Reversible charge transfer with unknown mass transfer. Characterization and modelling by using a mass-transfer operator determined from experimental data

Journal of Electroanalytical Chemktry, 375 (1994) 215-292

275

Reversible charge transfer with unknown mass transfer. Characterization and modelling by using a mass-transfer determined from experimental data F. Miomandre

operator

and E. Vieil

Laboratoire d’Electrochimie Mol&ulaire, Service d’Etudes des Systt?mes et Architectures Mol&ulaires, De’partement de Recherche Fondamentale sur la Mat&e Condensie, Centre d’Etudes Nuclt!aires de Grenoble, C.E.A., 38054 Grenoble Cedex 9 (France)

(Received 19 October 1993; accepted 1 March 1994)

Abstract When a charge transfer is reversible and when the two associated mass-transfer steps are of the same nature, the electrical current is formulated very simply as a result of the application of a single mass-transfer operator on a potential-dependent equilibrium constant. The charge-transfer equilibrium constant is then obtained from the current by inversion of this equation. With such a “reverse modelling” approach, the characteristics of this operator can be experimentally measured and the charge-transfer equilibrium constant can be determined for any kind of thermodynamic equilibrium, nemstian or not. Consequently, the response of an electrochemical mechanism to any perturbation function can be modelled without mathematical description of the mass transfer. It is sufficient to measure experimentally, and correct adequately for instrument limitations and double-layer charging, the response of the electrochemical system to a series of steps, and then to use the convolution tool, to obtain “theoretical” working curves. Conversely, analysis of an experimental response to any kind of perturbation can also be performed by convolution with experimentally determined operator characteristics. The main advantage of this new methodology is the ability to model or to analyse complex mechanisms for which a good mathematical description of the mass transfer is not available, or when the mass transfer mode is unknown. This powerful theory is extremely simple. Apart from the convolution product, the mathematics is easy and reduced to a minimum. The experimental verification of the proposed approach is performed without sophisticated instrumentation on the N-methylphenothiazine redox couple in acetonitrile with a mathematically poorly defined mass transport such as transient convection.

1. Introduction Modelling an electrochemical mechanism generally requires a detailed knowledge of each step composing the mechanism. Hypotheses concerning the dependence of the kinetic or equilibrium constants on the potential, time, hydrodynamic conditions, electrode geometry, and so on, need to be formulated beforehand to establish a model of the dependence of the electrical current on these parameters. The determination of the characteristics of each step from experimental data is only possible if the hypotheses are correct. For example, the determination of a standard or half-wave potential of a reversible redox couple requires the formulation of an equilibrium model, the Nernst equation, for the charge transfer, and a rigorous knowledge of the mass transfer; i.e. the electrode geometry, electrolyte properties, hydrodynamic regime, electrical field, transient or stationary perturbation. In most 0022-0728/94/$7.00 SSDZ 0022-0728(94)03415-Y

cases, such as simple redox couples soluble in liquid electrolytes or homogeneous rigid solids, numerous models were developed many years ago. The characterization of a charge-transfer step needs to be done in the framework of existing mass-transfer models, such as semi-infinite diffusion, restricted-space diffusion, micro electrodes and stationary convection, and for a few definite electrode geometries, such as disks, planes, bands, hemispheres, etc. This is the classical approach of direct modelling. However, when the electrochemical system does not fall into these delimited cases, as in the case of modified electrodes, redox or conducting polymer films, passive or intercalation layers, which involve complicated mass transfer, the question arises of having at one’s disposal an analytical tool able to overcome the lack of model. Without a model for producing a theoretical curve to compare with the experimental one, it is necessary to use a “reverse modelling” approach, 0 1994 - Elsevier Science S.A. All rights reserved

276

F. Miomandre, E. vied / Reversible charge transfer with unknown mass transfer

which consists in the processing of the experimental data to extract back the kinetic characteristic looked for. We have recently [l] modified the concept of the mass-transfer operator, already known in electrochemistry [2-41, by extending its scope to any kind of mass transfer. This extension opens new perspectives in the field of electrochemical modelling in enabling the treatment of ill-defined mass transfer. Classical kinetic equations can be employed in the modelling of any first-order mechanism by writing them with this operator instead of classical kinetic constants. For a class of linear mass transfer, defined by the fact that they obey linear differential equations, we have shown [5] that this operator was equivalent to a convolution with a function characteristic only of the mass transfer. The convolution tool has now become popular among electrochemists [6-161 and is recognized as a considerable improvement by allowing any temporal dependence of the perturbation, potential or current, applied to the electrode, to be taken into account. Nevertheless this tool has always been used with known theoretical functions, either for predicting theoretical current or potential responses or for analysing experimental data, that is to say in the framework of a known mass transfer. The present study extends the theory to the case where the mass transfer is unknown, without any model describing its characteristic function. It is assumed only that the mass transfer is linear and coupled to a reversible charge transfer, also that both members of the redox couple undertake the same kind of masstransfer process. Nevertheless, they may have unequal physical characteristics such as diffusion coefficients, mobilities, sizes, etc. The charge transfer does not need to be nernstian: it can be of any type, for example influenced by electrostatic interactions in a solid medium, and its characteristic can be determined from experimental data by following the proposed approach. This determination is an interesting consequence of the theory, but not the main purpose, which is to demonstrate that the characteristic functions of an unknown linear mass transfer can be evaluated experimentally by using classical step techniques, chronoamperometry and chronopotentiometry. These basic techniques have been known for a long time in electrochemistry under the name current - or potential sampling voltammetry, but only used in very well defined mass-transfer conditions [17], in contrast to the very general case presented here. This process does not require exact use of the convolution tool itself; only its justification relies on the convolution concept. Once the characteristic functions have been determined, and possibly corrected for double-layer charging and for instrumental imperfections of the step signals, they can

be used for convoluting experimental curves, as is done classically. To demonstrate the theory experimentally, we have chosen to study N-methylphenothiazine in acetonitrile, known to be a reversible system on a platinum electrode [ 181:

Redox system of N-methylphenothiazine

The redox potential E” of this couple in acetonitrile is 0.36 V vs. Ag I 10-l M Ag+. (All potentials will be referred to this electrode.) The cation radical is stable in this medium and can be further oxidized at a potential of 0.91 V [19]. As an example of unknown mass transfer, we have chosen the transient convection on a rotating electrode, which is not totally unknown, but modelled only with some approximations [20-221. In fact we have not used any theoretical expression describing this mass transfer. Alternatively we could have taken simpler ones such as the classic semi-infinite diffusion without invoking any square-root dependence with time, but the demonstration might have been less convincing. In the first part of this article, we introduce the convolution formalism as well as the two characteristic functions that take into account any mass-transfer mode. We explain the procedure that enables the experimental determination of these functions, and that leads, on the one hand, to the prediction of the current response to any potential signal or of the potential response to any current perturbation, and on the other hand, to the determination of the chargetransfer features from experimental potential or current response to any perturbation. We give the example of cyclic voltammetry in this ill-defined mass transfer, for which all functions involved in the convolution can be measured from chronoamperometric and chronopotentiometric data. In the second part, we apply this methodology to our electrochemical system and compare the results obtained with the experimental response. Incidentally, a means of identifying purely diffusional temporary behaviour during the evolution of mass transfer will be also presented. 2. Theory 2.1. The mass-transfer operator A key idea in our approach is the use of the same formalism for all transfer steps of any kind. The mass

F. Miomandre, E. vieil / Reversible charge transfer with unknown mass transfer

transfer will be considered in the same way as the charge transfer, i.e. the two states Co and C* of the same species C related by a mass transfer will be written like a reaction step:

__!c

co

c*

1*

The superscript “0” refers to a species on the electrode (zero distance) and the superscript “*” refers either to the bulk (infinite distance) in the case of mass transport (spatial displacement) or to the initial concentration (zero time) in the case of an accumulationconsumption process on the electrode. By mass transfer we mean a larger category of processes than mass transport, by adding to it the purely temporal process of accumulation or consumption, which does not imply any spatial displacement of matter, since this process occurs on an electrode site. The rate constant ti used in this formalism, in contrast to usual transfers, is not a scalar, but is an operator, which is the reason for the tilde placed over it. It expresses the fact that a mass transfer is not a purely dissipative process, as is a resistive process such as charge transfer, but a complex process involving possibly accumulation of mass or variation of potential energy. An analogy can be made with electronic transport through an electrical impedance. The electrical admittance operator Y, the reciprocal of the impedance 2, allows Ohm’s law, which relates the electrical potential difference V, - V2 to the current I, to be written in a generalized form: I I 4--

+--El-f 1

I=?(V, -

V,)

2

When the impedance device is a resistor, there only dissipation of energy and the admittance is scalar, the conductance l/R. When it is a capacitor, the admittance is a differential operator, since there accumulation or consumption of potential energy

$-II& 1

I=C-&-V,)

is a c, is

(2)

2

The mass-transfer operator also allows us to write an equivalent of Ohm’s law based on the kinetic equation of an electrochemical charge transfer, which relates the species concentration c to the flux density J J = hoc0 - ti*c*

(3) In contradistinction to Ohm’s law, the transfer rate is generally not the same in both directions of the transfer, and since for a transport process we speak about directions in space, this non-equivalence is termed anisotropy of the mass transfer. The main cause of anisotropy in a mass transfer is the influence

277

of an electric field upon charged species. It creates electrical migration whose rate is not identical in the two directions. Since in electroanalytical chemistry it is common to neglect the influence of migration owing to the presence of a supporting indifferent electrolyte, we shall assume in this article that all mass transfers are isotropic, that is to say they have equivalent rates in both directions: J J 4--

-j-p-’

jy

J=?h(cO-c”)

(4)

This assumption presents the advantage of a slight simplification of the discussion, since it strengthens the analogy with Ohm’s law, but there is really no difficulty in doing otherwise. As with the admittance operator, whose reciprocal is the impedance operator, the preceding relationship can be inverted: co-c* =ht-‘J (5) The choice of the symbol m for the mass-transfer rate has been dictated by the classical usage of this symbol in electrochemical textbooks 1231for expressing the mass-transfer rate in stationary convective mass transport (which is equal to the diffusion coefficient divided by the thickness of the diffusion layer). The units for m, for all mass transfer conditions, are those of a velocity, i.e. cm s-l. More details can be found in a previous publication [l], in particular some examples of mass-transfer operators for the various mass-transfer modes, stationary, transient, temporal (ultra-thin layer), or evolving modes such as transient-convective or spatially restricted diffusion. It must be recalled that the exact purpose of the present study is not to elaborate a theoretical mathematical expression for the mass-transfer operator, but precisely to show that an opposite approach can be worth undertaking. A first advantage of having defined such a concept, without any further specification, is the possibility of handling it like any algebraic quantity to build models of electrochemical mechanisms easily; as simply as if all steps were first-order chemical reactions or charge-transfer steps. A second advantage is the possibility of determining it experimentally by a deductive treatment of the data. What we need is to define more precisely how this can be done, by entering a little more deeply into the properties that this operator must possess. We have shown [ll and proven [5] that, provided the mass transfer obeys linear differential relationships, the mass transfer rate operator hi could be written in a very convenient way by using a temporal convolution product by a characteristic function as follows: tii=gi(t)*

(6)

F. Miomandre, E. vieil / Reversible charge transfer with unknown mass transfer

278

or for the inverse operator: &i-l1

=*gi(

t) *

imposing a step perturbation on the mass transfer. This is illustrated in Fig. 1, where a concentration step Aci applied at t = 0 induces a flux density J(t) proportional to the integral GJt) of the characteristic function gi( t). The step can be mathematically represented with the Heaviside function Z’(t) which is equal to 0 for negative times and to 1 for positive times:

(7)

where gi(t) and *gJt) are characteristic time-dependent functions of the mass-transfer mode and of the transferred species, gi(t> is the direct characteristic function and *gi(t) is the reciprocal one. The characteristic functions are independent of perturbation, concentration variation or flux variation: they depend only on the nature of the mass transfer, i.e. the transfer mode (stationary, transient, temporal, etc.), the physical characteristics of the species, such as the diffusion coefficient, the geometry of the medium in which the transfer takes place and the boundary conditions such as the length of the transport or the thickness of a layer. This formulation of the mass-transfer operator hi in terms of an operation with a characteristic function means that all the specificity and characteristics of a given mass transfer are included within the characteristic function gi(t> itself. Consequently, to characterize a mass transfer signifies knowing the dependence of the characteristic function with time and possibly with the other mass-transfer parameters mentioned. The temporal convolution product is defined by the integral of the product of the two functions considered, one of them (it does not matter which, since the product is commutative) being taken with an inverted time axis:

Ac,( t) = Ac,X(

t)

(9) Since we have, as observed previously for a constant function (for t > 0):

gi(t) * z(t)

= Gi(t)

(10)

with, by definition, Gi( t> = joki( A) dh

(11)

We obtain for the flux density variation given by J(t)

=tiAc,(t)

(12)

the scalar product J(t)

=AciGi(t)

The reciprocal characteristic function obtained by imposing a current step J(t)

(13) can also be

=J&P(t)

(14)

which produces a concentration

variation

Aci =J*Gi(t) with an integral of the reciprocal tion:

The convolution product has many properties, described elsewhere [5], but all that we need to observe in the meantime is that when one of the members of the convolution product is a constant function, the result of the convolution is the integral of the other function. This is a very useful property which gives a practical way of identifying a characteristic function gi(t) by

characteristic

*Gi( t) = /‘*gi( A) dh

func-

(15)

0

We can summarize this presentation by saying that any linear mass transfer can be represented by the operation of convolution with a characteristic function of the mass transfer, independently of the perturbation

J(f)/nmol cti's"

Aci(t)

ACi

IO-

5MASS 0

Aci . Gi(t)

=

TRANSFER

,

0 0

1

2

3

4

0

1

2

3

4

5

tls Fig. 1. Example of flux variation chronoamperometric measurement

with time produced by a concentration step. The shape on the electrochemical system and conditions described

of the flux density later.

curve has been deduced

from a

F. Miomandre, E. vieil / Reversible charge transfer with unknownmass transfer

279

used to provoke it. The characteristic function has two representations, a direct one and a reciprocal one, which can be determined experimentally by imposing a step perturbation in concentration or current. Once determined, these functions can be used for convoluting any perturbation signal used in another electroanalytical technique, as will be shown later.

with c*, the total initial (bulk) concentration of A and B, here equal to cz, and we have chosen the positive difference

2.2. The E mechanism Let us consider a charge transfer between two redox species A and B:

E(n) = Cri ‘tin KCEj

A&B To indicate whether this reaction is a reduction or an oxidation, we shall make use of an algebraic convention for the number of electrons n exchanged per molecule: n > 0 e reduction n < 0 = oxidation This overall reaction indeed involves mass transfer of both electroactive species between the electrode (superscript 0) and the bulk (superscript *> and must be written as a three-step sequential process: A* + A0 k, BO \ %3 B* kB &3 By adopting a sign convention making a flux density positive when the species flows away from the electrode, and by assuming that there are no kinetic complications such as side accumulation of intermediate species, one can write that the flux densities are the same for the three steps and, in particular, for both mass-transfer steps: J = Iji*( c; - c;> = tin&

k, kr3

= K@,

(17)

equilibwhere KCE, represents a potential-dependent rium constant. The combination of eqns. (16) and (17) leads to the following expression: c:(E) -=___ c*

(19)

to represent the concentration variation. In eqn. (18) an apparent equilibrium constant is used: (20)

When the mass-transfer mode is the same for both species A and B, i.e. when the physical nature of both mass transfers is identical, the mass-transfer operators z mA and ht, differ only by a scaling factor, which depends on the species characteristics such as diffusion coefficient, viscosity, size, etc. Then the product ti, %a is a scalar (generally close to 1 when the characteristics of the two species are not very different) and g is proportional to K. This proportionality assumption will be made to simplify the following consideration regarding the time-dependence of the apparent equilibrium constant. From eqn. (21) the expression of the current for a charge transfer of n electrons and for any kind of mass transfer is: Z(t) = -nFSJ = -nFSc*h,Ac,

(21)

in which F is the Faraday constant and S the electrode area. The minus sign is used to respect the convention of a positive current in oxidation. This is the equation to use in the potentiodynamic case, when the potential is applied to the electrochemical system and the electrical current is measured. Inversion of eqn. (21) by using the inverse operator fi i -’ leads to:

(16)

Species B is not supposed to be present initially in the system (i.e. ci = O), as is usually the case in an electroanalytical measurement. If the charge transfer between A and B is assumed to be reversible, an equilibrium relationship between both concentrations of A and B can be written: 4 -=4

AC, = c; - cl

K(E) 1 + E@)

(18)

(22) This equation (22) is used in the galvanodynamic case, when the current is imposed on the electrochemical system and the electrical potential is measured. Figure 2 summarizes the proposed approach in showing the path that must be followed to relate the response of a reversible electrochemical mechanism to the imposed perturbation. This scheme works either in the potentiodynamic case (imposed potential and measured current), from left to right, or in the opposite case of galvanodynamic experiment, from right to left. The left rectangle represents the charge-transfer characteristic used to relate the E(t) signal with the c,(t) signal (by using eqn. (18) and by choosing a function describing &,), whereas the right rectangle represents the mass-transfer characteristic which relates the c,(t) signal with the current 10) through eqns. (211, (22).

F. Miomandre, E. Vieil / Reversiblecharge transferwithunknown mass transfer

280

r

MASS

CHARGE

1UUNSFER

TRANSFEF t

,c Cf

E

!zl 1 9-o*

II -*-

i

1

-*-

y’p

1

POTENTIOSTAT

- GALVANOSTAT : measured i

E measured

f-

GALVANODYNAMIC

t

POTENTIODYNAMIC

)

imposed

_Ij

Fig. 2. General scheme illustrating the relationship between the perturbation and the response for a mechanism composed of a reversible charge transfer and identical mass-transfer type for the two species. The shapes of the various curves are arbitrary in this example, and should be adapted to each perturbation signal and to each peculiar case of charge and mass transfer. The operationsshown are: 0, the composition of signals (the potential scale is replaced by a time scale according to its time dependence); *, temporal convolution. 2.3. The potentiodynamic

case

By using the formulation of & with a convolution product, eqn. (12) can be reformulated in the following way:

J(t) -

nFSc*

=g,q(t) * AC,

(23)

This equation is the fundamental equation describing the mechanism response upon application of a potential, which we are going to use in various ways. Firstly we are going to use it for the experimental measurement of characteristic functions by choosing peculiar potential signals; secondly, we shall use the knowledge of g*(t) for measuring the equilibrium constant of the charge transfer. This allows us to carry out the determination of the mechanism as we proposed to do. A last use will be the checking of this equation in the reverse way, i.e. knowing a characteristic function and an equilibrium constant, and their dependence with time, to convolute them to produce a theoretical variation of the current with time. The current appears as the convolution product of two independent and well-specialized components: the first one is specific only to the mass transfer, the characteristic function g*(t), and the second one is specific only to the charge transfer, an expression of its equilibrium constant. Even in the case of a stationary mass transport, the function g,(t) depends on the time, but not on the potential, unless the influence of

the electric field complicates the mass transfer, which we have not assumed in this work, as already stated. The second component of the convolution product depends on the potential and, in consequence, on the time when the potential is not stationary. As we have already discussed, the separation between the two contributions can easily be done by using concentrationstep signals which convert the convolution product into a scalar product. However, the concentration cannot be directly controlled, but the electrode potential can be. The problem is that a concentration step can be realized by a classical electrochemical potentiostat only if the charge transfer is totally reversible. When the charge transfer has a limited reversibility or is completely irreversible, the electrode concentration depends not only on the electrode potential but also on the electrical current, preventing the signal from following the shape of a step. In the following paper we shall treat that important question [24], but now we shall assume that the charge transfer is perfectly reversible. Consequently, by choosing a potential step as input signal E(t)

=q+

(Ek-Ei)zyt)

(24)

with Ei the initial potential and E, the final potential, we obtain a step signal also in the case of a reversible charge transfer: AC*(E) =Ac,(&)z(t)

(25)

F. Miomandre, E. Vieil / Reversible charge transfer with unknown mass transfer

To simplify this equation it has been assumed that cA = 0 at potential Ei, (this is equivalent to saying that species B is not present at t = 0). Applying eqn. (23) leads to the following expression for the chronoamperometric current Istep: Z,,,,(~,~,)

= -nFSc*Ac,(ZQG,(t)

(26)

in which G,(t) is the integral of gA(t) as previously defined in eqn. (11). Equation (26) shows that the chronoamperometric current is the result of the product of two functions, each one depending on only one variable, respectively t and E,. So, by varying each variable independently, it is possible to measure both functions from the chronoamperometric current. If the potential is fixed at a value E,, then the characteristic function g*(t) appears to be proportional to the derivative of the chronoamperometric current. The measurement of this latter for a given value of potential allows g*(f) to be obtained for any mass-transfer mode. Now we are able to use eqn. (23), transformed into eqn. (26), for the determination of the characteristic function in the case of an unknown mass transfer. We are going to use this function for the determination of the charge-transfer equilibrium constant dependence on the potential through eqns. (18),(26). Let us consider the experiment consisting in repeating the chronoamperometry several times, with a dif-

281

ferent value of the potential E, each time. We obtain a set of chronoamperometric currents Z,,,,(t,E,) which all have the same shape when plotted vs. time as predicted by eqn. (26). This is shown by step 1 in Fig. 3. Each curve is proportional to the same curve -nFSG,(t), with a proportionality coefficient equal to AC,; it means that all curves have the same horizontal scale but are vertically homothetic. By sampling the curves of Istep vs. t at different times t,, . . . .,t, (step 2 in Fig. 31, we can plot Istep vs. E, for different values of t. This operation leads to a set of vertically homothetic Z-E, curves, with similarity coefficients proportional to the integral of the characteristic function G,(t) (see eqn. (26)). By dividing each curve by this coefficient (step 3 in Fig. 3), it is possible to superimpose all the curves to obtain AC, with a scaling factor equal to -nFSG,(t) which will be determined later, and from there the equilibrium constant Z&, itself. The determination of the scaling factor - nFSG,(t) is simple when a limiting current condition can be reached, i.e. when the concentration variation AC, reaches its limiting value c*, or in other words when the product -nE approaches a sufficiently high value for K(n) to be considered as infinite. This is the case shown in Fig. 3, where the observed plateau in the Z-E curves allows the replacement of AC, by c* at high potentials. The integral G,(t) of the characteristic function can then be obtained directly from the limit-

Fig. 3. Scheme illustrating the method used to determine the characteristic functions of unknown charge transfer and mass transfer from a set of chronoamperometric data. Step 1, measurement of a set of experimental chronoamperometric curves produced by different potential steps from Ei up to the values El, . . ,E,. Step 2, sampling of the set of experimental chronoamperometric curves at different times tr, ,t, leading to a set of homothetic Z-E curves. Step 3, division of each Z-E curve by the corresponding limiting current, leading to the superimposition of all the curves: the resulting curve is proportional to the concentration profile. Step 2’, selection of chronoamperometry for a given potential E, on a time scale equal to 2ta: the resulting curve represents GA(t), which is the integral of the direct characteristic function g,+(t).

282

F. Momandre,

E. vied / Reversible charge transfer with unknown mass transfer

ing current by a simple scale change, as comes from eqn. (26):

4tep.d t1

GA(t, = _n~sc*

(27)

The knowledge of the bulk concentration c* allows the AC, variation to be scaled correctly through eqn. (18). However, when the limiting current is not available, because of interference with another redox couple, or because of the existence of non-nernstian equilibrium owing to electrostatic interactions (for instance), the scaling factor, i.e. the knowledge of cA at least at one potential, has to be determined by another experiment such as spectrometry, gravimetry, etc. It should be remarked that the determination of G,(t) does not imply necessarily the repeating of the chronoamperometry at different potential step values. A single step at a value where the limiting current is reached may suffice, repetition at other step values being necessary only for the measurement of AC,. However, the correctness of the concentration step produced by the potential step needs to be checked, otherwise an appreciable error in the characteristic function can be difficult to detect. This must be done by checking the reversibility of the charge transfer, since it is the compulsory condition for obtaining a true concentration step. The simultaneous determination of both GA(t) and AC,(E) by multistep chronoamperometry is a convenient way to make this check. The last way to use eqn. (23) is the prediction of the current variation from known characteristic functions of the mass transfer and equilibrium constants of the charge transfer. This can be done for any potential perturbation and time dependence, from measured characteristic functions and equilibrium constants or from theoretical ones. We shall take the example of a measured characteristic function and of a theoretical equilibrium constant. Let us take the previous characteristic function experimentally measured from a set of chronoamperometric data. If we assume that a limiting current can be measured, we can directly replace the characteristic limiting function by Istep,,, the chronoamperometric current. In these special conditions, the expression of the current deriving from eqns. (23),(26) is the following:

(28) We can remark that eqn. (28) is similar to the expressions of current given by Tallman and co-workers, under integral formalism instead of convolution [25,26]. In their derivation Z_,(t) is defined as a diffusionlimited current that must be computed theoretically.

They established their equation in the restricted case of a purely diffusional mass transfer, but with several electrode geometries. They restricted its validity to equal diffusion coefficients for species A and B, which was made necessary by the complex treatment they used, but not at all compulsory, as we can deduce from our development. They also remarked that an equivalent equation was produced by Aoki et al. [12] but with a theoretical function instead of the chronoamperometric limiting current. Concerning the expression we propose (eqn. (28)), it must be emphasized that it can be applied both to experimental and theoretical currents for all mass transfers, and its validity extends far beyond the simple E mechanism we consider here. This last statement means in particular that the reversibility of the charge transfer is not a limiting condition for eqn. (28). In the example of a soluble redox couple following classical behaviour, the equilibrium constant Z?(n) is given by the Nernst equation: &n, = exp[ ( - nF/RT)(

E - E’)]

(2%

E” is the standard potential of the redox couple involved, and F,R and T have their usual meanings. Equations (28),(29) are valid for any dependence of E upon time, i.e. for any electroanalytical technique, without any restriction. The example given here concerns linear-sweep voltammetry, for which the potential is a linear function of time: E = Ei + ut with Ei is the initial potential and u is the scan rate. In these conditions, the linear-sweep voltammetric current will be given by Z,,(t) =-

d dt

Zste&) 1

* l+exp[-(-nF/RT)(vt+Ei-E’)]

(30)

To conclude, the different steps leading to the obtaining of the characteristic functions of both charge and mass transfers are recalled in Fig. 3: both terms involved in the convolution of eqn. (23) can be determined from experimental chronoamperometric data. By this method, the modelling of the electrochemical response to any potential signal can be performed for any mass transfer, without using a theoretical model for it. 2.4. lYhegalvanodynamic case A similar development as in the potentiodynamic case can be followed in this case. Equation (22) is

F. Miomandre, E. Vieil / Reversible charge transfer with unknown mass transfer translated into a convolution product by using eqn. (7):

/lea =*

gA(t)* I(t) -- nFSc*

(31)

The way to individualize the behaviour of the two terms of the convolution product is to impose a series of current steps I 1,I2. . . . . , I k

I(t) = Ik,~(t )

(32)

Applying the property that a product of convolution with the Heaviside function is equivalent to an integration (see eqn. (10)), eqn. (31) becomes a simple scalar product ACA = * G A ( t ) - n F S c *

(33)

between the integral of the reciprocal characteristic function defined in eqn. (15) and the value of the current step I~ that must be chosen in an adequate range and applied during the time necessary for measuring a potential variation corresponding to the redox couple studied. This is the classical chronopotentiometry technique. The resulting set of E - t curves at fixed I is sampled at chosen times tx,t 2 . . . . t, producing a set of I k - E points at fixed times. Since the product *GA(t)i k must be constant for the same potential, the quantity *GA(t)/(-nFSc*) is deduced for each time from the

I

.n

E

't

•

.

. t2...tn

283

various scaling factors used to superimpose all i k - E curves upon the one chosen as the limiting curve. This limiting current is, from eqn. (33):

I 1= - n F S c * / *

GA(t)

(34)

The *GA(t) curve can then be obtained, knowing the bulk concentration and the other parameters. Simultaneously the c A variation as a function of potential is obtained as given by eqn. (33), from which other representations such as K ' ( E ) = f ( E ) or log(K'(E))=f(E) can be deduced through eqn. (18). Figure 4 schematizes the process leading to the determination of these characteristic functions. An interesting feature of this new approach is the possibility of determining a reversible half-wave potential, or a "half-limiting current" potential, without any hypothesis regarding the charge-transfer model. From eqns. (18),(33),(34) and from the definition of the halfwave potential E l l 2 we deduce ( E = E l ~ 2 ~ I = 1 , / 2 ) ~ K(E,/2) = 1

(35)

Naturally, the measurement of the reciprocal characteristic function opens as many possibilities as in the potentiodynamic case. For example, eqn. (31) can be utilized to predict the concentration response c A or the potential response to any time-dependence of the applied current. Another example is the mixing of both approaches, potentiodynamic and galvanodynamic,

•

I~

I1

t

....

: E

y *Gill

4' zC[_ ....

x\\\\\\\\\x\\\\\~

ac c )

't

0

t

E

Fig. 4. Scheme illustrating the method used to determine the characteristic functions of unknown charge transfer and mass transfer from a set of chronopotentiometric data. Step I, measurement of a set of experimental chronopotentiometric curves produced by different current steps up to the values ll, ...,I k. Step 2, sampling of the set of experimental chronopotentiometric curves at different times t~. . . . . t,, leading to a set of homothetic I-E curves. Step 3, division of each I-E curve by the corresponding limiting current, leading to the superimposition of all the curves: the resulting curve is proportional to the concentration profile. Step 3', limiting current curve equal to the coefficients used for the superimposition of all I-E curves upon the same one. Step 4', inversion of the limiting current and multiplication by -nFSc* : the resulting curve represents *GA(t), which is the integral of the reciprocal characteristic function *gA(t).

284

F. Miomandre, E. tiei

G-

)

*J/

*

/ Reversible charge transfer with unknown mass transfer

!$

)

r

l

@O

t

$

t

Fig. 5. Convolution of the integral of the direct characteristic function G,(t) with the integral of the reciprocal characteristic function *Gi(t), followed by a derivation, gives the step function (Heaviside function).

which consists in applying eqn. (31) to a measured current Z(t) produced by imposing a potential perturbation: a direct determination of the equilibrium constant KCE, as a function of the imposed potential can thus be done easily. 2.5. Validity check The correct correspondence between gi(t) and *gi(t) can be checked by convoluting the two curves together. The unit element of a convolution product is the Dirac distribution, which is the peak function equal to 0 for f # 0 and equal to 03 at t = 0:

f(t) *

$( ;) =f(t>

The correct unit for this function arbitrary time 7. From the definitions istic functions given in eqns. (6),(7), it their convolution product should give 1

?iqIil; ‘=logi(t)**gi(t)=-s,

is given by an of the charactercan be seen that the unit element

(4.i

7

7

(37)

When one of these functions is integrated, one obtains the integral of the Dirac distribution, the Heaviside function S?(t) (see ref. 5 for explanations): Gi(t)* *gi(t)

=gi(t)**Gi(t)

=X(t)

(38)

112sp51 trarWent temporal

‘P=l temporal

(restnc,ed (acc”m”a,mo d,lluUO”,

mor”mprro”,

This relationship, illustrated in Fig. 5, provides an interesting way to check the accuracy of the measurement of the characteristic functions. 2.6. Space-time

symmetry and diffusion coefficient

The point we are discussing now is just a small detail that is a side-effect of the new methodology presented. In the previous theoretical work underlying this presentation [l], it has been shown that a remarkably simple relationship exists between the characteristic functions of two symmetrical mass-transfer modes: (39)

In eqn. (39), D is a diffusion coefficient and the subscript p refers to the mass-transfer mode, which is a quantity in the range 0 to 1, as shown in Fig. 6. Three characteristic p values describe the fundamental mass transfer modes: 0 for the stationary one, l/2 for linear semi-infinite diffusion and 1 for temporal mass transfer, i.e. accumulation-consumption without transport. Intermediate but fixed values correspond to fractal mass transfers, whereas variable values correspond to evolving mass transfers, as is the present case of transient-convective ones. We had interpreted p physically, as expressing a ratio of time component, and 1 -p a ratio of the distance components of the mass transfer; the linear semi-infinite diffusion being the only one having equal space and time components. This means that when p = l/2, eqn. (39) leads to the definition of a characteristic diffusion coefficient as the ratio of two characteristic functions, for example:

p= l/2

tramlent scm,“fi”lk ,,“eard,flurra”

transientstatmary ,rranaent convec,rve, /y

7 stahonary p=o )

distance/ Fig. 6. Representation in the time and space plane of various mass transfers according to their mode p.

where Gl-‘j(t) is the integral of G,(t), i.e. the double integral of the direct characteristic function g,(t). Another ratio involving derivatives is also convenient. The practical consequence of this relationship is to give an extremely powerful means of identifying purely diffusional behaviour when it occurs during an evolving mass transfer, by simply comparing its two characteristic functions. When, for a certain time, a proportionality is detected between the characteristic functions, according to eqn. (40), it means indeed that

F. Momandre,

E. Vieil / Reversible charge transfer with unknown mass transfer

the mass transfer at that time obeys the linear semi-infinite diffusion, and that the scaling factor is then the diffusion coefficient. To conclude this parenthesis, it should be said that the determination of the diffusion coefficient is not at all necessary in the present work, but simply gives a supplementary checking possibility.

I/PA

285

40

a

3. Experimental details

b c

A solution of HCIO, (about 5 x lo-* M) in acetonitrile was prepared, in which N-methylphenothiazine (C,,H,,NS, from Kodak) was dissolved (concentration c* = 10e3 M). The solution obtained was deoxygenated by argon bubbling for 30 min. The working electrode was a rotating electrode from Tacussel with a Pt disk of 2 mm diameter. The speed regulation of the rotating electrode was provided by a control unit CONTROVIT from Tacussel, calibrated with a frequency counter. In such conditions the product -nFSc* is equal to 3.03 mC cm-’ for a single charge transfer in oxidation (n = - 1). The reference electrode was Ag 10.1 M Agf, 5 x lo-* M HClO, in acetonitrile, and the auxiliary electrode was a Pt wire. Acquisition was performed with a potentiostat EG&G PAR273, driven by a personal computer.

e

d

4. Results To check the validity of our method, we have chosen to study a reversible redox system (N-methylphenothiazine in acetonitrile) in non-classical conditions of mass transfer such as transient convection. The reversibility of our redox system was checked by record-

f n 2

1

0

3

5

4 t IS

Fig. 8. Set of chronoamperometric curves of N-methylphenothiazine in acetonitrile on rotating electrode (60 rev min-‘1. Potential steps go from 0 V (rest time of 0.5 s) to the following values (in V vs. Ag/O.l M Ag+): curve (a), 0.6; curve (b), 0.5; curve (cl, 0.45; curve (d), 0.4; curve (e), 0.35; curve (f), 0.3; curve (g), 0.25.

ing cyclic voltammograms in transient-convective mode at different scan rates (10 mV s-l + 2 V s-l) for an angular velocity of 60 rev min-’ (Fig. 7). N-Methylphenothiazine shows good reversibility at scan rates up to about 200 mV s-l for this rotation rate: indeed, the distance between anodic and cathodic peaks is independent of the scan rate below this value. Beyond this limit, anodic and cathodic peaks are shifted so that their distance increases dramatically. This implies that the prediction or the analysis of the voltammetric response using convolution should be performed at scan rates less than or equal to 200 mV s-l for this system. 4.1. Chronoamperometric determination A set of chronoamperometric curves for increasing step potentials was recorded on the rotating electrode

t I____’ : IS

A

0.1

v

0.15 0.2 0.26 0.3 0.4

0

v . 0

1

A i

d

”

x 0

0’ n

0

0.1

0.2

0.3

0.4

0.5 E IV

0.6

0.7

vs. Ag/Ag+

Fig. 7. Experimental cyclic voltammetries of N-methylphenothiazine in acetonitrile on rotating electrode (60 rev min-‘). Scan rates are as follows (in mV s-t): curve (a), 2000; curve (b), 1000; curve Cc), 400; curve Cd), 200; curve (e), 40; curve (f), 20.

t 0.2

0.25

0.3

0.35

0.4

/

0.45 EN

0.5 vs.

L

0.55 AglAg

+

Fig. 9. Set of I-E curves resulting from current values of curves of Fig. 8 at the following times: r = 0.1, 0.15, 0.2, 0.25, 0.3 and 0.4 s.

286

F. Miomandre, E. vieil / Reversible charge transfer with unknown mass transfer

l2

Q /MC

11 10 9 a 7

0.6 1

6 5 4 3 0.2 1

0.1

0

0.3

0.2

0.4

0.6

0.6

E N KS AglAg+

Fig. 10. Division of I-E curves of Fig. 9 by the corresponding limiting current. The full line is a nernstian interpolation of the resulting points with the following parameters: - nF/RT = 38 V-l; E = 0.325 V vs. Ag IO.1 M Ag+.

Fig. 12. Integral Q at very short time of the chronoamperometric current of Fig. 11 vs. square root of time. The solid line corresponds to a theoretical purely diffusive behaviour.

at 60 rev min-’ (transient-convective mode, see Fig. 8). It is noticeable that a saturation in the increase of limiting currents occurs for applied potentials above 0.45 V vs. Ag 10.1 M Ag+. As a result, the chronoamperometric curve used to determine the integral GA(t) of the characteristic function should correspond to a potential of at least 0.45 V. The set of Fig. 8 can be re-drawn into a set of current vs. potential curves, for arbitrary chosen sample times (Fig. 9). The I-E curves obtained by this method appear to be vertically homothetic, that is to say that each curve is proportional to any other along the current scale - as predicted by eqn. (26). This enables us to calculate similarity coefficients, allowing

all the curves to be superimposed. These coefficients are simply the limiting currents at high potentials, since the potential-dependent term in eqn. (26) becomes very close to unity. Consequently, each curve of Fig. 9 has been divided by the corresponding limiting current, leading to the superimposition of all the curves. The result of such an operation is shown in Fig. 10. The superimposition observed for a given potential is evidence of the good reversibility of our redox system in the experimental conditions chosen. The interpolation of the resulting points of Fig. 10 was made using spline functions and then compared with a nernstian wave fit. As predicted by eqn. (30), a very good agreement was found, when using the following values for the nernstian wave:

0.06

-nF/RT

G4 Iem 8

-' 0.06

0.06 0.05

0.64

0.04 0.03

0.03

0.02 0.01 0

0.02

0.01

n

0

0.1

6

6

0.2

0.3

-nF/RT

0.4

3

4

7

6

9

10

0.5

11

= 38.9 V-’

E” = 0.34 V

for a mono-electronic transfer at 298 K. The good agreement between experimental and theoretical values proves the validity of the method used to calculate the potential-dependent function corresponding to our system:

i 12

E” = 0.325 V

which can be compared with the theoretical and known values [ 191:

0

0

= 38 V-’

12

13

14

t/s

Fig. 11. Integral G,(t) of the direct characteristic curve g*(t) of the mass transfer. This curve is simply obtained by dividing the chronoamperometric limiting current by - nFSc*. The potential was stepped from 0.0 V to 0.6 V vs. AglO.1 M Ag+ and the current recorded during 2t, = 14 s. The inset represents the first half-second of the transient.

&)

= exp{38.0[( E/V)

- 0.3251)

Since in the system studied the species concentration difference AC, reaches a limiting value for high potentials, eqn. (27) can be used to obtain the integral G,(t) of the characteristic function of the mass transfer as shown in Fig. 11, i.e. by simply dividing the limiting current by -nFSc*.

F. Miomandre, E. vieil / Reversiblecharge transfer withunknown mass transfer

In Fig. 11 the current has been recorded at a potential sufficiently high for complete independence of the potential to be observed, and for a longer time than in the previous experiment. The condition E > 0.45 V was found previously from the chronoamperometric curves obtained. A potential of 0.6 V, for which the equilibrium constant becomes very small, was chosen. A certain care must be exercised owing to the imperfections of both the electrochemical system and the instrument used for recording such a curve. These imperfections are several: the non-ideality of the potential step, which leads to a finite initial current, whose value may be fluctuating; the existence of a capacitive current at very short times owing to charging of interfaces; the imperfect reversibility of the redox system in the conditions chosen; the resistivity of the electrolyte, which creates a non-negligible ohmic drop; the limited bandwidth of the instrument, which cannot accept an infinite value of transient current. To check these hypotheses, the integral Q of the chronoamperometric current is plotted vs. the square root of time (Fig. 121, for the very short times of the experiment (t < 0.4 s). During this period, the mass transfer is assumed to be completely dominated by pure linear diffusion, so that the former curve should be linear (see the space-time symmetry and diffusion coefficient question discussed later). Nevertheless we notice a slight deviation from linear behaviour for the very first points of the recording. A correction of these points allows the effects of a perturbation occurring at the beginning of the experiment (capacitive current, non-ideal step, instrument bandwidth) to be taken into account. This correction, concerning only a very small

287

‘1

I

0

0.1

0.2

0.3

0.4

0.5

0.6 E N

vs. Ag/Ag+

Fig. 14. Set of Z-E curves resulting from curve sampling of Fig. 13 at the indicated values of t (0.5 s, 0.8 s, then 1 to 4 s by 0.5 s steps).

number of points, does not challenge the validity of the method, but is necessary when an accurate use of the characteristic curve is required, as will be shown later in the convolutive approach. 4.2. Chronopotentiometric determination The current steps were imposed for 5 s after a 0.5 s conditioning of the cell at I = 0, by increasing step values from 0.2 PA to 10 PA. The recorded potential is represented in Fig. 13. In this time window, only the oxidation of the N-methylphenothiazine is seen for current steps up to 8 PA. Beyond this value, the second redox system consisting in the oxidation of the radical cation [19] begins to be observed. Sampling of this set of curves for times ranging between 0.5 s and 4 s produces the I,-E set of points of Fig. 14.

0.5 0.4

0.3 0.2

-

NERNST

0.1

J

0 0

0.1

0.2

0.3

0.4

0.5

0.6 EN

Fig. 13. Set of E - 1 chronopotentiometric curves of N-methylphenothiazine in acetonitrile on rotating electrode (60 rev min-‘1. Current starts from a 0 PA rest period of 0.5 s and steps to the indicated values (0.2 @A, 0.5 PA, then 1 to 10 PA by 1 PA steps).

vs. Ag/Ag

+

Fig. 15. Concentration AC, of species at the electrode, normalized to the bulk concentration c*, as a function of potential deduced from the scaling factors used to superimpose the Z-E curves of Fig. 14. The full line represents the theoretical nernstian wave with - nF/ RT = 39 V-’ and E = 0.35 V vs. Ag 10.1 M Ag+.

F. Momandre,

288

E. vieil / Reversible charge transfer with unknown mass transfer

The proportionality factors AC*, as indicated by eqn. (331, can be determined exactly for all potential values: they are represented on Fig. 15 only for the lowest values of the time for the sake of readability. The knowledge of 0 c* allowed this figure to be scaled directly in concentration. The half-wave value, according to the statement of eqn. (39, gives: a

E 1/Z = 0.35 v

The curve representing also drawn in Fig. 15 with -nF/RT

= 39 V-i

a nernstian

equilibrium

is

E” = 0.35 V

This demonstrates that the chronopotentiometric determination and the chronoamperometric one produce equivalent results. The fact that both approaches converge towards the same behaviour is in itself an excellent demonstration of the reversibility of the system, since the basic definition of the reversibility is the invariance of the system upon reversal of its input and output. The next step of the treatment is to take the reciprocal l/Z, that gives values proportional to *GA(t)/ (-nFSc*), the proportionality factor being AC*, as mentioned above. Since it has been shown in the preceding chronoamperometric determination that a limiting current was reached above 0.45 V, the proportionality factor is equal to one for the highest potentials recorded. The values of *GA(t) measured for these potentials are obtained by multiplying l/Z, by -nFSc* and are shown in Fig. 16 together with other curves corresponding to the pure linear semi-infinite diffusion case that will be discussed just after. It can be seen that the transient convective mass transfer we measure becomes close to the linear semi-infinite diffusion for short times, below 1 s, and reaches a limiting value close to the purely stationary convective mode, as expected [20,22]. The points of Fig. 16 can easily be interpolated to produce a continuous *GA(t) curve that can be used for computing convolutions if necessary, but it is obvious that a denser sampling than the nine-point one that we have carried out should be undertaken for better accuracy. The expected behaviour of *GA(t) when the mass transfer follows a semi-infinite diffusive mass transport for a part of the time has also been plotted in Fig. 16. This behaviour was discussed in the theoretical section about the space-time symmetry, where it has been shown that the two characteristic functions *G,(t) and GA-“(t) were proportional in that case (eqn. (40)). It is exactly what we observe at short times, as can be seen in Fig. 16 where the GA-‘)(t)/D curve has been plotted with the best D value found: 2 X 10e5 cm* s-l.

0

1

2

3

4 t/s

Fig. 16. Curve (a): integral *GA(t) of the reciprocal characteristic curve *g*(t) of the mass transfer as a function of time, obtained by taking the reciprocal of the current at fixed potential (from Fig. 14) and by multiplication by - nFSc *: the solid line is the experimental one interpolated by spline functions without any correction. Curve (b): the dotted line is the theoretical characteristic function for the simple case of linear semi-infinite diffusion 2(t/pD)‘/* with D = 2 X10-’ cm’ s-t. Curve (c): the integral Go-‘) of the direct characteristic curve G,(t) previously determined (Fig. 11) divided by the same value of D.

This diffusion coefficient corresponds to the medium acetonitrile and perchloric acid; it does not differ from the one that can be extracted, but through a cottrellian model, from the linear asymptote of the charge vs. t ‘1’ representation made in Fig. 12. (Note that Q = -nFSc*G),-‘)(t).) In Fig. 16 has also been plotted the cottrellian model with the same diffusion coefficient value. This is not very far from the value known for N-methylphenothiazine in acetonitrile containing tetraethylammonium perchlorate, (1.1 f 0.2) X 10e5 cm* s-l, determined by EPR, and in their work the authors [27] show that the chronoamperometric determination always produces higher values. We have determined by four different electrochemical techniques (stationary rotating disk electrode, classical chronoamperometry, chronoamperometry on micro-disk electrode and linear sweep voltammetry; in pure diffusion conditions for the last three), an average value of (2.1 f 0.3) x 10P5 cm* s-l. It should be recalled that the determination of the diffusion coefficient is optional and simply gives a supplementary checking of the very short time measurements. 4.3. The convolutive approach Until now we have based our approach only on step techniques, potentiostatic and galvanostatic, because of the simple property of the convolution product of becoming equivalent to a scalar multiplication with the Heaviside function. This statement has given us two

F. Miomandre, E. vieil / Reversible charge transfer with unknown mass transfer

parallel procedures for determining the characteristic functions, direct and reciprocal, gi(t> and *gi(‘) through their integrals. Now we are going to use the convolution tool for two things: firstly for checking the correspondence between the two characteristic functions, secondly for appreciating their usefulness and their accuracy by using them in the case of the cyclic voltammetry technique. The correct correspondence between gi(t) and *gi(t> can be checked by convolving the two curves together, as explained in the theoretical section. The important point is that the two curves must have the same time scale. The reciprocal characteristic curve *G,(t) has been extrapolated to the same time, 14 s, as that of the g*(t) direct curve, by assuming linear behaviour at long times, as should be the case in a stationary mass transfer. Nevertheless the other choice of the shorter time scale of the reciprocal curve could have been made, but it is much more interesting to deal with long-time characteristic curves which are easy to shorten when necessary. Figure 17 shows the result of this operation made with the uncorrected experimental characteristic curves as determined before. Except at the very beginning of the curve, the expected result is obtained fairly well: the value of the curve remains constant and approximately equal to one, this value meaning that the scaling factors were correctly measured. The departure from the constant behaviour observed for short times is the result of the imperfections of the step techniques, as already mentioned. The same operation made with corrected characteristic curves obviously gives an almost perfect result that we have not represented. It can be remarked that this checking can be used as a

‘Gdt) * h(t) : :: t

lr

0.8

1

0.6

1

0.4 t 0.2

1

Fig. 17. Uncorrected convolution product of the experimental integral reciprocal characteristic curve *G,_,(t) (the interpolated curve *G,(t) in Fig. 16) with the experimental direct characteristic curve g*(t) (derivative of the curve G,(t) in Fig. 11). This product remains the same when the derivative is taken on the reciprocal characteristic curve: *GA(t)* g,&)=*g,&)* G,(t).

289

11

9 7 5 3 1 -1 -3

-51 0

“1’,,,1,“,1’,“1,,,,1’,” 0.1 0.2

I 0.3

0.4

0.5 E/V

0.6

0.7

VS. Ag/Ag +

Fig. 18. Cyclic voltammograms of N-methylphenothiazine in acetonitrile on a rotating electrode (60 rev min-‘1 at v = 100 mV s-l: -0-o-, experimental Z,; -, resulting from convolution of curves AcA/c* of Fig. 15 and the derivative g*(t) of GA(t) of Fig. 11 and multiplied by - nFSc*; curve (a), G,(t) taken without correction; curve (b), G,(t) taken after correction of the initial transient as shown on the chronoamperometry in Fig. 12.

criterion for undertaking an appropriate correction of one of the characteristic curves when the other is correct. The second checking procedure consists in changing the perturbation signal from the simple step perturbation already used. We have chosen to use cyclic voltammetry, i.e. an up-and-down potential ramp varying linearly with time between an initial potential Ei and a final one Ef. For a given scan rate U, the time required for one complete cycle is:

This value must then be taken as the time scale of the characteristic curves. The values chosen for the voltammetric parameters are as follows: Ei = 0.0 V

E,=O.7

v=lOOmVs-’

t,=7s

v

Identically, the nernstian wave function c,(E)/c* previously determined must be developed along the time, that is to say made symmetrical around t,, to obtain both positive and negative scans of cyclic voltammetry. The wave function developed, Ac,(t)/c*, involved in the derivative convolution of eqn. (28) is shown in a figure that will be discussed later (Fig. 19). Convolution of the uncorrected direct characteristic curve G,(t) of Fig. 11 with the time evolution of the concentration ratio Ac,/c*, followed by a derivation and a multiplication by -nFSc*, leads to the “theoretical” cyclic voltammogram of Fig. 19, which is also compared with the experimental one made at 100 mV s-l with a 60 rev min-’ rotation rate.

290

Ac,/c*

F. Miomandre, E. Keil / Reversible charge transfer with unknown mass transfer

ously is sufficiently good to lead to the conclusion that a more accurate convolution with a corrected reciprocal characteristic curve is pointless and to demonstrate the validity of the proposed approach.

I.’ 1 0.9 0.6 0.7 0.6

5. Conclusion

0.6 0.4 0.3 0.2 0.1 0 -0.1 0

2

3

4

6

6

6

9

10

11

12

I3

t/II

Fig. 19. Concentration cA of species at the electrode normalized to the bulk concentration c* as a function of time during a linear potential scan at 100 mV s-l. The solid line is the nernstian function of Fig. 10 made symmetrical around tR = 7 s. The dotted line (-O-O-) is the convolution product of the experimental current I, of Fig. 18 with the reciprocal characteristic curve *gA(t) (derivative of the interpolated *GA(t) in Fig. 16) multiplied by - nFSc*.

We note good agreement concerning the peak positions and the limiting currents. However, there is a noteworthy difference between the respective peak heights: the intensity of the peak current of the calculated voltammetry seems to be underestimated. Using the corrected direct characteristic curve, as explained in connection with Fig. 12, brings a real improvement in the peak height adjustment as observed, leading to very good similarity between the two curves. So, thanks to the convolution method, it is possible to predict the current response to a given potential signal, whatever the mass transfer mode might be. This can be performed from a set of experimental chronoamperometric data alone, without adjustable parameters or theoretical functions. Very good precision is obtained when applying a correction to the short time points, in order to take into account the perturbations due to capacitive currents or instrument limitations such as non-ideal potential steps. A more interesting point is the possibility of analysing an experimental voltammetric curve by using the convolution in the other way, in convolving the current response as a function of time with the reciprocal characteristic curve. Equation (31) shows that the result is AC*, which gives the equilibrium constant & variation through eqn. (18). The result of the convolution of the experimental cyclic voltammetric current Z&t) of Fig. 18 with the uncorrected reciprocal characteristic curve *G,(t) of Fig. 16, followed by a derivation and a division by -nFSc*, is shown in Fig. 19. The agreement with the nernstian curve determined by the chronoamperometric approach described previ-

In the simple case of a reversible charge transfer between two redox species undertaking the same type of mass transfer, the convolution concept was proven very successful in two ways; by predicting the current response corresponding to any potential signal and by analysing the experimental current response to any potential perturbation, even when the mass-transfer mode is unknown. Moreover it has been shown that the current response can be determined from experimental chronoamperometric data alone, without postulating theoretical equations. No model of mass transfer was used in any way in this study. For example, the rotating-disk electrode has been utilized without any recourse to a mathematical model, such as the Levich formula for the stationary state or the known approximate models for the transient state. The validity of this general method has been proven in the case of a soluble redox species (N-methylphenothiazine), showing mono-electronic reversible charge transfer and obeying a non-classical mass-transfer mode (transient convection). Good agreement has been found between the predicted and experimental cyclic voltammetries, especially after applying a slight correction to the very first times of the chronoamperometric curve. The necessity of this correction might be interpreted by the existence of instrumental limitations and imperfections at the beginning of the potential step. Concerning the opposite approach, the analysis of cyclic voltammetric current has been revealed to be very efficient even without any correction. The determination of the characteristic functions of mass transfer has been done through a step technique because it is the most direct and the simpler method. It is worth mentioning that this is not the only possibility: any perturbation shape can be used provided a deconvolution procedure is used to discriminate between the charge and the mass transfer. The use of small periodic signals is also perfectly possible, the correspondence between characteristic functions and mass-transfer impedance being obvious. This methodology opens new possibilities in the electroanalytical field in rendering the necessity for an accurate mathematical model less stringent for quantitative and precise mechanism characterization. It is now possible to determine accurately a kinetic equilibrium constant of a charge transfer with a non-centred and badly defined active area of an irregularly rotating

F. Momandre,

E. vied / Reversible charge transfer with unknown mass transfer

electrode in a stirred and non-isothermal medium, for example. There is however a condition: the reproducibility of the experiment. The repetition of several perturbation steps is made on the assumption that the mechanism does not change in between. This might be the most severe limitation of the proposed methodology. In this study, the characteristic functions of the unknown mass transfer have been determined without a priori knowledge of the charge-transfer equilibrium, to outline the powerfulness and the generality of the method. It is obvious that when the charge transfer is perfectly known, for example a nernstian one with known redox potential and slope, the determination of the characteristic functions is greatly enhanced and facilitated. This means that the mass transfer within an unknown electrochemical cell or material can be characterized with a standard redox couple independently of the analysis of ‘a more complex mechanism. The efficiency of our method has been proven only in the case of a reversible charge transfer. It must be outlined that the method itself carries the verification of this condition, preventing a possible artefact when this condition is not completely fulfilled. Nevertheless, reversible mechanisms are certainly not the most interesting field among the wide variety of electrochemical systems. A further development will be the study of irreversible systems, where the kinetic limitation due to the charge transfer must be taken into account [241. This could allow the analysis of complicated electrochemical systems to be performed, for which masstransfer mode as well as kinetics is imperfectly known. This is, for example, the case of the mechanisms involved in the redox behaviour of conducting polymers, which will be the scope of our next studies.

E E0 E l/2 Ei

4 4 F

giCt>

* giCt)

Gi(t)

*Gi(t)

G$-'j(t)

at> kv

ZLSV Zstep

initial species of an electrochemical reacA tion A*, B*, CT species, or species concentration, in the bulk of the electrolyte (far from the charge transfer site) or in its initial state species, or species concentration, on the A’, B”, c; charge transfer site (electrode) species produced by an electrochemical B reaction species volume concentration (M) ci C* total initial concentration of species by volume (M) D diffusion coefficient (cm2 s- ‘>

electrode potential (V) formal or standard redox potential (V) half-wave electrode potential (V) initial electrode potential (of a scan or a step) W> value of the electrode potential imposed during the k th step (V) final electrode potential (of a scan) (V) Faraday constant ( = 96 487 C mol- ‘) direct characteristic function of the mass transfer of species i. This is the function by which the concentration is convolved to give the flux density (cm sm2) reciprocal characteristic function of the mass transfer of species i. This is the function by which the flux density is convolved to give the concentration (cm-‘) time integral of the direct characteristic function gi(t>. This is the function to which the current is proportional in chronoamperometry (cm s-l> time integral of the reciprocal characteristic function *gi(‘>. This is the function to which the concentration of the produced species on the charge transfer site c; is proportional in chronopotentiometry (cm s-l) time integral of the integral Gi(t) of the direct characteristic function gJt>, i.e. the double time integral of gi(t> Heaviside function (= 0 for t < 0, 1 for t > 0)

Z

Zk Symbols

291

Zstep,1 4

J k‘4 43 K(E,

electrical current (I > 0 in oxidation) (A) electrical current in cyclic voltammetry (A) value of the electrical current imposed during the kth step (A) electrical current in linear sweep voltammetry (A) electrical current in chronoamperometry (potential step) (A) electrical limiting current in chronoamperometry (potential step) (A) electrical limiting current in any technique (A) flux density (mol cmP2 s-l) charge transfer rate constant for the reaction A + B (cm s-l> charge transfer rate constant for the reaction B +A (cm s-l) equilibrium constant of charge transfer (potential-dependent or time-dependent when the potential varies)

292

F. Momandre,

E. vieil / Reversible charge transfer with unknown mass transfer

K@,

n

P

Q

R s T t tk tR

V i i *

apparent equilibrium constant of a charge transfer. This is the effective equilibrium constant of a charge transfer associated with the - two mass transfers of its reactants: KcE, = rii+zB KcEj mass-transfer rate operator for species i, defined by Ji = ~ici (cm s-t> inverse of the mass-transfer rate operator for species i, defined by Ci = liiiJi (S cm-t) algebraic number of exchanged electrons per mol’ecule of substrate (n > 0, reduction; IZ< 0, oxidation) mass transfer mode (0 QP G 1) electrical charge (0 gas constant (- 8.3145 J mol-’ K-l) geometric electrode area (cm’> temperature (K) time (s) sampling time for the k th step (s) time of the reversal of a potential scan 6) electrical potential (V) potential scan rate (V s-l> admittance operator 6) impedance operator (0) temporal convolution operator. The temporal convolution of two functions f(t) and g(t) is defined by (this is a commutative operator: the order of the functions is not important) f(t)* g(t) = j,‘f(h)g(t A) dh

Greek letters Dirac distribution s,w positive difference in concentration for AC* the initial species, = cz - cl integration variable (temporal integraA tion) (s) Pythagoras’ number ( = 3.14159) 7r arbitrary time 6) 7

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