ORIGINAL PAPER

Staircase, cyclic and differential voltammetriesof the nine-member square scheme at microelectrodesof any geometry with arbitrary chemical stabilizationof the three redox states

Ángela Molina1 & Eduardo Laborda1 & José María Gómez-Gil1 & Richard G. Compton2

Received: 10 May 2016 /Revised: 22 June 2016 /Accepted: 25 June 2016 /Published online: 29 July 2016# Springer-Verlag Berlin Heidelberg 2016

Abstract Simple analytical expressions are deduced for thecyclic (staircase) and differential cyclic (staircase)voltammetric responses of the nine-member square schemeat the most widely used microelectrode geometries: disc,(hemi)spheres, cylinders and bands. The generality of the re-action scheme considered enables us to tackle many differentand common electrochemical situations where the oxidized,intermediate and/or reduced species of a two-electron transferalso take part in homogeneous chemical equilibria such asprotonations, complexations, or ion pairings. The effect ofthe coupled chemical processes on the voltammograms is an-alyzed at electrodes of different size and shape under a varietyof conditions depending on which redox state is chemicallystabilized most significantly. Also, working curves for thedetermination of the formal potentials and the chemical equi-librium constants are given for macroelectrodes, microelec-trodes and ultramicroelectrodes in cyclic voltammetry anddifferential cyclic voltammetry.

Keywords Nine-member square scheme .Microelectrodes .

Chemical stabilization . Cyclic voltammetry . Differentialcyclic voltammetry

Introduction

Analytical solutions are deduced for the staircase (SCV) andcyclic (CV) voltammetries of two-step electrode reactions atelectrodes of any size and shape in media where the redoxspecies are stabilized to different extents by reaction equilibriain solution such as complexations [1], protonations [2, 3] andion associations [4, 5]. This is found in a large number ofprocesses [6] impacting on highly diverse fields such aselectrocatalysis [5], biocide and drug development [7], bio-medicine (electrochemistry of DNA, metalloproteins, en-zymes, neurotransmitters, vitamins, etc.) [8, 9] andnanoscience [10]. The electrochemical response of these sys-tems depends on the stability and reactivity of the differentoxidation states in solution. Understanding them is essentialfor the optimization of energy conversion and molecular elec-tronic devices and the design of new compounds, materialsand analytical methods.

A general analytical theory for single-pulse and cyclicvoltammetries of the so-called nine-member square scheme[11, 12] with reversible electron transfers is developed here.Simple explicit solutions are deduced applicable to any elec-trode geometry, including microelectrodes of different shapes:discs, hemispheres, bands and cylinders. This is of great inter-est recognising the experimental value of microelectrodes [13]and the complexity of the theoretical modelling of diffusion atcertain geometries, most notably the ubiquitously employedmicrodiscs. Thus, the theory presented here provides physicalinsight into the influence of mass transport on these electro-chemical systems.

The features of the cyclic voltammograms of the nine-member square mechanism are investigated at electrodes ofdifferent geometry under transient and steady-state conditionsfor different values of the formal potentials and for distinctly

* Ángela Molinaamolina@um.es

1 Departamento de Química Física, Facultad de Química, RegionalCampus of International Excellence “Campus Mare Nostrum”;,Universidad de Murcia, 30100 Murcia, Spain

2 Department of Chemistry, Physical and Theoretical ChemistryLaboratory, University of Oxford, South Parks Road, OX13QZ Oxford, UK

J Solid State Electrochem (2016) 20:3239–3253DOI 10.1007/s10008-016-3308-2

different chemical stabilization of the three redox states.Depending on the electrode geometry and the difference be-tween the formal potentials, appropriate procedures based oncyclic voltammetry [14, 15] and differential cyclic voltamm-etry (DCV) [16] are established to elucidate the chemistry ofthe redox species in solution and to determine the correspond-ing equilibrium constants.

Theory

Consider an electrode process that follows a nine-membersquare mechanism as shown in Scheme 1. The concentrationsof the participating species after applying a potential-controlled perturbation E(t) obey the following diffusion-reaction differential equation system:

δ̂cXL j ¼ ϒ XL j

oX≡O; I ;R; j ¼ 0; 1; 2 ð1Þ

where δ̂ is the diffusion operator [17, 18] and ϒ XL j q; tð Þ cor-responds to the sum of kinetic terms of all the homogeneouschemical reactions a given species (XLj) takes part in, includ-ing cross-redox reactions such as disproportionation-comproportionation reactions, which are not shown inScheme 1 [19, 20]. Note that species L is not considered inEq. (1) because its concentration is assumed to be constant (i.e., cL(q, t) = cL

* ∀ q, t) for being buffered (e.g. protons) or in alarge excess with respect to the concentration of electroactivespecies (cL

*/cOT* > 10, [21]).

The boundary conditions of the above problem assumingthat only the oxidized species are initially present and that theelectrode reactions are reversible (i.e., k0qG/D > 10 with k0

being the standard heterogeneous rate constant and qG givenin the caption of Table 1 [18]) are given by

t ¼ 0; q≥qs

t > 0; q→∞

� cO q; tð Þ ¼ cO*; cOL q; tð Þ ¼ cOL* ; cOL2 q; tð Þ ¼ cOL2*

cI q; tð Þ ¼ cIL q; tð Þ ¼ cIL2 q; tð Þ ¼ 0cR q; tð Þ ¼ cRL q; tð Þ ¼ cRL2 q; tð Þ ¼ 0

8<:

ð2Þt > 0, q = qS:

DOL j

∂cOL j q; tð Þ∂qN

� �q¼qS

þ DIL j

∂cIL j q; tð Þ∂qN

� �q¼qS

þ DRL j

∂cRL j q; tð Þ∂qN

� �q¼qS

¼ 0g j ¼ 0; 1; 2

ð3Þ

cOL j qS; t

� � ¼ cIL j qS; t

� �eηOL j=IL j

cIL j qS; t

� � ¼ cRL j qS; t

� �eηIL j=RL j

�j ¼ 0; 1; 2 ð4Þ

where ηOL j=IL jand ηIL j=RL j

are defined as

ηOL j=IL j¼ F

RTE−E0

0OL j=IL j

� �ηIL j=RL j

¼ FRT

E−E00

IL j=RL j

� �9>=>; j ¼ 0; 1; 2 ð5Þ

qN is the coordinate normal to the electrode surface; qS is thevalue of the coordinates at the surface of the electrode; and F,R and T have their usual meanings.

Next, we assume that the chemical kinetics is much fasterthan the mass transport by diffusion so that total equilibriumconditions hold at any point in solution and time of the exper-iment such that

KO;1 ¼ kO;1kO;−1

¼ ceqOL q; tð ÞceqO q; tð Þ ¼ Kc

O;1c*L

KI ;1 ¼ kI ;1kI ;−1

¼ ceqIL q; tð ÞceqI q; tð Þ ¼ Kc

I ;1c*L

⋮

KR;2 ¼ kR;2kR;−2

¼ ceqRL2 q; tð ÞceqRL q; tð Þ ¼ Kc

R;2 c*L

9>>>>>>>>=>>>>>>>>;∀q; t ð6Þ

Scheme 1 Nine-member square scheme mechanism assuming all the

chemical steps as complexation or dissociation. E00

OL j=IL jVð Þ andE0

0

IL j=RL j

Vð Þ (j = 0, 1, 2) are the formal potentials of the redox couples OLj/ILj andILj/RLj, respectively. kX,n (s−1) X≡O; I ;R; n ¼ 1; 2ð Þ is the pseudo-

first order association rate constant where the concentration of species Lis assumed to be the same at any time and point of the electrolyte solution,and kX,− n (s−1) the first-order dissociation rate constant

3240 J Solid State Electrochem (2016) 20:3239–3253

where KX,n and KX,nc are the apparent equilibrium constants

and the equilibrium constants based on concentrations, respec-tively, and cL

* the concentration of species L. Also, it will beconsidered that all the electroactive species have the samediffusion coefficient DO ¼ DI ¼ ⋯ ¼ DRL2 ¼ Dð Þ. Thesetwo assumptions are reasonable for a number of frequentchemical processes (including protonations, ion pairings andcomplexations with small ligands [1–4]) that are generally fastand do not lead to any significant alteration of the moleculesize.

Under the above conditions (reversible electron transfers,equal diffusion coefficients of the electroactive species andhomogeneous chemical reactions at equilibrium conditions),it is convenient to define the total concentration of the pseudo-species XT (≡OT, IT, RT):

cOT q; tð Þ ¼ cO q; tð Þ þXn¼2

j¼1

cOL j q; tð Þ

cIT q; tð Þ ¼ cI q; tð Þ þXn¼2

j¼1

cIL j q; tð Þ

cRT q; tð Þ ¼ cR q; tð Þ þXn¼2

j¼1

cRL j q; tð Þ

9>>>>>>>>>>>=>>>>>>>>>>>;

ð7Þ

The differential equation system (1) becomes

δ̂cOT q; tð Þ ¼ 0

δ̂cIT q; tð Þ ¼ 0

δ̂cRT q; tð Þ ¼ 0

9>=>; ð8Þ

The boundary conditions become

t ¼ 0; q≥qSt > 0; q→∞

�cOT q; tð Þ ¼ c*OT ¼ c*O þ c*OL þ c*OL2

cIT q; tð Þ ¼ cRT q; tð Þ ¼ 0ð9Þ

t > 0, q = qS:

∂cOT q; tð Þ∂qN

� �q¼qS

þ ∂cIT q; tð Þ∂qN

� �q¼qS

þ ∂cRT q; tð Þ∂qN

� �q¼qS

¼ 0

ð10ÞcOT qS; t

� � ¼ cIT qS; t� �

eηOT=IT

cIT qS; t� � ¼ cRT qS; t

� �eηIT=R1T

�ð11Þ

ηOT/IT and ηIT/RT are defined as

ηOT=IT ¼ FRT

E−E00

OT=IT

� �ηIT=RT ¼ F

RTE−E0

0

IT=RT

� �9>=>; ð12Þ

E00

OT=IT and E00

IT=RT are the apparent formal potentials of thefirst and second electron transfers given by

E00

OT=IT ¼ E00

O=I−RT

Fln

1þX2

n¼1

βO;n

1þX2

n¼1

βI ;n

0BBBBB@

1CCCCCA

E00

IT=RT ¼ E00

I=R−RT

Fln

1þX2

n¼1

βI ;n

1þX2

n¼1

βR;n

0BBBBB@

1CCCCCA

ð13Þ

where βX,n corresponds to the overall formation equilibriumconstant

βX ;n ¼ ∏n

v¼1KX ;v ¼ cXLn q; tð Þ

cX q; tð Þ ;X≡O; I ;R ð14Þ

and KX ;n is the conditional formation constant for species XLn.

Table 1 Expressions for functions fG(ϑm,p, ξG) and fG,micro for the most usual electrode geometries considered

Electrode fG(ϑm,p, ξG) fG,micro

Disc (radius rd, Area Ad = πrd2)

4π ξd 0:7854þ 0:44315

ξdffiffiffiffiffiffiϑm;p

p þ 0:2146exp − 0:39115ξd

ffiffiffiffiffiffiϑm;p

p� �� �

; with ξd ¼ffiffiffiffiffiffiffiffiffiDr2d

RTFv

q 4π ξd

Sphere (radius rs, Area = 4πrs2)

ξs þ 1ffiffiffiffiffiffiffiffiffiπϑm;p

p with ξs ¼ffiffiffiffiffiffiffiffiffiDr2s

RTFv

q ξs

Band (height w, length l, Area =wl)ξb þ 1ffiffiffiffiffiffiffiffiffi

πϑm;p

p if (ξb2ϑm,p) < 0.4

0:25ffiffiffiffiffiffiπ

ϑm;p

qe−0:4

ffiffiffiffiffiffiffiffiπϑm;p

pξb þ πξb

ln 5:2945þ5:9944ffiffiffiffiffiffiϑm;p

pξbð Þ if (ξb

2ϑm,p) ≥ 0.4 with ξb ¼ffiffiffiffiffiffiffiffiffiffiDw2

RTFv

q2πξb

ln 64ξ2bϑm;p½ �

Cylinder (radius rc, length l, Area = 2πrcl)1ffiffiffiffiffiffiffiffiπϑm;p

p e−0:1ffiffiffiffiffiffiffiffiπϑm;p

pξc þ ξc

ln 5:2945þ1:4986ffiffiffiffiffiffiϑm;p

pξcð Þ with ξc ¼

ffiffiffiffiffiffiffiffiffiDr2c

RTFv

q 2ξcln 4ξ2cϑm;p½ �

Where qG is given by qG = rd for discs; qG = rs for spheres or hemispheres; qG = rc for cylinders and qG =w for bands. ϑm,p is given in Eq. (35)

J Solid State Electrochem (2016) 20:3239–3253 3241

It is worth noting that from Eqs. (8)–(10) it is immediatelydeduced that, under the conditions considered here, the totalconcentration of all the redox species remains constant overthe experiment at any point in solution whatever the electrodegeometry [18, 22],

cT ¼ cOT þ cIT þ cRT ¼ c*OT ; ∀q; t ð15Þ

which, together with the Nernstian conditions (Eqs. (11) and(12)), leads to the following expressions for the surface con-centrations of the three redox states:

cSOT ¼ c*OTeηOT=IT eηIT=RT

1þ eηIT=RT þ eηOT=IT eηIT=RT

cSIT ¼ c*OT eηIT=RT

1þ eηIT=RT þ eηOT=IT eηIT=RT

cSRT ¼ c*OT1þ eηIT=RT þ eηOT=IT eηIT=RT

ð16Þ

These are independent of time and only dependent on theapplied potential.

By defining the following linear combination,

W q; tð Þ ¼ 2cOT þ cIT ð17Þthe boundary value problem becomes (see Eqs. (8)–(12))

δ̂W q; tð Þ ¼ 0 ð18Þ

t≥0; q→∞t ¼ 0; q ¼ qS

�→W q; tð Þ ¼ W* ¼ 2c*OT

t > 0; q≥qS→W qS ; t� � ¼ WS ¼ c*OT

2eηOT=IT þ 1ð ÞeηIT=RT1þ 1þ e

ηOT=IT

� �eηIT=RT

ð19Þand the current response, which is not affected by the presenceof cross-redox reactions, is given by

I ¼ FAD 2∂cOT∂qN

� �q¼qS

þ ∂cIT∂qN

� �q¼qS

" #¼ FAD

∂W∂qN

� �q¼qS

ð20Þ

Single-pulse techniques

For this problem, it has been demonstrated [18, 23] that theinfluence of time and electrode geometry can be separatedfrom that of the applied potential, independently of the shapeand size of the electrode considered. Thus, the current in sin-gle potential step techniques such as chronoamperometry ornormal pulse voltammetry can be expressed as

I 1½ � ¼ FAGDf G qSG; t� �

W*−WS � ð21Þ

This can be rewritten in the following normalized form, whichis independent of the shape and size of the electrode

I 1½ �N ¼ I 1½ �

ID¼ W*−WS

c*OTð22Þ

ID is the diffusion-controlled limiting current

ID ¼ FAGDc*OT f G qSG; t� � ð23Þ

and fG(qGS , t) is a function dependent on time and on the ge-

ometry of the electrode (see [17, 18]). In the Appendix, theanalytical solutions for single-pulse techniques of the mostcommon particular mechanisms that can be derived fromScheme 1 are given.

Staircase cyclic voltammetry and cyclic voltammetry

Usually, in multipulse techniques, a sequence of potentialsteps E1, E2, …, Ep of the same length τ is applied and thecurrent is measured at a certain time, typically at the end ofeach pulse. Given that the superposition principle is applicableto the mechanism considered here (Eq. (16)) [17], the expres-sion for the current of the p-th potential pulse can be written as

I pð Þ ¼ FAGD∂W pð Þ q; tð Þ

∂qN

� �q¼qs

¼ FAGDXp

m¼1

W m−1ð Þ;s−W mð Þ;s� �

f G tm;p; qSG� �h i( )

ð24Þwhere

W mð Þ;s ¼ c*OT2eη

mð ÞOT=ITeη

mð ÞIT=RT þ eη

mð ÞIT=RT

1þ eηmð ÞIT=RT þ eη

mð ÞOT=ITeη

mð ÞIT=RT

m ¼ 1; 2;…; p

W 0ð Þ;s ¼ W* ¼ 2c*OT

ð25Þ

η pð ÞOT=IT ¼

F E pð Þ−E pð ÞOT=IT

� �RT

η pð ÞIT=RT ¼

F E pð Þ−E pð ÞIT=RT

� �RT

ð26Þ

tm;p ¼ p−mþ 1ð Þτ ð27Þ

and fG (tm,p, qG) is given in [17, 18]. From Eq. (24), the re-sponse in staircase cyclic voltammetry (SCV) [16] can becalculated, taking into account that the potential perturbationfor this technique is given by

Em tð Þ ¼ Einit− m−1ð ÞΔE for m ¼ 1; 2; …N=2Em tð Þ ¼ Efin þ m−p=2ð ÞΔE for m ¼ N=2;…N

�ð28Þ

3242 J Solid State Electrochem (2016) 20:3239–3253

where ΔE is the step width potential, m refers to the pulseapplied and N refers to the total number of pulses defined as

N ¼ 2 Einit−Efin�� ��

ΔEð29Þ

Note that the perturbation in cyclic voltammetry (CV) cor-responds to the limit of the SCV response (Eq. (28)) when thestep amplitudeΔE is very small (<0.01 mV [24]). Under suchconditions, the pulsed waveform behaves as a ramp with theapplied potential being a continuous function of time,

E tð Þ ¼ Einit − vt for t≤ tinvE tð Þ ¼ Efin þ vt for t > tinv

�ð30Þ

where tinv is the time at which the scan direction is inverted,and the scan rate is v = dE/dt. Then, the value of tm,p (Eq. (31))can be defined for CV as a function of the scan rate and theapplied potential as follows:

tCVm;p ¼jEp−mþ1−Einitj

v; m ¼ 1; 2;…p ð31Þ

such that the equation for the CV current at electrodes of anygeometry and size can be rewritten as

Ψ pð Þ;CV ¼ I pð Þ

FAGc*OTffiffiffiffiffiffiDa

p

¼ 1

c*OT

Xp

m¼1

W m−1ð Þ;s−W mð Þ;s� �

f G ϑm;p; ξG� �h i( )

ð32Þwhere

a ¼ FvRT

ð33Þ

ξd ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiDq2G

RTFv

sð34Þ

ϑm;p ¼FRT

Ep−mþ1−Einit

� �m≤N=2

FRT

Ep−mþ1−Einit þ 2Efin

� �m > N=2

8><>: ð35Þ

and fG (ϑm,pCV, ξG) is given in Table 1 and (N/2) is the number of

pulses of each (forward and reverse) scan.

Results and discussion

Effect of the chemical stabilization of the different redoxstates

In the absence of coupled chemical processes and/or kineticconstraints, the voltammetric response of two-electron elec-trode processes is governed by the thermodynamic stability of

the intermediate (i.e., the difference between the two formalpotentials) [1, 25–30]). Nevertheless, the experimentalvoltammetric response may differ from the above due to the‘stabilization’ of a particular redox state as a result of slowelectrode kinetics [31, 32] and/or coupled chemical reactionsin solution [1, 29, 33]. The ‘chemical stabilization’ effects forthe case of very fast kinetics can be studied with the analyticalexpressions reported in the “Theory”; section that enable us toenvisage and cover many different particular situations de-pending on whether one or more redox states undergo a sig-nificant chemical stabilization that affects the value of thecorresponding apparent formal(s) potential(s) and the magni-tude of such effect. According to Eqs. (13) and (14), this isquantified by the values of GO, GI and GR defined as

GO ¼ βO;1 þ βO;2GI ¼ βI ;1 þ βI ;2GR ¼ βR;1 þ βR;2

ð36Þ

such that the difference between the apparent formal potentialthat defines the behaviour of the nine-member square schemecan be written as

ΔE00

app ¼ E00

I=R−E00

O=I

� �þ RT

Fln

1þ GOð Þ 1þ GRð Þ1þ GIð Þ2

( )ð37Þ

Then, it is clear thatΔE00

app < ΔE00when the stabilization

of the intermediate (GI) is dominant, whereas if the GO- and/orGR-terms predominate with respect to GI, it follows that

ΔE00

app > ΔE00and an apparent potential inversion or coop-

erative [23, 29] situation may arise.Figure 1 shows the cyclic voltammograms at a microdisc

electrode of two-electron processes where the oxidized or re-duced species take part in chemical equilibria in solution andnot the intermediate, which would be highly stable in the

absence of chemical effects: ΔE00 ¼ −200mV. The value of

the surface concentrations is also calculated to show thechange in the stability of the redox states as a function of theapplied potential and of the GX value (X≡O, R). Note that thelatter can be modified experimentally by varying the concen-tration of species L (the larger the cL

* value, the larger GX) andthat it takes a null value in the absence of L.

In the case of the CEE or the CCEE mechanisms(Fig. 1a, b), only the oxidized species O is involved in homo-geneous chemical equilibria (GI =GR = 0) that stabilize it suchthat the depletion of the OLi species by the electrode reactiontakes place at more negative potentials (see Fig. 1b) whereasthe effect on the surface concentration of the reduced speciesis negligible (not shown). The stability of the intermediate isalso strongly affected. As a result of the homogeneous chem-ical processes (i.e., as GO increases), the “life potential range”;of the intermediate (defined as the potential range where the

J Solid State Electrochem (2016) 20:3239–3253 3243

surface concentration of the intermediate takes significantvalues) and its surface concentration are smaller. Moreover,the beginning of the potential region where ILj species aresomewhat stable is situated at more negative potentials (seeFig. 1b and “Determination of the stability constants and formalpotentials”, section). These effects give rise to the shift of thefirst electroreduction peak, the second one being scarcely af-fected in such a way that, eventually, they may merge into onein spite of the large difference between the formal potentials(ΔE0

0 ¼ −200mV ).

The “mirror image”; situation where it is the reduced spe-cies R that undergoes a strong chemical stabilization in solu-tion is considered in Fig. 1c, d; this would correspond to theEEC or the EECC mechanisms (GO = GI = 0). The intermedi-ate surface concentration decreases when GR increases and ILj“disappears”; at less negative potentials. With regard to thevoltammetric response, the two signals move closer as cL

* isincreased and they can merge into one. Nevertheless, thechange of the cL

* value only has a significant effect on theposition of the second signal in the EEC(C) mechanisms,

Fig. 1 Influence of coupled chemical equilibria involving only theoxidized species (a, b, GO values indicated on the graphs) or only the

reduced species (c, d,GR indicated on the graphs) forΔE00 ¼ −200 mV:

Cyclic voltammograms (a, c, Eq. (32)) and normalized surfaceconcentrations (b, d, Eq. (16)) at a microdisc electrode (rd ≈ 30 μm),v = 100 mV s−1, D = 10−5 cm2 s−1, |ΔE| = 0.01 mV, T = 298K

3244 J Solid State Electrochem (2016) 20:3239–3253

which can be easily distinguished from the (C)CEE one de-

pending on which curve shifts as cL* is modified.

When the formal potentials are not very different and asingle voltammogram is obtained (Fig. 2), the occurrence ofthe chemical processes can be detected in the change of theposition and shape of the voltammogram as cL

* is varied. Forthe two types of mechanism considered in Fig. 2, an increase

of cL* leads to largerΔE0

0

app values, to larger peak currents and

smaller peak-to-peak separations. On the other hand, the shift

of the position of the voltammograms with cL* enables us to

discriminate between the (C)CEE and EEC(C) cases giventhat it takes place in opposite directions. Also, note that under

these conditions (i.e., ΔE00 ¼ 0mV ) the “life potential

range”, of all the species is affected by GO or GR.The situation where it is only the intermediate species that

is involved in homogeneous chemical equilibria (GO = GR =

0) is studied in Fig. 3 for bothΔE00 ¼ −200mV and 0 mV. In

this situation, the increase of cL* (and so of GI) leads to the

Fig. 2 Influence of coupled chemical equilibria involving only theoxidized species (a, b, GO values indicated on the graphs) or only the

reduced species (c, d, GR indicated on the graphs) for ΔE00 ¼ 0 mV:

Cyclic voltammograms (a, c, Eq. (32)) and normalized surfaceconcentrations (b, d, Eq. (16)) at a microdisc electrode (rd ≈ 30 μm),v = 100 mV s−1, D = 10−5 cm2 s−1, |ΔE| = 0.01 mV, T = 298K

J Solid State Electrochem (2016) 20:3239–3253 3245

decrease of ΔE00

app (Eq. (37)) with the first signal shifting to

more positive potentials and the second one towards morenegative E-values in a symmetric way with respect to the

average apparent formal potential E00

app ¼E0

0OT=ITþE0

0IT=RT

2 . As a

result, when the cL* value is increased, the CV curve tends to

split in two (Fig. 3a) and the two peaks separate from eachother (Fig. 3a, c).

According to the results shown in Figs. 1, 2 and 3, one canconclude that the increase of the concentration of L has dif-ferent effects on the voltammograms depending on whetherthe corresponding chemical equilibria mostly affect the oxi-dized, reduced or intermediate species. In the latter case, thesplit of the voltammograms takes place whereas the signalstend to merge into one when the oxidized and/or the reducedstates show a more significant chemical stabilization.

Electrode geometry

Microelectrodes offer very valuable advantages for accurateexperimental studies by minimizing non-faradaic and ohmicdrop effects [13]. However, the theoretical treatment of theelectrochemical response at microelectrodes is, in general,more complicated than for macroelectrodes as a consequenceof the non-uniform accessibility to the electrode surface of theelectroactive species at some geometries such as microdiscs.The theory presented in the “Theory”, section enables us toovercome these difficulties and calculate the response in cy-clic voltammetry at the most common ones. Thus, Fig. 4shows the CV curves calculated from Eq. (32) at electrodesof very di fferent s ize (f rom macroelec trodes toultramicroelectrodes) and shape (disc, (hemi)spherical, bandand cylindrical).

Fig. 3 Influence of coupled chemical equilibria involving only the

intermmediate species (GI values indicated on the graphs) for ΔE00 ¼ 0

mV (a, b) and ΔE00 ¼ −200 mV (c, d). Cyclic voltammograms (a, c,

Eq. (32)) and normalized surface concentrations (b, d, Eq. (16)) at amicrodisc electrode (rd ≈ 30 μm), v = 100 mV s−1, D = 10−5 cm2 s−1,|ΔE| = 0.01 mV, T = 298K

3246 J Solid State Electrochem (2016) 20:3239–3253

As can be observed, the electrode geometry affectsthe shape of the voltammograms and the magnitude ofthe cur rent dens i ty, d i f fe rences be ing nul l a tmacroelectrodes (linear diffusion) and increasingly ap-parent as the electrode size shrinks. Under transient

conditions, a peak shape is obtained whatever the elec-trode shape with the current density increasing in theorder: cylinder < band < (hemi)sphere < disc. Atultramicroelectrodes, a peak-shaped pseudo-stationaryresponse is observed at microcylinders and microbands

Fig. 4 Influence of the electrode size R0 ¼ qGffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiFv= DRTð Þp� �

andshape on the CV response (Eq. (32)) for the most usual geometries: disc

(⋯), spherical (––), band (− ⋅ ⋅ −) and cylindrical (− − −) and for ΔE00

¼ 0 mV (a–c ) and ΔE00 ¼ −200 mV (d– f ) , v = 100 mV s−1 ,

D = 10−5 cm2 s−1, |ΔE| = 0.01 mV, T = 298K. At ultramicroelectrodes(c, f), the DCV responses for disc and spherical electrodes are alsoshown as insets: ΨDCV = Ψ(p + 1),CV − Ψ(p),CV, GO = 10, GI = 100,GR = 1000

J Solid State Electrochem (2016) 20:3239–3253 3247

Fig. 5 Variation of thedimensionless forward peakcurrent (a), peak-to-peakseparation (b) and forward peakpotential (c) in cyclicvoltammetry with the differencebetween the apparent formalpotentials at macroelectrodes(solid line), microdiscs

(R0 ¼ qGffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiFv= DRTð Þp ¼ 2and 1, dashed line) and micro(hemi)spheres (dotted line). Values obtained from

Eq. (31) and corresponding to the first signal in the region where two waves are observed. T = 298 K,

Evertex−E00

OT=IT ¼ −600 mV , ΔE= 0.01 mV

3248 J Solid State Electrochem (2016) 20:3239–3253

Fig. 6 Variation of thedimensionless forward peakcurrent (a), half-peak width (b)and forward peak potential (c) indifferential cyclic voltammetrywith the difference between theapparent formal potentials atmicrodiscs (dashed line) andmicro(hemi)spheres (dotted line)for R0 ¼ qG

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiFv= DRTð Þp ¼ 0:1

and 0:05. Values obtained fromEq. (32) and corresponding to thefirst peak in the region where twopeaks are observed. Otherconditions as in Fig. 5

J Solid State Electrochem (2016) 20:3239–3253 3249

(see Fig. 4c, f) [18] in contrast to the sigmoid steady-s ta te response at ta ined at ul t ramicrodiscs andultramicro(hemi)spheres where the “total”; limiting cur-

rent is 2 × ID independently of the ΔE00

app value. For

studies with ultramicroelectrodes, the use of a differen-tial technique like D(S)CV is recommended since it of-fers well-defined, peak-shaped signals that enable easierand more accurate quantitative analysis (see insets inFig. 4c, f and “Determination of the stability constantsand formal potentials”, section). Also, note that D(S)CVis particularly useful for any electrode geometry whenthere is partial overlapping of the CV curves of thetwo-electron transfers since differential techniques gen-erally provide better resolution of the two signals [31,34, 35].

Determination of the stability constants and formalpotentials

The determination of the reaction mechanism and the valuesof the equilibrium constants and formal potentials can be per-formed by fitting the variation of the apparent formal poten-tials with the concentration of species L with Eqs. (13) and(14). As discussed in “Effect of the chemical stabilization ofthe different redox states”, section, the change of cL

* will affect

in a different way the ΔE00

app value and so the shape of the

voltammogram, depending on which species takes part in ho-mogeneous chemical equilibria and the stoichiometry of suchprocess(es).

The value of ΔE00

app can be determined from the for-

ward peak current (Ψf,p, Fig. 5a) and the peak-to-peakseparation (ΔEpeak, Fig. 5b) of the cyclic voltammogram,

and (then) the individual values of E00

OT=IT and E00

IT=RT can

be obtained from the peak potential(s) (Fig. 5c). In theregion where two signals are observed, the values ofΨf,p, ΔEpeak and Ef,p in Fig. 5 refer to the first one.

Under transient conditions, Ψf,p increases continuously

withΔE00

app at both macroelectrodes and microelectrodes from

the value corresponding to a one-electron transfer to that of an“apparently simultaneous”, two-electron transfer. Atmacroelectrodes, the well-known lower and upper limit Ψf,p

values are 0.446 and 1.26 [14, 15] and empirical equations formicrodiscs and micro(hemi)spheres were reported in [35].

With respect toΔEpeak, the variation withΔE00

app also depends

on the geometry of the electrode in such a way that largerpeak-to-peak separations are observed at microelectrodes.Thus, for example, at 298 K, theΔEpeak value ranges betweenca. 57 and 28.5 mVat macroelectrodes for “independent”, and“simultaneous”, two-electron transfers respectively [14, 15],

whereasΔEpeak varies from ca. 160 to 70mVat a microdisc of

R0 ¼ qG

ffiffiffiffiffiffiffiFv

DRT

q¼ 1 (with qG being the electrode radius).

The position of the voltammogram(s) enables the determina-

tion of E00

OT=IT andE00

IT=RT . Aswas shown in Fig. 5c, the enhance-

ment of the diffusive mass transport (i.e., the decrease of theelectrode size and the use of discs rather than hemispheres)shifts the peaks to more negative potentials. Thus, for very

negative ΔE00

app values, the forward peak potential (Ef,p) of

the first peak takes the well-known value for one-electron re-

ductions E00

OT=IT−28 mV at macroelectrodes [14, 15] and E f ;p

¼ E00

OT=IT−81 mV at a microdisc electrode with R0 = 1. Also,

the two CV curves show a centre of symmetry (see Figs. 1 and3), the abscissa value of which coincides with the average

apparent formal potential E00

app ¼E0

0OT=IT

þE00

IT=RT

2 . For very positive

ΔE00

app values, the forward peak potential of the single wave is

given by E f ;p ¼ E00

app−14 mV at macroelectrodes and by

E f ;p ¼ E00

app−34mV at a microdisc with R0 = 1.

For R0 < 1, the system approaches steady-state condi-tions and the peak of the cyclic voltammograms becomesless defined. Eventually, when the steady-state response isachieved at microdiscs and micro(hemi)spheres (R0 < 0.1),the use of D(S)CV is more convenient (see “Electrodegeometry”, section) and the difference between theapparent formal potentials can be obtained from the peak

current and half-peak width (Fig. 6a, b). The E00

OT=IT and

E00

IT=RT values can be determined from the peak poten-

tial(s), which coincides with the apparent formal potential

for ΔE00

app < −140 mV and with their average value E00

app

forΔE00

app > −70 mV [23]. Note that the shape and position

of the D(S)CV signal (that is, the w1/2 and ΔEpeak

values) are independent of the electrode geometry,which is very convenient since the data obtained arenot affected by uncertainties related to the electrode sizeand shape.

Conclusions

A simple analytical expression has been deduced for the re-sponse of the nine-member square scheme in cyclic voltamm-etry and differential cyclic voltammetry at microdiscs,micro(hemi)spheres, microbands and microcylinders. Thishas enabled us to investigate the effects of physical (masstransfer) and chemical (coupled homogeneous reactions) phe-nomena on the voltammetric response.

3250 J Solid State Electrochem (2016) 20:3239–3253

When the coupled chemical equilibria mainly stabilize theoxidized and/or reduced species, there is an apparent increaseof the difference between the formal potentials so that the “lifepotential range”, of the intermediate redox state decreases andthe two voltammograms tend to merge into one. The oppositebehaviours are observed when the intermediate species are themost stabilized by chemical processes in solution such that thesignals tend to split in two.

The determination of the formal potentials and the equilibriumconstants can be performed from the variation with the concen-tration of species L of the values of the two apparent formalpotentials. In order to obtain the latter, working curves have beengiven for microdiscs and micro(hemi)spheres under transientconditions based on the peak current, peak potential(s) andpeak-to-peak separation of the cyclic voltammograms. Thevalues of all the abovemagnitudes are dependent on the electrodegeometry such that the well-known criteria for macroelectrodesdo not apply at microelectrodes. At disc and hemispherical

ultramicroelectrodes under steady-state conditions, the use ofdifferential cyclic voltammetry is beneficial and the values ofthe apparent formal potentials can be obtained from the peakcurrent, peak potential(s) and half-peak width. The last two mag-nitudes are independent of the ultramicroelectrode shape andsize, and then possible geometry-related uncertainties do not af-fect the accuracy of the quantitative analysis.

Acknowledgments The authors greatly appreciate financial supportprovided by the “Fundación Séneca de la Región de Murcia”, (Projects19887/GERM/15 and 18968/JLI/13) as well as by the “Ministerio deEconomía y Competitividad”, of the Spanish Government (ProjectsCTQ-2015-71955-REDT and CTQ-2015-65243-P-Newtwork ofExcellence “Sensors and Biosensors”,). EL also thanks the “Ministeriode Economía y Competitividad”, for the contract “Juan de la Cierva-Incorporación 2015”, and JMGG the University of Murcia and BancoSantander for the fellowship under the programme “ContratosPredoctorales FPU 2016”,.

Appendix

Particular reaction schemes

The analytical solution for single-pulse techniques of the mostcommon particular cases derived from Scheme 1 are presentedin Appendix Tables 2 and 3

Table 2 Particular cases derived from the nine-member square mechanism where an (apparent) one-electron transfer takes place

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