Tafel−Volmer Electrode Reactions: The Influence of Electron-TransferKineticsChuhong Lin,† Christopher Batchelor-McAuley,† Eduardo Laborda,‡ and Richard G. Compton*,†

†Department of Chemistry, Physical and Theoretical Chemistry Laboratory, University of Oxford, South Parks Road, OX1 3QZOxford, United Kingdom‡Departamento de Química Física, Facultad de Química, Universidad de Murcia, 30100 Murcia, Spain

*S Supporting Information

ABSTRACT: A typical Tafel−Volmer electrode reaction, embracing apreceding chemisorption step and a following electron transfer, is exploredby simulation. Two different electron-transfer formalisms, Butler−Volmerand a Marcus−Hush-like approach, are utilized in interpreting thepotential-dependent kinetics of the electrochemical process. Undernonreversible electron-transfer conditions, the steady-state voltammetricresponse at a microdisk electrode is sensitive to the underlying electron-transfer kinetics, such that with weak adsorption conditions, thediscrepancies between the two-electron-transfer models are morepronounced. At high overpotentials, a non-diffusion-limited current mayeither arise from slow adsorption or be a result of Marcus−Hush type kinetics.

1. INTRODUCTION

The Butler−Volmer (BV) equation, although phenomenolog-ical in its description of electron transfer, has historically beenand continues to be successful in describing a wide variety ofelectrochemical data.1 However, significant work has beenfocused upon both providing a more physically correctdescription of the interfacial electron-transfer process2,3 anddetermining experimental conditions under which the Butler−Volmer equation is found to be inadequate in describing theelectron-transfer process.4−6

It was with the publication of Chidsey’s work in 1991 thatthe first direct evidence for the limitations of the Butler−Volmer equation was obtained, highlighting how at highoverpotentials the rate of electron transfer is significantly belowthat predicted by the BV equation.5 This work utilized a surfacebound ferrocene group as the electroactive species. Impor-tantly, it is the use of the such a “diffusionless” system thatfacilitated the interrogation of the electron-transfer kinetics atpotentials significantly away from the formal potential for theredox couple, evidencing the validity of the Marcus−Hush(MH) description of electron-transfer kinetics. For an outer-sphere diffusional (solution phase) electrochemical system,evidencing distinctions between models of electron transferbecomes inherently challenging in part due to the rate ofelectrochemical reaction commonly being under mass-transport(diffusion) control at high overpotentials. Recent advancementsin the theory of interfacial electron transfer have demonstratedhow, for outer-sphere diffusional redox processes, deviationsfrom Butler−Volmer electron-transfer kinetics are predicted tobe observable under conditions of extremely high mass-transport.7 Such conditions may be achievable with the use ofan electrode with nanoscopic dimensions; however, under most

experimental conditions the electrochemical response is foundto be relatively insensitive to the mechanism of electron transfer(BV vs MH).8

Between the limits of the two above cases (surface boundand outer-sphere diffusional) exists a third class of interfacialelectrochemical reactions, inner-sphere heterogeneous elec-trode reactions.9 Here for inner-sphere reactions, the electro-active species is present in the solution phase; however,electron transfer proceeds via one or more surface adsorbedintermediates. This class of reactions encompasses a diverserange of systems including some of the most technological andindustrially relevant.10 Hence, the question arises: are “inner-sphere” heterogeneous reactions sensitive to the electron-transfer mechanism (for example BV vs MH) and under whatconditions can discrepancies between the different models ofelectron transfer be observed? Addressing these questionspresents two significant difficulties, first an appropriateelectrochemical mechanism needs to be selected and secondthe form of the kinetic formalism requires consideration. Interms of the overall operative electrochemical mechanism, ahost of possible reaction routes may be considered; however,within this work the Tafel−Volmer reaction mechanism11 istaken as a paradigmatic example of an inner-sphereheterogeneous electrochemical process. This reaction path isone route by which hydrogen may be oxidized on someelectrodes12,13 and so is both a “simple” example of an inner-sphere heterogeneous reaction and is of further academicinterest. The question as to the “appropriate” form of Marcus−Hush electron-transfer model is a far more challenging

Received: August 18, 2015Published: September 14, 2015

Article

pubs.acs.org/JPCC

© 2015 American Chemical Society 22415 DOI: 10.1021/acs.jpcc.5b08044J. Phys. Chem. C 2015, 119, 22415−22424

problem. Specifically, the majority of derivations assume thatthe distance between the electrode and the redox species issignificantly larger than ionic radius of the analyte and does notchange during the course of the electron-transfer event.14,15

However, for an ion-transfer process this approximation doesnot hold; consequently, inclusion of the molecule/interfacedistance into the model results in a potential energy surfaceover which the reaction must proceed.16 Calculation of thepotential energy surface at a single voltage has been previouslyachieved; such results usefully gain from a theoretical stand-point insight into the relative catalytic properties of differentmaterials.17 However, consideration of the reaction pathway asa function of potential is more challenging due partially to theinfluence of the distribution of the electrostatic potential in thedouble layer.On returning to the question of whether the electrochemical

response of an inner-sphere heterogeneous reaction is more orless sensitive to the electron-transfer mechanism, within thiswork the Marcus−Hush model of electron transfer is applied,such that the deviation from the Butler−Volmer equation issuitably described by the model. While being aware of thehigher complexity of the theoretical description of electro-adsorption/desorption processes,10 in this paper we aim toexamine the impact of the electron-transfer kinetics on thevoltammetric response in view of the non-diffusion-limitedcurrents recently reported for the hydrogen oxidation reaction(HOR) at high overpotentials.11 As a first approach, an MH-like model will be employed. A potential energy curve methodwith a single reaction coordinate and a (symmetric) Marcusiandescription of the energy profile; that is, a quadratic variation ofthe energy with the reaction coordinate that results in aquadratic relationship between the activation energy and theapplied potential. This quadratic relationship is in contrast withthe linear dependence predicted by the semiempirical BVmodel. However, strict physical interpretation of the potentialenergy curve utilized in this work would require inclusion of anunderstanding of the influence of the inter analyte/electrodedistance upon the reaction coordinate. Therefore, application ofthe Marcus−Hush model within the Tafel−Volmer electro-chemical mechanism provides an objective route by which thesensitivity of this paradigmatic “inner-sphere” reaction pathwaytoward the electron-transfer mechanism can be assessed. Fromthis work it is demonstrated that although the Tafel−Volmerreaction is sensitive to the electron-transfer model, i.e.,differences between the MH and BV processes are clearlydemonstrated, the influence of the electron-transfer kinetics onthe electrochemical response is not unique and may arise fromother kinetic limitations of the reaction.

2. THEORY AND SIMULATION

In this study, the electrode reaction, which includes a precedingadsorption step and a following electron-transfer step is takeninto consideration. The hydrogen oxidation reaction is utilizedas an example to explore this type of inner-sphere electrodereaction. Two-electron-transfer models, the Butler−Volmer anda Marcus−Hush-like approach, are applied in the description ofthe kinetics of the electron-transfer process.2.1. Tafel−Volmer Mechanism. As demonstrated in

previous works,11−13 the hydrogen oxidation reaction eq 1 onPt electrodes can be regarded as the combination of a chemicaladsorption step (the Tafel reaction, eq 2 and an electron-transfer step (the Volmer reaction, eq 3):

⇄ ++ −H 2H 2e2 (1)

+ ⇄H 2M 2H(ads)k

k

2d

a

(2)

+ ++ −X YoooH(ads) H e Mk

k

red

ox

(3)

where H(ads) is an adsorbed hydrogen atom, M is an availableadsorption site on the electrode surface, ka (m s−1) and kd (molm−2 s−1) are the adsorption and desorption rate constants forthe Tafel reaction (eq 2), and kox (mol m

−2 s−1) and kred (ms−1) are the oxidative and reductive electron-transfer rateconstants for the Volmer reaction (eq 3).The values of kox and kred are dependent on the applied

electrode potential E and the formal potential of the electron-transfer step Ef,H

+/H(ads), which is not identical with the formal

potential of the whole reaction Ef,H+/H2

. From the Gibbs energy

balance, Ef,H+/H(ads) can be expressed in terms of Ef,H

+/H2

and theadsorption equilibrium constant Kads (unitless):

= ++ +E ERT

FK

2ln( )f,H /H(ads) f,H /H ads2 (4)

=°

Kk ckadsa

d (5)

where the standard concentration c° = 1 mol dm−3.According to eqs 1−3, the fluxes of the electroactive species

on the electrode surface can be described as

= = − =Γ

Γ+

ΓΓ

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥j r z t k c r z t

r tk

r t( , 0, ) ( , 0, )

( , ) ( , )H a H

M

max

2

dH(ads)

max

2

2 2

(6)

∂Γ∂

= =Γ

Γ−

ΓΓ

−Γ

Γ+

ΓΓ

=+

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

r t

tk c r z t

r tk

r t

kr t

kr t

c r z t

( , )2 ( , 0, )

( , )2

( , )

( , ) ( , )( , 0, )

H(ads)a H

M

max

2

dH(ads)

max

2

oxH(ads)

maxred

M

maxH

2

(7)

= =Γ

Γ− =

ΓΓ

+ +

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥j r z t k

r tk c r z t

r t( , 0, )

( , )( , 0, )

( , )H ox

H(ads)

maxred H

M

max

(8)

Γi(r,t) is the surface coverage (mol m−2) and Γi(r,t)/Γmax is the

fractional surface coverage (unitless). Γmax is the maximumsurface coverage at the electrode surface and Γmax = ΓH(ads)(r,t)+ ΓM(r,t). More details about the differences between applyingΓi(r,t) or Γi(r,t)/Γmax in the rate equations can be found inSupporting Information I. The parameters r, z are thecylindrical coordinates used to model a microdisk electrodeas shown in Figure 1, and t is the time of applying the electricalperturbation. For linear sweep voltammetry, t and the appliedelectrode potential E follow:

< = +

> = − −

t t E E vt

t t E E v t t( )

switch ini

switch switch switch (9)

where Eini and Eswitch are the initial potential and the switchingpotential, tswitch is the time at which the direction of sweep isswitched, and v is the scan rate.The fluxes in eqs 6 and 8 follow Fick’s first law:

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= −∂

∂j r z t D

c r z t

z( , , )

( , , )H H

H

2 2

2

(10)

= −∂

∂+ +

+j r z t D

c r z tz

( , , )( , , )

H HH

(11)

Adsorption is considered to follow a Langmuir isotherm. Thus,the initial surface coverage fractions (Γi(r,t=0)/Γmax) forH(ads) and M are

Γ =Γ

=*

+ *

r t c k k

c k k

( , 0) /

1 /

H(ads)

max

H a d

H a d

2

2 (12)

Γ =Γ

=+ *

r t

c k k

( , 0) 1

1 /M

max H a d2 (13)

where cH2* is the bulk concentration of the hydrogen molecule.

In addition to the reaction on the electrode surface, thediffusion process of the reactant H2 and the product H

+ are alsoconsidered in the simulation. Given the electrode geometryshown in Figure 1, the diffusion equations for solution speciescan be described as

∂∂

=∂

∂+

∂∂

+∂

∂

⎛⎝⎜⎜

⎞⎠⎟⎟

c r z t

tD

c r z t

r r

c r z t

r

c r z t

z

( , , ) ( , , ) 1 ( , , )

( , , )

HH

2H

2H

2H

2

2

2

2 2

2

(14)

∂∂

=∂

∂+

∂∂

+∂

∂

++

+ +

+

⎛⎝⎜

⎞⎠⎟

c r z tt

Dc r z t

r rc r z t

r

c r z tz

( , , ) ( , , ) 1 ( , , )

( , , )

HH

2H

2H

2H

2(15)

2.2. Electron-Transfer Theories. In the Butler−Volmermodel, kox, kred are dependent on the standard heterogeneousrate constant k0, the transfer coefficient α or β (α + β = 1),18,19

and the applied potential E:

β= ° − +⎜ ⎟⎛⎝

⎞⎠k k c

FRT

E Eexp ( )oxBV

0 f,H /H(ads)(16)

α= − − +⎜ ⎟⎛⎝

⎞⎠k k

FRT

E Eexp ( )redBV

0 f,H /H(ads) (17)

Ef,H+/H(ads) is the formal potential associated with the Volmer

reaction eq 3. E − Ef,H+/H(ads) is the overpotential of the

electron-transfer step, which is different from the overpotentialfor the whole reaction E − Ef,H

+/H2

. Here notice that eq 16 isnot the common expression for the oxidative electron-transferrate, which reflects the stoichiometry of the reaction and thedifferent standard states defined for solution species andadsorbates on the left and right sides of the electron-transferreaction eq 3. More details about the derivation can be found inthe Supporting Information II.In the Marcus−Hush model, for an electrochemical reaction

RX Yoook

k

red

oxP + e−, the expression for a rate constant k is derived from

a transition-state theory approach:2,20,21

= −Δ *⎛

⎝⎜⎜

⎞⎠⎟⎟k A

Gk T

expox/redMH ox/red

B (18)

λ ελ

Δ * = ∓− ++⎛

⎝⎜⎞⎠⎟G

e E E

41

( )ox/red

f,H /H(ads)2

(19)

where A is the pre-exponential term, which is assumed to beindependent of E and ε. ΔGox/red* (eV) is the activation energyfor the oxidative (−) or the reductive (+) process of theelectrochemical reaction. ε (eV) is the energy level of the metalelectrode. λ (eV) is the reorganization energy. It is worthnoting that, within the present context, the “reorganizationenergy” λ has a parametric character, its value being related tothe curvature/shape of the energy profile with the reactioncoordinate. Thus, in this case the physical meaning of λ is likelyassociated not only with the reorganization of the configurationof the electroactive species and the solvation shell (as for outer-sphere processes) but also with interactions with the electrode,(partial) desolvation, and Coulombic forces that the moleculeundergoes along the reaction pathway.10

For the electron transfer between a reactant and a metalelectrode, as there is not a single energy level but an energyband for the metal electrode, the overall rate constant should bethe sum of all energy levels, that is, k = ∑k(ε). Because thereare only small intervals kBT among these energy levels, theprobability of this transfer varies very little over two nearbydifferent energy levels in the metal electrode. Therefore, therate constant expression can be written as5

∫ ελ

ελ

ε

=+ *

− *

−− * + *

**

−∞

+∞

+

⎛⎝⎜⎜

⎛⎝⎜

⎞⎠⎟

⎞⎠⎟⎟

k A

E E

11 exp( )

exp4

1( )

2d

oxMH

fH /H(ads)2

(20)

∫ ελ

ελ

ε

=+ − *

− *

+− * + *

**

−∞

+∞

+

⎛⎝⎜⎜

⎛⎝⎜

⎞⎠⎟

⎞⎠⎟⎟

k A

E E

11 exp( )

exp4

1( )

2d

redMH

f,H /H(ads)2

(21)

where fox/red(ε*) = 1/(1 + exp(±ε*)) is the Fermi−Diracdistribution for the electron occupancy (+) or the vacancy (−).

Figure 1. Illustration of the microdisk electrode considered in thesimulation. The microdisk electrode (black) is surrounded by aninsulating sheath (white). ρ is the radius of the electrode. z and r arethe cylindrical coordinates.

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For convenience, the dimensionless parameters are applied ineqs 18 and 19:

λ λ* = F RT/( ) (22)

ε ε* = F RT/( ) (23)

− * = −+ +E E E E F RT( ) ( ) /( )f,H /H(ads) f,H /H(ads) (24)

Considering k0 is the standard rate constant at the formalpotential, for the Volmer reaction, the MH rate constants canbe expressed in a more general way:7

∫

∫

ε

ε

= °

×− − *

− − *

ελ ε

ελ ε

−∞

+∞+ *

* − * + *

−∞

+∞+ *

* *

+

⎜ ⎟

⎜ ⎟⎛⎝⎜

⎛⎝

⎞⎠

⎞⎠⎟

⎛⎝

⎞⎠( )

k k c

exp 1 d

exp 1 d

E E

oxMH

0

11 exp( ) 4

( )

2

2

11 exp( ) 4 2

2

f,H /H(ads)

(25)

∫

∫

ε

ε

=

×− + *

− + *

ελ ε

ελ ε

−∞

+∞+ − *

* − * + *

−∞

+∞+ − *

* *

+

⎜ ⎟

⎜ ⎟⎛⎝⎜

⎛⎝

⎞⎠

⎞⎠⎟

⎛⎝

⎞⎠( )

k k

exp 1 d

exp 1 d

E E

redMH

0

11 exp( ) 4

( )

2

2

11 exp( ) 4 2

2

f,H /H(ads)

(26)

Similar to koxBV, k0 has been replaced by k0c

0 in the expression ofkoxMH.2.3. Numerical Simulation. Dimensionless parameters are

applied in the simulation, as listed in Table 1. Under thedimensionless format, the diffusion equations for the reactivespecies in the solution can be described as

∂∂

=∂

∂+

∂∂

+∂

∂

⎛⎝⎜⎜

⎞⎠⎟⎟

C R Z T

Td

C R Z T

R R

C R Z T

R

C R Z T

Z

( , , ) ( , , ) 1 ( , , )

( , , )

HH

2H

2H

2H

2

2

2

2 2

2

(27)

∂∂

=∂

∂+

∂∂

+∂

∂

++

+ +

+

⎛⎝⎜

⎞⎠⎟

C R Z TT

dC R Z T

R RC R Z T

R

C R Z TZ

( , , ) ( , , ) 1 ( , , )

( , , )

HH

2H

2H

2H

2(28)

The dimensionless boundary conditions for solving diffusionequations can be expressed as

= ∀> → ∞

= * = = * =+ +

⎫⎬⎭T R Z

T R ZC C C C

0, ,

0, ,1, 0H H H H2 2

(29)

= = ≤ ≤ Θ =+

T Z RK K

K K0, 0, 0 1:

/

1 /H(ads)a d

a d

(30)

> = ≤ ≤

−∂∂

= Θ − − Θ=

⎛⎝⎜

⎞⎠⎟

T Z R

dC

ZK K C

0, 0, 0 1:

(1 )Z

HH

0d H(ads)

2a H(ads)

2H2

2

2

(31)

−∂∂

= Θ − − Θ=

++

+⎛⎝⎜

⎞⎠⎟d

CZ

K K C(1 )Z

HH

0ox H(ads) red H(ads) H

(32)

γ γ

γ γ

∂Θ∂

= − Θ − Θ

+ − Θ − Θ+

⎛⎝⎜

⎞⎠⎟T

K C K

K C K

2 (1 ) 2

(1 )

H(ads)a H(ads)

2H d H(ads)

2

red H(ads) H ox H(ads)

2

(33)

> = >

∂∂

=∂∂

== =

+⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

T Z R

C

ZC

Z

0, 0, 1:

0; 0Z Z

H

0

H

0

2

(34)

∂∂

=∂∂

== =

+⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

C

RC

R0; 0

R R

H

0

H

0

2

(35)

where the parameter γ is generated from the dimensionlessprocess, equal to cH2

* ρ/Γmax. From Table 1, it is found that γ isthe only constant parameter that contains Γmax (Θi = Γi/Γmaxworks as a variable not a constant parameter during thesimulation). Under steady-state conditions, as ∂ΘH(ads)/∂T = 0,parameter γ cancels out in eq 33 such that the response of thesystem is independent of its value. It also indicates that fortransient voltammetry where ∂ΘH(ads)/∂T ≠ 0, the value of γcannot be canceled out from eq 33 and consequently thevoltammogram will be influenced by the magnitude of Γmax.The resulting problem was solved numerically by means of

the Newton−Raphson method and according to the alternatingdirection implicit (ADI) method, the details of which can befound in the textbook.22 The simulation was written in C++

Table 1. Dimensionless Parameters Employed in theSimulations

SI unit parameters dimensionless parameters

cj Cj = cj/cH2*

Γi/Γmax Θi = Γi/Γmax

Dj dj = Dj/DH2

r R = r/ρz Z = z/ρt T = DH2

t/ρ2

E θ = (F/RT)(E − Ef,H+/H2

)

Ef,H+/H(ads) θf,H+

/H(ads) = (F/RT)(Ef,H+/H(ads) − Ef,H

+/H2

)

v σ = (ρ2/DH2)(F/RT)v

I J = I/(2πρFDH2cH2* )

ka Ka = kaρ/DH2

kd Kd = kdρ/(DH2cH2* )

Kads = kac°/kd Kads = Kac°/(KdcH2* )

k0 K0 = k0ρ/DH2

kox Kox = koxρ/(DH2cH2* )

kred Kred = kredρ/DH2

λ* λF/(RT)ε* εF/(RT)

γ = cH2* ρ/Γmax

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with OpenMP for multithreading and simulations wereperformed using an Intel(R) Xeon(R) 3.60G CPU and theruntime was approximately 10 min per voltammogram.

3. RESULTS AND DISCUSSIONIn this section, the empirical Butler−Volmer model of theelectron transfer is compared with the Marcus−Hush-likemodel with respect to their effects on the Tafel−Volmermechanism and associated voltammetry. The steady-statevoltammetric responses at microdisk electrodes of the two-electron-transfer models are examined under different electron-transfer kinetics and adsorption behaviors.3.1. Influence of the Electron-Transfer Reversibility.

Figure 2 shows the variation of the steady-state voltammogramswith a series of standard dimensionless heterogeneous rateconstants K0. The electron-transfer models used in Figure 2aand 2b are described by the BV and the MH formulas,respectively. The value of the adsorption equilibrium constantKads is always set to unity such that the formal potential of the

electron-transfer reaction θf,H+/H(ads) coincides with the formal

potential of the whole hydrogen oxidation reaction θf,H+/H2

(eq

4). Also, the adsorption and desorption rate constants Ka, Kd

are considered to be so fast as not to be the limiting factor ofthe overall reaction. The value of K0 varies from 108 to 10−3,covering the transition from electrochemical reversibility toslow (“non-reversible”) electron transfer. As expected, for theelectrochemically reversible reactions (i.e., K0 = 108 and K0 =103), the voltammograms obtained with the two kinetic modelscoincide because they both follow the Nernst equation thatalong with the diffusive mass transport defines the voltammo-grams. On the contrary, important differences between the BVand MH models are found for electrochemically nonreversibleprocesses. Thus, under the same diffusion conditions, slowelectrode kinetics affects the steady-state voltammogram byincreasing the overpotential required to drive the current. Also,a decrease of the limiting current is predicted by the MHformalism.

Figure 2. Effects of the dimensionless standard heterogeneous rate constants K0 on the voltammmograms obtained with (a) the Butler−Volmermodel and (b) the Marcus−Hush model and (c) the comparison of the electron-transfer constants between the two kinetic models. Thedimensionless current is normalized by the diffusion-limited current Jdiff. Simulation parameters for the adsorption step are Ka = Kd = 108. In the BVmodel, α = β = 0.5; in the MH model, λ* = 20. θ is the dimensionless overpotential for the overall reaction.

Figure 3. Effects of the “reorganization energy” λ* on (a) steady-state voltammograms and (b) oxidative and reductive electron-transfer rates.Dashed lines correspond to results obtained from the MH model, and solid lines are from the BV model. K0 = 10−2 and Ka = Kd = 108. θ is thedimensionless overpotential for the overall reaction.

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22419

In the BV model (Figure 2a), for the reaction with small K0

values, the voltammograms shift to higher overpotentials whilethe limiting current is the same as the reversible value, which isthe diffusion-limited current Jdiff. This is because the electro-oxidation rate Kox

BV increases ad infinitum with overpotential(Figure 2c). Therefore, by increasing the overpotential, it isalways possible to achieve electron-transfer kinetics that is notrate limiting and only a potential shift is observed in thevoltammograms. Compared with the changes in the BVvoltammograms in Figure 2a, the decrease of K0 in the MHmodel in Figure 2b leads to much more apparent changes. Theoverall reaction rates under the MH model are obviously slowerthan those under the BV models as shown in Figure 2c,reflected in both the decline of the limiting current and theincrease of the overpotential. This can be explained as aconsequence of a limiting electron-transfer rate constant at highoverpotentials predicted by the MH model. As shown in Figure2c, these two models predict similar electron-transfer rateconstants only at the overpotentials near θf,H+

/H(ads), which isequal to θf,H+

/H2in this case. Kox/red

BV varies exponentially with the

overpotential whereas Kox/redMH tends to reach a limiting value at

high overpotentials, which kinetically controls the maximumrate of the whole reaction. This trend for the Tafel−Volmertype of reaction is consistent with that reported for a simpleouter-sphere one-electron-transfer reaction.7,23

In the MH kinetics, the parameter λ* plays a key role as well.Figure 3 shows the voltammetric responses of a Tafel−Volmerreaction for different λ* values. As shown in Figure 3a, thedecrease of λ* leads to the decline in the limiting current.Figure 3b provides details about the electron-transfer ratesvarying with different λ*. It is found that the maximum rate

constant for the electron-transfer step falls with decreasing λ*,which explains the voltammograms in Figure 3a.

3.2. Influence of Adsorption on Reversible ElectronTransfer. In this subsection, the influence of adsorption isstudied under reversible electron-transfer conditions (K0 ≫ 1).Kads is not constrained to being unity and Ka, Kd changes over alarge range. When the electron-transfer step is regarded as fast(reversible, K0 ≫ 1), the surface concentrations of the reactivespecies should follow the Nernst equation and hence the BVand MH formulas lead to identical voltammetric responses. Themain features of a steady-state voltammogram (that is, thelimiting current Jlim and the half-wave overpotential θ1/2) arecontrolled by either the diffusion or the adsorption/desorptionrates.11−13,24 Figure 4 shows the voltammograms of anelectrochemically reversible Tafel−Volmer reaction for differ-ent adsorption/desorption rate constants. In Figure 4a, thedecrease of Kd shifts the wave to higher overpotentials. WhenKa varies from 103 (shown by the solid lines) to 107 (shown bythe open circles), changes in Ka (and so Kads) have no influenceon the position of the voltammograms. Similarly, in Figure 4b,for Kd equal to 107 (solid lines) and 10−1 (open circles), thenormalized limiting current is only dependent on the value ofKa. Under the electrochemically reversible case, the limitingcurrent can be written as a function of the adsorption rateconstant Ka (see Supporting Information III for the derivationand validation check):

=+

π

πJ

J

K

K1lim

diff

4 a

4 a (36)

Figure 5 provides a general view on the change of thenormalized limiting current Jlim/Jdiff and the half-wave over-potential θ1/2 under various adsorption/desorption rate

Figure 4. Voltammograms of the electrochemically reversible HOR under different adsorption conditions: (a) half-wave potential shift withdecreasing Kd (solid lines, Ka = 103; open circles, Ka = 107); (b) normalized limiting current decline with decreasing Ka (solid lines, Kd = 107; opencircles, Kd = 10−1). K0 = 108. θ is the dimensionless overpotential for the overall reaction.

Figure 5. Voltammetric features: (a) normalized limiting current; (b) half-wave overpotential, under different adsorption conditions for reversibleelectron transfer. The heterogeneous standard rate constant K0 = 108. θ is the dimensionless overpotential for the overall reaction.

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constants. From eq 36 and Figures 4 and 5, it can be seen thatfor the electrochemically reversible Tafel−Volmer reaction, thenormalized limiting current Jlim/Jdiff is only dependent on thevalue of Ka. Figure 5b shows the half-wave overpotential varyingwith different Ka and Kd. In general, on the basis of thethermodynamic arguments, the half-wave overpotential shouldnot be smaller than the reversible value. However, consideringthat small adsorption rate constants can cause a decrease in thelimiting current, this leads to an artificial shift of the half-wavepotential. Except for this interference from slow adsorption rateconstants, the half-wave potential is mainly determined by thevalue of Kd rather than Kads, indicating that changing thethermodynamics of the preceding adsorption step will notcause any significant differences on the voltammogram for anelectrochemically reversible Tafel−Volmer reaction.3.3. Influence of Adsorption on Nonreversible

Electron Transfer. For the electrochemically nonreversiblecase (small K0), apart from the effects from slow adsorption/

desorption kinetics (Ka, Kd), the whole reaction is expected tobe sensitive to the adsorption thermodynamics (Kads) as well.Although the formal potential of the electron-transfer stepθf,H+

/H(ads) is always dependent on Kads, only when K0 is small, isthe change of formal potential (and hence the overpotential)expected to have an apparent influence on the overall reactionrate.Figure 6 shows the effects of the adsorption thermodynamics

(Kads) on the voltammograms simulated in the BV model(Figure 6a) and the MH model (Figure 6b). To distinguish thiscase from the electrochemically reversible reactions discussedabove, the standard heterogeneous constant K0 is selected as10−2 and λ* used in the MH model is 20. As shown in Figure6a, in the BV model, the normalized limiting current is alwaysunity (diffusion controlled) but the overpotential changes withKads. Both strong (large Kads) and weak (small Kads) chemicaladsorptions make the overpotential shift positively. In the MHmodel, strong adsorption leads to a similar overpotential shift

Figure 6. Voltammograms of electrochemically nonreversible Tafel−Volmer reaction with different adsorption conditions: (a) Butler−Volmerelectron-transfer model (K0 = 10−2 and α = β = 0.5); (b) Marcus−Hush electron-transfer model (K0 = 10−2 and λ* = 20). The parameters of theadsorption step used in the simulation are listed in Table S2. θ is the dimensionless overpotential for the overall reaction.

Figure 7. Voltammetric features under different adsorption conditions and electron-transfer models for nonreversible electron transfer: (a)Normalized limiting current under the BV model; (b) overpotential when J = 0.01Jdiff under the BV model; (c) normalized limiting current under theMH model; (d) overpotential when J = 0.01Jdiff under the MH model. For the BV model, K0 = 10−2 and α = β = 0.5; for the MH model, K0 = 10−2

and λ* = 20. θ is the dimensionless overpotential for the overall reaction.

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whereas weak adsorption significantly decreases the limitingcurrent, which is visibly different from that predicted by the BVmodel (Figure 6a). Here to avoid any additional effect fromslow adsorption/desorption, the values of the applied Ka and Kdfor Figure 6 are relatively fast, as listed in SupportingInformation IV (Table S2). Compared with Figure 5, it isfound that these Ka, Kd applied in Figure 6 should not lead toany extra potential shift. Therefore, the changes of thewaveshape in Figure 6 are only attributed to the value of Kads.Figure 7 provides a general view on the effect of the Tafel

step under electrochemically nonreversible conditions, coveringthe adsorption and desorption rate constants from 10−2 to 108.The results are calculated from the BV model (K0 = 10−2 and α= β = 0.5) and the MH model (K0 = 10−2 and λ* = 20). Unlikein Figure 6, the additional effect of slow adsorption anddesorption is also taken into consideration in Figure 7 wherethe variation of the normalized limiting current Jlim/Jdiff and theoverpotential corresponding to 0.01Jdiff (θ0.01Jdiff) are studied asthe characteristic magnitudes of the voltammetric responses.The reason for the use of θ0.01Jdiff rather than θ1/2 in Figure 7(and also Figure 8) is that when the adsorption thermody-namics effect (Kads) on the electron-transfer kinetics isexplored, the comparison of the overpotential at the samecurrent corresponding to the foot of the wave can clearly revealthe change of the formal potential of the electron-transfer step,without the interference of the adsorption/desorption kineticsdiscussed in Figure 5. On the contrary, in Figure 5 (and Figure9 as well), the kinetics of the adsorption step (Ka, Kd) isexplored as well as its effect on the whole voltammogram, suchthat the half-wave overpotential is chosen as the measurementof the wave position. From Figure 7a,b simulated in the BVmodel, it can be seen that the limiting current of the BV modelis only related to the value of Ka, similarly to that observed for

reversible electron transfers in Figure 5a. On the contrary, theoverpotential shift is not solely determined by Kd but insteadmainly dependent on the value of Ka/Kd (Kads). Parts c and d ofFigure 7 are simulated under the MH model. In contrast to theBV model, the limiting current of the MH model is determinedby both Ka and Kd and higher overpotential is needed to obtainthe same current in the MH model.In both the BV and MH models, when the electron-transfer

step is not reversible, the value of θ0.01Jdiff changes parabolicallywith Kads (Figure 7b,d), which implies that both strong andweak adsorption can impede the reaction. According to theSabatier principle,25 optimal electrocatalysis happens when theadsorption is neither strong nor weak. In the Tafel−Volmermechanism, strong adsorption makes the formal potential forthe electron-transfer step θf,H+

/H(ads) move positively and hencethe overpotential has to increase to drive the electrochemicalreaction. Therefore, the voltammetric wave shifts to higherpotentials when the interaction between the hydrogen and theelectrode is stronger. For weak adsorption, although theelectron-transfer reaction is more favorable thermodynamically,the voltammogram also shifts to higher overpotentials as aresult of the low availability of adsorbed hydrogen on theelectrode surface. Moreover, compared with the BV model, inthe MH model, the limiting current decreases and higheroverpotential is required to obtain the same current, especiallyunder weak adsorption conditions (small Ka/Kd). This can beexplained as follows: the amount of adsorbed hydrogenbecomes insufficient for the Volmer reaction and the lack ofH(ads) accentuates the limitation by the electron-transfer rateconstants Kox

MH. Interestingly, the BV and MH models givealmost the same voltammetric responses when the adsorption isstrong (large Ka/Kd), indicating that the discrepancies betweenthe two kinetic models can be more clearly observed under

Figure 8. Voltammetric features of electrochemically nonreversible HOR with different adsorption conditions: (a) normalized limiting current; (b)overpotential where J = 0.01Jdiff. Ka, Kd are the same as listed in Table S2. θ is the dimensionless overpotential for the overall reaction.

Figure 9. Diffusion effects of the adsorption and the electron-transfer-limited reactions: (a) normalized limiting current; (b) half-wave potential. Theadsorption-limited reaction is simulated in the BV model with K0 = 10−2, Ka = 60, and Kads = 1. The electron-transfer-limited reaction is simulated inthe MH model with K0 = 10−2, λ* = 20, and Kads = 1 (Ka = Kd ≫ 1). The radii of the electrode change from 50.0 to 0.1 μm. θ is the dimensionlessoverpotential for the overall reaction.

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weak adsorption conditions whereas strong adsorption canpartially compensate the loss of the reaction rate caused by slowelectron-transfer rate at high overpotentials. Thus, themaximum reaction rate for a Tafel−Volmer reaction is directlylinked with the electron-transfer rate, given by KoxΘH(ads) athigh overpotentials. For the Tafel−Volmer reaction with a slowMH electron-transfer kinetics, although Kox reaches a limitingvalue, there are more adsorbed hydrogen atoms on theelectrode surface under strong adsorption conditions ratherthan weak adsorption, as the initial amount of adsorbedhydrogen is determined by Kads eq 30.Figure 8 shows the variation of the limiting current and the

overpotential corresponding to 0.01Jdiff with the value of Kadsfor λ* = 10, 20, 30, 40. In Figure 8a, as the applied Ka and Kdare the same as in Figure 6, it is clear that under any adsorptionconditions, the values of the limiting current simulated from theBV model are equal to the diffusion-limited current. For theMH model, as discussed in Figure 3, the values of the limitingcurrents calculated from small reorganization energies are muchlower than the diffusion-limited value. It is also shown in Figure8a that the decrease of Jlim caused by nonreversible electrontransfer can be partially compensated by a strong adsorptionstep. Here the adsorption equilibrium constant Kads at whichthe electron-transfer limitation of the MH model starts to bedetectable is dependent on the reversibility of the electron-transfer step. When the electron transfer becomes moreirreversible (e.g., small K0 and/or small λ*), for the MHmodel, the deviation from the diffusion-limited current can beobserved for stronger adsorption.3.4. Influence of Diffusion on Different Limiting

Factors. In the BV model, the slow adsorption rate constant(small Ka) can determine the maximum current of the wholereaction, whereas in the MH model, even with fast adsorption(large Ka), the reorganization energy has similar effects asshown in Figure 6. Therefore, for the Tafel−Volmer type ofreaction, if the limiting current of a steady-state voltammogramis less than the diffusion-controlled value, it can be limited byeither a slow preceding adsorption or a sluggish electron-transfer step that follows a MH type kinetics. To examine ifthese two limiting factors can be experimentally distinguished,voltammograms for the two situations are simulated underdifferent diffusion conditions. In experiments this correspondsto the use of electrodes of different sizes. In Figure 9, anadsorption-limited reaction in the BV model with K0 = 10−2

and Ka = 60 and an electron-transfer-limited reaction in theMH model with K0 = 10−2 and λ* = 20 are taken as examples,the variations of Jlim/Jdiff and θ1/2 with the radius of theelectrode being explored. Note that to clearly show the results,the real radius with units of μm is considered. Other parametersare still dimensionless to remain consistent with other figures.At “large” microdisk electrodes (i.e., ρ = 50.0 μm), due to therelatively slow diffusion, the reaction rate is controlled by masstransport and the two limiting factors above-mentioned havelittle discrepancies in the voltammetric features. With thedecrease of the electrode size, diffusion becomes more efficientand the limiting factors have different behaviors: The limitingcurrent decreases faster in the MH model (electron-transfercontrolled) than in the BV model (adsorption controlled) andthe half-wave overpotential shifts more positively for the MHmodel as well. However, the general trends of their responsesto the diffusion change are the same for limitations associatedwith both adsorption and electron-transfer kinetics. Thus, whenthe reaction is not electrochemically reversible, it is difficult to

determine if the decrease of the limiting current is caused by aslow adsorption step or a sluggish MH electron-transfer step.

■ CONCLUSIONS

The effects of surface adsorption on Butler−Volmer andMarcus−Hush electron-transfer theories have been explored inthe context of the Tafel−Volmer mechanism. The voltammetricresponses to the change of various kinetic and thermodynamicparameters (K0, λ, Ka, Kd, Kads) were examined for both BV andMH electron-transfer models. When the preceding adsorptionstep is fast (Ka, Kd ≫ 1), a slow electron-transfer step (small K0

and/or small λ*) changes the voltammogram of the Tafel−Volmer reaction consistently as it does for a simple unity-stoichiometry, one-electron-transfer reaction.When the preceding adsorption is taken into consideration as

a limiting factor of the overall reaction, different electron-transfer kinetics show distinct voltammetric responses, depend-ing on the reversibility of the electron-transfer step and kineticmodels used to describe the electron-transfer step. For anelectrochemically reversible Tafel−Volmer reaction (K0 ≫ 1),the limiting current can be expressed as a function of theadsorption rate constant Ka whereas it is independent of thethermodynamics of the adsorption step (Kads).For a nonreversible electron-transfer step (relatively small

K0), if the electron-transfer step is described in terms of the BVmodel, the limiting current is only dependent on diffusion andthe value of Ka. But for the MH model, the limiting current isdetermined by both Ka and Kads, especially under weakadsorption conditions (small Kads). Therefore, when analyzingthe voltammograms of a Tafel−Volmer reaction with a weakpreceding adsorption step, it is important to consider thekinetic description of the electron transfer to correctlyunderstand the mechanism. It is also found that fornonreversible electron transfer, the overpotential changesparabolically with the adsorption equilibrium constant,indicating that both strong and weak adsorption can cause adecrease on the overall reaction rate for the Tafel−Volmerreaction.This work has also concluded out that the non diffusion

control of the limiting current of steady-state voltammograms isconsistent, not only with slow adsorption kinetics but also withthe deviation of the potential variation of the rate constantsfrom the BV formalism. Indeed, the variation of the rateconstants with the applied potential predicted by the MHmodel leads to non-diffusion-controlled limiting currents forslow electron-transfer kinetics and/or very small electrodes.Such evidence calls for further theoretical and experimentalassessment of the kinetics of the Volmer reaction as a possibledetermining factor of the HOR voltammetry and other similarTafel−Volmer processes.

■ ASSOCIATED CONTENT

*S Supporting InformationThe Supporting Information is available free of charge on theACS Publications website at DOI: 10.1021/acs.jpcc.5b08044.

Section I shows the differences between using surfacecoverages and fractional surface coverages in the rateequations. Section II provides the derivation of eq 16.Section III is the derivation of eq 36. Section IV gives thesimulation parameters used in Figure 6 (PDF)

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■ AUTHOR INFORMATION

Corresponding Author*R. G. Compton. E-mail: richard.compton@chem.ox.ac.uk.

NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTS

The research is sponsored by the funding from the EuropeanResearch Council under the European Unions SeventhFramework Programme (FP/2007-2013)/ERC Grant Agree-ment n. [320403]. E.L. also thanks the funding received fromthe European Union Seventh Framework Programme-MarieCurie COFUND (FP7/2007-2013) under UMU IncomingMobility Programme ACTion (U-IMPACT) Grant Agreement267143 and by the Fundacion Seneca de la Region de Murciaunder the III PCTRM 2011-2014 Programme (Project 18968/JLI/13).

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