Journal of

www.elsevier.com/locate/jelechem

Journal of Electroanalytical Chemistry 583 (2005) 184–192

ElectroanalyticalChemistry

Linear sweep voltammetric and chronopotentiometriccharge/potential curves for non reversible redox monolayers

Joaquın Gonzalez *, Angela Molina

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

Received 14 March 2005; received in revised form 2 June 2005; accepted 3 June 2005

Available online 12 July 2005

Abstract

A simple equation for the charge/potential (Q/E) dependence of an electroactive monolayer is presented. This equation is valid

for any chronopotentiometric or voltammetric electrochemical technique, whatever the degree of reversibility of the molecular film.

Particular expressions corresponding to linear sweep and cyclic voltammetry and to chronopotentiometry with a power time pro-

grammed current have been deduced, and analogies with the current/potential curves obtained in electrochemical systems under

semi-infinite diffusion mass transport control under transient and steady state conditions has been established. Reversible Q/E

curves present a time-independent and universal behaviour, which does not depend on the electrochemical technique employed

to obtain them. The irreversible charge/potential curves exhibit different behaviour depending on which electrochemical method

has been used. Easy methods for estimating thermodynamic and kinetic parameters of the electroactive film in both chronopoten-

tiometric and voltammetric techniques are proposed.

� 2005 Elsevier B.V. All rights reserved.

Keywords: Cyclic voltammetry; Linear sweep voltammetry; Chronopotentiometry; Programmed current; Quasi-irreversible monolayer; Diffusionless

systems

1. Introduction

Cyclic voltammetry and cyclic reciprocal derivative

chronopotentiometry have been found to be very useful

for studying the electrochemical behaviour of diffusion-less systems like modified electrodes and electroactive

monolayers such as those formed by thiols, antraqui-

none and viologen derivatives, inorganic polymers and

biomolecules (peptides, nucleic acids and proteins),

among others [1–12]. These systems present a great

interest on account of the fact that they enable a molec-

ular-level control of both the nature of the chemical

functional groups attached to the electrode surface andalso their topology [2,11,12].

0022-0728/$ - see front matter � 2005 Elsevier B.V. All rights reserved.

doi:10.1016/j.jelechem.2005.06.001

* Corresponding author. Tel.: +34 68 367429; fax: +34 68 364148.

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

In previous papers [6,7,13–15], the general theoretical

treatment of these systems, applicable when the elec-

trode is polarised by the application of a programmed

current or by the application of a time variable poten-

tial, has been carried out by considering even the contri-bution of nonfaradaic effects.

In order to analyze the behaviour of the charge/po-

tential (Q/E) responses of electroactive ultrathin layers,

we present in this paper a simple equation which is

applicable independently of the degree of reversibility

of the charge transfer process, and is valid when any

electrochemical technique (voltammetric or chronopo-

tentiometric) is used. From this equation we have de-duced that the expression of the Q/E dependence

corresponding to a reversible process is independent of

the electrochemical technique employed, in agreement

with previous results reported in [13–15]. We have also

deduced the charge/potential curves corresponding to

J. Gonzalez, A . Molina / Journal of Electroanalytical Chemistry 583 (2005) 184–192 185

slow charge transfer processes when using linear sweep

and cyclic voltammetry and also chronopotentiometry

with programmed current, and we show the differences

between both techniques.

Moreover, from the above, the following has been

established:

� for reversible reactions, the dependence of the norma-

lised interfacial charge with the potential is similar to

that found in: (a) the voltammetric measured norma-

lised current in semi-infinite diffusion (planar, spher-

ical or cylindrical) both in transient and steady state

conditions [3,16–18]; (b) the chronopotentiometric

normalised applied current under mass transportcontrol conditions [19].

� for totally irreversible processes, this equivalence can-

not be established, although the E/ln((QF � Q)/Q)

curves for electroactive films obtained from chrono-

potentiometric data, are formally similar to those

corresponding to the transient voltammetric totally

irreversible waves for semi-infinite planar diffusion

[3,20,21], and for steady state conditions [3,17] (E/ln((Id � I)/I) curve).

On the basis of the above, we propose easy methods

for estimating thermodynamic and kinetic parameters of

the electroactive film, i.e., the surface formal potential,

the surface rate constant and the charge transfer coeffi-

cients. These methods are, in both the above mentioned

extreme cases, similar to the well known ones used forreversible and totally irreversible charge transfer reac-

tions under mass transport control conditions [3,21,22].

2. Theory

2.1. General charge/potential curves corresponding to

electroactive monolayers for any reversibility degree of

the electron transfer process

We will analyse the following surface electrode pro-

cess taking place in an electroactive monolayer,

AðadsorptedÞ þ ne� ¡kred

koxBðadsorptedÞ. ðIÞ

We assume for the adsorbate monolayer that the

Langmuir isotherm 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 timescale of the experiment. We will not consider the pres-

ence of nonfaradaic effects.

In these conditions, the following assumptions are

fulfilled:

� The total excess CT is constant during the whole

experiment, in such a way that,

fCT ¼ CA;0 þ CB;0 ¼ CAðtÞ þ CBðtÞ; ð1Þwith Ci,0 and Ci(t), i = A or B, being the initial surfaceexcesses and the surface excesses corresponding to a

time t of i species, respectively.

� The adsorption coefficients of both electroactive spe-

cies and the maximum surface coverage are indepen-

dent of the potential.

The Faraday condition for the electroactive mono-

layer can be expressed as,

IðtÞ ¼ �QF

dfAdt

; ð2Þ

with I(t) being the cathodic applied current in a chrono-potentiometric technique, or the cathodic measured cur-

rent in the case of a voltammetric technique such as

cyclic voltammetry. Moreover, QF is the maximum far-

adaic charge given by,

QF ¼ nFSCT; ð3Þ

with S being the electrode area, and fi being the fractionof surface coverage of i species,

fi ¼Ci

CT

i ¼ A or B. ð4Þ

The relationship between the faradaic charge con-

verted at a time t, Q(t), and the current I(t) is,

IðtÞ ¼ dQdt

. ð5Þ

By taking into account Eq. (5), we can integrate Eq. (2)

and obtain,

fA ¼ fA;0 � QðtÞQF

fB ¼ fB;0 þ QðtÞQF

9=;; ð6Þ

with fi,0(=Ci,0/CT), i = A or B, being the surface coverage

fraction of the i species corresponding to the beginning

of the application of the electrical perturbation (currentor potential). For the sake of simplicity, we will consider

that fA,0 @ 1 and fB,0 @ 0, which corresponds to the most

frequent experimental situation.

The current–potential relationship for the reduction

of the surface bound molecules is given by [4],

IðtÞQFk

0¼ fA e�ag � fB eð1�aÞg; ð7Þ

g ¼ nFRT

ðE � E00 Þ; ð8Þ

k0, a and (1 � a) are the rate constant and the cathodic

and anodic charge transfer coefficients of the surface

charge transfer reaction, respectively, whereas E00 is

the formal potential of the surface process.By taking into account Eqs. (5) and (6), we can re-

write Eq. (7) in the following way:

186 J. Gonzalez, A . Molina / Journal of Electroanalytical Chemistry 583 (2005) 184–192

d lnQdt

1

k0eag ¼ QF � Q

Q� eg. ð9Þ

Eq. (9) is completely general such that it can be used for

any particular voltammetric or chronopotentiometric

electrochemical technique. From this equation we can

obtain the following particular cases:

(a) Reversible process (k0 ! 1)In this case, the left hand side of Eq. (9) becomes neg-

ligible and we obtain,

QQF

¼ 1

1þ eg; ð10Þ

which can be written as:

E ¼ E00 þ RTnF

lnQF � Q

Q

� �. ð11Þ

Eqs. (10) or (11) coincide with Eq. (9) in [14], and are of

great importance since they are valid for any electro-

chemical technique, i.e., the charge/potential responsefor a reversible process is universal. The Q/E curves gi-

ven by Eqs. (10) and (11) present a sigmoidal-type fea-

ture which allows us to obtain the QF value, and

therefore the total surface excess CT, at sufficiently neg-

ative potentials of the cathodic response. Moreover, the

half wave potential of these curves, deduced by making

Q = QF/2 in Eqs. (10) and (11), coincides with the formal

surface potential of the electroactive couple, E00 (seeTable 1).

Note that the Q/E dependence shown in Eqs. (10) or

(11) for reversible processes taking place in a diffusion-

less system is identical to the following responses:

� the voltammetric normalised current/potential curve

((I/Id)–E) corresponding to a charge transfer process

under mass transport control for linear, sphericaland cylindrical semi-infinite diffusion transient and

steady state conditions (see for example, Eqs.

(5.4.22), (8:7:9) and (9) of Chapter VII in [3], [17]

and [20], respectively);

Table 1

Expressions for the cathodic ðE1=2c Þ and anodic ðE1=2

a Þ half wave potentials

processes

Reversible processes Totally irreve

Any electrochemical

technique

Chronopoten

programmed

form I(t) = ±

Cathodic E00 ð1Þ E00 þ RTanF

ln v

Anodic

E00 � RTð1� aÞn

� the chronopotentiometric normalised time/potential

curve ((t/s)1/2/E) corresponding to the application of

a constant current to a planar electrode (see for

example, equation (8.3.1) in [3]);

� the chronopotentiometric normalised applied cur-

rent/potential curve ((JuI(t)/Id)/E) corresponding tothe application of a power current to a spherical elec-

trode of any radius (see Eq. (6) in [19]).

This equivalence is due to two main reasons: firstly, in

an electroactive monolayer the normalised charge is pro-

portional to the difference between the total and reactant

surface excesses ((Q/QF) � (CT � CA)), and in electro-

chemical systems under mass transport control, thevoltammetric normalised current and/or the chronopo-

tentiometric normalised applied current are proportional

to the difference between the bulk and surface concentra-

tions (ðI=IdÞ / ðc�A � cAðsurface;EÞÞ and JuIðtÞ=Id /ðc�A � cAðsurface; tÞÞ, respectively) [3,19,20]. Secondly, areversible diffusionless system fulfils the following

conditions,

CA þ CB ¼ constant

CA=CB ¼ enF ðE�E00 Þ=RT

); ð12Þ

and the same conditions must be fulfilled by the concen-

trations cA and cB when the process takes place undermass transport control, both in voltammetric [16,18],

and in chronopotentiometric techniques [19,22], when

the diffusion coefficients of both species are equal

(DA = DB).

(b) Irreversible process (k0 � 1 s�1)

In this situation, from Eq. (9) we deduce,

d lnQdt

1

k0eag ¼ QF � Q

Q. ð13Þ

As can be observed from this equation, the charge/

potential response for irreversible monolayers containsa temporal dependence through the term d lnQ/dt. The

particular form of this term is strongly influenced by

the type of electrochemical technique employed and,

therefore, Eq. (13) leads to Q/E curves which depend

on the chronopotentiometric or voltammetric technique

employed.

of the Q/E curves corresponding to reversible and totally irreversible

rsible processes

tiometry with a

current of the

I0tu

Cyclic voltammetry

p;1=2 ð2Þ E00 � RTanF

lna ln 2jmj

� �ð3Þ

Fln vp;1=2 ð4Þ E00 þ RT

ð1� aÞnF lnð1� aÞ ln 2

m

� �ð5Þ

J. Gonzalez, A . Molina / Journal of Electroanalytical Chemistry 583 (2005) 184–192 187

Note that, although we have obtained the charge/po-

tential expressions given by Eqs. (9)–(11) and (13), for a

cathodic process, no new analysis is needed to obtain

those corresponding to an anodic one since no cumula-

tive depletion effects have to be considered in the treat-

ment. Therefore, for a subsequent oxidation processafter the complete reduction of the substrate, Q changes

to negative, and we must interchange a by �(1 � a), andeg by e�g in Eqs. (9), (10) and (13).

2.2. Linear sweep and cyclic voltammetric Q/E curves

In this technique, the electrode is polarised by a po-

tential ramp of slope, or voltammetric sweep rate,equal to v = �(dE/dt), being the current the magnitude

measured. In this case, the charge can be easily ob-

tained as:

Q ¼Z E

Ei

IvdE; ð14Þ

with Ei being the initial potential of the voltammetric

sweep which will be considered as a very positive va-

lue for the cathodic sweep. In this case, it is fulfilled

that:

d lnQdt

¼ vd lnQdE

. ð15Þ

By introducing Eq. (15) in Eq. (9), and taking into ac-

count Eq. (8), we deduce:

dQdg

þ mQðe�ag þ eð1�aÞgÞ ¼ m e�agQF; ð16Þ

where m is the voltammetric dimensionless surface rate

constant which, for a cathodic sweep, is [4,23]:

m ¼ � k0

jvjRTnF

. ð17Þ

Eq. (16) is a first order linear differential equation

from which we obtain the following nonexplicit expres-

sion for the charge as a function of the applied potential:

QQF

¼ m emae

�age�

m1�ae

ð1�aÞgZ g

gi

e�mae

�age

m1�ae

ð1�aÞge�ag dg; ð18Þ

with gi ¼ nF ðEi � E00 Þ=ðRT Þ.For a reversible process jmj � 1 and, in these condi-

tions, Eq. (18) behaves in an identical way to Eq. (10)

since, as has been mentioned above, the reversible Q/E

curve is universal.For totally irreversible processes jmj � 1 (kf � kb),

and for the usual experimental situation at which we

use a very positive initial potential in the cathodic sweep

(i.e., we suppose that gi ! 1), Eq. (18) becomes,

QQF

¼ 1� emae

�ag; ð19Þ

which can be written as,

E ¼ E0 � RTanF

lnajmj

� �� RTanF

ln lnQF

QF � Q

� �� �.

ð20ÞAccording to Eqs. (19) and (20), the irreversible voltam-metric Q/E curve presents a Gompertz-type feature [24],

which allows us to obtain the value of QF at sufficiently

negative potentials of the cathodic charge potential

curve. Moreover, the half wave potentials of these

curves can be deduced by making Q = QF/2 in Eqs.

(10) and (19) corresponding to reversible and totally

irreversible processes, respectively, with their values

being given in Table 1. Note that for reversible processesthe half wave potential E1/2 coincides logically with the

chronopotentiometric value.

For irreversible processes, it is not possible to estab-

lish any formal analogy between the normalised

charge/potential curves obtained from cyclic or linear

sweep voltammetry (see Eq. (18)), and the voltammet-

ric current/potential ones obtained for mass transport

control as in the chronopotentiometric case (seeabove).

The voltammetric capacitative/potential curves (C/E

curves), can be obtained by differentiating (Q/QF) with

respect to the potential.

Thus, for reversible processes, the C/E dependence is

given by

1

QF

C ¼ dðQ=QFÞdE

�������� ¼ nF

RTeg

1þ egð Þ2. ð21Þ

This expression coincides with Eq. (12) deduced in [14],

since under these conditions, both the charge and the

capacitance of the monolayer are universal responses.

For an irreversible charge transfer process, we deducefrom Eq. (19) the following,

1

QF

C ¼ dðQ=QFÞdE

�������� ¼ nF

RTme

mae

�age�ag. ð22Þ

This expression coincides, within known multiplicative

constants, with the current obtained for an irreversible

monolayer in linear sweep voltammetry (see Eq. (18)

in [23]).

2.3. Chronopotentiometric Q/E curves by using a

programmed current of the form I(t) = I0tu

In this case, a programmed current which varies

with a power of time is the electrical perturbation ap-

plied and, by integrating Eq. (5), we deduce the fol-

lowing relationship between the applied current and

the charge:

QðtÞ ¼ I0tuþ1

ðuþ 1Þ ; ð23Þ

and therefore,

188 J. Gonzalez, A . Molina / Journal of Electroanalytical Chemistry 583 (2005) 184–192

d lnQdt

¼ uþ 1

t. ð24Þ

By substituting Eq. (24) into Eq. (9) we obtain:

QQF

¼ 1

1þ eg þ ð1=vpÞeag; ð25Þ

where vp is the chronopotentiometric dimensionless sur-

face rate constant, which is given by

vp ¼ ðk0tÞ=ðuþ 1Þ. ð26Þ

Note that all the above expressions, corresponding to

the application of a power time current of the form

I(t) = I0tu, are also applicable when a constant current

I(t) = I0 is employed by making u = 0 in Eqs. (23), (24)

and (26).

The general expression for the charge/potential curve

(see Eq. (25)), takes two simpler forms, depending onthe value of vp.

For a reversible process, vp � 1, and Eq. (25) for the

charge/potential curves becomes identical to Eq. (10).

Contrarily, when vp � 1 the process behaves as to-

tally irreversible and Eq. (25) simplifies to:

QQF

¼ 1

1þ ð1=vpÞeag; ð27Þ

which can be written as:

E ¼ E1=2irr;p þ

RTanF

lnQF � Q

Q

� �; ð28Þ

with

E1=2irr;p ¼ E00 þ RT

anFln vp. ð29Þ

Note that the totally irreversible E/Q relationship shown

in Eq. (28) is formally similar to that corresponding to

the voltammetric E/I one obtained for semi-infinite dif-

fusion controlled conditions (see for example Eq.

(3.21a) in [21]), or to that corresponding to steady state

conditions (see for example Eqs. (5.5.48) and (8:7:15) in

[3] and [17], respectively).Despite the similarity between the totally irreversible

voltammetric I/E responses obtained in systems under

mass transport control and the Q/E ones corresponding

to electroactive monolayers, there exists an important

difference between them due to the fact that the time t

is constant in the voltammetric current/potential curves,

whereas t varies through the whole experiment in Chro-

nopotentiometry. Therefore, the plots of E vs.ln((QF � Q)/Q) (see Eq. (28)), will not be linear. In order

to obtain easily the value of the surface rate constant k0

from linear plots, it is necessary to manipulate Eq. (28)

in the way proposed in Section 3.

Note also that from Eq. (25) for the Q/E curve, the

capacitative/potential (C/E) one can be deduced by dif-

ferentiating (Q/QF) with respect to the potential.

Thus, for reversible processes, the C/E dependence

is also given by Eq. (21), whereas for an irreversible

charge transfer process, by taking into account the

expression of the potential time curve when a power

time current I(t) = I0tu is applied (see Eq. (9) in [7]),

we can deduce:

1

QF

C ¼ dðQ=QFÞdE

�������� ¼ ðuþ 1Þ2

sk0anFRT

T ð2uþ1Þ=ðuþ1Þeag

T þ u; ð30Þ

with T = (t/s)u+1 and s being the transition time of the

chronopotentiogram [7].

Eqs. (21) and (30), obtained for a power time currentof the form I(t) = I0t

u, are also applicable when a con-

stant current I(t) = I0 is employed, by making u = 0.

Thus, whereas Eq. (21) for a reversible process remains

unaffected, Eq. (30) becomes,

1

QF

C

����IðtÞ¼I0

¼ 1

sk0anFRT

eag. ð31Þ

Note that Eqs. (21) and (30), (31) corresponding to the

capacitance potential responses, are identical to those

corresponding to reciprocal derivative chronopotenti-

ometry ((dT/dE)/E curves), with a power of time currentI(t) = I0t

u, which are given in Eqs. (14) and (24) of [7],

within known multiplicative constants. It is also of inter-

est to highlight that, for a totally irreversible process, the

C/E curve does not present a peak when a constant cur-

rent is applied (u = 0, see Eq. (31)).

3. Results and discussion. Behaviour of the voltammetricand chronopotentiometric charge/potential curves

In Fig. 1(a) we have plotted the normalised (Q/QF)–

(E � E0) curves corresponding to Chronopotentiometry

with a programmed current of the form I(t) = ±I0tu,

with u = 1/4. These curves have been obtained by calcu-

lating value of the charge for each value of time from

Eq. (23) and then, by substituting it in Eq. (25), the cor-responding potential is numerically determined. From

the experimental point of view, we must obtain first

the E/t curve or chronopotentiogram, and later we de-

duce the charge for each experimental value of the po-

tential from Eq. (23).

In Fig. 1(b) we have plotted the normalised (Q/QF)–

(E � E0) curves corresponding to cyclic voltammetry,

calculated by solving numerically Eq. (18) for each valueof the applied potential. In order to obtain the experi-

mental voltammetric Q/E responses, we must integrate

the ratio between the measured current and the sweep

rate with respect to the potential, since I/v = dQ/dE

(see Eq. (14)).

The curves in Fig. 1(a) and (b) have been calculated

for different values of the dimensionless surface rate con-

stant, which in the chronopotentiometric case we have

Fig. 1. (Q/QF)–(E � E0) curves corresponding to Chronopotentiome-

try with a programmed current of the form I(t) = ±I0tu, with u = 1/4

(Fig. a, see Eq. (25)), and to cyclic voltammetry (Fig. b, see Eq. (18)).

In the chronopotentiometric case I0 = 1.0 lA s�1/4 and QF = 0.25 lC.a = 0.5, n = 1 and T = 298 K. The values of the dimensionless

chronopotentiometric (vp,1/2) and voltammetric (m) surface rate

constants appear on the curves.

J. Gonzalez, A . Molina / Journal of Electroanalytical Chemistry 583 (2005) 184–192 189

defined as vp,1/2 = (k0t1/2/(u + 1)), with t1/2 being the time

at which the charge is equal to QF/2. This value can be

easily obtained from Eq. (23) and is given by

t1=2 ¼ðuþ 1ÞQF

2I0

� �1=ðuþ1Þ

. ð32Þ

In cyclic voltammetry, the dimensionless surface rateconstant is given by m = (RTk0)/(nFv). We have ob-

tained both chronopotentiometric and voltammetric sets

of charge/potential curves for the same values of the

dimensionless surface rate constants, and also for

a = (1 � a) = 0.5.

The (Q/QF)–(E � E0) curves in these figures show

that, for reversible processes (see curves with vp,1/2 =m = 102), both cathodic and anodic curves present thesame half wave potential, which coincides with E00 in

line with Eq. (1) in Table 1, independently of the electro-

chemical technique employed, as we have previously

pointed out [13–15]. When the dimensionless surface

rate constant, and therefore the reversibility degree of

the process, decreases, both cathodic and anodic curves

are shifted towards more negative and positive poten-

tials, respectively, in such a way that the separation be-

tween them increases. If we compare the effect of thedecrease of vp,1/2 and of m on these responses, we can

see that the cathodic and anodic voltammetric (Q/QF)–

(E � E0) curves of Fig. 1(b) appear, respectively, at

more positive and negative potentials than the chrono-

potentiometric ones of Fig. 1(a) (compare, for example,

the curves with vp,1/2 = m = 0.01).

We can better understand this fact by considering

that, for example, when m = 1, jvj = 0.026 V s�1 ifn = 1 and T = 298 K, and therefore the temporal win-

dow of the experiment is of approximately 20 s for a

total sweep of 0.5 V, whereas for the chronopotentio-

metric case, when vp,1/2 = 1, the temporal window of

the experiment, given by the transition time value, is

of approximately 0.4 s under these conditions (see [7]),

i.e., the voltammetric time window is nearly fifty times

greater than the chronopotentiometric one.It is possible to establish the limit values of the chro-

nopotentiometric and voltammetric dimensionless rate

constants, vp,1/2 = (k0t1/2/(u + 1)) and m = (RTk0)/(nFv),

respectively, for which the electrode process behaves as

reversible, quasi-irreversible or totally irreversible, by

following the variation of the cathodic and anodic half

wave potentials, E1=2c and E1=2

a , with the logarithm of

vp,1/2 and m. In Fig. 2 we have plotted the differencesðE1=2

c � E0Þ and ðE1=2a � E0Þ, calculated by numerically

solving Eqs. (18) and (25), which correspond to cyclic

voltammetry (dashed lines), and to Chronopotentiome-

try with a programmed current of the form I(t) = ±I0tu

(solid lines), respectively, for the particular case

Q = QF/2 with a = (1 � a) = 0.5.

From the curves in this figure we can see that for

low values of the dimensionless rate constant,E1=2c and E1=2

a vary linearly with log vp;1=2 or logm(logm < �1.25, see dashed lines, and log vp;1=2 <�0.57, see solid lines), a fact which indicates that the

process behaves as totally irreversible, in agreement

with Eqs. (2)–(5) in Table 1, and under these condi-

tions equations (19) and (27) can be used for treating

the voltammetric or chronopotentiometric responses,

respectively. For intermediate values of the dimension-less rate constants (�1.25 < logm < 0.65, see dashed

lines, and �0.57 < log vp;1=2 < 0.90, see solid lines),

the variation of both half wave potential with

log vp;1=2 or logm deviates from linearity. Therefore,

in this interval the process behaves as quasi-irreversible

and general equations (18) and (25) must be used for

describing the charge–potential responses of the

system. Finally, for high values of vp,1/2 or m

(logm > 0.65, see dashed lines, and log vp;1=2 > 0.90,

see solid lines), both half wave potentials coincide with

Fig. 3. (a) ðQ=QF Þ–ðE � E00 Þ curves corresponding to a totally

irreversible process in Chronopotentiometry with a programmed

current of the form I(t) = ±I0tu, calculated from Eqs. (23) and (27).

(b) ðE � E00 Þ– lnððQF � QÞ=Qu=uþ1Þ curves obtained for the charge/

potential ones of Fig. 3(a). k0 = 0.1 s�1, I0 = 1.0 lA s�u. The values of

the exponent u appear on the curves. Other conditions as in Fig. 1.

Fig. 2. Dependence of ðE1=2c � E00 Þ and ðE1=2

a � E00 Þ with log(vp,1/2) inChronopotentiometry with a programmed current of the form

I(t) = ±I0tu (calculated from Eq. (25) for the particular case Q = QF/2,

solid lines), and with log(m) in cyclic voltammetry (calculated from Eq.

(18) for the particular caseQ = QF/2, dashed lines). Other conditions as

in Fig. 1.

190 J. Gonzalez, A . Molina / Journal of Electroanalytical Chemistry 583 (2005) 184–192

E0 with an error of less than 3 mV and the electrode

process behaves as reversible in line with Eq. (1) in Ta-

ble 1 and, in this case, simplified Eq. (10) can be used

to characterise the electrode process. The difference be-

tween cyclic voltammetry and chronopotentiometry

can be clearly seen in this figure and is related withthe limit values of the dimensionless rate constant for

which we can establish a particular reversibility behav-

iour, in such a way that for a given value of vp,1/2 = m,

the process will appear as more reversible under vol-

tammetric conditions (compare dashed and solid lines).

In Figs. 3 and 4 we have plotted the chronopotentio-

metric Q/E curves corresponding to a totally irreversible

process (vp,1/2 6 0.1, see Eq. (27)) when a programmedcurrent which varies with a power of time, I(t) = ±I0t

u,

is applied. We have obtained these curves for different

values of the exponent u in the programmed current

with I0 constant (Fig. 3(a)), and also for different values

of I0 with a fixed value of u (Fig. 4(a)). From these fig-

ures it can be seen that the decrease of u or the increase

of I0 shifts both cathodic and anodic responses towards

more negative and positive potentials, respectively, inline with Eqs. (2) and (4) in Table 1.

Moreover, the decrease of u distorts the charge/po-

tential curves of Fig. 3(a) in such a way those corre-

sponding to the application of a current step (see

curves with u = 0), appear as truncated sigmoids.

We have previously discussed that for totally irrevers-

ible processes, the chronopotentiometric Q/E curves can

be described by Eq. (19). If we take into account that the

relationship between the charge and the time of applica-

tion of the programmed current is (see Eq. (23)),

t ¼ ðuþ 1ÞQI0

� �1=ðuþ1Þ

; ð33Þ

we can write Eq. (28) in the following way,

E¼ E0þ RTanF

lnk0

ððuþ 1ÞuI0Þ1=uþ1

!þ RTanF

lnQF �Q

Qu=ðuþ1Þ

� �.

ð34ÞTherefore, the plots of the chronopotentiometric mea-

sured potential versus ln((QF � Q)/Qu/(u+1)) should be

linear. Thus, in Figs. 3(b) and 4(b) we have shown theplots corresponding to the curves in Figs. 3(a) and

4(a), respectively. Note that, according to Eq. (34), for

the cathodic curves we obtain parallel lines of slope

Fig. 5. (a) ðQ=QFÞ–ðE � E00 Þ curves corresponding to a totally

irreversible process in cyclic voltammetry calculated from Eq. (19).

(b) ðE � E00 Þ– lnðlnðQF=ðQF � QÞÞÞ curves obtained for the charge/

potential ones of Fig. 5(a). The values of the voltammetric sweep rate v

(in V s�1) appear on the curves. Other conditions as in Fig. 3.

Fig. 4. (a) ðQ=QFÞ–ðE � E00 Þ curves corresponding to a totally

irreversible process in Chronopotentiometry with a programmed

current of the form I(t) = ±I0tu, u = 1/10, calculated from Eqs. (23)

and (27). (b) ðE � E00 Þ– lnððQF � QÞ=Qu=uþ1Þ curves obtained for the

charge/potential ones of Fig. 4(a). The values of I0 (in lA s�1/10)

appear on the curves. Other conditions as in Fig. 3.

J. Gonzalez, A . Molina / Journal of Electroanalytical Chemistry 583 (2005) 184–192 191

RT/(anF) and intercepts which depend on the values of u

and I0, although the influence of the exponent u is smal-

ler than that exerted by I0.

By combining the expressions for the cathodic and

anodic charge/potential curves, the values of E0 and

k0, and also those corresponding to a and (1 � a) canbe immediately obtained independently, in a similar

way to that previously reported in [22]. Thus, if thereare kinetic complications, the sum of a + (1 � a) will

be different from unity and such complications can

therefore be detected.

Finally, In Fig. 5(a) we have plotted the Q/E curves

obtained in cyclic voltammetry for a totally irreversible

process (m 6 0.05, see Eq. (19)) with different values of

the sweep rate v. From these curves it can be deduced

that the increase of v shifts both cathodic and anodic re-sponses towards more negative and positive potentials.

For cyclic voltammetry, the logarithm analysis of

totally irreversible Q/E curves can only be carried out

by using a very positive initial potential in the case

of a cathodic sweep, and a very negative one in thecase of an anodic sweep. Under these conditions, the

plots of the voltammetric applied potential versus

ln(ln(QF/(QF � Q))) should also be linear in line with

Eq. (20). In Fig. 5(b) we have plotted these curves

obtained from the voltammetric Q/E curves of Fig.

5(a). From the intercepts and slopes of both plots the

values of a, 1 � a, E0 and k0 can be immediately ob-

tained in a similar way to that previously discussedfor Chronopotentiometry.

4. Conclusions

We have obtained a simple equation which allows us

to deduce the charge/potential (Q/E) dependence of an

electroactive monolayer for any chronopotentiometricor voltammetric electrochemical technique and whatever

the degree of reversibility of the molecular film. From

192 J. Gonzalez, A . Molina / Journal of Electroanalytical Chemistry 583 (2005) 184–192

this general equation we have deduced particular expres-

sions corresponding to Linear Sweep and cyclic voltam-

metry and to chronopotentiometry with a power time

programmed current, and we have highlighted the differ-

ences between both techniques.

The Q/E curves corresponding to reversible processesdo not depend on the electrochemical technique em-

ployed to obtain them, and they present a similar depen-

dence on the potential to that shown by the current

obtained under mass transport control.

For nonreversible processes, the Q/E curves contain a

temporal dependence which is strongly dependent of the

type on electrical perturbation employed and therefore,

these responses will exhibit different behaviours, depend-ing on which electrochemical method has been used.

The chronopotentiometric Q/E curves corresponding

to totally irreversible processes are formally identical to

the totally irreversible voltammetric I/E waves obtained

in systems under mass transport control, although the

time t plays a very different role in voltammetric and

chronopotentiometric conditions.

Finally, we propose easy methods for determining ki-netic and thermodynamic parameters of the monolayer

from voltammetric and chronopotentiometric data from

linear plots.

Acknowledgements

The authors greatly appreciate the financial supportprovided by the Direccion General de Investigacion

Cientıfica y Tecnica (Project number BQU2003-04172)

and by the Fundacion SENECA (Expedient number

PB/53/FS/02).

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