Charge�/potential and capacitance�/potential curves corresponding toreversible redox monolayers

Joaquin Gonzalez *, Angela Molina

Facultad de Quımica, Departamento de Quımica Fısica, Universidad de Murcia, Espinardo, Murcia 30100, Spain

Received 26 November 2002; received in revised form 8 May 2003; accepted 22 May 2003

Journal of Electroanalytical Chemistry 557 (2003) 157�/165

www.elsevier.com/locate/jelechem

Abstract

General expressions for the charge�/potential (Q /E ) and capacitance�/potential (C /E ) curves corresponding to reversible redox

monolayers are deduced. These responses present time-independent and universal behaviour, which does not depend on the

voltammetric or chronopotentiometric technique employed to obtain them. This holds even in the more complex case in which non-

faradaic effects are taken into account. The simultaneous availability of both Q /E and C /E curves allows the total characterisation

of the reversible electroactive monolayer. The initial presence of both species in the monolayer can be detected only from the Q /E

curves which present a cross potential from which the ratio of initial excesses can be obtained. Experimental C /E data are easily

available from chronopotentiometric or voltammetric measurements. Nevertheless, to obtain the experimental Q /E curve,

chronopotentiometric data results are more suitable.

# 2003 Elsevier B.V. All rights reserved.

Keywords: Redox monolayers; Reversible processes; Interfacial charge; Interfacial capacitance; Derivative chronopotentiometry; Voltammetry

1. Introduction

The electrochemical literature reflects a growing

interest in the study of the redox behaviour of electro-

active monolayers such as those formed by thiols,

anthraquinone derivatives, bioactive peptides or nucleic

acids strongly adsorbed at mercury or carbon electrodes

[1�/7]. The study of the electrochemical behaviour of

surface bound molecules continues to be decisive for

their characterisation. A large number of electrochemi-

cal methods are currently employed, among which the

most used are cyclic voltammetry (CV), in the case of

the study of the kinetic behaviour [1,4,6,8�/12], and

potentiometric stripping analysis (PSA) also called

stripping chronopotentiometry with constant current,

in the case of analytical determination of surface

excesses [13�/18].Recent publications have also been devoted to the

development of the theory corresponding to the applica-

tion of other electrochemical methods for the study of

these processes. These include alternating current linear

sweep voltammetry (LSV) [19,20] and derivative chron-

opotentiometry with programmed currents of the form

I (t )�/I0 evt and I(t)�/I0tu [21�/23].

In this work, we will analyse the theoretical behaviour

of the charge�/potential responses (Q /E curves, with Q

being the total interfacial charge and E the potential

applied or measured, according to the electrochemical

method employed), and also that of the capacitance�/

potential responses (C /E curves, with C being the total

interfacial capacitance) obtained for surface bound

molecules. When the redox reaction is reversible, the

above Q /E and C /E curves present behaviour which is

independent of the electrochemical technique from

which they have been obtained. In other words, from

the I /E data obtained in voltammetric techniques (such

as LSV, CV, or alternating current LSV), or from the

E /t data obtained in chronopotentiometric techniques

(such as PSA, chronopotentiometry and derivative

chronopotentiometry with programmed currents), we

can obtain time-independent and universal sigmoidal Q /

E and peak-shaped (sigmoidal derivative) C /E curves.

* Corresponding author. Tel.: �/34-968-36-74-33; fax: �/34-968-36-

41-48.

E-mail address: josquin@um.es (J. Gonzalez).

0022-0728/03/$ - see front matter # 2003 Elsevier B.V. All rights reserved.

doi:10.1016/S0022-0728(03)00368-1

As far as we know, this behaviour has not been

previously pointed out in the electrochemistry literature

related to electroactive monolayers. However, it repre-

sents an interesting and particular characteristic, whichis exhibited only by reversible redox monolayers, even in

the more complex case in which non-faradaic effects are

taken into account [21,24]. Hence, the use of Q /E and

C /E curves by following the procedures indicated in this

work is of great interest in the total characterisation of a

reversible redox monolayer.

When the surface charge transfer reaction does not

present reversible behaviour, the temporal dependence,introduced by the form of the electric perturbation used,

cannot be eliminated in the charge�/potential and

capacitance�/potential responses, which will exhibit

different behaviour depending on which electrochemical

method has been used.

Firstly, we will show the theoretical behaviour of a

reversible redox monolayer in order to show that from

the E /t chronopotentiometric curve we can deduce theQ /E and C /E responses, which in the absence (Qa/E and

Ca/E curves, see Section 2.1) or in the presence (Qs/E

and Cs/E curves, see Section 2.2) of non-faradaic effects

are not dependent on the form of the perturbation

(current�/time function applied). Subsequently, in Sec-

tion 3, we will generalise the above results to the

electrochemical techniques most usually employed in

the analysis of molecular films.

2. Theoretical study of the electrochemical behaviour of

redox monolayers under the application of any

programmed current I (t )�/9/I0f(t)

We will consider the following surface electrode

process taking place in a reversible redox monolayer

A(adsorbed)�n e� ?kred

kox

B(adsorbed) (I)

in order to obtain the equations for the response

corresponding to the application of several successive

programmed currents of the form I(t)�/9/I0f(t), with I0

being the current amplitude of the programmed currentsapplied and f (t ) being any function of time. We assume

for the adsorbate monolayer that the Langmuir iso-

therm is obeyed. The redox couple is not present in the

solution and the adsorption is very strong to the extent

that the desorption is negligible in the time scale of the

experiment. Thus, the total excess GT is constant during

the whole experiment and it is fulfilled that [6]

GT�GA;0�GB;0�GA(t)�GB(t) (1)

with Gi ,0 and Gi (t ), i�/A or B, being the initial surfaceexcesses and the surface excesses corresponding to a

time t of i species, respectively. Moreover, the adsorp-

tion coefficients of both electroactive species and the

maximum surface coverage are independent of the

potential.

Since electron transfer reversibility is assumed, the

cathodic and anodic curves are the mirror images ofeach other. We will show the treatment for the cathodic

curve only and we will generalise the response obtained

for any number of programmed currents applied.

We will consider firstly that all the current is used in

the faradaic process given by Eq. (I) and then, secondly,

introduce the influence of non-faradaic effects by

considering the existence of an interfacial potential

distribution (IPD), according to the model proposedby Smith and White [24].

2.1. Application of current�/time functions of the form

I(t)�/I0f(t) in the absence of non-faradaic effects:

charge�/potential and capacitance�/potential curves

We will consider in this section that the appliedcurrent I (t ) is employed only on the surface charge

transfer process given by Eq. (I). Under these conditions

the following is fulfilled [6]:

I(t)

nFA��

dGA(t)

dt(2)

By integrating Eq. (2) between t�/0 and t�/t we

obtain the following expression for the surface excess of

oxidised species A:

GA(t)�GA;0�Qa(t)

nFA(3)

with Qa(t ) being given by

Qa(t)�gt

0

I(t?) dt? (4)

Qa(t ) is the charge at a time t corresponding to the

application of the programmed current I(t), equivalent

in this case to the faradaic reduction charge of the

surface bound molecules. The expression of Qa(t)

depends, following Eq. (4), on the particular form ofthe programmed current applied. As an example we will

show the expressions of Qa(t) for three different

current�/time functions which have been analysed pre-

viously in Refs. [21�/23,25,26]:

Constant current: I(t)�I0; Qa(t)�I0t (5a)

Power current: I(t)�I0tu; Qa(t)�I0tu�1

u � 1(5b)

Exponential current: I(t)�I0 evt;

Qa(t)�I0(evt � 1)

v

(5c)

The general current�/potential relationship for the

charge transfer reaction of the surface bound molecules

is given by [6]:

J. Gonzalez, A. Molina / Journal of Electroanalytical Chemistry 557 (2003) 157�/165158

I(t) eah

nFAk?0�GA(t)�eh(GT�GA(t)) (6)

with

h�nF

RT(E(t)�E?0) (7)

with k ?0 and a being the rate constant of the surface

electrochemical reaction, in s�1, and the charge transfer

coefficient of the cathodic surface reaction, respectively.

E ?0 is the standard potential of the surface process for

the Langmuir isotherm.Under reversible conditions for the surface charge

transfer process the Nernst equation is fulfilled. Thus,

Eq. (6) is simplified to:

eh�GA(t)

GT � GA(t)(8)

By taking into account Eq. (3) we can re-write Eq. (8)

in the following way:

Qa(E)�nFAGT

1 �V eh

(1 �V)(1 � eh)(9)

where V is given by:

V�GB;0

GA;0

(10)

Note that, in order to obtain this charge�/potentialresponse when both species are initially present, it is also

necessary to apply an anodic programmed current I(t).

Eq. (9) gives us the charge�/potential response (i.e. the

Qa/E curve) obtained when any programmed current

I (t) is applied to an electrode coated with a redox

monolayer of an electroactive reversible redox couple,

and non-faradaic effects are not present. This equation

indicates that the relationship between the reductioncharge Qa and the potential E has the same functional

dependence, independently of the type of programmed

current used.

By making Eq. (9) equal to zero we deduce the

adsorption cross potential, Eacp, which is given by:

Eacp�E?0�RT

nFln V (11)

which can exist only when both A and B species are

initially adsorbed. Therefore, when V"/0, the Qa/E

curve presents an intercept point with the potential axis

which corresponds to Eacp, according to Eq. (11).

Note that for reversible surface electrode processes,the Qa/E dependence given by Eq. (9) is identical to the

I /E one corresponding to reversible processes when

both electroactive species are soluble in the solution and

a constant potential is applied (see Ref. [2]).

It is interesting to obtain the derivative of this

response with respect to the potential, i.e. the (dQa/

dE)/E curve, which in line with Eq. (9) is also a general

relationship independent of the form of the appliedcurrent.

Thus, by differentiating Eq. (9) with respect to the

potential we obtain

j dQa

dE j�Ca(E)�n2F 2AGT

RT

eh

(1 � eh)2(12)

where Qa(E ) is given by Eqs. (4), (5a), (5b) and (5c) and

Ca(E) is the adsorption faradaic capacitance of thesurface bound molecules.

Observe that Eq. (12) corresponds to the

capacitance�/potential Ca(E )/E response of a reversible

electroactive monolayer.

The Ca(E )/E curve given by Eq. (12) presents a peak

whose potential is equal to the standard potential of the

surface process (Epeak�/E ?0) and whose capacitance is

equal to (Ca)peak�/n2F2AGT/4RT .In order to demonstrate the general behaviour of the

Qa/E and Ca/E curves, we show in Fig. 1a the potential�/

time response corresponding to the application of three

types of programmed currents (current step: I (t )�/I0,

current which varies with a power of time: I(t)�/I0tu ,

and current that varies exponentially with time: I(t)�/

I0 evt). We also show in Fig. 1b the relationship between

the charge Qa and the application time of the current. Ascan be seen in these figures, both E /t and the Qa/t curves

depend on the type of current employed, while the Qa/E

and Ca/E responses shown in Figs. 1c and d, respec-

tively, present general behaviour independent of the

form of the applied current I (t), according to Eqs. (9)

and (12).

2.2. Consideration of non-faradaic effects based on the

IPD model of Smith and White for any programmed

current applied: charge�/potential and capacitance�/

potential curves

In this section we will consider the Smith and White

model for IPD, which takes into account the distortion

due to non-faradaic effects [21,24]. Thus, in this case the

applied current I(t) is employed both with the surface

charge transfer process given by Eq. (I) and with the

charging process. This condition can be expressed in

terms of the total interfacial charge, Qs, correspondingto the application of a programmed current I(t)

I(t) dt�dQs��nFA dGA(t)�Cdl(E) dE (13)

with Cdl(E ) being the non-faradaic capacitance given by

[24]:

J. Gonzalez, A. Molina / Journal of Electroanalytical Chemistry 557 (2003) 157�/165 159

Cdl(E)�1�C�11 �

�o0oSkD cosh(ze(fP � fS))

2kT

��1

��

1�n2F 2GT

RTC1

f (1�f )

�(14a)

C1�o0o1

d1

(14b)

with

f �GA(t)

GT

(15)

and o0 being the permittivity of the free space, o1 and oSthe dielectric constant of the surface bound molecules

and of the solvent, respectively, d1 the thickness of the

monolayer, kD the inverse Debye length, fP and fS the

electrostatic potential at the electron transfer plane and

in the bulk solution, respectively, z the ion charge of the

supporting electrolyte, k the Boltzmann constant, and e

the electron charge.

The Nernst equation takes the following expression inthese conditions:

ehP �f

1 � f(16)

with

hP�nF

RT(E�E?0�(fP�fS)) (17)

By combining Eqs. (13) and (16) it is possible to

obtain an explicit expression for the coverage of

Fig. 1. (a) (E�/E ?0)/t curves (see Eq. (8)), (b) Qa/t curves (see Eq. (4)), (c) Qa/(E�/E ?0) curves (see Eq. (9)) and (d) Ca/(E�/E ?0) curves (see Eq. (12))

for a reversible redox Langmuir monolayer corresponding to the application of the following programmed currents: I (t )�/I0 (dashed and dotted

line), I (t )�/I0tu (solid line), and I (t )�/I0 evt (dashed line). I0�/1 mA, GT�/10�10 mol cm�2, A�/0.031 cm2, n�/1, u�/0.1, v�/0.75 s�1, T�/298.15

K.

J. Gonzalez, A. Molina / Journal of Electroanalytical Chemistry 557 (2003) 157�/165160

oxidised species, f , as a function of the charge. Thus, by

following the procedure indicated in Ref. [21] we obtain

f �C1RT

n2F 2GT

ln

�f

1 � f

�

�f0�C1RT

n2F 2GT

ln

�f0

1 � f0

��

Qs

nFAGT

(18)

with Qs being the total interfacial charge and

f0�GA;0

GT

(19)

The Qs/E response corresponding to the application

of any programmed current I (t) can be obtained by

taking into account Eqs. (16) and (18)

Qs(E)�nFAGT

�1 �V ehP

(1 �V)(1 � ehP )�

C1RT

n2F 2AGT

� (ln V�hP)

�(20)

According to Eq. (20) the total interfacial charge Qs

can be expressed as the sum of two terms, the first of

which is related to the faradaic process given by Eq. (I) (/

Qa�nFAGT(1�V ehP )=(1�V)(1�ehP )); while the sec-

ond is linked to non-faradaic effects (/

Qdl��C1RT(ln V�hP)=nF ):/Eq. (20) corresponds to the general charge�/potential

response obtained when any programmed current is

applied to an electrode coated with a reversible electro-active monolayer when IPD is taken into account.

Eq. (20) is greatly simplified when the condition

@fP=@E�1 is fulfilled (i.e. the non-faradaic capacitance

is practically independent of the electrode potential

during the reduction/oxidation of the film, Cdl(E )$/

C1). According to Smith and White [24] this condition

is governed by the following requirements:

FGT

(2000o0osRTcs)1=2

�1

C1

(2000o0oscs=RT)1=2�1

�(21)

with cs being the molar concentration of the supporting

electrolyte.

In this case, the charge�/potential response corre-

sponding to the application of any programmed current

I (t) is simplified to:

Qs(E)�nFAGT

�1 �V eh

(1 �V)(1 � eh)�

C1RT

n2F 2AGT

��

ln V�nF

RT(E�E?0)

��(22)

with h given by Eq. (7).

The Cs/E curve can be easily deduced by differentiat-

ing Eq. (20) with respect to the potential. Thus, we

obtain

j dQs

dE j�Cs(E)

��

n2F 2AGT

RT

ehP

(1 � ehP )2�C1

��1�

@fP

@E

�(23)

Eq. (23) corresponds to the capacitance�/potential

response, with Cs(E ) being the total interfacial capaci-

tance which, therefore, is also independent of the form

of the programmed current applied.

In the case in which the non-faradaic capacitance ispractically independent of the electrode potential during

the reduction/oxidation of the film (Cdl(E )$/C1), we

obtain from Eq. (23) the following expression for the

capacitance�/potential curve

Cs(E)�n2F 2AGT

RT

eh

(1 � eh)2�C1�Ca(E)�C1 (24)

with Ca(E ) given by Eq. (12).

It can be observed from Eq. (24) that, in this

simplified case, the total interfacial capacitance Cs(E )

is the summation of the capacitance Ca(E ) related to the

surface charge transfer given by Eq. (I), and a constant

non-faradaic contribution, C1, which does not affect the

peak potential. The peak capacitance in this case is

Cs(E)peak��

n2F 2AGT

4RT�C1

�� (Ca)peak�C1 (25)

Under these conditions, the non-faradaic contribution

to Cs/E is a constant value C1. This is analogous to that

observed under the same conditions in CV, a technique

for which the capacitative contribution of the ICV/E

curve is a constant which constitutes the so-called‘‘baseline’’ of the voltammogram [2,24].

In Fig. 2 we have plotted the charge�/potential (Qs/E ,

Qa/E and Qdl/E curves, Figs. 2a and b) and

capacitance�/potential responses (Cs/E , Ca/E and Cdl/E

curves, Fig. 2c) calculated from Eqs. (20) and (23),

respectively, corresponding to a reversible redox mono-

layer for two different values of the ratio of initial

excesses of both species V . By comparing Figs. 2a and bit can be observed that when both species are initially

adsorbed (Fig. 2b), the Qs/E curve presents an intersect

with the potential axis which coincides with the adsorp-

tion cross potential, Eacp, when non-faradaic effects are

not present. However, the Cs/E curve shown in Fig. 2c is

not affected by the ratio GB,0/GA,0 and the capacitance

peak is dependent only on the total excess GT (see Eqs.

(23)�/(25)).Moreover, from these figures it can be also observed

that the non-faradaic effects on the Qs/E curve are

practically linear with the potential (see curves for Qdl/E

J. Gonzalez, A. Molina / Journal of Electroanalytical Chemistry 557 (2003) 157�/165 161

in Figs. 2a and b), and are practically constant in the Cs/

E curves (see dotted line in Fig. 2c). Thus, in order to

obtain the total excess GT we must subtract that

constant contribution of the capacitance�/potential

response.

3. Results and discussion

3.1. Equivalence between voltammetric and

chronopotentiometric techniques with any programmed

current in the treatment of reversible redox monolayers:

charge�/potential and capacitance�/potential curves

In previous sections and in previous Refs.

[21,23,26,27], we have pointed out that in the case of

electroactive reversible monolayers, both the chronopo-

tentiometric (E /t curve) and the reciprocal derivativechronopotentiometric ((dt /dE )/E curve) responses pos-

sess different features when different current�/time

functions are applied (i.e. constant current I (t )�/I0,

exponential current I(t)�/I0 evt , power time current

I (t )�/I0tu ). Taking as our basis the above mentioned

results, we have demonstrated in this paper that the

charge�/potential (Qs/E curve) and the capacitance�/

potential (Cs/E curve) responses are both independentof the type of programmed current used I(t) (see Eqs.

(9), (12), (20) and (23)).

In this section we will show that the Qs/E and Cs/E

curves corresponding to reversible redox monolayers are

independent of whether the electrochemical technique

employed was under controlled potential conditions or

under controlled current conditions and that, therefore,

they show a universal behaviour. In other words, thetemporal independence of the Qs/E and Cs/E curves

yield to unique responses independently of their having

been obtained from voltammetric, I /E , or from chron-

opotentiometric, E /t , techniques.

Moreover, we must highlight the fact that the above

mentioned universal character of the Qs/E and Cs/E

curves for reversible redox monolayers is maintained

even when non-faradaic contributions are considered inthese responses through IPD model of Smith and White

[24] (see Eqs. (20) and (23)).

We will demonstrate below how both the voltam-

metric I /E and the chronopotentiometric E /t responses

Fig. 2

Fig. 2. (a, b) Charge�/potential and (c) capacitance�/potential curves

corresponding to a reversible redox Langmuir monolayer which

undergoes the reaction A��e�?B; for two values of the ratio of

initial excesses V . In (a, b), Qs (solid line), Qa�nFAGT(1�V ehP )=(1�V)(1�ehP ) (dashed line) and Qdl��C1RT(ln V�hP)=nF

(dotted line) have been calculated from Eq. (20). In (c), Cs (solid line),

Ca�n2F 2AGT(ehP=(1�ehP )2)Cdl(E)=RTC1 (dashed line) and Cdl

(dotted line) have been calculated from Eq. (23). o1�/5, o3�/78, d1�/

1 nm, kD�/3.3�/109 mV�1, z�/1, GT�/5�/10�11 mol cm�2, n�/1,

E ?0�/0.20 V, T�/298.15 K. In order to calculate (fP�/fS) we have

used Eqs. (10)�/(13) in Ref. [24]. The values of V are: 0 (a, c) and 1 (b,

c). Both E and E ?0 are referred to Epzc (zero charge potential, see Ref.

[24]).

J. Gonzalez, A. Molina / Journal of Electroanalytical Chemistry 557 (2003) 157�/165162

do indeed convert into Qs/E and Cs/E curves which are

independent of the electrochemical technique employed,

according to Eqs. (9), (12), (22) and (23). At this point,

we must take into account that the charge Qs can bedetermined experimentally by integrating the measured

current for a given value of the potential E in the case of

voltammetric techniques, or by integrating the applied

current in the case of chronopotentiometric ones.

Therefore, to obtain the experimental Q /E data, chron-

opotentiometric techniques are more suitable than

voltammetric ones (see Refs. [13,15�/18,21�/23,27]).

The total interfacial capacitance as a function of thepotential can be determined by using Eqs. (Ia)�/(Va)

shown in Table 1, depending on the electrochemical

technique used.

In the following discussion C and Q are equal to the

adsorption faradaic capacitance and charge, Ca and Qa,

respectively, when non-faradaic effects are not present,

whereas they refer to the total interfacial capacitance

and charge, Cs and Qs, respectively, when the IPDmodel is considered.

3.2. Potentiometric stripping analysis

PSA is a frequently used technique in the study of the

electrochemical behaviour of redox monolayers [13�/18].

It was developed by Jagner and coworkers [28,29] and itconsists of the plotting of the (dt /dE )/E response when a

constant current I(t)�/I0 is applied to the electrode. For

this technique, the following response is obtained for a

reversible redox monolayer (see Eq. (18) in Ref. [30]):

dt

dE�

n2F 2AGT

RTI0

eh

(1 � eh)2(26)

If we take into account that for this type of current

applied C�(dQ=dE)I(t)�I0�I0(dt=dE) (see Eq. (5a),

and Eq. (Ia) in Table 1), with Q given by Eq. (Ib) in

Table 1, then it is evident that in PSA the (dt /dE )

response is the interfacial capacitance divided by the

current amplitude I0.

3.3. LSV and CV

CV is frequently used as the first technique in the

study of the behaviour of redox monolayers [1,6,8�/12].The voltammetric response corresponding to a reversi-

ble monolayer (ICV/E curve) was deduced by Laviron

(see for example, Ref. [31]), and is given by the following

equation:

ICV�n2F 2AGTv

RT

eh

(1 � eh)2(27)

By taking into account that v is the voltammetric

sweep rate, which is given by v�/jdE /dt j in CV, it is

fulfilled that (see Eqs. (IIa) and (IIb) in Table 1)

C(E)�ICV

v

Q(E)�gE

Ei

ICV

vdE

�(28)

with Ei and E being the initial and the variable

potentials of the voltammetric sweep.

Eq. (28) clearly shows that the ratio between the

voltammetric current ICV and the sweep rate is equal to

the interfacial capacitance [5]. The integral of the

(ICV/v )/E curve for a given value of E correspond to

the interfacial charge Q (E ), whose potential dependenceis given by Eq. (9) or Eq. (23).

3.4. a.c. voltammetry

In a sinusoidal a.c. linear sweep voltammetric experi-

ment the potential applied to the electrode at any time is

given by [2,19,20]:

E(t)�Ed:c:�Ea:c:

Ed:c:�Ei�vd:c:t

Ea:c:�DE sin(vt)

�(29)

with Ei being the initial potential, vd.c. the constant

Table 1

Expressions for the Charge (Q ) and Capacitance (C ) corresponding to a reversible electroactive Langmuir monolayer for different electrochemical

techniques: PSA, CV, cyclic staircase voltammetry (CSV), a.c. LSV, derivative chronopotentiometry with exponential current I (t )�/I0 evt (DCE)

and derivative chronopotentiometry with a power time current I (t )�/I0tu (DCP)

Electrochemical techniques

PSA CV and CSV ACV DCE DCP

C�/dQ /dE I0(dt /dE ) (Ia) ICV/v (IIa) Ia.c./vTotal (IIIa) (I0/v )(d(evt�/1)/dE ) (IVa) (I0/(u�/1))(dtu�1/dE ) (Va)

Q I0t (Ib) /fE

Ei(ICV=v) dE (IIb) /f

E

Ei(Ia:c:=vTotal) dE (IIIb) (I0/v )(evt�/1) (IVb) (I0/(u�/1))tu�1 (Vb)

C and Q are equal to the adsorption faradaic capacitance and charge, Ca and Qa, respectively, when non-faradaic effects are not present (see Eqs.

(9) and (12)), or are equal to the total interfacial capacitance and charge, Cs and Qs, respectively, when the IPD model is considered (see Eqs. (18) and

(19)). v is the constant sweep rate in CV, vTotal is the total sweep rate in ACV given by Eq. (31). In Eqs. (IIb) and (IIIb) Ei and E are the initial and

variable potentials of the voltammetric sweeps. I0 is the current amplitude of the different programmed currents considered.

J. Gonzalez, A. Molina / Journal of Electroanalytical Chemistry 557 (2003) 157�/165 163

sweep rate (vd.c.�/jdEd.c./dt j), DE the a.c. peak ampli-

tude and v the angular frequency. For small values of

DE , the a.c. total current can be written as the sum of a

d.c. current Id.c. and a current corresponding to the firstharmonic of the a.c. current Iv (see Eqs. (16)�/(24) in

Ref. [20])

Ia:c:�Id:c:�Iv

� [vd:c:�vDE cos(vt)]n2F 2AGT

RT

eh

(1 � eh)2(30)

The term in brackets in the right hand side of Eq. (30)

is the time derivative of the potential applied to the

electrode (see Eq. (29)), and it can be considered as the

‘‘total sweep rate’’ for a.c. voltammetry, vTotal,

vTotal�dE(t)

dt�

dEd:c:

dt�

dEa:c:

dt

�vd:c:�vDE cos(vt) (31)

Thus, according to Eq. (31), the interfacial capaci-

tance and the interfacial charge are given by (see Eqs.

(IIIa) and (IIIb) in Table 1)

C(E)�Ia:c:

vTotal

Q(E)�gE

Ei

Ia:c:

vTotal

dE

�(32)

From Eq. (32) we can conclude that, as in the case

corresponding to CV and SCV, the ratio between the

current Ia.c. and the total sweep rate vTotal is equal to the

interfacial capacitance, and the integral of the (Ia.c./

vTotal)/E curve for a given value of E corresponds to theinterfacial charge Q (E ), which is given as a function of

the potential by Eq. (9) or Eq. (23).

3.5. Derivative chronopotentiometry with an exponential

current of the form I(t)�/I0 evt

In Refs. [21,22] we study the application of an

exponential current of the form I (t )�/I0 evt to deriva-

tive chronopotentiometry with a programmed currents

technique for studying reversible and irreversible elec-

troactive monolayers. In these references we plotted the

(devt /dE )/E response, which for the case of a reversible

redox monolayer is given by (see Eq. (18) in Ref. [21])

devt

dE�

n2F 2AGTv

RTI0

eh

(1 � eh)2(33)

If we take into account that for this type of current

applied it is fulfilled that C�(dQ=dE)½I(t)�I0 evt �(I0=v)(d(evt�1)=dE)�(I0=v)(devt=dE) (see Eq. (5b),and Eq. (IVa) in Table 1), with Q given by Eq. (IVb) in

Table 1, then it is evident that in derivative chronopo-

tentiometry with exponential currents the devt /dE curve

(or d(evt�/1)/dE ) is equivalent to the interfacial capa-

citance divided by the factor (I0/v).

3.6. Derivative chronopotentiometry with power current

of the form I(t)�/I0tu

In Refs. [21,23] we also study the application of a

power current of the form I (t)�/I0tu to derivative

chronopotentiometry technique in order to study rever-

sible and irreversible electroactive monolayers. In this

case, we used the (dtu�1/dE )/E derivative, which for the

case of a reversible redox monolayer is given by (see Eq.

(9) in Ref. [21])

dtu�1

dE�

n2F 2AGT(u � 1)

RTI0

eh

(1 � eh)2(34)

As in the previous case, we must take into account

that for this programmed current it is fulfilled that

C(E)�(dQ=dE)½I(t)�I0tu �(I0=(u�1))(dtu�1=dE) (seeEq. (5c), and Eq. (Va) in Table 1), with Q given by

Eq. (Vb) in Table 1. Thus, we can conclude that in

derivative chronopotentiometry with power time cur-

rents, the dtu�1/dE curve is equivalent to the interfacial

capacitance divided by the factor (I0/(u�/1)).

Hence, according to Eqs. (26)�/(34) obtained in

different electrochemical techniques for a redox mono-

layer which exhibits a reversible behaviour, we obtainthe expressions given in Eqs. (9) and (12) for the Qa/E

and Ca/E responses, respectively, when non-faradaic

effects are not present, whereas when IPD is taken into

account, Qs and Cs are given by Eqs. (20) and (23),

respectively, whatever be the electrochemical technique

used.

Note that when non-faradaic effects are taken into

account, the Qs/E curve given by Eq. (20) presents agreat sensitivity to the initial conditions through the

term ln V (with ln V��(nF=RT)(Einitial�E?0�(fP�fS)) and Einitial being the initial applied or measured

potential in the voltammetric or chronopotentiometric

technique used, respectively).

When the charge transfer reaction is not reversible,

both the Q /E and the C /E curves are time-dependent

and, therefore, the charge�/potential and thecapacitance�/potential responses depend on the tem-

poral sequence in which a specific perturbation has been

applied to the system, and hence different expressions

for the Q /E and C /E curves are found for different

electrochemical techniques.

4. Conclusions

Eqs. (9), (11), (20) and (23) show that the Qs/E and

the Cs/E responses are independent of the electroche-

mical technique (voltammetric or chronopotentio-

J. Gonzalez, A. Molina / Journal of Electroanalytical Chemistry 557 (2003) 157�/165164

metric), from which these curves have been obtained, i.e.

they present an identical variation with E , the double

layer parameters C1, hP and @fP/@E , the total surface

excess GT and the ratio of initial excesses V . Therefore,

the charge�/potential and capacitance�/potential curves

in Fig. 1 (without non-faradaic effects) and in Fig. 2

(with non-faradaic effects) are universal curves for

reversible redox monolayers.The simultaneous availability of both Qs/E and Cs/E

curves allows the total characterisation of a reversible

surface process. Thus, the standard potential of the

surface process E ?0 and the total excess GT can be easily

determined from the peak potential of the Cs/E curve

(see Figs. 1 and 2). The Qs/E curves allow us to

determine whether only one or both of the redox species

are initially adsorbed by means of a simple visual

inspection of the sigmoidal curves when both cathodic

and anodic currents are applied, because in the second

case the sigmoid shows an intersect with the potential

axis. When non-faradaic effects are not present, this

intersect corresponds to the adsorption cross potential,

Eacp, in agreement with Eq. (11), from which we can

calculate the initial surface excesses of A and B species,

GA,0 and GB,0, respectively. Experimental Cs/E data are

easily available from chronopotentiometric or voltam-

metric measurements. Nevertheless, for the obtention of

the experimental Qs/E data, chronopotentiometric tech-

niques are more adequate.

Acknowledgements

The authors greatly appreciate the financial support

provided by the Direccion General de Investigacion

Cientıfica y Tecnica (Project number BQU2000-0231)

and the Fundacion SENECA (Expedient number 00696/

CV/99).

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