-F) for a reversible CV. Concentrations of the oxidized (solid circles) and reduced form (open circles) are shown as a function of distance from the electrode.

Distance/pm Figure 2 8 (Contimud)

THE CYCLIC VOLTAMMETRIC EXPERIMENT

43

Criterionfor the E, Mechum'sm Several criteria can be util~zedto confirm a single, reversible, electron transfer. 1. The difference in cathodic [E,,3 and anodic {E,,,)peak potentials is around 57-60mV (depending on the switching potential): AE,

= absLE,,

- E,,,] w 58mV

In actual experiments the expected 58mV is rarely observed because of small, distortions due to solution resistance effects and electronic or mathematical "smoothing" of data. The result i s that AE, is often 60-70mV for reversible electron transfer. For reversible multielectron transfer, the peak separation is 60/n&1~. 2. The hfference between the initial sweep peak and half-peak potentialsbf the forward sweep is 56 mV/n. 3. The shifted ratio of the cathodic to anodic currents is unity: i,,/i:+, = 1. In the "shifted ratio," the anodic peak current is measured from a baseline that is moved to a value that can be predicted from the decaying portion of the cathodic peak. In this part of the cyclic voltammogram the current can be predicted by assuming an inverse square root dependence of the current of the time. The baseline is assumed to be the current that would be obtained if the forward sweep were continued for the same amount of time that it takes to reach the reverse peak. This procedure is illustrated in Figure 2-10. The absolute ratio (with zero current as a baseline) depends on switching potential and can be simulated (see Chapter 5). 4 . The forward scan peak current should be proportional to the square root of the scan rate. This criterion is used to distinguish "diffusion-controlled" processes from processes featuring the adsorption of the dectroactive s p i e s onto the electrode (in which case a h e a r current-scan rate relationship is observed). A plot of the Iog i, versus log v is linear, with a slope of 0.5 for a diffusion peak and a slope of 1 f o an ~ adsorption peak. Intermediate values of the slope are sometimes observed, suggesting a "mixed" diffusion-adsorption peak. Adsorption mechanisms are discussed in more detail in Chapter 3. > \.

Figure 2-9 Three half-cycles. Note the slight diminution in the second cathodic p a k

current.

The initial sweep peak current for a reversible electron transfer is:

At 25°C this is: i, = 2.686 x 105 n 3

/ 2 ~ ~ ~ ~ 1 / 2 ~ " 2 A

Y

The reverse peak current is a function of switching potential A reversible CV is a diffusion-controlled process; that is, the rate of electron transfer is controlled by the rate of supply of material to the electrode by di8usion. Multiple sweeps give similar results (Figure 2-9 shows a CV with three halfcycles), with slightly smaller currents (as the initial condition of [Ox] and [Red] near the electrode is nearly but not quite the same for each successive cycle) While for simple one-electron transfer no information is to be gained by scanning more than a single cyde, if chemical reactions are coupled to the be informative. . . electron transfer, more than a complete cycle may The peak potential (for the forward seeep) is*:

-

.+

* a1'..'.-

,

The Reduction Potential and Number of EIeckons ,I. The reduction potential is given by: ~ 0= '

2

--

specks are 8qual. ,.--.,--

,,

& .

(4)

2 If up has been determined to be 60 mV, and the reduction has been shown to be diffusion controlled, then the number of electrons n = 1, and the diffusion coefficient can be calculated from Equation (1). The geometric electrode area can be found from Equation (I), or the effective area can be calibrated with a standard with known D and n (e.g, ferrocene). Even if the diffusion coefficient can be only roughly estimated, a reliable value of n can be found. This i s

-

Assuming that the difhsion macients of the oxidized d reduced dausion d & t s give the exact result.

ED.,+ %,a

. "

..

.L....-LA

,

.,

THE CYCLIC VOLTAMMETRIC EXPERZMENT

CYCLIC VOLTAMMETRY 12,

I

2.3.2 Electrochemical Reversibility

TT

I

45

Let us consider the electrochemical rate equation and its limiting case of very fast rates, the Nernst equation: I

-= k,C,

- krC,,,

kf = g -

[g(E

nFA

kr

-8L-l

i

-10

-0.1

0.0

P .a

, I

-0.Z

1

-0.4

-0.3

= erp

- P~)]

If the rate of electron transfer is slow with respect to the time scale of the experiment, then non-Nernstian concentrations will exist at the electrode . surface. The qualitative effect is to shift the peak wave to more negative potentials in the case of reduction and to more positive potentials in the case of an oxidation. The proximity to equilibrium is termed "reversibility." Increasing the scan rate is equivalent to increasing the rate of Musion of Red ay from the electrode (for a reductive, Ox + e - = Red, process). At faster scan rates, steeper concentration gradients are produoed, increasing the rate of diffusion. Red will diffuse away from the electrode faster as the scan rate is increased. The hffusion of Red away from the electrode is a process that competes with the oxidation of Red. Thus for the E mechanism, the greatest sibility is observed at slow scan rates. In this case reduction potentials are most accurately measured at slower scan rates, where reversibility is greatest (and distortions least). Three cyclic voltammograms with various degrees of .reversibility are shown in Figure 2-1 1. In the reversible region the CV response

i

-0.5

Potential (V)

,

I

-80 02

0.0

-0.2

-04

-0.6

-

-0.R

Potential/v

Figure 2-10 Characteristics of the Er mechanism: (a) peak potentials and currents and (b) measurement of the shifted peak current ratio.

because the current has a square root dependence on D but is proportional to n3I2. Conversely, if n is known, the Do, can be determined.

,

,

The determination of the number of electrons involved in an elect roc he mica^ mechanism is crucial but not always as straightforward as the discussion above might suggest. This is because the electron transfer is not reversible. The number of electrons can be determined utilizing potential step experiments (see Section 2.8). Bulk electrolysis (exhaustive electrolysis of all the electroactive species present) is sometimes helpful, but here w e must be careful in i n t e r p ~ tation because time scaies differ widely Thin-layer bulk electrolysis brio@ the more in line with the CV time scale.'' -1ysis A

,

---

-

-

0.0

-0.2

---0.4

-0.0

-0.8

Potential (V)

ure 2-11 Effect ofthe heterogeneous rate constants, with the transfer coefficients set 0.5: circles (reversible), dashed lines (P= 0.01cm/s), squares (kO=0.001cm/s), and ne (kO 0.0001cws).

-

--

.

.

L

-.

. ,-,

-.. -., ,

.

.,-

CYCLIC VOLTAMMETRY

46

THE C Y C I C VOLTMMETIUC EXPERIMENT lor--

is called Nernstian, because the forward and reverse electron transfers are fast and occur simultaneously. In the irreversible region, the ca~hodicwave and anodic peaks are well separated, and in each case electron transfer occurs in one direction. The intermediate region is "quasi-reversible." Qualitatively, the demarcation point separating quasi-reversible from irreversible has been reached when there is no (potential) overlap between the cathodic and anodic peak shapes. The following criteria have been suggested for evaluation of reversibility, with cm2/s and T = 298 K": D=

1

Reversible: k0 > 0.3 v l J 2 cm/s Quasi-reversible: kO> 2 x 10Totally irreversible: k0 < 2 x

cmjs v'I2cm/s

vli2

The experimentalist's attitude toward reversibility or lack of it is shaped by his or her goals. When one desires to obtain information about either reduction potentials or the rates and mechanisms of following chemical reactions, a greater degree of reversibility of electron transfer allows for the use of a wide range of scan rates, and interpretations are simpler. However, studies of the fundamental act of electron transfer are often accomplished with totally irreversible systems. In these cases, the transfer coefficient a and the forward rate of electron transfer are easily measured. The transfer coefficient, being in the exponent of the Butler-Volmer equatiotl, and reflecting the symmetry of the barrier to electron transfer, is simply given by:

Potential (Y)

Figure 2-1 2 Effec~of transfer coefficients for a heterogeneous rate constant k0=O.OM1 wilh n scan rate of 1V/s.

where E,,, = is the half-peak potential, and n, is the number of electrons in the rate-determining step. In most circumstances n, = 1. Note that the symmetry of the barrier is reflected in the "steepness" of the cathodic wave (Figure 2-L1).The anodic transfer coefficient fi is determined by the same equation applied to an irreversible anodic wave. For quasi-reversible systems, the reduction potential can be obtained by curve-fitting methods." If the peaks are symmetrically shaped, then n = 0.5, and EO * (E,., - E,,J/2.

2.3.3 The EC Mechanism Upon oxidation or reduction, an electroactive species may be activated toward bond cleavage, as in the case of rnethykobalamin reduction (Figure 2-13). The addition of an electron activates the molecule toward dissociation of a methyl radical. The rate constant of the following reaction is 600s-' at -30°C in a DMF/propanoI solvent n-lix~ure.The cathodic transfer coefficient a for methylcobalamin reduction is 0.78.At even lower temperatures, the anodic peak grows in as a result of the lowering of the dissociatiot~rate. Generally speaking, the EC mechanism is recogmized by a diminished reverse

+ (a) Methylcobalamin. fi) Mechanism of methylcobalamin reduc olution. I

L,,,-..

, ,A,

A , .

..

. , . .

,. .

. .....

-.

.,. .

I...,..

....

I-.&1

'.

CYCLIC VOLTAMMETRY

48

THE CYCLIC VOLTAMMETRIC EXFERMENT

peak or the lack of one. Both the chemical and electrode reactions may have different degrees of reversibility, which affect the waveshape in characteristic ways. However, at the outset, we will consider the case of a reversible electron transfer followed by an irreversible chemical reaction. In this (limiting) case, the CV peak potential will l x detennined by the E0 of the electroactive species and the rate of the following chemical reaction. Consider the case of a reductive EC mechanism. Let us keep the parameters that describe the experimental conditions and properties of the electroactive s p 5 e s the same as in the "E" example, adding only a following chemical reaction.

Experimental Settings E,

= 0.0,E,

= -0.5,E, = 0.0 v

Area of electrode A

=

1 V/S

= 0.01 cm2

Figure 2-14 CVs for E,Ci mechanism lor a series of increasing solution chemical rate constants a t constant scan rate. The same effect would be observed with a constant chemical rate constant and decreasing scan rate.

Cell temperature T = 298 K

Electroactiue Species Parameters Ox

+e

-

= Red, EO' = -0.25 V

2. The peak potential of the forward scin (when k,,,RT/nFv

> 4) is given by:

Red =. Products k,,,, Cox= 1.00rnM. C,,,= O.UmM

Diffiisional coefficients D,, = Dred= I x

lo-' cm2/s

The following chemical reaction removes the reduced form Cronl solution as it is produced near the electrode. The effect is to shiR the peak to more positive values* and to increase the peak current slightly (Figure 2- 14). Thew effects can be understood qualitatively o n the basis of the Nernst equation (renloving Red makes the reaction more favorable and increases the net rate of reduction). Increasing the scan rate has the effect of making the C V more "chemically reversible,'' until, at high scan rates, the chemical reaction is "frozen" on the time scale of the experiment and the CV is completely reversible. The main effects of the E,C, mechanism on the CV waveshape are summarized next.

Chmcteristics of ErCi Mechanism 1. The ratio of cathodic to anodic peak current is a function of chemical rate constant and scan rate. The cyclic voliarnmogram can be analyzed to give

an estimate of the chcrnical rate constant. A l %>mepoinl, however, the ele~lrodereaction will be pushd Into the "'
Chapter 6.

Several methods have been utilized to determine the rate of the following chemical reaction from a series of CVs at different scan rates. The simplest involves a comparison of i , , and i:,,. The cathodic peak current is measured from the zero current baseline, while the anodic current baseline is established by the current at whch the potential is switched. The experimental peak current ratios can then be compared to a previously calculated theoretical "working" curve to find the rate constant (for a first-order or pseudo-first-order reaction.' Parker has emphasized the use of working curves based on derivative cyclic voltammetry, which dscriminates to some d e g r ~against capacitive background current.14 Our preference in analyzing the EC mechanism is a simulation-based rnetllod in which the following chemical rate constant is determined by double potential step chronoamperometry. Subsequently, CV simulation analysis can be utilized to determine EOand a and k0 for the quasi-reversible case. Generally, we are not fortunate enough to be presented with the E,C, limiting case. A simplified two-hmensional "zone" diagram illustrates the situation (Figure 2-15). AIong one dimension are changes in the heterogeneous rate constant, and along the other dimension are changes in the homogeneous rate constant. Cyclic voltammograms for several limiting cases are shown. 1. Irreversible. The electrode kinetics are slow and the chemical kinetics me slow-the CV is irreversible. T!ae forward peak shifts - 59mV per tenfold change in scan rate.

CYCLIC VOLTAMMETRY

I

THE CYCLIC VOLTAMMETRIC EXPERIMENT

51

(through the Butler-Volmer equation) over the entire cyctic voItarnmogram. A more detailed discussion of this is deferred until Chapter 6. s:

!E;

2.3.4 Scan Rate and the Role of Diffusion We have seen that for the E, mechanism, the CV waveshape becomes more irreversible with higher scan rates. This happens because increasing the scan rate is equivalent to increasing the rate of diffusion of the reduced material from the electrode. As a consequenq the diffusion process comwtes with the back electron transfer. In fact we can write the E rnechnni~m mnrp 6Vllrl(l,,J o-n*+nIlx~ 'La -. .. "-.follows: -- - -

OX,,, f e-

g Homogeneous Rate

YY..~ ---.--

c , . , 7 - I C 7nnpdiaorfim forr: l. gul~ r-xd UIUP.

he shapes of vollamm~grams ---- EC - - mechanism. Typical --- - -

are given for limiting cases.

2. Reuersible. The electrode kinetics are fast and the chemical kinetics are slow. The peak potential is not a function of scan rate, and the CC appears as m the E, mechanism. 3. The electrode kinetics are fast and the chemical hnetics are fast. The peak potential shifts -30mV per tenfold change in scan rate. 4. Irreversible. The electrode kinetics are slow and the chemical kinetics are fast.

I

i

1

A more detailed analysis can locate intermediate regions. Reversibility ih the zone diagram has an inverse square root dependence on scan rate in the "heterogeneous" axis and an inverse scan rate dependence in the "homogeneous" axis. The former case obtains because increasing the scan rate is equivalent to increasing the rate of diffusion, while in the latter case increasing the scan rate gives less time for the chemical reaction to occur. Thus, all else being equal, one can move around in the zone diagram by changing the scan rate. Klinger and Koclu have introduced a quantitative measure of reversibility in electrochemistry as the deviation of the surface concentrations from the values that would exist under Nernstian conditions.15 Reversibility thus defined is a function of position along the cyclic voltammogram (i-e., potential). This is because the heterogeneous rates are modulated several orders of magnitude

I

1

Il.YI1

00

=k,kr Red,,,

Tlus is shown schematically in Figure 2-16. We have aIready examined in detail the orign of the forward and reverse electron transfer rate constants. In this scheme a diffusion rate constant k, is introduced. To understand the origin of k,, we take a closer look at diffusion. Cyclic voltammetric experiments are performed under conditions of so-called semi-infinite linear diffusion. This means that the electrode dimensions are larger than the thickness of the diffusion layer (Figure 2-1 7) and that there is a bulk solution far enough from the electrode that the concentrations of all species remain unchanged throughout the experiment. The symmetry of diffusion under these conditions means that we need consider only diffusion perpendicular to the electrode. No net diffusion takes place in the y or z directions, parallel to the electrode surface. Diffusion is described by Ficks' first law of diffusion (Equation 7), which states that the number of particIes diffusing through a cross-sectional area per

Figure 2-16 Schematic of the EC mechanism.

CYCLIC VOLTAMMETRY

THE CYCLIC VOLTAMMETRIC EXPERIMENT

53

dependence on time and on the di5usion coefficient. The average net displacement of a particle by diffusion can be shown to be: Equativn ( 8 ) can be used to estimate the diffusion layer thickness in a CV experirnenf. Let us now return to the original observation of this section. namely that diffusion rate is related to scan rate and competes with the reverse rate of elwirun transfer. The concentration grahents of the oxihzed species (at the cathodc peak) is shown in Fiyre 2-18. The surface flux of both species is indicated, and by conservation of material we have:

We can take the irreversible case to make an estimate of the diffusion rate constant k:. Tile peak current is given by:

The terms multiplying C,, can be taken as the diffusion rate constant, which has the same units as the heterogeneous rate constant kO(a= 0.5and T = 25°C).

Figure 2-17

v5'

Representations of (a) linear and (b) spherical difftsion.

unit time (the flux, 3) is proportional to the concentration difference across the constant, which describes the inherent selected area. The mobility of the particles. i s called the dltfusion coeficient (D) and is expressed in square centimeters per second.

For D = l O - k d / s , we can estimate limiting cases of rcversibility, as when one rate constant is two orders of magnitude larger than the other. In this way we arrive at the 'quasi-reversible" region ranging from 0.5 vlt2 (the reversible

'"

1.2

t

/

Gradient =

+

Taking the derivative of J with respect to distance indicates how the flux changes with distance (the difference between particles coming in and going out) and as a consequence describes the net accumulation of particles at a point (Ficks' second law). Fick's laws follow from the random nature of the motion of particles." For linear diffusion, the net displacement of particles by diffusionhas a square root

* See Bard and Faulknw (EIc.ctrockmica1 Methods) or Rieger (Ehcfrochemistry) For a more detailed look at diffusion.Rereren~es11 and 18. respzctivdy.

Figure 2-18 Concenlrarion gradient near the electrode. The tangent at a point determines the direction and rate or diffusion.

THE CYCLIC: VOLTAMMETRIC EXPElUMENT

CYCLIC VOLTAMMETRY

54

case) to 5 x loL5vl/' (the irreversjble case), in good agreement with the regions suggested earlier. 2.3.5

55

The three components of an electrochemical mechanism (disregarding adsorption) arc electrode reaction, diffusion,and chemical reaction. Thcse rates of these processes can k compared in a semiquantitative manner. As a consequence of the effect of scan rate on the C V result Tor both homogeneous and heterogeneous processes, theoretical presentations often employ dmensionless clusters. For instanm, for the E, mechanism, the factor:

Competition between Heterogenous and Homogenous Reactions

It is also useful to be able to compare heterogeneous nnd hamogeneous rate constants. Irreversibility in the electrode reactjon can be induced by the following chemical reaction by removing the reduced species at a rate faster than the reverse electron transfer. The heterogeneous rate constant can be converted to an equivalent homogeneous rate constant by introducing a third dimension, which converts the surface reaction rate [mol cmP2s-') to a volume reaction (mol L - l s- I). A reasonable choice is the diffusion layer thickness at one second (the time unit of the rate constant), shown in Figun 2-19. Thus, the eflective volume for the surface reaction is 1cm x 1cm x (2D x 1 s)'J2 crn. The heterogeneous rate can be converted to an equivalent homogeneous rate by multiplyin_e by (20)-'I2. A factor of 1OOO takes the result from cubic centimeters to liters. We have, finally.

controls the result, and for the E,C, mechanism the coritrolling factor is: kch,mRT Fv

As the mechanisms become more complicated, the number of dimensionless factors needed to characterize the CV response also increases. Whle this approach is powerful in showing the combined effectof all parameters at once, it can become extremely abstract.

2.3.6 Visualizing the Meaning of Reversibility with Cprof I

For D = 10-'cm3/s, kVbi = 2.2 x 105k,,,. Homogenecrul; reactions must be rathcc fast to compete with heterogeneous reactions. Even though cyclic voltammograms may appear to be irreversible (no reverse peak is apparent) the competition be~weenheterogeneous and homogeneous processes makes itself apparent in more subtle ways (see Chapter 6).

G: :

The program Cprof (provided on the diskette) simulates and graphs the concentration profiles near the electrode for a reductive EC mechanism.

Ox + e - = Red

E*', k" (n = 0.5)

Red * Product k,,,,,

In this way it is possible to v i s u ~ l i uthe effect of heterogeneous rate constants and rates of following chemical reactions on the concentrations of the oxidized i e s , the r e d u d species, and the product of the following chemical reaction. tE s p Cprof is for illustrative purposes only and has not been optimized. The scan rate should be set to at least 1volt/sec, and the rate constant of the following i chemical rertction can be practically set from 0 to 500s- It is sugsested that the reader simulate a reversible CV using the follo\l.jng parameters:

1

'.

Initial Potential = 0.0V

1 I'

Final Potential = - 0.3 v Scan rate = 1 V s - I

Figure t 1 9

Following this, simulations can be performed varying the k0 (making it smaller) and k,, (making it larger) to visualize the meaning of reversible, quasireversible, and irreversible.

Volume of a heterogeneous reaction. -..

.

CYCLIC VOLTAMMETRY

56

2.4 Distortions of the Faradaic Response There are several experimental realities that tend to distort the observed CV waveshape. Through a combination of experimental and theoretical methods it is often possible to alleviate these problems to the extent that a reliable analysis becomes possible. An electrical circuit model for the electrochemical a l l illustrates the origin of these problems (Figure 2-20). The electrode-solution interface is modeled as a capacitor in series with a resistance. The formation of the double layer as potential is changed gives rise to an analogue of an elechical capacitor. So far we have considered in detail only the "ideal" or faradaic current due to electrode processes. A complete model of the CV experiment must indude the current due to capacitive charging as the potential changes. A resistan= element models the resistance between the working electrode and the reference electrode. The potentiostat is controlling and monitoring the potential of the working electrode with respect to the reference electrode. To the extent that there is an IR drop across the solution, the potential of the WE will be misrepresented by the monitored potential.

THE CYCLIC VOLTAMMETRIC EXPERIMENT

57

to shift peak potentials in a negative direction. For negative (anodic) current, the

shift is in the positive direction. Thus for a reversible eIectron transfer, AE, > 58 mV in an experiment that has even a relatively small IR drop. Typical experiments involving 5 pA of current in nonaqueous solution with resistance of approximately 100052 can result in peak sepuatiou about 70mV. Use of the equation is a reasonable way to correct measured peak potentials in solution with moderate resistance problems. Note that the correction for IR drop tracks the current (the anodc and cathodic peak currents differ in sign and in magnitude). The effect of IR drop can be included in simulation treatments. It is best to make reasonable effort to minimize the solution IR drop by:

Optimizing the electrolyte concentration. Placing WE and RE close together. Using the feedback compensation method, available as a automated option in some commercial units. Using smaller electrodes.

--

2.4.2 Capacitive Current 2.4.1 TR Drop Theoretical treatments of cyclic voltammetry usually assume a cyclic linear potential sweep at the working electrode. However, the solution resistance causes a potential drop to exist across the working electrode and the reference electrode. Thus, the potential at the working electrode is really the applied potential plus the solution IR drop: E W E = E,,,

Anvnher djsrorfion of the CV from the ideal waveshape arises because the electrode-solution interface acts as a capacitor in series with the solution res~stance.The most familiar capacitive current occurs in a potential step (to a potential at which no faradaic current occurs). In this case, the current is given by:

+ IR,,l

(This equation is sometimes written with R, to represent "uncompensated" resistance, indicating the possibility of partla1 instmental compensation of the solution resistance). Consider the effect on a typical CV. For positive (cathodic) current, the actual WE potential is less negative than the applied [measured) potential, I b e eBen is

where the total capacitance is ,

Cdl= C/cm2 x area

and AE is the size of the potential step (V), t is time, and R is resistance. A plot of In(0 versus time has a slope of -(RC,,)-I and an intercept In(AE/R). In this way the capacitance and resistance can be measured (Figure 2-21). The C / ~ r n -(double ~ layer capacitance) for most electrodes falls in the region 10-20 pF/crn2. For a cyclic linear sweep, the situation is more complex. The double-layer capacitance is in eneral a function of potential. However, the general features can be described th constant capacitance. In t h s case, the capacitive current rises (starting at E,),with a characteristic time that levels off.Upon the reverse in scan direction, the capacitive current changes sign. The plateau current is equal to the,,i = Cd, x v (Figure 2-22), while the faradaic current is proportional to -thesquare root of the scan rate. Thus,the distorting effect of capacitive current is largest at higher scan rates (Figure 2-23). The ratio of faradaic to capacitive current is independent of electrode size.

ri

Figure 2 2 0

Equivalent circuit for an electrochentical cell.

THE CYCLIC VOLTAMMETRIC EXPERIMENT

Correcting for Capacitive Current There are two approaches to correcting capacitive current: background subtraction and theoretical treatment. In background subtraction, the current obtained from a CV collected in the absence of electroactive species is subtracted from current collected in the presence of electroactive species. T h s procedure assumes that the double layer remains the same in the two experiments, which is generally true because the double-layer properties are most strongly related to the excess electrolyte. - A background subtraction also eliminates the distortion due 'to residual faracaic current from electrolyte and solvent impurities. Theoretical treatment is more difficult, since capacitance is a undetermined (usually) function of potential. The capacitive current and I R drop are interrelated. Since capacitive current depends on sweep rate, and I R drop causes distortions in linearity of sweep, the capacitance will not follow exactly from theory of linear potential sweep. T h s effect can be incorporated into finite difference simulations. ,

4.01 0.0

'

0.1

1

'

0.2

I

I

0.3

0.4

0.5

0.6

0.7

Tirne/mSec

Figure 2-21 Procedure for measuring resistance and capacitana by a potential step experiment.

2.5

Figure 2-22 Capacitive response lor a cyclic sweep, assuming constant capacitance

,

Microelectrodes and Fast Scan Voltammetry

ElectrodEdiameter d is an important experimental variable. For a typical planar disk WE and reference probe, solution resistance is proportional to 1/d,I6 whereas the current i s proportional to the area, or d2. As a consequence, I R drop is proportional to d The capacitive rise time decreases as the electrode area is made smaller. (This property is of particular consequence in potential step 1 experiments.) With the appearance of microelectrodes and fast potentiostatic 1 circuits, the time scale of CV has k e n extended into the submicrosecond : range.'' As a consequence, faster following chemical reactions can be examined. Note that for fast scan experiments to give useful results, electron transfer must be rather facile. (See, e.g., Figure 2-24). If the heterogeneous rate constant is too

I

- 4 5 0 1 , 0.0

,

,

,

-0.1

-0.2

-0.3

, I -0.4

-,0.5

~ot~ntial/V

Figure 2-23 CVs with capacitive current, assuming typical values for electrode area and capacitance: solid curve, 100; open circles, 500; and solid circles, 850V/s.

E (V)

'Figure2 2 4 Example of last scan CV. The reduction of 9-bromoanthracene(IOmM) at a scan rate of 113,000V/~. (Adapted from Reference 18.)

I

CYCLIC VOLTAMMTRY

60

small, the peak currents can be shifted out of the bounds of the potential window at fast scan rates. In using microelectrodcff one must take into accvu~ltthe lower limit of scan rate for which linear dinusion predorninatzs. As the sca~rrate becomes lower, the predominant mode of diffusion changes over from linear to spherical (Figure 217b). The current response is ddferent for spherjcai symmetry, because the rate of diffusion increases. Microelectrode (slow scan) studies can take advantage of this to measure fast heterogeneous rate constants. The spherical diffusion-limiting current can be compared to the peak current in CV to obtain a lower limit of useful scan rates for studies in the linear diffusion regime. For a one-electron transfere we have:

THE CYCLIC VOLTAMMETRIC EXPERIMENT

61

able 2-3 Usable Scan Rates for EIeclmdes of Different Sizes (I mM solution)a

--

Diameter (crn)

Lower limit (V/s)

Upper limit (V/s) icaplirsr

0.1

0.005 0.001 0.0005 - .

0.02-0.2 8-SO 200-2,000

800-8,000

30-600 1,350-15,000

2.5

8,ooo-s5,ooo 16,000-175,000

30

14 45

" The lower limit criterion is that ~ h spherical e Nusion current is l a s thao 5% of the iinear dausion c u r m l (value shown for D = IO-'- 10-6cm2/s. The upper limit criter~on1s based on an IR drop of 10/100mV(assuming R = I,000Q). The ratlo of capacitive to faradaic current is also given for the higher scan rate. Note that at rbe highest scan rates it is advisable to use higher wncentrations, about 001 M.

The uppw and lower scan rate bounds for an electrode of a given diameter for use in the linear diffusion regime are listed in Table 2-3, based on a I or 5% current contribution from spherical diffusion, and a 0.005 or 0.01,* V IR drop. As a consequence of the short time involved in gast scan experiments, total electrolysjs is tnuch smaller at microelecirodes, and fouling of electrodes by electrolysis products is minimized.

2.6 Potential Step Methods

and Cyclic Voltammetry

Cyclic voltammetry is a powerful technique because the CV waveshape is sensitive to dl the parameters of the electrochemical mechanism. For the same reasons, however, a full quantitative analysis with CV can be dillicult, Tt is usually helpful if qualitative or quantitative information can be obtained from other sources. Potential step methods can play a complementary role to CV in the analysis of electrochemical mechanisms. This is because they can be performed under conditions of a fast and irreversible forward heterogeneous rate constant.

2.6.1 Double Potential Step Chronoamperornetry The potential waveform imposed at the working electrode for the double potential step (DPS) is shown in Figure 2-25a. The characteristics of the waveform are the initial potential E , the step potential Es, we step time z, and the find potential E,. Here, we consider only the case of Ei = E,. Normally, * Sine we am looking for an estimte, i l wouldn't matter much if we took the reversible pe& current.

Figure 2-25 (a) Wavefonn and (b) current response for a double potential step experiment.

THE CYCLIC VOLTAMhfETRIC EXPERIMENT

CYCLIC V O L T A M T R Y

62

electrolysis occurs at the initial potential setting, and the potential is stepped to a voltage well beyond the formal reduction potential of the principal electroactive specles that undergoes reduction or oxidation. This arrangement ensures that the electrode reaction will be fast and irreversible, as can be seen by considering the effect on the Butler-Volmer equation and the Nernst equation. The exact amount of voltage beyond the reduction potential necessary to obtain t h s situation depends on k0 and a; usually 200mV will suf6ce. The DPS current response (a current-time curve) for a simple electron transfer is shown in Figure 2-25b. For the first step, the current is given by:

and after the step the current is given by*: Figure 2-26 DPS response for the EC mechanism, for values of the Following reaction rate constant of 50, 100, and 500s-'. The reverse step current decays Faster as the rate constant is increased.

The current response from singlerlpotentjal step is often used to determine diffusioncoefficients.In combination with cyclic voltammetry, both n and D can be determined. 2.6.2 DPS

coefficient on the first step, and the chemica1 reaction rate constant by fitting the reverse step data to theory. Programs for the simulation of DPS experiments ase included with this book and are described in the appendix.

Analysis of the EC Mechanism

The DPS response i s shown tor the case of a reduction with a following irreversible first-order chemical reaction in Figure 2-26, for a step of 1 ms and homogeneous rate constants ranging from 0 to lo4 s-'. The current on the first step is unaffected by the chemical reaction (so that the first step result can be used to determine the diffusion coefficient of the electcoactive species). The chemical reaction removes the reduced species, resulting in reverse current that is diminished according to the rate constant of the followjng chemical reaction. A solution for this problem is available in a series solutions involving modified Rzssel ftmctionsi8; two or three terms are commonly taken:

t

'

S i d e potential step experiments are invariably performed with the goal of h o m p l e t i n g some type d quantitative analysis, it is imperative that both capacitive current and background currents be taken into acmunt. Several I

.: ;

$ where

2.6.3 Capacitive and Background Currents

'

1

i

i

approaches are possible with regard to the capacitive current. Data points can be ignored until the ratio of faradaic to capacitive current is targe. This is a common approach, particularly with microelectcodes, where the capacitive decay can be very fast. Data can be taken in rapions where i x tl" is constant. It is possible to try to improve time resolution by including the theoretical capacitive current in the analysis. In DPS work the rninlfization of background currents is particularly important; background subtraction sllould bc used if unavoidable background faradaic currents occur.

L

1 and I, represents the modified Bessel functions. Experimental DPS data can be analyzed by determining the diffusion

*

Such large IR drops can be effmively dealt with by mulatian methods.

:: ; '

2.6.1 Time-Resolved Potential Step with Microelectrodes Improved time resolution is possible with a combination of microelectrodes and a fast potentiostat such as that described klow (Section 2.7). Figure 2-27 shows the DPSexperiment for ferricyanide, utilizing~acornmerciallyavailablepotentiostat with a time resolution of 0.1ms. and the same experiment performed with the "home-built" instrument which has a time resolution of about 0.25 ps.

CYCLIC VUL'TAMMETRY

THE CYCLIC VOLTAMhIETRIC EXPERLMENT

Figure 2-28

Operational amplifier.

- V-), where the open loop gain A i s usually greater than lo*. In feedback mode,an immediate consequence of this condition is that V+ x V - . Consider why this is so (refer to Figure 2-29). If there is a initial positive difference in (I/+ - V-)= +1 V, a large positive output voltage will Appear and feed back intv the inverting input. This feedback will decrease the value of V+ - V - . The i d b a c l i process ensures that the value of V- is very nearly equal to V + .The particular feedback loop shown also ensures that the voltage output will be equal to voltage input. 2. OAs have large input resistance. In other words, no current is allowed to flow through the - or t inputs. 3. OAs have a low output resistance. Large currents can be drawn from the output.

I. V, = A(V+

Ti!,>c (~,-,?v<)

Figure 2-27 Increasing the time resolutio~lor DPS. Solid circles are with best time resolution with BAS-100A (one point per 0.1 ms), dotted Iinr consists of theoretical points, and the line represents data taken with a fast potmriosiat (one point per 0.004ms). i

Fast potential step experiments have a great advantage over cyclic voltarnmetric methods with regard to the analysis of the EC mechanism. Analysis by CV is made difficult at high scan razes; data are distorted as a result of separations due to IR drop and slow electrode kinetics. DPS has neither of these problems.

2.7 Construction of a Fast Potentiostat At the time of writing the available commercial instruments do not allow for the Full range of time scales for CV or DPS experiments. In particular, they are generally limited to less than 1000V/s or to time steps exceeding 1 ms. Most likely this situation will soon change, as the utility of such experiments becomes apparent. Electrochemical instrumentation that can perform high speed experiments can k constructed rather simply and at relatively small cost. Potentiostatic circuits make use of operutional urnpl$ers (0-4) to controi voltage and to measure current. Operational amplifiers have a number of inputs, but in circuit diagrams the power supply connections are understood and only three connections are considered (Figure 2-28). Based upon their action, these connections are the inverting input (-1, the noninverting input (+), and the output. In normal use, OAs are used in a feedback mode; that is, feedback loop is introduced from the output or from some other source to the inverting input. The action of the OA in electrical currents can be understood largely on the basis of a few characteristics.

I

: b

S

i E i /

;

1 :

Besides the voltage follower configuration, which we have mentioned, the current follower and voltage amplifier configurations are important in designjng a potentiostatic circuit. In the current follower mode (Figure 2-30), the current flows around h e oop amp through a resistance. The voltage monitored at the output is then, with respect to ground, V,,, = iR. The voltage amplifier (Figure 2-31) has resistance elements at the inverting input and in the feedback loop. Since the voltage at the inverting input is zero with respect to ground, the incoming current is i,, = l,;,,,'R,,. The current in the loop is - 1/,,,/RP Since there is no current flowing at the inverting input, the

I

Figure 2-29

Op amp in feedback mode (voltage follower).

CYCLIC VOLTAMMETRY

THE CYCLIC VOLTAMMETRIC EXPERIMENT

%s

Figure 2-30

Op amp in current follower modc.

incoming cwrent must flow around the loop, and i, proportional to the ratio of the resistances:

= i,,,.

The output voltage is

A full potentiostatic circuit is shown in Figure 2-32. We assume that a voltage input (a potential sweep or step) is available from a function generator. The voltages measured versus ground in this circuit can be determined by consideration ofthe action of the three OAs. The voltage at the inverting input of op amp 1can be calculated according to the constraint that no current may flowto the imputs. This means that ii = - i,; and as a consequence, if R , = R,, then V, = - V2.The output of op amp 1 is connected to the auxiliary electrode. Note that the reference electrode can be considered in a feedback loop to opamp I , and a voltage follower is inserted between the reference and the inverting input of op amp 1. This design allows the reference to maintain a stable potential by ensuring that no current is drawn from it. To complete the picture, we must examine the working electrode. It i s connected in a current follower mode so that current through the working electrode i s shunted around op amp 3. In a feedback mode, the inputs are equal,

Figure 2 3 1 Op amp used as voltage amplifier.

i axis Figure 2-32

Complete potentiostatic circuit.

and the working electrode is pinned to the ground established by the control amplifier. Although by convention, voltages in a circuit are measured with respect to ground (assigning the common ground = O.OV), it is also true that the working electrode's Fermi level (and thus its potential) is changing with respect to vacuum. Finally, a voltage amplifier (op amp$) can be added to step up the output voltage to an acceptable value for a recording instrument. A quick check on the circuit operation can be made by inserting resistances between the AE and W E and the RE and WE (to emulate an electrochemical ce!l). Upon imposition of a voltage sweep, a linear output voltage can be obtained that should agree with that calculnted from the circuit. The time resolution of the circuit is Ijmited by the time response of the operational amplifiers. Op amps with a time response of tens of voIks per microsecond can be purchased at low cost. Noise can be m i n i n l i d by wrapping

THE CYCLlC VOLTAMMETR~CEXPERLMENT

CYCLIC VOLTAMMETRY

69

Experiments can be performed in both regions with the same electrode by using dimerent time scales or different sized electrodes. The equations for i, and S can be solved to find n and D. For instance, Barallski et report thc ColIowing for the reduction of 9.64 mM benzoquinone in DM SO: in = 30.0 x

Amp (r = 8.8 pm)

Solving for n and D results in n = 1.05 and D = 8.7 x 10-bc~nZ s- I , in good agreement with the values found by Baranski et al. using a slightly different caIculation with a ferrocene reference. M @ d Waveform Generator

References 1. Reichardt, R., Solvents und Solvent Effects in Organic Chmisiry. VCH Publishers: New York, 1%-

2 Fry, A.; Britton, W. E. In Lahorutory Echniques in EleciroanaIytical Chnnisrry, 1st ed., Kissenger, P.T.;Heineman, W. R., Eds. Marcel Dekker: New York, 1984, Chapter 13.

u

3. Rusling, J. Auc. Chem. Rex 1991,24, 75. 4.

Figare 2-33 Construction of a tirneresolvd electrochemical instrument.

Gosser, D. K., Jr.: Huang, Q. J . Ebctruunub Chem. 1990,278,399.

5. Gosser, D. K., Jr.; Huang, Q.; Rieger, P. H. J. Electroanal. Chem. 19W, 136,285.

6. Frat, U.; Iwasita, T.; Schmickler, W.; Stimming, U. J . Phys. C h m . 1985.89, 1059.

a11 connections with shielding tape. Leads to the electrodes should be as short as possible to minimize capacitan~,to achieve the best time response. A fast rise time potentiostat system based on the circuit described is shown in Figure 2-33. Tlus system features a progammable waveform generator board for general simulation of waveforms and a waveform analyzer to digtize and store the data. The present setup features cyclic staircase generation (2.5mV/step) and double potential step experiments. The effect of IR drop can also be ehminated by modifications of the potentiostatic circuit. Performance of such a circuit under very high scan rates has been discussed by Amatore et d.l0

7. Bond, A. M.; Zongpeng L. J. Eleetrounal. Chem. 1989,259. 321.

8. Wightman, R. M., Wipl D. 0.Acc. Chcm. Rch. 1990.23,64 9. Wiedemann, D J.: Kawagw. K. T.; Kennedy, R. T.; Ciolowski. E. L.; Wightman, R. M. Anul. Chem 1991,63, 2965. 10. Kissenger, P. T. In Lahoratorv Techniques in E l e ~ ~ r u u n a ~ ~Ch&,tt~brry, tic~l 1st ed., Kissenger, P. T.; Heineman, W. R . Eds. Marcel Dekker New York, 1984. Chapter 12 11. Bard, A. J.; Faulkner. L. R Elrrtmt.lu~riicul Mt'thodh 1st 4.John Wiley Sr Sons: New York, 1980.

12. Gosser, D. K., Jr.; Zhang. F. filunra 1991.39. 715. 13. Schwartq W. M.; Shain. I. I . Pl:,vs. Chcrrr. 19h5, 69, 30.

2,8 Determination of the Number of Electrons

14. Parker, V.; Britton, W. E. und Fry. A. J. Edr Topics in Orguttic E l e u r r w h ~ ~ m i s tIrst ~ ,ed. Plenum Press: New York, 1986.

Several methods involving rnicmelsctrodes have been suggested for unarnbiguously determining a, the number of electrons in an electrochemical mechanism and D, the diffusion coefficient. While this information cannot be determined from a cyclic voltammetric experiment alone, w and D can be found utilizing experiments that combine li~lcarand nonlinear diffusion2'. Using the standard inlaid disk electrode geometry that has been described, the equations in the planar and nonlinear diffilsion limit are:

15. Klingr, R. J.; Kochi, J . K. J . P l r ~ s .Cllrm. 1981,86, 1731. 16. Greef, R.; Peat, R.; Peter. L. M.: Fletcher, D.; Robinson, J. lnsrrvmmtul Merhods i r ~Electrochemistry. Ellis A o r w o d : Cllich~ter,1985.

17. Andrieux, C. P.; Hapiot, P.; Saveant, J. M. Chem. Rev. 1990, 90,723. 18. Rieger, P. H. Elcctrochemislry Prentice-Hall: Englewood Cliffs, NJ, 1987. 19. Schwarz, W. M.; Shain, I. J . Phys. 1981, 85, 1731.

20. Amatore, C.; Lefrou, C.; Pluger, F. J. EIecrroonuI. Chem. 1989, 43.

i, = n ~ ~ D ~ ~ ~ / ( . r r r ) ' ~ " (a plot of i vs t-It'

21. Aoki, K.; Osteryoung, J. J . Elcctrounrrl. Chem. 1981,122, 19.

gives S = ~ F A D ' ' ~ C / ~ ' ~ ~ )

times2') im = *nFDC.. .(at . .long -.

22. Baranski, A. S.; Fawcett. W.R, and Giibert, C . M.Awl. Chm., 1985,57, 166. -..,A

.

>

CHAPTER

3 A Survev of Electrochemical ~ e c h a n i s m s

arc of such practicat or theore tical interest, that they have become recognizable "standards." Often they are components of larger, more complex mechanisms. A reversible chemical reaction can precede an electrode reaction, shfting the peak potential and altering the waveshape. Electron transfer is often associated with a change in motecular structure. An examination of the voltammetric response indicates whether the structural change is concurrent with electron transfer or occurs in a following, activated step. In aqueous solution, protonations can be at equilibrium with respect to the electrode processes. In recent years. such catalytic transfer c h i # (EX7 catalysis mechanisms involving electron transfera~~lrctron and homogeneous redox catalysis ( H R C ) have g o w n in itnportat~ce.Cyclic voltammetric experiments have be11 pivot 33 in understanding and utilizing these mechanisms. This ckapter presents examples of CV analysis For more complex mechanisms. The examples illustrate not only the wide range of chemistry amenable ta a CV study, but aIso the various ways in whicl~sjnlulatjon methods have aided in the analysis. The theoretical basis of simulation methods i s discussed in Chapter 4. Chapter 5 uses several of the mechanisms discussed here to illustrate the use of the simulation program CVSIM. In these cases, the mechanism parameters obtain4 from simulation analysis are listed lor comparison with the examples in Chapter 5. Certain zl~ctrochemicalmechanisms reappear frequently, or

CYCLIC VOLTAMMETRY

72

A SURVEY OF ELECTROCHEMICAL MECHANISMS

3.1 The CE Mechanism The reductive CE mechanism is formulated as follows: k!

Pe o x k-,

ox+e-=Red

EO',~O,U

Experimentally, the initial solution contains the species P and Ox at equilibrium concentrations. During the reductive sweep, Ox is reduced to Red, and P starts to convert to Ox according to the chemical kinetic rate constants. If the experiment is started with initial concentration of the species Red only, the oxidative CV is an EC mechanism. The cyclic voltammetric response for this system (with the electrode reaction reversible, a CE, mechanism) is determined by K c , and ( k , + k-,)/v. The limiting cases of the CE mechanism are (I) the chemical step is fast and at equilibrium, and (2) the forward chemical reaction is slow with respect to the time scale of the experiment. In case 1, the CV has an E, determined by the EO' and K,,. Tn case 2, the cyclic voltammogram will be reversible and will reflect only EO' and the initial concentration of species Ox. Let us examine the transition from case 1 to case 2, which can be effected by a change in scan rate. As the scan rate is increased, the kinetics of the chemical reaction are apparent: the cathodic peak/Jv decreases because Ox does not maintain equilibrium concentrations on the time scale of the experiment. The cathodic peak potential also shifts negative relative to the peak in case 1. At large scan rates, the reversible case 2 is achieved. A quantitative study of the equilibrium and kinetics of an CE/EC-type mechanism was recently reported by Lee et al.' This study utilized a combination of potential step and cyclic voltammetric experiments in an examination of the reduction of the nitronium ion and the reverse process, the oxidation of nitrogen dioxide.

As we have noted, potential step methods are particularly attractive for the determination of chemical rate constants in electrochemical mechanisms because the potential can be stepped to a potential at which the forward electron transfer is fast and irreversible, so that the current response depends only on the rates and mechanism of coupled chemical reactions. A complete quantitative evaluation of the mechanism was achieved by combining the potential step results with a series of simulations. The chemical reaction rate constants were determined by single-step experiments (the oxidation of NO2).Early in the step, the single-step response is determined by the equilibrium concentration of NO,. At later times, the response reflects the rate of conversion of N,O, to NO,. Simulated potential step response curves could be compared to experimental data to extract the K , , and k , and k - ,(see Figure 3-1).

Time (mScc)

Figure 31 Simulated single potential step response for the oxidation of NO, (CE mechanism). Compare the response for an E mechanism (dashed line), with concentration cqual to the (NO, N,O,) concentration.

+

Cyclic voltammograms for the reduction of NO: and the oxidation of NO, are shown in Figure 3-2. Since the chemical reaction rates were rather low, the reduction potential could be obtained at modest scan rates by taking the average of the cathohc and anodic peak potentials. At the lower scan rates, the kinetics of the chemical reaction i n f l u e n ~ sthe voltammograms significantly. Simulations of these experiments are shown in Figure 3-3. From CV simulations of the faster scan rate experiments, the heterogeneous parameters k0 and cc were determined. Figure 3-4 iltustrates the importance of scan rate as an experimental variable. Several mechanisms may give similar current responses at a particular scan rate. However, they arc less tikely to give similar current responses over a wide range of scan rates (unless, of course, the mechanisms are kinetically equivalent). The good match between simulated peak currents versus scan rate is an important validation of the proposed mechanism. This is our first encounter with the use of simulation to analyze CV results. Through the theory of simulation (Chapters 4-6), a cyclic vollammetric or potential step response can be calculated for any electrochemical mechanism, given the parameters that describe the experiment (scan rate, scan range, electrode area) and the mechanism (reduction potentials, electrode kinetics, chemical reaction kinetics, and lffusion coefficients of all chemical species). The unknown parameters of the electrochemical mechanism can be varied until a simulation is obtained that closely resembles the experimental result. The CV,'potential step analysis was carried out in a series of other solvents. The variation in the reduction potential (measured against a ferrocene standard in each solvent) was shown to be best correIated with the solvent donor number (rather than the dielectric constant), indicating a specific interaction of the solvent with the nitronium ion. The kinetics of the electrode process were much .

.. . ,

-.--.-

.A

..

.

.

I . . . . . . .

---"-i

CYCLIC VOLTAMMETRY

A SURVEY OF ELECTROCHEMICAL MECHANISMS

-

I .e

I .I

1.0

E,V us SCE

- -

1.1

1.4 B V w m

1.0

1.1

1.4

J .O

6,VnSCE

Figure 3-2 Cyclic voltammograms of (a) NO: in acetonitrile at scan rates 0.1 and 7 V/s and (b) N 2 0 , in acetonitrile at scan rates of 0.5 and 7 V/s. (Adapted from Reference 1.)

faster than could be expected on the basis of Marcus theory of electron transfer, including both solvent reorganizatian and internal, molecular reorganization. Lee et al. interpreted the unusually high electron transfer rate as being indicative of an "~nnersphere" type of mechanism, whereby the nitraniurn ion can form a complex with electrode surface and thereby lower the activation energy for electron transfer. -.The simulation analysis resulted in the following parameters for the CE mechanism (in acetonitrile):

c

Figure 3-3 Simulated cyclic voltammograms: (a) corresponds to Figure 3-2a; (b) corresponds to Figure 3-Zb. (Adapted from Refcrence I.)

C = chemical: k l = 5.0s-I, k L i = 6 x 1 0 4 ~ - ' s - ' E = electrode: EO' = 1.32, k0 = 0.031, = 0.48

3.2 Multielectron Transfer Formulas for AE,, and peak current include n, the total number of electrons concurrently transferred. However, straightforward application of such formulas may not be possible when pa exceeds 1. The second electron transfer might take place akpotentials negative to the first, giving rise to two overlapped oneelectron peaks. Often there is a chemical reaction upon reduction or oxidation, giving rise lo an ECE process with an apparent rnultielectron transfer. From theoretical considerations, we expect concurrent two-electron transfer to be rareU2 We start by considering a reversible two-electron transfer: 1+ e - ~ 2

Ey

A SURVEY

CYCLIC VOLTAMMETRY

76

OF ELECTROCHEMICAL MECHANISMS

(al

O

u -I

o

logy,

I

vs4

from the simulation analysis. The scan rate dependence of the anodic peak current for different concentrations of N,O, is compared to that predictd by the simulation analysis

J0

-

r---.

0.1

0.0 2

-0.1 -0.2 0 -0.3 1- 0 4 - 0 5h -0 8 -0.7l

Poiental ( V )

(smooth curves). (Adapted from ReFerence 1.)

The shape of the cyclic voltammetric wave will depend on the relative values of Ey' and ET (i.e.,whether 2 is stable with respect to reduction at the reduction potential of 1). Figure 3-5 shows the cyclic voltammetric result expected tor different values of AE,-, = (ET - E y ) . If the second reduction potential exceeds 180mV positive with respect to the first (AE,-,> 180mV),then the C V takes on the characteristic n = 2 process, where the overall formal potential is the average of the two reduction potentials and the AE, = 29 mV. Otherwise, the CV will reflect two overlapped one-elsctron transfers, with the extent of overlap depending on the relative peak potentials. A recent report3 on the twodectron reduction of (tl-C6Me,),Ru2+ demonstrated the resolution of two-electron processes even when the second is favored thermodynamically. This is a possibility when the second electron transfer is slower than the first. In this case an increase in the scan rate shifts the peak due to the second electron transfer to more negative potentials, eventually resulting in a peak splitting (Figure 3-6). The values of the reduction potentials and electrode rate constants were estimated by simulation analysis:

Potenlial ( V )

Figure $5

CVs lor two-electron transfer for (a) El > E,: and (b) E l < E,.

metal) is concurrent with the second electron transfer (Figure 3-7), resulting in a small heterogeneous rate constant. It is a little puzzling, however, that the second is exactly 0.5; since a significant structural change is postulated, one might expect the second electron transfer to show a transfer coefficient smaller than 0.5. TO obtain a good fit, it was necessary to include the kinetics of the disproportionation reaction (3 + 1 9 2 + 2). Note that the equilibrium constant IK,, = k,/k,) is constrained by the reduction potentials to be k,, = 0.5:

:

3.3 Protonations at

Often protonations are coupled to electron translers. It is expected that in this the el~trochemicalmechanism will be sensitive to the availability of protons (pH and nature of solvent). case

I

It appears that a change in hapticity (the character of arene bonding to the

Equilibrium

A SURVEY OF ELECTROCHEMICAL MECHANISMS

a

-I volt rs. PC

The reduction pathways open to quinones and many other similar systems (nitraso compounds. azobenzenes, etc.) can be represented by the "square scheme" in Figure 3-8. It has been known for a long time that pH affects the kinetics of the reduction of the bznzoquinones. The reduction generally becomes more irreversible at higher pH values. Otl this basis it has been proposed that the reduction pathway of BQ is HeHe (proton-electron-proton-electron) at low pH and eHeH at higher pH. The pK,, the EO',and the rates of the individual steps in the square scheme will detern~incthe preferred pathway for reduction. Lavjron has further developed the theory for quinonc-like systems.$ Assuming that the protunations were at equilibrium (fast with respect to the electron transfers), and that the transfer coefficients for the electrode reactions were all 0.5, he was able to show that such systems behaved exactly like two sequential one-electron transfers, with apparent heterogeneous rate constants:

-2 /

79

PC+

and apparent reduction gotentjals:

0

- 1 Volt rr. PC

-

Cyclic vultamn~ograrnsof (qb - C,Me,),Ru: flower). (Adapted from Reference 3.1 Figurr $6

2

/ 'CF

This approach has been utilized by Dznkjn and Wjphtman in analyzing the pathway for the oxidation of catecIlvl to o-q~inones.~

0.2Vls (upper) and 10V/s

Figure 3-7 Proposed change in bonding to ruthenium upon the second electron transfer. (Adapted from Reference 3.)

1

Figure 3-8

The square scheme for benzoquinone reduction.

CYCLIC VOLTAMMETRY

80

The cyclic voltammograms in Figure 3-9 are simulations of BQ reduction at pH 0 and 6 based on Laviron's values for the apparent reduction potentiais and apparent rate constants. The quinone electrochemistry story becomes even more complex, and the role of adsorption and solution chemical reactions are still being actively investigated: this topic has been reviewed by Charnben6

3.4 Catalytic Mechanisms Catalytic processes have k e n studied to great advantage by cyclic voltammetry. These include catalysis of chemical reactions initiated by electron transfer at the electrode and catalysis of redox processes by solution chemical reactions.

A SURVEY OF ELECTROCHEMICAL MECHANISMS

81

We can approach this mechanism by examinins the eflect on cyclic voltammetry of increasing thc first-order rate constant k (Figure 3-10). Of course, when k, = 0, only the first reduction occurs, and the cyclic voltammogram corresponding to a single electron transfer is ilhtaioed. As k, increases, several characteristic changes occur. The lizight of the peak due to the reduction of 1 decreases. This is s result of the fornlatilsn of 3, which is immediately oxidized to 4 at po~entialswhere 1 is reduced. 4 is reduced back to 3 at a more negative potential. The peak associated with this process increases with the rate of production OF 3. Thc production of 4 occurs through a catalytic chain electron transfer. For large rate constants k , , a stnall atriount of charge passed results in con~plctccnnversion of 1 to 4 near the electrode, The reverse peaks that correspond to the reoxidation of 3 and 2, also are related to the production/consumption of these species. A similar catalysis can bc initiated in solution. If a small amount of reducing agent is added to a solution containing species I, a llornogeneous redox reaction:

,

3.4.1 Electron Transfer Chain (ETC) Catalysis Consider an ECE rncchanism where the product of the intervening chemical reaction is itself part of a redox couple:

kl

K Cq = = exp k-,

4+e- = 3

has the same net result as the electrode reactions. Many nucleophilic substitution reactions have been catalyzed by the electron transfer chain mechanism; reviews have appeared of ETC applications in organic' and organometallic8 chemistry, as well as in general.' OF course, there is nothing stopping the homogeneous redox reaction from taking place during the electrochemicat experiment. However, the effect is rather small as is illustrated for CVs simulated with k, constant and tuning the homogeneous redox reactioi~upward'' as shown in Figure 3-11. The ETC mechanism is sometimes written in a picturesque square scheme (Figure 3-12). An early electrochemical study of the ETC catalysis of ligand suhsrituriun in metal carbonyls is still one of the most thorough. In this work. Kochi et sl. studied the oxidative catalysis of ligand substitution in manganese cyclopentadienyl complexes. A typical reaction was the ETC reaction for trjphenylphosphine substitution:

Ey'

where E;' < E?.

g5 - MeCpMn(CO),L =

Me

= MeCN,

-

MnMe

P

=

KL

PPh,

-

e-=MnMct'

Ey

,

= 0.19, ky = 0.048, u = 0.5

Pot e r l l i a l / V

Figure 3-9 Simulations of benzoquinone reduction (dashed).

at

pH values ill 0 (solid) and 6

For every mde of charge passed, I013 moles of starting material was

A SURVEY OF ELECTROCHEMICAL MECHANISMS

I'otcntial ( V )

Figure 3 1 1 Effect or the homogeneous redox reactinn on the ECE ttiecl~at~ist~~: = 105111-'s-' (soIid line), compare with 3-10b (dashed line).

k

-6 0.1

0.0 -0.1

-0.2 -0.3 -0.4

-0.5 -0.6 -0.7

Potential ( V )

consumed, and the substitution reaction is complete in a manner of minutes. The thermal reaction is not meawlreable on this time scale. Cyclic vultammogran~silh~strafingvarious facets of this study Figure 3-13 shows the cyclic voltammo_eram for thc oxidation of the substrate cyclopentad~enylcclrnplex (3-13). The disappearance of the original peak and the appearance of a new peak positive to the original are suggestive of an ETC mccha~iism.Use of lower conce~ltrations allows the observation of intermediate results. In this case, the rate of product formation is reflected by the size of the product peak. Cvmparjson with simulation permitted an estimate of the thermodynamics and kinetic parameters of the entire mechanism (Figure 3-14) for a variety of lipands.

Potcnt ial ( V )

Simulations of the ECE mechanism for increasing value of the rate constant for the case E , < E,: {a) k , = OsL1, (b) k, = 10s-', and (c) k , = 1000s-'. Figure $10

Figure %12 The ECE mechanism written as a square scheme.

CYCLIC VOLTAMMETRY

A SURVEY OF ELECTROCHEMICAL MECHANISMS

3.4.2 Homogeneous Redox Catalysis (HRC) With the ETC mechanism, the electrode reaction initiates a chain catalysis reaction. It is also possible for a sequence of solution chcmical reactions to drive an electrode reaction (e.g., to cause a reduction at a potential positive to its formal reduction potential). Saveant and co-workers have developed the theory of the eIectrochemical response for HRC schemes. In a reductive HRC (Figure 3-15) based on an EC mechanism, the mediator, which has a reduction potential positive to the substrate, is reduced at the electrode. The reduced mediator (species 2) diffuses into solution and reduces the substrate, an uphill thermodynamic reaction. This unfavorable reaction is driven by an irreversible reaction of the reduced substrate:

'

POTENTIAL,

V

vs SCE

I

+e

-

=2

E!' (fast)

Figure 3-13 The reversible oxidation of MnMe in acetonitrile. (Adapted from Reference 11.)

The llon~ogzrleousredox reaction recycles species 1 back to the electrode according to the kinetics of the system and the relative concentration of 3 to 1 (Saveant terms this the excess factor, q = [3]/[l]). The HRC mechanisms arc agnaled by two events: 1. An increase in current compared to the reduction of 1 alone, due to the recycling of 1 to the electrode. 2, A decrease in the reverse peak (2 - e - = I), which occurs because of the

irreversibility of the overall process. A steady state treatment of the solution chemical react ions results in the following expression for the observed rate of recycling of 1:

POTENTIAL,

VSSCE

Figure $14 Experimental and Grnulated CVr for the oxidation of 1.7 x l o - ' M MnMe in the presenoe of various mt~larratios of phosphine ligand and maaganesz complex. (Adapted from Reference I I .)

,.*.<

Figure $15

Schematic representation of an H R C process.

A

CYCLIC VOLTAMMETRY

86

SURVEY OF ELECTROCHEMICAL MECHANISMS yi.0

This rate can be controlled by the homogeneous redox step or by the irreversible chemical reaction, depending an the ratio of k , to k - , to k - [I].

I

,

Case 1.

k-,[I]

<< kkz

The catalysis is controlled by the forward step of the homogeneous redox reaction. 111 this case, thc rntto of the peak catalytic current to the current obtained by reducing 1 alone, !,,,;ips,, can be analyzed to give the rare constant k , . This analysis can be done lusing published workirlg curves or by creating curves by simulation. Figlire 3-16 shows simulated CVs indicating the effect clf several values of the rate constant k , for an excess factor of 10.

I'otcntial ( V )

The rate-determining step of the catalysis is the irreversible reaction. The value afli,,k, can be f w n d from simulation analysis. IT, under the circumstances of the experiment, k - , can be shown to be diffusion controlled (because of the very large driving force for the revel st: reaction) thcn k , can bc estimated (I;, having k e n determined from case 1). Rate constants up to lo9 s - can be measured in this manner* The reduction potential of the 3/4 couple, normally undetermined because of the fast following reaction, can he estimated. If the rate constarlt k , for a series of mediator coupIes PQ is measured by the HRC (case 1) method, an application of "linearized" Marcus theory (simple LFER) Ieads to a plot of the measured rate constant versus the driving force as shown in Figure 3-17. The plot has three regions:

Figure 3-16 Simulations ui HRC lor L

use

1 ( h ,[PI x I,), illurtrat~ngthe effect or a

change in the forward homogeneous electron transfer rat? constant.

An excess facior

r

(3:1)=lO.k,=1~.2.5~10~,dndIO~M-~s-~.

!

{FEH) mediated rednetion of glucose oxidase (GO)." The mechanisn~ is expressed as follows:

'

I . The diflusion-controlled region. Here, the driving force for the forward redox ~ractionis largc. and the rate k , i s diffusion controlled. Nn experimental points can be obtained in this region, but it can be estimated from rate theory. 2. The artiuation-controlled region. Here k, is determined by the following LFER-type equation:

: I

I

GO,,,

r

GO,,

:

ki + 2 FEH+ = Gnu, + I FFH k.. f glucose GO,,, + gluconate -+

and is shown schemotieallyin Figure 3-18 Tlre cyc11cvoltammograms for the

I

P

I

:r

Dffusion

Activahon

where P is a standard rate canstant referred to zero driving force and a is t h ~

transfer coefficient referred to the ho~nogeneoureaction. 3. The '.counterd~$usion" region. Here the rate of the reverse electron transfer has reached the diffusion limit, and the forward rate is determined by this value and the overall equiiibriunl cuostant.

The sylnmetry of the graph requires that !he point of intersection of the two diffusion lines (extrapolated) be at the reduction potential of the substrate. The analysis ofmore complicated examples of HRC mechanisn~sis discussed in Reference 12. An example of an HRC-type mechanism involves a ferroceae carboxylic acid

\\

-

cou,,, Diffusion

I

EO

Figure 3-17

The measured rate constant k , versus dr~vingforce for the ARC scheme.

I

A SURVEY OF ELECTROCHEMICAL MECHANISMS

CYCLIC VOLTAMMETRY

88

GO(red) /

-2e-

C/

, ' J'

A,

/' /' /'

2FeH'

"

*

GO(ox)

++

Gluconate

/'

Figure 318 A two-electron HRC schrtns: the fzrrircrm carboxylic acid catalyzed reduction of glucose oxidse.

Figure 3-20 Working curve based on a simulation analysis of the G O catalys~s.Tllr simulations took account of the large difference in diffusion coeficients betweer1fcrroacr~c. carboxyhc acid and GO. (Adapted from Reference 14.)

FEH alone and in the presence of glucose oxidase and glucosc are shown in Figure 3-19. In this case GO was in large excess and the catalytic action was cvntrolled by the homogeneous redox reaction. A simulation study using the full description of a second-order reaction and actual diffusion coefficients resulted in the determination of the rate constant for clectron transfer between FEH and GO, k = 1.06 x I05LM-'s-'. The values obtained were further verified by examining the scan rate dependence of the results (Figure 3-20). In a more recent study, the mediating electron transfer reactions of glucose oxidase have been reexamined in great detail using simulation methods. Several outstandmg problerns wcre rcsolved regarding the pH dependence of the

Figure 3-19 CVs of ferrowne carbnxlyjc acid done (solid line) and din the presence or glucose oxida~cand g l u c ~ s e(dottcd line). [Adapted from Reference 13.)

reaction, and the mistaken assumption that the homogeneous redox reaction obtains pseudo first conditions.14

3.5

The Reduction of Nitrobenzoic Acid

A study that coill bjned cyclic voltammet ry with a time-resolved, surfaceenhanced Rarnan spectroscopy ISERS) examination of the electroreduction of pnitrclbenzoic acid (PNBA)provided a more complete description of this complex me~hanisrn.'~ The redactive cyclic voltammogram to - 1.4V is shown in Figure 3-21 (pH = 1I), for smooth and roughened Ag electrodes.15 The use of roughened dectrvdes is necessary for the SERS experiment. While an adsorption prepeak was indicated on the roughened electrodes, the overall cyclic voltammetric responses for both electrodes wcre remarkably similar. In such cases, adsorption equilibrium is probabIy obtained, and qualitative comparisons between SERS studies on roughened electrodes and CV studies on smooth electrodes are possi bie. In Figure 3-21, the first peak (A), occurring at about -0.65 V, could be assigned to a four-electron reduction of p-nitrobenzoic acid to the corresponding hydroxyamine:

,

The second cathodic peak (B), assigned to the further reduction of the hydroxyamine to p-aminobetlzoic acid, is not considered in more detail here. On the reverse scan, the anodic pzak ( C ) is due to the oxidation of the p(hydroxyamino)benzoate to nitrosobenzoate, a two-elsctron process. Direct observation of the nitrosobenzoate proveded by SERS performed during the CV experiment provided strong evidence for this assignment (Figures 3-22).

CYCLIC: VOLTAMMETRY

A SURVEY OF ELECTROCHEMICAL MECHANISMS

91

On the third half-cycle of the CV experiment, the couple to C is observed, peak D. However the peak is smaller than peak C , particularly at $low scan rates. This result indicates a following chemical reaction which is thought to be the coupling of the nitrosobenzoate with the hydroxyamine.

A careful consideration of the charac?cristics of the CV experiments, including the dependencies of the peak currents and potentials on scan rate, resulted in the detailed mechanism shown in Figure 3-23, with the following values for the mechanistic parameters:

Potential,V vs. SCE

Figure 3-21 CV of 511 m M paranitrobenzoic acid {PNBA) on smooth Ag electrodes. Aqueous solution, 0.1 M Na,SO,; scan rate, 50rnV/s. (Adapted from Reference 15.)

Sequenoe of SERS spectra on a roughened Ag electrode, identifying the nitroso intermedialc in the electrode process. (Adapted from Reference 15.)

NOH

Figure %22

Figure 3-23

W

H

Proposed scheme for the mechanism of reduct~onuf PNBA.

CYCLIC VOLTAMMETRY

92

As has been discussed. 3e-/2H+ PToCeSSCS can be modeled as two-electron processes with transfer cueficients equal to 0.5. Values for the reduction potentials, heterogeneous rate parameters, and chemical rate constants were obtaind from the best fit obtained between simulation and experiment, a process that required more than 500 simulations. The value of k,,,, - is a lower bound. Figure 3.24 shows a sinlulation of the CV with the experimental data. The variation of sunulated and observed peak potentials and currents with scan rate, shown in Figure 3-15, are in good agreement. In this case 14 parameters have been obtained from a visual fit of the simulation to the data. It is important to note that different sections of the CV waveshape are sensitive to different parameters. For instance, the first cathodic peak (A) and its scan rate dependency is related to 7 parameters (the first two reductions and a following chemical reaction rate), The position and shape of peaks C and D are sensitive to the second two reduction potentials and heterogeneous parameters, while their relative size is determined by k,,,-,. In general, the 7 parameters obtained from the single first peak are less credible and serve to allow a fit to be obtained. However, much more confidence can be placed in the individual parameters obtained from the second and third half-cycles of the CV.

,

3.6 Reduction of the Nitrosoniurn Cation and Its Complexes Lee et al. studed the reduction of the nitrosonium cation (NOt BF, ) to nirric oxide." The CV OC nitrosonium cation in acetonitrile is shown in Figure 3-26. A nearly reversible reduction is observed, from which thermodynamic (EO' = - 1.18) and kinetic parameters (kO = 0.005 crnls, a = 0.5) were obtained. I

I

Figure 3-25 Working curves based on paramelera arrived at by a simulation analysis. (Adapted from Reference 15.)

Pctentlal, V us. SCE

Figure S 2 4 Simulated versus experimental data for the reduction of PNBA. (Adapted from Referace 15.)

Figure S 2 6 CVs of NO'BF; Reference 16.)

E, V vs SCE in acetonitrile at 0.3. 0.5, and 0.7V/s. (Adapted from

A SURVEY OF ELECTROCHEMICAL MECHANlSMS

CYCLIC VOLTAMMETRY

3,7 Reactivity of 17-,18-,and 19-Electron

Tungsten Complexes The oxidation of tricarbony l(rncsityleneltungsten was examined by 1R spectroelectrochemical, N M R . and C V e u p z r i m z ~ ~This t s ~ ~study probed the reactivity of the odd-electron species generated by oxidizing the 18-electron starting material. The oxidative CV in acelonitrile (TBAPF,) is shown in Figure 3-29. A superficial examination of the CV might lcad one to conclude that this CV is the result of a one-electron oxidation and that the resulting 17-electron species is being reduced on the reverse sweep, with the peaks separated because of a slow heterogeneous charge transfer. However, a more detailed examination shows that this cannot be the case. In particular, CVs such as those in Figure 3-39 displayed the following characteristics:

-

t6

v

vs

y

v us SCE

SCE

\I

0s

us SCE

(B) 5.0, and (C) lOmM hexaFigure 3-27 CVs or IOmMNO'BF; with (A) 2.0, methylbenzent. at a scan rate of 0.5 V/s.

In the presence of beramethylbenzene [Ar] a second wave appeared and was assigned to the reduction of the char9e transfer complex [NO,AB, and the thermodynamics and kinetics of the formation of this complex could also be determined. (see Figure 3-27 lor the experimental CVs and Figure 3-28 for simulations). The entire mechanism could hc formulated as follows:

NO+ + e - % N O k,

volt8

Figure 5 2 8

positive.

1

Simulation analysis showcd that a mechanism consistent with all these

5

by a fast addition of solvenl (S), a 1Pelectron adduct, which is further oxidized,

; observations is one in which the initial one-electron oxidation of W is followed

E?'

NO+ + Ar(Me), e[NO', ArIMe)bl k -, [NOf,Ar(Me),]+e~[NO,Ar(Me),l

1. The oxidative current (peak A) corresponds to a two-electron process at scan rates up to at least 10V/s at 228 K. 2. Both the observed waves are irreversible at scan rates 11p to 10,0B)V/s. 3. Wl~cnthe switchng potential is 1.1V, the B/A pcak curren? ratio i s collstatlt and equal to 0.5. 4. The B/A current ratio increases as the switchins potential is made morc

Eq'

volts

Simulations of (Hj and (C)of Figure 3-27. (Adapted from Reference 16.)

1

Figure 3-19 Experimental and simulated CYs of the oxidation of 1 rnM trisarbonyl(rncsitylenc~tungaten(W) in acetonitrile at 298 K. (Adapted from Reference 14.)

A S m V E Y OF ELECTROCHEMICAL MECHANISMS

resulting in an overall two-electron process. Peak B then corresponds to the one-electron reduction of the diuatjon:

W ++ S e w s +

k1 K 1 =k-

E q'

WS2+ + e - g WS'

K,=I

I

k,=io5s-'

= 0.08V

The dependence of the peak currcnt ratio (B/A) could be matched by simulation only if the equilibrium constant for the adduct formation was approximate1y unity and the rate constant k , % 105s-'. The oxidation of W in the presence of mdlimolar concentrations of P(OBU)~ is shown in Figure 3-30. Peak B, the reduction of W2+, i s seen to decrease, particularly at low scan ratcs. The new peak, C, which occurs at morc negative potentials, was assigned to the one-electron reduction WPM. Two routes are available to produce WP2 +: the addition of P to W, followed by oxidation and the exchange of P with S in WSZ+.Only when all these steps were included could simulations nlin~icthe experimental results, including the scan rate dependence or thc peak currents. The complete mechanism, in the presence of

Figure 3-31 Simulations or the experimental CVs ill Figure 3-30. (Adapted Reference 14.)

,

from

the ligand, involves he following additior~alsteps: W + t P+ W P *

wP"

WS"

K,

=

+ e & WP* E"'2 + p=WP2' + S

ioJ

M-IS-'

K , = 10,k3=S

MIS-'

10, k ,

=2 x

- -0.74V -

Simulations of lhc CV oC W with added phosphite ligand are shown in Figure 3-31.

3.8

1

0.5

0

w

5

-1

-1.5

vs. AdAgCl

Figure 3-30 CVs of 1 rnM W in the presence of 3.7 m M P(ORu),lP) at 228 K. (Adaptcd horn Reference 14.)

Mechanisms Involving Adsorption

I t often happens that an electroactive species h:~ssignificant interaction with the electrode surface. I n such cases, the possibility exists an adsorbed species will undergo electron transfer processes (Figure 3-32). While few quantitative mechanistic studies involving adsorkd species have been attempted with CV. several common characteristics of the CV response for redox processes invojving adsorbed species are readily recognizable. The CV response involving only adsorbed species is characterized by symmetric cathodic and anodic waveshapes. The relative positions of the two

CY CLlC VOLTAMMETRY la)

Figure 3-32 An eiectrochemical rncchanistt~can include participation of an adsorbed species.

Potential ( V )

depend on the relative adsorption energies, af the oxiiii~edand reduced species (Figure 3-33). The equation for the current volts2e is:

[&I

n2F2vArox

exp

[$-

IE - E")]

where the reduction poren tials refer to the unadsorbed species and

r,, = surface coverage (mol/cm2) h,, = exp( - AG:ds} Potential ( V )

The peak curretlt and potential are:

Note that rather than the aquare root dependence found for the diffusioncontrolled response. here we have the peak current proportional to the scan rate. A lug-log plot of peak currcnt versus scan rate will give a slope = 0.3 for diffusion and 1 for adsorption. This is the primary quantitativc method by which simple adsorption peaks are distinguished from diffusion peaks. To avoid misleading results, however, care must be taken to subtract capacitive and Faradajc background currents. Adsorption peaks can occur along with diffusion peaks. The extent of overlap is determined by the relative strength of adsorption of the oxidized and reduced species. For strong interactions, adsorption pre- or postwaves can be clearly

Figure 333 CVs in which both the oxidized and reduced forms,arestrongly adsorbsd: AGor.rda = AGrrdads. Ib) *Goa,,h < AGdad,, and (c) AGma, > A C ,, .

(a) L..

.

100

CYCLIC VOLTAMMETRY

distinguished, depending on the relative strength ofthe interactions.la However, diffusion and adsorption peaks may be overlapped and difficult to distinguish. A superficial analysis may result in the adsorption component being overlooked. Quantitative treatments of adsorption require a knowledge of the adsorption isotherm, which may be potential dependent." An electrochemical study of the mechanism of action of the naturally occurjng antimalarial artemisinin, shown in (Figure 3-34a) illustrates how a cyclic voltammetric diagran~of a process involving adsorption was useful in developing an understanding of a pharmacological mechanism." Isolated in 1971, artemisinin has recently been widely used in Asia for the treatment of malaria. I t has been suggested that intraparasitic hemin (Figure 3-34b) mediates the antimalarial activity of artemisinin. Malaria parasites contain precipitated hemin, known as hemozoin. In vivo, a reaction between hemin and artemisinin appears to generate free radicals, some of which appear to alkylate hemin." It is reasonable to suggest that hemin is involved in a reductive decomposition of arternisinin, in analogy with known action of hemin on inorganic peroxide. To test this idea, Zhang et al.19 examined the cyclic valtrzmmetric responses of hemin and artemisinin separately and in the same solution. The results supported the earlier models for the action of artemisinin and showed that electron transfer is a viable mechanism Tor the interaction of hzmin and artemisinin. Artemisinin shows an irreversible, diffusi on-controlled reduction peak at around -1 V at moderate scan rates (Figure 3-35) The cathodic transfer coefficient, determined with Equation 2.5. is x .-- 0.35, indicating substantial structural change upon reduction. This structural change is most likely the cleavage of the Ubond. Hemin has an quasi-reversible reduction at -0.38V, resulting Cram the reduction of an adsorbed species (Figure 3-36). The log-log plot of current versus scan rate has unit slope (Figure 3-37).

A SURVEY OF ELECTROCHEMICAL MECHANISMS

-10

--

-0.3

-0

6

-0.9

-1.2

-1 j

Potential I V )

Figure 3-35 CV or 5 rnM concentration of artemisinin at 0.3V/s.

When artemisj~linis in thc presence of hemin concentrations abovc 0.01 mM. the - 1V arternisj~linpeak disappears completely and a new peak appears at around -0.45 V (Figure 3-38]. This peak can be interpreted as resulting from the catalytic reduction of artemisitlit1 mediated by the adsorbed hemin. The mechanism is sketched in Figure 3-39. Presumably, the coordination of the iron to the dioxygen acts as a kind of ir~nersphere (conduit for reduction of artemisinin.* In the abscnce of the Fe, tlie reduction is extremely slow and takes

Poteritial ( V )

Figure 3-36 The reversible reduction of adsorbed hemin, 0.3 V/s.

Figure 134 (a) Artemisinin and (M hemin.

* Iron compounds have long k n known decnmpo~ition.'~

to

function as catalysts for hydrogen peroxide

1

A SURVEY OF ELECTROCHEMICAL MECHANISMS

CYCLIC VOLTAMMETRY

Figure 3-39 The catalytic reduction of artemisinin mediated by adsorbed hemin. Log of Scan Rate in V/Sec

Figure 3 3 7 Test for adsorption: a log-log plot of the peak current versus swn rate lor the reduction o l hemin is linear with a slope of 1.

References I. Lm, K. Y.; Amatore, C.; Kochi, J. K. J. Phys. Chem. 1991,95, 1285. 1. Pross. A. 4cc. Chcm. Res. 1985,18, 212.

place by an outer sphere electron transfer between the carbon electrode and Ihe artemisinin. The hcmin-mediated decomposition of artemisin in vivo can be understood to take place via an inner sphere similar to that suggested by the CV results. In the ---- case -. .- . of the CV exweriments, the electrode is the source of electrons, whereas -. the participation of thiols such as glutathione as electron donors is suspected jn the physiological reacfion. The slow, outer sphere, reduction in the abance d -.exposed" Fe also explains the low toxicity of artemisinin.

3. Pierce, D. T.;Geiger, W. E. J . Am. Uhem.Soc. 1989, 111, 7636. 4. Laviron, E. J. ELctrounaI.

!

1[ 1 i

I1

6. Chambers, J. Q. In The Chemistry uf thc Quinoid Compnunds; Val. 2: Patai, 3.. Rappoprt, Z., Eds., John Wiley & Sons: New York, 1988, pp. 719-756.

7. Saveant,J. M. Acc. Chem. Rt7.r.IW, 13, 323.

8. Astruc, D. Anqew. Chem., Int. Ed. E~rgl.1988, 27, 643.

9. Channon, M.; Tobe, M. L. Ange~..Chn~r..I v ; . Ed. Engl. 1981, d l , 1 -86. 10. Gosser, D. K.,, Jr.; Rieger, P. H. .Jriul. Chrrrl. 1 9 s . 60. 1159. 11. Hershberger, J.

I I

;

Am. Clrrm

13. Rusling. J. F.; Ito,

14. Bourdillon, C.; Dernaile, C.; Moiroux. J.: Snvbnt, J. M . J . Amer. Chem. Soc. 1W3,115, 2. 15. Chi, C.; Zhang,

W.;Birke, R. L.; Gosser, D K..Jr.: Lombardi, J. R . J . Phyr Chrm. 1991, Q5,

6276. 16.

k K. Y.;Kuch~nka,D. I.Kochi, J

1;. I

m g Chin 1% 29,4196.

17. Zhang Y.;Gasser, D. K.,Jr.; Rieger, P H , Swiegart, D. A. J . Am. Chem. Sot. 1991,113,4062

i

1

2 0 ~ d n i c kS? R+;

C

.Yvc. 19W, 105. 71.

K.Anul. Chim. Acru 1991.2.72. 23.

18. Bard, A. J.; Faulkner, L. R. Elecrroclrt~miral Merhdt, 1980.

e1. Figure 3-38 The C V of I mM artemisinin and 0.W6mM hemin. 0.3V/S.

W.;Klinger, R. J.; knchi. .I. K.J .

12. Andrieux, C. P.;Hapiot, P.;Savdani, J. M. ChtPnr.R ~ T .1990. 90,723.

F

Poten0;rl I V )

Chem. 1984,164, 213.

5. Dcakin, M. B.; Wightman, R. M. J . Electrounal. Chem. 1%,2116, 167

151

e d . John Wiley & Sons: New York,

19. Zhang, F.; Gosser, D.K, Jr.; Meshnick, S. R. Bio. Pharm. 1992. 181.

A ; R ~ WA.; X& c M.;Pan,n.Z. M

d

B i m h n Pormifo,. Im, 49,

CHAPTER

The Simulation of Electrochemical Experiments

i

' :

There are many different approaches to solving the mathematical equations that dcscribe electrochen~ici~l experiments. Because electrochemical mechanisms that involve coupled chemical reactions can be quite complicated and varied, our primary requirement 1.; that the method of solution be general. The method of simulation by explicit finite differences (EFD)' lends itself to a quite general trea~rnent.~ The EFD method can also be optimized to improve computational speed without compromising its generality. The straightforward manner in which the physical problem is translated into a numerical method also makes the EFD method attractive to study. More detailed treatments than that presented here can be found in the chapter by Maloy3 and the book by Britz?

1

;

1.1 The Discretized Difirinn Equation

I

r

!

! 1 \

i

Figure 4-1 shows a generalized electrochemical experiment that includes diffusion of compounds in solution, reduction or oxidation at the electrode, and chemical reactions in solution. Although adsorption processes can be included in the framework of a simulation analysis, they are not considered here. The fclllowing partial differential equation describes what is going on in solution:

THE SIMULATION OF ELECTROCHEMICAL EXPEIUMENTS

CYCLIC VOLTAMMETRY

mur*-/ -

ox,",,

--

~ e d

Re$,,.

Butler -Volmer i?,quation

k

.-.Red,

DIffuslon

107

Ths is the discrete version of the right-hand side of Fick's second law, where i i s an index that keeps track of spatial points. Considering only the first two terms of the forward series. and regarding the variables y and z as concentration and time (and introducing the time index j). we have:

Products

the left-hand side of Fick's law. Equations (4) and ( 5 ) can be rearranged to give the discrete version of Fick's law:

Chemical Reaction

ac/at=~a'c/ag ac/atNRl

Figure 4 1 Model for the electrode process includes diffusion (Fick's laws), electrode

kinetics (the Butler-Volmer equation), and chemical reaction kinetics.

where

C, = concentration of s p e c k q r = time x = distance

kg = chemical rate constant for the nth solution chemical reaction C,,, (.,,= concentrations for the first and second species for the nth solution chemical reaction rate expression This is Fick's second law modified by a term to account for a generalized set of first- and/or second-order chemical reactions. Because of the symmetry oft he situation, we need consider urily drffusion that is normal to the electrode. Near the electrode the concentrations will be determined in part by the potential of the elect rode (through the Butler-Volmer equation). At a distance beyond the diffusivn layer. the concentrations will he at bulk values. These are the boundary conditions for Equation (1). The Fjck's law part of Equalion ( 1 ) can be coriverted to hscrete form through an application of Taylor serics apyrtlxi~nationsof a function around a point. The "forward" and "backward" series are given in Equations (2) and (3) respectively.

'

i

where the concentration in the nest tune elzmetlt is giveti by the preceding concentration and its neighbors. Equation (6) is the framework of an explicit finite difference it-~ulation. The e~ectrochemicalexperiment can be descrikd by a discrete time (of the cxperiment) and space (distance from the electrode) grid (Figure 4-2), where t = 0 at the beginning of the experiment and x = 0 at the electrode surface. If the concentration of every species is known for every space and time grid point, then the experiment is completely described. A point on the grid represents the concentration for an entire volume element, the boundaries of whch are the midpoints between the grid points. Equation (6)can be said to describe the exchange of concentrations between

Adding the two series, and regarding the variables y and z as concentration and distance, respectivsly. we iirrive at:

a2C - 5

axZ

D [ C i - , - 2 C i + C ,+t]

AX^

f

(4) L

Figwe 4 2 The time (or experime~~t) and space {distance horn electrode surf=) that is defined by the explicit finite diierence approach. -.

-

. .

. ----*---

,,

grid

THE SIMULATION OF ELECTROCHEMICALEXPEWENTS

neighboring volume elements. We can take a closer Look at how the space and time grid i s related to the discretizd diffusion equation. In any one time increment dt, diffusion propagates only to neighboring volume elements. This brings in a constraint to Equation (6).that is:

109

tration gradients of the oxidized and reduced spenes by the Butler-Volmer

equation:

The e x ~ r i m c o ttakes place over a lotal time t,,, so that the value of 6t will

depend on the n u r n b r of time increments nt:

value of D,if nt is chosen, then Sx is fixed by the constraint of For a equation (7). In addition, t,, fixes the total size X of the space grid. A safe estimate to include the entire-diffusion layer is:

X

=

where the heterogeneous rate constants k are in centimeters per second. It is also true that J,, = -J,,, Solving equations (1 1) and (12) for C,, and C,,, and substituting these values into the heterogeneous rate equation results in the following expression for the flux:

qDi*xp)1/2

(about 4 Gmes thc mean square diffusion length). Then the nurnkr of spatial elements ns is:

where the rate constants are given by:

4.2 Evaluation of the Boundary CondiLiurm Consider the spatial cells near thc electrode (Figure 4-3). The distance between the center of the first spatial element (the first grid point) and the surface of the electrode is

fdx. The current flux, J = -f/nFA, is determined b y the concen-

from the Butler-Volmer equat ton. This approach can k extended to two or more consscutive electron transfers. The flux cxprcssion can be incorporated into an expression similar to Equation (6).The flux (mol cmL2s-l) is converted to a concentration change by multipiying through by the appropriate dimensional factors:

4.3 Dimensionless Units The use of dimensionless units simplifies the construction of programs for simulation. All the variables we have discussed can be made dimensionless by normalizing them t o a standard value. For instance (denotitlg the dimensior~less unit with a double dagger), we can write:

1

-3

A d

~ look r at the space gnd near the surface of the electrode.

-

.

THE SIMULATION OF ELECTROCHEMICAL EXPERIhlENTS

CYCLIC VOLTAhfMETRY

I10

111

Rocedure input. {Input all the initial conditions, experimental parameters, a d electrochemical mechanisms. Convert to dimensionless units when appropriate.)

Procedure Electrude, {Calculare the flux due to electron rrander at the electrode, according to the Butkt-Vdrner equation. Output the simulated current potential To a data file.)

where dx a d 6t are the time and space illcrerncnts of the simnlat,inn, C , is a chosen normalizing concentration (typically the co~~centratinn of the principal electroactive species), and 0,is a chosen normalizing diffusion coefficient. The heterogeneous rate constant I S converted to dimensionless form hy multiplying through by the appropriate simulation units. The equation for diffusiol~ from the first spatial cell:

Procedure DiRusron;

{Calculate the drffusion of each specles according to the discrete version of Fick's law. Modlfy concentrations In each spatial element.) Procedure Reaction; [Calculate the extent of reamion for all species and modify accordrngly rhe concentrarrons

calculated earlier in the diffusion procedure.) Main P?ogram Begin Input; For I = 1 to nr do

is written in dimensionless units as follows.

Beg~n

Ekctrode: D~ffusion; I, '-

Reaction, end; End.

I f all caIcdations are made in dimensionless units, how would one get back to real units? With dimensional analysis. it can be swn that:

I-,'

A program in Turbo Pascal to simulate a cyclic voltammogram for an EC mechanism follows.

where J is needed in units of r n ~ l c m - ~ s -and ' C , is r n o l / ~ m - ~ . FinaIly flit current (A) is i = FA.1

Program C'J;

i

(A program to illustrate simulation by expllcit fin~tedifferences-Simulates an EC mechan~sm) '

+

(SN )

4.4 Solution Chemical Kinetics

Uses Crt;

Homogeneous chemical kinetics can be treated in a number of ways in the context of simula~ion.~ Thc simplest way is to use a simple differential approximation. For instance, the change in concentrati~ndue to first-order chemical kinetics, A 3 B, can be calculated as follows:

sc = - kf[C,]dt

{enables math c~processor)

(25)

nl, ns, k. a, b. r, s : longint; Pol, ipot. spot, ipot. T, X: extended

delx. delp, delt, E, smnr, area .extended: Current, khet, kf, kr, kchem :extended:

I

C : arrayrl . .3,1 ..ZOO] of extended; Ctemp: arrayrl . 3.1 . . 2001 of exrended; J. array[l . . 3 ] of extended; outfile:text;

This approximation will be accurate for kdt i0.1. This means that for large k, we will be f o r d to use a small 6 t and thus many time increments.

Procedure Setup;

4.5

begin

A Sample Simulatiun Program

write ('What 1s 1l1ereducliurl pulrrllial ill volts 7 :'); readln (E); writein;

A recipe for a Pascal program that will sinlulate cyclic voltammograms based on the previous discussion is listed here. Prwram Asimulation:

i

write { ' W M is the hefwwneous readln(khet); wrifeln: . . . writelnl:

...

..,

.

,

rate constant T ) ,

CYCLIC VOLTAMMETRY

112 writeln; write ('What is the k

(sec -1) of the following reaction ? :');

readln (kchsm);

THE SMULATION OF ELECTROCHEMICAL EXPERIMENTS assign (outfile.'data.pas'): rewrite (outfile);

end; {end of procedure setup)

writeln;

write {'What is the initial potential ? :'): readln(ipo1): writeln; write ('What is the switching potential ? :'); readln (spot); writeln; writeln ('final potential = initial potent~al'); fpot : = ipof;

writeln; write ('What is the scan rate ? :'I; readln (scanr); T :2 abs(spot-inpot)/scanc (time of experiment) X = 2 sqrt(1E-5 T); {assuming D = 1 E-51

--

nt := 150, {A default number of time increments] area : = 0.01; delt := T/nt;

if kc hen > 1 00 then

Procedure Electrode; Begin

-

kf := khet exp ( - 19.46*(pot-E)); kr := khet exp (1 9.46=(pot-E)):

.

.

Current : = J [I ] (delxldelt) 96484 1E-6 writeln(outf1le,pot~9:5,~'.current: 1 2); end: {end of procedure Electrode;

.

area: {This is i/Area)

Procedure Diffus~on;

Beg~n Fork:= 1 to3do begin Ctemp[k.l!

begin dell . = 0,3/kchem:

. = C[k.l ]

+ 0.45.

(C[k,2] - C[k,l 1)

- J [k];

For b := 2 tons do

nt := trunc(T1delt);

end:

Ctemp[k,b] := C[k.b]

C[k.b1 +C[kb+l]);

.

kchem := kchem delt: {make it dimensionless] delx := sqrt(1 E - 5 delt/0.45): ns : = tiunc (X/delx). khet := khet

deltldelx;

For a := 1 to ns

+I

+ 0.45 - (C[k,b-I] - 2.0

end; end; end: {end of procdure Diffusionj Procedure Chemreact;

do If kchem > 0.00 then begin Fork := 1 to nsdo begin

Ctemp[l ,a] := 1.00; Ctemp [2.a] := 0.00; Ctemp[3,a] : = 0.00; end; end; end; {end of procedure Chemreact]

end;

for a : = 1 t 0 3 d 0 begin j[a] := 0.0

begin

end;

- ivot)/nt;

Delp := 2

(spot

pot := ipot

+ delp;

Writeln('CV Simulation of EC Mechanism'); writeln;

CYCLIC VOLTAMMETRY

CHAPTER

setup:

5

begin

CVSIM: A General Program for the Simulation of Cyclic VoltammetryExperiments

begin

for r : = 1 to ns do begin

end; end: electrode: diffusion: chernreact;

+

if a -r nt12 then pot := got delp; if a > = nt12 then pot := pot - delp;

end; close(outfile); writeln; writeln('DATA FILE IN DATA.PAS'); writeln('H1T RETURN TO EXIT'); readln;

5.1 An Overview of CVSIM

end.

References 1. Feldberg. S. W. A Generul Meihodl;,r Simulation, Vol. 3 in ~le&roanal~tical Chemistry Series. Maml Dekkec New York, 1969. 2. Gosser, D. K..Jr.; Rieger, P. H.Aml. Chem. 1988,611, 1159- 1 167. I

3. Maloy, J . T. In Lnhorarury Techniques in Electroanulyticnl Chemistry, 1st HI Kissenger, .; P. T.; Heineman, W. R.. Eds Marcel Dekker: New York, 1984, pp. 417-461. 4. Brilz. D. Digitdl Sinitllutiurr i. Eluctruchrmistry. 2nd ed. Springer-Verlag:Berlin, 1988.

1

,

i $L

Computer simulation by the explicit finite difference method has proven to be a powerful tool for the analysis of cyclic voItammetric experiments. The shape of a cyclic voltammetric curve reflects both electron transfer at the electrode and solution chemical reactions that are coupled to the electron transfer. Thus, through a cyclic voltammetric study, one can often deduce a great deal of information regarding the electrode processes and chemical reactions in solution that are initiated by electron transfer at the electrode. Simulations are often helpful in the preliminary stages of a cyclic voltammetric study, assisting in predicting what possible mecbanism(s) can give rise to the cyclic voltnmmograms one observes. Once a particular mechanism has been decided on, rnte and equilibrjum parameters of the chosen mechanism cat1 he extracted by comparing expenmental results with successive simu1ations. Much attention has been focused on ~mprovingthe explicit fimte d~fferencemethod outlined in Chapter 4. Notable advances have been the description of an expanding space grid to save computation t~me.the inclusion of the mutually interrelated effects of IR drop and capacitive current,' and the development of general methods to treat solution chemical reactions in the context of the explicit finite difference method.' This chapter describes the structure and use of the computer program CVSIM.3 This general-purpose simulation program can simulate the CV response or a great variety ofelectrode mechanisms. Also described is a graphics -

-_

1-

--

I/

3 16

CYCLIC VOLTAMMETRY

CVSIM: SIMULATING CYCr.IC VCI7.TAMMETRY EXFERMENTS

program CVGRAF, for the visual comparison of two data sets (i.e., experimental and simulated). T h e executable codes for CVSIM and CVGRAF are included on a diskette with this hook. Hardware required is a PC-compatible (80x86)computer. A math coprocessor will speed up the calculations considerably. Chapter 6 describes another program, CVFIT, which incorporates a madified version of CVSIM for the least-squares analysis of zxperu~~etitnlCV data. Finally, the appendix describes DSTEP, a general program for the simulation of double potential step experiments. CVSIM is a PC-based program that enables the experimentalist to simulate the CV for nearly any desired mechanism and to compare the result with experiment. The desired mechanism is built up in a simple manner with a full screen display, and there are options for varying the dflusion coefficients of individual species. The value of such an approach lies in the elimination of the need to solve by analytiml or numerical means the partial differential equation I hat models each new rrlechar~isrn.Thus, more timc i s available €or the consideration of prnblems of a chemical nature, and a large amount OF repetitious work is avoided. In addition, by taking advantage of the irlcreasirig computational power OP personal computers, a generalized program can be an accessible toot for thc mechanistic analysis ofcyclic voltammograrns. Together, the programs CVSIM, CVFIT, and CVGRAF can be used for the efficientanalysis of CV and DPS data. The programs incorporate the following qualities. 1. Mechariislic gener.a/ify.The program CVSIM mes a modular structure with a general solution of the homogeneous chemical kinetics. This means that the user can simulate virtually any electrochemical mechanism that crtn be formulated as a combination of electron transfers at the electrode and homogeneous chemical reactions. Diffusion coefficients for each species can be specified. 2. Speed of computation. An expanhng space grid is used to minimize computation time. In addition, the program is written in Turbo Pascal 6.0, which can take advantage of the math coprocessors commonly used in personal computers. 3. Ease d u e . A Cull screen display for input of mechanistic and experiaentai parameters with options for corrections and changes is utilized. The paraIneters are entercd for thc most part in common experimental usage. not in dimensionless units. A graphic analysis program is provided for on-screen viewing of simulated or experimental files. Input is -error corrected"; that is, if a "fatal" error is made in the input (such as entry of an real instead of an integer), the program will re-request the information. 4. Inclusion of nonideul factors. The effects of 1R drop and capacitive current can be incorporated in the simulation. 5 . Quanrlratiw cumpurison oJexpt~rime~~ral and simulated dc~tir.A simplex routine is used to optimize the fit between experimental and simulated cyclic

117

voltammograms (CVFIT). Experimental and simulated results can be quickly compared visually (CVGRAF).

5.2 Extensions of the Simulation Method

The partial diflerential equation for linear diffusion and solution chemical kinetics is:

The solution of this equation in terms of the EFD method w a i described in Chapter 4. A discrete form of the diffusion term on the right-hand side of Equation (1) is

where i i s spatial subscript and j is a time subscript. Equation (2) is equivalent to the creation of a space and time grid (see Figure

.

:

4-2).Computation time can be decreased by utilizing an expanding space grid or an expanding space-time grid. In this work, an expanding spatial grid was utilized:an expanding time grid can cause inaccurate calculations of the solution chemical kmetics. The expanding grid was constructed such that the spatial increment doubles 111 size every fourth grid point. Near ihe electrode the wnc.t?ntrarionswill be determined by the potential (i.e., through the equations of electrochemical kinetics) and diffusion, and at "infinite" &stance the concentrations will be the same as the initial conditions.These facts constitute the boundary cnndition~for Equation (I). The Aux at the electrode was calculated according to the method descritd in Chapter 4, which takes into account the flux of species at the surface of the electrode due to both the heterogeneous kinetics (as described by the Butler- Volmer equalion) and diffusion.The potential E In the Butler- Volrner equation can be corrected far I R drop in a number of ways. Following the method of Evans et a].,' the IR term is inclu&d by reading an experimental current file and using a meissored tor estimated) resistance to calculate IR.

The capacitive cument is also calculated as in Reference f , using the I R drop corrected potential in calculating the capacitive charging current at each time step in the simulation. ~ l t h o u g hdiffusion and chemical reaction arc concurrent p r w c s in the explicit finite differencemethad, they are calculated separately. This procedure,

118

CVSIM: sTMULATING' CYCLIC V O L T A W E T R Y EXPERIMENTS

CYCLlC VOLTAMMETRY

OWION- CHANGE I . 2.3 OR 4 OR DEFAULT PARAMETERS (TEMPERATURE OR DIFFUSION COEFFICIENT} OPTION: READ PARAMETERS NECESSARY TO INCLUDE 1R DROP OR CAPACITIVE CURRENT (RESISTANCE, AREA, AND CURRENT)

which is valid if the time increments are small enough, leads to the possibility of cry general treatments of the chemical kinetic term in Equation (2). Simple numerical methods such as an iterative modified Euler method can be used to provide general solutions for the differential equations that model chemical kinetics. Thus, a simple algorithm based on the following equation provides a general solution chemical reaction kinetics:

END

PROCEDURE ELECTRODE BEGlN FOR I = 1 TO NCOUP DO :LOOP OVER ELECTRODE COUPLES) BEGIN CALCULATE THE CURRENT FOR EACH ELECTRODE REACTION END OPTION, CALCULATE CAPACITIVE CURRENT CALCULATE TOTAL CURRENT OUTPUT POTENTIAL/CURRENT VALUES TO SCREEN AND 50 DATA FILES END

where the changes in concentrations are obtained from a differential approximation for the kinetics using the initial concentration. For example, for the kirletic scheme

PROCEDURE DIFFUSION BEGlN FOR I = 1 TO NSPEC DO {LOOP OVER SPECIES] BEGIN FOR I = J TO NS DO ;LOOP OVER SPATIAL INCREMENTS; BEGIN CALCULATE CONCENTRATION CHANGE DUE TO DIFFUSION FOR SPECIES I END END END

the ~ t concentrations h for A and B are calculated as follows:

:

PROCf DURE CHEMREACT BEGIN FOR J = 1 TO NS DO (LOOP OVER SPATIAL INCREMENTS) BEGlN USE MODlFIED EULER METHOD TO CALCULATE CONCENTRATION CHANGES DUE TO SOLUTlON CHEMICAL REACTIONS. END END

k

BEGIN

1

The inclusion of ali the features described above in a general program for the simulation of cyciic voltammetric experiments is a matter of program structure. In general, the solution of this problem involves assigning index numbers to each species in the mechanism, to each electrode reaction, and to each solution chemical reaction. The use of program loops to include all reactions for all species is then implemented within the usual scheme for simulation by explicit finite differences. This is illustrated by the following skeleton code of the Pascal

i

t

program. I

VARIABLE DECLARATIONS {DECLARE ALL VARIABLES) PROCEDURE SETUP BEGlN READ THE FOLLOWING INPUT INFORMATION 1. NUMBER OF SPEClES (NSPEC) AND CONCENTRATIONS OF EACH 2. NUMBER OF ELECTRODE COUPLES {NCOUPJ. AND THERMODYNAMIC A N D KINETIC PARAMETERS FOR EACH COUPLE 3. NUMBER OF CHEMICAL REACTIONS {NREACT) A N D FORWARD A N D REVERSE RATE CONSTANTS FOR EACH REACTION 4. EXPERIMENTAL PARAMETERS {SCAN RANGES AND SCAN RATE] CALCULATE: FROM MAXIMUM CHEMICAL RATE CONSTANT AND EXPERIMENTAL PARAMETERS THE NUMBER OF TlME INCREMENTS {NT] AND NUMBER OF SPACE INCREMENTS (MS)

1

im***--*=***+*

M A I N PROGRAM

==-====a*-=.*

SETUP: FOR I=: 1 TO NT {LOOP OVER TlME INCREMENTS) INCREMENT THE ELECTRODE POTENT1AL ACCORDlNG TO SCAN RANGE, SCAN RATE AND AS OPTION CURRENT A N D RESISTANCE {IR DROP] BEGlN ELECTRODE; DIFFUSION CHEMREACT: END; WRITE OUT SIMULATION PARAMETERS TO DATA FILE ANOTHER SIMULATION? YES-GOT0 SETUP NO (FINISH SIMULATION) END.

CYCLIC: VULTAMMBTRY

120

53

Accuracy of Simulations

As discussed in Chapter 4, the number of space-time grid points (increments) used in a simulation must be carefully considered. As with any numerical procedure based on the construction of a grid, the mote grid points used, the more accurate the results of the sunulation (assuming no roundoff errors). However, the more grid points used, the longer the cornputation time. Thus, one desires to find the least stringent criterion for number of grid points necessary to result in accurate simulations. The Turbo Pascal variable type extended, used throughout for all real numbers, has about 20 significant figures and a maximum red number OF 1.1 x Errors due to roundoff error or linlitations of magnitude do not commonly occur. Simulation of electrochemical experiments by the finite difference method is a rather complicated numerical procedure: far more than, say, the Runge-Kutta (R-K) method for solving ordinary differential equations. In fact, the method (as presented here) is a coupling of the finite difference method for solving the diffusion equation (using an expanding spatial grid) with a second-order R-K method for solving homogeneous chemical reaction equations. A formal error analysis has not been performed; only ad hoc suggestions have been made. An approach taken by Feldberg, for instance, suggests a criterion based on the size of the reaction layer4:

CVSIM: SIMULATING CYCLIC VOLTAMMETRY EXPElUMENTS

121

Simulations of several mechanisms are shown t hat illust~atzthe difference in simulation time and the difference in the result when the two contrasting constraints are used. The simulation of an EC mechanism according to both the Feldberg criterion and our criterion shows no visible difference (Figure 5-1). Indeed. the peak currents d~fferby less than 0.2%. An ECE simulation is shown in Figure 5-2. Here again, there is nu sipnificanl difference. Finally, we consider a catalytic mechanism, where the electrode reactant is regenerated by a second-order reaction (k = lob m - ' s- I), with a 20-[old excess of the catalyst (Figure 5-3). This is considered to be a more severe test than the others, because the reactant is generated at the electrode surface. It is important to relate the accuracy to the goals of the experiment. For instance, this type of catalytic wave is usually used to measure rate cot~stants by comparing the diffusion wave (with only the substrate present) to the catalytic wave. Invariably, this procedure itself is not accurate to more than a few percent, and rate constants obtained ate usually reported to one or two significant f i g ~ r e s Tncidentally, .~ as a consequence of the autocatalytic nature of the mechanism, pseudo-first-order conditions are not achieved until a large excess of the ca~al ysi exists (200: 1). : I

(For a first-order chemical reaction).

Feldkrg has suggested that p = 56x; that is, the reaction layer should bc at least 5 times the grid space size. In terms of the number of time increments, we have:

and for fl = 0.45,

The number of time increments needed for a simulation then becomes nt = time of experinlent x 50 x kc,,,.

However, our experience in using the program CVSlM is that the constraint: nt

=4

x time of experiment x kc,,,

is sufficient to obtain desirable accuracy under most circumstances. The result is a time savings of a little more than an order of magnitude. The reasons for this particular constraint have been investigated in some detail,',' and it appears that because of the complex, n ~ u l t i s t enature ~ of the simulation, error estimates cannot be ohtined by focusing on one asp& (i.e., the reaction layer).

Potential (V)

Figure 5 1 Simulation or an EL' mechanism (R,,,, = 1000 s- ') with t k numhr or time increments (NT)set to NT = 4 x time x k,,,, (c~rcles). and NT = 50 x time x kc,,, (line).

CYCLIC VOLTAMMETRY

r

CVSlM: SlMULATlNG CYCLIC VOLTAMMETRY EXPERIMENTS

123

Generally, highly accurate results for all mechanisms have been obtained ( ~ 0 . 5 %error). In any case, it is important to remember that the number or time increments, which determines the accuracy, can be set b j the user. For any new mechanism, a test simulation can be done using a more rigorous constraint (use 10 times more time increments than the default) to check for accuracy.

I

I -

5.4 Installation and Use of CVSIM and CVGRAF All the files on the diskette provided should be copied onto the bard disk to the directory C:\SIMULATE. (First create the directory; then copy the files to the directory.) All the programs should then be run from C:\SIMULATE.

CVSIM is initiated by typing CVSIM and hitting the enter key. The experiment and the mechanism are described in response to a series of inquiries presented in ii ti111screen display.

Potential (V)

Figure 5-2 Simulation OF an ECE mechanism, with the number of time iltcrenlenls set as in Figure 5-1.

Number of Species in the Mechanism An integer equal to the total number of species in the mechanism is entered. Although the screen input is designed for up to eight spccics, more can be entered if necessary.* I I

i

;

i

i

Concentrations of Each Species The initial concentrativn of each species is entered in normalized units. That is, the concentration oi each species is divided by a chosen narn~alizingconcentration (Cnorm),typicall) the species present with the highcst concentration (excluding of course the eleclrulyte). Each species is now associated with an integer, in the order of entry. Su hsequent entries. which describe electrode reactions and chemical react ions. utilize these number idznt ificatiotis.

Number and Description of Redox Couples The number of redox couples in the mechanism is entered. Then the integer index of the oxidized and reduced Forms of each redox couple are entered, followed by the reduction potential ot each couple (in volts), I he heterogeneous rate constant (cmjs),and the cathodic transfer coefficient a. For an electron transfer that is considered to be reversible, a value of lOcm/s will ensure reversibility under most circumstances. If a two-electron transfer (where one of Figure 5-3 Simulation of a second-order catalytic reaction, with k,,,,, = 106M-'s and 20-fold excess of reagent. Tune increments set as in Figure 5- 1.

* line.

If the enter key is accidentally fiit without first entering data, simply enter data o n the next

CYCLIC VOLTAMMETRY

124

the redox species is shared by t w o couples) is among the electrode reactions, these should be entered in sequence when written as sedudions. For instance:

l+e--2

2+e--3

Solution Chemical Reactions Each chemical reaction (first or second order) is described by entering two indexes for the reactants and two for the products. (If the reaction is first order, zero is entered in the space reserved for the second species of a second-order reaction.) First-order rate constants are entered in reciprocal seconds. Secondorder rate constants are entered in units of M-'s-'*Cnorm.

Experimental Parameters The number of half-cycles ( 1 . 1 or 31, the initial potential. switching potential(s), final potential (V) and scan rate (Vls) are entered. Asymmetric potential waveforms, which are often useft~l,are possible in this schztne.

CVSIM: SIMULATING CYCLIC VOLTAMMETRY EXPERIMENTS

125

Option for LR Drop The effect of the 1R drop can be included as well. It i s necessary to provide an experimental data file of current-potential values (1 point per millivolt) and to enter the resistance and electrode area. The experimental current is read and the simulation potentials are corrected accordingly.* During the simulation, the potential-current values every f OOmV appear on the screen. This provides a feeling for the time the simulation will take and aIso serves as a guide to the progress of the simulation.

A simple graphics program for PC compatibles (CGA, VGA, and Hercules graphics) is provided for the simultaneous examination of two data sets. It is intended primarily for comparison of experimental and simulated data. The input is quite simple: initial potential, final potential (V), current scale (pA), and the names of the two data files are requested. Default values are provided. If only one data file is to be examind, a blank file named tdata.pas should be created. Movement of a crosshair cursor is controlled with the arrow keys, and potential-current values are displayed at the top of the screen. A sample CVGRAF display is shown in Figures 5-4.

Changes in Input At this point, the user can inspect the entire input screen and change selected parameters. It is also possible to change the default temperature (298.1S°K) or the default diffusion coeficient for each species (1.00 x 10-5cm2/s). The diffusion coefficients are changed by entering a multiplier of the default vduc (e.g., enter 0.5 for a diffusion coefficient of 5.00 x loz6cm2/s). If it is desired to simulate a mechanism with a diffusion coefficient larger than the default coefficient, the default diffusion coefficient itself can be changed.

I

,

,

I

5.5 Examples Figures 5-5 through 5-1 1 show CVs that exhibit mechanisms discussed in Chapters 2 and 3. Each figure is preceded by one or more exampIes that illustrate thc use and capabilities of CVSIM.

N m e s of Data and Record Files The names of the simulated potential/current data file and file that saves the entire simulation parameter set are requested. The current is output in amperes and the potential in volts. This format is required of data sets to be graphed with the CVGRAF program or to be used by the curve-fitting program CVFTT.

The Normalizing Concenimtion The choice of normalizing concentration is entered in rnillimolar units (rnM).

Option for Capacitive Current If it is desired to include a capacitive current in the simulation, then the electrode area (an2),double-layer capacitance (pF/cm2), and solution resistance (kQ) must be entered.

Figure 5-4 Example of the CVGRAF graphic analysis program. Two data sets can b superimposed (circles vs harsh for comparison. c he format of the data fik is deshibed on page 138.

I

I

PoLential (")

Figure 5 8 NO; reduction, an EC mechanism.

Figure 5.5 Simulation of ferricyanide reduction.

-2

-1.0

I

-1.2

i

!

-1.6

-10

-1.4

mechanism (see Figure 5-6 Simulation of the reduction of methylctrbalmin, a" EC Cha~ter6).

--

00

Fotential (V)

-U4

-013

-I 2

-1

6

-2U

Potctll~al;L

Figure 1 9 SimuIat~onof a two-electron transfer (see Chapter 3).

Potential (V)

1

CVSM: SIMULATING CYCLIC VOLTAMMETRY EXPERIMENTS

129

EXAMPLE 2: Methylwbalaimin reduction (Chapter 6). Number of Chemical Species: 3 Concentrations in normalized units: 1: 1 .oo 2: 0.00 3: 0.00

Figure 5-11 Example of slrnulation of fast sweep voltammetric analysis or an EC mrzhan~sm with a fast lollowlng reactidti.

Ox

+ e-

1

+e

RX1

2

--

Number of Electrode Reactions: 1 E khet alpha

Red

2

-1.529 0.012 0.78

Number of Chemical Reactions: 1 PR2 k-forward k-reverse 0 580.0 0.0

-

+ RX2 * P R t + + 0 3 +

EXAMPLE 1: Ferncyanide reduction.

Number of half-cycles ?: 2

p

r

Initial potential Switching potential Final potential Scan rate (V/sec)

Number of Chemical Species: 2 Concentrations in normalized units:

1: 1-00 2: 0.00

Number of Electrode Reactions: 1 Ox 1

1

+ e-

9

+e

9

1 1

Red E khet alpha 2 0.253 0.044 0.5

I

Number of Chemical Reactions: 0

Number of half-cycles 1: 2 : 0.5

I

I

Area of electrode (cm2) Normalizing concentration (mM)

II

IR option (YIN): N

I

II

1

I

:0.01 :1

I 1 1

Capacitive cllrrent (Y/N): N

Change # (Choose 0-8.0= no change): 5 Enter normalized diffusion coefficients 0.760 2: 0.630

1:

I

I I

: -1.0 0.300

:

Area of electrode (cm2) Normalizing concentration ( m M ) IR option (Y/N):

: 0.019 : 2.00

N

Change # (Choose 1-8. 0 = no changes): 6 Enter the detault diffusion coefflc~ent D = I .7E-6

Initial potential Switching potential :0.0 Final potential :0.5 Scan rate (v/s~L.) : 1.0

(

: -1.0 : - 1.65

Change lnput or Default Value: (5) Diffusion coefficients ( = I ) (I ) Concentrations ( 2 ) Electrode reactions (6) Default diff. coeff.(D= 1E - 5 ) (3) Chemical reactions (7) Temperature (T = 298.15 ) (4) Exp. parameters (8) MT ( # of time increments) I ..

A

"1,

Change l n p u t or D e f a u l t V a i u ~ s ;

I I I

I (1) Concentrations (2) Electrode reactions (3) Chemical reactions (4) Exp. parameters

(5) Diffusion coefficients ( = I ) (6)Default diff. coeff. {D = 1 E-5)

(7) Temperature (T = 298.1 5 ) ( 8 ) NT I # of time increments)

CVSIM: S M ~ A T I N C CYCLIC VOLTAMMETRY EXPERIMENTS EXAMPLE 4: NO; reduction.

EXAMPLE 3: CE mechanism (NO, oxidation).

r

7

Number of Chemical Species: 3

Number of Chemicel Species: 3

Concentrztions b normalized units: I : 1 .OO 2: 0.740 3; 0.00

Concentrations rn normalized units: 1: 1 .OO 2: 0.740 3: 0.00 Ox

1

RX1

2

-

+ e- * Red +e 2

-

Number of 8 e c t r o d e Reactions: 1 E khet alpha

1

1.32 0.03 0.48

Switching potential Final potential Scan rate (Vlsec)

+ +

0

100.0

5.00

Number of half-cycles 7: 2 1.10 : 1.6 : 1.1

1.32 0.03 0.48

Number of half-cycles 7: 2 1.60 Switching potential : 1 .I Final potential : 1.6 Scan rate (V/sec) : 0.10

.

Inilia1 potential

: 0.50

Normalir~ngconcentratian (mMj

Number of Electrode Reactions: 1 E khet alpha

RX1 -t RX2 PRI I- PR2 k-forward k-reverse 2 + 2 - 3 + 0 120.0 2.50

1:

Area of electrode (crn2)

+ e'c;. Red +e s 2

Number of Chemical Reactions:

Number of Chemical Reactions: 1 P R 2 k-forward k-reverse

+ RX2 u PR1 + 2 3

Initial porential

)

Ox

Area of electrode (cm2)

: 0.008 : 1.6

Norrnaliang concentration ( m M )

N

1R aprion (Y/N): N

IR option (Y/N):

Capacitive current (Y/N): N

Capacitive current (Y/N):

/

Change

1I

Enter the default diffusion coefficient D = 1.8E-5

#f

(Choose 1-8.0 = no changes):

6

I

'I

i !

I

I

N

Change # (Choose 1-8,0 = no changes): 6 Enter the default diffusion coefficient D = 1.8E-5

Change Input or Default Values:

Change Input or Default Values:

(1 ) Concentrations ( 2 ) Electrode reactions (3) Chemical reactions (4) Exp, parameters

I

: 0.008 : 2.0

(5) Diffusion coefficients ( = I ) (6) Default diff. coeff. (D = 7 E-5) (7) Temperature (T = 298.15) (8) NT ( # of time increments)

-

(1) Conrentratians ( 2 ) Electrode reactions (3) Chemical reactions (4) EXP. parameters

(5) Diffusion coefficients ( = 1) (6) Default diff. coeff. (D = 1E-5) (7) Temperature (T = 298.15 ) (81 NT ( # of time increments)

131

CVSIM: SIMULATING CYCLIC VOLTAmETRY EXPERIMENTS

CYCLIC VOLTAMMETRY

132

EXAMPLE 5: Two-electron transfer. I

i

1

Number of Chemical Species: 3 Concentrations in normalized units: 1: 1.OO 2: 0.00 3 0.00

Number of Chemical Species: 3 Concentrations in normalized units: 1: 1 .oo 2: 0.00 3: 0.00

Number of Electrode Reactions: 2 Ox + e - 9Red E khet alpha 1 + e u 2 -1.421 0.50 0.5 2 + "-3 - 1.403 0.0006 0.5

Ox 1

I RX1

3

+ +

9

2

+

Initial potential Switching potential Final potential Scan rate (Vlsec)

1 (

2

39.30

78.60

I

: 1.8E-3 : f .3E-3

1I

I

Capacitive current (YIN): N Change # (Choose 1-8,0 = no changes): 6

I

9

9

Number of Electrode Reactions: 2 E khet alpha 2 0.0980 503.0 0.5 3 0.5690 0.021 0.5

Red

Number of Chemical Reactions: 0

: : : :

Number o f half-cycles 7: 2 0.700 - 0.200 1.200 0.20

Area of electrode (cmZ) Normalizing concentration (mM)

=

no changes): 5

IR option (Y/NJ: N Capacitive current (Y/N): N

Change # (Choose 1-8, 0 = no changes): 6

!

Change Input or Default Values:

(5) Diffusion coefficients ( = 1) (6) Default diff. coeff. (0= 1 E-5) (7) Temperature (T = 298.15 ) (8)NT ( # of time increments)

I

l ( 1 ) Concentrations (2) Electrode reactions (3) Chemical reactions (4) Exp. Parameters

I

: 0.01 : 'I

Change Input or Default Values:

Change # (Choose 1-8, 0

L

-

D = 5E-6

= 2E-5

(1 ) Concentrations (2) Electrode reactions (3) Chemical reactions (4) Exp. parameters

+ +

ee e-

Initial potential Switching potential Final potential Scan rate (V/sec)

Number of half-cycles7: 2 -0.20 : - 2.0 : -0.20 : 10.0

IR option (Y/N): N

+

I

:

Area of electrode (cm2) Normalizing concentration {mM)

D

2

Number of Chemical Reactions: 1 R X 2 9 P R I + PR2 k-forward k-reverse 1

133

EXAMPLE 6: Protonations at equilibrium.

(5) Diffusion coefhcients ( = 1 ) (6) Default diff. coeH. (D = 1 E-5) (7) Temperature (1= 298.1 5) (8) NT ( # of time increments)

I

I

I

-___

CVSIM: SIMULATING CYCLIC VOLTAMMETRY EXPERIMENTS

CYCWC VOLTAMMETRY EXAMPLE7: ETC mechanism.

134 -

EXAMPLE 8: Fast sweep voltalnmetry.

_--

Number of Chemical Species: 4 Concentrations in normalized units: 1: 0.330 2: 0.000 3: 1.000 4: 0.000

Ox 2 4

-+ + +

+

2

+ +

4

I

1

--

RX2 3 1

c7 G

1

1

I

+ +

Number of Electrode Reactions: 1 e- o Red E khet alpha e s 2 -1.68 2.90 0.5

+

+

Number of Chemical Reactions: 1 + PR2 k-forward k-reverse 0 8.OE+5 0.00

RK2 + PR1 0 a 3

+

Number of half -cycles7: 2

1

Initial potential Switchirlg potential F ~ n apotentiai l Scan rate (V/sec)

:

:

-1.2 - 2.0

: -1.2 2,28E+5

:

Area uf electrode (cmZ) Normalizing concentration (mM)

: 0.005 : 0-3

: 1.OE-2 : 1

IR option (Y/N): N Resistance (kohm : 1

LR option (YIN): N

Capacitive current {Y/N): Y Double-layer capacitance (fiF cm-2): 20 Resistance (kohm): 55

Capacitive current (Y/N): N = no

1

2

Number of half-cyclw 1: 2 Initial potential = 0.0000 Switching potential= 0.7000 Final potential = 0.0000 Scan rate = 0.8000

C h a n ~ e# (Choose 1-B.

Or

RX4

Number of Chemical Reactions: 2 PR1 -I- PR2 k-fonrvard k-reverse 39.000 0.000 4 4 0 2 + 5 1500.000 0.000

Area of electrode (cma) Normalizing concentration (mM)

Number of Chemical Species: 3 Concentrations in normalized units: 1: 1.oo 2: 0.00 3: 0.00

I

Number of Electrode Reactions: 2 e- e Red E khet alpha e1 0.190 0.048 0.50 e5 0.520 0.0230 0.50

RXZ

135

changes): 7 T = 200

Change # (Choose 1-8, 0 = no changes): 6

D = 5E-6

Change lnput or Default Values:

1I 1

Change lnput or Default Values: (1 ) Concentrations (2) Electrode reactions (3) Chemical reactions ( 4 ) Exp. parameters

( 5 ) Diffusion coefficients ( = t ) (6) Default diff. coeff. (D = 1 E - 5 ) (7) Temperature (T = 298.15) 18) NT ( # of time increments)

1

CYCLIC VOLTAMMETRY

CHAPTER References 1. Bowjrer,W.J.; Engelman,

E. E.; Evans, D. H. J . Elcctrt.tt~nal.Chk,)~~. 1989, 241. 67.

2. Gosser. D. K..Jr.. Rieger.

P. H. Anal. Chum. 1988, 60, I159.

CVFIT: S i m ~ l e xData Analyisis w&h CVSIM

3, Gomr. D.K..Jr.: Zhang, F. Talanta 1991,38, 715. 4. Feldberg. S. W . J . Electruanal. Chem. 1W,290, 49. 5. Britz. 0. Digital Sirnulotinn in Eiedrochemistry, 2nd ed. Springer-Verlag:Derlin, 1981.

6. Andrieux, C.P.: Hapiot, P., Saveant, J. M. Chem. Reu. 1990, 90,723.

6.1 CVFIT: Simplex Data Analysis with CVSM

I

I

;

k

i

i

Simulation analysis of cyclic voltammograrns has typically been done by visual comparison of experimental and simulated data. Many simulations are performed, varying unknown parameters (reduction potentiab, rate constants, etc.), untii a good visual fit is obtained. An alternative, more rigorous, approach is to make a quantitative comparison of the entire cyclic voltammogram with t h a ~ predicted by theory. The program CVFIT quantitatively compares cxpcrimental and simulated data in a systematic manner.' It uses the Welder-Mead simplex algorit hm2 to minimize the least-squares difference between the simulated and experimentnl cyclic voltammograms. In prjnciplc, CVFTT permits the analyuis of any cyctic voltammogram that can be simulated with CVSTM. However, as with Any fitting procedure, interpretation must be tempered with an understanding of the method. The first requirement is the cullecrion of lugh quality data. Care *nust be taken to avoid adsorption of electroactive species and to properly correct background currents. Data analysis should be performed on independent data sets to e n w e the reliability of the parameters obtained. In some cases the computation time required will make the use of CVFIT impractical. However, despite i t s limitations, data analysis with CVFIT can expand the horizons of cyclic voltammetry. This chapter provides instructions for the use of CVFIT, presents a prototype analysis, and offers several suggestions regarding the analysis of cyclic volt ammograms.

CVFlT: SIMPLEX D A T A ANALYSIS WITH W S [ M

CYCLIC VOLTAMMETRY

138

6.2 Instru~tionsfor the use

of CVFIT

6.2.1 Preparation of the Experimental Data File The data file should be an ASCII file (with filename "data.pas")of potential (in volts or millivolts) and current (in amps). Each voltage/current data point should be separated by at least one space and should occupy one line. Experiential notation is permissible. For example, Tor an initial potential of 0.000 V, a switching potential of -0.500V,and a final potential equal to 0.000V, the data file could Iook like the following.

l'otcntial ( V )

Figure 6 1 An example u l a fit (line) done on simutated data with added noise. The data wcre originated from a simulation with E0 = -0.200V, ko = O.O2cm/s, 2 = 0.5,;md area = 0.1 cmZ.The fitted parameters were E0 = -0.1996 V, k 0 = 0.02096cm/s, o: = 0.5. and area = 0.0994cmz. The initial guesses were E = -0.25, k = 0.01, z = 0.3, and area = 0.2. The four-parameter fit took I68 simplex iterations.

Note that the potential starts at an jn~rement of 1 m V from the initihl potential, and the last potential is exactly the final potential. Strictly adherence to this form is necessary for the proper operation of CVFIT. The experimental data fiks should follow the form of the simulated data files. For comparison of experimental data with simulated data with the program CVGRAF, the potential should be in units of volts.

current scale of the experimental data. Typical values of tolerances range from i E-8 to I E- 12. Finally, CVFIT asks for the total number of parameters to be fit. The simplex procedure requires at least two parameters. In principle there is no upper limit. However. a practical limit is about five or six. A good way to bzcarne acquainted with CVFIT is to generate data with CVSIM and fit the data with CVFIT. Such an example is shown in Figure 6-1, where simulated data with noise added were fit for a reversible CV.*

6.2.2 R u d n g the Program

6.3 A Prototype Analysis: The EC Mechanism

The actual operation of CVFIT is quite similar to CVSIM, and the aspects that are the same need not be repeated in detail. The mechanism is described and the parameters entered as with CVSIM, except that each parameter must be identified as a constant or as a parameter that will be optimized during the fitting procedure. This is done by entering 0 or 1 when a parameter i s requested, prior to entering thc parameter. Zero signifies that the parameter will remain constant, and 1 signifies that the parameter will be optimized. The following parameters must be identified iti this way: diffusion coefficients of individual species. if different from the default value; reduction potentials, heterogeneous rate constants and transfer coefficietlts, homogeneous rate constants, and electrode area. 111 the program CVFlT a tolerance n~uslbe stipulated. The tolerance is a criterion fur ending the fitting procedure. it is defined here as the largest difference between the least-squares difference between any two simulations based o n the Np -+ 1 parameter sets that are continually generated by the simplex procedure (Np = the number of parameters to be fitted). The tolerance is bnsed on the current data sets, which are in amperes. The tolerance required for a reasonable fit is somewhat a matter of trial and error, and it depends on the -. -. ,,

,

,.-.-

I

_ ....

_ . .

.

,-

' 1, I

,

1

Reduction potentials are most commonly obtained by cyclic voltammetry for a simple E mechanism. When a fast following chemical reaction occurs, no reverse wave is observed and the peak potential and waveshape are both affectcd by both the heterogeneous and homogeneous rates. It is sometimes possible to measure reduction potentials using fast scan rates. However, slow hcterogeneous rate constants and growing distortions due to I R drop and capacitive current work against this strategy. An alternative melhod is to combine double potential step chronoamperometry with the CV simulation analysis. In this way a complete characterization of the EC mechanism is possib1ev3 The reduction of methylcobalamin is a typical EC mechanism in which a n electron transfer is foIIowed by a fast and irreversible chemical reaction.'' The

*

See page 152 for more information on using CVFIT. For a review of B,, electrochemistry see Reference 4. More m Referen= 5.

.

i work mn be found

in

140

CYCLIC VOLTAbfmTRY

mechanism of the metbylcobalnmin reduction wus suggcstcd by Lexa and Saveant using rapid sweep cyclic voltammetry in 1 : I ratio of DMF and 1propanol.6Based on these and other results, (7- lo), it appears that the chemical reaction that follows reduction is a cobalt -carbon bond cleavage. A working electrode that gives iacjle charge transfer is essential in the characterization of thc electrochemical properties of a system by simulation and fitting analysis. Methylcobalamin usually shows adsorption on common electrode materials. After an exhaustive examination of electrode materials, it was found that a silver amalgam electrode gives the best electrode response. The proper solvent mixture also required some detailed consideration. Methylcobalamin has limited solubility in DMF, which bas a higher freezing point than methanol. On the other hand, methanol has a lower dielectric constant than DMF, which causes higher resistance, particularly at Iow temperatures. The optimum solvent mixture was 40: 60, DMFjmethanol. The uncompensated resistance was analyzed by potential step chronoamperometry. The initial potential was set the same as the initial potential in cyclic voltammetry experiments, and then a step of - 50 mV to the initial potential was applied. The analysis of the RC decay curve was performed as described by He and Faulkner" and in Chapter 2. The rate coustant of the following chemical reaction was measured by doublc' potential step cbronoamperometry. The double potential steps were carried out with a time resolution 100ps at a silver amalgam electrode with a diameter of 0.25 mm. The results were fitted to the theoretical curve developed by Schwarz and Shain" and as discussed in Chapter 2. The first few points of data due to double-layer charging current were ignored. The usable time window was confim~zdby the Cottrell equation,13 where constancy of it1I2 was observed. The catllodic current was fitted first within time 0 it < z,where r is the period of pulse applied. Then k,,,, was adjusted when thc parameters (n, A, C:, D,) were kept constant within time z < t d 2.r until the anodic current could be fitted by the double potential step relation for the EC mechanism. A general experimental and fitting result is shown in Figure 6-Za, and Figure 6-2b gives the detail, with the error range indicated. The results fall within 5-15% of an average value. Cyclic voltammogran~s(with background subtracted) at scan rates 50, 100, and 300 mV/s are shown in Figure 6-3. These voltammograms are characterized by a cathodic wave showing a well-defined current maximum but no anodic wave on the reverse scan. The digital simulation and fitting results from the CVFIT program are shown on the same voltammogcams. The input parameters of the program included the diffusion coeficient Do, the number of electron transfers n, the rate of the followin9 chemical reaction k,,,,, resistance R, and experimental conditions. Then CVFlT gives an initial set OF guess paratneters for the formal reduction potential EO', the heterogeneous charge transfer constant kO, the charge transfer coefhcim~a. and a proportionate cor~stant (nominally the area) that includes small errors in concentration of methylcobularnin, the electrode area, diffusion coefficients. and so on. The difFi~sion coefficient used at -30°C was measured by double potential step chrono-

CVFIT: SIMPLEX DATA ANALYSIS WITH CVSIM

Figure 6-2 Double potentiai step chronoamperometry experiment and htting result of methylcobalarnin in mixed solvent (40% DMF, 60°/, methanol) at - 301'C. Silver amalgam electrode, 0.25 mm in diameter, methylcobalarnin = I I ~ MTBAF , = 0.30M. (a) Result fitted by kdcm = 5 9 0 s ' : solid points are experimental resulfr. (b) Detdil o l tho fitting with error range, where the data close to noise background (at the end of the time period) are ignord; open points are experimental results. Curve s wit11 = 590s- '. b with 690s-', and c with 490sL'. (Adapted from Referenoe 3.)

amperametry in the time window ofconstancy it1I2. The iR calcalntjon vption u l CVFIT was utilized, which theoreticalfy corrects for iR drop effects on the CV waveshipe. The standard reduction potential, the heterogeneous rate constanr, and thc transfer coefficient obtained from the CVFIT program are showll in Table 6-1. For repetitive experiments, and for different scan rates, the fitted parameters show only small variations. This is a good inlcation both that the mechanism is correct and that the parameters obtained are meaningfd and not the result of Iocal minima- Furthermqre, the reduction potential is consistent with that measured under A h - , , , > ,>

<

I

--.-

Current ( u ~ m p s )

- 1.529

0.004

+

0.015 0.012 0.002

S m c a : Adapted from Rr!crcow 3

-

0.lx16

---

7.0

0.02

1.9 1 .9

2.1

2.2

1.O 1.1

0.75

0.78

10- "A2!

7.9

% I (x

1.9

2.0 2.0

1.9 1.9

-,

1vm3

A 10-

- --

(

--

Q77

L1.80 0.010 0012

.-

079 0.75 0.80

. -

r

0.012 0.014 0.010

-,

-.

" 1 1 crzul~mce mcasurad and used in the CVFIT program IF. 165061.

- 1.529

- 1.532

- 1.530

- 1.523

- 1.52:

-1.553

> -

-

Iv)

----

kc

(em's)

EO'

----

Table 6-1 Simulation!Fitting Results for Methylcobalamin Wectrochemical Parametersu at -30°C

Currant 4)rArnpcl

--

.-

- -

95 100 120

72 102

129

timza

-- -

iteration

Simplex

144

CYCLIC VOLTMMETRY

interesting to compare the reduction potential determined in this way to the estimate obtained by assuming a reversible electron transfer (Equation 6, Chapter 2). The difference is about 3 0 m V at the lowest scan rate. To estimate the EO' measurement error, which is caused by an error in the measurement of the lolluwing chemical reaction k,,,,, a series of cyclic voltammetry simulatjons was performed with different scan rate and k,,,,. It was found that the deviation in k,,,, would not affect EO' measurement in our electrochemical system. At a scan rate of 300mV/s, the peak potential shift is less than 1mV as k,,,, chatlgs from 490 s -'to 690 s-l; and the shift is 1mV in the same kchemchange at a scan rate of 50 mV/s. The rather large value of the cathodic transfer coefficient, a = 0.78, is indicative of a transition state for electron transfer very close to the reactants. This result supports t h e proposed mechanism, where the transition state for [BLJ reduction is expected to have the similar structure of the reactant B,,,. Although the cyclic voltammograms appear to be irreversible, they should not be. In the case of a totally irreversible system, we cannot expcct to determine the reduction potential. To understand this apparent anon-laly more fully, a study of the reversibility of the system was performed. The electrochemical current corresponds to the difference between the rate of the forward electron transfer k f [ B 1 2 Jand its rate k,[B;,,] for the reverse process. The reversibility factor f;12 for the methylcobalamin system is described as follows:

where [ E l 2,]o and [B;2a], are the corresporthtlg concentrations on the electrode surface. These concentrations can be obtained from thc digital simulation for the first spatial grid ofthe system, where the concentration points of the first several simulation grid points are extrapolated to the surface. Tllr variation of the reversibility factor 1,as the function of the heterogeneous charge transfer constant 1s shown in Figure 6-4. These reversibility factors are evaluated at the peak potential of the first spatial grid in the digital simulation. The simulations of the cyclic voltammetry for methylcobalamin at - 30°C result in reversibility factors f,. .- 0.18 at a scan rate of 300 mV/s and f, = 0.21 at 50 mVls. Both factors are close to the root of the curve, but they are not completety irreversible even tho ugh their cyclic voltammograms appear "irreversible." For a totally irreversible system, a is given by

I

CVFIT: SIMPLEX DATA ANALYSIS WITH CVSlM

Figure 6 4 The relationship of the rcversibiliry factor j; and the heterogeneous charge transfer Constant kO. The concentrations of the electroactive species in the electrode surface were evaluated a l the peaks or cyclic voltammogram by digital simulation with E0 = - 1.529 V, 3 = 0.78. and Do = 1.8 x l O 6 mz/sat - 30°C.The reversibility factor fr wascdculrtcd through E q ~ ~ a l ~ ( I o).~(a) l 30OrnV/s, k,., = 590 s-l; (b) SOmV/s, .ck = 5 9 0 s '; (c) 30On1Vls. kc... = 0 s '; and (d) SOmV/s, kc,em = 0 s ' . (Adapted from Reference 3.)

I

The small error in potsntials used in Equation (2) due to iR drop also results in errors of a values. The potential was corrected for iR drop as suggested in Chapter 2.

where i, is the peak current and i,,, is the half-wave-hdpht peak current. The values of a,,,,,, from measurements in cyclic voltammc.pram are listed in Table

6-2. These results are consistent with the values of n obtained by simulation. Figure 6-5 shows the influence o f f , on o! values calculalrd by estimation from simulation and through the graphical estimations ah,, and a,,,,,,rr. The error is also consistent with the experimental measurements. Valucs in a,,,,,u, are slightly larger than thaf in cqi, because the system is not complrtely irreversiMe.

6.4 Some Final Comments on Simulation Analysis where E, is the peak potential and E,,, is the potential of half-peak. We expect that E., obtained in this manner will be a reasonable value even though the system 1s not totally irreversible. Kochi's results and the digital simulation data also show the influence off, to asp, values by Equation (3).Tlre larger a,, value will have the smaller error if the system has similar irreversibility. Since a = 0.78 in this system, we expect a small error estimating a through relation (2) i f f , is in the foot area of the curves shown in Figure 6-3a and b.

J

1. Ensure, as far as is possible, that the electrode

controlled.

,2

p

are diffusi n -.y

Y

CYCLIC VOLTAMMETRY

Table 6 2 The Various Measurements of x -

R~tc (mv$)

%:

L

0.76

0.18

0.83 0.84 0.84 a84

0.8(1

0.20

0.77

0.21

0.P 1

0.85 0.86 0.P6

0.81

0.86

a ,, . '

~,pp.cor: .,, - - .

0.72 0.7 1 0.7 1 0.71 0.76 0.7s

3M

ICO

0.75 0.78 0.8 1 0.81

50

-

-

0.80

I

CVFIT: SIMPLEX DATA ANALYSIS WITH CVSM

backgound or capantive current when the double-layer capacitance is a function of potential. 3. Prepare correct data fiIcs, using 0.001 V per point, and start w ~ t hthe initial potential plus 0.001 V. End with the final potential. The units of current

should be amperes. 4. If computation time is an issue, use moderate to fast scan rates. Simulation time 1s roughly proportional to experimental time. A simulation that take:: an hour. with a scan rate of 0.1 V/s will take less than a rn~nutevcith a scan rate of

u.80

0.90 Tc80

10 v/s. 5 Use the smallest scan range possible. Don't include lots of tlat baseline, which absorbs much computation time without ylelding any meaningful mformation. 6. ColIect data sets over several orders of magnitude of scan rates. The time scale should correlate with the time scale of processes being investigated, for instance, if chemical processes are involved. 7. For sol~itionswith moderate resistance and moderate scan rates, don't use IR compensation Rathcr, mzasurr: the resistance and use the I R calculation in the fitting program. 8. FOTcomplex electrochemical mwhanisms, collect data on as mdny subsystems as possible. Tbe fits of these wilI give independent numbers and also will reduce the overall fitting time. For an example, see Wandlowski et aLI4

-

-.*-

"

Measured from experimwtal results. Expwimmtal ]mulls conecld Ibrougb E q ~ a t i m(2). ~ c rkuln a~" l l u ~ a~r i u ~ / f~uf i ~ ~the ~ gcyclic ~ vulrarrllllugvr~~a d with input EO = - 1 . 5 3 V. kg = (I.01?cm/s. Do= 1.8 x : 0 - 6 ~ m 2 ~and . k,, = 590s-' at -30°C.

I

II

SOIITCB:Adapted lrom Reference 3.

147

I

1. Develop a reasonable mechanism using CVSIM. Try various ~nechanis~l~s to see which one follows the experimental trcnds clo5ely. 1. Use CVFlT with as many independently determined parameters as possible [e.g., diffusion coefkients, rate constants of following chemical steps as measured by potenha1 step experiments). 3. Let the area of the electrode Roat in each fit. This can amount far small errors in diffusioll coeficients or concentrations. 4. Fit the same methanism for data collected at dderent scan rates. This is a very good indication of the correctness of the a.isurlied mechanism. It a equivalent to a working curve approach. A divergence of a fitted parameter indicates an incomplete or incorrect mechanism. O n rhe other hand, fitted parameters that are constant over scan rate indicate that the mechanism is consistent with experiment. Error estimates on the parameters can be obtnincd by standard deviation of parameters obtained for diKerent data, sets {both at the same scan rate and varying scan rate). 5. Fit cxp.rimen~sdone as a result d sumestion 8 in Section 6.4.1. T J J w i l l , provide independent parameter estimates, avoiding many pardter fits, , which take a long time and become less reliable. 6. When first starting, choose a high value far the tokrro&rf.the pro-

I

1

/

i

a \

ml'P

-. F~&E6 5 The relationship of the reverstbility factor jrand the apparent transfer ~ ea,,, The~solid pbint t rcprencnts tbesimdarion and fitting result, then esLim.trd I l ~ r o u Equation ( 2 ) .( ~ d a p k d from Reference 3.)

to start a ne .

..

-

CYCLIC VOLTAMMETRY

148

References

APPENDIX

1. Gosser, D. K., Jr.; Zhang, F. Talunta 1991,38, 71 5-722.

2. Press, W.H.: Flannery, B. P.; Teukolsky, S. A,; Vetterling, W. T. Numrical Recipes The Art of Scienrific Compurirrg: Cambridge Univemity Press: Cambridge, 1985. 3. Huang Q.; G e r , D. K..Jr. Talanta IW2.3 9 0 , 1155. 4. Lexa, D.; Savant. J. M. Acc. Chern. Res. 1983,16,235. 5. Zhou,D. L;Tinernban, 0.; Scheffol4

R.;Wal&r, L. Hdv. Chim. Acta $994 71,2225.

Lem D.; Saveant, J. M,J. Am. Chcm. Sw. 1978, I#, 3220. 7. Kenyhera T. M.;DeAngelis, T.P.;Norris. 6. J.; Heineman, W. R.;Mark, H. B. I. Am. Chem

6.

Soc. 1976. 98,2469.

Summary of Instructions for CVSIM, CVGRAF, CVFIT, and DSTEP

8. Lexa, D.: Saveanr, J. M.;Zickler, J. J . Am Chem. Soc. 1977,99, 2786. 9. Rubinson, 10. Kim,

1I.

K. A; habashi, E.; Mark, H. B. Inurg, Chem. 1482,21, 3571.

M.H.; Birke, R. L. I. Elecsroanal. Cbem. 1983, 144, 331.

He,P.; Faulkner, L. R. Anal. Ckm.1986,58, 517. M.;Shain, I. J. Phys. Chem. 1=, 69,30.

12. Schwarz, W.

L3. Klingler, R. J.; Kochi, J.

K. 3. Phys. Chem. 1981, 85, 1731.

I 4 Wandlowski T.; Gosser, D.

K.,jr.; Akinele, E.; de Levle. R.: Horxk.

V. Tulaotcl 1993,JU. 1789.

Installation CVSIM, CVFIT, and CVGRAF, and DSTEP A11 the files on the diskette provided should be copied onto the hard disk to the directory C:\SIMULATE (first create the directory; then copy the files to the directory). All the programs should then be run from C:\SIMULATE.

CVSIM The program is initiated by typing CVSIM and hitting the enter key. The experi~nentansthe mechanism are described in response to a series of inquiries przserlted in a fuil screen display.

Number of Species in the Mechanism An integer equal to the total number of species in the mechanism is entered. T h e screen input is designed for up to eight species; mon: can b entered if necessary.

Concentrations of Each Species

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CYCLIC V O L T A M T R Y

150

(excluding of course the elsctrolylyte). Each species is now associated with an integer, in the order of entry. Following entries, which describe electrode reactions and chemical reactions. utilize these number identifications.

Number and Description of Redox Couples The number of redox couples in the mechanism is entered. Next the integer index is entered for tlic oxidized and reduced forms of each redox couple, followed by the reduction potential of each couple (in volts), the heterogeneous rate constant (cm/s), and the cntt~odictransfer coefficient u. For an electron transfer that is considered to be reversible, a value of lOcm/s will ensure reversibility under most circumstances. IF a two-electron transfer (where one of the redox species is shared by two couples) is among the electrode reactions, these should be entered in sequence when written as reductions. For instance:

APPENDIX

advised to make the default ddf~eioncoefficient equal to the largest diffusion coefficient of the nlechanism. It is also possible to increase the number of time increments used in the aimulution, to check for convergence (and thus accuracy) of the simulation.

Names of Data and Record Files The names of the simulated potential-current data filc and a Ale that saves the entire simulation parameter set are requested. The current is output in amperes and the potential in volts. This format i s required of data sets to be graphed with the CVGRAF program or to be used by the curve-fitting program CVFIT.

Normalizing Concentration The choice of normalizing concentratjon is entered in millimular units (mM).

Option for Capacitive Current

Numbner and Description of Solution Chemical Reactions The number of solution reactions in the mechanistn is entered. Each chemical reaction (first or second order) is described by entering two indexes for the reactants and two for the products. (Ti the reaction is first order. 0 is entered in the space reserved fur the second species of a second-order reaction.) First-order rate constants are elltered in units of reciprocal seconds. Second-order rate constants arc entered in units of M - 's-l*Cnorm.

Experimental Parameters The number of half-cycles (1,2, or 31, the initial potential, switching potential(s), final potential (V). and scdn rate (Via) are entered. Asymmetric potential waveforms, whch are often useful, are possible in this scheme.

Changes in Input At this point, the user can inspect the entire input screen and change selected parameters by entering the number associated with the parameters (ie-, 1 = concentrations, 2 = electrode reactions. etc.). It is also possible to change the default temperature (298.351(), the default diffusion coefficient (1.00 x 1 0 - 5 c ~ 2 / s ) , or the individual diffusion coefficients. The individual diffusion coefficients are changed by entering a multiplier of the default value (e.g., enter 0.5 for a diffusion coefficient of 5.00 x lo-' crnz/s). If it is desired to simulate a mechanism with a dlBvsion coefficient larger than the default coefficient, the default diflusion m f i c i c n t itself can be changed. lt is always

Ifit is desired to include a capacitive current in the simulation, then the electrow area (em2), double-layer capacitance (pF/cm2), solution resistance (kG) must be entered.

Option for IR Drop The effect of the I R drop can he included as well. It is necessary to provide an experimental data fileof current-potential values (A/V, 1 point per O.DOl V), and to enter the resistance, electrode area. The experimental current is read and the simulation potentials are corrected accordingly See texf fur detailed description of the data file.

CVGRAF A simple graphics program for PC con~patibles(CGA, VGA, and Hercules graphics) is provided for the simultaneour examinntiotl of two d a h sets. It is intended primarily for comparison of experinlcr~tul and simulated data. C V G R A F is started by typing CVGRAF. The input is quite simple: initial potential and find potential (V), current scale (pAmps), and the names of the two data fiIes are requested. Default values are provided. If only one data file is ro be exnmined, a blank file named tdata.pas should be created. Altemativd~, the second file can have the same name as -the firdt. Movement: of a crosshair cursor is controlled with the arrow keys, ,and pokhti81-current v a l m am :. displayedatthetopofthescreen. -

152

CYCLIC VOLTAMMETRY

CVFIT is similar to CVSIM. Lt is necessary to identify parameters to be fitted, provide an experimental data filc (data.pas), and input a tolerance. Parameters capable of being fitted must be preceded by a 0 (constant) or 1 (fitted). The history oi the fitting including the tstimitted parameters and the error term for each simplex iteration is in a filc irstdatxpas. The error term listed is actually the sum of the least squares errors * lo6. The final fitted parameters are in a file called paramtr.pas and a simulation using the fitted parameters in a ,file segl.pas. An examplc of paramtr.pas is as follows:

Index

n {no cap current) n (no IR drop correction) 10 {program code) 2 {number of fitting parameters) 1.OE-0005 {standard Diffusion Coefficient) 1.00000000000000E-0010 {tolerance) 0.00 {fixed) 0.010000000 (area] 0.000001 00 {normalizingconcenlrat~on,rnol/mlj 298.150 {temperature) 3 {number of species) 1.000 (normalized cancentration of first species] 0.000 {normalized concentration of second species) 0.000 {norrnal~zed concentration of third species) 2 {two half cycles) 0.000 (init pot) -0.300 {switch pot) 0.000 {final pot) 1.0000 (scan rate; 1.000 {First diffusion coefficientj 0.000 {fixed) 1.000 {Second diffusion coefficient) 1.000 {Third diffusion coefficient) 0.000 (fixed] {electrode &uple) 2 1.OE+0000 (fit] -0.1860 (Red Poi] 1.00 (ht) 0.0095 (khet) 0.00 (fixed] 0.50 {chemical reaction] 0 3 D 0.00 {fixed) I.OE+0.002 {kf) 0.000 {fixed) 0.OE +0000 (kr)

Absolurt energy scale, 2 Adsorption. 43, 97-99 Acceptor number, 32 Artemisinin, 99- 102

0.000 {fixed) 1 1 1 2

(alihha)

DSTEP is a program that simulate single or double potential step experiments. The entry mechanism is the same as CVStM. The experimental parameters are initial potential, step potential, find potential, and time d step. The output is lime in milliseconds, and current is in amperes.

Benzoquinone reduction. 7- 10, 1 1 - 12, 79-80 Boiling point, 32t Born model of solvation. 9- I I Capacitive current, 57- 59 in double step experiments. 63 simulation of, 116- 17 Catalysis of artemisinin reduction, 1M)-2 electron transfer chain. 80-84 of glucose oxidase oxidation, 87-88 homogeneous redox, 85-89 Current follower, 65-66 Dielectric constant, 18, 32r, 73 DiKusion, 51-53 simulation of, 105-8 visualization of, 55 Dimensionless units, 109-1 10

Double layer, 23-23 Electrochemical e l l , 5, 28, 30-31 equivalent circuit for, 56 Electrodes luggin aippilfary, 30-31 refercne, 34-35, 66-67 silver atnalgam, 33 work~ng.32-34.66-67

Electrolytes, 33 Electron trnnsfer, Butler-Voirner eqi~atiot~, 14,45, 109 free energy relationship. for, 19-21 inner sphere, 16.-17. 19-2 1 Marcus theory, 17- 19 mullieleclron transfer, 75-77, 127, 132 ouler sphere, 16- 17 Entropy of reduction, 11

Faradaic current, 20 Fermi level energy, 1-3, 67 Ferncyanide reduction, I cyclic voitammetry, 30 simulation, 126, 128 Fick's law. 51-52 105-7

Graphics, 125 Heterogeneous ratc constant. I?. 14, 45, 46 Hydrogen reference, 3-4

Inner sphere complex, 16, 14, 74, 101 -2 1R drop. 56-57 simulation oT, 1 16-7 Koopman's theorem, 9

Potential electrical, 3, 5 electrode, 3-5, 6-7, 27 window, 32, 35 of zero charge, 22, 24 Potential step chronoamperometry, 61-63

of methylcobalamin, I40-41 of nitrogen dioxide, 72-73 Potentiostat, 64-68 Protonations, 77-80, 133

Linear free energy relationship, 12- 15 Mean square difiusion, 38

Melting point, 32t Methylcobalamin reduction, 3 4 4 7 , 139- 146 simulation, 126, 129 Microetectrodes, 59 used to determine electron number, 59-69

Nerna Equation, 3-445 Nitric oxidc, 92-94 Nitrobenzoic acid. 90- 92 Nitrogen dioxide. 72-75 simulation, 126- 117. 130- 13 1

Operational amplifiers, M-66 Peak current reversible one clectron transfer, 43,44 shifted r:itio. 43 Peak potcntial Reversible one electron transfer, 42, 44

Reduction potential, 1-6 Tor E,C mechanism, 139-46 lor E, mechanism, 43 for E,C mechanism, 49 Reversibility, 36: 35-46, 49-50, 14--45 Scan rate, 25

and diffi~siori.5 1 - 52 Solvation. 2. 8, 9- 11 Solvatochromatic, j 2 t

Solvents, 3 I - 32 properties, 32t

Tempo oxidation, I0 Transfer coefficient, 14-16,47, 1.15-46 Tricarbonyl(mesitylene)tungsten oxidation. 95-97 Work function, 1 Voltage amplifier, 65-66 Voltage follower, 65