4894 Chem. Soc. Rev., 2013, 42, 4894--4905 This journal is c The Royal Society of Chemistry 2013

Cite this: Chem. Soc. Rev.,2013,42, 4894

Asymmetric Marcus–Hush theory for voltammetry

Eduardo Laborda, Martin C. Henstridge, Christopher Batchelor-McAuley andRichard G. Compton*

The current state-of-the-art in modeling the rate of electron transfer between an electroactive species and an

electrode is reviewed. Experimental studies show that neither the ubiquitous Butler–Volmer model nor the

more modern symmetric Marcus–Hush model are able to satisfactorily reproduce the experimental voltammetry

for both solution-phase and surface-bound redox couples. These experimental deviations indicate the need for

revision of the simplifying approximations used in the above models. Within this context, models encompassing

asymmetry are considered which include different vibrational and solvation force constants for the electroactive

species. The assumption of non-adiabatic electron transfer is also examined. These refinements have provided

more satisfactory models of the electron transfer process and they enable us to gain more information about

the microscopic characteristics of the system by means of simple electrochemical measurements.

Key learning points� The ubiquitous Butler–Volmer model cannot account for the kinetic behaviour of electroactive monolayers.� The symmetric Marcus–Hush (SMH) model works well for electroactive monolayers, but often only if cathodic and anodic data are considered separately.� The SMH model performs poorly for many redox couples dissolved in solution.� Experimental data suggests that a model for the kinetics of electron transfer should consider the possibility of an asymmetric process.� The newly developed asymmetric Marcus–Hush model has proven successful in modeling the behaviour of both surface-bound and solution phase systems.

1 Introduction

Since its inception in the early 1930s, the model due to Butler1

and Volmer2 has become the standard formalism for interfacialelectron transfer (ET) kinetics in the absence of bonds beingbroken or formed. It is an empirical model of three independentparameters: the formal potential (E�Jf ), the standard hetero-geneous rate constant (k0) and the transfer coefficient (a or b).†The last of these parameters gives an indication of the ‘position’of the transition state and, for the transfer of a single electron,typically takes values of approximately 1

2.Within the Butler–Volmer model the rate constants for

reduction and oxidation (kred and kox) are exponentially depen-dent upon the applied potential (E) thus:

kred ¼ k0 exp �aFRT

E � E�Jf� �� �

(1)

kox ¼ k0 exp þbFRT

E � E�Jf� �� �

(2)

where F, R and T represent, respectively, the Faraday constant,the gas constant and the absolute temperature.

This model is easily implemented, is computationallyinexpensive and, more importantly, has been enormouslysuccessful in providing a quantitative description of the kineticsfor a vast number of electrochemical systems over the past80 years. Consequently, the parameterisation of experimentaldata in terms of {E�Jf , k0, a} has become standard practice.

However, despite its ubiquity, this model has its limitations.The transfer coefficient is generally understood in qualitative,rather than quantitative, manner and as such the model is oflimited use as a predictive tool in terms of the molecularprocesses taking place during electron transfer. Further, themodel naively suggests that the rate constants should increaseexponentially with overpotential ad infinitum.

In addition to these theoretical concerns, there is experimentalevidence that the Butler–Volmer model is not universally applic-able. As early as 1975,4 Saveant and Tessier observed a potential-dependent transfer coefficient for the reduction of tert-nitrobutane,which is not consistent with the Butler–Volmer model.

Department of Chemistry, Physical & Theoretical Chemistry Laboratory, Oxford

University, South Parks Road, Oxford, OX1 3QZ, UK.

E-mail: richard.compton@chem.ox.ac.uk; Fax: +44 (0)1865 275410;

Tel: +44 (0)1865 275413

Received 29th November 2012

DOI: 10.1039/c3cs35487c

www.rsc.org/csr

† The notation for transfer coefficients is not uniform across the world, see forexample Bockris.3 The notation used in this review is as per eqn (1) and (2).

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Chidsey provided further evidence while studying thechronoamperometric response of the ferrocene/ferroceniumcouple tethered via an alkylthiol chain to a gold electrode. Heobserved a markedly curved Tafel plot which could not beaccounted for by the Butler–Volmer model, it was necessaryto employ the ‘‘contemporary theory of electron transfer’’ dueto Marcus5 and Hush.6

2 The ‘symmetric’ Marcus–Hush model2.1 Background

Originally developed in the 1950s by Marcus5 for outer-spherehomogeneous electron transfer reactions, this theory rationa-lises the rate of electron transfer in terms of the reaction Gibbsenergy (DG�J ) and a ‘reorganisation energy’ (l), defined as theenergy required to distort the atomic configurations ofthe reactant molecule and its solvation shell to those of theproduct in its equilibrium configuration. Electron transfer isconsidered to take place subsequent to this molecular reorga-nisation in a radiationless transition according to the Franck–Condon principle.7

Earlier work by Randles8 was instrumental in the develop-ment of this theory, which was also subsequently extended byHush.6

The ‘symmetric’ Marcus–Hush (SMH) model assumes theGibbs energy surfaces of species Red and Ox to be parabolicand, significantly, to have equal curvature. The energy of thetransition state, and hence the activation energy, is thereforedetermined by the intersection of these two curves.‡

DGz ¼ l4

1þ DG�J

l

� �2

(3)

The rate of electron transfer is then related to the activationenergy (DG‡) via an Arrhenius-like expression:

k ¼ A exp�DGzRT

� �(4)

where k is a rate constant and A is a pre-exponential factor.These ideas are summarised in Fig. 1 which shows a schematic of

the parabolic Gibbs energy curves for a homogeneous electrontransfer, that is electron transfer between two species dissolved insolution.

For heterogeneous electron transfer (i.e. electron transferbetween a donor and acceptor which are in difference phases)between a molecule in solution and a metallic electrode (suchas that in eqn (5)), the overall rate of electron transfer willconsist of contributions from each of the energy levels in theconduction band of the metal.

Oxþ e� Ðkred

koxRed (5)

For each level, i, of the metal conduction band the activationenergy is given by:

DGzred=oxðEiÞ ¼

l4

1� FðEi � E�Jf Þl

� �2(6)

Fig. 1 Schematic of the parabolic Gibbs energy curves of the reactants andproducts as given by Marcus theory for a homogeneous electron transfer.

Eduardo Laborda carried out his doctoral research from 2007 to2011 in the group of Theoretical and Applied Electrochemistry ofthe University of Murcia (Spain) under the supervision of Prof.Molina and Prof. Martinez-Ortiz. His PhD thesis was devoted to thekinetic and mechanistic study of complex electrode processes withvoltammetric pulse techniques. In March 2011 he joined the groupled by Prof. Compton in Oxford University where his postdoctoralresearch has included the assessment and development of kineticmodels for electron transfer reactions and the study of theelectrocatalytic properties of metal nanoparticles.

Martin C. Henstridge completed his doctoral research in computa-tional electrochemistry under the supervision of Prof RichardCompton at Oxford University. His thesis focused on the compar-ison of Butler–Volmer and Marcus models of electrode kinetics. Heis now a postdoctoral researcher in the Compton group.

Christopher Batchelor-McAuley undertook his MChem and D. Philat the University of Oxford under the guidance of Prof. Compton.Having been awarded an EPSRC doctoral prize he is currentlycontinuing work as a post-doc.

Richard G Compton is Professor of Chemistry and AldrichianPraelector at Oxford University where he is Tutor in Chemistry atSt John’s College. He has published more than 1000 papers (h = 68;Web of Science, March 2013), 5 books and numerous patents. The2nd edition of his graduate textbook ‘Understanding Voltammetry’(with C E Banks) was published in late 2010 by Imperial CollegePress. He is a Fellow of the RSC and of the ISE and is the FoundingEditor and Editor-in-Chief of the journal Electrochemistry Commu-nications (current IF = 4.86) published by Elsevier.

‡ This relies upon the assumption that the electron transfer is non-adiabatic,which will be discussed further in Section 7.

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where FEi is the energy of the ith energy level. Where two signsappear, the upper sign refers to reduction and the lower signrefers to oxidation. The overall rate of electron transfer is then asum of the rates of electron transfer for each electronic state ofthe electrode, weighted by the probability of occupancy ( f+) orvacancy ( f�) of that state:9

kred=ox ¼Xi

f�ðEiÞAðEiÞ exp�DGz

red=oxðEiÞRT

" #(7)

where f�ðEiÞ is given by the Fermi–Dirac distribution:

f�ðEiÞ ¼ 1þ exp � F

RTEi � Eð Þ

� �� ��1(8)

We then assume that the term AðEiÞ may be approximated asconstant with respect to potential. This implicitly assumes thatboth the density of metallic states (r) and coupling between thereactant molecule and the electrode are also independent ofpotential—these assumptions have nevertheless proven to beconsistent with experiments.9,10

Further, considering the energy levels of the metal form acontinuum, we may re-write eqn (7) as an integral:

kred=ox ¼ A

Z 1�1

f�ðEÞexp�DGz

red=oxðEÞRT

" #dE (9)

We then introduce k0 as the rate constant at the formalpotential and obtain the following general expressions for theelectron transfer rate constants:

kred ¼ k0Sredðy;LÞSredð0;LÞ

(10)

kox ¼ k0Soxðy;LÞSoxð0;LÞ

(11)

where y is dimensionless potential and L is the dimensionlessanalogue of the reorganisation energy (assuming units of eV):

y ¼ F

RTE � E�Jf Þ�

(12)

L ¼ F

RTl (13)

and Sred/ox(y, L) is an integral of the form:

Sred=oxðy;LÞ ¼Z 1�1

exp �DGzsym;red=oxðxÞ

.RT

h i1þ exp½ � x� dx (14)

where

DGzsym;red=oxðxÞRT

¼ L4

1� yþ x

L

� �2

(15)

and x is a dimensionless integration variable:

x ¼ F

RTðE� EÞ (16)

The rate constants given in eqn (10) and (11) take a convenientform whereby the rate of electron transfer at a given potential

can be expressed in terms of three independent variables: {E�Jf ,k0, l}. In reality k0 is a function several variables, as will bediscussed in Section 7. However, this form permits easy com-parison with the Butler–Volmer model, which shares two ofthese characteristic variables.

2.2 Characteristics of the model

With these expressions for the rate constants we can analyse thebehaviour of the SMH model of electrode kinetics. Fig. 2(b) showsthe variation of kred and kox as a function of potential for a range ofreorganisation energies (plotted as ln k against overpotential). Themost striking feature of this model is that the rate constants reach alimiting value at large overpotentials. This is in stark contrast to theButler–Volmer model for which the rate constants continue toincrease exponentially ad infinitum (Fig. 2(a)).

The potential at which the rate constants begin to level offdecreases as the reorganisation energy decreases. More quanti-tatively, Oldham has shown that the rate constant reaches halfof its limiting value when y = L.11 Thus when the reorganisa-tion energy is large, the rate constants continue to increaseeven for large overpotentials. Feldberg12 has shown that in thelimit L - N the rate constants given by the symmetricMarcus–Hush model become exactly equivalent to those givenby the Butler–Volmer model for a = 1

2. This relationship canclearly be seen in Fig. 2(b).

Fig. 2 Plot of ln(k/k0) against overpotential for (a) the Butler–Volmer model for arange of transfer coefficients (a = 0.3, 0.4, 0.5, 0.6, 0.7) and (b) the symmetric Marcus–Hush model for a range of reorganisation energies (l = 5, 2, 1 and 0.5 eV). The dottedline shows the Butler–Volmer model for a = 0.5 for comparison. Reductive rateconstants are shown in blue and oxidative rate constants are shown in red.

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Clearly the value of the reorganisation energy affects the rateconstants of the SMH model in a different manner to the value ofthe transfer coefficient within the BV model. This is highlighted inFig. 3 which shows simulated cyclic voltammetry for a diffusionalsystem for varying values of a and l in the two models.

Within the BV model varying a from 12 results in one peak

becoming taller and sharper, while the other peak becomesbroader and shorter. From looking at Fig. 3 we see that as ldecreases both peaks become shorter and broader. This iscaused by the limiting behaviour of the SMH model: as ldecreases the rate constants begin to level off at smaller over-potentials and therefore reach smaller limiting values.

It has been shown that the SMH model does not conform tothe Randles–Sevcık equation which, for an irreversible diffu-sional redox couple, predicts a square root dependence of peakcurrent with voltage scan rate.13 The Randles–Sevcık equationis derived assuming Butler–Volmer kinetics and at high scanrates the SMH model yields a peak current which is smallerthan predicted, with the deviation becoming more pronouncedas the reorganisation energy decreases.

A similar deviation from classical theory has been demon-strated for electroactive monolayers, for which the peak currentis predicted to vary linearly with scan rate. This feature hasbeen verified experimentally for surface-bound systems.14

3 Evaluation of the SMH model3.1 Electroactive monolayers

Since Chidsey’s original work9 the symmetric Marcus–Hushmodel has been employed successfully in modelling the kinetics

of many other surface-bound redox couples15 due to its ability toreproduce the experimentally observed curved Tafel plots.

There are, however, documented examples of experimentalsystems which show deviations from the symmetric Marcus–Hush model (see ref. 14). These systems exhibit Tafel plotswhich are curved, but whose anodic and cathodic brancheshave differing slopes in the vicinity of the formal potential. Thisfeature cannot be accounted for either by the symmetricMarcus–Hush model, for which a Tafel plot is always symmetricalabout the formal potential, or by the Butler–Volmer model whichcannot reproduce the curvature of such plots. Some researchershave treated the anodic and cathodic branches entirelyseparately using the symmetric Marcus–Hush model with twodifferent values for the standard rate constant and reorganisa-tion energy, but this approach is not consistent with the Nernstequation (since kox/kred a ey).

3.2 Diffusional redox systems

While the SMH model has been routinely used in the analysisof the kinetics of electroactive monolayers since Chidsey’s workin the early 1990s, the model has rarely been employed forsolution-phase couples until very recently.

An early example is the study of the electron transferdynamics of decamethyl ferrocene at low temperature byRichardson et al.16 This work revealed significant deviations fromclassical Butler–Volmer kinetic behaviour at low temperatureswhen the overpotential exceeds about 40% of the reorganisationenergy. The SMH model, however, was able to accurately reproducethe broadened peaks evident in the experimental voltammetry overa range of temperatures.

Feldberg examined the implications of the choice of kineticmodel on the steady-state voltammetric response at a microdiscelectrode.12 Starting from the premise that the Marcusianmodel was ‘correct’ he simulated steady-state voltammetryusing the SMH model for a range of values of k0 and l andthen attempted fitting of this data using the Butler–Volmermodel. He observed that for a typical value of reorganisationenergy (B1 eV) the value of k0 required to achieve the samesteady-state response using the BV model could differ fromthat used within the SMH model by a factor of 2 for a quasi-reversible system, or by as much as a factor of 200 for anextremely irreversible system. He also notes that, due to itslimiting rate constant behaviour, the SMH model can yield asteady-state current which is significantly smaller than thediffusion-limited current when both k0 and l are small. Thissituation, however, seems improbable since, according toeqn (3), a small reorganisation energy leads to a small activationenergy which, in turn, leads to a high rate constant througheqn (4). Thus it is unlikely that both parameters will take smallvalues simultaneously.

Recently there have been several experimental studies comparingthe Butler–Volmer and symmetric Marcus–Hush models in theirability to fit experimental voltammetry. Using a fast-flow channelsystem in combination with very high scan rates (B1 kV s�1)Suwatchara et al. studied the oxidation of 9,10-diphenylanthracene(DPA) and the reduction of 2-nitropropane (2NP) at a platinum

Fig. 3 Cyclic voltammetry for a diffusional system comparing the effects of (a) awithin the BV model (a = 0.3, 0.4, 0.5, 0.6 and 0.7) and (b) l within the SMHmodel (l = 10, 1, 0.1 and 0.01 eV).

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microband electrode.17 The increased rate of mass transportafforded by this set-up, in which the solution flows past theworking electrode at velocities of up to 20 m s�1, allowscharacterisation of systems with fast electrode kinetics. Never-theless, due to the high electrochemical reversibility of the DPAsystem, no detectable differences were observed between the twokinetic models.

The reduction of 2NP, however, displayed much slowerelectrode kinetics (k0 was found to be B10�1 cm s�1) and didyield noticeable differences in the behaviour of the two models.For this system the outer-sphere reorganisation energy wascalculated to be approximately 2.5 eV, but in order to achievea good fit to experiment using the SMH model a much smallerreorganisation energy (1.0 eV) was required. Furthermore,while the peak separation was reproduced, neither peak currentnor waveshape was accurately reproduced.

A good fit was achieved using the Butler–Volmer model,although it was necessary to relax the requirement that a + b = 1,which was justified because the oxidative and reductive peaksoccur at very different potentials (DEp C 1.5 V).

Compton et al. have also studied the one-electron reductionsof Eu3+, 2-methyl-2-nitropropane (MeNP) and cyclooctatetraene(COT) under diffusion-only conditions at a mercury microhemi-sphere electrode.18–21 In all cases the systems and experimentalconditions where selected such that the voltammetric responsewas informative about the electrode kinetics and any possibledistorting effects (such as adsorption, ion pairing or coupledchemical reactions) were minimized. Moreover, high concen-trations of supporting electrolyte were added, low chargedelectroactive species were preferred and mercury microhemi-spheres were employed as working electrodes. This enabledaccurate quantitative analysis of the electrode kinetics and thereduction of double layer, migrational and ohmic drop effects.

The possibility of distortion due to ohmic drop was consid-ered through simulations using a model which considers theeffects of migration as well as diffusion.22 For the experimentalconditions used, however, such effects were found to benegligible.

Finally the systems were studied using both cyclic voltam-metry and cyclic square wave voltammetry that enable thesimultaneous examination of the reduction and oxidationreactions to ensure the consistency of the results for differentelectrochemical techniques.

In each case the experimental cyclic voltammetry was best fitusing the Butler–Volmer model. In general it was possible to fiteither the forward peak or the back peak using the SMH model,but it was not possible to fit both peaks simultaneously.

For the reduction of MeNP, the SMH model achieved its bestfit using a reorganisation energy which is significantly smallerthan the theoretical value, as was the case for the reduction of2NP. In this instance the outer-sphere reorganisation energywas calculated to be 1.6 eV and the inner-sphere contributionhas been reported as 0.9 eV.18 The best fit value, however, wasfound to be 0.5 eV. Fig. 4 shows a typical experimental cyclicvoltammogram, along with the best fits for both the BV andSMH models.

The differences between the two models are shown even moreclearly when using reverse scan square wave voltammetry, forwhich it is possible, under certain experimental conditions,19 toobserve two peaks in one potential sweep for systems with sluggishelectrode kinetics (see Fig. 5). The two peaks correspond to theforward and back reactions and the characteristics of these peaks(peak–peak separation and ratio of peak currents) can be used todifferentiate between kinetic models. For the reduction of MeNPthe BV model was able to accurately fit both of these characteristicssimultaneously19 with k0 = 2.9 � 10�3 cm s�1 and a = 0.39, whichare consistent with the values previously obtained from cyclicvoltammetry.18 By contrast, using the SMH model it was notpossible to fit both the peak–peak separation and the ratio of peakcurrents simultaneously.

Further, it has been shown23 that the differential pulsevoltammetry of quasi-reversible redox systems can display apeak splitting if the transfer coefficient deviates significantlyfrom 1

2. This phenomenon has been experimentally verified24

for the reduction of Zn(II), yet no set of kinetic parameters existssuch that the SMH model is able to reproduce this feature.

Fig. 4 Experimental cyclic voltammetry (solid lines) for the reduction of MeNP inacetonitrile at a 25 mm mercury hemisphere electrode recorded at 5 V s�1. Thebest fits achieved using the BV (black squares) and SMH (white circles) models areshown for comparison.

Fig. 5 Schematic SWV voltammetry showing the emergence of a second peakin the reverse scan for a slow electron transfer process.

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4 Refinement of the kinetic models

As discussed before, despite the simplicity of the Butler–Volmermodel and the ground-breaking contribution of the Marcus–Hush–Chidsey approach, neither is able to fully describe thecomplexity of heterogeneous electron transfer processes anddeviations from both kinetic models have been documented.The reasons behind the discrepancy between experimental andtheoretical results using the SMH formalism can be found inthe simplifying hypotheses assumed throughout the model.

In the next sections the suitability of some of these hypotheseswill be revised together with theoretical models that enable us tojustify and overcome the deviations reported. These models leadto a more realistic picture of the process and provide furtherphysical and molecular insight into the electron transfer eventwithout significant increase of the numerical effort.

4.1 Asymmetric Gibbs energy curves

Within the SMH model it is assumed that the Gibbs energycurves of the oxidized and reduced species are symmetricalparabolae (Fig. 1). Thus, since the transition state is given by theintersection of two parabolae, if the parabolae have the samecurvature then at the formal potential the transition state islocated midway between the oxidized and reduced configurations.

Equal curvature for both Gibbs curves means that theintramolecular vibrations and interactions with the solvent ofboth electroactive species are the same, which seems unlikelygiven the structural and solvation changes that often accompanyelectron transfer processes. For example, significant changes inthe frequencies of molecular vibrations have been reported fortransition metal complexes and more advanced solvation modelspredict that the interaction of an ion with the solvent moleculeswill be dependent on its charge. Moreover, changes in the ionicatmosphere of the electroactive species can also be important andthey are considered in the charge fluctuation model developed byFletcher.25

As shown in Fig. 6, different force constants lead to Gibbsenergy curves with different curvatures. The greater the forceconstant, the tighter the energy curve of the electroactivespecies. The configuration of the transition state is also affectedsuch that it is more similar to the electroactive species with thetighter Gibbs curve. This enables us to establish a qualitativeconnection between asymmetric potential energy curves and thetransfer coefficient, which is classically interpreted as indicatorof an ‘early’ or ‘late’ transition state. Thus, a o 0.5 indicates a‘late’ transition state and thus a greater force constant for thereduced species. The opposite applies for a > 0.5.26–33

Several authors have studied the effects of unequal Gibbscurves on the kinetics and voltammetry of heterogeneous ETreactions.26,28–36 Their approaches provide a first evaluation ofthe effects of asymmetry although, obviously, more sophisticatedmodels requires more complex mathematical and physicalformulations that limit their applicability and generality.Nevertheless, there are some ‘manageable’ results in the litera-ture where asymmetric kinetic models provide closed-formexpressions for the activation energy or the Gibbs energy curves.

Their numerical implementation is easy and fast and the con-nection of the kinetic parameters to the molecular properties ofthe electrochemical system is well established.

Recently, two of these models have been applied successfullyto the quantitative fitting of the voltammetry of electrontransfer reactions that diverge from the SMH model. Theyrationalize these deviations in terms of differences betweenthe inner-sphere (asymmetric Marcus–Hush model) and outer-sphere (Matyushov model) force constants.

5 The asymmetric Marcus–Hush (AMH) model

Results derived from spectroscopic measurements and quantum-mechanical calculations show significant differences between thevibrational modes of the reduced and oxidized species for transitionmetal complexes and clusters. For example, the average stretchingfrequency of the metal–oxygen bond in aquacomplexes of chro-mium and europium is dependent on the oxidation state of themetal, being ca. 2.0 times higher for [Cr(OH2)6]3+ than for[Cr(OH2)6]2+ and ca. 1.6 times in the case of the redox couple[Eu(OH2)n]3+/2+ 36,37 Similarly, inner-sphere asymmetries have beenreported for chromium and cobalt chelate complexes.29

Originally, the theory developed by Marcus34 includes thecase of unequal inner-sphere reorganization energies. However,for the sake of simplicity only the symmetrized version of theMarcus model had been employed. Given the number ofsystems that cannot be described properly by the SMH model,Compton et al. have recently adopted the asymmetric version ofthe Marcus model for homogeneous ET reactions and applied itto the study of electrode processes.14,30,31,38–40 The asymmetricMarcus–Hush model provides simple analytical expressions forthe oxidation and reduction rate constants to connect theexperimental data to differences between the force constantsof the redox species.

While the SMH model has proven to be inconsistent withexperimental observation for several systems, the AMH modelis able to accurately reproduce all of the features of these

Fig. 6 Gibbs energy curves according to the harmonic oscillator approximationfor the electroactive species as a function of the reaction coordinate, q. Thecontinuum of electrode energy levels is not shown for clarity.

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experiments. A summary of the kinetic parameters used to fitseveral systems are given in Table 1.

In the asymmetric version of the Marcus model, the follow-ing expression is obtained for the activation energy ofreduction/oxidation:

DGzasym;red=oxðxÞRT

¼ L4

1� yþ x

L

� �2

þg yþ x

4

� �1� yþ x

L

� �2( )

þ g2L16

(17)

where the new parameter g accounts for the differencesbetween the inner-shell force constants of oxidized andreduced species:§

g ¼ lilhlsi ¼

lil

Ps

ks Dq0s� �2

lsPs

ks Dq0s� �2 (18)

with li being the inner-sphere reorganization energy, Dq0s the

difference of the equilibrium values for the s-th normal modecoordinate of reactants and products, and ks and ls symmetricand antisymmetric combinations of the force constants of thes-th mode of reactants and products, f Ox

s and f Reds , respectively:

ks ¼2f Red

s f Oxs

f Reds þ f Ox

s

(19)

ls ¼f Oxs

�f Reds � 1

f Oxs

�f Reds þ 1

(20)

According to the above definitions, the absolute values of hlsi,and thus g, increase with the disparity of the values of the forceconstants. A positive g value is associated with greater forceconstants for the oxidized species and the opposite for negativeg values. Note that the the symmetric form of the Marcus model

can be viewed as a particular case of the more general asym-metric model for equal average force constants: hlsi = 0 and g = 0.

For the sake of generality and simplicity, only first terms inexpansion of hlsi are considered in the derivation of eqn (17).This enables us to take into account the case of different forceconstants by including only one additional kinetic parameter:g. On the other hand, it limits the application of eqn (17), whichcannot be applied for very large differences between the forceconstants (i.e. large |g|) and/or very large overpotentials. Inaddition the limits of the integral in eqn (9) cannot be extendedindefinitely and they must be restricted to the range of x-valuesfor which the integrand has a significant value, typically �50.14

The accuracy of the model has been evaluated by compar-ison with the complete solution given by eqn (A6)–(A8) inAppendix IV of ref. 34 (assuming a coordination number of 6).It is difficult to establish general criteria given that this isdependent not only on the difference between the force con-stants but also on the l value, the ‘‘weight’’ of the inner-spherereorganization energy to this value (li/l), the number of vibra-tional modes and the overpotential (y). Thus, the accuracy isfound to decrease as the difference between force constants andthe li/l and |y| values increase, and as l decreases. Consideringthe most demanding situation where li = l and typical l values(1–2 eV), the rate constants calculated from eqn (9) and (17) areaccurate within 5% for |y| o 9–11 and |g| o 0.35. Thiscorresponds to force constants differing by a factor of less than2 and overpotentials smaller than ca. 230–280 mV at 25 1C. Thisestablishes a conservative range of validity given that the errordecreases significantly with the ratio li/l.

Under conditions where the AMH model is not suitable,higher-order terms in hlsi are necessary in eqn (17) and then apriori information regarding the number and characteristics ofthe vibrational modes of the electroactive species are requiredfor simulation.

The effect of unequal force constants on the variation withpotential of the rate constants and transfer coefficient areshown in Fig. 7A. The asymmetric model predicts asymmetric,curved Tafel plots such that kred(y) a kox(�y). When the forceconstants of the oxidized species are greater, g > 0, the cathodicbranch is steeper than the cathodic one and vice versa.

Table 1 Kinetic parameters obtained from the fitting of experimental voltammetry at mercury hemispherical table, disc microelectrodes with the asymmetricMarcus–Hush model platinum. MeNP: 2-methyl-2-nitropropane; NPent: 1-nitropentane; NPh: 3-nitrophenol; COT: cyclooctatetraene; TPE: tetraphenylethylene; TBAP:tetrabutylammonium perchlorate; TEABr: tetraethylammonium bromide; TBAPF6: tetrabutylammonium hexafluorophosphate; MeCN: acetonitrile; DMSO: dimethyl-sulfoxide; DCM: dichloromethane

Redox couple Microelectrode Conditions Kinetic parameters

MeNP/MeNP�� Hg (25.0 mm) 0.1 M TBAP, MeCN k0 = 3.0 � 10�3 cm s�1

25.0 � 0.2 1C g = �0.31NPent/NPent�� Hg (28.5 mm) 0.1 M TBAP, MeCN k0 = 1.3 � 10�2 cm s�1

26.0 � 0.1 1C g = �0.14NPh�/NPh2� Hg (23.0 mm) 0.1 M TBAP, DMSO k0 = 2.0 � 10�2 cm s�1

24.0 � 0.1 1C g = �0.42COT/COT�� Hg (50.0 mm) 0.1 M TEABr, DMSO k0 = 1.1 � 10�2 cm s�1

25.0 � 0.2 1C g = �0.22TPE+/TPE Pt (25.0 mm) 0.15 M TBAPF6, DCM k0 = 0.15 cm s�1

25.0 � 0.2 1C g = +0.55Eu3+/Eu2+ Hg (50.0 mm) 0.4 M KCl, H2O k0 = 1.7 � 10�4 cm s�1

25.0 � 0.2 1C g = +0.55

§ In several early publications the parameter g has been written as the ratio b/l,in which the parameter b indicated the degree of asymmetry. This notation hassince been revised, however, and the symbol g has been used in order to avoidconfusion with either the oxidative transfer coefficient or the tunneling factor.

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Regarding the transfer coefficient (a) the asymmetric modelpredicts that this parameter depends on the applied potentialand the reorganization energy but also on the symmetry para-meter as can be observed in Fig. 7B. The following relationshipenables us to connect both kinetic parameters, a(E�Jf ) and g:

a E�Jf Þ ¼1

2þ g

1

4� 1:267

Lþ 3:353

� ��(21)

which is accurate to within 0.1% for reorganization energiesgreater than 0.5 eV. Thus, the value of the transfer coefficient atthe formal potential, a(E�Jf ), is predicted to differ from 0.5 forga 0, being greater than 0.5 for positive g values (i.e. f Ox

s > f Reds )

and less than 0.5 for negative g values (i.e. f Oxs o f Red

s ).In addition, the deviation from 0.5 is more apparent as the

reorganization energy decreases and the g absolute valueincreases. These results support the qualitative interpretationof a as an indicator of the relative symmetry of the Gibbs curvesof the electroactive. Moreover, they provide a quantitativerelationship when the asymmetry arises from differencesbetween the inner-sphere force constants.

According to the above discussion, the effect of the asym-metric parameter on the voltammetric curves is expected to be

analogous to that for a. As can be observed in Fig. 8, whereasthe reorganization energy affects the reductive and oxidativepeaks of cyclic voltammograms similarly (see Fig. 3), the g valuehas an effect on the relative anodic/cathodic peak heights andthe peak potentials. The reductive peak increases in height andshifts to less negative potentials as g takes more positive valuesand the opposite applies for negative g values. This points outthe greater flexibility of the AMH model for quantitative fittingthrough the new kinetic parameter g.

Indeed, the asymmetric model quantitatively fits the voltam-metry of the redox systems for which low quality fittings werereported using the SMH model. The kinetic parameters obtainedfor the different redox systems studied using this model areshown in Table 1.31,38–40 Note that in all cases microelectrodeswere employed as the working electrode together with a highconcentration of supporting electrolyte (Z0.1 M), which reducesgreatly possible distortions due to ohmic drop and migrationeffects. Indeed, diffusion–migration simulations were performedaccording to the procedure detailed in ref. 22 and the occurrenceof such undesirable effects was discarded.

The fitting of the voltammmetry of the systems included inTable 1 highlighted that the evaluation of the reorganizationenergy is not accurate since the voltammograms are relativelyinsensitive to this parameter. By contrast the g value is acces-sible and in all cases is consistent with a higher force constantfor the electroactive species with the higher charge. Thus, for allthe electroreductions of neutral species into radical anionsnegative g values were obtained, which corresponds to tighterGibbs energy curve for the reduced species. On the other hand,positive g values were obtained for the electroreduction of Eu3+

and the electrooxidation of tetraphenylethylene where theoxidized species has a higher charge.

The g value obtained for the electroreduction of Eu3+ sug-gests a large average ratio between the force constants of theEu(III) and Eu(II) aquo-complexes of f Ox/f Red

Z 3.4, which issignificantly larger than that deduced from spectroscopic data( f Ox/f Red = 1.6). According to the above discussion on thelimits of applicability of the AMH model, the high g value

Fig. 7 (A) Variation of the heterogeneous rate constants with the appliedpotential according to the asymmetric Marcus–Hush for different g values. l =2 eV. (B) Variation of the value of the transfer coefficient at the formal potentialwith the reorganization energy and the parameter g.

Fig. 8 Effect of the parameter g on the cyclic voltammetry of diffusional systemsas predicted by the asymmetric MH model. l = 2 eV, k0 = 10�3 cm s�1, scan rate =1 V s�1.

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obtained for this redox couple must be taken with caution.Nevertheless, the discrepancy with respect to spectroscopic resultsmay indicate that there are other factors contributing to theasymmetry besides those affecting the inner-sphere reorganizationenergy. The factors associated with the outer-sphere reorganizationenergy will be examined in the following section.

While the AMH model has proven successful in modeling theexperimental voltammetry of each of the systems shown in Table 1,the simulations were typically indistinguishable from those pro-duced using the Butler–Volmer model. Indeed through eqn (21) wesee that these two models are related and in the limit of very largereorganisation energy the two models become exactly equivalent.31

It has been recommended that, for the analysis of solution-phaseredox systems, the BV model should continue to be used due to itsgreater ease of application with the resulting transfer coefficientable to be converted into a g value through eqn (21).31

A theoretical study of the AMH model for the fast-flowchannel electrode41 came to the same conclusion—for typicalexperimental kinetic parameters the BV and AMH models arepractically indistinguishable. This study also found that, whenemploying the AMH model with small values of both k0 and l,the steady-state current could be significantly smaller than thetransport-limited current given by the Levich equation.41 Thisobservation is in line with that reported by Feldberg12 for thesteady-state response at a microdisc electrode, however thenecessary experimental conditions are equally unlikely to occur.

While the BV model has been shown to be adequate for themodeling of solution-phase systems under both diffusion-only31

and convective41 mass transport, it has proven inadequate formodeling the electron transfer kinetics of electroactive mono-layers.9,14 As discussed above, the SMH model has also proved tobe inadequate for some systems which demonstrate asymmetricTafel plots. In this regard the AMH model has been shown to bebetter able to reproduce the features of experimental voltammetryincluding not only asymmetric Tafel plots (see Fig. 9), but also thevariation of peak current and peak potential with scan rate.

6 Nonlinear solvation

In the Marcus model a continuum dielectric model for the solvent isemployed such that the outer-sphere component of the reorganiza-tion energy for heterogeneous electron transfer reactions is given by:

lo ¼e2

8pe0

1

a0� 1

2d

� �1

eop� 1

es

� �(22)

where e is the elementary charge, e0 the permittivity of free space, a0

the radius of the reactant, eop the optical permittivity, es the staticpermittivity and d the distance from the reactant to the metalsurface.

Eqn (22) predicts that lo is independent of the charge of thespecies in solution. Accordingly the AMH model does not considerasymmetric effects arising from the charge/solvent fluctuationsupon the electron transfer reaction but identical solute–solventinteractions for both electroactive species are assumed.

More sophisticated theories have challenged this view byincluding nonlinear solvation effects that provide a morerealistic picture of the effect of the nature of the solvent onthe electron transfer. In this context the model developed byMatyushov33 has recently been applied to electrode processes.Unlike other nonlinear models, the Matyushov theory offersclosed-form, analytical equations for bilinear solute–solventcoupling from which the Gibbs energy curves and rate constantscan be calculated without much computational effort. So, the effectsof nonlinear solvation can be easily evaluated, although it can onlybe applied quantitatively to systems where the inner-spherecomponent is expected to be negligible, like in the electro-reduction/oxidation of some aromatic hydrocarbons.

Similarly to the AMH model, the Matyushov model para-meterizes the asymmetric nonlinear effects with only oneadditional parameter that has a direct correspondence withthe transfer coefficient.32,33 Thus, greater solvent force con-stants for the oxidized species correspond to a tighter Gibbsenergy curve for this species and to values a(E�Jf ) > 0.5. On theother hand, a(E�Jf ) o 0.5 relates to greater solvent frequenciesfor the reduced species. The greater the nonlinear effects are,the greater the deviation of the transfer coefficient from 0.5.Reconsidering the data shown in Table 1 in light of theseresults it can be inferred that the interaction with the solventis stronger for more charged species. This behaviour isexpected once Coulombic interactions are considered.42

The effect on the voltammetry is also similar to thatdescribed for the AMH model in such a way that the cathodicpeak increases and shifts towards lower overpotentials for f Ox >f Red, whereas the anodic peak decreases and moves to morepositive potentials for f Ox o f Red.

In summary, the results obtained with the Matyushov modelshow that the deviations from SMH reported (i.e. asymmetricTafel plots, transfer coefficients different from 0.5 and poorquantitative fitting of voltammetry) can also be justifiedby nonlinear solvation effects. Consequently, the origin ofasymmetric ET potential curves can be due to several andinteractive factors and it must be analyzed in terms of the

Fig. 9 Plot of rate constant vs. overpotential for a surface-bound Osmium aquocomplex from ref. 14. Shown are the experimental data (squares) and thetheoretical fit using AMH kinetics (solid line) with g = �0.18.

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physical–chemical properties of the system, ideally assisted byspectroscopic data and quantum-mechanical calculations.

7 Adiabaticity: variable temperature kineticstudies

Both the SMH and AMH models assume non-adiabatic electrontransfer, that is they assume a very weak interaction betweenthe electroactive species and the electrode. Therefore, thesplitting of the potential energy curves due to electronic cou-pling of the electroactive species with the electrode is supposedto be negligible such that the energy barrier is not affected.However, this is not necessarily the case and this matter iscurrently subject to intense scientific debate.43

The asymmetry of the reactant and product energy curves can beaffected by the degree of adiabaticity and Matyushov has reportedthat the asymmetric effects arising from nonlinear solvation aremore apparent as the process becomes more adiabatic.33

In general terms, the degree of adiabaticity of the processwill depend on the electronic properties of the electrode andelectroactive molecules, the distance across which ET occursand the medium. The modeling of the effect of the electroniccoupling on the activation energy is tackled through two mainformalisms: potential energy curves (pec) and density of states(dos) methods.44 These approaches converge in the non-adiabaticlimit (weak coupling) but they differ when the electronic couplingbetween electrode and reactant species is significant.

According to all the above it is important to have at ourdisposal experimental procedures for the evaluation of theadiabaticity of electrode processes. This will provide furtherphysical insight into the electron transfer event and it will alsoenable us to evaluate the suitability of the different theoreticalmodels. A first evaluation of the degree of adiabaticity can becarried out through variable temperature kinetic studies. Thisstrategy also enables the extraction of the value of the reorga-nization energy that is not accurately accessible from voltam-metry at constant temperature.

The standard rate constant, k0, is given by the product ofthree factors:

k0 = nnkelkn(E = E�Jf ) (23)

where nn is the nuclear vibration frequency factor (dependent onthe solvent reorientation dynamics), kel the electronic transmis-sion coefficient and kn(E = E�Jf ) the nuclear reorganization factorat the formal potential that can be expressed as:¶

kn E ¼ E�Jf� �

¼Z 1�1

exp �DGzred=oxðx0Þ

.RT

h i1þ exp �x0ð Þ dx0

� C exp � lF4RT

1þ g2

4

� �� � (24)

where x0 = F(e � E�Jf )/RT and the value of C may be consideredapproximately constant for a given system over a small range oftemperatures since it is only weakly dependent upon tempera-ture, it is mainly dependent on the reorganization energy. Thevalue of the electronic transmission coefficient is directly relatedto the electronic coupling between the electroactive species andthe energy levels of the electrode, such that kel-1 for stronglyadiabatic processes and kel { 1 in the non-adiabatic limit.Feldberg and Sutin gave an approximate expression for theevaluation of kel

45:

kel �1

btunln 1þ 4p2rMH0;AB

2

nnhffiffiffiffiffiffiffiffiffi4pLp

� �(25)

where btun is the tunneling factor, rM the density of electroniclevels in the electrode, H0,AB electronic coupling element at theplane of closest approach and h the Planck’s constant.

According to the above expressions it can be easily inferredthat the variation of the standard heterogeneous rate constantwith temperature is predicted to have an Arrhenius-like beha-viour:

lnðk0Þ ¼ lnðZÞ � lF4R

1þ g2

4

� �1

T(26)

where:

Z � Cnnbtun

ln 1þ 4p2rMH0;AB2

nnhffiffiffiffiffiffiffiffiffi4pLp

� �(27)

Therefore, by plotting ln k0 vs. 1/T we may determine thereorganization energy from the slope (once the g value is knownfrom the fitting of the voltammetry). The adiabatic character ofthe process can also be evaluated from the intercept througheqn (27) (see Fig. 10).

This approach had been applied traditionally to electro-active monolayers and more recently to diffusional processesproviding essential information for the understanding ofelectron transfer kinetics. Thus, the effect of the supportingelectrolyte cation on the strength of the electronic couplingthrough steric hindrance has been shown for the reduction of

Fig. 10 Variation of the intercept (Z) of the plot ln k0 vs. 1/T with the couplingconstant (H0,AB

2) according to eqn (27).

¶ Note that the effect of the electronic coupling on the energy barrier is assumedto be negligible. This assumption can be refined a posteriori through the dos orpec models for adiabatic processes.

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nitromesitylene at a mercury electrode in propylene carbonate,which indicates the non-adiabatic character of this process.46

Striking differences in terms of reorganization energies andadiabaticity between nitroaliphatic and nitroaromatic com-pounds have also been highlighted by means of variabletemperature kinetics studies (see Fig. 11).38,40

8 Further factors to consider

Other factors can also give rise to deviations with respect to themodels considered above, some of them directly related to thecharacteristics of the ET process and others to undesirabledistorting effects. Among the latter, the double layer and ohmicdrop effects can alter the mass transport and kinetics of thesystem giving rise to ‘‘apparent’’ deviations from the theoreticalmodels. In the case of solution-phase systems double layereffects have been demonstrated to be significantly mitigated athigh supporting electrolyte concentrations after consideringthat the electron transfer can take place not only at theHelmholtz plane but over a significant distance of severalangstroms.47,48 The experimental conditions required to mini-mize potential drop effects have also been studied,22 showingthat the concentration of the supporting electrolyte must bemore than 100 times than that of the electroactive species incyclic voltammetry at macroelectrodes.

The solvent dynamics is an important aspect to be consid-ered. As mentioned in Section 7 its influence can be consideredin a simplistic way through the nuclear factor of the standardrate constant such that nn p tL

�s where tL is the longitudinalrelaxation time of the solvent and s = 1 for adiabatic processesand its value decreases as the adiabaticity is weaker. However,solvent dynamical effects can be more complex and have aninfluence on the reaction energy barrier (through deviation of theET path from the saddle point) such that the potential-dependenceof the rate constants (and so the transfer coefficient) is affected.49

This may be important in measurements in solvents with highviscosity like room temperature ionic liquids.

Finally, in previous sections we have considered the case ofprocesses where the changes in bonds upon the electron transferare small such that the description by the harmonic approximationis satisfactory. On the other hand, this picture is not correct whenlarge structural changes occur with even bond formation or break-ing. In these situations anharmonic energy curves describe betterthe system (e.g., the Morse potential) which can have a significanteffect on the reorganization energy of the process.50,51

9 Conclusions

We have examined the limitations of the most commonly-employed kinetic models for electrode reactions: the empiricalButler–Volmer (BV) and symmetric Marcus–Hush (SMH)models. Several independent experimental deviations fromthese models indicate that they cannot describe the complexityof these processes.

Potential-dependent transfer coefficients and curved Tafelplots have been found experimentally, which are not predictedby the BV model. Likewise, asymmetric Tafel plots and transfercoefficients which deviate from 0.5 at the formal potentialcannot be explained by the SMH model. In addition, this modelgives rise to poor-quality fittings of the voltammetry of severalsolution-phase redox systems.

These results question the suitability of the simplifyinghypotheses considered in the development of the SMH model.One of these is the assumption that the Gibbs energy curves forthe reduced and oxidized species have equal curvature. Thissimplified view fails when the electroactive species showdifferent vibrational and/or solvation force constants due tostructural and solvation changes upon electron transfer. Theasymmetric Marcus–Hush (AMH) and Matyushov models offera simple evaluation of the effects of asymmetric Gibbs energycurves on the rate constants and voltammetry of electrodeprocesses.

The asymmetric MH model (AMH) accounts for differencesbetween the inner-sphere force constants and it is consistentwith all of the experiments reported in this review. In additionit provides satisfactory quantitative fitting of the voltammetryof those systems where the SMH model fails, with a qualitycomparable to that obtained with the BV model. Similarly thenonlinear Matyushov model rationalizes deviations from theSMH model in terms of different solvation force constants. Inboth models a direct correspondence can be established betweenthe asymmetric effects and the BV transfer coefficient, a.

For electroactive monolayers the use of the asymmetrickinetic models is necessary since neither the BV model nor theSMH model can reproduce both the asymmetry and curvatureevident in many Tafel plots documented in the literature. On theother hand, the simpler BV model can be recommended for thefitting of diffusional voltammetry with the a value as indicator ofthe difference between the force constants of the electroactivespecies (Ox, Red) such that: f Ox > f Red for a(E�Jf ) > 0.5, f Ox o f Red

for a(E�Jf ) o 0.5 and f Ox = f Red for a(E�Jf ) = 0.5.

Fig. 11 Arrhenius plot ln k0 vs. 1000 T�1 for 3-nitrophenolate in DMSO (fullcircles) and 1-nitropentane in acetonitrile (empty squares) containing 0.1 MTBAP. From the fitting of the best-fitting line the following parameters areobtained in each case: l = 0.6 eV, ln(Z/cm s�1) = 1.8 (3-nitrophenolate);l = 1.7 eV, ln(Z/cm s�1) = 13 (1-nitropentane).

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The asymmetric models examined here assume that theasymmetric effects are due exclusively to either the inner-sphere (AMH) or outer-sphere (Matyushov) components of thereorganization energy. However, the reality is likely morecomplex and different vibrational and solvation force constantscan occur simultaneously. Quantitative modeling of this situa-tion requires more complex models that take into account theinterplay between both components. Therefore, the origin ofthe asymmetries must be analyzed based on the properties ofthe particular system under scrutiny.

The use of these more advanced kinetic models in combi-nation with variable temperature kinetic studies provide acheap, fast and simple methodology to gain molecular informa-tion of the system. Besides the evaluation of the different forceconstants discussed above, the degree of adiabaticity and thereorganization energy of the process can be estimated easilyfrom the variation of the standard heterogeneous rate constant,k0, with temperature. This can assist in the existing debateabout the adiabatic character of ET reactions and its modeling.

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