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Electrochimica Acta 286 (2018) 374e396

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Electrochimica Acta

journal homepage: www.elsevier .com/locate/e lectacta

Detailed theoretical treatment of homogeneous chemical reactionscoupled to interfacial charge transfers

A. Molina*, E. LabordaDepartamento de Química Física, Facultad de Química, Regional Campus of International Excellence “Campus Mare Nostrum”, Universidad de Murcia,30100 Murcia, Spain

a r t i c l e i n f o

Article history:Received 14 June 2018Received in revised form16 July 2018Accepted 19 July 2018Available online 1 August 2018

Keywords:Mathematical modellingHeterogeneous charge transferCoupled chemical reactionsVoltammetryMicroelectrodes

* Corresponding author.E-mail address: amolina@um.es (A. Molina).

1 Charge transfer mechanisms including the adsorface are also frequent but they will not be considereda specific review.

https://doi.org/10.1016/j.electacta.2018.07.1420013-4686/© 2018 Elsevier Ltd. All rights reserved.

a b s t r a c t

The mathematical modelling of heterogeneous charge transfer processes complicated by the reactivity ofthe redox species in solution is revisited, elaborating on powerful and accurate theoretical approachesand simplifications that can facilitate and speed up the mathematical resolution, the solution imple-mentation, the design of experiments and the calculation and analysis of the electrochemical response.The physicochemical fundamentals and mathematical implications behind such approaches will bediscussed for frequent and paradigmatic reaction mechanisms involving homogeneous chemical equi-libria and (pseudo)first-order and second-order chemical kinetics. The cases considered will point outhow the suitable preliminary examination of the boundary value problem offers profound insights intothe electrochemical system that can assist the design of theoretical and experimental studies. Thisprovides criteria for the assessment of complex simulations (for example, by comparison with exactanalytical solutions for particular cases) and it can alert about meaningless results derived from blind useof approximations and/or ‘brute force’ approaches.

© 2018 Elsevier Ltd. All rights reserved.

1. Introduction

Electron [1e5] and ion [6,7] transfer processes are veryfrequently ‘complicated’ by chemical reactions in which the elec-troactive species take part. Electrochemical methods enable thedetection of such coupled homogeneous reactions and their char-acterization with the aid of appropriate theoretical models thattake into account the interplaying heterogeneous (interfacialtransfer) and homogeneous (mass transport and chemical re-actions) processes.1 Indeed, the use of electrochemical measure-ments for the elucidation of homogeneous chemical processes andthe determination of the corresponding rate and equilibrium con-stants can be found in many and very relevant fields. Moreover, thecombination of electrochemical and spectroscopic techniques canbe very advantageous for the identification of chemical species andreaction pathways [8e11]. Classical examples include protonations(like in the electro-reduction of quinones in protic media with

ption of species on the inter-here as they certainly deserve

implications in biological electron transfer processes, molecularredox catalysis, chemical synthesis and medicine [12]), complexa-tions with molecular [13], macromolecular [14] and colloidal [15]‘ligands’, ion pairing (affecting the redox activity of catalysts such aspolyoxometalates [16]), regeneration, isomerization, dimerization,[1e4]… that are relevant in the performance of natural and artifi-cial processes of energy conversion, electrocatalysis, electro-synthesis and electro-analysis, among others. The particular caseswhere one of the reactant species is a macromolecule or a colloidalparticle are complex since the ‘macro-reactant’ contains several‘reactive sites’ that can be equivalent or different, independent orinteracting, evenly- or inhomogeneously-distributed, and locatedwithin the particles or only on their surface.

The tremendous progress in the computational capabilities hasgreatly extended our ability to tackle the theoretical treatment ofthe above electrochemical problems, but it may have also led to the‘abuse’ of ‘brute force’ digital simulations, that is, to the directresolution of the boundary value problem without previousmathematical manipulations and/or suitable physicochemical ap-proximations that offer simpler and faster solutions as well as in-sights into the ‘natural’ variables of the system for the appropriatedesign of experiments and data analysis. Thus, the preliminarycomprehensive analysis of the physicochemical problem and of the

A. Molina, E. Laborda / Electrochimica Acta 286 (2018) 374e396 375

corresponding mathematical formulation is beneficial in severalways. Another important problem that can arise is the uncriticaluse of assumptions, approximations or simplifications, eitherrelated to the mechanism itself (total equilibrium conditions, first-order kinetics or constant concentration of one of the reactants)and/or to the resolution method (kinetic steady state with the re-action layer assumption, diffusive-kinetic steady state with diffu-sion and reaction layers, or total steady state with time-independent diffusion and reaction layers).

The aim of this Critical Review is to get into the theoreticaltreatment of several fundamental and paradigmatic charge transferprocesses including coupled chemical reactions, pointing out thebenefits of the preliminary scrutiny, mathematical manipulationand simplification of the boundary value problem (bvp) whatever

the resolution method employed (either analytical or numerical).At the very least, this will provide the electrochemist with a clearerview and a more profound understanding of the phenomena takingplace, as well as of the natural chief variables that define thebehaviour of the system; eventually, this will enable the suitableand rational design of experimental studies and the detection oferrors, misinterpretations or wrong predictions. From a morepractical point of view, the a priori analysis of the problem cangreatly simplify its resolution (by reducing the number of variablesand the complexity of the differential equations) leading to simplerexpressions and faster implementation and calculations.

Well-established theoretical simplifications will be analysed,paying attention to their fundamentals and to those situationswhere they are more accurate and beneficial. Thus, a greatsimplification of the problem can be achieved in the case of fastchemical kinetics at macroelectrodes, which can be extended tomore complex reaction schemes following the ideas here discussed.Also, simple approximate theoretical treatments are included forthe accurate modelling of the response at uniformly and non-uniformly accessible microelectrodes under transient conditions,as well as of the steady state response at ultramicroelectrodes.

2. Homogeneous chemical equilibria coupled to the chargetransfer reaction under voltammetric conditions

In the theoretical treatment of heterogeneous electron and iontransfers with very fast coupled chemical kinetics - such as pro-tonations, ion pairing or labile complexations - chemical equilib-rium conditions are commonly presumed so that the boundaryvalue problem greatly simplifies. Indeed, the mathematical prob-lem can be reduced to one formally-identical to that of chargetransfers without chemical complications where the diffusion co-efficients and formal potential(s) take conditional or apparentvalues (see below). This is also the basis of widely-used proceduresfor the experimental determination of chemical equilibrium con-stants [17].

As will be discussed below, mathematical inconsistencies arisein some situations when chemical equilibrium conditions areassumed a priori such that the differential equations or theboundary conditions of the original problem can lead to a new bvpthat does not fully reflect the original one. In spite of it, provided

that the chemical kinetics is fast enough, in general accurate resultscan be obtained for the current-potential response and also for theconcentration profiles that are only affected in a very small regionnext to the electrode surface.

2.1. One or both species under chemical equilibrium undergoingelectrode reactions: the CE, oSQ and EC mechanisms

In order to clarify the above, it is helpful to compare the caseswhere one or both species involved in the equilibrium are elec-troactive. Two general schemes will be considered where only one(C) or both (B and C) species involved in the chemical reaction areelectroactive at the applied potential:

with K ¼ c*Bc*C

and species Z being present in a large excess. The above

heterogeneous electron transfer mechanisms can also be found inion-transfer electrochemistry, the CE mechanism corresponding tothe so-called ACT mechanism (Aqueous Complexation followed byTransfer) [6,7] and the oSQ scheme to the so-called ACDT mecha-nism (Aqueous Complexation-Dissociation Coupled to Transfer, seeSection 2.4) [18,19]. As mentioned above, relevant examples of theCE mechanisms include the association of redox species with mo-lecular, (bio)macromolecular and colloidal species [13e15] and alsoion transfers inhibited by the presence of ligands such as hydro-philic cyclodextrins [20]; with regard to the oSQ and ACDTschemes, this can be found in liquidjliquid electrochemistry whenboth the free and complexed ion are transferred across theliquidjliquid interface [18,19].

The differential equations that describe the transient concen-tration profiles of the species participating in the chemical reactionupon the application of a perturbation are given by:

vcBvt� DBV

2GcB ¼ �k1cB þ k2cC

vcCvt� DCV

2GcC ¼ k1cB � k2cC

(1)

where V2G is the diffusion operator. Note that the diffusive-kinetic

differential equation of species Z is not considered since its con-centration is assumed to be constant for being in a large excess (orbuffered). If this is not the case, the diffusion (labile systems) andalso the chemical transformation (non-labile systems) of species Zare to be considered in the problem [21]. Otherwise, significanterrors can affect the determination of equilibrium [22,23] and rateconstants.

With respect to the surface conditions, a relevant differencebetween the CE and oSQ situations is that, in the first case, species Bis assumed to be electroinactive such that the surface flux is null:

DC

�vcCvqN

�qs

¼ �DD

�vcDvqN

�qs

(2)

A. Molina, E. Laborda / Electrochimica Acta 286 (2018) 374e396376

DB

�vcBvqN

�qs

¼ 0 (3)

in contrast with the oSQ case:

DC

�vcCvqN

�qs

¼ �DD

�vcDvqN

�qs

(4)

DB

�vcBvqN

�qs

¼ �DA

�vcAvqN

�qs

(5)

where qN refers to the spatial coordinate normal to the electrodesurface and qs to the value(s) of the spatial coordinate(s) at thesurface of the electrode; thus, for example, in the case of a sphericalelectrode of radius rs, then qN≡r and qs≡rs. For the resolution of theproblem, it is appropriate to sum the separate differential equationsof the species participating in the chemical reaction [23]:

vcBðq; tÞvt

þ vcCðq; tÞvt

¼ DBV2GcBðq; tÞ þ DCV

2GcCðq; tÞ (6)

that can be re-written in terms of the total concentration ofoxidized species (B and C):

c1ðq; tÞ ¼ cBðq; tÞ þ cCðq; tÞ (7)

as follows:

vc1ðq; tÞvt

¼ V2G

�Deff ðq; tÞc1ðq; tÞ

�(8)

where Deff ðq; tÞ can be viewed as the ‘effective diffusion coefficient’defined as the molar weighting of the diffusion coefficients of theindividual species:

Deff ðq; tÞ ¼ DBcBðq; tÞc1ðq; tÞ

þ DCcCðq; tÞc1ðq; tÞ

(9)

At this point, Deff ðq; tÞ is not a constant diffusion coefficient andEq. (8) does not correspond to a diffusion equation (see Appendix Ain Ref. [23]) since c1 does not refer to a real species subject todiffusion but to a formal species.

If total equilibrium conditions are assumed at any point in so-lution and time of the experiment, then the following relationshipsbetween the concentrations of species B and C hold:

cBðq; tÞc1ðq; tÞ

¼ K1þ K

;ccðq; tÞc1ðq; tÞ

¼ 11þ K

(10)

so that the ratio between the concentration of species B and C isconstant and it takes the value established by K according to Eq.(10). Introducing the above condition in the original boundaryvalue problem (Eqs. (1)e(5)), mathematical inconsistencies arisewhen the diffusion coefficients of the species involved in thechemical reaction are different (i.e., DBsDC). Thus, the differentialequations (1) become into:

vc1vt¼ DBV

2Gc1

vc1vt¼ DCV

2Gc1

(11)

that are only compatible when the two species involved in thechemical reaction show equal diffusivities (i.e., DB ¼ DC ¼ D) [24].At this point, in order to take into account the possible different

diffusivity of the species, it is very convenient to reconsider theconcept of the effective diffusion coefficient that under total equi-librium conditions does take a constant value (Eqs. (9) and (10)):

Deff ¼ DBK

1þ Kþ DC

11þ K

¼ KDB þ DC

1þ K(12)

Note that DB and DC are very different when the ligand is a largemolecule such as in host-guest chemistry [13] and in the associa-tion of redox molecules with (bio)macromolecules and nano-particles [14,15]. Thus, the differential equation for the variation ofc1 with time and distance (Eq. (8)) is given by Fick's second law:

vc1vt¼ DeffV

2Gc1 (13)

subject to the following surface flux condition for the CEmechanism:

Deff

�vc1vqN

�qs

¼ �DD

�vcDvqN

�qs

(14)

and for the oSQ mechanism:

Deff

�vc1vqN

�qs

¼ �DD

�vcDvqN

�qs

� DA

�vcAvqN

�qs

(15)

It is important to highlight that, even when the diffusion co-efficients are equal, in the CE mechanism the total equilibriumassumption for B is evidently non-compatible with the surface fluxcondition (3) since Eq. (10) leads to:�vcBv qN

�qs

¼ K1þ K

�vc1v qN

�qs

s0 (16)

In the oSQ mechanism, the total equilibrium condition is fullyapplicable when the species participating in the chemical equilib-rium show equal diffusion coefficients and the differential equa-tions (13) and (11) become identical.

The effect of the above questions on the theoretical concentra-tion profiles of the chemical species is illustrated in Fig. 1 thatshows the profiles for a very fast chemical kinetics (c ¼ ðk1þ k2Þt ¼1010) in a limiting-current chronoamperometric experiment at amacroelectrode. Numerical simulations without assuming totalequilibrium conditions (lines) are compared with the profiles ob-tained from the following analytical expressions derived undersuch assumption (points) [1e3]:

climC ðx; tÞ ¼c�1

1þ K

8><>:1� erfc

0B@ x

2ffiffiffiffiffiffiffiffiffiffiDeff t

p1CA9>=>;

climB ðx; tÞ ¼c�1K1þ K

8><>:1� erfc

0B@ x

2ffiffiffiffiffiffiffiffiffiffiDeff t

p1CA9>=>;

(17)

with c�1 ¼ c�B þ c�C and Deff being given by Eq. (12). As can be seen,the total equilibrium expressions (17) describe satisfactorily theconcentration profiles except in a very small region of the solutionfrom the electrode surface to a few nanometres away from it in thecase of the CEmechanism (see inset in Fig. 1A). For this mechanism,it is obvious that the null surface flux condition for B (Eq. (3)) is notmathematically consistent with the hypothesis of equilibratedconcentrations. As such condition does not apply in the case of theoSQ mechanism, differences in the concentration profiles are muchsmaller (negligible) as can be seen in the inset of Fig. 1B.

Fig. 1. Theoretical concentration profiles of the species involved in the chemical reaction in limiting current chronoamperometry of the CE mechanism (A) and the open-SQmechanism (B) obtained with (points) and without (lines) the assumption of total equilibrium conditions (Eq. (10)). Deff ¼ 10�5cm2s�1, DB¼ 5DC(¼DD), c ¼ 1010, K¼ 0.5.

A. Molina, E. Laborda / Electrochimica Acta 286 (2018) 374e396 377

Extending the comparison between the total equilibrium andrigorous solutions to the current-potential response of the CE andoSQmechanisms, it can also be concluded that very accurate valuesare obtained with the total equilibrium assumption even withchemical kinetics much slower than the diffusion-controlled upperlimit of the chemical rate constants (corresponding to k�1010 s�1)[25]. Thus, for example, with K¼ 1 and equal diffusion coefficientsthe error is smaller than 1% for ðk1þ k2Þt >20. Discrepancies in-crease for small K-values and as the species diffusion coefficientsdiffer (DBsDC), but in any situation accurate results are achievedfor realistically fast chemical kinetics (that is, c<1010).

In the case of the EC mechanism:

Aþ e� ��������������!k0 > >1Bhk ¼ k1 þ k2C

EC mechanism

The key voltammetric parameter that reflects the effect of thecoupled reaction is the half-wave potential (EEC1=2) that for total

equilibrium conditions, EEC1=2 is predicted to take the value:

EEC1=2;eq ¼ E00 þ RT

Fln�1þ KK

�(18)

where:

K ¼ c�Bc�C

(19)

The values predicted by Eq. (18) in the range 0:01<K <2 areachieved for ðk1 þ k2Þt >106 to ðk1þ k2Þt >102: the larger the K-value, the faster the kinetics must be to reach the total equilibriumresponse [26]. Obviously, in the case of the chemically irreversibleEC mechanism (K/0), the condition of chemical equilibrium doesnot make any sense.

2.2. Multiple coupled chemical equilibria: the ladder mechanism

The redox species can take part in multiple successive chemicalreactions (such as complexations and protonations) as in thefollowing ladder mechanism where species L is assumed to be in alarge excess (or buffered) so that its concentration can be consid-ered constant:

O ����������������!ðþLÞ

ð�LÞOL… ����������������!ðþLÞ

ð�LÞOLn

hE00

O=R hE00

OL=RL hE00

OLn=RLn

R ����������������!ðþLÞ

ð�LÞRL… ����������������!ðþLÞ

ð�LÞRLn Laddermechanism

Provided that the reaction kinetics are rapid enough (see Section2.1), total equilibrium conditions can be assumed such that themathematical problem reduces to that of a simple electron transfer

(E mechanism) with the following apparent formal potential (E00

app)and effective diffusion coefficient [27]:

E00

app ¼ E00� RT

Fln

0BBB@1þPn

i¼1bO;i

1þPni¼1

bR;i

1CCCA

DOeff ¼

DO þXni¼1

DOLibi

1þPni¼1

bi

; DReff ¼

DR þXni¼1

DRLibi0

1þPni¼1

bi0

(20)

where bi and b0i correspond to the overall formation equilibrium

constant:

bi ¼Yiv¼1

KOLv ¼cOLi ðq; tÞcOðq; tÞ

b0i ¼

Yiv¼1

KRLv ¼cRLiðq; tÞcRðq; tÞ

(21)

with KXLv (X≡O; R) being the conditional formation constant forspecies XLv:

KXLv ¼ K0XLvc

*L ¼

cXLv ðq; tÞcXLv�1ðq; tÞ

ðX≡O;RÞ (22)

where c*L is the concentration of species L and K0XLv are the

concentration-based equilibrium constants:

A. Molina, E. Laborda / Electrochimica Acta 286 (2018) 374e396378

K0XLv ¼

cXLvðq; tÞcXLv�1ðq; tÞc*L

(23)

In the particular case of a hypothetical large redox moleculewith n identical and independent reactive sites, the multipleKXLv-values of a given oxidation state (O or R in the above scheme)can be related via a Scatchard analysis [28]. Thus, any partialequilibrium constant of species X in the ladder mechanism, KXLv ,could be defined from the so-called intrinsic constant (Kint; X) asfollows:

KcXLv ¼

cXLvcXLv�1c

*L¼�n� vþ 1

v

�Kint;X ðX≡O;RÞ (24)

such that the system can be characterized simply by an equilibriumconstant per oxidation state (Kint; X) and by the number of reactivesites (n). The former can be viewed as an averaged intrinsic ormicroscopic concentration stability constant for the binding of

ROi¼ k�1;Oi

cOiL1 � k1;Oic*LcOi

ði ¼ 1;2;3ÞRSETO1¼�k0;0�1c

2O2� k0;01 cO1

cO3

�þ�k0;1�1cO2

cO2L1 � k0;11 cO1cO3L1

�þ�k0;2�1cO2

cO2L2 � k0;21 cO1cO3L2

�RSETO2¼�k0;01 cO1

cO3� k0;0�1c

2O2

�þ�k0;11 cO1

cO3L1 � k0;1�1cO2cO2L1

�þ�k0;21 cO1

cO3L2 � k0;2�1cO2cO2L2

�þ�k1;01 cO1L1 cO3

� k1;0�1cO2cO2L1

�þ�k2;01 cO1L2 cO3

� k2;0�1cO2cO2L2

�RSETO3¼�k0;0�1c

2O2� k0;01 cO1

cO3

�þ�k1;0�1cO2

cO2L1 � k1;01 cO1L1 cO3

�þ�k2;0�1cO2

cO2L2 � k2;01 cO1L2 cO3

�(28)

2 In case that the redox species have different diffusion coefficients, the intro-duction of an effective diffusion coefficient for each oxidation state would lead tothe following differential equation system:

v cOiT ðq; tÞv t

¼ Deff ;OiTV2GcOiT ðq; tÞ þ RSET

OiT fi ¼ 1; 2;3

with:

Deff ;OiT ¼DO1þ DO1L1KOi ;1 þ DO1L2KOi ;1KOi ;2

1þ KOi ;1 þ KOi ;1KOi ;2

and:

RSETO1T ¼

RSETO1þ RSET

O1LKOi ;1 þ RSETO1L2KOi ;1KOi ;2

1þ KOi ;1 þ KOi ;1KOi ;2

As the RSETO1T terms depends on different equilibrium constants that, in general, take

different values, they do not cancel out by addition of Eqs. (27) such that a simpletheoretical treatment with effective diffusion coefficients of extended squareschemes cannot be performed unless the SET terms can be neglected.

species L to each and every potential binding site of the redoxspecies:

Kint; X ¼cX0LcX0c*L

ðX0≡O0;R0Þ (25)

where X0 are hypothetical redox species with a single binding sitefor L.

2.3. Multiple electron transfers coupled with chemical reactions eextended square schemes

Let us consider a reaction scheme with multiple electrontransfers and coupled chemical equilibria such as the 9-membersquare scheme [29] corresponding, for example, to the electro-reduction of quinones in protic media [12]. As species undergomultiple electron transfers, the possibility of the occurrence ofhomogeneous comproportionation-disproportionation reactions isto be considered in addition to the reactions explicitly shown in theabove scheme. Thus, the following solution-phase electron transfer

(SET) reactions can be envisaged:

O1Lj þ O3Lj0 %kj;j0

1

kj;j0�1

O2Lj þ O2Lj0�

j ¼ 0;1;2j0 ¼ 0;1;2 (26)

Hence, under diffusion-only mass transport conditions the dif-ferential equation system that describes the variation of the con-centrations with time (t) and position (q) is given by:

v cOiLjðq; tÞv t

� DOiLjV2GcOiLj ðq; tÞ ¼ ROiLj þ RSET

OiLj

�i ¼ 1; 2;3j ¼ 0; 1;2

(27)

where V2G is the diffusion operator, ROiLj refers to the kinetic terms

of the complexation reactions with species L and RSETOiLj to the kinetic

terms of the SET reactions; thus, for example, for species O1, O2 andO3:

If the diffusion coefficient of all the redox species are equal(DOiLj ¼ D; i ¼ 1; 2;3; j ¼ 0; 1;2),2 then the addition of all thedifferential equations in (27) leads to:

vcTðq; tÞvt

� DV2GcT ðq; tÞ ¼ 0 (29)

where cT ðq; tÞ refers to the total concentration of electroactivespecies:

cT ðq; tÞ ¼X3i¼1

cOiT ðq; tÞ (30)

A. Molina, E. Laborda / Electrochimica Acta 286 (2018) 374e396 379

where:

cOiTðq; tÞ ¼X2j¼0

cOiLj ðq; tÞ (31)

that is subject to the following boundary conditions:

bulk solution : cTðq; tÞ ¼ c*T ¼ c*O1þ c*O1L þ c*O1L2

electrode surface :

�vcTðq; tÞvqN

�qs¼ 0

(32)

Thus, given that none of the boundaries is a source or sink ofredox species, it is immediately inferred that:

cTðq; tÞ ¼ c*T (33)

If the reaction with species L is very fast compared to diffusion,then the following chemical equilibrium relationships for theconcentration of any species OiLj can be assumed (see Section 2.1):

cOiLjðq; tÞcOiLj�1ðq; tÞ

¼ KOi;j ¼ K 0Oi;jc*L ; c q; t

�i ¼ 1;2;3j ¼ 1;2 (34)

so that all the concentrations at a given i-th oxidation state (i¼ 1, 2,3) can be related through the total concentration of such oxidationstate as follows:

cOiLjðq; tÞ ¼ cOiTðq; tÞ

Y2n¼j

KOi;n

1þ KOi;1 þ KOi;1KOi;2; c q; t

�i ¼ 1;2;3j ¼ 0;1;2

(35)

where cOiT ðq; tÞ ¼ cOiðq; tÞ þ cOiLðq; tÞ þ cOiL2 ðq; tÞ and KOi ;0 ¼ 1.

fmacroðtÞ ¼ 1ffiffiffiffiffiffiffiffipDtp

fsphðrs; tÞ ¼1rsþ 1ffiffiffiffiffiffiffiffi

pDtp

fdiscðrd; tÞ ¼4p

1rd

�0:7854þ 0:44315

rdffiffiffiffiffiffiDtp þ 0:2146 exp

�� 0:39115

rdffiffiffiffiffiffiDtp

��(41)

Hence, the 9-variable problem can be reduced to a 3-variableproblem of the pseudo species cO1T , cO2T and cO3T , which is math-ematically equivalent to the problem of the EE mechanismwith thefollowing apparent formal potentials:

E00;app

1 ¼ E00

O1=O2� RT

Fln�1þ KO2;1 þ KO2;1KO2 ;2

1þ KO1;1 þ KO1;1KO1 ;2

�

E00;app

2 ¼ E00

O2=O3� RT

Fln�1þ KO3;1 þ KO3;1KO3 ;2

1þ KO2;1 þ KO2;1KO2 ;2

� (36)

The above treatment can be extended to k electron transfers andh coupled chemical equilibria [30] such that the problem becomesformally equivalent to that of k-electron transfers without chemicalcomplications with the apparent formal potentials being given by:

E00

app; n ¼ E00

Onþ RT

Fln

0BBBB@1þPh

j¼1bOnþ1;j

1þPhj¼1

bOnj

1CCCCA ðn ¼ 1;2; :::kÞ (37)

Under the above conditions (chemical and electrochemicalequilibria and equal diffusion coefficients), the response in anyvoltammetric technique can be calculated from:

IG ¼ FAGDXpm¼1

h�W ½m�1�;s �W ½m�;s

�fGqG; tm;p

i(38)

where W ½m�;s depends on the potential perturbation and on thethermodynamics of the electron transfers and chemical equilibria(i.e., on the reaction mechanism):

W ½0�;S ¼W* ¼ ðkÞc�T

W ½m�;S ¼ c�T

Pkn¼1

"ðk� nþ 1Þ

Ykb¼n

eh½m�app;b

#

1þ Pkn¼1

Ykb¼n

eh½m�app;b

;m ¼ 1; 2; :::; p

(39)

with

h½m�app; n ¼

FRT

�E½m� � E0

0app; n

�; n ¼ 1; 2; :::; k (40)

AG is the electrode area and fGðqG; tÞ is a function that dependson time and on the shape and size of the electrode [31]; thus, forexample, for planar, spherical and disc electrodes the followingexpressions apply, respectively:

Note that expression (38) is applicable to simpler mechanismsby using the corresponding values of the number of electrontransfers (k), coupled chemical equilibria (h) and equilibrium con-stants inW ½m�;s (Eq. (39)) and in E0

0app; n (Eq. (37)). Thus, for example,

for the ladder mechanism (k ¼ 1), W ½m�;s simplifies to:

W ½0�;S ¼W* ¼ c�T

W ½m�;S ¼ c�Teh½m�app;1

1þ eh½m�app;1

;m ¼ 1; 2; :::; p

(42)

The particular case of the simple E mechanism corresponds toh ¼ 0 and so E0

0app; 1 ¼ E0

0O1; as another example, for the ECeqE

A. Molina, E. Laborda / Electrochimica Acta 286 (2018) 374e396380

mechanism k ¼ 2, h ¼ 1, KO1 ;1/0 and KO3;1/∞.

2.4. The oSQ mechanism with different charge numbers transferred

In contrast with the reaction schemes studied in previous sec-tions, in the case of the oSQ mechanism where different chargenumbers are transferred in each electrochemical step, the intro-duction of the total equilibrium assumption (Eq. (10)) in the orig-inal bvp is not adequate since it leads towrong theoretical solutionsthat do not describe the electrochemical behaviour of the systemeven for hypothetical infinitely fast chemical reactions (k/∞).

The above is clearly illustrated when studying the oSQ schemefor ion transfers (also referred as the ACDT mechanism [18,19])where the transfer of the target ionic species between the aqueous(w) and organic (o) phases can take place in its free (Mz1) andcomplexed (Mz2) forms:

z1 z2

Species M and M can have equal or different charges. Theformer is found in the case of neutral ligands (L≡water, ammonia,pyridine, ethylenediamine) and the latter when species L is charged,such as in protonations and complexations with charged ligands.

Fig. 2 shows the theoretical current-potential response innormal pulse voltammetry of the ACDT mechanism with a mono-cation Mþ, with species L being neutral (z2 ¼ z1 ¼ þ 1, Fig. 2B) orpositively-charged (z2 ¼ 2z1 ¼ þ 2, Fig. 2A) and notably hydro-philic such that the transfer of the complex MLz2 is thermody-namically less favourable: Df00

ML � Df00M ¼ 200 mV. The variation of

the I-E curves as the chemical kinetics is faster (i.e., as c increases) isshown and compared with the response predicted when totalequilibrium conditions are presumed (grey lines in Fig. 2).

As can be seen, in the case of equal charge numbers z2 ¼ z1 ¼ z,the electrochemical response reaches the total equilibrium limit(grey solid lines in Fig. 2B) for realistically fast chemical kinetics(c>100). Under such conditions, both the rigorous and totalequilibrium solutions predict a single voltammetric wave with thelimiting current being given by:

Iz2¼z1lim;eq ¼ Iz2¼z1lim ðc/∞Þ ¼ Idc*1; z1

¼ z FAc*1

ffiffiffiffiffiDpt

r(43)

Fig. 2. Normal pulse voltammetry of the ACDT mechanism as a function of the chemical kinetheir charge is different (z2 ¼ þ 2 ; z1 ¼ þ 1) (A) or equal (z2 ¼ z1 ¼ þ 1) (B).

When z2sz1 (Fig. 2A), the NPV response of the ACDT mecha-nism shows two waves whatever the chemical kinetics when nosimplifying assumptions are made [19]. In the limit of infinitely fastkinetics, the limiting current takes the value:

Iz2sz1lim ðc/∞Þ ¼ z2

z1Idc*1; z1

(44)

which reflects that, at very positive potentials, the transfer of themost charged species (ML2þ) is more favourable and, given that theinterconversion from Mþ to ML2þ is ‘instantaneous’, the responsebehaves as the transfer of a dication of total concentration c*1. This isnot predicted by the total equilibrium solution (grey line) accordingto which a single wave would be expected with a limiting currentthat reflects the simultaneous transfer of the two species (Mþ andML2þ, not only ML2þ) according to their equilibrium concentrationsas follows (note that K ¼ 1 in Fig. 2):

Iz2sz1lim;eq ¼

1þ z2z1K

1þ KIdc*1; z1

sIz2sz1

lim ðc/∞Þ (45)

Thus, total chemical equilibrium conditions cannot be assumeddue to that the heterogeneous electrochemical reactions of Mz1 andMLz2 ‘force’ the predominant transfer of one of them (the one withthe highest charge).

3. First-order or pseudofirst order chemical kinetics

The treatment of homogeneous chemical kinetics coupled to theheterogeneous charge transfer is an important and challengingtopic in theoretical electrochemistry. The corresponding diffusive-kinetic problems involve, in general, more variables (more chemi-cal species) and more complex differential equations (including thecorresponding kinetic terms) than the cases discussed in Section 2.This generally means that the analytical or numerical resolution ismore complex, involving more awkward analytical expressions(when possible) or resolution algorithms that slow down theimplementation in computer programs and the calculation times.Moreover, in the case of numerical methods (adaptive) unequalgrids are to be used to address fast chemical reactions [32e34].

The above difficulties can be mitigated by a previous carefulanalysis of the mathematical problem that enable the number ofvariables to be reduced [35] as well as by the use of suitable ap-proximations. These aspects will be reviewed in the following

tics (c ¼ kt) when species MLz2 is less lipophilic than Mz1 (Df00ML � Df00

M ¼ 200 mV) and

A. Molina, E. Laborda / Electrochimica Acta 286 (2018) 374e396 381

sections for (pseudo)first-order kinetic mechanisms, also reportingrigorous and approximate analytical expressions for the voltam-metric response of the most common reaction schemes.

3.1. The special case of the first order catalytic mechanism

The simplest electrochemical problem including homogeneouschemical kinetics is the (pseudo)first-order catalytic mechanismwhere the redox reactant is regenerated chemically in solution:

where Z and P are electroinactive species and their concentration isin large excess so that they can be assumed constant and thepseudo-first order rate constants k1 ¼ k

01c*Z and k2 ¼ k

02c*P can be

defined. The above reaction scheme underlies the molecularcatalysis of performance-limiting redox reactions of energy con-version [36] and electroanalytical devices, which justifies theenduring interest in the above EC0 mechanism [37e49] as well as inother more complex catalytic pathways [39,50] with coupledchemical and electrochemical reactions such as:

C þ e�#BB # AA/Dþ :::Dþ e�#C

or different stoichiometric coefficients [51,52]:

a C þ a e�#b BB/h C þ :::

The mathematical treatment of the first-order EC' scheme im-plies the resolution of a diffusive-kinetic two-variable problemgiven by:

vcCv t� DV2

GcC ¼ k1cB � k2cC

vcBv t� DV2

GcB ¼ �k1cB þ k2cC

(46)

t ¼ 0;q � qst>0;q/∞

�cCðq; tÞ ¼ c*C ; cBðq; tÞ ¼ c*B (47)

t>0; q ¼ qs : D�vcCvqN

�qs

¼ �D�vcBvqN

�qs

(48)

csC ¼ ehcsB (49)

with:

h ¼ FRT

�E � E0

0�(50)

and where it is assumed that the electrode reaction is reversibleand that the diffusion coefficients are the same for the reduced andoxidized species; qN refers to the spatial coordinate normal to theelectrode surface and qs to the value(s) of the spatial coordinate(s)at the surface of the electrode. The rigorous resolution of the aboveproblem greatly simplifies with the introduction of the following

variable changes [2,53]:

c1 ¼ cC þ cB (51)

f ¼ ðcB � K cCÞec (52)

with

c ¼ ðk1 þ k2Þt (53)

and:

K ¼ c�Bc�C

(54)

With the variable changes (51) and (52), the differential equa-tions (46) become into:

vc1v t� DV2

Gc1 ¼ 0 ;vf

v t� DV2

Gf ¼ 0 (55)

that are formally identical to the differential equations in theabsence of chemical complications. Moreover, it can be easilydemonstrated that c1 remains constant at any point in solution andtime of the experiment regardless of the reversibility of the elec-trode reaction and of the electrode geometry [35]:

c1ðq; tÞ ¼ c*1 ¼ c*B þ c*C ðcq; tÞ (56)

Therefore, the boundary value problem of the (pseudo)first-or-der catalytic mechanism with equal diffusion coefficients reducesto the following single-variable problem:

vf

v t� DV2

Gf ¼ 0 (57)

t ¼ 0; q � qst >0; q/∞

�fðq; tÞ ¼ 0 (58)

t >0; q ¼ qs: fs ¼ ec1� Keh

1þ ehc�1 (59)

with fs being the value of variable f at the electrode surface andthe current response being given by:

ICat

FAD¼�vcCvqN

�qS¼ � e�c

1þ K

�vf

vqN

�qS

(60)

3.1.1. Exact analytical solutions for single- and multi-pulsevoltammetric techniques

When the electrode reaction is reversible, the transient currentresponse upon the application of a single potential pulse E is givenby [42,43,54]:

ICat

FADc*1¼ 1� Keh

ð1þ KÞð1þ ehÞfGCatðc; qGÞ (61)

where fGðc; qGÞ is a function of the catalytic kinetics (c) and thecharacteristic dimension of the electrode (qG), the mathematicalform of which depends on the electrode geometry. In the case ofplanar (macroelectrodes) and spherical electrodes, fGðc; qGÞ isgiven by:

A. Molina, E. Laborda / Electrochimica Acta 286 (2018) 374e396382

f macroCat ðcÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik1 þ k2

D

r "e�cffiffiffiffiffiffiffiffip cp þ erf ð ffiffifficp Þ

#

f spheCat ðc; rsÞ ¼1rsþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik1 þ k2

D

r "e�cffiffiffiffiffiffiffiffip cp þ erf ð ffiffifficp Þ

# (62)

respectively. Very accurate expressions for fGðc; qGÞ at microdiscsare also available [47e49] (error in the current below 1% [54]), forexample:

f discCat ðrd;cÞ ¼1rdþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik1 þ k2

D

rTdiscðc; xÞ (63)

with:

Tdiscðc;xÞ¼ erf ð ffiffifficp Þþ 0:2732ffiffifficpx

Zx20

exp

"�0:39115ffiffiffi

up � c

x2u

#du þ

þexpð�cÞ�0:2732

xffiffifficp exp

��0:39115

x

�þ 1ffiffiffiffiffiffiffi

pcp

�(64)

and:

x ¼ffiffiffiffiffiffiffiD tp

rd(65)

From equation (61), the expression for the cathodic limiting

current (E< < E00) is immediately obtained by making eh/0:

ICatlim;c

FAD c*1¼ 1

1þ KfGCatðc; qGÞ (66)

whereas for E> > E00(i.e., eh/∞), the anodic limiting current is

ICatlim;a

FAD c*1¼ � K

1þ KfGCatðc; qGÞ (67)

such that it holds that:

Fig. 3. Cyclic voltammetry of the first-order catalytic mechanisms with K¼ 0 at a microdisc e

the scan rate (B, k1¼1s�1). JCV ¼I

FA c*1

ffiffiffiffiffiffiffiffiffiD Fv

RT

q .

���ICatlim;a

���ICatlim;c

¼ K (68)

Moreover, from equations (61), (66) and (67) it is deduced that,independently of the shape and size of the electrode, it is fulfilledthat:

E ¼ E00 þ RT

Fln

ICatlim;c � ICatICat � ICatlim;a

!(69)

so that the variation of the potential Ewith ln

ICatlim;c � ICatICat � ICatlim;a

!is linear

and identical to the case of a reversible single charge transfer re-action (the E mechanism).

From Eqs. (49) and (56), it can be proven that the surface con-centrations of species B and C are constant over time (and also overthe electrode surface even in the case of non-uniformly accessibleelectrodes) [54]. Hence, the superposition principle can be applied[55,56] such that the problem corresponding to the application of npotential pulses (typically of the same duration, t) can be treated asn independent single-pulse problems and the current is given bythe sum of the n responses of single potential steps. Thus, thefollowing general expression applies to any voltammetrictechnique:

ICatFAD c*1

¼Xnm¼1

�1

1þ ehm� 11þ ehm�1

�fGCatðcmn; qGÞ (70)

where eh0 ¼ 1=K and:

cmp ¼ ðk1 þ k2Þtmptmp ¼ ðp�mþ 1Þ t (71)

where the function fGCatðcmn; qGÞ is given by Eqs. (62)-(64). Solution(70) is exact in the case of uniformly accessible electrodes (mac-roelectrodes and spherical electrodes) and very accurate for non-uniformly ones with only the intrinsic error of the corresponding

expression for fGCatðcmn; qGÞ (see [47e49] for the case of micro-discs). Making use of Eq. (70), Fig. 3 shows the transition of thecyclic voltammograms of the first-order catalytic mechanism atmicrodiscs from transient to stationary conditions (that is, frompeak-shaped to sigmoid-shaped voltammograms) as the catalytickinetics is faster and/or the scan rate is decreased.

lectrode (rd ¼ 50 mm) as a function of the catalytic rate constants (A, v¼ 100mV/s) and

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3.1.2. The kinetic steady state (kss) treatmentFor large enough values of the dimensionless kinetic parameter

c, the function fGCatðc; qGÞ becomes time-independent; thus, forexample, in the case of macroelectrodes and hemispherical mi-croelectrodes (Eq. (62)), the exponential term e�c=

ffiffiffiffiffiffiffiffip cp

/0 whilethe error function erf ð ffiffifficp Þ/1 such that:

f macroCat;ss ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik1 þ k2

D

r

f spheCat;ss ¼1rsþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik1 þ k2

D

r (72)

and the steady state current-potential response can be written as:

ICat;ss

FAD c*1¼ 1� Keh

ð1þ KÞð1þ ehÞ1dr;G

¼ 11þ K

fss;s

dr;G(73)

where dr;Gð¼ 1=fGCat;ssÞ is the so-called linear reaction layer thick-ness [57,58]. The dr;G-value is an estimation of the magnitude of thesolution regionwhere chemical equilibrium conditions are broken;

hence, the faster the chemical reaction is, the smaller the dGr -valuebecomes, which also depends on the electrode geometry [59]. Also,note that for the first-order catalytic mechanism the kss assump-tion leads to a time-independent response, unlike for other reactionmechanisms (see Section 3.2).

The same result (Eq. (73)) is obtained by defining the equilib-rium perturbation function in the original problem (Eq. (57)-(59))as:

fss ¼ cB � K cC (74)

and assuming that this is independent of time:

vfss

v t¼ 0 ¼ D V2

Gfss � k fss (75)

which is the basis of the so-called kinetic steady state (kss)approximation. In the case of uniformly accessible electrodes, such

�dr;disc

� ¼ rdp

4

8<:0:7854

1þ rd

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik1 þ k2

D

r !þ 0:4292 r2d

k

D

Z∞0

u exp�� 0:39115ffiffiffi

up � r2d

k

Du2�du

9=;�1

(82)

as planar and (hemi)spherical electrodes, the direct integration ofequation (75) under (assumed) time-independent boundary con-ditions leads to the following closed-form expressions for thesteady-state fss-functions:

fssmacro ¼ f

ss;smacro e

� ffiffikD

px

fsssph ¼ f

ss;ssph

rsr

e�ffiffikD

p ðr�rsÞ (76)

from which it follows that:

�vfss

GvqN

�qs¼ f

ss;bulkG � f

ss;sG

dr;GðG≡macro; sphÞ (77)

where fss;bulkG ¼ 0 and dr;G is the thickness of the linear reaction

layer that depends on the geometry of the diffusion field:

dr;macro ¼ffiffiffiffiDk

r

dr;sph ¼�1rsþ

ffiffiffiffik

D

r ��1 (78)

Note that Eqs. (74)-(78) are only dependent on the chemicalkinetics (and not on preceding or subsequent transformations) sothat the expression for the thickness of the linear reaction layer isidentical for the catalytic, CE or EC mechanism [59].

Considering the surface condition:

fss;s ¼ 1� Keh

1þ ehc�1 (79)

and that the current response is given by:

ICatssFAD

¼ � 11þ K

�vfss

vqN

�qs

(80)

expression (73) is obtained.Extending the kss treatment to non-uniformly accessible mi-

croelectrodes (discs, bands, rings, …), it can be written that:

*�vfss

vqN

�qs

+¼ � fss;s�

dr;G� (81)

where the surface flux of fss and the linear reaction layer have anaverage character given that the flux of the electroactive species isnot uniform over the electrode surface. Note that this is not the caseof fss;s since the surface concentrations are constant as mentionedbefore (Eqs. (49) and (56)). Attending to (80) and (81), theexpression for hdr;Gi at each electrode geometry can be derivedfrom the expression for the steady-state current of the catalyticmechanism [59]. Thus, in the case of disc microelectrodes it is ob-tained that [48]:

3.1.3. Non-Nernstian electron transferGiven that condition (56) holds for any charge transfer kinetics,

the bvp of the first-order catalytic mechanism with a non-reversible electron transfer can also be reduced to a single-variable problem in terms of the equilibrium perturbation func-tion, fðq; tÞ. Thus, the rigorous treatment of this mechanism in-volves the resolution of the differential equation (57) subject to theinitial and bulk conditions (58) and to the surface condition:

t>0; q¼ qs:�

vf

vqN

�qs¼ðkoxþkredÞ

D

fs�ecc*1

þkoxDð1þKÞecc*1

(83)

where c*1 is given by Eq. (56) and kox and kred are the heterogeneousoxidation and reduction rate constants that according to the Butler-

A. Molina, E. Laborda / Electrochimica Acta 286 (2018) 374e396384

Volmer kinetic model are given by Refs. [1e3]:

kred ¼ k0e�ah

kox ¼ k0eð1�aÞh ¼ kredeh

(84)

with k0 being the heterogeneous standard rate constant and a thecharge transfer coefficient.

Under steady state conditions (kss treatment), the surface con-dition (83) is transformed into [60]:�vfss

vqN

�qs¼ kredð1þ ehÞ

D

fss;s � c*1

þ kredeh

Dð1þ KÞc*1 (85)

where fss is given by Eq. (74) and the steady-state solution for thecurrent-potential response of the first-order catalytic mechanismfor any electrode kinetics is:

ICatssFAD c*1

¼ 11þ K

�kredð1� K ehÞ

Dþ dr;Gkredð1þ ehÞ�

(86)

3.2. Non-catalytic (pseudo)first-order mechanisms

In general, the mathematical problem of the electrochemicalresponse of chemically-complicated charge transfer processes in-cludes as many variables as participating species, the concentrationof which are not only subject to mass transport but also to homo-geneous chemical kinetics. In this section, we will review pre-liminary considerations and variables changes of the boundaryvalue problems that reduce the number of variables as well as thecomplexity of the differential equation system while providingexact or very accurate results either by analytical or numericalmethods.

3.2.1. The CEC, CE and EC mechanismsCharge transfer mechanisms where the heterogeneous charge

transfer is preceded and/or followed by a homogeneous chemicalreaction are ubiquitous in electrochemical measurements and theirtheoretical modelling has been repeatedly considered in the liter-ature for electron-transfers [61e83] and also for ion-transfers[7,18,19,26,84e86]. The mathematical treatment of the CEC mech-anism, as well as the particular CE and EC cases, have been broadlyconsidered over years, considering different mathematicalmethods (Laplace transform [67], dimensionless parametermethod [64,65,69,87], numerical calculations [32,88,89]) andelectrochemical techniques. Exact solutions have been found forthe CE and ECmechanisms in chronoamperometry and single pulsetechniques at macroelectrodes and hemispherical microelectrodes[69,87]; under other conditions (i.e., CEC mechanism, multipulsetechniques or non-uniformly accessible microelectrodes), theintroduction of approximations or numerical methods (finite-dif-ferences or elements, numerical integration of integral solutions)are necessary.

A general view of the above situations can be studied throughthe CEC mechanism:

B ����������!k1

k2C; C þ e�%E; E ����������!k3

k4F CEC mechanism

where species B and F are electro-inactive in the whole range ofapplied potentials, and k1, k2, k3 and k4 are first- or pseudofirst-order rate constants such that the corresponding equilibriumconstants are defined as:

K1 ¼k2k1¼ c�B

c*C(87)

K2 ¼k4k3¼ c�E

c�F(88)

where c�i is the equilibrium concentration of species i (≡ B, C, E, F).Note that the so-called CE and EC mechanisms can be viewed asparticular cases where K2 > >1 or K1/0, respectively.

The differential equation system that describes the variation ofthe concentration profiles of the CEC mechanism when applying apotential perturbation is:

vcBvt� D1V

2GcB ¼ �k1cB þ k2cC

vcCvt� D1V

2GcC ¼ k1cB � k2cC

vcEvt� D2V

2GcE ¼ �k3cE þ k4cF

vcFvt� D2V

2GcF ¼ k3cE � k4cF

(89)

t ¼ 0; cqt >0; q/∞

�cBðq; tÞ ¼ c*B ; cCðq; tÞ ¼ c*CcEðq; tÞ ¼ 0 ; cFðq; tÞ ¼ 0

(90)

t >0; q ¼ qs; D1

�vcCvqN

�qs¼ �D2

�vcEvqN

�qs

(91)

�vcBvqN

�qs¼�vcFvqN

�qs¼ 0 (92)

csC ¼ eh csE (93)

where D1 and D2 are the diffusion coefficients of the oxidized (Band C) and the reduced (E and F) species, respectively.

Many theoretical approaches to the above mechanisms areperformed by direct resolution of the diffusive-kinetic differentialequation system considering all the chemical species. Nevertheless,as will be discussed here, the treatment can be simplified andspeeded up after some preliminary considerations and variablechanges. Thus, in the common situation where the diffusion co-efficients of all the species can be assumed as equal, it can bedemonstrated that for all these mechanisms, regardless of thecharge transfer kinetics, the total concentration of chemical speciesremains constant at any time of the experiment and in any region ofthe solution; that is, the chemical kinetics does not compromise the

A. Molina, E. Laborda / Electrochimica Acta 286 (2018) 374e396 385

so-called “principle of unchanging total concentration” [90]:

c1ðq; tÞ þ c2 ðq; tÞ ¼ c�1 þ c�2 (94)

where c�1 and c�2ð¼ 0Þ, hereafter refer to the total concentration ofoxidized and reduced species, respectively, in each mechanism.Note that condition (94) will enable us to reduce the number ofvariables by one (see below).

Equivalently to the case of the first-order catalytic mechanism,the following variable changes are convenient to simplify the formof the differential equations to Fick's second law:

c1 ¼ cB þ cC (95)

f1 ¼ ðcB � K1 cCÞec1 (96)

c2 ¼ cE þ cF (97)

f2 ¼ ðcE � K2 cFÞec2 (98)

where:

c1 ¼ ðk1 þ k2Þ tc2 ¼ ðk3 þ k4Þ t (99)

Taking into account Eqs. (94)-(99), the CEC problem simplifies tothe following 3-variable boundary value problem whatever theelectrode geometry and voltammetric technique:

vc1v t� D V2

Gc1 ¼ 0 ;vf1v t� D V2

Gf1 ¼ 0 ;vf2v t� D V2

Gf2 ¼ 0

(100)

t ¼ 0; q � qst>0; q/∞

�c1ðq; tÞ ¼ c*1; f1ðq; tÞ ¼ 0; f2ðq; tÞ ¼ 0

(101)

t>0; q ¼ qs : K1

�vc1vqN

�qs¼ �e�c1

�vf1vqN

�qs

(102)

�vc2vqN

�qs¼ e�c2

�vf2vqN

�qs

(103)

cs1

�1þ K2

1þ K1

1þ K2

�� fs

1e�c1 ¼ 1þ K1

1þ K2

�K2c

�1 þ fs

2e�c2

�eh

(104)

where c*1 ¼ c*B þ c*C.

3 In this case (linear diffusion), the surface concentrations fulfil that [35]:

ffiffiffiffiffiffiD1

pc1ð0; tÞ þ

ffiffiffiffiffiffiD2

pc2ð0; tÞ ¼

ffiffiffiffiffiffiD1

pc*1 þ

ffiffiffiffiffiffiD2

pc*2

where D1 and D2 are the diffusion coefficients of the oxidized and reduced species,respectively. From the above it can be demonstrated [35] that the bvp for unequaldiffusion coefficients are also given by Eqs. (109)-(111) and the analytical solution for

single pulse techniques by Eq. (116) by making h ¼ FRTðE � E1=2Þ where E1=2 ¼ E0

0 þFRT

lnðffiffiffiffiffiffiffiffiffiffiffiffiffiffiD2=D1

pÞ.

3.2.1.1. The kinetic steady state treatment: reduction to a one-variable problem. The linear reaction layer. The kinetic steadystate (kss) approximation [63,67,68,91,92] is a very powerfultheoretical approach that has been successfully applied to a num-ber of electron- and ion-transfer [93,94] mechanisms, enabling thededuction of analytical equations as well as very efficient numericalsimulations [35] with very accurate results provided that the ho-mogeneous chemical kinetics is not slow (less than 5% error in thecurrent for c>3 and less than 1% for c>10 at macroelectrodes, theerror decreasing with the electrode size [87]). As indicated inSection 3.1.2, on the basis of the kss treatment it can be assumedthat:

vfss1

v t¼ 0 ;

vfss2

v t¼ 0 (105)

where:

fss1 ¼ cB � K1 cC

fss2 ¼ cE � K2 cF

(106)

such that the corresponding differential equations become into:

D V2Gf

ss1 ¼ k1f

ss1 ; D V2

Gfss2 ¼ k2f

ss2 (107)

with k1 ¼ k1 þ k2 and k2 ¼ k3þ k4. Since the differential equation(107) are only dependent on spatial coordinate(s), in analogy to Eq.(81) it can be written that:

*�vfss

1vqN

�qs

+¼ �

�fss;s1

�DdGr;1

E*�

vfss2

vqN

�qs

+¼ �

�fss;s2

�DdGr;2

E(108)

where the expression for the linear reaction layer is only dependenton the geometry of the diffusion field. So, the equation for dr;G forthe first-order mechanisms considered in this section (CEC, CE, EC)is formally identical to that obtained for the catalytic mechanismfor each electrode geometry as explained in Section 3.1.2 (Eqs. (78)and (82)).

For the sake of simplicity, hereafter the case of uniformlyaccessible electrodeswill be considered for which the kss bvp of theCEC mechanism can be reduced to a single-variable problemwhatever the voltammetric technique (that is, regardless of thepotential perturbation). Thus, in the case of macroelectrodes it isobtained that3:

vc1v t� D

v2c1vx2

¼ 0 (109)

t ¼ 0; x � 0t � 0; x/∞

�c1ðx; tÞ ¼ c�1 (110)

t >0; x ¼ 0

�vc1ðx; tÞ

vx

�x¼0¼ G

hc1ð0; tÞ � c1ð0Þeq

i(111)

where c*1 ¼ c*B þ c*C and c1ð0Þeq is the surface concentration ofoxidized species under total equilibrium (electrochemical andchemical) conditions:

A. Molina, E. Laborda / Electrochimica Acta 286 (2018) 374e396386

c1ð0Þeq ¼c*1e

h

eh þ�1þK21þK1

�1K2

(112)

and:

G ¼ 1þ K2 þ ð1þ K1ÞK2eh

K1ð1þ K2Þ þ ehð1þ K1Þffiffiffiffik1k2

q ffiffiffiffiffik1D

r(113)

with h ¼ FRT ðE� E0

0 Þ. The solutions for the particular cases of the CEand EC mechanisms can be derived from the above equations bymaking K2 > >1 (CE):

GCE ¼1þ ð1þ K1Þeh

K1

ffiffiffiffiffik1D

r

cCE1 ð0Þeq ¼c*1e

h

eh þ 11þ K1

(114)

and K1/0 (EC):

GEC ¼1þ K2 þ K2e

h

eh

ffiffiffiffiffik2D

r

cEC1 ð0Þeq ¼c*1e

h

eh þ 1þ K2

K2

(115)

where c*1 ¼ c�C.According to the above, the kss approach offers a very general,

simple and efficient way of treating the response of chemically-complicated charge transfer reactions. Thus, for example, theanalytical resolution of (109)e(111) for the application of a constantpotential pulse leads to the following general expression for thecurrent-potential-time response of the CEC, CE and EC mechanismsat macroelectrodes4 [35]:

I ¼ Ieqffiffiffipp U

2eðU=2Þ

2

erfc ðU=2Þ (116)

where Ieq corresponds to the current under total electrochemicaland chemical equilibrium conditions:

Ieq ¼ FA

ffiffiffiffiffiffiDp t

r �c*1 � c1ð0Þeq

�(117)

and U is a dimensionless parameter that accounts for the chemicalkinetics:

U ¼ 2ffiffiffiffiffiffiffiD tp

G (118)

4 The expression is formally identical to the rigorous solution for non-reversiblesimple electron transfers where the definition of G accounts for the electrode ki-netics [35].

Also, Eqs. (109)-(111) with any waveform (EðtÞ) can be solvedvery easily by numerical methods in order to study the electro-chemical response of the CE, EC and CEC mechanisms as shown inFig. 4 in cyclic and square wave voltammetries.

3.2.1.2. The dkss treatment. Microelectrodes: towards the steadystate. The use of microelectrodes for the study of coupled chemicalreactions has been extensively considered [61,62,71,95]. Whilecapacitive and Ohmic drop effects are greatly reduced and very fastchemical kinetics can be quantified [61,95], the enhanced masstransport can mask slow and intermediate chemical kinetics [63].Also, direct diagnosis criteria of the reaction mechanism may beless conclusive than at macroelectrodes given the shape of thevoltammograms and that the influence of the time variable isgradually lost as approaching the steady state.

The so-called diffusive-kinetic steady state (dkss) treatment is avery powerful theoretical approach to reaction mechanisms thatprovides simple and accurate analytical expressions for micro-electrodes under transient and stationary conditions and for fastchemical reactions at macroelectrodes (see Refs. [63,64]). For theabove cases, numerical simulations are more complicated (andslow) for the depletion layer being very thin, which requires the useof (adaptive) unequal spatial grids and very short timesteps[32e34].

The dkss formalism includes two main assumptions. First, ki-netic steady state (kss) conditions are supposed (Eq. (105)). In thecase of non-uniformly accessible microelectrodes (i.e., microdiscs,microrings, microbands, …), the surface value of fss and thethickness of the linear reaction layer are not constant over theelectrode surface. Nevertheless, it has been proven for a variety offirst-order mechanisms [35] that accurate results can be obtainedby taking constant average values for the above magnitudes suchthat the average surface gradient of fss can be expressed as:��

vfss

vqN

��qs

¼ � hfss;si�dr;G� /

�dr;G� ¼ � hfss;siD�

vfss

vqN

�Eqs

(119)

The mathematical expression for hdr;Gi is independent of thereaction mechanism and, for a given microelectrode geometry, itcan be deduced from the corresponding analytical solution for thesteady-state chronoamperometric response of the (pseudo)first-order catalytic mechanism (see Eqs. (76) and (82)).

The second assumption in the dkss approach is that pseudo-species c1 and c2 are subject only to diffusive conditions (and notto kinetic ones) such that their surface gradients can be expressedin an approximate way as:*�

vc1vqN

�qN;S

+¼ c*1 �

�cs1��

dd;G�

*�vc2vqN

�qN;S

+¼ �

�cs2��

dd;G�

(120)

where dd;G is the thickness of the linear diffusion layer, which isonly dependent on the electrode size and shape, analogously todr;G. Also note that the values of the surface gradients, surfaceconcentrations and dd;G have an average character in the case ofnon-uniformly accessible microelectrodes. The mathematicalexpression for hdd;Gi is obtained from the chronoamperometric

response of the E mechanism at the corresponding electrode ge-ometry [96]:

Fig. 4. Influence of the chemical kinetics on the CV (A,B,C) and SWV (D,E, F) responses of different CE, EC and CEC schemes at planar electrodes. ESW ¼ 25 mV , ES ¼ 5 mV JSW ¼I

FA c*1ffiffiffiffiffiffiffiffiD

2fSW

q .

dd;macro ¼ffiffiffiffiffiffiffiffiffipDtp

dd;sph ¼�1rsþ 1ffiffiffiffiffiffiffiffiffi

pDtp

��1�dd;disc

� ¼ �4p

1rdð0:7854þ 0:44315

rdffiffiffiffiffiffiDtp þ 0:2146exp

�� 0:39115

rdffiffiffiffiffiffiDtp

����1(121)

A. Molina, E. Laborda / Electrochimica Acta 286 (2018) 374e396 387

A. Molina, E. Laborda / Electrochimica Acta 286 (2018) 374e396388

Taking into account Eq. (120) together with the kss relationships(108) and (119) in the resolution of the bvp of the CEC mechanism,the mathematical treatment greatly simplifies. Thus, for example,the following simple expression is obtained for the current-potential-time response at any electrode geometry G whenapplying a constant potential pulse [59]:

IdkssCECIlim;E

c*1 ¼ 1

1þ K1hdr;Gihdd;Gi þ

�hdr;Gihdd;Gi þ K2

��1þK11þK2

�eh

(122)

It is worth noting that the same expression is obtained from therigorous or kss solutions by making U> >1. Thus, the value offfiffiffipp U

2eðU=2Þ2erfc ðU=2Þ in Eq. (116) can be well-approximated byffiffiffi

pp U

2

1þ ffiffiffipp U

2

for U>20, leading to the dkss solution (Eq. (122)) [63].

From (122), it can be concluded that the current-potentialresponse of the CEC mechanism can be linearized in thefollowing form:

E ¼ ECEC1=2 þRTF

ln

ICEClim;c � IdkssCEC

IdkssCEC � ICEClim;a

!(123)

where the value of the half-wave potential (ECEC1=2 ) depends on the

equilibrium and rate constants:

ECEC1=2 ¼ E00 þ RT

Fln�1þ K2

1þ K1

�þ RT

Fln

0@1þ K1

hdr;Gihdd;Gi

hdr;Gihdd;Gi þ K2

1A (124)

Note that in the case of the catalytic mechanism, the lineariza-tion of the current-potential response is deduced from the rigoroussolution (Eq. (69)).

The relationship (123) also holds in the particular cases of the CEmechanism (K2 > >1 ) where the half-wave potential is given by:

ECE1=2 ¼ E00 þ RT

Fln�

11þ K1

�þ RT

Fln

1þ K1

�dr;G��

dd;G�!

(125)

and the EC mechanism (K1/0):

EEC1=2 ¼ E00 þ RT

Flnð1þ K2Þ �

RTF

ln

�dr;G��

dd;G�þ K2

!(126)

With the dkss treatment, simple expressions have also been

IdkssoSQ ¼FAD c�1�dd;G

�hdd;Gihdr;Gi ðe

hB=A þ KehC=DÞ þ ð1þ KÞehB=AehC=D

ehB=A

�K þ hdd;Gihdr;Gi

�þ ehC=D

�1þ K hdd;Gihdr;Gi

�þ ehB=AehC=Dð1þ KÞ þ hdd;Gihdr;Gi ð1þ KÞ

(127)

obtained for the open-square scheme:and for the square mechanism [59].

Total steady state (tss) conditions

When working with ultramicroelectrodes, a total steady statebehaviour can be attained where the concentration profiles of the

species are time-independent: vciv t ¼

vfssi

v t ¼ 0 ði≡1;2Þ. Under theseconditions, the analytical equations for the steady-state current-potential curves are identical to the dkss solutions (Eq. (122)) forrs < <

ffiffiffiffiffiffiffiffiffipDtp

and rd < <ffiffiffiffiffiffiffiffiffipDtp

in the case of ultramicro-hemispheresand discs, respectively:

dssd;sph ¼ rs

Ddssd;disc

E¼ p

4rd

(128)

while the expressions for the thickness of the linear reaction layerare logically the same as those indicated in Section 3.2.1.2 (Eq.(121)). Note that the solution obtained for each mechanism isapplicable to any potential waveform (i.e., to any electrochemicaltechnique).

3.2.1.3. Non-Nernstian electron transfer. When the electron transferis not reversible and the diffusion coefficients of the differentspecies are equal, relationship (94) still holds and the bvp of theCEC mechanism can also be reduced through the kss treatment to asingle-variable problem formally equivalent to that given by Eqs.(109)-(111). Hence, the current-potential response in single pulsetechniques is also given by Ref. [60]:

I ¼ Ieqffiffiffipp Uirrev

2eðUirrev=2Þ2erfc ðUirrev=2Þ (129)

with Uirrev ¼ 2ffiffiffiffiffiffiDtp

Girrev and:

Girrev ¼kred

1þ K1þ koxK2

1þ K2

Dþ dr;1kredK1

1þ K1þ dr;2

kox1þ K2

(130)

where dr;1 and dr;2 are given by Eq. (78) with k1 ¼ k1 þ k2 and k2 ¼k3þ k4, respectively. As indicated in Section 3.2.1.2, the dkss so-lution for the non-Nernstian CEC mechanism can be obtained by

makingffiffiffipp U

2 eðU=2Þ2erfc ðU=2Þ /

ffiffiffipp U

2

1þ ffiffiffipp U

2

in the kss solution

(129).

3.2.2. ECE mechanismIt is interesting to consider the modelling and influence of

coupled chemical reactions in multi-electron transfer mechanisms,

in particular the situationwhere two consecutive electron transfersare coupled through a homogeneous chemical reaction (the so-called ECE mechanism):

A. Molina, E. Laborda / Electrochimica Acta 286 (2018) 374e396 389

where E00

A=B and E00

C=D are the formal potentials of the redox couples

A/B and C/D, respectively, and k1s�1and k2

s�1are (pseudo-)

first order rate constants of the homogeneous chemical reaction.The corresponding boundary value problem assuming that theelectron transfers are reversible and that the species are onlysubject to diffusionwith equal diffusion coefficients (D) is given by:

vcAðq; tÞvt

� DV2GcA ¼ 0

vcBðq; tÞvt

� DV2GcB ¼ �k1cB þ k2cC

vcCðq; tÞvt

� DV2GcC ¼ k1cB � k2cC

vcDðq; tÞvt

� DV2GcD ¼ 0

(131)

t ¼ 0; cqt >0; q/∞

�cAðq; tÞ ¼ c*A; cBðq; tÞ ¼ cCðq; tÞ ¼ cDðq; tÞ ¼ 0

(132)

t >0; q ¼ qs

D�vcAvqN

�qs¼ �D

�vcBvqN

�qs

(133)

D�vcCvqN

�qs¼ �D

�vcDvqN

�qs

(134)

csA ¼ ehA=BcsB (135)

csC ¼ ehC=DcsD (136)

Note that the incidence of comproportionation-disproportionation reactions is not considered in the abovescheme and it will be discussed in Section 4.

The rigorous solution of the above problem (131)e(136) underthe application of a constant potential pulse at a macroelectrode(linear diffusion) have been deduced by means of Koutecký’sdimensionless parameter method [97,98]. Analogously to themechanisms considered in Sections 3.1 and 3.2, for this purpose it isvery convenient to introduce the following variable changes:

c1 ¼ cB þ cC

f ¼ ðcB � K cCÞek t(137)

where k ¼ k1 þ k2 and:

K ¼ k2k1¼ c*B

c*C¼ 1

Kcc*L(138)

with Kc being the concentration-based equilibrium constant.The kss and dkss treatments have also been applied to the

modelling of the ECE scheme, for which the steady-state equilib-rium perturbation function (fss) is assumed as time-independentand defined as:

fss ¼ cB � K cC (139)

Through the dkss approach, the problem is reduced to a singlesystem of linear equations that can be solved by standard methodsto obtain the following simple expression for the current responsethat yields accurate results (less than 5% error in the current) atultramicroelectrodes (r0 <

ffiffiffiffiffiffiDtp

), medium-size microelectrodes forc>12:5 and at macroelectrodes for c>30 [98]:

IdkssECE ¼FADc�A�dd;G

� 2�K�12 þ

hdd;Gihdr;Gi

�þehC=D

hdd;Gihdr;Gi ðKþ1Þ

ð1þehC=DÞ�hdd;Gihdr;Gi þehA=B

�þKðehA=Bþ1Þ

�hdd;Gihdr;Gie

hC=Dþ1�

(140)

where dd;G and dr;G are the thickness of the linear diffusion andreaction layers, respectively, which have an average character in thecase of non-uniformly accessible electrodes (see Section 3.1). Forthe ECE response in multipulse techniques, numerical simulationshave been employed [99,100].

As mentioned above, expression (140) is notably accurate forfast chemical reactions at macroelectrodes and when micro- andultramicro-electrodes are employed. Thus, Eq. (140) enables thederivation of expressions for the limit case of steady-state condi-tions by taking the corresponding expression for the time-independent linear diffusion layer hdd;Gi≡hdssd;Gi (see Eq. (128)).Also, the response under total equilibrium conditions is derived bymaking hdr;Gi/0 (i.e., k/∞):

IECeqE ¼FAD�dd;G

�c�A 2þ ð1þ KÞehC=D

1þ ð1þ KÞehC=D þ KehC=DehA=B(141)

The above expression is also deduced by solving the problemunder the hypothesis that chemical equilibrium conditions apply(see Section 2).

The coupled chemical reaction of the ECE mechanism is usuallycharacterized through the apparent number of electron transferred,the value of which depends on the features of the ‘interposed’chemical reaction. Hence, studying the cathodic limiting current for

both electron transfers Idksslim;ECE (i.e., hA=B/0 and hC=D/0) is ofparticular interest:

Idksslim;ECE ¼FAD�dd;G

�c�A2�K�12 þ hdd;Gihdr;Gi

�hdd;Gihdr;Gi þK

(142)

Note that when the chemical reaction is very fast (hdr;Gi/0), thecurrent is twice the value expected for a simple E mechanism:

Idksslim;ECEðk> >1Þ¼ 2FADdD

c�A¼ 2Ilim;E

(143)

and so 2 electrons are transferred per molecule of species A giventhat the chemical reaction does not establish any kinetic limitation.For slower chemical kinetics, the apparent number of electrons

Fig. 5. Variation of the apparent number of electrons of the ECE mechanism athemispherical microelectrodes as obtained from the value of the limiting current as afunction of the chemical kinetics and electrode size (i.e., of the dd=dr -value) and of theK-value (indicated on the curves).

A. Molina, E. Laborda / Electrochimica Acta 286 (2018) 374e396390

decreases tending to the value of 1 as the chemical reaction isslower (see Fig. 5).

As a summary of the above, Table 1 shows a simple overview ofsuitable variable changes for the theoretical treatment of chargetransfer mechanisms with (pseudo)first-order kinetics, indicatingboth rigorous and very accurate simplifications that can beconsidered prior to the resolution in order to reduce the number ofvariables.

The superscript in each variable means that it can be ‘avoided’ inthe theoretical treatment. Thus, in any rigorous treatment, vari-ables with superscript ‘1’ can be avoided attending to that, whenthe diffusion coefficients of the oxidized and reduced species areequal, it is fulfilled that c1ðq; tÞ þ c2ðq; tÞ ¼ c*1 þ c*2 whatever theelectrode geometry; at macroelectrodes, this can also be achievedwith unequal diffusion coefficients considering that:ffiffiffiffiffiffiD1

pc1ð0; tÞ þ

ffiffiffiffiffiffiD2

pc2ð0; tÞ ¼

ffiffiffiffiffiffiD1

pc*1 þ

ffiffiffiffiffiffiD2

pc*2 [35,60]. Moreover,

when employing the kss or dkss approaches, further reduction ofthe number of variables is possible and the variables marked withthe superscripts ‘1’ and ‘2’ can be avoided so that the bvp reduces toa single variable for the SQ, CEC, CE and EC mechanisms and to twovariables for the ECE and oSQ mechanisms [35,60]. Note that thekinetic steady state assumption is only valid, strictly speaking, touniformly-accessible electrodes although the extension to non-uniformly accessible electrodes by averaging the values of the

Table 1Suitable variable changes for the theoretical treatment of paradigmatic charge transfer m

Mechanism IFAD

¼ c1

ECE2�vcAvqN

�qSþ�vc1vqN

�qS

X

oSQ(z1¼ z2)

�vc1vqN

�qS

X

SQ�vc1vqN

�qS

X

CEC�vc1vqN

�qS

X

CE�vc1vqN

�qS

X

EC�vcAvqN

�qS

X

Catalytic�

vf

vqN

�qS

X1

reaction layer and surface value of f could be envisaged. Thediffusive-kinetic steady state approach can be applied whatever theelectrode geometry.

3.3. Application of current-time functions. chronopotentiometry

Chronopotentiometric methods have also been considered forthe investigation of homogeneous chemical reactions [101e103].The major differences between the theoretical treatment of vol-tammetric and chronopotentiometric techniques are that, in thelatter, the value of the surface flux is known/controlled and thereversibility of the electrode reaction is not a necessary surfacecondition to solve the problem.

The analytical expressions for the chronopotentiometricresponse under the application of a constant or a time-dependent[104] current function show differences between the diverse re-action mechanisms in an even more evident way than in voltam-metry. In order to illustrate this point, the solutions for thechronopotentiometric responses of the E, catalytic and CEC mech-anisms will be briefly reviewed in this section. Linear diffusionconditions will be considered for the sake of simplicity, althoughthe treatment and conclusions can be easily extrapolated touniformly-accessible microelectrodes [105,106]. Difficulties asso-ciated with the simulation of chronopotentiometry at non-uniformly accessible microelectrodes are discussed in Ref. [107].

3.3.1. The E mechanismUpon the application of a power time current the surface con-

centrations of the electroactive species A and D are given by:

cAð0; tÞc*A

¼ 1� NS SD (144)

cDð0; tÞc*C

¼ c*Dc*C� g NS SD (145)

where:

NS ¼2I0

FAD1=2A c�A

P2uþ1 ¼G1þ 2uþ 1

2

�Gðuþ 1Þ

(146)

echanisms.

c2 f1 f2 cA cD

e X2 e X X1

e X2 e X X1

X1 X2 X2 e e

X1 X2 X2 e e

e X2 e e X1

e X2 e X1 e

e X e e e

A. Molina, E. Laborda / Electrochimica Acta 286 (2018) 374e396 391

and in the case of IðtÞ ¼ I0 tu with u � � 1=2:

SD ¼ tuþ1=21

P2uþ1(147)

whereas for IðtÞ ¼ I0 eut:

SD ¼ euterf ðutÞ2ffiffiffiffiffiffiutp (148)

From Eq. (144), the so-called generalized Sand equation is ob-tained for the transition time t:

tuþ1=2 ¼ P2uþ1NS

(149)

Note that the above expression turns into the Sand equation foru ¼ 0 since P1 ¼

ffiffiffipp

.

3.3.2. The catalytic mechanismFor the first-order catalytic mechanism with equal diffusion

coefficients (DB ¼ DC ), the following expressions are deduced forthe surface concentrations of the electroactive species:

cCð0; tÞc*B þ c*C

¼ 11þ K

½1� NS SC �

cBð0; tÞc*B þ c*C

¼ 11þ K

½Kþ NS SC �(150)

where:

NS ¼2I0

FAD1=2C

c�B þ c�C

(151)

and TuðcÞ is a kinetic series function of the dimensionless param-eter c ¼ kt that in the case of IðtÞ ¼ I0 tu with u � � 1=2:

SC ¼ tuþ1=2TuðcÞ (152)

with:

TuðcÞ ¼ e�cX∞j¼0

cj

j!P2uþ2jþ1(153)

that for very fast chemical kinetics fulfils that:

Tuðc/∞Þ ¼ 12ffiffiffikp (154)

For IðtÞ ¼ I0 eut:

SC ¼ euterf ððuþ kÞtÞ2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðuþ kÞt

p (155)

fromwhich it can be demonstrated that the linear reaction layer inthe case of IðtÞ ¼ I0 tu takes the same form as obtained for apotential-controlled perturbation:

dr;macro ¼ffiffiffiffiDk

r(156)

It is worth noting that the above may not hold when othercurrent perturbations are applied. Thus, for example, whenapplying an exponential current of the form IðtÞ ¼ I0 ewt , theexpression for the linear reaction layer is given by:

dr;macro ¼ffiffiffiffiffiffiffiffiffiffiffiffiD

kþw

r(157)

3.3.3. The CEC, CE and EC mechanismsThe surface concentrations of the electroactive species in these

cases are given by:CEC mechanism

cCð0; tÞc*B þ c*C

¼ 11þ K1

½1� NS ðSD þ K1SCðc1ÞÞ �

cEð0; tÞc*B þ c*C

¼ 11þ K2

"K2

c*E þ c*Fc*B þ c*C

þ gNS ðK2SD þ SCðc2ÞÞ# (158)

and:

NS ¼2I0

FAD1=2C

c�B þ c�C

(159)

CE mechanism

cCð0; tÞc*B þ c*C

¼ 11þ K

½1� NS ðSD þ KSCðcÞÞ �

cDð0; tÞc*B þ c*C

¼ c*Dc*B þ c*C

þ gNS SD

(160)

with:

NS ¼2I0

FAD1=2C

c�B þ c�C

(161)

EC mechanism

cAð0; tÞc*A

¼ 1� NS SD

cBð0; tÞc*A

¼ 11þ K

"Kc*B þ c*C

c*Aþ gNSðKSD þ SCðcÞÞ

# (162)

and:

NS ¼2I0

FAD1=2C c�A

(163)

Note that in all the expressions of the surface concentrations inthe above mechanisms, the term SD is purely diffusive and identicalto that deduced for the E mechanism. With regard to the series SC ,this has a kinetic-diffusive character and it is identical to the seriesin the expression for the catalytic mechanism (Eq. (152) or (155)).

4. Second-order chemical kinetics

In the presence of chemical reactions following a second-orderkinetics the electrochemical problem becomes non-linear (thediffusive-kinetic differential equations include nonlinear terms)and the exact analytical resolution of the corresponding bvp is, ingeneral, not feasible except for very few cases [108]. Hence, thetheoretical modelling of these systems is mostly carried outthrough numerical resolution of the partial differential equationproblem; still, this approach is more complicated and slow than forfirst-order chemical kinetics since more complex and time-

A. Molina, E. Laborda / Electrochimica Acta 286 (2018) 374e396392

consuming resolution algorithms (such as the Newton-Raphsonmethod) are generally to be used.

4.1. Approximate analytical treatments

Some works can be found where the bvp of second order re-action mechanisms at ultramicroelectrodes has been treatedanalytically. Given the complexity of the problems, the steady statereaction layer concept [44,109,110] is generally invoked. Accord-ingly, concentrations are supposed to be at chemical equilibriumoutside a small region next to the interface (the reaction layer) andwithin such region the concentration of the electroinactive speciesis approximated as uniform. These simplifications have been re-ported to yield accurate results for fast chemical kinetics where thethickness of the reaction layer region is considerably smaller than

the diffusion layer thickness (ffiffiffiffiffiffiffiffiffiffiffiffiffikcsZ=D

q� dd;G=3 [111]).

The second-order catalytic mechanism with an irreversiblechemical reaction has been studied based on the above consider-ations, obtaining an approximate analytical solution for the steady-state current-potential response at hemispherical ultra-microelectrodes [112,113]:

C þ e�#B

Bþ Z/kC þ P 2nd­order EC0 mechanism

I ¼ Id1þ ðDC=DBÞeh

241� k r2sDCc*C

2DBDZ ½1þ ðDC=DBÞeh�

þ 12

k r2sDCc*C

DBDZ ½1þ ðDC=DBÞeh�þ 4k r2s c

*Z

DB

!1=235 (164)

where k is the second-order catalytic rate constant. The steady stateresponse at microcylinders [114], microdiscs and microrings [111]has also been analysed under the reaction layer approximation, aswell as non-catalytic second-order schemes, including dimeriza-tion [115], EC, ECE and DISP1 [71,111,112] mechanisms.

Fig. 6. Zone diagram showing the conditions under which the limiting current or the forwar

mechanism are affected significantly by the occurrence of disproportionation-compropo

chronoamperometry), K (Eq. (138)).

4.2. Numerical approaches

The numerical resolution of second-order kinetic-diffusive dif-ferential equation problems can be performed directly by dis-cretization and conventional iterative methods for nonlinearequations [32,88,116]. Through this approach, it has been possibleto study accurately a number of complex electrochemical situations(mainly via implicit methods), recent works includingdisproportionation-comproportionation reactions [99,117] anddifferent second-order catalytic schemes [118e120].

Alternatively, the effects of the chemical reactions and thediffusive mass transport can be uncoupled (the so-called sequentialmethod [32,33]), or the nonlinear kinetic terms can be linearized[32] such that very efficient simulations are achieved via directresolution algorithms [121]. Thus, for example, in the case of thesecond-order irreversible catalytic mechanism above consideredthe kinetic term within the backward implicit integration methodcan be approximated as [32]:

kctþdtR ctþdtZ zkctþdtR ctZ þ kctRctþdtZ � kctRc

tZ (165)

where ctZ and ctR are the known values of the concentrations at theprevious timestep such that nonlinear terms disappear.

4.3. ‘Minimizing’ second-order kinetic effects

Given the complexity of the modelling of electrochemical sys-tems involving second-order kinetics, the minimization of theirincidence on the electrochemical response by the selection ofsuitable experimental conditions is very convenient, wherepossible. Such conditions can be established theoretically fromsimulations.

One of the questions revisited over years is the excess concen-tration necessary to consider that a species concentration is con-stant such that no concentration gradients develop and the second-order kinetics can be well-approximated as pseudofirst order. Thiswas quantified for the catalytic mechanism finding that the error issmaller than 5% provided that the following relationship is fulfilled[46]:

d peak current in CV (Zones Ilim/peak) and the position of the wave (Zone E1/2) of the ECE

rtionation reactions. cdisp=comp ¼ kdisp=compc�ARTFv

(in CV), cdisp=comp ¼ kdisp=compc*At (in

Fig. 7. The zone diagram shows the situations where the rigorous resolution of the problem is necessary and when it can be simplified into a simpler form through accurateapproximations for heterogeneous charge transfer mechanisms with first-order chemical kinetics. Transition values are only indicative for errors in the current below 5% [63,87,98].t is a suitable ‘reference time’ that accounts for the duration of the electrical perturbation; for example, in the case of chronoamperometry, t corresponds to the total duration of thepotential pulse applied.

A. Molina, E. Laborda / Electrochimica Acta 286 (2018) 374e396 393

c*Zc*C

>6:7DC

DZ(166)

In the case of multi-electron transfer mechanisms, second-ordercomproportionation-disproportionation (COMP-DISP) reactionscan affect the voltammetric response. Thus, when studying the ECEmechanism (see Section 3.2.2) the inference of the followingsolution-phase redox reaction should be taken into account:

Bþ C ����������������!kdisp

kcomp

Aþ D

Kdisp ¼kdispkcomp

¼ exp

8<:F�E00

C=D � E00

A=B

�RT

9=; ¼ c�Ac

�D

c�Bc�C

(167)

such that the comproportionation or the disproportionation reac-tion is thermodynamically more or less favourable depending on

the DE0’ ¼�E00

C=D � E00

A=B

�-value.

Fig. 6 summarizes the experimental conditions where signifi-cant effects on the voltammetric curves are found and so theapplication of the theory for a simple ECE is not suitable forquantitative analysis. The so-called DISP1/COMP1 case(cdisp=comp > >1) [1e4] is included within the far-right zone of thediagrams, whereas the DISP2/COMP2 situation (smallcdisp=comp-values) [1e4] where the DISP/COMP-influence is negli-gible corresponds to the left-hand zone of the diagrams. Outsidethe grey zones in this figure, the limiting current in chro-noamperometry and the peak current in cyclic voltammetry areaffected less than 10% and the position of the wave less than 5mV.

Note that, as can be inferred from the dimensionlesscomproportionation-disproportionation rate constant (cdisp=comp),the DISP/COMP effects can be reduced experimentally bydecreasing c�A in order that cdisp <103 and ccomp <50 (see Fig. 6).

5. Conclusions

As a summary of the above, the zone diagram in Fig. 7 illustratesthe conditions where the rigorous treatment of the coupledchemical reactions must be used and, most importantly, thosewhere much simpler theoretical approaches can be employed. Thisis a function of two dimensionless parameters that define thesystem's response as depending on the ‘effective’ electrode size (R0)and chemical kinetics (c). The diagram also shows the regionwherethe electrochemical response informs about the chemical kineticsor, otherwise, this is not accessible either for being masked by theenhanced mass transport at microelectrodes or for being very fastcompared to diffusion (chemical equilibrium).

Acknowledgements

The authors greatly appreciate the financial support provided bythe Fundaci�on S�eneca de la Regi�on deMurcia (Project 19887/GERM/15) and by the Ministerio de Economía y Competitividad (ProjectsCTQ-2015-65243-P and CTQ-2015-71955-REDT).

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